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aam20120p
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Networks, Crowds, and Markets: Reasoning about a Highly Connected World
David Easley Dept. of Economics
Cornell University
Jon Kleinberg Dept. of Computer Science
Cornell University
Cambridge University Press, 2010 Draft version: June 10, 2010.
2
Contents
Preface i
1 Overview 1
1.1 Aspects of Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Central Themes and Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
I Graph Theory and Social Networks 21
2 Graphs 23
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Paths and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Distance and Breadth-First Search . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Network Datasets: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Strong and Weak Ties 47
3.1 Triadic Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 The Strength of Weak Ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Tie Strength and Network Structure in Large-Scale Data . . . . . . . . . . . 56
3.4 Tie Strength, Social Media, and Passive Engagement . . . . . . . . . . . . . 60
3.5 Closure, Structural Holes, and Social Capital . . . . . . . . . . . . . . . . . . 64
3.6 Advanced Material: Betweenness Measures and Graph Partitioning . . . . . 69
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 Networks in Their Surrounding Contexts 85
4.1 Homophily . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Mechanisms Underlying Homophily: Selection and Social Influence . . . . . . 90
4.3 A�liation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.4 Tracking Link Formation in On-Line Data . . . . . . . . . . . . . . . . . . . 97
4.5 A Spatial Model of Segregation . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3
4 CONTENTS
5 Positive and Negative Relationships 119 5.1 Structural Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.2 Characterizing the Structure of Balanced Networks . . . . . . . . . . . . . . 123 5.3 Applications of Structural Balance . . . . . . . . . . . . . . . . . . . . . . . 126 5.4 A Weaker Form of Structural Balance . . . . . . . . . . . . . . . . . . . . . . 129 5.5 Advanced Material: Generalizing the Definition of Structural Balance . . . . 132 5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
II Game Theory 153
6 Games 155 6.1 What is a Game? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.2 Reasoning about Behavior in a Game . . . . . . . . . . . . . . . . . . . . . . 158 6.3 Best Responses and Dominant Strategies . . . . . . . . . . . . . . . . . . . . 163 6.4 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.5 Multiple Equilibria: Coordination Games . . . . . . . . . . . . . . . . . . . . 168 6.6 Multiple Equilibria: The Hawk-Dove Game . . . . . . . . . . . . . . . . . . . 172 6.7 Mixed Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.8 Mixed Strategies: Examples and Empirical Analysis . . . . . . . . . . . . . . 179 6.9 Pareto-Optimality and Social Optimality . . . . . . . . . . . . . . . . . . . . 184 6.10 Advanced Material: Dominated Strategies and Dynamic Games . . . . . . . 186 6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7 Evolutionary Game Theory 209 7.1 Fitness as a Result of Interaction . . . . . . . . . . . . . . . . . . . . . . . . 210 7.2 Evolutionarily Stable Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.3 A General Description of Evolutionarily Stable Strategies . . . . . . . . . . . 216 7.4 Relationship Between Evolutionary and Nash Equilibria . . . . . . . . . . . . 218 7.5 Evolutionarily Stable Mixed Strategies . . . . . . . . . . . . . . . . . . . . . 220 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8 Modeling Network Tra�c using Game Theory 229 8.1 Tra�c at Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.2 Braess’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.3 Advanced Material: The Social Cost of Tra�c at Equilibrium . . . . . . . . 233 8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9 Auctions 249 9.1 Types of Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.2 When are Auctions Appropriate? . . . . . . . . . . . . . . . . . . . . . . . . 251 9.3 Relationships between Di↵erent Auction Formats . . . . . . . . . . . . . . . 252 9.4 Second-Price Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 9.5 First-Price Auctions and Other Formats . . . . . . . . . . . . . . . . . . . . 257 9.6 Common Values and The Winner’s Curse . . . . . . . . . . . . . . . . . . . . 258
CONTENTS 5
9.7 Advanced Material: Bidding Strategies in First-Price and All-Pay Auctions . 260 9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
III Markets and Strategic Interaction in Networks 275
10 Matching Markets 277 10.1 Bipartite Graphs and Perfect Matchings . . . . . . . . . . . . . . . . . . . . 277 10.2 Valuations and Optimal Assignments . . . . . . . . . . . . . . . . . . . . . . 282 10.3 Prices and the Market-Clearing Property . . . . . . . . . . . . . . . . . . . . 284 10.4 Constructing a Set of Market-Clearing Prices . . . . . . . . . . . . . . . . . . 288 10.5 How Does this Relate to Single-Item Auctions? . . . . . . . . . . . . . . . . 291 10.6 Advanced Material: A Proof of the Matching Theorem . . . . . . . . . . . . 293 10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
11 Network Models of Markets with Intermediaries 311 11.1 Price-Setting in Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 11.2 A Model of Trade on Networks . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.3 Equilibria in Trading Networks . . . . . . . . . . . . . . . . . . . . . . . . . 322 11.4 Further Equilibrium Phenomena: Auctions and Ripple E↵ects . . . . . . . . 326 11.5 Social Welfare in Trading Networks . . . . . . . . . . . . . . . . . . . . . . . 330 11.6 Trader Profits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 11.7 Reflections on Trade with Intermediaries . . . . . . . . . . . . . . . . . . . . 334 11.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
12 Bargaining and Power in Networks 339 12.1 Power in Social Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 12.2 Experimental Studies of Power and Exchange . . . . . . . . . . . . . . . . . 342 12.3 Results of Network Exchange Experiments . . . . . . . . . . . . . . . . . . . 344 12.4 A Connection to Buyer-Seller Networks . . . . . . . . . . . . . . . . . . . . . 348 12.5 Modeling Two-Person Interaction: The Nash Bargaining Solution . . . . . . 349 12.6 Modeling Two-Person Interaction: The Ultimatum Game . . . . . . . . . . . 352 12.7 Modeling Network Exchange: Stable Outcomes . . . . . . . . . . . . . . . . 355 12.8 Modeling Network Exchange: Balanced Outcomes . . . . . . . . . . . . . . . 359 12.9 Advanced Material: A Game-Theoretic Approach to Bargaining . . . . . . . 361 12.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
IV Information Networks and the World Wide Web 373
13 The Structure of the Web 375 13.1 The World Wide Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 13.2 Information Networks, Hypertext, and Associative Memory . . . . . . . . . . 378 13.3 The Web as a Directed Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 384 13.4 The Bow-Tie Structure of the Web . . . . . . . . . . . . . . . . . . . . . . . 388
6 CONTENTS
13.5 The Emergence of Web 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
14 Link Analysis and Web Search 397 14.1 Searching the Web: The Problem of Ranking . . . . . . . . . . . . . . . . . . 397 14.2 Link Analysis using Hubs and Authorities . . . . . . . . . . . . . . . . . . . 399 14.3 PageRank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 14.4 Applying Link Analysis in Modern Web Search . . . . . . . . . . . . . . . . 412 14.5 Applications beyond the Web . . . . . . . . . . . . . . . . . . . . . . . . . . 415 14.6 Advanced Material: Spectral Analysis, Random Walks, and Web Search . . . 417 14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
15 Sponsored Search Markets 437 15.1 Advertising Tied to Search Behavior . . . . . . . . . . . . . . . . . . . . . . 437 15.2 Advertising as a Matching Market . . . . . . . . . . . . . . . . . . . . . . . . 440 15.3 Encouraging Truthful Bidding in Matching Markets: The VCG Principle . . 444 15.4 Analyzing the VCG Procedure: Truth-Telling as a Dominant Strategy . . . . 449 15.5 The Generalized Second Price Auction . . . . . . . . . . . . . . . . . . . . . 452 15.6 Equilibria of the Generalized Second Price Auction . . . . . . . . . . . . . . 456 15.7 Ad Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 15.8 Complex Queries and Interactions Among Keywords . . . . . . . . . . . . . 461 15.9 Advanced Material: VCG Prices and the Market-Clearing Property . . . . . 462 15.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
V Network Dynamics: Population Models 481
16 Information Cascades 483 16.1 Following the Crowd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 16.2 A Simple Herding Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 485 16.3 Bayes’ Rule: A Model of Decision-Making Under Uncertainty . . . . . . . . . 489 16.4 Bayes’ Rule in the Herding Experiment . . . . . . . . . . . . . . . . . . . . . 494 16.5 A Simple, General Cascade Model . . . . . . . . . . . . . . . . . . . . . . . . 496 16.6 Sequential Decision-Making and Cascades . . . . . . . . . . . . . . . . . . . 500 16.7 Lessons from Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 16.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
17 Network E↵ects 509 17.1 The Economy Without Network E↵ects . . . . . . . . . . . . . . . . . . . . . 510 17.2 The Economy with Network E↵ects . . . . . . . . . . . . . . . . . . . . . . . 514 17.3 Stability, Instability, and Tipping Points . . . . . . . . . . . . . . . . . . . . 517 17.4 A Dynamic View of the Market . . . . . . . . . . . . . . . . . . . . . . . . . 519 17.5 Industries with Network Goods . . . . . . . . . . . . . . . . . . . . . . . . . 526 17.6 Mixing Individual E↵ects with Population-Level E↵ects . . . . . . . . . . . . 528 17.7 Advanced Material: Negative Externalities and The El Farol Bar Problem . 533
CONTENTS 7
17.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
18 Power Laws and Rich-Get-Richer Phenomena 543 18.1 Popularity as a Network Phenomenon . . . . . . . . . . . . . . . . . . . . . . 543 18.2 Power Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 18.3 Rich-Get-Richer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 18.4 The Unpredictability of Rich-Get-Richer E↵ects . . . . . . . . . . . . . . . . 549 18.5 The Long Tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 18.6 The E↵ect of Search Tools and Recommendation Systems . . . . . . . . . . . 554 18.7 Advanced Material: Analysis of Rich-Get-Richer Processes . . . . . . . . . . 555 18.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
VI Network Dynamics: Structural Models 561
19 Cascading Behavior in Networks 563 19.1 Di↵usion in Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 19.2 Modeling Di↵usion through a Network . . . . . . . . . . . . . . . . . . . . . 565 19.3 Cascades and Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 19.4 Di↵usion, Thresholds, and the Role of Weak Ties . . . . . . . . . . . . . . . 578 19.5 Extensions of the Basic Cascade Model . . . . . . . . . . . . . . . . . . . . . 580 19.6 Knowledge, Thresholds, and Collective Action . . . . . . . . . . . . . . . . . 583 19.7 Advanced Material: The Cascade Capacity . . . . . . . . . . . . . . . . . . . 587 19.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
20 The Small-World Phenomenon 611 20.1 Six Degrees of Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 20.2 Structure and Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 20.3 Decentralized Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 20.4 Modeling the Process of Decentralized Search . . . . . . . . . . . . . . . . . 619 20.5 Empirical Analysis and Generalized Models . . . . . . . . . . . . . . . . . . 622 20.6 Core-Periphery Structures and Di�culties in Decentralized Search . . . . . . 629 20.7 Advanced Material: Analysis of Decentralized Search . . . . . . . . . . . . . 631 20.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
21 Epidemics 645 21.1 Diseases and the Networks that Transmit Them . . . . . . . . . . . . . . . . 645 21.2 Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 21.3 The SIR Epidemic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 21.4 The SIS Epidemic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 21.5 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 21.6 Transient Contacts and the Dangers of Concurrency . . . . . . . . . . . . . . 662 21.7 Genealogy, Genetic Inheritance, and Mitochondrial Eve . . . . . . . . . . . . 666 21.8 Advanced Material: Analysis of Branching and Coalescent Processes . . . . . 672 21.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
8 CONTENTS
VII Institutions and Aggregate Behavior 689
22 Markets and Information 691 22.1 Markets with Exogenous Events . . . . . . . . . . . . . . . . . . . . . . . . . 692 22.2 Horse Races, Betting, and Beliefs . . . . . . . . . . . . . . . . . . . . . . . . 694 22.3 Aggregate Beliefs and the “Wisdom of Crowds” . . . . . . . . . . . . . . . . 700 22.4 Prediction Markets and Stock Markets . . . . . . . . . . . . . . . . . . . . . 704 22.5 Markets with Endogenous Events . . . . . . . . . . . . . . . . . . . . . . . . 708 22.6 The Market for Lemons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 22.7 Asymmetric Information in Other Markets . . . . . . . . . . . . . . . . . . . 714 22.8 Signaling Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 22.9 Quality Uncertainty On-Line: Reputation Systems and Other Mechanisms . 720 22.10Advanced Material: Wealth Dynamics in Markets . . . . . . . . . . . . . . . 723 22.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730
23 Voting 735 23.1 Voting for Group Decision-Making . . . . . . . . . . . . . . . . . . . . . . . 735 23.2 Individual Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 23.3 Voting Systems: Majority Rule . . . . . . . . . . . . . . . . . . . . . . . . . 740 23.4 Voting Systems: Positional Voting . . . . . . . . . . . . . . . . . . . . . . . . 745 23.5 Arrow’s Impossibility Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 748 23.6 Single-Peaked Preferences and the Median Voter Theorem . . . . . . . . . . 750 23.7 Voting as a Form of Information Aggregation . . . . . . . . . . . . . . . . . . 756 23.8 Insincere Voting for Information Aggregation . . . . . . . . . . . . . . . . . . 758 23.9 Jury Decisions and the Unanimity Rule . . . . . . . . . . . . . . . . . . . . . 761 23.10Sequential Voting and the Relation to Information Cascades . . . . . . . . . 766 23.11Advanced Material: A Proof of Arrow’s Impossibility Theorem . . . . . . . . 767 23.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
24 Property Rights 775 24.1 Externalities and the Coase Theorem . . . . . . . . . . . . . . . . . . . . . . 775 24.2 The Tragedy of the Commons . . . . . . . . . . . . . . . . . . . . . . . . . . 780 24.3 Intellectual Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 24.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786
Preface
Over the past decade there has been a growing public fascination with the complex
“connectedness” of modern society. This connectedness is found in many incarnations: in
the rapid growth of the Internet and the Web, in the ease with which global communication
now takes place, and in the ability of news and information as well as epidemics and financial
crises to spread around the world with surprising speed and intensity. These are phenomena
that involve networks, incentives, and the aggregate behavior of groups of people; they are
based on the links that connect us and the ways in which each of our decisions can have
subtle consequences for the outcomes of everyone else.
Motivated by these developments in the world, there has been a coming-together of mul-
tiple scientific disciplines in an e↵ort to understand how highly connected systems operate.
Each discipline has contributed techniques and perspectives that are characteristically its
own, and the resulting research e↵ort exhibits an intriguing blend of these di↵erent flavors.
From computer science and applied mathematics has come a framework for reasoning about
how complexity arises, often unexpectedly, in systems that we design; from economics has
come a perspective on how people’s behavior is a↵ected by incentives and by their expec-
tations about the behavior of others; and from sociology and the social sciences have come
insights into the characteristic structures and interactions that arise within groups and pop-
ulations. The resulting synthesis of ideas suggests the beginnings of a new area of study,
focusing on the phenomena that take place within complex social, economic, and technolog-
ical systems.
This book grew out of a course that we developed at Cornell, designed to introduce
this topic and its underlying ideas to a broad student audience at an introductory level.
The central concepts are fundamental and accessible ones, but they are dispersed across the
research literatures of the many di↵erent fields contributing to the topic. The principal goal
of this book is therefore to bring the essential ideas together in a single unified treatment,
and to present them in a way that requires as little background knowledge as possible.
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
i
ii PREFACE
Overview. The book is intended to be used at the introductory undergraduate level, and
as such it has no formal prerequisites beyond a level of comfort with basic mathematical
definitions at a pre-calculus level. In keeping with the introductory style, many of the ideas
are developed in special cases and through illustrative examples; our goal is to take concepts
and theories that are complex in their full generality, and to provide simpler formulations
where the essential ideas still come through.
In our use of the book, we find that many students are also interested in pursuing some
of these topics more deeply, and so it is useful to provide pathways that lead from the
introductory formulations into the more advanced literature on these topics. With this in
mind, we provide optional sections labeled Advanced Material at the ends of most chapters.
These advanced sections are qualitatively di↵erent from the other sections in the book; some
draw on more advanced mathematics, and their presentation is at a more challenging level
of conceptual complexity. Aside from the additional mathematical background required,
however, even these advanced sections are self-contained; they are also strictly optional, in
the sense that nothing elsewhere in the book depends on them.
Synopsis. The first chapter of the book provides a detailed description of the topics and
issues that we cover. Here we give a briefer summary of the main focus areas.
The book is organized into seven parts of three to four chapters each. Parts I and II
discuss the two main theories that underpin our investigations of networks and behavior:
graph theory, which studies network structure, and game theory, which formulates models
of behavior in environments where people’s decisions a↵ect each other’s outcomes. Part III
integrates these lines of thought into an analysis of the network structure of markets, and
the notion of power in such networks. Part IV pursues a di↵erent integration, discussing
the World Wide Web as an information network, the problem of Web search, and the de-
velopment of the markets that currently lie at the heart of the search industry. Parts V
and VI study the dynamics of some of the fundamental processes that take place within
networks and groups, including the ways in which people are influenced by the decisions
of others. Part V pursues this topic at an aggregate scale, where we model interactions
between an individual and the population as a whole. Part VI continues the analysis at the
more fine-grained level of network structure, beginning with the question of influence and
moving on to the dynamics of search processes and epidemics. Finally, Part VII considers
how we can interpret fundamental social institutions — including markets, voting systems,
and property rights — as mechanisms for productively shaping some of the phenomena we’ve
been studying.
Use of the Book. The book is designed for teaching, as well as for any reader who finds
these topics interesting and wants to pursue them independently at a deeper level.
iii
Several di↵erent types of courses can be taught from this book. When we teach from
it at Cornell, the students in our class come from many di↵erent majors and have a wide
variety of technical backgrounds; this diversity in the audience has served as our primary
calibration in setting the introductory level of the book. Our course includes a portion of the
material from each chapter; for the sake of concreteness, we provide the approximate weekly
schedule we follow below. (There are three 50-minute lectures each week, except that weeks
6 and 7 of our course contain only two lectures each. In each lecture, we don’t necessarily
include all the details from each indicated section.)
Week 1: Ch. 1; Ch. 2.1-2.3; Ch. 3.1-3.3,3.5,4.1
Week 2: Ch. 5.1-5.3; Ch. 6.1-6.4; Ch. 6.5-6.9
Week 3: Ch. 8.1-8.2; Ch. 9.1-9.6; Ch. 10.1-10.2
Week 4: Ch. 10.3; Ch. 10.4-10.5; Ch. 11.1-11.2
Week 5: Ch. 11.3-11.4; Ch. 12.1-12.3; Ch. 12.5-12.6
Week 6: Ch. 12.7-12.8; Ch. 13
Week 7: Ch. 14.1-14.2; Ch. 14.3-14.4
Week 8: Ch. 15.1-15.2; Ch. 15.3-15.4; Ch. 15.5-15.6,15.8
Week 9: Ch. 16.1-16.2; Ch. 16.3-16.4; Ch. 16.5-16.7
Week 10: Ch. 17.1-17.2; Ch. 17.3-17.5; Ch. 18
Week 11: Ch. 19.1-19.2; Ch. 19.3; Ch. 19.4,19.6
Week 12: Ch. 22.1-22.4; Ch. 22.5-22.9; Ch. 7.1-7.4
Week 13: Ch. 20.1-20.2; Ch. 20.3-20.6; Ch. 21.1-21.3
Week 14: Ch. 23.1-23.5; Ch. 23.6-23.9; Ch. 24
There are many other paths that a course could follow through the book. First, a number
of new courses are being developed at the interface of computer science and economics,
focusing particularly on the role of economic reasoning in the design and behavior of modern
computing systems. The book can be used for such courses in several ways, building on
four chapters as a foundation: Chapter 2 on graphs, Chapter 6 on games, Chapter 9 on
auctions, and Chapter 10 on matching markets. From here, a more expansive version of
such a course could cover the remainder of Parts II and III, all of Parts IV and V, Chapter
19, and portions of Part VII. A more focused and potentially shorter version of such a course
concerned principally with auctions, markets, and the on-line applications of these ideas
could be constructed from Chapters 2, 6, 9, 10, 13, 15, 17, 18, and 22, and drawing on parts
of Chapters 11, 12, 14, 16, and 19. When these courses are taught at a more advanced level,
the advanced sections at the ends of most of these chapters would be appropriate material;
depending on the exact level of the course, the text of many of these chapters could be used
to lead into the more advanced analysis in their respective final sections.
In a di↵erent but related direction, new courses are also being developed on the topic of
social computing and information networks. The book can be used for courses of this type by
iv PREFACE
emphasizing Chapters 2-6, 13-14, 17-20, and 22; many such courses also include sponsored
search markets as part of their coverage of the Web, which can be done by including Chapters
9, 10, and 15 as well. The advanced sections in the book can play a role here too, depending
on the level of the course.
Finally, portions of the book can serve as self-contained “modules” in courses on broader
topics. To pick just a few examples, one can assemble such modules on network algorithms
(Sections 2.3, 3.6, 5.5, 8.3, 10.6, 14.2-3, 14.6, 15.9, 20.3-4, and 20.7); applications of game
theory (Chapters 6-9, 11, Sections 12.9, 15.3-15.6, 19.2-19.3, 19.5-19.7, 23.7-23.9); social net-
work analysis (Chapters 2-5, 12.1-12.3, 12.5-12.8, 18-20); the role of information in economic
settings (Chapters 16, 22, Sections 23.6-23.10); and the analysis of large-scale network data
sets (Sections 2.3, 3.2-3.3, 3.6, 4.4, 5.3, 13.3-13.4, 14.2-14.5, 18.2, 18.5, 20.5). Most of these
modules use graphs and/or games as fundamental building blocks; for students not already
familiar with these topics, Chapters 2 and 6 respectively provide self-contained introductions.
Acknowledgements. Our work on this book took place in an environment at Cornell that
was particularly conducive to interaction between computing and the social sciences. Our
collaboration began as part of a project with Larry Blume, Eric Friedman, Joe Halpern,
Dan Huttenlocher, and Éva Tardos funded by the National Science Foundation, followed by
a campus-wide “theme project” on networks sponsored by Cornell’s Institute for the Social
Sciences, with a group that included Larry and Dan together with John Abowd, Geri Gay,
Michael Macy, Kathleen O’Connor, Je↵ Prince, and David Strang. Our approach to the
material in the book draws on perspectives — ways of thinking about these topics and ways
of talking about them — that we’ve learned and acquired from this interdisciplinary set of
colleagues, a group that includes some of our closest professional collaborators.
The course on which the book is based grew out of discussions that were part of the
Cornell theme project; the two of us had taught distinct portions of this material separately
in graduate courses that we had developed, and Michael Kearns’s Networked Life course at
Penn demonstrated the vibrancy and relevance this material could have for an introductory
undergraduate audience as well. We were intrigued by the prospect of combining di↵erent
perspectives that hadn’t previously appeared together — a process that would be educational
not only to the students in the course but to us as well. Creating and teaching this new
interdisciplinary course was made possible by the support of our departments, Computer
Science and Economics, and by support from the Solomon Fund at Cornell University.
Once the book had begun to take shape, we benefitted enormously from the feedback,
suggestions, and experiences of colleagues who taught from early drafts of it. In particu-
lar, we thank Daron Acemoglu (MIT), Lada Adamic (Michigan), Allan Borodin (Toronto),
Noshir Contractor (Northwestern), Jason Hartline (Northwestern), Nicole Immorlica (North-
western), Ramesh Johari (Stanford), Samir Khuller (Maryland), Jure Leskovec (Stanford),
v
David Liben-Nowell (Carleton), Peter Monge (USC), Asu Ozdaglar (MIT), Vijay Ramachan-
dran (Colgate), R. Ravi (CMU), Chuck Severance (Michigan), Aravind Srinivasan (Mary-
land), and Luis von Ahn (CMU). The graduate and undergraduate teaching assistants in
our own teaching of this subject have been very helpful as well; we thank Alex Ainslie, Lars
Backstrom, Jacob Bank, Vlad Barash, Burak Bekdemir, Anand Bhaskar, Ben Cole, Bistra
Dilkina, Eduard Dogaru, Ram Dubey, Ethan Feldman, Ken Ferguson, Narie Foster, Eric
Frackleton, Christie Gibson, Vaibhav Goel, Scott Grabnic, Jon Guarino, Fahad Karim, Ko-
ralai Kirabaeva, Tian Liang, Austin Lin, Fang Liu, Max Mihm, Sameer Nurmohamed, Ben
Pu, Tal Rusak, Mark Sandler, Stuart Tettemer, Ozgur Yonter, Chong-Suk Yoon, and Yisong
Yue.
In addition to the instructors who used early drafts, a number of other people provided
extensive comments on portions of the book, leading to many improvements in the text: Lada
Adamic, Robert Kerr, Evie Kleinberg, Gueorgi Kossinets, Stephen Morris, David Parkes,
Rahul Sami, Andrew Tomkins, and Johan Ugander. We also thank a further set of colleagues,
in addition to those already listed, who have provided very useful advice and suggestions on
this project as it has proceeded: Bobby Kleinberg, Gene Kleinberg, Lillian Lee, Maureen
O’Hara, Prabhakar Raghavan, and Steve Strogatz.
It has been a pleasure to be able to work with the editorial team at Cambridge University
Press. Lauren Cowles, our main point of contact at Cambridge, has been an amazing source
of advice and help, and we likewise very much appreciate the contributions of Scott Parris
and David Tranah to this project, and Peggy Rote and her colleagues at Aptara for their
work on the production of the book.
Finally, a profound thanks to our families, in continuing appreciation of their support
and many other contributions.
David Easley
Jon Kleinberg
Ithaca, 2010
vi PREFACE
Chapter 1
Overview
Over the past decade there has been a growing public fascination with the complex
“connectedness” of modern society. At the heart of this fascination is the idea of a network
— a pattern of interconnections among a set of things — and one finds networks appearing
in discussion and commentary on an enormous range of topics. The diversity of contexts in
which networks are invoked is in fact so vast that it’s worth deferring precise definitions for
a moment while we first recount a few of the more salient examples.
To begin with, the social networks we inhabit — the collections of social ties among friends
— have grown steadily in complexity over the course of human history, due to technological
advances facilitating distant travel, global communication, and digital interaction. The past
half-century has seen these social networks depart even more radically from their geographic
underpinnings, an e↵ect that has weakened the traditionally local nature of such structures
but enriched them in other dimensions.
The information we consume has a similarly networked structure: these structures too
have grown in complexity, as a landscape with a few purveyors of high-quality informa-
tion (publishers, news organizations, the academy) has become crowded with an array of
information sources of wildly varying perspectives, reliabilities, and motivating intentions.
Understanding any one piece of information in this environment depends on understanding
the way it is endorsed by and refers to other pieces of information within a large network of
links.
Our technological and economic systems have also become dependent on networks of
enormous complexity. This has made their behavior increasingly di�cult to reason about,
and increasingly risky to tinker with. It has made them susceptible to disruptions that
spread through the underlying network structures, sometimes turning localized breakdowns
into cascading failures or financial crises.
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
1
2 CHAPTER 1. OVERVIEW
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Figure 1.1: The social network of friendships within a 34-person karate club [421].
The imagery of networks has made its way into many other lines of discussion as well:
Global manufacturing operations now have networks of suppliers, Web sites have networks
of users, and media companies have networks of advertisers. In such formulations, the
emphasis is often less on the structure of the network itself than on its complexity as a large,
di↵use population that reacts in unexpected ways to the actions of central authorities. The
terminology of international conflict has come to reflect this as well: for example, the picture
of two opposing, state-supported armies gradually morphs, in U.S. Presidential speeches, into
images of a nation facing “a broad and adaptive terrorist network” [296], or “at war against
a far-reaching network of violence and hatred” [328].
1.1 Aspects of Networks
How should we think about networks, at a more precise level, so as to bring all these issues
together? In the most basic sense, a network is any collection of objects in which some pairs
of these objects are connected by links. This definition is very flexible: depending on the
setting, many di↵erent forms of relationships or connections can be used to define links.
Because of this flexibility, it is easy to find networks in many domains, including the ones
we’ve just been discussing. As a first example of what a network looks like, Figure 1.1 depicts
the social network among 34 people in a university karate club studied by the anthropologist
Wayne Zachary in the 1970s. The people are represented by small circles, with lines joining
the pairs of people who are friends outside the context of the club. This is the typical way
in which networks will be drawn, with lines joining the pairs of objects that are connected
1.1. ASPECTS OF NETWORKS 3
Figure 1.2: Social networks based on communication and interaction can also be constructed from the traces left by on-line data. In this case, the pattern of e- mail communication among 436 employees of Hewlett Packard Research Lab is su- perimposed on the o�cial organizational hierarchy [6]. (Image from http://www- personal.umich.edu/ ladamic/img/hplabsemailhierarchy.jpg)
by links.
Later in this chapter we’ll discuss some of the things one can learn from a network such as
the one in Figure 1.1, as well as from larger examples such as the ones shown in Figures 1.2–
1.4. These larger examples depict, respectively, e-mail exchanges among employees of a
company; loans among financial institutions; and links among blogs on the Web. In each
case, links indicate the pairs who are connected (specifically, people connected by e-mail
exchange, financial institutions by a borrower-lender relationship, and blogs through a link
on the Web from one to the other).
Simply from their visual appearance, we can already see some of the complexity that net-
work structures contain. It is generally di�cult to summarize the whole network succinctly;
there are parts that are more or less densely interconnected, sometimes with central “cores”
containing most of the links, and sometimes with natural splits into multiple tightly-linked
regions. Participants in the network can be more central or more peripheral; they can strad-
dle the boundaries of di↵erent tightly-linked regions or sit squarely in the middle of one.
Developing a language for talking about the typical structural features of networks will be
an important first step in understanding them.
4 CHAPTER 1. OVERVIEW
Figure 1.3: The network of loans among financial institutions can be used to analyze the roles that di↵erent participants play in the financial system, and how the interactions among these roles a↵ect the health of individual participants and the system as a whole The network here is annotated in a way that reveals its dense core, according to a scheme we will encounter in Chapter 13. (Image from Bech and Atalay [50].)
Behavior and Dynamics. But the structure of the network is only a starting point.
When people talk about the “connectedness” of a complex system, in general they are really
talking about two related issues. One is connectedness at the level of structure — who is
linked to whom — and the other is connectedness at the level of behavior — the fact that
each individual’s actions have implicit consequences for the outcomes of everyone in the
system.
This means that in addition to a language for discussing the structure of networks, we
also need a framework for reasoning about behavior and interaction in network contexts.
And just as the underlying structure of a network can be complex, so too can the coupled
behavior of its inhabitants. If individuals have strong incentives to achieve good outcomes,
1.1. ASPECTS OF NETWORKS 5
Figure 1.4: The links among Web pages can reveal densely-knit communities and prominent sites. In this case, the network structure of political blogs prior to the 2004 U.S. Presiden- tial election reveals two natural and well-separated clusters [5]. (Image from http://www- personal.umich.edu/ ladamic/img/politicalblogs.jpg)
then not only will they appreciate that their outcomes depend on how others behave, but they
will take this into account in planning their own actions. As a result, models of networked
behavior must take strategic behavior and strategic reasoning into account.
A fundamental point here is that in a network setting, you should evaluate your actions
not in isolation, but with the expectation that the world will react to what you do. This
means that cause-e↵ect relationships can become quite subtle. Changes in a product, a Web
site, or a government program can seem like good ideas when evaluated on the assumption
that everything else will remain static, but in reality such changes can easily create incentives
that shift behavior across the network in ways that were initially unintended.
Moreover, such e↵ects are at work whether we are able to see the network or not. When
a large group of people is tightly interconnected, they will often respond in complex ways
that are only apparent at the population level, even though these e↵ects may come from
implicit networks that we do not directly observe. Consider, for example, the way in which
new products, Web sites, or celebrities rise to prominence — as illustrated, for example, by
Figures 1.5 and 1.6, which show the growth in popularity of the social media sites YouTube
6 CHAPTER 1. OVERVIEW
Search volume for YouTube
1.0
2.0
Figure 1.5: The rapidly growing popularity of YouTube is characteristic of the way in which new products, technologies, or innovations rise to prominence, through feedback e↵ects in the behavior of many individuals across a population. The plot depicts the number of Google queries for YouTube over time. The image comes from the site Google Trends (http://www.google.com/trends?q=youtube); by design, the units on the y-axis are sup- pressed in the output from this site.
and Flickr over the past several years. What we see in these figures is a growing awareness
and adoption of a new innovation that is visible in aggregate, across a whole population.
What are the underlying mechanisms that lead to such success? Standard refrains are
often invoked in these situations: the rich get richer; winners take all; small advantages are
magnified to a critical mass; new ideas get attention that becomes “viral.” But the rich don’t
always get richer and small advantages don’t always lead to success. Some social networking
sites flourish, like Facebook, while others, like SixDegrees.com, vanish. To understand how
these processes work, and how they are realized through the interconnected actions of many
people, we need to study the dynamics of aggregate behavior.
A Confluence of Ideas. Understanding highly connected systems, then, requires a set
of ideas for reasoning about network structure, strategic behavior, and the feedback e↵ects
they produce across large populations. These are ideas that have traditionally been dis-
persed across many di↵erent disciplines. However, in parallel with the increasing public
interest in networks, there has been a coming-together of scientific fields around the topic of
network research. Each of these fields brings important ideas to the discussion, and a full
understanding seems to require a synthesis of perspectives from all of them.
One of our central goals in this book is to help bring about such a synthesis, combining
approaches that have traditionally been pursued separately. From computer science, ap-
plied mathematics, and operations research we draw on a language for talking about the
complexity of network structure, information, and systems with interacting agents. From
1.1. ASPECTS OF NETWORKS 7
Search volume for Flickr
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Figure 1.6: This companion to Figure 1.5 shows the rise of the social media site Flickr; the growth in popularity has a very similar pattern to that of other sites including YouTube. (Image from Google Trends, http://www.google.com/trends?q=flickr)
economics we draw on models for the strategic behavior of individuals who interact with
each other and operate as members of larger aggregates. From sociology — particularly the
more mathematical aspects concerned with social networks — we draw on a broad set of
theoretical frameworks for talking about the structure and dynamics of social groups.
And the overall picture can help fill in pieces that are arguably missing from the intel-
lectual landscape of each of these disciplines. Economics has developed rich theories for the
strategic interaction among small numbers of parties, as well as for the cumulative behavior
of large, homogeneous populations. The challenge it faces is that much of economic life
takes place in the complex spectrum between these extremes, with macroscopic e↵ects that
arise from an intricate pattern of localized interactions. Sociology has developed some of the
fundamental insights into the structure of social networks, but its network methodology has
been refined in the domains and scales where data-collection has traditionally been possible
— primarily, well-defined groups with tens to hundreds of people. The explosion of new con-
texts where we find network data and network applications — including enormous, digitally
mediated ones — leads to new opportunities for how we can pose questions, formulate theo-
ries, and evaluate predictions about social networks. Computer science, with the rise of the
Web and social media, has had to deal with a world in which the design constraints on large
computing systems are not just technological ones but also human ones — imposed by the
complex feedback e↵ects that human audiences create when they collectively use the Web for
communication, self-expression, and the creation of knowledge. A fully satisfactory theory
of network structure and behavior has the potential to address the simultaneous challenges
that all these fields are encountering.
A recurring theme underlying these challenges is the way in which networks span many
di↵erent levels of scale and resolution. There are interesting questions that reach from the
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Figure 1.7: From the social network of friendships in the karate club from Figure 1.1, we can find clues to the latent schism that eventually split the group into two separate clubs (indicated by the two di↵erent shadings of individuals in the picture).
scale of small groups, such as the 34-person social network in Figure 1.1, all the way up to
the level of whole societies or economies, or to the body of global knowledge represented by
the Web. We will think of networks both at the level of explicit structures, like those in
Figures 1.1–1.4, and at the level of aggregate e↵ects, like the popularity curves in Figures 1.5
and 1.6. As we look at networks of increasing scales, it becomes correspondingly more
appropriate to take aggregate models into account. But the ability to work with massive
network datasets has also enriched the picture, making it possible to study networks with
billions of interacting items at a level of resolution where each connection is recorded. When
an Internet search engine identifies the most useful pages from an index of the entire Web,
for example, it is doing precisely this in the context of a specific task. Ultimately, it is an
ongoing and challenging scientific problem to bridge these vastly di↵erent levels of scale, so
that predictions and principles from one level can be reconciled with those of others.
1.2 Central Themes and Topics
With this set of ideas in mind, we now introduce some of the main topics the book will
consider, and the ways in which these topics reinforce the underlying principles of networks.
We begin with the two main bodies of theory that we will be building on — graph theory
and game theory. These are theories of structure and behavior respectively: Graph theory
is the study of network structure, while game theory provides models of individual behavior
1.2. CENTRAL THEMES AND TOPICS 9
in settings where outcomes depend on the behavior of others.
Graph Theory. In our discussion of graph theory, we will focus particularly on some of
the fundamental ideas from social network analysis, framing a number of graph-theoretic
concepts in these terms. The networks in Figures 1.1 and 1.2 hint at some of these ideas. In
the corporate e-mail communication network from Figure 1.2, for example, we can see how
the communication is balanced between staying within small organizational units and cutting
across organizational boundaries. This is an example of a much more general principle in
social networks — that strong ties, representing close and frequent social contacts, tend to
be embedded in tightly-linked regions of the network, while weak ties, representing more
casual and distinct social contacts, tend to cross between these regions. Such a dichotomy
suggests a way of thinking about social networks in terms of their dense pockets of strong ties,
and the ways in which they interact with each other through weaker ties. In a professional
setting, it suggests a strategy for navigating one’s way through the social landscape of a
large organization, by finding the structural holes between parts of the network that interact
very little with each other. At a global scale, it suggests some of the ways in which weak
ties can act as “short-cuts” that link together distant parts of the world, resulting in the
phenomenon colloquially known as the six degrees of separation.
Social networks can also capture the sources of conflict within a group. For example,
latent conflicts are at work in the karate-club social network from Figure 1.1. The people
labeled 1 and 34 (the darker circles) are particularly central in the network of friendships,
with many connections to other people. On the other hand, they are not friends with each
other, and in fact most people are only friends with one or the other of them. These two
central people were, respectively, the instructor and the student founder of the club, and this
pattern of non-interacting clusters was the most visible symptom of a conflict between them
and their factions that ultimately splintered the group into two rival karate clubs, as shown
in Figure 1.7. Later, we will see how the theory of structural balance can be used to reason
about how fissures in a network may arise from the dynamics of conflict and antagonism at
a purely local level.
Game Theory. Our discussion of game theory starts from the observation that there
are numerous settings in which a group of people must simultaneously choose how to act,
knowing that the outcome will depend on the joint decisions made by all of them. One
natural example is the problem of choosing a driving route through a network of highways
at a time when tra�c is heavy. If you’re a driver in such a situation, the delays you experience
depend on the pattern of tra�c congestion arising not just from your choice of route, but
from the choices made by all other drivers as well. In this example, the network plays
the role of a shared resource, and the combined actions of its users can either congest this
10 CHAPTER 1. OVERVIEW
Figure 1.8: In a network representing international trade, one can look for countries that occupy powerful positions and derive economic benefits from these positions [262]. (Image from http://www.cmu.edu/joss/content/articles/volume4/KrempelPlumper.html)
resource or use it more e�ciently. In fact, the interactions among people’s behavior can lead
to counter-intuitive e↵ects here: for instance, adding resources to a transportation network
can in fact create incentives that seriously undermine its e�ciency, in a phenomenon known
as Braess’s Paradox [76].
Another example that will recur in several settings throughout the book is the problem
of bidding in an auction. If a seller is trying to sell a single item using an auction, then the
success of any one bidder in the auction (whether she gets the item, and how much she pays)
depends not just on how she bids but on how everyone else bids as well — and so an optimal
bidding strategy should take this into account. Here too there are counter-intuitive e↵ects at
work: for example, if the seller introduces more aggressive pricing rules into the auction, he
can make the strategic behavior of the bidders much more complex, and in particular induce
optimal bidding that o↵sets whatever gains he might have expected to make from the new
rules. We will find that auctions represent a basic kind of economic interaction that can be
1.2. CENTRAL THEMES AND TOPICS 11
Figure 1.9: In some settings, such as this map of Medieval trade routes, phys- ical networks constrain the patterns of interaction, giving certain participants an intrinsic economic advantage based on their network position. (Image from http://upload.wikimedia.org/wikipedia/commons/e/e1/Late Medieval Trade Routes.jpg.)
directly generalized to more complex patterns of interactions on networks.
As a general part of our investigation of game theory, we will abstract such situations
with inter-dependent behavior into a common framework, where a collection of individuals
must each commit to a strategy, thereby receiving a payo↵ that depends on the strategies
chosen by everyone. Interpreting our preceding examples in this light, the strategies available
to a driver on a set of highways consist of the di↵erent options for routes he can take, and
the payo↵ to this driver is based on his resulting travel time. In an auction, the strategies are
the di↵erent choices for how to bid, and the payo↵ to a bidder is the di↵erence between the
value of the goods she receives and the price she pays. This general framework allows us to
make predictions about how people will behave in a range of such situations. A fundamental
part of this framework will be the notion of equilibrium — a state that is “self-reinforcing,”
12 CHAPTER 1. OVERVIEW
in that it provides no individual with an incentive to unilaterally change his or her strategy,
even knowing how others will behave.
Markets and Strategic Interaction on Networks. Once we’ve developed graph theory
and game theory, we can combine them to produce richer models of behavior on networks.
One natural setting where we can explore this is in models of trade and other forms of
economic activity. The interactions among buyers and sellers, or pairs of counterparties to a
trade or loan, naturally forms a network. In Figure 1.3 we saw an example of such a network,
with links between banks engaging in a loan. Figure 1.8 shows another example: a network
representation of international trade among 28 countries [262], with the size of each country
depicting its total amount of trade, and the thickness of each link connecting two countries
indicating the amount of trade between them.
Where do these networks come from? In some cases, they are the traces of what happens
when each participant seeks out the best trading partner they can, guided by how highly
they value di↵erent trading opportunities. In other cases, they also reflect fundamental
underlying constraints in the market that limit the access of certain participants to each
other. In modern markets, these constraints could be institutional restrictions based on
regulations; in other settings, they could be based on physical constraints like geography.
For example, Figure 1.9 shows a map of trade routes in medieval Europe: when the physical
movement of goods is costly and di�cult, the economic outcome for di↵erent cities can
depend significantly on where they are located in the underlying transportation network.
In all these settings, then, the network structure encodes a lot about the pattern of
trade, with the success levels of di↵erent participants a↵ected by their positions in the
network. Having a powerful position, however, depends not just on having many connections
providing di↵erent options, but also on more subtle features — such as the power of the
other individuals to which one is connected. We will see that this idea of network positions
conferring power has been extended much more broadly, reaching beyond just economic
exchange to suggest how power imbalances in many forms of social relationships may have
their roots in the network patterns that the relationships form.
Information networks. The information we deal with on-line has a fundamental network
structure. Links among Web pages, for example, can help us to understand how these pages
are related, how they are grouped into di↵erent communities, and which pages are the most
prominent or important. Figure 1.4 illustrates some of these issues: it shows a network of
links among political blogs constructed by Lada Adamic and Natalie Glance in the period
leading up to the 2004 U.S. Presidential election [5]. Although the network is too large
here to be able to really see the detailed structure around individual blogs, the image and
its layout does convey the clear separation of the blogging network into two large clusters,
1.2. CENTRAL THEMES AND TOPICS 13
which turn out to closely correspond to the sets of liberal and conservative blogs respectively.
From more detailed analysis of the raw linkage data underlying the image, it is possible to
pick out the prominent blogs within each of these clusters.
Current Web search engines such as Google make extensive use of network structure in
evaluating the quality and relevance of Web pages. For producing search results, these sites
evaluate the prominence of a Web page not simply based on the number of links it receives,
but based on more subtle aspects of its position in the network. For example, a page can be
viewed as more prominent if it receives links from pages that are themselves prominent; this
is a circular kind of notion in which prominence is defined in terms of itself, but we will see
that this circularity can be resolved through careful definitions that are based on a kind of
equilibrium in the link structure.
The interaction between search engines and the authors of Web pages is also a compelling
example of a system where “connectedness” at the level of behavior produces interesting
e↵ects. Whenever a search engine introduces a new method for evaluating Web pages,
deciding which pages to rank highly in its results, the creators of Web content react to this:
they optimize what they put on the Web so as to try achieving a high rank under the new
method. As a result, changes to a search engine can never be designed under the assumption
that the Web will remain static; rather, the Web inevitably adapts to the ways in which
search engines evaluate content, and search methods must be developed with these feedback
e↵ects in mind.
This inherently game-theoretic interaction existed in latent form even in the early days of
the Web. Over time it became more explicit and formalized, through the design of markets
for advertising based on search, with advertising space allocated by auction mechanisms.
Today, such markets are a principal source of revenue for the main search engines.
Network Dynamics: Population E↵ects. If we observe a large population over time,
we’ll see a recurring pattern by which new ideas, beliefs, opinions, innovations, technologies,
products, and social conventions are constantly emerging and evolving. Collectively, we
can refer to these as social practices [382] (holding opinions, adopting products, behaving
according to certain principles) that people can choose to adopt or not. As we watch a
group or society over time, we’ll see that new practices can be introduced and either become
popular or remain obscure; meanwhile, established practices can persist or potentially fade
over time. If we think back to Figures 1.5 and 1.6, they show the adoption of particular
practices over time — the use of two very popular social media sites (taking the total number
of Google queries for these sites over time as proxies for their popularity). Figure 1.10 depicts
an analogous curve for the social-networking site MySpace, where we see a life cycle of rapid
adoption followed by a slower period of decline, as MySpace’s dominance was challenged by
newer competitors including Facebook.
14 CHAPTER 1. OVERVIEW
Figure 1.10: Cascading adoption of a new technology or service (in this case, the social- networking site MySpace in 2005-2006) can be the result of individual incentives to use the most widespread technology — either based on the informational e↵ects of seeing many other people adopt the technology, or the direct benefits of adopting what many others are already using. (Image from Google Trends, http://www.google.com/trends?q=myspace)
The way in which new practices spread through a population depends in large part on
the fact that people influence each other’s behavior. In short, as you see more and more
people doing something, you generally become more likely to do it as well. Understanding
why this happens, and what its consequences are, is a central issue for our understanding of
networks and aggregate behavior.
At a surface level, one could hypothesize that people imitate the decisions of others simply
because of an underlying human tendency to conform: we have a fundamental inclination
to behave as we see others behaving. This is clearly an important observation, but as an
explanation it leaves some crucial questions unresolved. In particular, by taking imitation as
a given, we miss the opportunity to ask why people are influenced by the behavior of others.
This is a broad and di�cult question, but in fact it is possible to identify multiple reasons
why even purely rational agents — individuals with no a priori desire to conform to what
others are doing — will nonetheless copy the behavior of others.
One class of reasons is based on the fact that the behavior of others conveys information.
You may have some private information on which to base a decision between alternatives,
but if you see many people making a particular choice, it is natural to assume that they too
have their own information, and to try inferring how people are evaluating di↵erent choices
from how they are behaving. In the case of a Web site like YouTube or Flickr, seeing a lot
of people using it can suggest that these people know something about its quality. Similarly,
seeing that a certain restaurant is extremely crowded every weekend can suggest that many
people think highly of it. But this sort of reasoning raises surprisingly subtle issues: as many
people make decisions sequentially over time, the later decisions can be based in complex
1.2. CENTRAL THEMES AND TOPICS 15
ways on a mixture of private information and inferences from what has already happened,
so that the actions of a large set of people can in fact be based on surprisingly little genuine
information. In an extreme form of this phenomenon we may get information cascades,
where even rational individuals can choose to abandon their private information and follow
a crowd.
There is a completely di↵erent but equally important class of reasons why people might
imitate the behavior of others — when there is a direct benefit from aligning your behavior
with that of others, regardless of whether they are making the best decision. Let’s go back
to our examples of social-networking and media-sharing sites. If the value of such sites is in
the potential to interact with others, to have access to a wide range of content, and to have
a large audience for the content you post, then these types of sites become more and more
valuable as people join them. In other words, regardless of whether YouTube had better
features than its competitors, once it became the most popular video-sharing site, there was
almost by definition an added value in using it. Such network e↵ects amplify the success of
products and technologies that are already doing well; in a market where network e↵ects are
at work, the leader can be hard to displace. Still, this type of dominance is not necessarily
permanent; as we will see, it is possible for a new technology to displace an old one if it
o↵ers something markedly di↵erent — and often when it starts in a part of the network
where there is room for it to take hold.
These considerations show how popularity — as a general phenomenon — is governed
by a “rich-get-richer” feedback process in which popularity tends to build on itself. It is
possible to build mathematical models for this process, with predictions for the distribution
of popularity that are borne out by empirical data — a picture in which society’s attention
is divided between a small number of prominent items and a “long tail” of more obscure
ones.
Network Dynamics: Structural E↵ects. As we’ve just seen, the question of how people
influence each other’s behavior is already quite subtle even when the actual structure of the
underlying network is left implicit. But taking network structure into account provides
important further insights into how such kinds of influence take place. The underlying
mechanisms — based on information and direct benefits — are present both at the level of
whole populations, and also at a local level in the network, between an individual and his
or her set of friends or colleagues. In many cases you care more about aligning your own
behavior with the behavior of your immediate neighbors in the social network, rather than
with the population as a whole.
When individuals have incentives to adopt the behavior of their neighbors in the network,
we can get cascading e↵ects, where a new behavior starts with a small set of initial adopters,
and then spreads radially outward through the network. Figure 1.11 shows a small example,
16 CHAPTER 1. OVERVIEW
Figure 1.11: When people are influenced by the behaviors their neighbors in the network, the adoption of a new product or innovation can cascade through the network structure. Here, e-mail recommendations for a Japanese graphic novel spread in a kind of informational or social contagion. (Image from Leskovec et al. [271].)
in which e-mail recommendations for a particular Japanese graphic novel spread outward
from four initial purchasers. By reasoning about the underlying network structure, we will
see how it becomes possible for a superior technology to displace a universally-used but
inferior one, if the superior technology starts in a portion of the network where it can
make progress incrementally, a few people at a time. We will also find that the di↵usion of
technologies can be blocked by the boundary of a densely-connected cluster in the network —
a “closed community” of individuals who have a high amount of linkage among themselves,
and hence are resistant to outside influences.
Cascading behavior in a network is sometimes referred to as “social contagion,” because
it spreads from one person to another in the style of a biological epidemic. Figure 1.12
reinforces this analogy; it shows the beginning of a tuberculosis outbreak [16] and forms a
visual counterpart to the social cascade in Figure 1.11. There are fundamental di↵erences
in the underlying mechanisms between social and biological contagion — social contagion
tends to involve decision-making on the part of the a↵ected individuals, whereas biological
contagion is based on the chance of catching a disease-causing pathogen through contact
with another individual. But the network-level dynamics are similar, and insights from the
study of biological epidemics are also useful in thinking about the processes by which things
spread on networks.
The act of spreading, which transmits both ideas and diseases, is just one kind of dynamic
1.2. CENTRAL THEMES AND TOPICS 17
Figure 1.12: The spread of an epidemic disease (such as the tuberculosis outbreak shown here) is another form of cascading behavior in a network. The similarities and contrasts between biological and social contagion lead to interesting research questions. (Image from Andre et al. [16].)
process that takes place on networks. A di↵erent process that we also consider is search —
the way people can explore chains of social contacts for information or referrals to others.
The surprising e↵ectiveness with which people are able to accomplish such tasks, confirmed
both by experiments and everyday experience, suggests characteristic patterns of structure
at the network level that help facilitate these types of activities.
Institutions and Aggregate Behavior. Once we have developed some of the basic forces
underlying networks and strategic behavior, we can ask how the institutions a society designs
can, in e↵ect, channel these forces to produce certain kinds of overall outcomes. Our notion
of an institution here is very broad — it can be any set of rules, conventions, or mechanisms
that serve to synthesize individual actions into a pattern of aggregate behavior. We’ve
already discussed particular examples of this process: for example, in the way in which a
particular auction mechanism leads to bidding behavior and hence prices; or the way in
which the Internet search industry has become a significant influence on how Web content
is created.
18 CHAPTER 1. OVERVIEW
100
90
80
70
60
50
40
30
20
10
0
Figure 1.13: Prediction markets, as well as markets for financial assets such as stocks, can synthesize individual beliefs about future events into a price that captures the aggregate of these beliefs. The plot here depicts the varying price over time for two assets that paid $1 in the respective events that the Democratic or Republican nominee won the 2008 U.S. Presidential election. (Image from Iowa Electronic Markets, http://iemweb.biz.uiowa.edu/graphs/graph PRES08 WTA.cfm.)
There are a number of settings in which this kind of analysis, applied to fundamental
social institutions, can be very informative. One such setting is to think about markets and
their role in aggregating and conveying information. In a financial market, for example,
the market price serves as an aggregator of individuals’ beliefs about the value of the assets
being traded. In this sense, the overall behavior of the market serves to synthesize the
information that is held by many participants; consequently, when people speak of what the
market “expects,” they are really referring to the expectations that can be read out of this
composite of information.
How this synthesis works depends on how the market is designed, and on the kind of
individual and aggregate behavior that results. Nor are such issues restricted to markets
for financial assets such as stocks. Recent work, for example, has explored the design of
1.2. CENTRAL THEMES AND TOPICS 19
prediction markets that use a market mechanism to provide predictions of future events such
as the outcomes of elections. Here, participants in the market purchase assets that pay a
fixed amount if a certain event takes place. In this way, the price of the asset reflects an
aggregate estimate for the probability of the event, and such estimates have been found to
be highly accurate in a number of cases — with the market’s aggregate predictions often
outperforming the opinions of expert analysts. Figure 1.13 shows an example from the
2008 U.S. Presidential Election: the upper curve depicts the price over time for an asset
that paid $1 in the event that the Democratic Party’s nominee won the election, and the
lower curve depicts the corresponding price for the Republican Party’s nominee. Note that
the market was already functioning before the identities of these nominees were known,
and it shows a clear aggregate reaction to certain events such as the contentious end of the
Democratic primary process between Obama and Clinton (in early May) and the Republican
National Convention (in early September), both of which brought the prices for the opposing
predictions close to equality, before they diverged once and for all as the actual election
neared.
Voting is another social institution that aggregates behavior across a population. While
markets and voting systems both seek a synthesis of individual beliefs or preferences, there
are some fundamental contrasts in the settings where they are generally applied. We have just
outlined a view of markets as aggregators of beliefs about the probabilities of future events.
In this view, each individual belief that forms an ingredient of the market’s consensus will
ultimately be confirmed as correct or incorrect, based on whether certain relevant future
events actually happen or not. Voting systems, on the other hand, are typically applied
to cases where each individual has a preference or prioritization over a set of arbitrary and
subjective choices for which there may be no eventual way to say that any one is “right” or
“wrong.” The question is then to synthesize a cumulative social preference that reconciles,
as well as possible, the conflicting priorities of the individuals in the population. In our
analysis of voting, we will explore a long history of work showing that the task of producing
such a social preference is fraught with unavoidable di�culties — results that formalize such
di�culties began with work of 18th-century French philosophers, and came fully into focus
with Arrow’s Impossibility Theorem in the 1950s.
This perspective on institutions is a natural one for social systems that are highly inter-
connected. Whenever the outcomes across a population depend on an aggregate of everyone’s
behavior, the design of the underlying institutions can have a significant e↵ect on how this
behavior is shaped, and on the resulting consequences for society.
Looking ahead. Examples, phenomena, and principles such as these will motivate the
ways in which we analyze networks, behavior, and population-level dynamics throughout the
book. Understanding whether a principle holds across many settings will involve formulating
20 CHAPTER 1. OVERVIEW
and reasoning about mathematical models, and also reasoning qualitatively about these
models and searching for their broader implications. In this way, we can hope to develop
a network perspective as a powerful way of looking at complex systems in general — a
way of thinking about social dynamics, economic interaction, on-line information, designed
technology, and natural processes, and approaching such systems with an eye toward their
patterns of internal structure and the rich feedback e↵ects that result.
Part I
Graph Theory and Social Networks
21
Chapter 2
Graphs
In this first part of the book we develop some of the basic ideas behind graph theory,
the study of network structure. This will allow us to formulate basic network properties in a
unifying language. The central definitions here are simple enough that we can describe them
relatively quickly at the outset; following this, we consider some fundamental applications
of the definitions.
2.1 Basic Definitions
Graphs: Nodes and Edges. A graph is a way of specifying relationships among a collec-
tion of items. A graph consists of a set of objects, called nodes, with certain pairs of these
objects connected by links called edges. For example, the graph in Figure 2.1(a) consists
of 4 nodes labeled A, B, C, and D, with B connected to each of the other three nodes by
edges, and C and D connected by an edge as well. We say that two nodes are neighbors if
they are connected by an edge. Figure 2.1 shows the typical way one draws a graph — with
little circles representing the nodes, and a line connecting each pair of nodes that are linked
by an edge.
In Figure 2.1(a), you should think of the relationship between the two ends of an edge as
being symmetric; the edge simply connects them to each other. In many settings, however,
we want to express asymmetric relationships — for example, that A points to B but not
vice versa. For this purpose, we define a directed graph to consist of a set of nodes, as
before, together with a set of directed edges; each directed edge is a link from one node
to another, with the direction being important. Directed graphs are generally drawn as in
Figure 2.1(b), with edges represented by arrows. When we want to emphasize that a graph
is not directed, we can refer to it as an undirected graph; but in general the graphs we discuss
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
23
24 CHAPTER 2. GRAPHS
B
A
C D
(a) A graph on 4 nodes.
B
A
C D
(b) A directed graph on 4 nodes.
Figure 2.1: Two graphs: (a) an undirected graph, and (b) a directed graph.
will be undirected unless noted otherwise.
Graphs as Models of Networks. Graphs are useful because they serve as mathematical
models of network structures. With this in mind, it is useful before going further to replace
the toy examples in Figure 2.1 with a real example. Figure 2.2 depicts the network structure
of the Internet — then called the Arpanet — in December 1970 [214], when it had only 13
sites. Nodes represent computing hosts, and there is an edge joining two nodes in this picture
if there is a direct communication link between them. Ignoring the superimposed map of the
U.S. (and the circles indicating blown-up regions in Massachusetts and Southern California),
the rest of the image is simply a depiction of this 13-node graph using the same dots-and-lines
style that we saw in Figure 2.1. Note that for showing the pattern of connections, the actual
placement or layout of the nodes is immaterial; all that matters is which nodes are linked
to which others. Thus, Figure 2.3 shows a di↵erent drawing of the same 13-node Arpanet
graph.
Graphs appear in many domains, whenever it is useful to represent how things are either
physically or logically linked to one another in a network structure. The 13-node Arpanet in
Figures 2.2 and 2.3 is an example of a communication network, in which nodes are computers
or other devices that can relay messages, and the edges represent direct links along which
messages can be transmitted. In Chapter 1, we saw examples from two other broad classes of
graph structures: social networks, in which nodes are people or groups of people, and edges
represent some kind of social interaction; and information networks, in which the nodes
are information resources such as Web pages or documents, and edges represent logical
2.2. PATHS AND CONNECTIVITY 25
Figure 2.2: A network depicting the sites on the Internet, then known as the Arpanet, in December 1970. (Image from F. Heart, A. McKenzie, J. McQuillian, and D. Walden [214]; on-line at http://som.csudh.edu/cis/lpress/history/arpamaps/.)
connections such as hyperlinks, citations, or cross-references. The list of areas in which
graphs play a role is of course much broader than what we can enumerate here; Figure 2.4
gives a few further examples, and also shows that many images we encounter on a regular
basis have graphs embedded in them.
2.2 Paths and Connectivity
We now turn to some of the fundamental concepts and definitions surrounding graphs. Per-
haps because graphs are so simple to define and work with, an enormous range of graph-
theoretic notions have been studied; the social scientist John Barnes once described graph
theory as a “terminological jungle, in which any newcomer may plant a tree” [45]. Fortu-
nately, for our purposes, we will be able to get underway with just a brief discussion of some
of the most central concepts.
26 CHAPTER 2. GRAPHS
LINC
CASE
CARN
HARV
BBN
MIT
SDC
RAND
UTAHSRI
UCLA
STANUCSB
Figure 2.3: An alternate drawing of the 13-node Internet graph from December 1970.
Paths. Although we’ve been discussing examples of graphs in many di↵erent areas, there
are clearly some common themes in the use of graphs across these areas. Perhaps foremost
among these is the idea that things often travel across the edges of a graph, moving from
node to node in sequence — this could be a passenger taking a sequence of airline flights, a
piece of information being passed from person to person in a social network, or a computer
user or piece of software visiting a sequence of Web pages by following links.
This idea motivates the definition of a path in a graph: a path is simply a sequence of
nodes with the property that each consecutive pair in the sequence is connected by an edge.
Sometimes it is also useful to think of the path as containing not just the nodes but also the
sequence of edges linking these nodes. For example, the sequence of nodes mit, bbn, rand,
ucla is a path in the Internet graph from Figures 2.2 and 2.3, as is the sequence case,
lincoln, mit, utah, sri, ucsb. As we have defined it here, a path can repeat nodes: for
example, sri, stan, ucla, sri, utah, mit is a path. But most paths we consider will not
do this; if we want to emphasize that the path we are discussing does not repeat nodes, we
can refer to it as a simple path.
Cycles. A particularly important kind of non-simple path is a cycle, which informally is a
“ring” structure such as the sequence of nodes linc, case, carn, harv, bbn, mit, linc
on the right-hand-side of Figure 2.3. More precisely, a cycle is a path with at least three
edges, in which the first and last nodes are the same, but otherwise all nodes are distinct.
There are many cycles in Figure 2.3: sri, stan, ucla, sri is as short an example as possible
according to our definition (since it has exactly three edges), while sri, stan, ucla, rand,
bbn, mit, utah, sri is a significantly longer example.
In fact, every edge in the 1970 Arpanet belongs to a cycle, and this was by design: it means
that if any edge were to fail (e.g. a construction crew accidentally cut through the cable),
there would still be a way to get from any node to any other node. More generally, cycles
2.2. PATHS AND CONNECTIVITY 27
(a) Airline routes (b) Subway map
(c) Flowchart of college courses (d) Tank Street Bridge in Brisbane
Figure 2.4: Images of graphs arising in di↵erent domains. The depictions of airline and subway systems in (a) and (b) are examples of transportation networks, in which nodes are destinations and edges represent direct connections. Much of the terminology surrounding graphs derives from metaphors based on transporta- tion through a network of roads, rail lines, or airline flights. The prerequisites among college courses in (c) is an example of a dependency network, in which nodes are tasks and directed edges indicate that one task must be performed before another. The design of complex software systems and industrial processes often requires the analysis of enormous dependency networks, with important consequences for e�cient scheduling in these settings. The Tank Street Bridge from Brisbane, Australia shown in (d) is an example of a structural network, with joints as nodes and physical linkages as edges. The internal frameworks of mechanical structures such as buildings, vehicles, or human bodies are based on such networks, and the area of rigidity theory, at the inter- section of geometry and mechanical engineering, studies the stability of such structures from a graph-based perspective [388]. (Images: (a) www.airlineroutemaps.com/USA/Northwest Airlines asia pacific.shtml, (b) www.wmata.com/metrorail/systemmap.cfm, (c) www.cs.cornell.edu/ugrad/flowchart.htm.)
28 CHAPTER 2. GRAPHS
F
G H
J
I
K
L
A
B
C
E
D
M
Figure 2.5: A graph with three connected components.
in communication and transportation networks are often present to allow for redundancy —
they provide for alternate routings that go the “other way” around the cycle. In the social
network of friendships too, we often notice cycles in everyday life, even if we don’t refer to
them as such. When you discover, for example, that your wife’s cousin’s close friend from
high school is in fact someone who works with your brother, this is a cycle — consisting
of you, your wife, her cousin, his high-school-friend, his co-worker (i.e. your brother), and
finally back to you.
Connectivity. Given a graph, it is natural to ask whether every node can reach every
other node by a path. With this in mind, we say that a graph is connected if for every pair of
nodes, there is a path between them. For example, the 13-node Arpanet graph is connected;
and more generally, one expects most communication and transportation networks to be
connected — or at least aspire to be connected — since their goal is to move tra�c from
one node to another.
On the other hand, there is no a priori reason to expect graphs in other settings to be
connected — for example, in a social network, you could imagine that there might exist two
people for which it’s not possible to construct a path from one to the other. Figures 2.5
and 2.6 give examples of disconnected graphs. The first is a toy example, while the second
is built from the collaboration graph at a biological research center [134]: nodes represent
2.2. PATHS AND CONNECTIVITY 29
Figure 2.6: The collaboration graph of the biological research center Structural Genomics of Pathogenic Protozoa (SGPP) [134], which consists of three distinct connected components. This graph was part of a comparative study of the collaboration patterns graphs of nine research centers supported by NIH’s Protein Structure Initiative; SGPP was an intermediate case between centers whose collaboration graph was connected and those for which it was fragmented into many small components.
researchers, and there is an edge between two nodes if the researchers appear jointly on a
co-authored publication. (Thus the edges in this second figure represent a particular formal
definition of collaboration — joint authorship of a published paper — and do not attempt to
capture the network of more informal interactions that presumably take place at the research
center.)
Components. Figures 2.5 and 2.6 make visually apparent a basic fact about disconnected
graphs: if a graph is not connected, then it breaks apart naturally into a set of connected
“pieces,” groups of nodes so that each group is connected when considered as a graph in
isolation, and so that no two groups overlap. In Figure 2.5, we see that the graph consists
of three such pieces: one consisting of nodes A and B, one consisting of nodes C, D, and E,
and one consisting of the rest of the nodes. The network in Figure 2.6 also consists of three
pieces: one on three nodes, one on four nodes, and one that is much larger.
To make this notion precise, we we say that a connected component of a graph (often
shortened just to the term “component”) is a subset of the nodes such that: (i) every node
in the subset has a path to every other; and (ii) the subset is not part of some larger set
with the property that every node can reach every other. Notice how both (i) and (ii)
30 CHAPTER 2. GRAPHS
are necessary to formalize the intuitive definition: (i) says that the component is indeed
internally connected, and (ii) says that it really is a free-standing “piece” of the graph, not
a connected part of a larger piece. (For example, we would not think of the set of nodes F ,
G, H, and J in Figure 2.5 as forming a component, because this set violates part (ii) of the
definition: although there are paths among all pairs of nodes in the set, it belongs to the
larger set consisting of F -M , in which all pairs are also linked by paths.)
Dividing a graph into its components is of course only a first, global way of describing
its structure. Within a given component, there may be richer internal structure that is
important to one’s interpretation of the network. For example, thinking about the largest
component from Figure 2.6 in light of the collaborations that it represents, one notices certain
suggestive features of the structure: a prominent node at the center, and tightly-knit groups
linked to this node but not to each other. One way to formalize the role of the prominent
central node is to observe that the largest connected component would break apart into three
distinct components if this node were removed. Analyzing a graph this way, in terms of its
densely-connected regions and the boundaries between them, is a powerful way of thinking
about network structure, and it will be a central topic in Chapter 3.
Giant Components. There turns out to be a useful qualitative way of thinking about
the connected components of typical large networks, and for this it helps to begin with the
following thought experiment. Consider the social network of the entire world, with a link
between two people if they are friends. Now, of course, this is a graph that we don’t actually
have explicitly recorded anywhere, but it is one where we can use our general intuitions to
answer some basic questions.
First, is this global friendship network connected? Presumably not. After all, connec-
tivity is a fairly brittle property, in that the behavior of a single node (or a small set of
nodes) can negate it. For example, a single person with no living friends would constitute
a one-node component in the global friendship network, and hence the graph would not be
connected. Or the canonical “remote tropical island,” consisting of people who have had
no contact with the outside world, would also be a small component in the network, again
showing that it is not connected.
But there is something more going on here. If you’re a typical reader of this book, then
you have friends who grew up in other countries. You’re in the same component as all these
friends, since you have a path (containing a single edge) to each of them. Now, if you consider,
say, the parents of these friends, your friends’ parents’ friends, their friends and descendants,
then all of these people are in the same component as well — and by now, we’re talking
about people who have never heard of you, may well not share a language with you, may
have never traveled anywhere near where you live, and may have had enormously di↵erent
life experiences. So even though the global friendship network may not be connected, the
2.2. PATHS AND CONNECTIVITY 31
component you inhabit seems very large indeed — it reaches into most parts of the world,
includes people from many di↵erent backgrounds, and seems in fact likely to contain a
significant fraction of the world’s population.
This is in fact true when one looks across a range of network datasets — large, complex
networks often have what is called a giant component, a deliberately informal term for a
connected component that contains a significant fraction of all the nodes. Moreover, when
a network contains a giant component, it almost always contains only one. To see why, let’s
go back to the example of the global friendship network and try imagining that there were
two giant components, each with hundreds of millions of people. All it would take is a single
edge from someone in the first of these components to someone in the second, and the two
giant components would merge into a single component. Just a single edge — in most cases,
it’s essentially inconceivable that some such edge wouldn’t form, and hence two co-existing
giant components are something one almost never sees in real networks. When there is a
giant component, it is thus generally unique, distinguishable as a component that dwarfs all
others.
In fact, in some of the rare cases when two giant components have co-existed for a long
time in a real network, their merging has been sudden, dramatic, and ultimately catastrophic.
For example, Jared Diamond’s book Guns, Germs, and Steel [130] devotes much of its
attention to the cataclysm that befell the civilizations of the Western hemisphere when
European explorers began arriving in it roughly half a millenium ago. One can view this
development from a network perspective as follows: five thousand years ago, the global
social network likely contained two giant components — one in the Americas, and one in
the Europe-Asia land mass. Because of this, technology evolved independently in the two
components, and perhaps even worse, human diseases evolved independently; and so when
the two components finally came in contact, the technology and diseases of one quickly and
disastrously overwhelmed the other.
The notion of giant components is useful for reasoning about networks on much smaller
scales as well. The collaboration network in Figure 2.6 is one simple example; another
interesting example is depicted in Figure 2.7, which shows the romantic relationships in an
American high school over an 18-month period [49]. (These edges were not all present at
once; rather, there is an edge between two people if they were romantically involved at any
point during the time period.) The fact that this graph contains such a large component is
significant when one thinks about the spread of sexually transmitted diseases, a focus of the
researchers performing the study. A high-school student may have had a single partner over
this time period and nevertheless — without realizing it — be part of this large component
and hence part of many paths of potential transmission. As Bearman, Moody, and Stovel
note in the paper where they analyze this network, “These structures reflect relationships
that may be long over, and they link individuals together in chains far too long to be
32 CHAPTER 2. GRAPHS
Figure 2.7: A network in which the nodes are students in a large American high school, and an edge joins two who had a romantic relationship at some point during the 18-month period in which the study was conducted [49].
the subject of even the most intense gossip and scrutiny. Nevertheless, they are real: like
social facts, they are invisible yet consequential macrostructures that arise as the product of
individual agency.”
2.3 Distance and Breadth-First Search
In addition to simply asking whether two nodes are connected by a path, it is also interesting
in most settings to ask how long such a path is — in transportation, Internet communication,
or the spread of news and diseases, it is often important whether something flowing through
a network has to travel just a few hops or many.
To be able to talk about this notion precisely, we define the length of a path to be the
number of steps it contains from beginning to end — in other words, the number of edges
in the sequence that comprises it. Thus, for example, the path mit, bbn, rand, ucla in
Figure 2.3 has length three, while the path mit, utah has length one. Using the notion of
2.3. DISTANCE AND BREADTH-FIRST SEARCH 33
you
distance 1
distance 2
distance 3
your friends
friends of friends
friends of friends
of friends
all nodes, not already discovered, that have an
edge to some node in the previous layer
Figure 2.8: Breadth-first search discovers distances to nodes one “layer” at a time; each layer is built of nodes that have an edge to at least one node in the previous layer.
a path’s length, we can talk about whether two nodes are close together or far apart in a
graph: we define the distance between two nodes in a graph to be the length of the shortest
path between them. For example, the distance between linc and sri is three, though to
believe this you have to first convince yourself that there is no length-1 or length-2 path
between them.
Breadth-First Search. For a graph like the one in Figure 2.3, we can generally figure
out the distance between two nodes by eyeballing the picture; but for graphs that are even
a bit more complicated, we need some kind of a systematic method to determine distances.
The most natural way to do this — and also the most e�cient way to calculate distances
for a large network dataset using a computer — is the way you would probably do it if you
34 CHAPTER 2. GRAPHS
LINC
CASE
CARN
HARV
BBN
MIT
SDC RAND
UTAH
SRI
UCLA STANUCSB
distance 1
distance 2
distance 3
Figure 2.9: The layers arising from a breadth-first of the December 1970 Arpanet, starting at the node mit.
really needed to trace out distances in the global friendship network (and had the unlimited
patience and cooperation of everyone in the world). This is pictured in Figure 2.8:
(1) You first declare all of your actual friends to be at distance 1.
(2) You then find all of their friends (not counting people who are already friends of yours),
and declare these to be at distance 2.
(3) Then you find all of their friends (again, not counting people who you’ve already found
at distances 1 and 2) and declare these to be at distance 3.
(...) Continuing in this way, you search in successive layers, each representing the next
distance out. Each new layer is built from all those nodes that (i) have not already
been discovered in earlier layers, and that (ii) have an edge to some node in the previous
layer.
This technique is called breadth-first search, since it searches the graph outward from a start-
ing node, reaching the closest nodes first. In addition to providing a method of determining
distances, it can also serve as a useful conceptual framework to organize the structure of a
graph, arranging the nodes based on their distances from a fixed starting point.
2.3. DISTANCE AND BREADTH-FIRST SEARCH 35
Of course, despite the social-network metaphor we used to describe breadth-first search,
the process can be applied to any graph: one just keeps discovering nodes layer-by-layer,
building each new layer from the nodes that are connected to at least one node in the previous
layer. For example, Figure 2.9 shows how to discover all distances from the node mit in the
13-node Arpanet graph from Figure 2.3.
The Small-World Phenomenon. As with our discussion of the connected components
in a graph, there is something qualitative we can say, beyond the formal definitions, about
distances in typical large networks. If we go back to our thought experiments on the global
friendship network, we see that the argument explaining why you belong to a giant compo-
nent in fact asserts something stronger: not only do you have paths of friends connecting
you to a large fraction of the world’s population, but these paths are surprisingly short.
Take the example of a friend who grew up in another country: following a path through this
friend, to his or her parents, to their friends, you’ve followed only three steps and ended up
in a di↵erent part of the world, in a di↵erent generation, with people who have very little in
common with you.
This idea has been termed the small-world phenomenon — the idea that the world looks
“small” when you think of how short a path of friends it takes to get from you to almost
anyone else. It’s also known, perhaps more memorably, as the six degrees of separation; this
phrase comes from the play of this title by John Guare [200], and in particular from the line
uttered by one of the play’s characters: “I read somewhere that everybody on this planet is
separated by only six other people. Six degrees of separation between us and everyone else
on this planet.”
The first experimental study of this notion — and the origin of the number “six” in the
pop-cultural mantra — was performed by Stanley Milgram and his colleagues in the 1960s
[297, 391]. Lacking any of the massive social-network datasets we have today, and with a
budget of only $680, he set out to test the speculative idea that people are really connected in
the global friendship network by short chains of friends. To this end, he asked a collection of
296 randomly chosen “starters” to try forwarding a letter to a “target” person, a stockbroker
who lived in a suburb of Boston. The starters were each given some personal information
about the target (including his address and occupation) and were asked to forward the
letter to someone they knew on a first-name basis, with the same instructions, in order to
eventually reach the target as quickly as possible. Each letter thus passed through the hands
of a sequence of friends in succession, and each thereby formed a chain of people that closed
in on the stockbroker outside Boston.
Figure 2.10 shows the distribution of path lengths, among the 64 chains that succeeded
in reaching the target; the median length was six, the number that made its way two decades
later into the title of Guare’s play. That so many letters reached their destination, and by
36 CHAPTER 2. GRAPHS
Figure 2.10: A histogram from Travers and Milgram’s paper on their small-world experiment [391]. For each possible length (labeled “number of intermediaries” on the x-axis), the plot shows the number of successfully completed chains of that length. In total, 64 chains reached the target person, with a median length of six.
such short paths, was a striking fact when it was first discovered, and it remains so today.
Of course, it is worth noting a few caveats about the experiment. First, it clearly doesn’t
establish a statement quite as bold as “six degrees of separation between us and everyone
else on this planet” — the paths were just to a single, fairly a✏uent target; many letters
never got there; and attempts to recreate the experiment have been problematic due to lack
of participation [255]. Second, one can ask how useful these short paths really are to people
in society: even if you can reach someone through a short chain of friends, is this useful to
you? Does it mean you’re truly socially “close” to them? Milgram himself mused about this
in his original paper [297]; his observation, paraphrased slightly, was that if we think of each
person as the center of their own social “world,” then “six short steps” becomes “six worlds
apart” — a change in perspective that makes six sound like a much larger number.
Despite these caveats, the experiment and the phenomena that it hints at have formed
a crucial aspect in our understanding of social networks. In the years since the initial
experiment, the overall conclusion has been accepted in a broad sense: social networks tend
to have very short paths between essentially arbitrary pairs of people. And even if your six-
2.3. DISTANCE AND BREADTH-FIRST SEARCH 37
Figure 2.11: The distribution of distances in the graph of all active Microsoft Instant Mes- senger user accounts, with an edge joining two users if they communicated at least once during a month-long observation period [273].
step connections to CEOs and political leaders don’t yield immediate payo↵s on an everyday
basis, the existence of all these short paths has substantial consequences for the potential
speed with which information, diseases, and other kinds of contagion can spread through
society, as well as for the potential access that the social network provides to opportunities
and to people with very di↵erent characteristics from one’s own. All these issues — and
their implications for the processes that take place in social networks — are rich enough
that we will devote Chapter 20 to a more detailed study of the small-world phenomenon and
its consequences.
Instant Messaging, Paul Erdös, and Kevin Bacon. One reason for the current em-
pirical consensus that social networks generally are “small worlds” is that this has been
increasingly confirmed in settings where we do have full data on the network structure. Mil-
gram was forced to resort to an experiment in which letters served as “tracers” through a
global friendship network that he had no hope of fully mapping on his own; but for other
kinds of social network data where the full graph structure is known, one can just load it
into a computer and perform the breadth-first search procedure to determine what typical
38 CHAPTER 2. GRAPHS
Figure 2.12: Ron Graham’s hand-drawn picture of a part of the mathematics collaboration graph, centered on Paul Erdös [189]. (Image from http://www.oakland.edu/enp/cgraph.jpg)
distances look like.
One of the largest such computational studies was performed by Jure Leskovec and Eric
Horvitz [273]. They analyzed the 240 million active user accounts on Microsoft Instant
Messenger, building a graph in which each node corresponds to a user, and there is an
edge between two users if they engaged in a two-way conversation at any point during a
month-long observation period. As employees of Microsoft at the time, they had access to
a complete snapshot of the system for the month under study, so there were no concerns
about missing data. This graph turned out to have a giant component containing almost
all of the nodes, and the distances within this giant component were very small. Indeed,
the distances in the Instant Messenger network closely corresponded to the numbers from
Milgram’s experiment, with an estimated average distance of 6.6, and an estimated median
2.3. DISTANCE AND BREADTH-FIRST SEARCH 39
of seven. Figure 2.11 shows the distribution of distances averaged over a random sample
of 1000 users: breadth-first search was performed separately from each of these 1000 users,
and the results from these 1000 nodes were combined to produce the plot in the figure.
The reason for this estimation by sampling users is a computational one: the graph was
so large that performing breadth-first search from every single node would have taken an
astronomical amount of time. Producing plots like this e�ciently for massive graphs is an
interesting research topic in itself [338].
In a sense, the plot in Figure 2.11 starts to approximate, in a striking way, what Milgram
and his colleagues were trying to understand — the distribution of how far apart we all are
in the full global friendship network. At the same time, reconciling the structure of such
massive datasets with the underlying networks they are trying to measure is an issue that
comes up here, as it will many times throughout the book. In this case, enormous as the
Microsoft IM study was, it remains some distance away from Milgram’s goal: it only tracks
people who are technologically-endowed enough to have access to instant messaging, and
rather than basing the graph on who is truly friends with whom, it can only observe who
talks to whom during an observation period.
Turning to a smaller scale — at the level of hundred of thousands of people rather than
hundreds of millions — researchers have also discovered very short paths in the collaboration
networks within professional communities. In the domain of mathematics, for example,
people often speak of the itinerant mathematician Paul Erdös — who published roughly
1500 papers over his career — as a central figure in the collaborative structure of the field.
To make this precise, we can define a collaboration graph as we did for Figure 2.6, in this
case with nodes corresponding to mathematicians, and edges connecting pairs who have
jointly authored a paper. (While Figure 2.6 concerned a single research lab, we are now
talking about collaboration within the entire field of mathematics.) Figure 2.12 shows a
small hand-drawn piece of the collaboration graph, with paths leading to Paul Erdös [189].
Now, a mathematician’s Erdös number is the distance from him or her to Erdös in this graph
[198]. The point is that most mathematicians have Erdös numbers of at most 4 or 5, and —
extending the collaboration graph to include co-authorship across all the sciences — most
scientists in other fields have Erdös numbers that are comparable or only slightly larger;
Albert Einstein’s is 2, Enrico Fermi’s is 3, Noam Chomsky’s and Linus Pauling’s are each 4,
Francis Crick’s and James Watson’s are 5 and 6 respectively. The world of science is truly
a small one in this sense.
Inspired by some mixture of the Milgram experiment, John Guare’s play, and a compelling
belief that Kevin Bacon was the center of the Hollywood universe, three students at Albright
College in Pennsylvania sometime around 1994 adapted the idea of Erdös numbers to the
collaboration graph of movie actors and actresses: nodes are performers, an edge connects
two performers if they’ve appeared together in a movie, and a performer’s Bacon number is
40 CHAPTER 2. GRAPHS
his or her distance in this graph to Kevin Bacon [372]. Using cast lists from the Internet
Movie Database (IMDB), it is possible to compute Bacon numbers for all performers via
breadth-first search — and as with mathematics, it’s a small world indeed. The average
Bacon number, over all performers in the IMDB, is approximately 2.9, and it’s a challenge
to find one that’s larger than 5. Indeed, it’s fitting to conclude with a network-and-movie
enthusiast’s description of his late-night attempts to find the largest Bacon number in the
IMDB by hand: “With my life-long passion for movies, I couldn’t resist spending many
hours probing the dark recesses of film history until, at about 10 AM on Sunday, I found an
incredibly obscure 1928 Soviet pirate film, Plenniki Morya, starring P. Savin with a Bacon
number of 7, and whose supporting cast of 8 appeared nowhere else” [197]. One is left with
the image of a long exploration that arrives finally at the outer edge of the movie world —
in the early history of film, in the Soviet Union — and yet in another sense, only 8 steps
from where it started.
2.4 Network Datasets: An Overview
The explosion of research on large-scale networks in recent years has been fueled to a large
extent by the increasing availability of large, detailed network datasets. We’ve seen examples
of such datasets throughout these first two chapters, and it’s useful at this point to step
back and think more systematically about where people have been getting the data that
they employ in large-scale studies of networks.
To put this in perspective, we note first of all that there are several distinct reasons
why you might study a particular network dataset. One is that you may care about the
actual domain it comes from, so that fine-grained details of the data itself are potentially
as interesting as the broad picture. Another is that you’re using the dataset as a proxy
for a related network that may be impossible to measure — as for example in the way
the Microsoft IM graph from Figure 2.11 gave us information about distances in a social
network of a scale and character that begins to approximate the global friendship network.
A third possibility is that you’re trying to look for network properties that appear to be
common across many di↵erent domains, and so finding a similar e↵ect in unrelated settings
can suggest that it has a certain universal nature, with possible explanations that are not
tied to the specifics of any one of the domains.
Of course, all three of these motivations are often at work simultaneously, to varying
degrees, in the same piece of research. For example, the analysis of the Microsoft IM graph
gave us insight into the global friendship network — but at a more specific level, the re-
searchers performing the study were also interested in the dynamics of instant messaging
in particular; and at a more general level, the result of the IM graph analysis fit into the
broader framework of small-world phenomena that span many domains.
2.4. NETWORK DATASETS: AN OVERVIEW 41
As a final point, we’re concerned here with sources of data on networks that are large.
If one wants to study a social network on 20 people — say, within a small company, or a
fraternity or sorority, or a karate club as in Figure 1.1 — then one strategy is to interview
all the people involved and ask them who their friends are. But if we want to study the
interactions among 20,000 people, or 20,000 individual nodes of some other kind, then we
need to be more opportunistic in where we look for data: except in unusual cases, we can’t
simply go out and collect everything by hand, and so we need to think about settings in
which the data has in some essential way already been measured for us.
With this in mind, let’s consider some of the main sources of large-scale network data
that people have used for research. The resulting list is far from exhaustive, nor are the
categories truly distinct — a single dataset can easily exhibit characteristics from several.
• Collaboration Graphs. Collaboration graphs record who works with whom in a specific setting; co-authorships among scientists and co-appearance in movies by actors and
actresses are two examples of collaboration graphs that we discussed in Section 2.3.
Another example that has been extensively studied by sociologists is the graph on
highly-placed people in the corporate world, with an edge joining two if they have
served together on the board of directors of the same Fortune 500 company [301]. The
on-line world provides new instances: the Wikipedia collaboration graph (connecting
two Wikipedia editors if they’ve ever edited the same article) [122, 246] and the World-
of-Warcraft collaboration graph (connecting two W-o-W users if they’ve ever taken part
together in the same raid or other activity) [419] are just two examples.
Sometimes a collaboration graph is studied to learn about the specific domain it comes
from; for example, sociologists who study the business world have a substantive in-
terest in the relationships among companies at the director level, as expressed via
co-membership on boards. On the other hand, while there is a research community
that studies the sociological context of scientific research, a broader community of
people is interested in scientific co-authorship networks precisely because they form
detailed, pre-digested snapshots of a rich form of social interaction that unfolds over a
long period of time [318]. By using on-line bibliographic records, one can often track
the patterns of collaboration within a field across a century or more, and thereby at-
tempt to extrapolate how the social structure of collaboration may work across a range
of harder-to-measure settings as well.
• Who-talks-to-Whom Graphs. The Microsoft IM graph is a snapshot of a large commu- nity engaged in several billion conversations over the course of a month. In this way,
it captures the “who-talks-to-whom” structure of the community. Similar datasets
have been constructed from the e-mail logs within a company [6] or a university [259],
as well as from records of phone calls: researchers have studied the structure of call
42 CHAPTER 2. GRAPHS
graphs in which each node is a phone number, and there is an edge between two if they
engaged in a phone call over a given observation period [1, 334]. One can also use the
fact that mobile phones with short-range wireless technology can detect other similar
devices nearby. By equipping a group of experimental subjects with such devices and
studying the traces they record, researchers can thereby build “face-to-face” graphs
that record physical proximity: a node in such a graph is a person carrying one of the
mobile devices, and there is an edge joining two people if they were detected to be in
close physical proximity over a given observation period [141, 142].
In almost all of these kinds of datasets, the nodes represent customers, employees, or
students of the organization that maintains the data. These individuals will generally
have strong expectations of privacy, not necessarily even appreciating how easily one
can reconstruct details of their behavior from the digital traces they leave behind
when communicating by e-mail, instant messaging, or phone. As a result, the style of
research performed on this kind of data is generally restricted in specific ways so as to
protect the privacy of the individuals in the data. Such privacy considerations have
also become a topic of significant discussion in settings where companies try to use this
type of data for marketing, or when governments try to use it for intelligence-gathering
purposes [315].
Related to this kind of “who-talks-to-whom” data, economic network measurements
recording the “who-transacts-with-whom” structure of a market or financial commu-
nity has been used to study the ways in which di↵erent levels of access to market
participants can lead to di↵erent levels of market power and di↵erent prices for goods.
This empirical work has in turn motivated more mathematical investigations of how
a network structure limiting access between buyers and sellers can a↵ect outcomes
[63, 176, 232, 261], a focus of discussion in Chapters 10—12.
• Information Linkage Graphs. Snapshots of the Web are central examples of network datasets; nodes are Web pages and directed edges represent links from one page to
another. Web data stands out both in its scale and in the diversity of what the nodes
represent: billions of little pieces of information, with links wiring them together. And
clearly it is not just the information that is of interest, but the social and economic
structures that stand behind the information: hundreds of millions of personal pages on
social-networking and blogging sites, hundreds of millions more representing companies
and governmental organizations trying to engineer their external images in a crowded
network.
A network on the scale of the full Web can be daunting to work with; simply manipu-
lating the data e↵ectively can become a research challenge in itself. As a result, much
network research has been done on interesting, well-defined subsets of the Web, includ-
2.4. NETWORK DATASETS: AN OVERVIEW 43
ing the linkages among bloggers [264], among pages on Wikipedia [404], among pages
on social-networking sites such as Facebook or MySpace [185], or among discussions
and product reviews on shopping sites [201].
The study of information linkage graphs significantly predates the Web: the field of
citation analysis has, since the early part of the 20th century, studied the network
structure of citations among scientific papers or patents, as a way of tracking the
evolution of science [145]. Citation networks are still popular research datasets today,
for the same reason that scientific co-authorship graphs are: even if you don’t have a
substantive interest in the social processes by which science gets done, citation networks
are very clean datasets that can easily span many decades.
• Technological Networks. Although the Web is built on a lot of sophisticated technology, it would be a mistake to think of it primarily as a technological network: it is really a
projection onto a technological backdrop of ideas, information, and social and economic
structure created by humans. But as we noted in the opening chapter, there has
clearly been a convergence of social and technological networks over recent years, and
much interesting network data comes from the more overtly technological end of the
spectrum — with nodes representing physical devices and edges representing physical
connections between them. Examples include the interconnections among computers
on the Internet [155] or among generating stations in a power grid [411].
Even physical networks like these are ultimately economic networks as well, represent-
ing the interactions among the competing organizations, companies, regulatory bodies,
and other economic entities that shape it. On the Internet, this is made particularly
explicit by a two-level view of the network. At the lowest level, nodes are individual
routers and computers, with an edge meaning that two devices actually have a physical
connection to each other. But at a higher level, these nodes are grouped into what are
essentially little “nation-states” termed autonomous systems, each one controlled by a
di↵erent Internet service-providers. There is then a who-transacts-with-whom graph
on the autonomous systems, known as the AS graph, that represents the data transfer
agreements these Internet service-providers make with each other.
• Networks in the Natural World. Graph structures also abound in biology and the other natural sciences, and network research has devoted particular attention to several
di↵erent types of biological networks. Here are three examples at three di↵erent scales,
from the population level down to the molecular level.
As a first example, food webs represent the who-eats-whom relationships among species
in an ecosystem [137]: there is a node for each species, and a directed edge from node
A to node B indicates that members of A consume members of B. Understanding
the structure of a food web as a graph can help in reasoning about issues such as
44 CHAPTER 2. GRAPHS
cascading extinctions: if certain species become extinct, then species that rely on them
for food risk becoming extinct as well, if they do not have alternative food sources;
these extinctions can propagate through the food web as a chain reaction.
Another heavily-studied network in biology is the structure of neural connections within
an organism’s brain: the nodes are neurons, and an edge represents a connection
between two neurons [380]. The global brain architecture for simple organisms like
C. Elegans, with 302 nodes and roughly 7000 edges, has essentially been completely
mapped [3]; but obtaining a detailed network picture for brains of “higher” organisms is
far beyond the current state of the art. However, significant insight has been gained by
studying the structure of specific modules within a complex brain, and understanding
how they relate to one another.
A final example is the set of networks that make up a cell’s metabolism. There are
many ways to define these networks, but roughly, the nodes are compounds that play
a role in a metabolic process, and the edges represent chemical interactions among
them [43]. There is considerable hope that analysis of these networks can shed light
on the complex reaction pathways and regulatory feedback loops that take place inside
a cell, and perhaps suggest “network-centric” attacks on pathogens that disrupt their
metabolism in targeted ways.
2.5 Exercises
1. One reason for graph theory’s power as a modeling tool is the fluidity with which
one can formalize properties of large systems using the language of graphs, and then
systematically explore their consequences. In this first set of questions, we will work
through an example of this process using the concept of a pivotal node.
First, recall from Chapter 2 that a shortest path between two nodes is a path of the
minimum possible length. We say that a node X is pivotal for a pair of distinct nodes
Y and Z if X lies on every shortest path between Y and Z (and X is not equal to
either Y or Z).
For example, in the graph in Figure 2.13, node B is pivotal for two pairs: the pair
consisting of A and C, and the pair consisting of A and D. (Notice that B is not
pivotal for the pair consisting of D and E since there are two di↵erent shortest paths
connecting D and E, one of which (using C and F ) doesn’t pass through B. So B
is not on every shortest path between D and E.) On the other hand, node D is not
pivotal for any pairs.
(a) Give an example of a graph in which every node is pivotal for at least one pair of
nodes. Explain your answer.
2.5. EXERCISES 45
B
A
C D
E
F
Figure 2.13: In this example, node B is pivotal for two pairs: the pair consisting of A and C, and the pair consisting of A and D. On the other hand, node D is not pivotal for any pairs.
(b) Give an example of a graph in which every node is pivotal for at least two di↵erent
pairs of nodes. Explain your answer.
(c) Give an example of a graph having at least four nodes in which there is a single
node X that is pivotal for every pair of nodes (not counting pairs that include
X). Explain your answer.
2. In the next set of questions, we consider a related cluster of definitions, which seek to
formalize the idea that certain nodes can play a “gatekeeping” role in a network. The
first definition is the following: we say that a node X is a gatekeeper if for some other
two nodes Y and Z, every path from Y to Z passes through X. For example, in the
graph in Figure 2.14, node A is a gatekeeper, since it lies for example on every path
from B to E. (It also lies on every path between other pairs of nodes — for example,
the pair D and E, as well as other pairs.)
This definition has a certain “global” flavor, since it requires that we think about paths
in the full graph in order to decide whether a particular node is a gatekeeper. A more
“local” version of this definition might involve only looking at the neighbors of a node.
Here’s a way to make this precise: we say that a node X is a local gatekeeper if there
are two neighbors of X, say Y and Z, that are not connected by an edge. (That is,
for X to be a local gatekeeper, there should be two nodes Y and Z so that Y and Z
each have edges to X, but not to each other.) So for example, in Figure 2.14, node
A is a local gatekeeper as well as being a gatekeeper; node D, on the other hand, is a
local gatekeeper but not a gatekeeper. (Node D has neighbors B and C that are not
connected by an edge; however, every pair of nodes — including B and C — can be
connected by a path that does not go through D.)
So we have two new definitions: gatekeeper, and local gatekeeper. When faced with
46 CHAPTER 2. GRAPHS
A
B
D
E
C F
Figure 2.14: Node A is a gatekeeper. Node D is a local gatekeeper but not a gatekeeper.
new mathematical definitions, a strategy that is often useful is to explore them first
through examples, and then to assess them at a more general level and try to relate
them to other ideas and definitions. Let’s try this in the next few questions.
(a) Give an example (together with an explanation) of a graph in which more than
half of all nodes are gatekeepers.
(b) Give an example (together with an explanation) of a graph in which there are no
gatekeepers, but in which every node is a local gatekeeper.
3. When we think about a single aggregate measure to summarize the distances between
the nodes in a given graph, there are two natural quantities that come to mind. One is
the diameter, which we define to be the maximum distance between any pair of nodes
in the graph. Another is the average distance, which — as the term suggests — is the
average distance over all pairs of nodes in the graph.
In many graphs, these two quantities are close to each other in value. But there are
graphs where they can be very di↵erent.
(a) Describe an example of a graph where the diameter is more than three times as
large as the average distance.
(b) Describe how you could extend your construction to produce graphs in which the
diameter exceeds the average distance by as large a factor as you’d like. (That is,
for every number c, can you produce a graph in which the diameter is more than
c times as large as the average distance?)
Chapter 3
Strong and Weak Ties
One of the powerful roles that networks play is to bridge the local and the global — to
o↵er explanations for how simple processes at the level of individual nodes and links can have
complex e↵ects that ripple through a population as a whole. In this chapter, we consider
some fundamental social network issues that illustrate this theme: how information flows
through a social network, how di↵erent nodes can play structurally distinct roles in this
process, and how these structural considerations shape the evolution of the network itself
over time. These themes all play central roles throughout the book, adapting themselves
to di↵erent contexts as they arise. Our context in this chapter will begin with the famous
“strength of weak ties” hypothesis from sociology [190], exploring outward from this point
to more general settings as well.
Let’s begin with some backgound and a motivating question. As part of his Ph.D.
thesis research in the late 1960s, Mark Granovetter interviewed people who had recently
changed employers to learn how they discovered their new jobs [191]. In keeping with earlier
research, he found that many people learned information leading to their current jobs through
personal contacts. But perhaps more strikingly, these personal contacts were often described
by interview subjects as acquaintances rather than close friends. This is a bit surprising:
your close friends presumably have the most motivation to help you when you’re between
jobs, so why is it so often your more distant acquaintances who are actually to thank for
crucial information leading to your new job?
The answer that Granovetter proposed to this question is striking in the way it links
two di↵erent perspectives on distant friendships — one structural, focusing on the way
these friendships span di↵erent portions of the full network; and the other interpersonal,
considering the purely local consequences that follow from a friendship between two people
being either strong or weak. In this way, the answer transcends the specific setting of job-
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
47
48 CHAPTER 3. STRONG AND WEAK TIES
B
A
C
G
F
E D
(a) Before B-C edge forms.
B
A
C
G
F
E D
(b) After B-C edge forms.
Figure 3.1: The formation of the edge between B and C illustrates the e↵ects of triadic closure, since they have a common neighbor A.
seeking, and o↵ers a way of thinking about the architecture of social networks more generally.
To get at this broader view, we first develop some general principles about social networks
and their evolution, and then return to Granovetter’s question.
3.1 Triadic Closure
In Chapter 2, our discussions of networks treated them largely as static structures — we take
a snapshot of the nodes and edges at a particular moment in time, and then ask about paths,
components, distances, and so forth. While this style of analysis forms the basic foundation
for thinking about networks — and indeed, many datasets are inherently static, o↵ering us
only a single snapshot of a network — it is also useful to think about how a network evolves
over time. In particular, what are the mechanisms by which nodes arrive and depart, and
by which edges form and vanish?
The precise answer will of course vary depending on the type of network we’re considering,
but one of the most basic principles is the following:
If two people in a social network have a friend in common, then there is an
increased likelihood that they will become friends themselves at some point in the
future [347].
We refer to this principle as triadic closure, and it is illustrated in Figure 3.1: if nodes B and
C have a friend A in common, then the formation of an edge between B and C produces
a situation in which all three nodes A, B, and C have edges connecting each other — a
structure we refer to as a triangle in the network. The term “triadic closure” comes from
3.1. TRIADIC CLOSURE 49
B
A
C
G
F
E D
(a) Before new edges form.
B
A
C
G
F
E D
(b) After new edges form.
Figure 3.2: If we watch a network for a longer span of time, we can see multiple edges forming — some form through triadic closure while others (such as the D-G edge) form even though the two endpoints have no neighbors in common.
the fact that the B-C edge has the e↵ect of “closing” the third side of this triangle. If
we observe snapshots of a social network at two distinct points in time, then in the later
snapshot, we generally find a significant number of new edges that have formed through this
triangle-closing operation, between two people who had a common neighbor in the earlier
snapshot. Figure 3.2, for example, shows the new edges we might see from watching the
network in Figure 3.1 over a longer time span.
The Clustering Coe�cient. The basic role of triadic closure in social networks has
motivated the formulation of simple social network measures to capture its prevalence. One
of these is the clustering coe�cient [320, 411]. The clustering coe�cient of a node A is
defined as the probability that two randomly selected friends of A are friends with each
other. In other words, it is the fraction of pairs of A’s friends that are connected to each
other by edges. For example, the clustering coe�cient of node A in Figure 3.2(a) is 1/6
(because there is only the single C-D edge among the six pairs of friends B-C, B-D, B-E,
C-D, C-E, and D-E), and it has increased to 1/2 in the second snapshot of the network in
Figure 3.2(b) (because there are now the three edges B-C, C-D, and D-E among the same
six pairs). In general, the clustering coe�cient of a node ranges from 0 (when none of the
node’s friends are friends with each other) to 1 (when all of the node’s friends are friends
with each other), and the more strongly triadic closure is operating in the neighborhood of
the node, the higher the clustering coe�cient will tend to be.
50 CHAPTER 3. STRONG AND WEAK TIES
BA
ED
C
Figure 3.3: The A-B edge is a bridge, meaning that its removal would place A and B in distinct connected components. Bridges provide nodes with access to parts of the network that are unreachable by other means.
Reasons for Triadic Closure. Triadic closure is intuitively very natural, and essentially
everyone can find examples from their own experience. Moreover, experience suggests some
of the basic reasons why it operates. One reason why B and C are more likely to become
friends, when they have a common friend A, is simply based on the opportunity for B and C
to meet: if A spends time with both B and C, then there is an increased chance that they
will end up knowing each other and potentially becoming friends. A second, related reason
is that in the process of forming a friendship, the fact that each of B and C is friends with
A (provided they are mutually aware of this) gives them a basis for trusting each other that
an arbitrary pair of unconnected people might lack.
A third reason is based on the incentive A may have to bring B and C together: if A is
friends with B and C, then it becomes a source of latent stress in these relationships if B
and C are not friends with each other. This premise is based in theories dating back to early
work in social psychology [217]; it also has empirical reflections that show up in natural but
troubling ways in public-health data. For example, Bearman and Moody have found that
teenage girls who have a low clustering coe�cient in their network of friends are significantly
more likely to contemplate suicide than those whose clustering coe�cient is high [48].
3.2 The Strength of Weak Ties
So how does all this relate to Mark Granovetter’s interview subjects, telling him with such
regularity that their best job leads came from acquaintances rather than close friends? In
fact, triadic closure turns out to be one of the crucial ideas needed to unravel what’s going
on.
3.2. THE STRENGTH OF WEAK TIES 51
BA
ED
C
F H
GJ K
Figure 3.4: The A-B edge is a local bridge of span 4, since the removal of this edge would increase the distance between A and B to 4.
Bridges and Local Bridges. Let’s start by positing that information about good jobs is
something that is relatively scarce; hearing about a promising job opportunity from someone
suggests that they have access to a source of useful information that you don’t. Now consider
this observation in the context of the simple social network drawn in Figure 3.3. The person
labeled A has four friends in this picture, but one of her friendships is qualitatively di↵erent
from the others: A’s links to C, D, and E connect her to a tightly-knit group of friends who
all know each other, while the link to B seems to reach into a di↵erent part of the network.
We could speculate, then, that the structural peculiarity of the link to B will translate into
di↵erences in the role it plays in A’s everyday life: while the tightly-knit group of nodes A, C,
D, and E will all tend to be exposed to similar opinions and similar sources of information,
A’s link to B o↵ers her access to things she otherwise wouldn’t necessarily hear about.
To make precise the sense in which the A-B link is unusual, we introduce the following
definition. We say that an edge joining two nodes A and B in a graph is a bridge if deleting
the edge would cause A and B to lie in two di↵erent components. In other words, this edge
is literally the only route between its endpoints, the nodes A and B.
Now, if our discussion in Chapter 2 about giant components and small-world properties
taught us anything, it’s that bridges are presumably extremely rare in real social networks.
You may have a friend from a very di↵erent background, and it may seem that your friendship
is the only thing that bridges your world and his, but one expects in reality that there will
52 CHAPTER 3. STRONG AND WEAK TIES
BA
ED
C
F H
GJ KS
S S
W
W S
W W W W
WS
S
S S
W W
S
SS
S S
S
Figure 3.5: Each edge of the social network from Figure 3.4 is labeled here as either a strong tie (S) or a weak tie (W ), to indicate the strength of the relationship. The labeling in the figure satisfies the Strong Triadic Closure Property at each node: if the node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them.
be other, hard-to-discover, multi-step paths that also span these worlds. In other words, if
we were to look at Figure 3.3 as it is embedded in a larger, ambient social network, we would
likely see a picture that looks like Figure 3.4.
Here, the A-B edge isn’t the only path that connects its two endpoints; though they may
not realize it, A and B are also connected by a longer path through F , G, and H. This kind
of structure is arguably much more common than a bridge in real social networks, and we
use the following definition to capture it. We say that an edge joining two nodes A and B
in a graph is a local bridge if its endpoints A and B have no friends in common — in other
words, if deleting the edge would increase the distance between A and B to a value strictly
more than two. We say that the span of a local bridge is the distance its endpoints would
be from each other if the edge were deleted [190, 407]. Thus, in Figure 3.4, the A-B edge is
a local bridge with span four; we can also check that no other edge in this graph is a local
bridge, since for every other edge in the graph, the endpoints would still be at distance two if
the edge were deleted. Notice that the definition of a local bridge already makes an implicit
connection with triadic closure, in that the two notions form conceptual opposites: an edge
is a local bridge precisely when it does not form a side of any triangle in the graph.
Local bridges, especially those with reasonably large span, still play roughly the same
3.2. THE STRENGTH OF WEAK TIES 53
role that bridges do, though in a less extreme way — they provide their endpoints with
access to parts of the network, and hence sources of information, that they would otherwise
be far away from. And so this is a first network context in which to interpret Granovetter’s
observation about job-seeking: we might expect that if a node like A is going to get truly
new information, the kind that leads to a new job, it might come unusually often (though
certainly not always) from a friend connected by a local bridge. The closely-knit groups that
you belong to, though they are filled with people eager to help, are also filled with people
who know roughly the same things that you do.
The Strong Triadic Closure Property. Of course, Granovetter’s interview subjects
didn’t say, “I learned about the job from a friend connected by a local bridge.” If we believe
that local bridges were overrepresented in the set of people providing job leads, how does
this relate to the observation that distant acquaintances were overrepresented as well?
To talk about this in any detail, we need to be able to distinguish between di↵erent levels
of strength in the links of a social network. We deliberately refrain from trying to define
“strength” precisely, but we mean it to align with the idea that stronger links represent closer
friendship and greater frequency of interaction. In general, links can have a wide range of
possible strengths, but for conceptual simplicity — and to match the friend/acquaintance
dichotomy that we’re trying to explain — we’ll categorize all links in the social network as
belonging to one of two types: strong ties (the stronger links, corresponding to friends), and
weak ties (the weaker links, corresponding to acquaintances).1
Once we have decided on a classification of links into strong and weak ties, we can take a
social network and annotate each edge with a designation of it as either strong or weak. For
example, assuming we asked the nodes in the social network of Figure 3.4 to report which
of their network neighbors were close friends and which were acquaintances, we could get an
annotated network as in Figure 3.5.
It is useful to go back and think about triadic closure in terms of this division of edges
into strong and weak ties. If we recall the arguments supporting triadic closure, based on
opportunity, trust, and incentive, they all act more powerfully when the edges involved are
1In addition to the di�culty in reducing a range of possible link strengths to a two-category strong/weak distinction, there are many other subtleties in this type of classification. For example, in the discussion here, we will take this division of links into strong and weak ties as fixed in a single snapshot of the network. In reality, of course, the strength of a particular link can vary across di↵erent times and di↵erent situations. For example, an employee of a company who is temporarily assigned to work with a new division of the company for a few months may find that her full set of available social-network links remains roughly the same, but that her links to people within the new division have been temporarily strengthened (due to the sudden close proximity and increased contact), while her links to her old division have been temporarily weakened. Similarly, a high-school student may find that links to fellow members of a particular sports team constitute strong ties while that sport is in season, but that some of these links — to the teammates he knows less well outside of the team — become weak ties in other parts of the year. Again, for our purposes, we will consider a single distinction between strong and weak ties that holds throughout the analysis.
54 CHAPTER 3. STRONG AND WEAK TIES
strong ties than when they are weak ties. This suggests the following qualitative assumption:
If a node A has edges to nodes B and C, then the B-C edge is especially likely
to form if A’s edges to B and C are both strong ties.
To enable some more concrete analysis, Granovetter suggested a more formal (and somewhat
more extreme version) of this, as follows.
We say that a node A violates the Strong Triadic Closure Property if it has strong
ties to two other nodes B and C, and there is no edge at all (either a strong or
weak tie) between B and C. We say that a node A satisfies the Strong Triadic
Closure Property if it does not violate it.
You can check that no node in Figure 3.5 violates the Strong Triadic Closure Property, and
hence all nodes satisfy the Property. On the other hand, if the A-F edge were to be a strong
tie rather than a weak tie, then nodes A and F would both violate the Strong Triadic Closure
Property: Node A would now have strong ties to nodes E and F without there being an
E-F edge, and node F would have strong ties to both A and G without there being an
A-G edge. As a further check on the definition, notice that with the labeling of edges as
in Figure 3.5, node H satisfies the Strong Triadic Closure Property: H couldn’t possibly
violate the Property since it only has a strong tie to one other node.
Clearly the Strong Triadic Closure Property is too extreme for us to expect it hold across
all nodes of a large social network. But it is a useful step as an abstraction to reality,
making it possible to reason further about the structural consequences of strong and weak
ties. In the same way that an introductory physics course might assume away the e↵ects of
air resistance in analyzing the flight of a ball, proposing a slightly too-powerful assumption
in a network context can also lead to cleaner and conceptually more informative analysis.
For now, then, let’s continue figuring out where it leads us in this case; later, we’ll return to
the question of its role as a modeling assumption.
Local Bridges and Weak Ties. We now have a purely local, interpersonal distinction
between kinds of links — whether they are weak ties or strong ties — as well as a global,
structural notion — whether they are local bridges or not. On the surface, there is no direct
connection between the two notions, but in fact using triadic closure we can establish a
connection, in the following claim.
Claim: If a node A in a network satifies the Strong Triadic Closure Property and
is involved in at least two strong ties, then any local bridge it is involved in must
be a weak tie.
In other words, assuming the Strong Triadic Closure Property and a su�cient number of
strong ties, the local bridges in a network are necessarily weak ties.
3.2. THE STRENGTH OF WEAK TIES 55
BA S
C
S
Strong Triadic Closure says the B-C edge must exist, but the definition of a local bridge
says it cannot.
Figure 3.6: If a node satifies Strong Triadic Closure and is involved in at least two strong ties, then any local bridge it is involved in must be a weak tie. The figure illustrates the reason why: if the A-B edge is a strong tie, then there must also be an edge between B and C, meaning that the A-B edge cannot be a local bridge.
We’re going to justify this claim as a mathematical statement – that is, it will follow
logically from the definitions we have so far, without our having to invoke any as-yet-
unformalized intuitions about what social networks ought to look like. In this way, it’s
a di↵erent kind of claim from our argument in Chapter 2 that the global friendship network
likely contains a giant component. That was a thought experiment (albeit a very convinc-
ing one), requiring us to believe various empirical statements about the network of human
friendships — empirical statements that could later be confirmed or refuted by collecting
data on large social networks. Here, on the other hand, we’ve constructed a small num-
ber of specific mathematical definitions — particularly, local bridges and the Strong Triadic
Closure Property — and we can now justify the claim directly from these.
The argument is actually very short, and it proceeds by contradiction. Take some net-
work, and consider a node A that satisfies the Strong Triadic Closure Property and is involved
in at least two strong ties. Now suppose A is involved in a local bridge — say, to a node
B — that is a strong tie. We want to argue that this is impossible, and the crux of the
argument is depicted in Figure 3.6. First, since A is involved in at least two strong ties,
and the edge to B is only one of them, it must have a strong tie to some other node, which
we’ll call C. Now let’s ask: is there an edge connecting B and C? Since the edge from A to
B is a local bridge, A and B must have no friends in common, and so the B-C edge must
not exist. But this contradicts Strong Triadic Closure, which says that since the A-B and
56 CHAPTER 3. STRONG AND WEAK TIES
A-C edges are both strong ties, the B-C edge must exist. This contradiction shows that our
initial premise, the existence of a local bridge that is a strong tie, cannot hold, finishing the
argument.
This argument completes the connection we’ve been looking for between the local prop-
erty of tie strength and the global property of serving as a local bridge. As such, it gives us
a way to think about the way in which interpersonal properties of social-network links are
related to broader considerations about the network’s structure. But since the argument is
based on some strong assumptions (mainly Strong Triadic Closure, since the other assump-
tion is very mild), it is also worth reflecting on the role that simplifying assumptions play in
a result like this.
First, simplifying assumptions are useful when they lead to statements that are robust
in practice, making sense as qualitative conclusions that hold in approximate forms even
when the assumptions are relaxed. This is the case here: the mathematical argument can
be summarized more informally and approximately as saying that in real life, a local bridge
between nodes A and B tends to be a weak tie because if it weren’t, triadic closure would
tend to produce short-cuts to A and B that would eliminate its role as a local bridge. Again,
one is tempted to invoke the analogy to freshman physics: even if the assumptions used
to derive the perfectly parabolic flight of a ball don’t hold exactly in the real world, the
conclusions about flight trajectories are a very useful, conceptually tractable approximation
to reality.
Second, when the underlying assumptions are stated precisely, as they are here, it becomes
possible to test them on real-world data. In the past few years researchers have studied the
relationship of tie strength and network structure quantitatively across large populations,
and have shown that the conclusions described here in fact hold in an approximate form.
We describe some of this empirical research in the next section.
Finally, this analysis provides a concrete framework for thinking about the initially sur-
prising fact that life transitions such as a new jobs are often rooted in contact with distant
acquaintances. The argument is that these are the social ties that connect us to new sources
of information and new opportunities, and their conceptual “span” in the social network
(the local bridge property) is directly related to their weakness as social ties. This dual role
as weak connections but also valuable conduits to hard-to-reach parts of the network — this
is the surprising strength of weak ties.
3.3 Tie Strength and Network Structure in Large-Scale Data
The arguments connecting tie strength with structural properties of the underlying social
network make intriguing theoretical predictions about the organization of social networks
3.3. TIE STRENGTH AND NETWORK STRUCTURE IN LARGE-SCALE DATA 57
in real life. For many years after Granovetter’s initial work, however, these predictions
remained relatively untested on large social networks, due to the di�culty in finding data
that reliably captured the strengths of edges in large-scale, realistic settings.
This state of a↵airs began to change rapidly once detailed traces of digital communication
became available. Such “who-talks-to-whom” data exhibits the two ingredients we need
for empirical evaluation of hypotheses about weak ties: it contains the network structure
of communication among pairs of people, and we can use the total time that two people
spend talking to each other as a proxy for the strength of the tie — the more time spent
communicating during the course of an observation period, the stronger we declare the tie
to be.
In one of the more comprehensive studies of this type, Onnela et al. studied the who-
talks-to-whom network maintained by a cell-phone provider that covered roughly 20% of a
national population [334]. The nodes correspond to cell-phone users, and there is an edge
joining two nodes if they made phone calls to each other in both directions over an 18-
week observation period. Because the cell phones in this population are generally used for
personal communication rather than business purposes, and because the lack of a central
directory means that cell-phone numbers are generally exchanged among people who already
know each other, the underlying network can be viewed as a reasonable sampling of the
conversations occurring within a social network representing a significant fraction of one
country’s population. Moreover, the data exhibits many of the broad structural features
of large social networks discussed in Chapter 2, including a giant component — a single
connected component containing most (in this case 84%) of the individuals in the network.
Generalizing the Notions of Weak Ties and Local Bridges. The theoretical formu-
lation in the preceding section is based on two definitions that impose sharp dichotomies
on the network: an edge is either a strong tie or a weak tie, and it is either a local bridge
or it isn’t. For both of these definitions, it is useful to have versions that exhibit smoother
gradations when we go to examine real data at a large scale.
Above, we just indicated a way to do this for tie strength: we can make the strength of
an edge a numerical quantity, defining it to be the total number of minutes spent on phone
calls between the two ends of the edge. It is also useful to sort all the edges by tie strength,
so that for a given edge we can ask what percentile it occupies this ordering of edges sorted
by strength.
Since a very small fraction of the edges in the cell-phone data are local bridges, it makes
sense to soften this definition as well, so that we can view certain edges as being “almost”
local bridges. To do this, we define the neighborhood overlap of an edge connecting A and
B to be the ratio
number of nodes who are neighbors of both A and B
number of nodes who are neighbors of at least one of A or B , (3.1)
58 CHAPTER 3. STRONG AND WEAK TIES
Figure 3.7: A plot of the neighborhood overlap of edges as a function of their percentile in the sorted order of all edges by tie strength. The fact that overlap increases with increasing tie strength is consistent with the theoretical predictions from Section 3.2. (Image from [334].)
where in the denominator we don’t count A or B themselves (even though A is a neighbor of
B and B is a neighbor of A). As an example of how this definition works, consider the edge
A-F in Figure 3.4. The denominator of the neighborhood overlap for A-F is determined by
the nodes B, C, D, E, G, and J, since these are the ones that are a neighbor of at least one
of A or F . Of these, only C is a neighbor of both A and F , so the neighborhood overlap is
1/6.
The key feature of this definition is that this ratio in question is 0 precisely when the
numerator is 0, and hence when the edge is a local bridge. So the notion of a local bridge
is contained within this definition — local bridges are the edges of neighborhood overlap 0
— and hence we can think of edges with very small neighborhood overlap as being “almost”
local bridges. (Since intuitively, edges with very small neighborhood overlap consist of nodes
that travel in “social circles” having almost no one in common.) For example, this definition
views the A-F edge as much closer to being a local bridge than the A-E edge is, which
accords with intuition.
3.3. TIE STRENGTH AND NETWORK STRUCTURE IN LARGE-SCALE DATA 59
Empirical Results on Tie Strength and Neighborhood Overlap. Using these defi-
nitions, we can formulate some fundamental quantitative questions based on Granovetter’s
theoretical predictions. First, we can ask how the neighborhood overlap of an edge depends
on its strength; the strength of weak ties predicts that neighborhood overlap should grow as
tie strength grows.
In fact, this is borne out extremely cleanly by the data. Figure 3.7 shows the neigh-
borhood overlap of edges as a function of their percentile in the sorted order of all edges
by tie strength. Thus, as we go to the right on the x-axis, we get edges of greater and
greater strength, and because the curve rises in a strikingly linear fashion, we also get edges
of greater and greater neighborhood overlap. The relationship between these quantities thus
aligns well with the theoretical prediction.2
The measurements underlying Figure 3.7 describe a connection between tie strength
and network structure at a local level — in the neighborhoods of individual nodes. It is
also interesting to consider how this type of data can be used to evaluate the more global
picture suggested by the theoretical framework, that weak ties serve to link together di↵erent
tightly-knit communities that each contain a large number of stronger ties. Here, Onnela et
al. provided an indirect analysis to address this question, as follows. They first deleted edges
from the network one at a time, starting with the strongest ties and working downward in
order of tie strength. The giant component shrank steadily as they did this, its size going
down gradually due to the elimination of connections among the nodes. They then tried the
same thing, but starting from the weakest ties and working upward in order of tie strength.
In this case, they found that the giant component shrank more rapidly, and moreover that
its remnants broke apart abruptly once a critical number of weak ties had been removed.
This is consistent with a picture in which the weak ties provide the more crucial connective
structure for holding together disparate communities, and for keeping the global structure
of the giant component intact.
Ultimately, this is just a first step toward evaluating theories of tie strength on net-
work data of this scale, and it illustrates some of the inherent challenges: given the size
and complexity of the network, we cannot simply look at the structure and “see what’s
there.” Indirect measures must generally be used, and since one knows relatively little about
the meaning or significance of any particular node or edge, it remains an ongoing research
challenge to draw richer and more detailed conclusions in the way that one can on small
datasets.
2It is of course interesting to note the deviation from this trend at the very right-hand edge of the plot in Figure 3.7, corresponding to the edges of greatest possible tie strength. It is not clear what causes this deviation, but it is certainly plausible that these extremely strong edges are associated with people who are using their cell-phones in some unusual fashion.
60 CHAPTER 3. STRONG AND WEAK TIES
3.4 Tie Strength, Social Media, and Passive Engage- ment
As an increasing amount of social interaction moves on-line, the way in which we maintain
and access our social networks begins to change as well. For example, as is well-known
to users of social-networking tools, people maintain large explicit lists of friends in their
profiles on these sites — in contrast to the ways in which such friendship circles were once
much more implicit, and in fact relatively di�cult for individuals even to enumerate or
mentally access [244]. What e↵ect does this have on social network structure more broadly?
Understanding the changes arising from these forms of technological mediation is a challenge
that was already being articulated in the early 1990s by researchers including Barry Wellman
[414, 413], as the Internet began making remote interaction possible for a broad public; these
issues have of course grown steadily more pervasive between then and now.
Tie strength can provide an important perspective on such questions, providing a lan-
guage for asking how on-line social activity is distributed across di↵erent kinds of links —
and in particular, how it is distributed across links of di↵erent strengths. When we see
people maintaining hundreds of friendship links on a social-networking site, we can ask how
many of these correspond to strong ties that involve frequent contact, and how many of these
correspond to weak ties that are activated relatively rarely.
Tie Strength on Facebook. Researchers have begun to address such questions of tie
strength using data from some of the most active social media sites. At Facebook, Cameron
Marlow and his colleagues analyzed the friendship links reported in each user’s profile, ask-
ing to what extent each link was actually used for social interaction, beyond simply being
reported in the profile [286]. In other words, where are the strong ties among a user’s friends?
To make this precise using the data they had available, they defined three categories of links
based on usage over a one-month observation period.
• A link represents reciprocal (mutual) communication, if the user both sent messages to the friend at the other end of the link, and also received messages from them during
the observation period.
• A link represents one-way communication if the user sent one or more messages to the friend at the other end of the link (whether or not these messages were reciprocated).
• A link represents a maintained relationship if the user followed information about the friend at the other end of the link, whether or not actual communication took place;
“following information” here means either clicking on content via Facebook’s News
Feed service (providing information about the friend) or visiting the friend’s profile
more than once.
3.4. TIE STRENGTH, SOCIAL MEDIA, AND PASSIVE ENGAGEMENT 61
All Friends
One-way Communication Mutual Communication
Maintained Relationships
Figure 3.8: Four di↵erent views of a Facebook user’s network neighborhood, showing the structure of links coresponding respectively to all declared friendships, maintained relation- ships, one-way communication, and reciprocal (i.e. mutual) communication. (Image from [286].)
Notice that these three categories are not mutually exclusive — indeed, the links classified
as reciprocal communication always belong to the set of links classified as one-way commu-
nication.
This stratification of links by their use lets us understand how a large set of declared
friendships on a site like Facebook translates into an actual pattern of more active social
interaction, corresponding approximately to the use of stronger ties. To get a sense of the
relative volumes of these di↵erent kinds of interaction through an example, Figure 3.8 shows
the network neighborhood of a sample Facebook user — consisting of all his friends, and all
links among his friends. The picture in the upper-left shows the set of all declared friendships
in this user’s profile; the other three pictures show how the set of links becomes sparser once
we consider only maintained relationships, one-way communication, or reciprocal communi-
62 CHAPTER 3. STRONG AND WEAK TIES
Figure 3.9: The number of links corresponding to maintained relationships, one-way com- munication, and reciprocal communication as a function of the total neighborhood size for users on Facebook. (Image from [286].)
cation. Moreover, as we restrict to stronger ties, certain parts of the network neighborhood
thin out much faster than others. For example, in the neighborhood of the sample user in
Figure 3.8, we see two distinct regions where there has been a particularly large amount of
triadic closure: one in the upper part of the drawing, and one on the right-hand side of the
drawing. However, when we restrict to links representing communication or a maintained
relationship, we see that a lot of the links in the upper region survive, while many fewer of
the links in the right-hand region do. One could conjecture that the right-hand region rep-
resents a set of friends from some earlier phase of the user’s life (perhaps from high school)
who declare each other as friends, but do not actively remain in contact; the upper region,
on the other hand, consists of more recent friends (perhaps co-workers) for whom there is
more frequent contact.
We can make the relative abundance of these di↵erent types of links quantitative through
the plot in Figure 3.9. On the x-axis is the total number of friends a user declares, and the
curves then show the (smaller) numbers of other link types as a function of this total. There
are several interesting conclusions to be drawn from this. First, it confirms that even for
users who report very large numbers of friends on their profile pages (on the order of 500),
3.4. TIE STRENGTH, SOCIAL MEDIA, AND PASSIVE ENGAGEMENT 63
Figure 3.10: The total number of a user’s strong ties (defined by multiple directed messages) as a function of the number of followees he or she has on Twitter. (Image from [222].)
the number with whom they actually communicate is generally between 10 and 20, and the
number they follow even passively (e.g. by reading about them) is under 50. But beyond this
observation, Marlow and his colleagues draw a further conclusion about the power of media
like Facebook to enable this kind of passive engagement, in which one keeps up with friends
by reading news about them even in the absence of communication. They argue that this
passive network occupies an interesting middle ground between the strongest ties maintained
by regular communication and the weakest ties from one’s distant past, preserved only in
lists on social-networking profile pages. They write, “The stark contrast between reciprocal
and passive networks shows the e↵ect of technologies such as News Feed. If these people
were required to talk on the phone to each other, we might see something like the reciprocal
network, where everyone is connected to a small number of individuals. Moving to an
environment where everyone is passively engaged with each other, some event, such as a new
baby or engagement can propagate very quickly through this highly connected network.”
Tie Strength on Twitter. Similar lines of investigation have been carried out recently on
the social media site Twitter, where individual users engage in a form of micro-blogging by
posting very short, 140-character public messages known as “tweets.” Twitter also includes
social-network features, and these enable one to distinguish between stronger and weaker
ties: each user can specify a set of other users whose messages he or she will follow, and each
user can also direct messages specifically to another user. (In the latter case, the message
64 CHAPTER 3. STRONG AND WEAK TIES
remains public for everyone to read, but it is marked with a notation indicating that it is
intended for a particular user.) Thus, the former kind of interaction defines a social network
based on more passive, weak ties — it is very easy for a user to follow many people’s messages
without ever directly communicating with any of them. The latter kind of interaction —
especially when we look at users directing multiple messages to others — corresponds to a
stronger kind of direct interaction.
In a style analogous to the work of Marlow et al., Huberman, Romero, and Wu analyzed
the relative abundance of these two kinds of links on Twitter [222]. Specifically, for each
user they considered the number of users whose messages she followed (her “followees”), and
then defined her strong ties to consist of the users to whom she had directed at least two
messages over the course of an observation period. Figure 3.10 shows how the number of
strong ties varies as a function of the number of followees. As we saw for Facebook, even
for users who maintain very large numbers of weak ties on-line, the number of strong ties
remains relatively modest, in this case stabilizing at a value below 50 even for users with
over 1000 followees.
There is another useful way to think about the contrast between the ease of forming
links and the relative scarcity of strong ties in environments like Facebook and Twitter. By
definition, each strong tie requires the continuous investment of time and e↵ort to maintain,
and so even people who devote a lot of their energy to building strong ties will eventually
reach a limit — imposed simply by the hours available in a day — on the number of ties
that they can maintain in this way. The formation of weak ties is governed by much milder
constraints — they need to be established at their outset but not necessarily maintained
continuously — and so it is easier for someone to accumulate them in large numbers. We
will encounter this distinction again in Chapter 13, when we consider how social networks
di↵er at a structural level from information networks such as the World Wide Web.
Understanding the e↵ect that on-line media have on the maintenance and use of social
networks is a complex problem for which the underlying research is only in its early stages.
But some of these preliminary studies already highlight the ways in which networks of strong
ties can still be relatively sparse even in on-line settings where weak ties abound, and how
the nature of the underlying on-line medium can a↵ect the ways in which di↵erent links are
used for conveying information.
3.5 Closure, Structural Holes, and Social Capital
Our discussion thus far suggests a general view of social networks in terms of tightly-knit
groups and the weak ties that link them. The analysis has focused primarily on the roles
that di↵erent kinds of edges of a network play in this structure — with a few edges spanning
di↵erent groups while most are surrounded by dense patterns of connections.
3.5. CLOSURE, STRUCTURAL HOLES, AND SOCIAL CAPITAL 65
B
F
A
E
D
C
Figure 3.11: The contrast between densely-knit groups and boundary-spanning links is re- flected in the di↵erent positions of nodes A and B in the underyling social network.
There is a lot of further insight to be gained by asking about the roles that di↵erent nodes
play in this structure as well. In social networks, access to edges that span di↵erent groups is
not equally distributed across all nodes: some nodes are positioned at the interface between
multiple groups, with access to boundary-spanning edges, while others are positioned in the
middle of a single group. What is the e↵ect of this heterogeneity? Following the expositional
lead of social-network researchers including Ron Burt [87], we can formulate an answer to
this question as a story about the di↵erent experiences that nodes have in a network like the
one in Figure 3.11 — particularly in the contrast between the experience of a node such as
A, who sits at the center of a single tightly-knit group, and node B, who sits at the interface
between several groups.
Embeddedness. Let’s start with node A. Node A’s set of network neighbors has been
subject to considerable triadic closure; A has a high clustering coe�cient. (Recall that the
clustering coe�cient is the fraction of pairs of neighbors who are themselves neighbors).
To talk about the structure around A it is useful to introduce an additional definition.
We define the embeddedness of an edge in a network to be the number of common neighbors
the two endpoints have. Thus, for example, the A-B edge has an embeddedness of two, since
A and B have the two common neighbors E and F . This definition relates to two notions
from earlier in the chapter. First, the embeddedness of an edge is equal to the numerator in
66 CHAPTER 3. STRONG AND WEAK TIES
the ratio that defines the neighborhood overlap in Equation (3.1) from Section 3.3. Second,
we observe that local bridges are precisely the edges that have an embeddedness of zero —
since they were defined as those edges whose endpoints have no neighbors in common.
In the example shown in Figure 3.11, what stands out about A is the way in which all
of his edges have significant embeddedness. A long line of research in sociology has argued
that if two individuals are connected by an embedded edge, then this makes it easier for
them to trust one another, and to have confidence in the integrity of the transactions (social,
economic, or otherwise) that take place between them [117, 118, 193, 194, 395]. Indeed, the
presence of mutual friends puts the interactions between two people “on display” in a social
sense, even when they are carried out in private; in the event of misbehavior by one of the
two parties to the interaction, there is the potential for social sanctions and reputational
consequences from their mutual friends. As Granovetter writes, “My mortification at cheat-
ing a friend of long standing may be substantial even when undiscovered. It may increase
when a friend becomes aware of it. But it may become even more unbearable when our
mutual friends uncover the deceit and tell one another” [194].
No similar kind of deterring threat exists for edges with zero embeddedness, since there
is no one who knows both people involved in the interaction. In this respect, the interactions
that B has with C and D are much riskier than the embedded interactions that A experiences.
Moreover, the constraints on B’s behavior are made complicated by the fact that she is
subject to potentially contradictory norms and expectations from the di↵erent groups she
associates with [116].
Structural holes. Thus far we have been discussing the advantages that accrue to node A
in Figure 3.11 from the closure in his network neighborhood, and the embedded edges that
result from this. But a related line of research in sociology, catalyzed by influential work of
Burt [86], has argued that network positions such as node B’s, at the ends of multiple local
bridges, confer a distinct set of equally fundamental advantages.
The canonical setting for this argument is the social network within an organization or
company, consisting of people who are in some ways collaborating on common objectives and
in other ways implicitly competing for career advancement. Note that although we may be
thinking about settings in which there is a formal organizational hierarchy — encoding who
reports to whom — we’re interested in the more informal network of who knows whom, and
who talks to whom on a regular basis. Empirical studies of managers in large corporations has
correlated an individual’s success within a company to their access to local bridges [86, 87].
At a more abstract level, the central arguments behind these studies are also supported by
the network principles we have been discussing, as we now explore further.
Let’s go back to the network in Figure 3.11, imagining the network to represent the
interaction and collaboration among managers in a large company. In Burt’s language,
3.5. CLOSURE, STRUCTURAL HOLES, AND SOCIAL CAPITAL 67
node B, with her multiple local bridges, spans a structural hole in the organization — the
“empty space” in the network between two sets of nodes that do not otherwise interact
closely. (Unlike the term “local bridge,” which has a precise mathematical definition in
terms of the underlying graph, we will keep the term “structural hole” somewhat informal in
this discussion.) The argument is that B’s position o↵ers advantages in several dimensions
relative to A’s. The first kind of advantage, following the observations in the previous
section, is an informational one: B has early access to information originating in multiple,
non-interacting parts of the network. Any one person has a limited amount of energy they
can invest in maintaining contacts across the organization, and B is investing her energy
e�ciently by reaching out to di↵erent groups rather than basing all her contacts in the same
group.
A second, related kind of advantage is based on the way in which standing at one end of a
local bridge can be an amplifier for creativity [88]. Experience from many domains suggests
that innovations often arise from the unexpected synthesis of multiple ideas, each of them on
their own perhaps well-known, but well-known in distinct and unrelated bodies of expertise.
Thus, B’s position at the interface between three non-interacting groups gives her not only
access to the combined information from these groups, but also the opportunity for novel
ideas by combining these disparate sources of information in new ways.
Finally, B’s position in the network provides an opportunity for a kind of social “gate-
keeping” — she regulates the access of both C and D to the tightly-knit group she belongs
to, and she controls the ways in which her own group learns about information coming from
C’s and D’s groups. This provides B with a source of power in the organization, and one
could imagine that certain people in this situation might try to prevent triangles from form-
ing around the local bridges they’re part of — for example, another edge from C or D into
B’s group would reduce B’s gatekeeping role.
This last point highlights a sense in which the interests of node B and of the organization
as a whole may not be aligned. For the functioning of the organization, accelerating the flow
of information between groups could be beneficial, but this building of bridges would come
at the expense of B’s latent power at the boundaries of these groups. It also emphasizes that
our analysis of structural holes is primarily a static one: we look at the network at a single
point in time, and consider the e↵ects of the local bridges. How long these local bridges last
before triadic closure produces short-cuts around them, and the extent to which people in an
organization are consciously, strategically seeking out local bridges and trying to maintain
them, is less well understood; it is a topic of ongoing research [90, 188, 252, 259].
Ultimately, then, there are trade-o↵s in the relative positions of A and B. B’s position at
the interface between groups means that her interactions are less embedded within a single
group, and less protected by the presence of mutual network neighbors. On the other hand,
this riskier position provides her with access to information residing in multiple groups, and
68 CHAPTER 3. STRONG AND WEAK TIES
the opportunity to both regulate the flow of this information and to synthesize it in new
ways.
Closure and Bridging as Forms of Social Capital. All of these arguments are framed
in terms of individuals and groups deriving benefits from an underlying social structure or
social network; as such, they are naturally related to the notion of social capital [117, 118,
279, 342, 344]. Social capital is a term in increasingly widespread use, but it is a famously
di�cult one to define [138]. In Alejandro Portes’s review of the topic, he writes, “Consensus
is growing in the literature that social capital stands for the ability of actors to secure benefits
by virtue of membership in social networks or other social structures” [342].
The term “social capital” is designed to suggest its role as part of an array of di↵erent
forms of capital, all of which serve as tangible or intangible resources that can be mobilized
to accomplish tasks. James Coleman and others speak of social capital alongside physical
capital — the implements and technologies that help perform work — and human capital —
the skills and talents that individual people bring to a job or goal [118]. Pierre Bourdieu o↵ers
a related but distinct taxonomy, considering social capital in relation to economic capital —
consisting of monetary and physical resources — and cultural capital — the accumulated
resources of a culture that exist at a level beyond any one individual’s social circle, conveyed
through education and other broad social institutions [17, 75].
Borgatti, Jones, and Everett [74], summarizing discussions within the sociology commu-
nity, observe two important sources of variation in the use of the term “social capital.” First,
social capital is sometimes viewed as a property of a group, with some groups functioning
more e↵ectively than others because of favorable properties of their social structures or net-
works. Alternately, it has also been considered as a property of an individual; used in this
sense, a person can have more or less social capital depending on his or her position in the
underlying social structure or network. A second, related, source of terminological variation
is based on whether social capital is a property that is purely intrinsic to a group — based
only on the social interactions among the group’s members — or whether it is also based on
the interactions of the group with the outside world.
A view at this level of generality does not yet specify what kinds of network structures
are the most e↵ective for creating social capital, and our discussion earlier in this section
highlights several di↵erent perspectives on the question. The writings of Coleman and oth-
ers on social capital emphasize the benefits of triadic closure and embedded edges for the
reasons discussed above: they enable the enforcement of norms and reputational e↵ects, and
hence can help protect the integrity of social and economic transactions. Burt, on the other
hand, discusses social capital as a tension between closure and brokerage — with the former
referring to Coleman’s conception and the latter referring to benefits arising from the ability
to “broker” interactions at the interface between di↵erent groups, across structural holes.
3.6. ADVANCED MATERIAL: BETWEENNESS MEASURES AND GRAPH PARTITIONING69
In addition to the structural distinctions between these perspectives, they also illustrate
di↵erent focuses on groups versus individuals, and on the activity within a group versus
its contacts with a larger population. The contrasts are also related to Robert Putnam’s
dichotomy between bonding capital and bridging capital [344]; these terms, while intended
informally, correspond roughly to the kinds of social capital arising respectively from con-
nections within a tightly-knit group and from connections between such groups.
The notion of social capital thus provides a framework for thinking about social structures
as facilitators of e↵ective action by individuals and groups, and a way of focusing discussions
of the di↵erent kinds of benefits conferred by di↵erent structures. Networks are at the heart
of such discussions — both in the way they produce closed groups where transactions can be
trusted, and in the way they link di↵erent groups and thereby enable the fusion of di↵erent
sources of information residing in these groups.
3.6 Advanced Material: Betweenness Measures and Graph Partitioning
This is the first in a series of sections throughout the book labeled “Advanced Material.”
Each of these sections comes at the end of a chapter, and it explores mathematically more
sophisticated aspects of some of the models developed earlier in the chapter. They are
strictly optional, in that nothing later in the book builds on them. Also, while these sections
are technically more involved, they are written to be completely self-contained, except where
specific pieces of mathematical background are needed; this necessary background is spelled
out at the beginnings of the sections where it is required.
In this section, we will try formulating more concrete mathematical definitions for some of
the basic concepts from earlier in the chapter. The discussion in this chapter has articulated
a way of thinking about networks in terms of their tightly-knit regions and the weaker ties
that link them together. We have formulated precise definitions for some of the underlying
concepts, such as the clustering coe�cient and the definition of a local bridge. In the process,
however, we have refrained from trying to precisely delineate what we mean by a “tightly-knit
region,” and how to formally characterize such regions.
For our purposes so far, it has been useful to be able to speak in this more general,
informal way about tightly-knit regions; it helps to be flexible since the exact characterization
of the notion may di↵er depending on the di↵erent domains in which we encounter it. But
there are also settings in which having a more precise, formal definition is valuable. In
particular, a formal definition can be crucial if we are faced with a network dataset and
actually want to identify densely connected groups of nodes within it.
This will be our focus here: describing a method that can take a network and break it
down into a set of tightly-knit regions, with sparser interconnections between the regions.
70 CHAPTER 3. STRONG AND WEAK TIES
Figure 3.12: A co-authorship network of physicists and applied mathematicians working on networks [322]. Within this professional community, more tightly-knit subgroups are evident from the network structure.
We will refer to this as the problem of graph partitioning, and the constituent parts the
network is broken into as the regions arising from the partitioning method. Formulating a
method for graph partitioning will implicitly require working out a set of definitions for all
these notions that are both mathematically tractable and also useful on real datasets.
To give a sense for what we might hope to achieve from such a method, let’s consider
two examples. The first, shown in Figure 3.12, depicts the co-authorships among a set of
physicists and applied mathematicians working on networks [322]. Recall that we discussed
co-authorship networks in Chapter 2 as a way of encoding the collaborations within a profes-
sional community. It’s clear from the picture that there are tightly-knit groups within this
community, and some people who sit on the boundaries of their respective groups. Indeed it
resembles, at a somewhat larger scale, some of the pictures of tightly-knit groups and weak
ties that we drew in schematic form earlier, in examples such as Figure 3.11. Is there a
general way to pull these groups out of the data, beyond using just our visual intuition?
3.6. ADVANCED MATERIAL: BETWEENNESS MEASURES AND GRAPH PARTITIONING71
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Figure 3.13: A karate club studied by Wayne Zachary [421] — a dispute during the course of the study caused it to split into two clubs. Could the boundaries of the two clubs be predicted from the network structure?
A second example, in Figure 3.13, is a picture of the social network of a karate club studied
by Wayne Zachary [421] and discussed in Chapter 1: a dispute between the club president
(node 34) and the instructor (node 1) led the club to split into two. Figure 3.13 shows the
network structure, with the membership in the two clubs after the division indicated by the
shaded and unshaded nodes. Now, a natural question is whether the structure itself contains
enough information to predict the fault line. In other words, did the split occur along a weak
interface between two densely connected regions? Unlike the network in Figure 3.12, or in
some of the earlier examples in the chapter, the two conflicting groups here are still heavily
interconnected. So to identify the division in this case, we need to look for more subtle
signals in the way in which edges between the groups e↵ectively occur at lower “density”
than edges within the groups. We will see that this is in fact possible, both for the definitions
we consider here as well as other definitions.
A. A Method for Graph Partitioning
Many di↵erent approaches have been developed for the problem of graph partitioning, and
for networks with clear divisions into tightly-knit regions, there is often a wide range of
methods that will prove to be e↵ective. While these methods can di↵er considerably in their
specifics, it is useful to identify the di↵erent general styles that motivate their designs.
72 CHAPTER 3. STRONG AND WEAK TIES
General Approaches to Graph Partitioning. One class of methods focuses on iden-
tifying and removing the “spanning links” between densely-connected regions. Once these
links are removed, the network begins to fall apart into large pieces; within these pieces,
further spanning links can be identified, and the process continues. We will refer to these as
divisive methods of graph partitioning, since they divide the network up as they go.
An alternate class of methods starts from the opposite end of the problem, focusing on the
most tightly-knit parts of the network, rather than the connections at their boundaries. Such
methods find nodes that are likely to belong to the same region and merge them together.
Once this is done, the network consists of a large number of merged chunks, each containing
the seeds of a densely-connected region; the process then looks for chunks that should be
further merged together, and in this way the regions are assembled “bottom-up.” We refer
to these as agglomerative methods of graph partitioning, since they glue nodes together into
regions as they go.
To illustrate the conceptual di↵erences between these two kinds of approaches, let’s con-
sider the simple graph in Figure 3.14(a). Intuitively, as indicated in Figure 3.14(b), there
appears to be a broad separation between one region consisting of nodes 1-7, and another
consisting of nodes 8-14. Within each of these regions, there is a further split: on the left
into nodes 1-3 and nodes 4-6; on the right into nodes 9-11 and nodes 12-14. Note how this
simple example already illustrates that the process of graph partitioning can usefully be
viewed as producing regions in the network that are naturally nested: larger regions poten-
tially containing several smaller, even more tightly-knit regions “nested” within them. This
is of course a familiar picture from everyday life, where — for example — a separation of
the gobal population into national groups can be further subdivided into sub-populations
within particular local areas within countries.
In fact, a number of graph partitioning methods will find the nested set of regions indi-
cated in Figure 3.14(b). Divisive methods will generally proceed by breaking apart the graph
first at the 7-8 edge, and subsequently at the remaining edges into nodes 7 and 8. Agglom-
erative methods will arrive at the same result from the opposite direction, first merging the
four triangles into clumps, and then finding that the triangles themselves can be naturally
paired o↵.
This is a good point at which to make the discussion more concrete, and to do so we
focus on a particular divisive method proposed by Girvan and Newman [184, 322]. The
Girvan-Newman method has been applied very widely in recent years, and to social network
data in particular. Again, however, we emphasize that graph partitioning is an area in which
there is an especially wide range of di↵erent approaches in use. The approach we discuss is
an elegant and particular widely-used one; however, understanding which types of methods
work best in di↵erent situations remains a subject of active research.
3.6. ADVANCED MATERIAL: BETWEENNESS MEASURES AND GRAPH PARTITIONING73
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(b) Tightly-knit regions and their nested structure
Figure 3.14: In many networks, there are tightly-knit regions that are intuitively apparent, and they can even display a nested structure, with smaller regions nesting inside larger ones.
The Notion of Betweenness. To motivate the design of a divisive method for graph
partitioning, let’s think about some general principles that might lead us to remove the 7-8
edge first in Figure 3.14(a).
A first idea, motivated by the discussion earlier in this chapter, is that since bridges and
local bridges often connect weakly interacting parts of the network, we should try removing
these bridges and local bridges first. This is in fact an idea along the right lines; the problem
is simply that it’s not strong enough, for two reasons. First, when there are several bridges,
it doesn’t tell us which to remove first. As we see in Figure 3.14(a), where there are five
bridges, certain bridges can produce more reasonable splits than others. Second, there can
be graphs where no edge is even a local bridge, because every edge belongs to a triangle —
and yet there is still a natural division into regions. Figure 3.15 shows a simple example,
where we might want to identify nodes 1-5 and nodes 7-11 as tightly-knit regions, despite
74 CHAPTER 3. STRONG AND WEAK TIES
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3
5 7
9
10
8
6
41 11
Figure 3.15: A network can display tightly-knit regions even when there are no bridges or local bridges along which to separate it.
the fact that there are no local bridges to remove.
However, if we think more generally about what bridges and local bridges are doing,
then we can arrive at a notion that forms the central ingredient of the Girvan-Newman
method. Local bridges are important because they form part of the shortest path between
pairs of nodes in di↵erent parts of the network — without a particular local bridge, paths
between many pairs of nodes may have to be “re-routed” a longer way. We therefore define
an abstract notion of “tra�c” on the network, and look for the edges that carry the most of
this tra�c. Like crucial bridges and highway arteries, we might expect these edges to link
di↵erent densely-connected regions, and hence be good candidates for removal in a divisive
method.
We define our notion of tra�c as follows. For each pair of nodes A and B in the graph
that are connected by a path, we imagine having one unit of fluid “flow” along the edges from
A to B. (If A and B belong to di↵erent connected components, then no fluid flows between
them.) The flow between A and B divides itself evenly along all the possible shortest paths
from A to B: so if there are k shortest paths from A and B, then 1/k units of flow pass
along each one.
We define the betweenness of an edge to be the total amount of flow it carries, count-
ing flow between all pairs of nodes using this edge. For example, we can determine the
betweenness of each edge in Figure 3.14(a) as follows.
• Let’s first consider the 7-8 edge. For each node A in the left half of the graph, and each node B in the right half of the graph, their full unit of flow passes through the
7-8 edge. On the other hand, no flow passing between pairs of nodes that both lie in
the same half uses this edge. As a result, the betweenness of the 7-8 edge is 7 · 7 = 49.
• The 3-7 edge carries the full unit of flow from each node among 1, 2, and 3 to each
3.6. ADVANCED MATERIAL: BETWEENNESS MEASURES AND GRAPH PARTITIONING75
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(b) Step 2
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(c) Step 3
Figure 3.16: The steps of the Girvan-Newman method on the network from Figure 3.14(a).
node among 4-14. Thus, the betweenness of this edge is 3 · 11 = 33. The same goes for the edges 6-7, 8-9, and 8-12.
• The 1-3 edge carries all the flow from 1 to every other node except 2. As a result, its betweennness is 12. By strictly symmetric reasoning, the other edges linked from 3, 6,
9, and 12 into their respective triangles have betweenness 12 as well.
• Finally, the 1-2 edge only carries flow between its endpoints, so its betweenness is 1. This also holds for the edges 4-5, 10-11, and 13-14.
Thus, betweenness has picked out the 7-8 edge as the one carrying the most tra�c.
In fact, the idea of using betweenness to identify important edges draws on a long history
in sociology, where most attribute its first explicit articulation to Linton Freeman [73, 168,
169]. Its use by sociologists has traditionally focused more on nodes than on edges, where
the definition the same: the betweenness of a node is the total amount of flow that it carries,
when a unit of flow between each pair of nodes is divided up evenly over shortest paths. Like
edges of high betweenness, nodes of high betweenness occupy critical roles in the network
76 CHAPTER 3. STRONG AND WEAK TIES
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5 7
9
10
8
6
41 11
(a) Step 1
2
3
5 7
9
10
8
6
41 11
(b) Step 2
2
3
5 7
9
10
8
6
41 11
(c) Step 3
2
3
5 7
9
10
8
6
41 11
(d) Step 4
Figure 3.17: The steps of the Girvan-Newman method on the network from Figure 3.15.
structure — indeed, because carrying a large amount of flow suggests a position at the
interface between tightly-knit groups, there are clear relationships of betweenness with our
earlier discussions of nodes that span structural holes in a social network [86].
The Girvan-Newman Method: Successively Deleting Edges of High Betweenness.
Edges of high betweenness are the ones that, over all pairs of nodes, carry the highest volume
of tra�c along shortest paths. Based on the premise that these are the most “vital” edges
for connecting di↵erent regions of the network, it is natural to try removing these first. This
is the crux of the Girvan-Newman method, which can now be summarized as follows.
(1) Find the edge of highest betweenness — or multiple edges of highest betweenness, if
there is a tie — and remove these edges from the graph. This may cause the graph
to separate into multiple components. If so, this is the first level of regions in the
partitioning of the graph.
(2) Now recalculate all betweennesses, and again remove the edge or edges of highest be-
tweenness. This may break some of the existing components into smaller components;
if so, these are regions nested within the larger regions.
(...) Proceed in this way as long as edges remain in graph, in each step recalculating all
betweennesses and removing the edge or edges of highest betweenness.
Thus, as the graph falls apart first into large pieces and then into smaller ones, the method
naturally exposes a nested structure in the tightly-knit regions. In Figures 3.16 and 3.17
3.6. ADVANCED MATERIAL: BETWEENNESS MEASURES AND GRAPH PARTITIONING77
we show how the method operates on the graphs from Figures 3.14(a) and 3.15 respectively.
Note how smaller regions emerge from larger ones as edges are successively removed.
The sequence of steps in Figure 3.17 in fact exposes some interesting points about how
the method works.
• When we calculate the betweennesses in the first step, the 5-7 edge carries all the flow from nodes 1-5 to nodes 7-11, for a betweenness of 25. The 5-6 edge, on the other hand,
only carries flow from node 6 to each of nodes 1-5, for a betweenness of 5. (Similarly
for the 6-7 edge.)
• Once the 5-7 edge is deleted, however, we recalculate all the betweennesses for the second step. At this point, all 25 units of flow that used to be on this deleted edge
have shifted onto the path through nodes 5, 6, and 7, and so the betweenness of the
5-6 edge (and also the 6-7 edge) has increased to 5 + 25 = 30. This is why these two
edges are deleted next.
In their original presentation of the method, Girvan and Newman showed its e↵ectiveness
at partitioning a number of real network datasets into intuitively reasonable sets of regions.
For example, on Zachary’s karate club network in Figure 3.13, when the method is used to
remove edges until the graph first separates into two pieces, the resulting partition agrees
with the actual split that occurred in the club except for a single person — node 9 in the
figure. In real life, node 9 went with the instructor’s club, even though the graph partitioning
analysis here would predict that he would join the president’s club.
Zachary’s original analysis of the karate club employed a di↵erent approach that also
used the network structure. He first supplemented the network with numerical estimates of
tie strength for the edges, based on his empirical study of the relationships within the karate
club. He then identified a set of edges of minimum total strength whose removal would place
node 1 and node 34 (the rival leaders) in di↵erent connected components, and he predicted
this as the split. The approach Zachary used, deleting edges of minimum total strength so
as to separate two specified nodes, is known as the problem of finding a minimum cut in a
graph, and it has the been the subject of extensive research and applications [8, 164, 253]. On
the karate-club network, this minimum-cut approach produced the same split as the Girvan-
Newman method: it agreed with the split that actually occurred except for the outcome
of node 9, an alignment of predictions that emphasizes how di↵erent approaches to graph
partitioning can produce corresponding results. It is also interesting to note that Zachary
traced the anomalous nature of node 9 to a fact that the network structure could not capture:
at the time of the actual split, the person corresponding to node 9 was three weeks away
from completing a four-year quest to obtain a black belt, which he could only do with the
instructor (node 1).
78 CHAPTER 3. STRONG AND WEAK TIES
B
E
D
F
KG
I
H
C
A
J
(a) A sample network
B ED
F
K
G
I
H
C
A
J
(b) Breadth-first search starting at node A
Figure 3.18: The first step in the e�cient method for computing betweenness values is to perform a breadth-first search of the network. Here the results of breadth-first from node A are shown; over the course of the method, breadth-first search is performed from each node in turn.
Among the other examples discussed by Girvan and Newman, they provide a partition
of the co-authorship network from Figure 3.12, with the top level of regions suggested by
the di↵erent shadings of the nodes in that figure.
Ultimately, it is a challenge to rigorously evaluate graph partitioning methods and to
formulate ways of asserting that one is better than another — both because the goal is hard
to formalize, and because di↵erent methods may be more or less e↵ective on di↵erent kinds
of networks. Moreover, a line of recent work by Leskovec et al. has argued that in real social-
network data, it is much easier to separate a tightly-knit region from the rest of the network
when it is relatively small, on the order of at most a few hundred nodes [275]. Studies on
a range of di↵erent social and information networks suggest that beyond this size, sets of
nodes become much more “inextricable” from the rest of the network, suggesting that graph
partitioning approaches on this type of data may produce qualitatively di↵erent kinds of
results for small networks and small regions than for large ones. This is an area of ongoing
investigation.
In the remainder of this section, we address a final important issue: how to actually
compute the betweenness quantities that are needed in order to make the Girvan-Newman
method work.
3.6. ADVANCED MATERIAL: BETWEENNESS MEASURES AND GRAPH PARTITIONING79
B. Computing Betweenness Values
In order to perform the Girvan-Newman method, we need a way to find the edges of highest
betweenness in each step. This is done by computing all the betweennesses of all edges and
then looking for the ones with the highest values. The tricky part is that the definition
of betweenness involves reasoning about the set of all the shortest paths between pairs of
nodes. Since there could be a very large number of such shortest paths, how can we e�ciently
compute betweenness without the overhead of actually listing out all such paths? This is
crucial for implementing the method on a computer to work with datasets of any reasonable
size.
In fact, there is a clever way to compute betweennesses e�ciently [77, 317], and it is
based on the notion of breadth-first search from Section 2.3. We will consider the graph
from the perspective of one node at a time; for each given node, we will compute how the
total flow from that node to all others is distributed over the edges. If we do this for every
node, then we can simply add up the flows from all of them to get the betweennesses on
every edge.
So let’s consider how we would determine the flow from one node to all other nodes in
the graph. As an example, we’ll look at the graph in Figure 3.18(a), focusing on how the
flow from node A reaches all other nodes. We do this in three high-level steps; below we
explain the details of how each of these steps works.
(1) Perform a breadth-first search of the graph, starting at A.
(2) Determine the number of shortest paths from A to each other node.
(3) Based on these numbers, determine the amount of flow from A to all other nodes that
uses each edge.
For the first step, recall that breadth-first search divides a graph into layers starting at a
given node (A in our case), with all the nodes in layer d having distance d from A. Moreover,
the shortest paths from A to a node X in layer d are precisely the paths that move downward
from A to X one layer at a time, thereby taking exactly d steps. Figure 3.18(b) shows the
result of breadth-first search from A in our graph, with the layers placed horizontally going
downward from A. Thus, for example, some inspection of the figure shows that there are
two shortest paths (each of length two) from A to F : one using nodes A, B, and F , and the
other using nodes A, C, and F .
Counting Shortest Paths. Now, let’s consider the second step: determining the number
of shortest paths from A to each other node. There is a remarkably clean way to do this, by
working down through the layers of the breadth-first search.
80 CHAPTER 3. STRONG AND WEAK TIES
B ED
F
K
G
I
H
C
A
J
1 1 1 1
2 1 2
3 3
6
# shortest A-K paths = # shortest A-I paths + # shortest A-J paths
# shortest A-J paths = # shortest A-G paths + # shortest A-H paths
# shortest A-I paths = # shortest A-F paths + # shortest A-G paths
Figure 3.19: The second step in computing betweenness values is to count the number of shortest paths from a starting node A to all other nodes in the network. This can be done by adding up counts of shortest paths, moving downward through the breadth-first search structure.
To motivate this, consider a node like I in Figure 3.18(b). All shortest-paths from A to
I must take their last step through either F or G, since these are the two nodes above it
in the breadth-first search. (For terminological convenience, we will say that a node X is
above a node Y in the breadth-first search if X is in the layer immediately preceding Y , and
X has an edge to Y .) Moreover, in order to be a shortest path to I, a path must first be a
shortest path to one of F or G, and then take this last step to I. It follows that the number
of shortest paths from A to I is precisely the number of shortest paths from A to F , plus
the number of shortest paths from A to G.
We can use this as a general method to count the number of shortest paths from A to
all other nodes, as depicted in Figure 3.19. Each node in the first layer is a neighbor of A,
and so it has only one shortest path from A: the edge leading straight from A to it. So
we give each of these nodes a count of 1. Now, as we move down through the BFS layers,
we apply the reasoning discussed above to conclude that the number of shortest paths to
3.6. ADVANCED MATERIAL: BETWEENNESS MEASURES AND GRAPH PARTITIONING81
B ED
F
K
G
I
H
C
A
J
1 1 1 1
2 1 2
3 3
6
1/2 1/2
1 1/2 1/2 1
1 1 2 1 1
2 2 4 2
Figure 3.20: The final step in computing betweenness values is to determine the flow values from a starting node A to all other nodes in the network. This is done by working up from the lowest layers of the breadth-first search, dividing up the flow above a node in proportion to the number of shortest paths coming into it on each edge.
each node should be the sum of the number of shortest paths to all nodes directly above it
in the breadth-first search. Working downward through the layers, we thus get the number
of shortest paths to each node, as shown in Figure 3.19. Note that by the time we get to
deeper layers, it may not be so easy to determine these number by visual inspection — for
example, to immediately list the six di↵erent shortest paths from A to K — but it is quite
easy when they are built up layer-by-layer in this way.
Determining Flow Values. Finally, we come to the third step, computing how the flow
from A to all other nodes spreads out across the edges. Here too we use the breadth-first
search structure, but this time working up from the lowest layers. We first show the idea in
Figure 3.20 on our running example, and then describe the general procedure.
• Let’s start at the bottom with node K. A single unit of flow arrives at K, and an equal number of the shortest paths from A to K come through nodes I and J, so this unit
82 CHAPTER 3. STRONG AND WEAK TIES
of flow is equally divided over the two incoming edges. Therefore we put a half-unit of
flow on each of these edges.
• Now, working upward, the total amount of flow arriving at I is equal to the one unit actually destined for I plus the half-unit passing through to K, for a total of 3/2. How
does this 3/2 amount of flow get divided over the edges leading upward from I, to F
and G respectively? We see from the second step that there are twice as many shortest
paths from A through F as through G, so twice as much of the flow should come from
F . Therefore, we put one unit of the flow on F , and a half-unit of the flow on G, as
indicated in the figure.
• We continue in this way for each other node, working upward through the layers of the breadth-first search.
From this, it is not hard to describe the principle in general. When we get to a node
X in the breadth-first search structure, working up from the bottom, we add up all the
flow arriving from edges directly below X, plus 1 for the flow destined for X itself. We then
divide this up over the edges leading upward from X, in proportion to the number of shortest
paths coming through each. You can check that applying this principle leads to the numbers
shown in Figure 3.20.
We are now essentially done. We build one of these breadth-first structures from each
node in the network, determine flow values from the node using this procedure, and then sum
up the flow values to get the betweenness value for each edge. Notice that we are counting
the flow between each pair of nodes X and Y twice: once when we do the breadth-first search
from X, and once when we do it from Y . So at the end we divide everything by two to
cancel out this double-counting. Finally, using these betweenness values, we can identify the
edges of highest betweenness for purposes of removing them in the Girvan-Newman method.
Final Observations. The method we have just described can be used to compute the
betweennesses of nodes as well as edges. In fact, this is already happening in the third step:
notice that we are implicitly keeping track of the amounts of flow through the nodes as well
as through the edges, and this is what is needed to determine the betweennesses of the nodes.
The original Girvan-Newman method described here, based on repeated removal of high-
betweenness edges, is a good conceptual way to think about graph partitioning, and it
works well on networks of moderate size (up to a few thousand nodes). However, for larger
networks, the need to recompute betweenness values in every step becomes computationally
very expensive. In view of this, a range of di↵erent alternatives have been proposed to identify
similar sets of tightly-knit regions more e�ciently. These include methods of approximating
the betweenness [34] and related but more e�cient graph partitioning approaches using
3.7. EXERCISES 83
divisive and agglomerative methods [35, 321]. There remains considerable interest in finding
fast partitioning algorithms that can scale to very large network datasets.
3.7 Exercises
1. In 2-3 sentences, explain what triadic closure is, and how it plays a role in the formation
of social networks. You can draw a schematic picture in case this is useful.
2. Consider the graph in Figure 3.21, in which each edge — except the edge connecting
b and c — is labeled as a strong tie (S) or a weak tie (W).
According to the theory of strong and weak ties, with the strong triadic closure as-
sumption, how would you expect the edge connecting b and c to be labeled? Give a
brief (1-3 sentence) explanation for your answer.
a
b c
d
e f
W
W W
S
S
S
S
?
Figure 3.21:
3. In the social network depicted in Figure 3.22, with each edge labeled as either a strong
or weak tie, which nodes satisfy the Strong Triadic Closure Property from Chapter 3,
and which do not? Provide an explanation for your answer.
4. In the social network depicted in Figure 3.23 with each edge labeled as either a strong
or weak tie, which two nodes violate the Strong Triadic Closure Property? Provide an
explanation for your answer.
5. In the social network depicted in Figure 3.24, with each edge labeled as either a strong
or weak tie, which nodes satisfy the Strong Triadic Closure Property from Chapter 3,
and which do not? Provide an explanation for your answer.
84 CHAPTER 3. STRONG AND WEAK TIES
A
B
D
C
E
S
S
SW
S
S W
W
Figure 3.22:
A
B C
S W
E
S
D
S
S
W
W S
Figure 3.23: A graph with a strong/weak labeling.
A
B
D
C
S S
E
W
W
S
S
Figure 3.24:
Chapter 4
Networks in Their Surrounding Contexts
In Chapter 3 we considered some of the typical structures that characterize social net-
works, and some of the typical processes that a↵ect the formation of links in the network. Our
discussion there focused primarily on the network as an object of study in itself, relatively
independent of the broader world in which it exists.
However, the contexts in which a social network is embedded will generally have signif-
icant e↵ects on its structure, Each individual in a social network has a distinctive set of
personal characteristics, and similarities and compatibilities among two people’s characteris-
tics can strongly influence whether a link forms between them. Each individual also engages
in a set of behaviors and activities that can shape the formation of links within the network.
These considerations suggest what we mean by a network’s surrounding contexts: factors
that exist outside the nodes and edges of a network, but which nonetheless a↵ect how the
network’s structure evolves.
In this chapter we consider how such e↵ects operate, and what they imply about the
structure of social networks. Among other observations, we will find that the surrounding
contexts a↵ecting a network’s formation can, to some extent, be viewed in network terms as
well — and by expanding the network to represent the contexts together with the individuals,
we will see in fact that several di↵erent processes of network formation can be described in
a common framework.
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
85
86 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
4.1 Homophily
One of the most basic notions governing the structure of social networks is homophily — the
principle that we tend to be similar to our friends. Typically, your friends don’t look like a
random sample of the underlying population: viewed collectively, your friends are generally
similar to you along racial and ethnic dimensions; they are similar in age; and they are also
similar in characteristics that are more or less mutable, including the places they live, their
occupations, their levels of a✏uence, and their interests, beliefs, and opinions. Clearly most
of us have specific friendships that cross all these boundaries; but in aggregate, the pervasive
fact is that links in a social network tend to connect people who are similar to one another.
This observation has a long history; as McPherson, Smith-Lovin, and Cook note in
their extensive review of research on homophily [294], the underlying idea can be found in
writings of Plato (“similarity begets friendship”) and Aristotle (people “love those who are
like themselves”), as well as in proverbs such as “birds of a feather flock together.” Its role
in modern sociological research was catalyzed in large part by influential work of Lazarsfeld
and Merton in the 1950s [269].
Homophily provides us with a first, fundamental illustration of how a network’s sur-
rounding contexts can drive the formation of its links. Consider the basic contrast between
a friendship that forms because two people are introduced through a common friend and
a friendship that forms because two people attend the same school or work for the same
company. In the first case, a new link is added for reasons that are intrinsic to the network
itself; we need not look beyond the network to understand where the link came from. In
the second case, the new link arises for an equally natural reason, but one that makes sense
only when we look at the contextual factors beyond the network — at some of the social
environments (in this case schools and companies) to which the nodes belong.
Often, when we look at a network, such contexts capture some of the dominant fea-
tures of its overall structure. Figure 4.1, for example, depicts the social network within a
particular town’s middle school and high school (encompassing grades 7-12) [304]; in this
image, produced by the study’s author James Moody, students of di↵erent races are drawn
as di↵erently-colored circles. Two dominant divisions within the network are apparent. One
division is based on race (from left to right in the figure); the other, based on age and school
attendance, separates students in the middle school from those in the high school (from top
to bottom in the figure). There are many other structural details in this network, but the
e↵ects of these two contexts stand out when the network is viewed at a global level.
Of course, there are strong interactions between intrinsic and contextual e↵ects on the
formation of any single link; they are both operating concurrently in the same network. For
example, the principle of triadic closure — that triangles in the network tend to “close”
as links form between friends of friends — is supported by a range of mechanisms that
range from the intrinsic to the contextual. In Chapter 3 we motivated triadic closure by
4.1. HOMOPHILY 87
Figure 4.1: Homophily can produce a division of a social network into densely-connected, homogeneous parts that are weakly connected to each other. In this social network from a town’s middle school and high school, two such divisions in the network are apparent: one based on race (with students of di↵erent races drawn as di↵erently colored circles), and the other based on friendships in the middle and high schools respectively [304].
hypothesizing intrinsic mechanisms: when individuals B and C have a common friend A,
then there are increased opportunities and sources of trust on which to base their interactions,
and A will also have incentives to facilitate their friendship. However, social contexts also
provide natural bases for triadic closure: since we know that A-B and A-C friendships
already exist, the principle of homophily suggests that B and C are each likely to be similar
to A in a number of dimensions, and hence quite possibly similar to each other as well. As
a result, based purely on this similarity, there is an elevated chance that a B-C friendship
will form; and this is true even if neither of them is aware that the other one knows A.
The point isn’t that any one basis for triadic closure is the “correct” one. Rather, as we
take into account more and more of the factors that drive the formation of links in a social
88 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
Figure 4.2: Using a numerical measure, one can determine whether small networks such as this one (with nodes divided into two types) exhibit homophily.
network, it inevitably becomes di�cult to attribute any individual link to a single factor.
And ultimately, one expects most links to in fact arise from a combination of several factors
— partly due to the e↵ect of other nodes in the network, and partly due to the surrounding
contexts.
Measuring Homophily. When we see striking divisions within a network like the one in
Figure 4.1, it is important to ask whether they are “genuinely” present in the network itself,
and not simply an artifact of how it is drawn. To make this question concrete, we need to
formulate it more precisely: given a particular characteristic of interest (like race, or age),
is there a simple test we can apply to a network in order to estimate whether it exhibits
homophily according to this characteristic?
Since the example in Figure 4.1 is too large to inspect by hand, let’s consider this question
on a smaller example where we can develop some intuition. Let’s suppose in particular that
we have the friendship network of an elementary-school classroom, and we suspect that it
exhibits homophily by gender: boys tend to be friends with boys, and girls tend to be friends
with girls. For example, the graph in Figure 4.2 shows the friendship network of a (small)
hypothetical classroom in which the three shaded nodes are girls and the six unshaded nodes
are boys. If there were no cross-gender edges at all, then the question of homophily would
be easy to resolve: it would be present in an extreme sense. But we expect that homophily
should be a more subtle e↵ect that is visible mainly in aggregate — as it is, for example, in
the real data from Figure 4.1. Is the picture in Figure 4.2 consistent with homophily?
There is a natural numerical measure of homophily that we can use to address questions
4.1. HOMOPHILY 89
like this [202, 319]. To motivate the measure (using the example of gender as in Figure 4.2),
we first ask the following question: what would it mean for a network not to exhibit ho-
mophily by gender? It would mean that the proportion of male and female friends a person
has looks like the background male/female distribution in the full population. Here’s a
closely related formulation of this “no-homophily” definition that is a bit easier to analyze:
if we were to randomly assign each node a gender according to the gender balance in the
real network, then the number of cross-gender edges should not change significantly relative
to what we see in the real network. That is, in a network with no homophily, friendships are
being formed as though there were random mixing across the given characteristic.
Thus, suppose we have a network in which a p fraction of all individuals are male, and
a q fraction of all individuals are female. Consider a given edge in this network. If we
independently assign each node the gender male with probability p and the gender female
with probability q, then both ends of the edge will be male with probability p2, and both
ends will be female with probability q2. On the other hand, if the first end of the edge is
male and the second end is female, or vice versa, then we have a cross-gender edge, so this
happens with probability 2pq.
So we can summarize the test for homophily according to gender as follows:
Homophily Test: If the fraction of cross-gender edges is significantly less than
2pq, then there is evidence for homophily.
In Figure 4.2, for example, 5 of the 18 edges in the graph are cross-gender. Since p = 2/3
and q = 1/3 in this example, we should be comparing the fraction of cross-gender edges to
the quantity 2pq = 4/9 = 8/18. In other words, with no homophily, one should expect to
see 8 cross-gender edges rather than than 5, and so this example shows some evidence of
homophily.
There are a few points to note here. First, the number of cross-gender edges in a random
assignment of genders will deviate some amount from its expected value of 2pq, and so
to perform the test in practice one needs a working definition of “significantly less than.”
Standard measures of statistical significance (quantifying the significance of a deviation below
a mean) can be used for this purpose. Second, it’s also easily possible for a network to have a
fraction of cross-gender edges that is significantly more than 2pq. In such a case, we say that
the network exhibits inverse homophily. The network of romantic relationships in Figure 2.7
from Chapter 2 is a clear example of this; almost all the relationships reported by the high-
school students in the study involved opposite-sex partners, rather than same-sex partners,
so almost all the edges are cross-gender.
Finally, it’s easy to extend our homophily test to any underlying characteristic (race,
ethnicity, age, native language, political orientation, and so forth). When the characteristic
can only take two possible values (say, one’s voting preference in a two-candidate election),
then we can draw a direct analogy to the case of two genders, and use the same formula
90 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
2pq. When the characteristic can take on more than two possible values, we still perform a
general version of the same calculation. For this, we say that an edge is heterogeneous if it
connects two nodes that are di↵erent according to the characteristic in question. We then ask
how the number of heterogeneous edges compares to what we’d see if we were to randomly
assign values for the characteristic to all nodes in the network — using the proportions from
the real data as probabilities. In this way, even a network in which the nodes are classified
into many groups can be tested for homophily using the same underlying comparison to a
baseline of random mixing.
4.2 Mechanisms Underlying Homophily: Selection and Social Influence
The fact that people tend to have links to others who are similar to them is a statement about
the structure of social networks; on its own, it does not propose an underlying mechanism
by which ties among similar people are preferentially formed.
In the case of immutable characteristics such as race or ethnicity, the tendency of people
to form friendships with others who are like them is often termed selection, in that people
are selecting friends with similar characteristics. Selection may operate at several di↵erent
scales, and with di↵erent levels of intentionality. In a small group, when people choose
friends who are most similar from among a clearly delineated pool of potential contacts,
there is clearly active choice going on. In other cases, and at more global levels, selection
can be more implicit. For example, when people live in neighborhoods, attend schools, or
work for companies that are relatively homogeneous compared to the population at large,
the social environment is already favoring opportunities to form friendships with others like
oneself. For this discussion, we will refer to all these e↵ects cumulatively as selection.
When we consider how immutable characteristics interact with network formation, the
order of events is clear: a person’s attributes are determined at birth, and they play a
role in how this person’s connections are formed over the course of his or her life. With
characteristics that are more mutable, on the other hand — behaviors, activities, interests,
beliefs, and opinions — the feedback e↵ects between people’s individual characteristics and
their links in the social network become significantly more complex. The process of selection
still operates, with individual characteristics a↵ecting the connections that are formed. But
now another process comes into play as well: people may modify their behaviors to bring
them more closely into alignment with the behaviors of their friends. This process has
been variously described as socialization [233] and social influence [170], since the existing
social connections in a network are influencing the individual characteristics of the nodes.
Social influence can be viewed as the reverse of selection: with selection, the individual
characteristics drive the formation of links, while with social influence, the existing links in
4.2. MECHANISMS UNDERLYING HOMOPHILY: SELECTION AND SOCIAL INFLUENCE91
the network serve to shape people’s (mutable) characteristics.1
The Interplay of Selection and Social Influence. When we look at a single snapshot
of a network and see that people tend to share mutable characteristics with their friends, it
can be very hard to sort out the distinct e↵ects and relative contributions of selection and
social influence. Have the people in the network adapted their behaviors to become more like
their friends, or have they sought out people who were already like them? Such questions can
be addressed using longitudinal studies of a social network, in which the social connections
and the behaviors within a group are both tracked over a period of time. Fundamentally, this
makes it possible to see the behavioral changes that occur after changes in an individual’s
network connections, as opposed to the changes to the network that occur after an individual
changes his or her behavior.
This type of methodology has been used, for example, to study the processes that lead
pairs of adolescent friends to have similar outcomes in terms of scholastic achievement and
delinquent behavior such as drug use [92]. Empirical evidence confirms the intuitive fact that
teenage friends are similar to each other in their behaviors, and both selection and social
influence have a natural resonance in this setting: teenagers seek out social circles composed
of people like them, and peer pressure causes them to conform to behavioral patterns within
their social circles. What is much harder to resolve is how these two e↵ects interact, and
whether one is more strongly at work than the other. As longitudinal behavior relevant to
this question became available, researchers began quantifying the relative impact of these
di↵erent factors. A line of work beginning with Cohen and Kandel has suggested that while
both e↵ects are present in the data, the outsized role that earlier informal arguments had
accorded to peer pressure (i.e. social influence) is actually more moderate; the e↵ect of
selection here is in fact comparable to (and sometimes greater than) the e↵ect of social
influence [114, 233].
Understanding the tension between these di↵erent forces can be important not just for
identifying underlying causes, but also for reasoning about the e↵ect of possible interventions
one might attempt in the system [21, 396]. For example, once we find that illicit drug use
displays homophily across a social network — with students showing a greater likelihood to
use drugs when their friends do — we can ask about the e↵ects of a program that targets
certain high-school students and influences them to stop using drugs. To the extent that the
observed homophily is based on some amount of social influence, such a program could have
a broad impact across the social network, by causing the friends of these targeted students
to stop using drugs as well. But one must be careful; if the observed homophily is arising
instead almost entirely from selection e↵ects, then the program may not reduce drug use
1There are other cognitive e↵ects at work as well; for example, people may systematically misperceive the characteristics of their friends as being more in alignment with their own than they really are [224]. For our discussion here, we will not focus explicitly on such e↵ects.
92 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
beyond the students it directly targets: as these students stop using drugs, they change
their social circles and form new friendships with students who don’t use drugs, but the
drug-using behavior of other students is not strongly a↵ected.
Another example of research addressing this subtle interplay of factors is the work of
Christakis and Fowler on the e↵ect of social networks on health-related outcomes. In one
recent study, using longitudinal data covering roughly 12,000 people, they tracked obesity
status and social network structure over a 32-year period [108]. They found that obese and
non-obese people clustered in the network in a fashion consistent with homophily, according
to the numerical measure described in Section 4.1: people tend to be more similar in obesity
status to their network neighbors than in a version of the same network where obesity status
is assigned randomly. The problem is then to distinguish among several hypotheses for why
this clustering is present: is it
(i) because of selection e↵ects, in which people are choosing to form friendships with others
of similar obesity status?
(ii) because of the confounding e↵ects of homophily according to other characteristics, in
which the network structure indicates existing patterns of similarity in other dimensions
that correlate with obesity status? or
(iii) because changes in the obesity status of a person’s friends was exerting a (presumably
behavioral) influence that a↵ected his or her future obesity status?
Statistical analysis in Christakis and Fowler’s paper argues that, even accounting for e↵ects of
types (i) and (ii), there is significant evidence for an e↵ect of type (iii) as well: that obesity is
a health condition displaying a form of social influence, with changes in your friends’ obesity
status in turn having a subsequent e↵ect on you. This suggests the intriguing prospect that
obesity (and perhaps other health conditions with a strong behavioral aspect) may exhibit
some amount of “contagion” in a social sense: you don’t necessarily catch it from your
friends the way you catch the flu, but it nonetheless can spread through the underlying
social network via the mechanism of social influence.
These examples, and this general style of investigation, show how careful analysis is
needed to distinguish among di↵erent factors contributing to an aggregate conclusion: even
when people tend to be similar to their neighbors in a social network, it may not be clear
why. The point is that an observation of homophily is often not an endpoint in itself, but
rather the starting point for deeper questions — questions that address why the homophily
is present, how its underlying mechanisms will a↵ect the further evolution of the network,
and how these mechanisms interact with possible outside attempts to influence the behavior
of people in the network.
4.3. AFFILIATION 93
Anna
Daniel Karate
Club
Literacy
Volunteers
Figure 4.3: An a�liation network is a bipartite graph that shows which individuals are a�liated with which groups or activities. Here, Anna participates in both of the social foci on the right, while Daniel participates in only one.
4.3 A�liation
Thus far, we have been discussing contextual factors that a↵ect the formation of links in
a network — based on similarities in characteristics of the nodes, and based on behaviors
and activities that the nodes engage in. These surrounding contexts have been viewed,
appropriately, as existing “outside” the network. But in fact, it’s possible to put these
contexts into the network itself, by working with a larger network that contains both people
and contexts as nodes. Through such a network formulation, we will get additional insight
into some broad aspects of homophily, and see how the simultaneous evolution of contexts
and friendships can be put on a common network footing with the notion of triadic closure
from Chapter 3.
In principle we could represent any context this way, but for the sake of concreteness we’ll
focus on how to represent the set of activities a person takes part in, and how these a↵ect
the formation of links. We will take a very general view of the notion of an “activity” here.
Being part of a particular company, organization, or neigborhood; frequenting a particular
place; pursuing a particular hobby or interest — these are all activities that, when shared
between two people, tend to increase the likelihood that they will interact and hence form a
link in the social network [78, 161]. Adopting terminology due to Scott Feld, we’ll refer to
such activities as foci — that is, “focal points” of social interaction — constituting “social,
psychological, legal, or physical entit[ies] around which joint activities are organized (e.g.
workplaces, voluntary organizations, hangouts, etc.)” [161].
A�liation Networks. As a first step, we can represent the participation of a set of people
in a set of foci using a graph as follows. We will have a node for each person, and a node
for each focus, and we will connect person A to focus X by an edge if A participates in X.
94 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
John
Doerr
Amazon
Apple
Disney
General
Electric
Al Gore
Shirley
Tilghman
Susan
Hockfield
Arthur
Levinson
Andrea
Jung
Steve
Jobs
Figure 4.4: One type of a�liation network that has been widely studied is the memberships of people on corporate boards of directors [301]. A very small portion of this network (as of mid-2009) is shown here. The structural pattern of memberships can reveal subtleties in the interactions among both the board members and the companies.
A very simple example of such a graph is depicted in Figure 4.3, showing two people (Anna
and Daniel) and two foci (working for a literacy tutoring organization, and belonging to a
karate club). The graph indicates that Anna participates in both of the foci, while Daniel
participates in only one.
We will refer to such a graph as an a�liation network, since it represents the a�liation of
people (drawn on the left) with foci (drawn on the right) [78, 323]. More generally, a�liation
networks are examples of a class of graphs called bipartite graphs. We say that a graph is
bipartite if its nodes can be divided into two sets in such a way that every edge connects a
node in one set to a node in the other set. (In other words, there are no edges joining a pair
of nodes that belong to the same set; all edges go between the two sets.) Bipartite graphs
are very useful for representing data in which the items under study come in two categories,
and we want to understand how the items in one category are associated with the items
in the other. In the case of a�liation networks, the two categories are the people and the
foci, with each edge connecting a person to a focus that he or she participates in. Bipartite
4.3. AFFILIATION 95
graphs are often drawn as in Figure 4.3, with the two di↵erent sets of nodes drawn as two
parallel vertical columns, and the edges crossing between the two columns.
A�liation networks are studied in a range of settings where researchers want to un-
derstand the patterns of participation in structured activities. As one example, they have
received considerable attention in studying the composition of boards of directors of major
corporations [301]. Boards of directors are relatively small advisory groups populated by
high-status individuals; and since many people serve on multiple boards, the overlaps in
their participation have a complex structure. These overlaps can be naturally represented
by an a�liation network; as the example in Figure 4.4 shows, there is a node for each person
and a node for each board, and each edge connects a person to a board that they belong to.
A�liation networks defined by boards of directors have the potential to reveal interesting
relationships on both sides of the graph. Two companies are implicitly linked by having
the same person sit on both their boards; we can thus learn about possible conduits for
information and influence to flow between di↵erent companies. Two people, on the other
hand, are implicitly linked by serving together on a board, and so we learn about particular
patterns of social interaction among some of the most powerful members of society. Of
course, even the complete a�liation network of people and boards (of which Figure 4.4
is only a small piece) still misses other important contexts that these people inhabit; for
example, the seven people in Figure 4.4 include the presidents of two major universities and
a former Vice-President of the United States.2
Co-Evolution of Social and A�liation Networks. It’s clear that both social networks
and a�liation networks change over time: new friendship links are formed, and people
become associated with new foci. Moreover, these changes represent a kind of co-evolution
that reflects the interplay between selection and social influence: if two people participate in
a shared focus, this provides them with an opportunity to become friends; and if two people
are friends, they can influence each other’s choice of foci.
There is a natural network perspective on these ideas, which begins from a network
representation that slightly extends the notion of an a�liation network. As before, we’ll
have nodes for people and nodes for foci, but we now introduce two distinct kinds of edges
as well. The first kind of edge functions as an edge in a social network: it connects two
2The structure of this network changes over time as well, and sometimes in ways that reinforce the points in our present discussion. For example, the board memberships shown in Figure 4.4 are taken from the middle of 2009; by the end of 2009, Arthur Levinson had resigned from the board of directors of Google (thus removing one edge from the graph). As part of the news coverage of this resignation, the chair of the U.S. Federal Trade Commission, Jon Leibowitz, explicitly invoked the notion of overlaps in board membership, saying, “Google, Apple and Mr. Levinson should be commended for recognizing that overlapping board members between competing companies raise serious antitrust issues, and for their willingness to resolve our concerns without the need for litigation. Beyond this matter, we will continue to monitor companies that share board members and take enforcement actions where appropriate” [219].
96 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
AnnaBob
Claire
Daniel Karate
Club
Literacy
Volunteers
Figure 4.5: A social-a�liation network shows both the friendships between people and their a�liation with di↵erent social foci.
people, and indicates friendship (or alternatively some other social relation, like professional
collaboration). The second kind of edge functions as an edge in an a�liation network: it
connects a person to a focus, and indicates the participation of the person in the focus. We
will call such a network a social-a�liation network, reflecting the fact that it simultaneously
contains a social network on the people and an a�liation network on the people and foci.
Figure 4.5 depicts a simple social-a�liation network.
Once we have social-a�liation networks as our representation, we can appreciate that
a range of di↵erent mechanisms for link formation can all be viewed as types of closure
processes, in that they involve “closing” the third edge of a triangle in the network. In
particular, suppose we have two nodes B and C with a common neighbor A in the network,
and suppose that an edge forms between B and C. There are several interpretations for
what this corresponds to, depending on whether A, B, and C are people or foci.
(i) If A, B, and C each represent a person, then the formation of the link between B and
C is triadic closure, just as in Chapter 3. (See Figure 4.6(a).)
(ii) If B and C represent people, but A represents a focus, then this is something di↵erent:
it is the tendency of two people to form a link when they have a focus in common. (See
Figure 4.6(b).) This is an aspect of the more general principle of selection, forming
links to others who share characteristics with you. To emphasize the analogy with
triadic closure, this process has been called focal closure [259].
(iii) If A and B are people, and C is a focus, then we have the formation of a new a�liation:
B takes part in a focus that her friend A is already involved in. (See Figure 4.6(c).)
This is a kind of social influence, in which B’s behavior comes into closer alignment
4.4. TRACKING LINK FORMATION IN ON-LINE DATA 97
C
A
B
person
person
person
(a) Triadic closure
C
B
person
focus
person
A
(b) Focal closure
A
B
person
person
focus
C
(c) Membership closure
Figure 4.6: Each of triadic closure, focal closure, and membership closure corresponds to the closing of a triangle in a social-a�liation network.
with that of her friend A. Continuing the analogy with triadic closure, we will refer to
this kind of link formation as membership closure.
Thus, three very di↵erent underlying mechanisms — reflecting triadic closure and aspects
of selection and social influence — can be unified in this type of network as kinds of closure:
the formation of a link in cases where the two endpoints already have a neighbor in common.
Figure 4.7 shows all three kinds of closure processes at work: triadic closure leads to a new
link between Anna and Claire; focal closure leads to a new link between Anna and Daniel;
and membership closure leads to Bob’s a�liation with the karate club. Oversimplifying the
mechanisms at work, they can be summarized in the following succinct way:
(i) Bob introduces Anna to Claire.
(ii) Karate introduces Anna to Daniel.
(iii) Anna introduces Bob to Karate.
4.4 Tracking Link Formation in On-Line Data
In this chapter and the previous one, we have identified a set of di↵erent mechanisms that
lead to the formation of links in social networks. These mechansisms are good examples
98 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
AnnaBob
Claire
Daniel Karate
Club
Literacy
Volunteers
Figure 4.7: In a social-a�liation network containing both people and foci, edges can form under the e↵ect of several di↵erent kinds of closure processes: two people with a friend in common, two people with a focus in common, or a person joining a focus that a friend is already involved in.
of social phenomena which are clearly at work in small-group settings, but which have
traditionally been very hard to measure quantitatively. A natural research strategy is to
try tracking these mechanisms as they operate in large populations, where an accumulation
of many small e↵ects can produce something observable in the aggregate. However, given
that most of the forces responsible for link formation go largely unrecorded in everyday life,
it is a challenge to select a large, clearly delineated group of people (and social foci), and
accurately quantify the relative contributions that these di↵erent mechanisms make to the
formation of real network links.
The availability of data from large on-line settings with clear social structure has made
it possible to attempt some preliminary research along these lines. As we emphasized in
Chapter 2, any analysis of social processes based on such on-line datasets must come with
a number of caveats. In particular, it is never a priori clear how much one can extrapolate
from digital interactions to interactions that are not computer-mediated, or even from one
computer-mediated setting to another. Of course, this problem of extrapolation is present
whenever one studies phenomena in a model system, on-line or not, and the kinds of mea-
surements these large datasets enable represent interesting first steps toward a deeper quan-
titative understanding of how mechanisms of link formation operate in real life. Exploring
these questions in a broader range of large datasets is an important problem, and one that
will become easier as large-scale data becomes increasingly abundant.
Triadic closure. With this background in mind, let’s start with some questions about
triadic closure. Here’s a first, basic numerical question: how much more likely is a link to
4.4. TRACKING LINK FORMATION IN ON-LINE DATA 99
AnnaBob
Claire
Daniel Karate
Club
Literacy
Volunteers Esther
Frank
Grace
Figure 4.8: A larger network that contains the example from Figure 4.7. Pairs of people can have more than one friend (or more than one focus) in common; how does this increase the likelihood that an edge will form between them?
form between two people in a social network if they already have a friend in common? (In
other words, how much more likely is a link to form if it has the e↵ect of closing a triangle?)
Here’s a second question, along the same lines as the first: How much more likely is an
edge to form between two people if they have multiple friends in common? For example,
in Figure 4.8, Anna and Esther have two friends in common, while Claire and Daniel only
have one friend in common. How much more likely is the formation of a link in the first of
these two cases? If we go back to the arguments for why triadic closure operates in social
networks, we see that they all are qualitatively strengthened as two people have more friends
in common: there are more sources of opportunity and trust for the interaction, there are
more people with an incentive to bring them together, and the evidence for homophily is
arguably stronger.
We can address these questions empirically using network data as follows.
(i) We take two snapshots of the network at di↵erent times.
(ii) For each k, we identify all pairs of nodes who have exactly k friends in common in the
first snapshot, but who are not directly connected by an edge.
(iii) We define T (k) to be the fraction of these pairs that have formed an edge by the time
100 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
0 2 4 6 8 10
Number of common friends
0
0.001
0.002
0.003
0.004
0.005
0.006
p ro
b . o f li n k f o rm
a ti o n
Figure 4.9: Quantifying the e↵ects of triadic closure in an e-mail dataset [259]. The curve determined from the data is shown in the solid black line; the dotted curves show a compar- ison to probabilities computed according to two simple baseline models in which common friends provide independent probabilities of link formation.
of the second snapshot. This is our empirical estimate for the probability that a link
will form between two people with k friends in common.
(iv) We plot T (k) as a function of k to illustrate the e↵ect of common friends on the
formation of links.
Note that T (0) is the rate at which link formation happens when it does not close a triangle,
while the values of T (k) for larger k determine the rate at which link formation happens
when it does close a triangle. Thus, the comparison between T (0) and these other values
addresses the most basic question about the power of triadic closure.
Kossinets and Watts computed this function T (k) using a dataset encoding the full history
of e-mail communication among roughly 22,000 undergraduate and graduate students over
a one-year period at a large U.S. university [259]. This is a “who-talks-to-whom” type of
dataset, as we discussed in Chapter 2; from the communication traces, Kossinets and Watts
constructed a network that evolved over time, joining two people by a link at a given instant
if they had exchanged e-mail in each direction at some point in the past 60 days. They then
determined an “average” version of T (k) by taking multiple pairs of snapshots: they built
a curve for T (k) on each pair of snapshots using the procedure described above, and then
4.4. TRACKING LINK FORMATION IN ON-LINE DATA 101
averaged all the curves they obtained. In particular, the observations in each snapshot were
one day apart, so their computation gives the average probability that two people form a
link per day, as a function of the number of common friends they have.
Figure 4.9 shows a plot of this curve (in the solid black line). The first thing one notices
is the clear evidence for triadic closure: T (0) is very close to 0, after which the probability
of link formation increases steadily as the number of common friends increases. Moreover,
for much of the plot, this probability increases in a roughly linear fashion as a function
of the number of common friends, with an upward bend away from a straight-line shape.
The curve turns upward in a particularly pronounced way from 0 to 1 to 2 friends: having
two common friends produces significantly more than twice the e↵ect on link formation
compared to having a single common friend. (The upward e↵ect from 8 to 9 to 10 friends is
also significant, but it occurs on a much smaller sub-population, since many fewer people in
the data have this many friends in common without having already formed a link.)
To interpret this plot more deeply, it helps to compare it to an intentionally simplified
baseline model, describing what one might have expected the data to look like in the presence
of triadic closure. Suppose that for some small probability p, each common friend that two
people have gives them an independent probability p of forming a link each day. So if two
people have k friends in common, the probability they fail to form a link on any given day is
(1�p)k: this is because each common friend fails to cause the link to form with probability 1 � p, and these k trials are independent. Since (1 � p)k is the probability the link fails to form on a given day, the probability that it does form, according to our simple baseline
model, is
T baseline
(k) = 1 � (1 � p)k.
We plot this curve in Figure 4.9 as the upper dotted line. Given the small absolute e↵ect of
the first common friend in the data, we also show a comparison to the curve 1 � (1 � p)k�1, which just shifts the simple baseline curve one unit to the right. Again, the point is not to
propose this baseline as an explanatory mechanism for triadic closure, but rather to look at
how the real data compares to it. Both the real curve and the baseline curve are close to
linear, and hence qualitatively similar; but the fact that the real data turns upward while the
baseline curve turns slightly downward indicates that the assumption of independent e↵ects
from common friends is too simple to be fully supported by the data.
A still larger and more detailed study of these e↵ects was conducted by Leskovec et
al. [272], who analyzed properties of triadic closure in the on-line social networks of LinkedIn,
Flickr, Del.icio.us, and Yahoo! Answers. It remains an interesting question to try under-
standing the similarities and variations in triadic closure e↵ects across social interaction in
a range of di↵erent settings.
102 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
0 1 2 3 4 5
number of common foci
0
0.0001
0.0002
0.0003
0.0004
0.0005
p ro
b . o f li n k f o rm
a ti o n
Figure 4.10: Quantifying the e↵ects of focal closure in an e-mail dataset [259]. Again, the curve determined from the data is shown in the solid black line, while the dotted curve provides a comparison to a simple baseline.
Focal and Membership Closure. Using the same approach, we can compute probabil-
ities for the other kinds of closure discussed earlier — specifically,
• focal closure: what is the probability that two people form a link as a function of the number of foci they are jointly a�liated with?
• membership closure: what is the probability that a person becomes involved with a particular focus as a function of the number of friends who are already involved in it?
As an example of the first of these kinds of closure, using Figure 4.8, Anna and Grace have
one activity in common while Anna and Frank have two in common. As an example of the
second, Esther has one friend who belongs to the karate club while Claire has two. How do
these distinctions a↵ect the formation of new links?
For focal closure, Kossinets and Watts supplemented their university e-mail dataset with
information about the class schedules for each student. In this way, each class became a
focus, and two students shared a focus if they had taken a class together. They could then
compute the probability of focal closure by direct analogy with their computation for triadic
closure, determining the probability of link formation per day as a function of the number of
shared foci. Figure 4.10 shows a plot of this function. A single shared class turns out to have
roughly the same absolute e↵ect on link formation as a single shared friend, but after this the
4.4. TRACKING LINK FORMATION IN ON-LINE DATA 103
0
0.005
0.01
0.015
0.02
0.025
0 5 10 15 20 25 30 35 40 45 50
p ro
b a
b il it y
k
Probability of joining a community when k friends are already members
Figure 4.11: Quantifying the e↵ects of membership closure in a large online dataset: The plot shows the probability of joining a LiveJournal community as a function of the number of friends who are already members [32].
curve for focal closure behaves quite di↵erently from the curve for triadic closure: it turns
downward and appears to approximately level o↵, rather than turning slightly upward. Thus,
subsequent shared classes after the first produce a “diminishing returns” e↵ect. Comparing
to the same kind of baseline, in which the probability of link formation with k shared classes
is 1 � (1 � p)k (shown as the dotted curve in Figure 4.10), we see that the real data turns downward more significantly than this independent model. Again, it is an interesting open
question to understand how this e↵ect generalizes to other types of shared foci, and to other
domains.
For membership closure, the analogous quantities have been measured in other on-line
domains that possess both person-to-person interactions and person-to-focus a�liations.
Figure 4.11 is based on the blogging site LiveJournal, where friendships are designated by
users in their profiles, and where foci correspond to membership in user-defined communities
[32]; thus the plot shows the probability of joining a community as a function of the number
of friends who have already done so. Figure 4.12 shows a similar analysis for Wikipedia [122].
Here, the social-a�liation network contains a node for each Wikipedia editor who maintains
a user account and user talk page on the system; and there is an edge joining two such editors
if they have communicated, with one editor writing on the user talk page of the other. Each
104 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
Figure 4.12: Quantifying the e↵ects of membership closure in a large online dataset: The plot shows the probability of editing a Wikipedia articles as a function of the number of friends who have already done so [122].
Wikipedia article defines a focus — an editor is associated with a focus corresponding to a
particular article if he or she has edited the article. Thus, the plot in Figure 4.12 shows the
probability a person edits a Wikipedia article as a function of the number of prior editors
with whom he or she has communicated.
As with triadic and focal closure, the probabilities in both Figure 4.11 and 4.12 increase
with the number k of common neighbors — representing friends associated with the foci. The
marginal e↵ect diminishes as the number of friends increases, but the e↵ect of subsequent
friends remains significant. Moreover, in both sources of data, there is an initial increasing
e↵ect similar to what we saw with triadic closure: in this case, the probability of joining a
LiveJournal community or editing a Wikipedia article is more than twice as great when you
have two connections into the focus rather than one. In other words, the connection to a
second person in the focus has a particularly pronounced e↵ect, and after this the diminishing
marginal e↵ect of connections to further people takes over.
Of course, multiple e↵ects can operate simultaneously on the formation of a single link.
For example, if we consider the example in Figure 4.8, triadic closure makes a link between
Bob and Daniel more likely due to their shared friendship with Anna; and focal closure also
makes this link more likely due to the shared membership of Bob and Daniel in the karate
club. If a link does form between them, it will not necessarily be a priori clear how to
attribute it to these two distinct e↵ects. This is also a reflection of an issue we discussed
4.4. TRACKING LINK FORMATION IN ON-LINE DATA 105
in Section 4.1, when describing some of the mechanisms behind triadic closure: since the
principle of homophily suggests that friends tend to have many characteristics in common,
the existence of a shared friend between two people is often indicative of other, possibly
unobserved, sources of similarity (such as shared foci in this case) that by themselves may
also make link formation more likely.
Quantifying the Interplay Between Selection and Social Influence. As a final
illustration of how we can use large-scale on-line data to track processes of link formation,
let’s return to the question of how selection and social influence work together to produce
homophily, considered in Section 4.2. We’ll make use of the Wikipedia data discussed earlier
in this section, asking: how do similarities in behavior between two Wikipedia editors relate
to their pattern of social interaction over time? [122]
To make this question precise, we need to define both the social network and an underlying
measure of behavioral similarity. As before, the social network will consist of all Wikipedia
editors who maintain talk pages, and there is an edge connecting two editors if they have
communicated, with one writing on the talk page of the other. An editor’s behavior will
correspond to the set of articles she has edited. There are a number of natural ways to define
numerical measures of similarity between two editors based on their actions; a simple one is
to declare their similarity to be the value of the ratio
number of articles edited by both A and B
number of articles edited by at least one of A or B , (4.1)
For example, if editor A has edited the Wikipedia articles on Ithaca NY and Cornell Uni-
versity, and editor B has edited the articles on Cornell University and Stanford University,
then their similarity under this measure is 1/3, since they have jointly edited one article
(Cornell) out of three that they have edited in total (Cornell, Ithaca, and Stanford). Note
the close similarity to the definition of neighborhood overlap used in Section 3.3; indeed,
the measure in Equation (4.1) is precisely the neighborhood overlap of two editors in the
bipartite a�liation network of editors and articles, consisting only of edges from editors to
the articles they’ve edited.3
Pairs of Wikipedia editors who have communicated are significantly more similar in their
behavior than pairs of Wikipedia editors who have not communicated, so we have a case
where homophily is clearly present. Therefore, we are set up to address the question of selec-
tion and social influence: is the homophily arising because editors are forming connections
with those who have edited the same articles they have (selection), or is it because editors
are led to the articles of those they talk to (social influence)?
3For technical reasons, a minor variation on this simple similarity measure is used for the results that follow. However, since this variation is more complicated to describe, and the di↵erences are not significant for our purposes, we can think of similarity as consisting of the numerical measure just defined.
106 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
Selection: rapid
increase in similarity
before first contact
Social influence:
continued slower
increase in similarity
after first contact
Figure 4.13: The average similarity of two editors on Wikipedia, relative to the time (0) at which they first communicated [122]. Time, on the x-axis, is measured in discrete units, where each unit corresponds to a single Wikipedia action taken by either of the two editors. The curve increases both before and after the first contact at time 0, indicating that both selection and social influence play a role; the increase in similarity is steepest just before time 0.
Because every action on Wikipedia is recorded and time-stamped, it is not hard to get
an initial picture of this interplay, using the following method. For each pair of editors A
and B who have ever communicated, record their similarity over time, where “time” here
moves in discrete units, advancing by one “tick” whenever either A or B performs an action
on Wikipedia (editing an article or communicating with another editor). Next, declare time
0 for the pair A-B to be the point at which they first communicated. This results in many
curves showing similarity as a function of time — one for each pair of editors who ever
communicated, and each curve shifted so that time is measured for each one relative to
the moment of first communication. Averaging all these curves yields the single plot in
Figure 4.13 — it shows the average level of similarity relative to the time of first interaction,
over all pairs of editors who have ever interacted on Wikipedia [122].
There are a number of things to notice about this plot. First, similarity is clearly increas-
ing both before and after the moment of first interaction, indicating that both selection and
4.5. A SPATIAL MODEL OF SEGREGATION 107
social influence are at work. However, the the curve is not symmetric around time 0; the
period of fastest increase in similarity is clearly occurring before 0, indicating a particular
role for selection: there is an especially rapid rise in similarity, on average, just before two
editors meet.4 Also note that the levels of similarity depicted in the plot are much higher
than for pairs of editors who have not interacted: the dashed blue line at the bottom of the
plot shows similarity over time for a random sample of non-interacting pairs; it is both far
lower and also essentially constant as time moves forward.
At a higher level, the plot in Figure 4.13 once again illustrates the trade-o↵s involved in
working with large-scale on-line data. On the one hand, the curve is remarkably smooth,
because so many pairs are being averaged, and so di↵erences between selection and social
influence show up that are genuine, but too subtle to be noticeable at smaller scales. On the
other hand, the e↵ect being observed is an aggregate one: it is the average of the interaction
histories of many di↵erent pairs of individuals, and it does not provide more detailed insight
into the experience of any one particular pair.5 A goal for further research is clearly to
find ways of formulating more complex, nuanced questions that can still be meaningfully
addressed on large datasets.
Overall, then, these analyses represent early attempts to quantify some of the basic
mechanisms of link formation at a very large scale, using on-line data. While they are
promising in revealing that the basic patterns indeed show up strongly in the data, they
raise many further questions. In particular, it natural to ask whether the general shapes of
the curves in Figures 4.9–4.13 are similar across di↵erent domains — including domains that
are less technologically mediated — and whether these curve shapes can be explained at a
simpler level by more basic underlying social mechanisms.
4.5 A Spatial Model of Segregation
One of the most readily perceived e↵ects of homophily is in the formation of ethnically and
racially homogeneous neighborhoods in cities. Traveling through a metropolitan area, one
finds that homophily produces a natural spatial signature; people live near others like them,
and as a consequence they open shops, restaurants, and other businesses oriented toward the
populations of their respective neighborhoods. The e↵ect is also striking when superimposed
on a map, as Figure 4.14 by Möbius and Rosenblat [302] illustrates. Their images depict the
4To make sure that these are editors with significant histories on Wikipedia, this plot is constructed using only pairs of editors who each had at least 100 actions both before and after their first interaction with each other.
5Because the individual histories being averaged took place at many distinct points in Wikipedia’s history, it is also natural to ask whether the aggregate e↵ects operated di↵erently in di↵erent phases of this history. This is a natural question for further investigation, but initial tests — based on studying these types of properties on Wikipedia datasets built from di↵erent periods — show that the main e↵ects have remained relatively stable over time.
108 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
(a) Chicago, 1940 (b) Chicago, 1960
Figure 4.14: The tendency of people to live in racially homogeneous neighborhoods produces spatial patterns of segregation that are apparent both in everyday life and when superim- posed on a map — as here, in these maps of Chicago from 1940 and 1960 [302]. In blocks colored yellow and orange the percentage of African-Americans is below 25, while in blocks colored brown and black the percentage is above 75.
percentage of African-Americans per city block in Chicago for the years 1940 and 1960; in
blocks colored yellow and orange the percentage is below 25, while in blocks colored brown
and black the percentage is above 75.
This pair of figures also shows how concentrations of di↵erent groups can intensify over
time, emphasizing that this is a process with a dynamic aspect. Using the principles we’ve
been considering, we now discuss how simple mechansisms based on similarity and selection
can provide insight into the observed patterns and their dynamics.
The Schelling Model. A famous model due to Thomas Schelling [365, 366] shows how
global patterns of spatial segregation can arise from the e↵ect of homophily operating at a
local level. There are many factors that contribute to segregation in real life, but Schelling’s
model focuses on an intentionally simplified mechanism to illustrate how the forces leading to
segregation are remarkably robust — they can operate even when no one individual explicitly
wants a segregated outcome.
4.5. A SPATIAL MODEL OF SEGREGATION 109
X X
X
X
X
X
X X
XXX
O
O
O
O O
O O O
O O
O
(a) Agents occupying cells on a grid.
O
O O
X
O
X O
X
X
X
XO
O
O
X X
XX
X
O O O
(b) Neighbor relations as a graph.
Figure 4.15: In Schelling’s segregation model, agents of two di↵erent types (X and O) occupy cells on a grid. The neighbor relationships among the cells can be represented very simply as a graph. Agents care about whether they have at least some neighbors of the same type.
The general formulation of the model is as follows. We assume that there is a population
of individuals, whom we’ll call agents; each agent is of type X or type O. We think of the
two types as representing some (immutable) characteristic that can serve as the basis for
homophily — for example, race, ethnicity, country of origin, or native language. The agents
reside in the cells of a grid, intended as a stylized model of the two-dimensional geography
of a city. As illustrated in Figure 4.15(a), we will assume that some cells of the grid contain
agents while others are unpopulated. A cell’s neighbors are the cells that touch it, including
diagonal contact; thus, a cell that is not on the boundary of the grid has eight neighbors.
We can equivalently think of the neighbor relationships as defining a graph: the cells are the
nodes, and we put an edge between two cells that are neighbors on the grid. In this view,
the agents thus occupy the nodes of a graph that are arranged in this grid-like pattern, as
shown in Figure 4.15(b). For ease of visualization, however, we will continue to draw things
using a geometric grid, rather than a graph.
The fundamental constraint driving the model is that each agent wants to have at least
some other agents of its own type as neighbors. We will assume that there is a threshold t
common to all agents: if an agent discovers that fewer than t of its neighbors are of the same
type as itself, then it has an interest in moving to a new cell. We will call such an agent
unsatisfied with its current location. For example, in Figure 4.16(a), we indicate with an
asterisk all the agents that are unsatisfied in the arrangement from Figure 4.15(a), when the
threshold t is equal to 3. (In Figure 4.16(a) we have also added a number after each agent.
This is simply to provide each with a unique name; the key distinction is still whether each
agent is of type X or type O.)
110 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
X1* X2*
X3
X4
X6*
X5
X7 X8
X11X10X9*
O1*
O3
O6
O7 O8
O9 O10 O11*
O4 O5*
O2
(a) An initial configuration.
X1
X3
X4
X6
X5
X7 X8
X11X10X9
O1
O3
O6
O7 O8
O9 O10*
O11
O4
O5
O2
X2
(b) After one round of movement.
Figure 4.16: After arranging agents in cells of the grid, we first determine which agents are unsatisfied, with fewer than t other agents of the same type as neighbors. In one round, each of these agents moves to a cell where they will be satisfied; this may cause other agents to become unsatisfied, in which case a new round of movement begins.
4.5. A SPATIAL MODEL OF SEGREGATION 111
The Dynamics of Movement. Thus far, we have simply specified a set of agents that
want to move, given an underlying threshold; we now discuss how this gives the model its
dynamic aspect. Agents move in a sequence of rounds: in each round, we consider the
unsatisfied agents in some order, and for each one in turn, we have it move to an unoccupied
cell where it will be satisfied. After this, the round of movement has come to an end,
representing a fixed time period during which unsatisfied agents have changed where they
live. These new locations may cause di↵erent agents to be unsatisfied, and this leads to a
new round of movement.
In the literature on this model, there are numerous variations in the specific details of
how the movement of agents within a round is handled. For example, the agents can be
scheduled to move in a random order, or in an order that sweeps downward along rows of
the grid; they can move to the nearest location that will make them satisfied or to a random
one. There also needs to be a way of handling situations in which an agent is scheduled to
move, and there is no cell that will make it satisified. In such a case, the agent can be left
where it is, or moved to a completely random cell. Research has found that the qualitative
results of the model tend to be quite similar however these issues are resolved, and di↵erent
investigations of the model have tended to resolve them di↵erently.
For example, Figure 4.16(b) shows the results of one round of movement, starting from
the arrangement in Figure 4.16(a), when the threshold t is 3. Unsatisfied agents are scheduled
to move by considering them one row at a time working downward through the grid, and
each agent moves to the nearest cell that will make it satisfied. (The unique name of each
agent in the figure allows us to see where it has moved in Figure 4.16(b), relative to the initial
state in Figure 4.16(a).) Notice that in some concrete respects, the pattern of agents has
become more “segregated” after this round of movement. For example, in Figure 4.16(a),
there is only a single agent with no neighbors of the opposite type. After this first round of
movement, however, there are six agents in Figure 4.16(b) with no neighbors of the opposite
type. As we will see, this increasing level of segregation is the key behavior to emerge from
the model.
Larger examples. Small examples of the type in Figures 4.15 and 4.16 are helpful in
working through the details of the model by hand; but at such small scales it is di�cult to
see the kinds of typical patterns that arise. For this, computer simulation is very useful.
There are many on-line computer programs that make it possible to simulate the Schelling
model; as with the published literature on the model, they all tend to di↵er slightly from
each other in their specifics. Here we discuss some examples from a simulation written by
Sean Luke [282], which is like the version of the model we have discussed thus far except
that unsatisfied agents move to a random location.
In Figure 4.17, we show the results of simulating the model on a grid with 150 rows and
112 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
(a) A simulation with threshold 3. (b) Another simulation with threshold 3.
Figure 4.17: Two runs of a simulation of the Schelling model with a threshold t of 3, on a 150-by-150 grid with 10, 000 agents of each type. Each cell of the grid is colored red if it is occupied by an agent of the first type, blue if it is occupied by an agent of the second type, and black if it is empty (not occupied by any agent).
150 columns, 10, 000 agents of each type, and 2500 empty cells. The threshold t is equal to
3, as in our earlier examples. The two images depict the results of two di↵erent runs of the
simulation, with di↵erent random starting patterns of agents. In each case, the simulation
reached a point (shown in the figures) at which all agents were satisfied, after roughly 50
rounds of movement.
Because of the di↵erent random starts, the final arrangement of agents is di↵erent in
the two cases, but the qualitative similarities reflect the fundamental consequences of the
model. By seeking out locations near other agents of the same type, the model produces
large homogeneous regions, interlocking with each other as they stretch across the grid. In
the midst of these regions are large numbers of agents who are surrounded on all sides by
other agents of the same type — and in fact at some distance from the nearest agent of
the opposite type. The geometric pattern has become segregated, much as in the maps of
Chicago from Figure 4.14 with which we began the section.
Interpretations of the Model. We’ve now seen how the model works, what it looks
like at relatively large scales, and how it produces spatially segregated outcomes. But what
broader insights into homophily and segregation does it suggest?
The first and most basic one is that spatial segregation is taking place even though no
4.5. A SPATIAL MODEL OF SEGREGATION 113
X X O
X
O
O
O
O O
XXO
X
X
O
X OX
O O X
X O
O
O X X
O X X
X X
O
X X X
Figure 4.18: With a threshold of 3, it is possible to arrange agents in an integrated pattern: all agents are satisfied, and everyone who is not on the boundary on the grid has an equal number of neighbors of each type.
individual agent is actively seeking it. Sticking to our focus on a threshold t of 3, we see that
although agents want to be near others like them, their requirements are not particularly
draconian. For example, an agent would be perfectly happy to be in the minority among
its neighbors, with five neighbors of the opposite type and three of its own type. Nor are
the requirements globally incompatible with complete integration of the population. By
arranging agents in a checkerboard pattern as shown in Figure 4.18, we can make each agent
satisfied, and all agents not on the boundary of the grid have exactly four neighbors of each
type. This is a pattern that we can continue on as large a grid as we want.
Thus, segregation is not happening because we have subtly built it into the model —
agents are willing to be in the minority, and they could all be satisfied if we were only able
to carefully arrange them in an integrated pattern. The problem is that from a random
start, it is very hard for the collection of agents to find such integrated patterns. Much
more typically, agents will attach themselves to clusters of others like themselves, and these
clusters will grow as other agents follow suit. Moreover, there is a compounding e↵ect
as the rounds of movement unfold, in which agents who fall below their threshold depart
for more homogeneous parts of the grid, causing previously satisfied agents to fall below
their thresholds and move as well — an e↵ect that Schelling describes as the progressive
“unraveling” of more integrated regions [366]. In the long run, this process will tend to
cause segregated regions to grow at the expense of more integrated ones. The overall e↵ect
is one in which the local preferences of individual agents have produced a global pattern that
none of them necessarily intended.
114 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
(a) After 20 steps (b) After 150 steps
(c) After 350 steps (d) After 800 steps
Figure 4.19: Four intermediate points in a simulation of the Schelling model with a threshold t of 4, on a 150-by-150 grid with 10, 000 agents of each type. As the rounds of movement progress, large homogeneous regions on the grid grow at the expense of smaller, narrower regions.
4.5. A SPATIAL MODEL OF SEGREGATION 115
This point is ultimately at the heart of the model: although segregation in real life is
amplified by a genuine desire within some fraction of the population to belong to large
clusters of similar people — either to avoid people who belong to other groups, or to acquire
a critical mass of members from one’s own group — we see here that such factors are not
necessary for segregation to occur. The underpinnings of segregation are already present in
a system where individuals simply want to avoid being in too extreme a minority in their
own local area.
The process operates even more powerfully when we raise the threshold t in our examples
from 3 to 4. Even with a threshold of 4, nodes are willing to have an equal number of
neighbors of each type; and a slightly more elaborate checkerboard example in the spirit of
Figure 4.18 shows that with careful placement, the agents can be arranged so that all are
satisfied and most still have a significant number of neighbors of the opposite type. But now,
not only is an integrated pattern very hard to reach from a random starting arrangement —
any vestiges of integration among the two types tends to collapse completely over time. As
one example of this, Figure 4.19 shows four intermediate points in one run of a simulation
with threshold 4 and other properties the same as before (a 150-by-150 grid with 10, 000
agents of each type and random movement by unsatisfied agents) [282]. Figure 4.19(a) shows
that after 20 rounds of movement, we have an arrangement of agents that roughly resembles
what we saw with a lower threshold of 3. However, this does not last long: crucially, the
long tendrils where one type interlocks with the other quickly wither and retract, leaving
the more homogeneous regions shown after 150 rounds in Figure 4.19(b). This pulling-back
continues, passing through a phase with a large and small region of each type after 350
rounds (Figure 4.19(c)) eventually to a point where there is only a single significant region
of each type, after roughly 800 rounds (Figure 4.19(d)). Note that this is not the end of the
process, since there remain agents around the edges still looking for places to move, but by
this point the overall two-region layout has become very stable. Finally, we stress that this
figure corresponds to just a single run of the simulation — but computational experiments
show that the sequence of events it depicts, leading to almost complete separation of the two
types, is very robust when the threshold is this high.
Viewed at a still more general level, the Schelling model is an example of how character-
istics that are fixed and unchanging (such as race or ethnicity) can become highly correlated
with other characteristics that are mutable. In this case, the mutable characteristic is the
decision about where to live, which over time conforms to similarities in the agents’ (im-
mutable) types, producing segregation. But there are other, non-spatial manifestation of
the same e↵ect, in which beliefs and opinions become correlated across racial or ethnic lines,
and for similar underlying reasons: as homophily draws people together along immutable
characteristics, there is a natural tendency for mutable characteristics to change in accor-
dance with the network structure.
116 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
As a final point, we note that while the model is mathematically precise and self-
contained, the discussion has been carried out in terms of simulations and qualitative obser-
vations. This is because rigorous mathematical analysis of the Schelling model appears to
be quite di�cult, and is largely an open research question. For partial progress on analyzing
properties of the Schelling model, see the work of Young [420], who compares properties of
di↵erent arrangements in which all agents are satisfied; Möbius and Rosenblat [302], who
perform a probabilistic analysis; and Vinković and Kirman [401], who develop analogies to
models for the mixing of two liquids and other physical phenomena.
4.6 Exercises
b
e
c
a
d
Figure 4.20: A social network where triadic closure may occur.
1. Consider the social network represented in Figure 4.20. Suppose that this social net-
work was obtained by observing a group of people at a particular point in time and
recording all their friendship relations. Now suppose that we come back at some point
in the future and observe it again. According to the theories based on empirical studies
of triadic closure in networks, which new edge is most likely to be present? (I.e. which
pair of nodes, who do not currently have an edge connecting them, are most likely to
be linked by an edge when we return to take the second observation?) Also, give a
brief explanation for your answer.
2. Given a bipartite a�liation graph, showing the membership of people in di↵erent social
foci, researchers sometimes create a projected graph on just the people, in which we
join two people when they have a focus in common.
(a) Draw what such a projected graph would look like for the example of memberships
on corporate boards of directors from Figure 4.4. Here the nodes would be the
4.6. EXERCISES 117
seven people in the figure, and there would be an edge joining any two who serve
on a board of directors together.
(b) Give an example of two di↵erent a�liation networks — on the same set of people,
but with di↵erent foci — so that the projected graphs from these two di↵erent
a�liation networks are the same. This shows how information can be “lost” when
moving from the full a�liation network to just the projected graph on the set of
people.
A
X
Y
Z
D
B
C
F
E
Figure 4.21: An a�liation network on six people labeled A–F , and three foci labeled X, Y , and Z.
3. Consider the a�liation network in Figure 4.21, with six people labeled A–F , and three
foci labeled X, Y , and Z.
(a) Draw the derived network on just the six people as in Exercise 2, joining two
people when they share a focus.
(b) In the resulting network on people, can you identify a sense in which the triangle
on the nodes A, C, and E has a qualitatively di↵erent meaning than the other
triangles that appear in the network? Explain.
118 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
BA
C D
E
F
Figure 4.22: A graph on people arising from an (unobserved) a�liation network.
4. Given a network showing pairs of people who share activities, we can try to reconstruct
an a�liation network consistent with this data.
For example, suppose that you are trying to infer the structure of a bipartite a�liation
network, and by indirect observation you’ve obtained the projected network on just
the set of people, constructed as in Exercise 2: there is an edge joining each pair of
people who share a focus. This projected network is shown in Figure 4.22.
(a) Draw an a�liation network involving these six people, together with four foci that
you should define, whose projected network is the graph shown in Figure 4.22.
(b) Explain why any a�liation network capable of producing the projected network
in Figure 4.22 must have at least four foci.
Chapter 5
Positive and Negative Relationships
In our discussion of networks thus far, we have generally viewed the relationships con-
tained in these networks as having positive connotations — links have typically indicated
such things as friendship, collaboration, sharing of information, or membership in a group.
The terminology of on-line social networks reflects a largely similar view, through its em-
phasis on the connections one forms with friends, fans, followers, and so forth. But in most
network settings, there are also negative e↵ects at work. Some relations are friendly, but
others are antagonistic or hostile; interactions between people or groups are regularly beset
by controversy, disagreement, and sometimes outright conflict. How should we reason about
the mix of positive and negative relationships that take place within a network?
Here we describe a rich part of social network theory that involves taking a network
and annotating its links (i.e., its edges) with positive and negative signs. Positive links
represent friendship while negative links represent antagonism, and an important problem
in the study of social networks is to understand the tension between these two forces. The
notion of structural balance that we discuss in this chapter is one of the basic frameworks for
doing this.
In addition to introducing some of the basics of structural balance, our discussion here
serves a second, methodological purpose: it illustrates a nice connection between local and
global network properties. A recurring issue in the analysis of networked systems is the way
in which local e↵ects — phenomena involving only a few nodes at a time — can have global
consequences that are observable at the level of the network as a whole. Structural balance
o↵ers a way to capture one such relationship in a very clean way, and by purely mathematical
analysis: we will consider a simple definition abstractly, and find that it inevitably leads to
certain macroscopic properties of the network.
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
119
120 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
5.1 Structural Balance
We focus here on perhaps the most basic model of positive and negative relationships, since
it captures the essential idea. Suppose we have a social network on a set of people, in which
everyone knows everyone else — so we have an edge joining each pair of nodes. Such a
network is called a clique, or a complete graph. We then label each edge with either + or �; a + label indicates that its two endpoints are friends, while a � label indicates that its two endpoints are enemies.
Note that since there’s an edge connecting each pair, we are assuming that each pair
of people are either friends or enemies — no two people are indi↵erent to one another,
or unaware of each other. Thus, the model we’re considering makes the most sense for a
group of people small enough to have this level of mutual awareness (e.g. a classroom, a
small company, a sports team, a fraternity or sorority), or for a setting such as international
relations, in which the nodes are countries and every country has an o�cial diplomatic
position toward every other.1
The principles underlying structural balance are based on theories in social psychology
dating back to the work of Heider in the 1940s [216], and generalized and extended to
the language of graphs beginning with the work of Cartwright and Harary in the 1950s
[97, 126, 204]. The crucial idea is the following. If we look at any two people in the group
in isolation, the edge between them can be labeled + or �; that is, they are either friends or enemies. But when we look at sets of three people at a time, certain configurations of +’s
and �’s are socially and psychologically more plausible than others. In particular, there are four distinct ways (up to symmetry) to label the three edges among three people with +’s
and �’s; see Figure 5.1. We can distinguish among these four possibilities as follows.
• Given a set of people A, B, and C, having three pluses among them (as in Figure 5.1(a)) is a very natural situation: it corresponds to three people who are mutual friends.
• Having a single plus and two minuses in the relations among the there people is also very natural: it means that two of the three are friends, and they have a mutual enemy
in the third. (See Figure 5.1(c).)
• The other two possible labelings of the triangle on A, B, and C introduce some amount of psychological “stress” or “instability” into the relationships. A triangle with two
pluses and one minus corresponds (as in Figure 5.1(b)) to a person A who is friends
with each of B and C, but B and C don’t get along with each other. In this type of
situation, there would be implicit forces pushing A to try to get B and C to become
1Later, in Section 5.5, we will consider the more general setting in which not every pair of nodes is necessarily connected by an edge.
5.1. STRUCTURAL BALANCE 121
A
B C
+ +
+
(a) A, B, and C are mutual friends: balanced.
A
B C
+ +
-
(b) A is friends with B and C, but they don’t get along with each other: not balanced.
A
B C
+ -
-
(c) A and B are friends with C as a mutual en- emy: balanced.
A
B C
- -
-
(d) A, B, and C are mutual enemies: not bal- anced.
Figure 5.1: Structural balance: Each labeled triangle must have 1 or 3 positive edges.
friends (thus turning the B-C edge label to +); or else for A to side with one of B or
C against the other (turning one of the edge labels out of A to a �).
• Similarly, there are sources of instability in a configuration where each of A, B, and C are mutual enemies (as in Figure 5.1(d)). In this case, there would be forces motivating
two of the three people to “team up” against the third (turning one of the three edge
labels to a +).
Based on this reasoning, we will refer to triangles with one or three +’s as balanced, since
they are free of these sources of instability, and we will refer to triangles with zero or two
+’s as unbalanced. The argument of structural balance theorists is that because unbalanced
triangles are sources of stress or psychological dissonance, people strive to minimize them in
their personal relationships, and hence they will be less abundant in real social settings than
122 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
A
C D
B
A
C D
B
+
+
- -
- -
- -
-
+
+ +
balanced not balanced
Figure 5.2: The labeled four-node complete graph on the left is balanced; the one on the right is not.
balanced triangles.
Defining Structural Balance for Networks. So far we have been talking about struc-
tural balance for groups of three nodes. But it is easy to create a definition that naturally
generalizes this to complete graphs on an arbitrary number of nodes, with edges labeled by
+’s and �’s. Specifically, we say that a labeled complete graph is balanced if every one of its triangles
is balanced — that is, if it obeys the following:
Structural Balance Property: For every set of three nodes, if we consider the three
edges connecting them, either all three of these edges are labeled +, or else exactly
one of them is labeled +.
For example, consider the two labeled four-node networks in Figure 5.2. The one on
the left is balanced, since we can check that each set of three nodes satisfies the Structural
Balance Property above. On the other hand, the one on the right is not balanced, since among
the three nodes A, B, C, there are exactly two edges labeled +, in violation of Structural
Balance. (The triangle on B, C, D also violates the condition.)
Our definition of balanced networks here represents the limit of a social system that has
eliminated all unbalanced triangles. As such, it is a fairly extreme definition — for example,
one could instead propose a definition which only required that at least some large percentage
of all triangles were balanced, allowing a few triangles to be unbalanced. But the version
with all triangles balanced is a fundamental first step in thinking about this concept; and
5.2. CHARACTERIZING THE STRUCTURE OF BALANCED NETWORKS 123
mutual friends
inside X
mutual friends
inside Y
set X set Y
mutual
antagonism
between
sets
Figure 5.3: If a complete graph can be divided into two sets of mutual friends, with complete mutual antagonism between the two sets, then it is balanced. Furthermore, this is the only way for a complete graph to be balanced.
as we will see next, it turns out to have very interesting mathematical structure that in fact
helps to inform the conclusions of more complicated models as well.
5.2 Characterizing the Structure of Balanced Networks
At a general level, what does a balanced network (i.e. a balanced labeled complete graph)
look like? Given any specific example, we can check all triangles to make sure that they
each obey the balance conditions; but it would be much better to have a simple conceptual
description of what a balanced network looks like in general.
One way for a network to be balanced is if everyone likes each other; in this case, all
triangles have three + labels. On the other hand, the left-hand side of Figure 5.2 suggests
a slightly more complicated way for a network to be balanced: it consists of two groups of
friends (A, B and C, D), with negative relations between people in di↵erent groups. This is
actually true in general: suppose we have a labeled complete graph in which the nodes can
be divided into two groups, X and Y , such that every pair of nodes in X like each other,
every pair of nodes in Y like each other, and everyone in X is the enemy of everyone in
Y . (See the schematic illustration in Figure 5.3.) You can check that such a network is
balanced: a triangle contained entirely in one group or the other has three + labels, and a
triangle with two people in one group and one in the other has exactly one + label.
So this describes two basic ways to achieve structural balance: either everyone likes
each other; or the world consists of two groups of mutual friends with complete antagonism
124 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
between the groups. The surprising fact is the following: these are the only ways to have
a balanced network. We formulate this fact precisely as the following Balance Theorem,
proved by Frank Harary in 1953 [97, 204]:
Balance Theorem: If a labeled complete graph is balanced, then either all pairs
of nodes are friends, or else the nodes can be divided into two groups, X and Y ,
such that every pair of nodes in X like each other, every pair of nodes in Y like
each other, and everyone in X is the enemy of everyone in Y .
The Balance Theorem is not at all an obvious fact, nor should it be initially clear why it
is true. Essentially, we’re taking a purely local property, namely the Structural Balance
Property, which applies to only three nodes at a time, and showing that it implies a strong
global property: either everyone gets along, or the world is divided into two battling factions.
We’re now going to show why this claim in fact is true.
Proving the Balance Theorem. Establishing the claim requires a proof: we’re going to
suppose we have an arbitrary labeled complete graph, assume only that it is balanced, and
conclude that either everyone is friends, or that there are sets X and Y as described in the
claim. Recall that we worked through a proof in Chapter 3 as well, when we used simple
assumptions about triadic closure in a social network to conclude all local bridges in the
network must be weak ties. Our proof here will be somewhat longer, but still very natural
and straightforward — we use the definition of balance to directly derive the conclusion of
the claim.
To start, suppose we have a labeled complete graph, and all we know is that it’s balanced.
We have to show that it has the structure in the claim. If it has no negative edges at all,
then everyone is friends, and we’re all set. Otherwise, there is at least one negative edge,
and we need to somehow come up with a division of the nodes into sets of mutual friends
X and Y , with complete antagonism between them. The di�culty is that, knowing so little
about the graph itself other than that it is balanced, it’s not clear how we’re supposed to
identify X and Y .
Let’s pick any node in the network — we’ll call it A — and consider things from A’s
perspective. Every other node is either a friend of A or an enemy of A. Thus, natural
candidates to try for the sets X and Y would be to define X to be A and all its friends, and
define Y to be all the enemies of A. This is indeed a division of all the nodes, since every
node is either a friend or an enemy of A.
Recall what we need to show in order for these two sets X and Y to satisfy the conditions
of the claim:
(i) Every two nodes in X are friends.
(ii) Every two nodes in Y are friends.
5.2. CHARACTERIZING THE STRUCTURE OF BALANCED NETWORKS 125
A
B
C E
D
+
+
-
-
?
?
?
friends of A enemies of A
Figure 5.4: A schematic illustration of our analysis of balanced networks. (There may be other nodes not illustrated here.)
(iii) Every node in X is an enemy of every node in Y .
Let’s argue that each of these conditions is in fact true for our choice of X and Y . This will
mean that X and Y do satisfy the conditions of the claim, and will complete the proof. The
rest of the argument, establishing (i), (ii), and (iii), is illustrated schematically in Figure 5.4.
For (i), we know that A is friends with every other node in X. How about two other
nodes in X (let’s call them B and C) — must they be friends? We know that A is friends
with both B and C, so if B and C were enemies of each other, then A, B, and C would
form a triangle with two + labels — a violation of the balance condition. Since we know
the network is balanced, this can’t happen, so it must be that B and C in fact are friends.
Since B and C were the names of any two nodes in X, we have concluded that every two
nodes in X are friends.
Let’s try the same kind of argument for (ii). Consider any two nodes in Y (let’s call them
D and E) — must they be friends? We know that A is enemies with both D and E, so if D
and E were enemies of each other, then A, D, and E would form a triangle with no + labels
— a violation of the balance condition. Since we know the network is balanced, this can’t
happen, so it must be that D and E in fact are friends. Since D and E were the names of
any two nodes in Y , we have concluded that every two nodes in Y are friends.
Finally, let’s try condition (iii). Following the style of our arguments for (i) and (ii),
consider a node in X (call if B) and a node in Y (call it D) — must they be enemies? We
know A is friends with B and enemies with D, so if B and D were friends, then a, B, and
126 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
D would form a triangle with two + labels — a violation of the balance condition. Since
we know the network is balanced, this can’t happen, so it must be that B and D in fact
are enemies. Since B and D were the names of any node in X and any node in Y , we have
concluded that every such pair constitutes a pair of enemies.
So, in conclusion, assuming only that the network is balanced, we have described a
division of the nodes into two sets X and Y , and we have checked conditions (i), (ii), and
(iii) required by the claim. This completes the proof of the Balance Theorem.
5.3 Applications of Structural Balance
Structural balance has grown into a large area of study, and we’ve only described a simple
but central example of the theory. In Section 5.5, we discuss two extensions to the basic
theory: one to handle graphs that are not necessarily complete, and one to describe the
structure of complete graphs that are “approximately balanced,” in the sense that most but
not all their triangles are balanced.
There has also been recent research looking at dynamic aspects of structural balance
theory, modeling how the set of friendships and antagonisms in a complete graph — in other
words, the labeling of the edges — might evolve over time, as the social network implicitly
seeks out structural balance. Antal, Krapivsky, and Redner [20] study a model in which we
start with a random labeling (choosing + or � randomly for each edge); we then repeatedly look for a triangle that is not balanced, and flip one of its labels to make it balanced. This
captures a situation in which people continually reassess their likes and dislikes of others, as
they strive for structural balance. The mathematics here becomes quite complicated, and
turns out to resemble the mathematical models one uses for certain physical systems as they
reconfigure to minimize their energy [20, 287].
In the remainder of this section, we consider two further areas in which the ideas of struc-
tural balance are relevant: international relations, where the nodes are di↵erent countries;
and on-line social media sites where users can express positive or negative opinions about
each other.
International Relations. International politics represents a setting in which it is natural
to assume that a collection of nodes all have opinions (positive or negative) about one another
— here the nodes are nations, and + and � labels indicate alliances or animosity. Research in political science has shown that structural balance can sometimes provide an e↵ective
explanation for the behavior of nations during various international crises. For example,
Moore [306], describing the conflict over Bangladesh’s separation from Pakistan in 1972,
explicitly invokes structural balance theory when he writes, “[T]he United States’s somewhat
surprising support of Pakistan ... becomes less surprising when one considers that the USSR
5.3. APPLICATIONS OF STRUCTURAL BALANCE 127
GB
Fr
Ru
AH
Ge
It
(a) Three Emperors’ League 1872– 81
GB
Fr
Ru
AH
Ge
It
(b) Triple Alliance 1882
GB
Fr
Ru
AH
Ge
It
(c) German-Russian Lapse 1890
GB
Fr
Ru
AH
Ge
It
(d) French-Russian Alliance 1891– 94
GB
Fr
Ru
AH
Ge
It
(e) Entente Cordiale 1904
GB
Fr
Ru
AH
Ge
It
(f) British Russian Alliance 1907
Figure 5.5: The evolution of alliances in Europe, 1872-1907 (the nations GB, Fr, Ru, It, Ge, and AH are Great Britain, France, Russia, Italy, Germany, and Austria-Hungary respec- tively). Solid dark edges indicate friendship while dotted red edges indicate enmity. Note how the network slides into a balanced labeling — and into World War I. This figure and example are from Antal, Krapivsky, and Redner [20].
was China’s enemy, China was India’s foe, and India had traditionally bad relations with
Pakistan. Since the U.S. was at that time improving its relations with China, it supported
the enemies of China’s enemies. Further reverberations of this strange political constellation
became inevitable: North Vietnam made friendly gestures toward India, Pakistan severed
diplomatic relations with those countries of the Eastern Bloc which recognized Bangladesh,
and China vetoed the acceptance of Bangladesh into the U.N.”
Antal, Krapivsky, and Redner use the shifting alliances preceding World War I as another
example of structural balance in international relations — see Figure 5.5. This also reinforces
the fact that structural balance is not necessarily a good thing: since its global outcome is
often two implacably opposed alliances, the search for balance in a system can sometimes
be seen as a slide into a hard-to-resolve opposition between two sides.
128 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
Trust, Distrust, and On-Line Ratings. A growing source for network data with both
positive and negative edges comes from user communities on the Web where people can
express positive or negative sentiments about each other. Examples include the technology
news site Slashdot, where users can designate each other as a “friend” or a “foe” [266],
and on-line product-rating sites such as Epinions, where a user can express evaluations of
di↵erent products, and also express trust or distrust of other users.
Guha, Kumar, Raghavan, and Tomkins performed an analysis of the network of user
evaluations on Epinions [201]; their work identified an interesting set of issues that show
how the trust/distrust dichotomy in on-line ratings has both similarities and di↵erences
with the friend/enemy dichotomy in structural balance theory. One di↵erence is based on a
simple structural distinction: we have been considering structural balance in the context of
undirected graphs, whereas user evaluations on a site like Epinions form a directed graph.
That is, when a user A expresses trust or distrust of a user B, we don’t necessarily know
what B thinks of A, or whether B is even aware of A.
A more subtle di↵erence between trust/distrust and friend/enemy relations becomes ap-
parent when thinking about how we should expect triangles on three Epinions users to
behave. Certain patterns are easy to reason about: for example, if user A trusts user B,
and user B trusts user C, then it is natural to expect that A will trust C. Such triangles
with three forward-pointing positive edges make sense here, by analogy with the all-positive
(undirected) triangles of structural balance theory. But what if A distrusts B and B dis-
trusts C: should we expect A to trust or to distrust C? There are appealing arguments
in both directions. If we think of distrust as fundamentally a kind of enemy relationship,
then the arguments from structural balance theory would suggest that A should trust C:
otherwise we’d have a triangle with three negative edges. On the other hand, if A’s distrust
of B expresses A’s belief that she is more knowledgeable and competent than B — and if
B’s distrust of C reflects a corresponding belief by B — then we might well expect that A
will distrust C, and perhaps even more strongly than she distrusts B.
It is reasonable to expect that these two di↵erent interpretations of distrust may each
apply, simply in di↵erent settings. And both might apply in the context of a single product-
rating site like Epinions. For example, among users who are primarily rating best-selling
books by political commentators, trust/distrust evaluations between users may become
strongly aligned with agreement or disagreement in these users’ own political orientations.
In such a case, if A distrusts B and B distrusts C, this may suggest that A and C are
close to each other on the underlying political spectrum, and so the prediction of structural
balance theory that A should trust C may apply. On the other hand, among users who are
primarily rating consumer electronics products, trust/distrust evaluations may largely reflect
the relative expertise of users about the products (their respective features, reliability, and
so forth). In such a case, if A distrusts B and B distrusts C, we might conclude that A is
5.4. A WEAKER FORM OF STRUCTURAL BALANCE 129
mutual friends
inside W
mutual friends
inside X
set W set X
mutual
antagonism
between
all sets
mutual
friends
inside Y
set Y
mutual
friends
inside Z
set Z
mutual
friends
inside V
set V
Figure 5.6: A complete graph is weakly balanced precisely when it can be divided into multiple sets of mutual friends, with complete mutual antagonism between each pair of sets.
far more expert than C, and so should distrust C as well.
Ultimately, understanding how these positive and negative relationships work is impor-
tant for understanding the role they play on social Web sites where users register subjective
evaluations of each other. Research is only beginning to explore these fundamental questions,
including the ways in which theories of balance — as well as related theories — can be used
to shed light on these issues in large-scale datasets [274].
5.4 A Weaker Form of Structural Balance
In studying models of positive and negative relationships on networks, researchers have also
formulated alternate notions of structural balance, by revisiting the original assumptions we
130 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
used to motivate the framework.
In particular, our analysis began from the claim that there are two kinds of structures on
a group of three people that are inherently unbalanced: a triangle with two positive edges
and one negative edge (as in Figure 5.1(b)); and a triangle with three negative edges (as in
Figure 5.1(d)). In each of these cases, we argued that the relationships within the triangle
contained a latent source of stress that the network might try to resolve. The underlying
arguments in the two cases, however, were fundamentally di↵erent. In a triangle with two
positive edges, we have the problem of a person whose two friends don’t get along; in a
triangle with three negative edges, there is the possibility that two of the nodes will ally
themselves against the third.
James Davis and others have argued that in many settings, the first of these factors may
be significantly stronger than the second [127]: we may see friends of friends trying to recon-
cile their di↵erences (resolving the lack of balance in Figure 5.1(b)), while at the same time
there could be less of a force leading any two of three mutual enemies (as in Figure 5.1(d)) to
become friendly. It therefore becomes natural to ask what structural properties arise when
we rule out only triangles with exactly two positive edges, while allowing triangles with three
negative edges to be present in the network.
Characterizing Weakly Balanced Networks. More precisely, we will say that a com-
plete graph, with each edge labeled by + or �, is weakly balanced if the following property holds.
Weak Structural Balance Property: There is no set of three nodes such that the
edges among them consist of exactly two positive edges and one negative edge.
Since weak balance imposes less of a restriction on what the network can look like, we
should expect to see a broader range of possible structures for weakly balanced networks
— beyond what the Balance Theorem required for networks that were balanced under our
original definition. And indeed, Figure 5.6 indicates a new kind of structure that can arise.
Suppose that the nodes can be divided into an arbitrary number of groups (possibly more
than two), so that two nodes are friends when they belong to the same group, and enemies
when they belong to di↵erent groups. Then we can check that such a network is weakly
balanced: in any triangle that contains at least two positive edges, all three nodes must
belong to the same group. Therefore, the third edge of this triangle must be positive as well
— in other words, the network contains no triangles with exactly two + edges.
Just as the Balance Theorem established that all balanced networks must have a simple
structure, an analogous result holds for weakly balanced networks: they must have the
structure depicted in Figure 5.6, with any number of groups.
Characterization of Weakly Balanced Networks: If a labeled complete graph is
weakly balanced, then its nodes can be divided into groups in such a way that
5.4. A WEAKER FORM OF STRUCTURAL BALANCE 131
A
B
C E
D
+
+
-
-
?
?
friends of A enemies of A
Figure 5.7: A schematic illustration of our analysis of weakly balanced networks. (There may be other nodes not illustrated here.)
every two nodes belonging to the same group are friends, and every two nodes
belonging to di↵erent groups are enemies.
The fact that this characterization is true in fact provided another early motivation for
studying weak structural balance. The Cartwright-Harary notion of balance predicted only
dichotomies (or mutual consensus) as its basic social structure, and thus did not provide
a model for reasoning about situations in which a network is divided into more than two
factions. Weak structural balance makes this possible, since weakly balanced complete graphs
can contain any number of opposed groups of mutual friends [127].
Proving the Characterization. It is not hard to give a proof for this characteriza-
tion, following the structure of our proof for the Balance Theorem, and making appropriate
changes where necessary. Starting with a weakly balanced complete graph, the characteriza-
tion requires that we produce a division of its nodes into groups of mutual friends, such that
all relations between nodes in di↵erent groups are negative. Here is how we will construct
this division.
First, we pick any node A, and we consider the set consisting of A and all its friends.
Let’s call this set of nodes X. We’d like to make X our first group, and for this to work, we
need to establish two things:
(i) All of A’s friends are friends with each other. (This way, we have indeed produced a
group of mutual friends).
132 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
(ii) A and all his friends are enemies with everyone else in the graph. (This way, the people
in this group will be enemies with everyone in other groups, however we divide up the
rest of the graph.)
Fortunately, ideas that we already used inside the proof of the Balance Theorem can be
adapted to our new setting here to establish (i) and (ii). The idea is shown in Figure 5.7.
First, for (i), let’s consider two nodes B and C who are both friends with A. If B and C
were enemies of each other, then the triangle on nodes A, B, and C would have exactly two
+ labels, which would violate weak structural balance. So B and C must indeed be friends
with each other.
For (ii), we know that A is enemies with all nodes in the graph outside X, since the
group X is defined to include all of A’s friends. How about an edge between a node B in X
and a node D outside X? If B and D were friends, then the triangle on nodes A, B, and D
would have exactly two + labels — again, a violation of weak structural balance. So B and
D must be enemies.
Since properties (i) and (ii) hold, we can remove the set X — consisting of A and all his
friends — from the graph and declare it to be the first group. We now have a smaller complete
graph that is still weakly balanced; we find a second group in this graph, and proceed to
remove groups in this way until all the nodes have been assigned to a group. Since each
group consists of mutual friends (by property (i)), and each group has only negative relations
with everyone outside the group (by property (ii)), this proves the characterization.
It is interesting to reflect on this proof in relation to the proof of the Balance Theorem —
in particular, the contrast reflected by the small di↵erences between Figures 5.4 and 5.7. In
proving the Balance Theorem, we had to reason about the sign of the edge between D and
E, to show that the enemies of the set X themselves formed a set Y of mutual friends. In
characterizing weakly balanced complete graphs, on the other hand, we made no attempt to
reason about the D-E edge, because weak balance imposes no condition on it: two enemies
of A can be either friends or enemies. As a result, the set of enemies in Figure 5.7 might not
be a set of mutual friends when only weak balance holds; it might consist of multiple groups
of mutual friends, and as we extract these groups one by one over the course of the proof,
we recover the multi-faction structure illustrated schematically in Figure 5.6.
5.5 Advanced Material: Generalizing the Definition of Structural Balance
In this section, we consider more general ways of formulating the idea of structural balance
in a network. In particular, our definition of structural balance thus far is fairly demanding
in two respects:
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE133
1
2 3
6
11
1312
9
4
8
5
10
7
+ +
+
-
-
-
+
+
-
-
-
+
+
-
+
-
+
-
14
15
-
- -
Figure 5.8: In graphs that are not complete, we can still define notions of structural balance when the edges that are present have positive or negative signs indicating friend or enemy relations.
1. It applies only to complete graphs: we require that each person know and have an
opinion (positive or negative) on everyone else. What if only some pairs of people
know each other?
2. The Balance Theorem, showing that structural balance implies a global division of the
world into two factions [97, 204], only applies to the case in which every triangle is
balanced. Can we relax this to say that if most triangles are balanced, then the world
can be approximately divided into two factions?
In the two parts of this section, we discuss a pair of results that address these questions. The
first is based on a graph-theoretic analysis involving the notion of breadth-first search from
Chapter 2, while the second is typical of a style of proof known as a “counting argument.”
Throughout this section, we will focus on the original definition of structural balance from
Sections 5.1 and 5.2, rather than the weaker version from Section 5.4.
134 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
1
25
4 3
-
+
+
-
+
(a) A graph with signed edges.
1
25
4 3
-
+
+
-
+
-
-
-
-
+
(b) Filling in the missing edges to achieve balance.
1
25
4 3
-
+
+
-
+
X
Y
(c) Dividing the graph into two sets.
Figure 5.9: There are two equivalent ways to define structural balance for general (non-complete) graphs. One definition asks whether it is possible to fill in the remaining edges so as to produce a signed complete graph that is balanced. The other definition asks whether it is possible to divide the nodes into two sets X and Y so that all edges inside X and inside Y are positive, and all edges between X and Y are negative.
A. Structural Balance in Arbitrary (Non-Complete) Networks
First, let’s consider the case of a social network that is not necessarily complete — that is,
there are only edges between certain pairs of nodes, but each of these edges is still labeled
with + or �. So now there are three possible relations between each pair of nodes: a positive edge, indicating friendship; a negative edge, indicating enmity; or the absence of an edge,
indicating that the two endpoints do not know each other. Figure 5.8 depicts an example of
such a signed network.
Defining Balance for General Networks. Drawing on what we’ve learned from the
special case of complete graphs, what would be a good definition of balance for this more
general kind of structure? The Balance Theorem suggests that we can view structural balance
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE135
in either of two equivalent ways: a local view, as a condition on each triangle of the network;
or a global view, as a requirement that the world be divided into two mutually opposed sets
of friends. Each of these suggests a way of defining structure balance for general signed
graphs.
1. One option would be to treat balance for non-complete networks as a problem of filling
in “missing values.” Suppose we imagine, as a thought experiment, that all people
in the group in fact do know and have an opinion on each other; the graph under
consideration is not complete only because we have failed to observe the relations
between some of the pairs. We could then say that the graph is balanced if it possible
to fill in all the missing labeled edges in such a way that the resulting signed complete
graph is balanced. In other words, a (non-complete) graph is balanced if it can be
“completed” by adding edges to form a signed complete graph that is balanced.
For example, Figure 5.9(a) shows a graph with signed edges, and Figure 5.9(b) shows
how the remaining edges can be “filled in” to produce a balanced complete graph:
we declare the missing edge between nodes 3 and 5 to be positive, and the remaining
missing edges to be negative, and one can check that this causes all triangles to be
balanced.
2. Alternately, we could take a more global view, viewing structural balance as implying
a division of the network into two mutually opposed sets of friends. With this in mind,
we could define a signed graph to be balanced if it is possible to divide the nodes into
two sets X and Y , such that any edge with both ends inside X or both ends inside Y
is positive, and any edge with one end in X and the other in Y is negative. That is,
people in X are all mutual friends to the extent that they know each other; the same
is true for people in Y ; and people in X are all enemies of people in Y to the extent
that they know each other.
Continuing the example from Figure 5.9(a), in Figure 5.9(c) we show how to divide
this graph into two sets with the desired properties.
This example hints at a principle that is true in general: these two ways of defining balance
are equivalent. An arbitrary signed graph is balanced under the first definition if and only
if it is balanced under the second definition.
This is actually not hard to see. If a signed graph is balanced under the first definition,
then after filling in all the missing edges appropriately, we have a signed complete graph to
which we can apply the Balance Theorem. This gives us a division of the network into two
sets X and Y that satisfies the properties of the second definition. On the other hand, if
a signed graph is balanced under the second definition, then after finding a division of the
nodes into sets X and Y , we can fill in positive edges inside X and inside Y , and fill in
136 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
1
25
4 3
-
-
+
-
+
X
X
Y Y
X
label as
X or Y
Figure 5.10: If a signed graph contains a cycle with an odd number of negative edges, then it is not balanced. Indeed, if we pick one of the nodes and try to place it in X, then following the set of friend/enemy relations around the cycle will produce a conflict by the time we get to the starting node.
negative edges between X and Y , and then we can check that all triangles will be balanced.
So this gives a “filling-in” that satisfies the first definition.
The fact that the two definitions are equivalent suggests a certain “naturalness” to the
definition, since there are fundamentally di↵erent ways to arrive at it. It also lets us use
either definition, depending on which is more convenient in a given situation. As the example
in Figure 5.9 suggests, the second definition is generally more useful to work with — it tends
to be much easier to think about dividing the nodes into two sets than to reason about filling
in edges and checking triangles.
Characterizing Balance for General Networks. Conceptually, however, there is some-
thing not fully satisfying about either definition: the definitions themselves do not provide
much insight into how to easily check that a graph is balanced. There are, after all, lots of
ways to choose signs for the missing edges, or to choose ways of splitting the nodes into sets
X and Y . And if a graph is not balanced, so that there is no way to do these things suc-
cessfully, what could you show someone to convince them of this fact? To take just a small
example to suggest some of the di�culties, it may not be obvious from a quick inspection
of Figure 5.8 that this is not a balanced graph — or that if we change the edge connecting
nodes 2 and 4 to be positive instead of negative, it becomes a balanced graph.
In fact, however, all these problems can be remedied if we explore the consequences of
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE137
the definitions a little further. What we will show is a simple characterization of balance in
general signed graphs, also due to Harary [97, 204]; and the proof of this characterization
also provides an easy method for checking whether a graph is balanced.
The characterization is based on considering the following question: what prevents a
graph from being balanced? Figure 5.10 shows a graph that is not balanced (obtained from
Figure 5.9(a) and changing the sign of the edge from node 4 to node 5). It also illustrates a
reason why it’s not balanced, as follows. If we start at node 1 and try to divide the nodes
into sets X and Y , then our choices are forced at every step. Suppose we initially decide
that node 1 should belong to X. (For the first node, it doesn’t matter, by symmetry.) Then
since node 2 is friends with node 1, it too must belong to X. Node 3, an enemy of 2, must
therefore belong to Y ; hence node 4, a friend of 3, must belong to Y as well; and node
5, an enemy of 4, must belong to X. The problem is that if we continue this reasoning
one step further, then node 1, an enemy of 5, should belong to Y — but we had already
decided at the outset to put it into X. We had no freedom of choice during this process —
so this shows that there is no way to divide the nodes in sets X and Y so as to satisfy the
mutual-friend/mutual-enemy conditions of structural balance, and hence the signed graph
in Figure 5.10 is not balanced.
The reasoning in the previous paragraph sounds elaborate, but in fact it followed a simple
principle: we were walking around a cycle, and every time we crossed a negative edge, we had
to change the set into which we were putting nodes. The di�culty was that getting back
around to node 1 required crossing an odd number of negative edges, and so our original
decision to put node 1 into X clashed with the eventual conclusion that node 1 ought to be
in Y .
This principle applies in general: if the graph contains a cycle with an odd number of
negative edges, then this implies the graph is not balanced. Indeed, if we start at any node A
in the cycle and place it in one of the two sets, and then we walk around the cycle placing the
other nodes where they must go, the identity of the set where we’re placing nodes switches
an odd number of times as we go around the cycle. Thus we end up with the “wrong set”
by the time we make it back to A.
A cycle with an odd number of negative edges is thus a very simple-to-understand reason
why a graph is not balanced: you can show someone such a cycle and immediately convince
them that the graph is not balanced. For example, the cycle back in Figure 5.8 consisting of
nodes, 2, 3, 6, 11, 13, 12, 9, 4 contains five negative edges, thus supplying a succinct reason
why this graph is not balanced. But are there other, more complex reasons why a graph is
not balanced?
In fact, though it may seem initially surprising, cycles with an odd number of negative
edges are the only obstacles to balance. This is the crux of the following claim [97, 204].
Claim: A signed graph is balanced if and only if it contains no cycle with an odd
138 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
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2 3
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11
Figure 5.11: To determine if a signed graph is balanced, the first step is to consider only the positive edges, find the connected components using just these edges, and declare each of these components to be a supernode. In any balanced division of the graph into X and Y , all nodes in the same supernode will have to go into the same set.
number of negative edges.
We now show how to prove this claim; this is done by designing a method that analyzes the
graph and either finds a division into the desired sets X and Y , or else finds a cycle with an
odd number of negative edges.
Proving the Characterization: Identifying Supernodes. Let’s recall what we’re try-
ing to do: find a division of the nodes into sets X and Y so that all edges inside X and Y are
positive, and all edges crossing between X and Y are negative. We will call a partitioning
into sets X and Y with these properties a balanced division. We now describe a procedure
that searches for a balanced division of the nodes into sets X and Y ; either it succeeds, or
it stops with a cycle containing an odd number of negative edges. Since these are the only
two possible outcomes for the procedure, this will give a proof of the claim.
The procedure works in two main steps: the first step is to convert the graph to a reduced
one in which there are only negative edges, and the second step is to solve the problem on
this reduced graph. The first step works as follows. Notice that whenever two nodes are
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE139
A
B
+
+
++
+
+
-
Figure 5.12: Suppose a negative edge connects two nodes A and B that belong to the same supernode. Since there is also a path consisting entirely of positive edges that connects A and B through the inside of the supernode, putting this negative edge together with the all-positive path produces a cycle with an odd number of negative edges.
connected by a positive edge, they must belong to the same one of the sets X or Y in a
balanced division. So we begin by considering what the connected components of the graph
would be if we were to only consider positive edges. These components can be viewed as a
set of contiguous “blobs” in the overall graph, as shown in Figure 5.11. We will refer to each
of these blobs as a supernode: each supernode is connected internally via positive edges, and
the only edges going between two di↵erent supernodes are negative. (If there were a positive
edge linking two di↵erent supernodes, we should have combined them together into a single
supernode.)
Now, if any supernode contains a negative edge between some pair of nodes A and B,
then we already have a cycle with an odd number of negative edges, as illustrated in the
example of Figure 5.12. Consider the path of positive edges that connects A and B inside
the supernode, and then close o↵ a cycle by including the negative edge joining A and B.
This cycle has only a single negative edge, linking A and B, and so it shows that the graph
is not balanced.
If there are no negative edges inside any of the supernodes, then there is no “internal”
problem with declaring each supernode to belong entirely to one of X or Y . So the problem
is now how to assign a single label “X” or “Y ” to each supernode, in such a way that
these choices are all consistent with each other Since the decision-making is now at the
level of supernodes, we create a new version of the problem in which there is a node for each
140 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
Figure 5.13: The second step in determining whether a signed graph is balanced is to look for a labeling of the supernodes so that adjacent supernodes (which necessarily contain mutual enemies) get opposite labels. For this purpose, we can ignore the original nodes of the graph and consider a reduced graph whose nodes are the supernodes of the original graph.
supernode, and an edge joining two supernodes if there is an edge in the original that connects
the two supernodes. Figure 5.13 shows how this works for the example of Figure 5.11: we
essentially forget about the individual nodes inside the supernodes, and build a new graph
at the level of the large “blobs.” Of course, having done so, we can draw the graph in a less
blob-like way, as in Figure 5.14.
We now enter the second step of the procedure, using this reduced graph whose nodes are
the supernodes of the original graph.
Proving the Characterization: Breadth-First Search of the Reduced Graph. Re-
call that only negative edges go between supernodes (since a positive edge between two su-
pernodes would have merged them together into a single one). As a result, our reduced graph
has only negative edges. The remainder of the procedure will produce one of two possible
outcomes.
1. The first possible outcome is to label each node in the reduced graph as either X or
Y , in such a way that every edge has endpoints with opposite labels. From this we
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE141
A
EB
C D
-
-
-
-
-
F
G
-
-
-
Figure 5.14: A more standard drawing of the reduced graph from the previous figure. A negative cycle is visually apparent in this drawing.
can create a balanced division of the original graph, by labeling each node the way its
supernode is labeled in the reduced graph.
2. The second possible outcome will be to find a cycle in the reduced graph that has an
odd number of edges. We can then convert this to a (potentially longer) cycle in the
original graph with an odd number of negative edges: the cycle in the reduced graph
connects supernodes, and corresponds to a set of negative edges in the original graph.
We can simply “stitch together” these negative edges using paths consisting entirely
of positive edges that go through the insides of the supernodes. This will be a path
containing an odd number of negative edges in the original graph.
For example, the odd-length cycle in Figure 5.14 through nodes A through E can be
realized in the original graph as the darkened negative edges shown in Figure 5.15.
This can then be turned into a cycle in the original graph by including paths through
the supernodes – in this example using the additional nods 3 and 12.
In fact, this version of the problem when there are only negative edges is known in graph
theory as the problem of determining whether a graph is bipartite: whether its nodes can be
divided into two groups (in this case X and Y ) so that each edge goes from one group to the
other. We saw bipartite graphs when we considered a�liation networks in Chapter 4, but
there the fact that the graphs were bipartite was apparent from the ready-made division of
the nodes into people and social foci. Here, on the other hand, we are handed a graph “in
142 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
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Figure 5.15: Having found a negative cycle through the supernodes, we can then turn this into a cycle in the original graph by filling in paths of positive edges through the inside of the supernodes. The resulting cycle has an odd number of negative edges.
the wild,” with no pre-specified division into two sets, and we want to know if it is possible
to identify such a division. We now show a way to do this using the idea of breadth-first
search from Chapter 2, resulting either in the division we seek, or in a cycle of odd length.
We simply perform breadth-first search starting from any “root” node in the graph,
producing a set of layers at increasing distances from this root. Figure 5.16 shows how this
is done for the reduced graph in Figure 5.14, with node G as the starting root node. Now,
because edges cannot jump over successive layers in breadth-first search, each edge either
connects two nodes in adjacent layers or it connects two nodes in the same layer. If all
edges are of the first type, then we can find the desired division of nodes into sets X and
Y : we simply declare all nodes in even-numbered layers to belong to X, and all nodes in
odd-numbered layers to belong to Y . Since edges only go between adjacent layers, all edges
have one end in X and the other end in Y , as desired.
Otherwise, there is an edge connecting two nodes that belong to the same layer. Let’s
call them A and B (as they are in Figure 5.16). For each of these two nodes, there is a path
that descends layer-by-layer from the root to it. Consider the last node that is common to
these two paths — let’s call this node D (as it is in Figure 5.16). The D-A path and the
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE143
G
D F
C E
B A
An odd cycle is
formed from two
equal-length paths
leading to an edge
inside a single layer.
Figure 5.16: When we perform a breadth-first search of the reduced graph, there is either an edge connecting two nodes in the same layer or there isn’t. If there isn’t, then we can produce the desired division into X and Y by putting alternate layers in di↵erent sets. If there is such an edge (such as the edge joining A and B in the figure), then we can take two paths of the same length leading to the two ends of the edge, which together with the edge itself forms an odd cycle.
D-B path have the same length k, so a cycle created from the two of these plus the A-B
edge must have length 2k + 1: an odd number. This is the odd cycle we seek.
And this completes the proof. To recap: if all edges in the reduced graph connect nodes
in adjacent layers of the breadth-first search, then we have a way to label the nodes in the
reduced graph as into X and Y , which in turn provides a balanced division of the nodes in
the original graph into X and Y . In this case, we’ve established that the graph is balanced.
Otherwise, there is an edge connecting two nodes in the same layer of the breadth-first
search, in which case we produce an odd cycle in the reduced graph as in Figure 5.16. In
this case, we can convert into this to a cycle in the original graph containing an odd number
of negative edges, as in Figure 5.15. Since these are the only two possibilities, this proves
the claim.
144 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
B. Approximately Balanced Networks
We now return to the case in which the graph is complete, so that every node has a positive
or negative relation with every other node, and we think about a di↵erent way of generalizing
the characterization of structural balance.
First let’s write down the original Balance Theorem again, with some additional format-
ting to make its logical structure clear.
Claim: If all triangles in a labeled complete graph are balanced, then either
(a) all pairs of nodes are friends, or else
(b) the nodes can be divided into two groups, X and Y , such that
(i) every pair of nodes in X like each other,
(ii) every pair of nodes in Y like each other, and
(iii) everyone in X is the enemy of everyone in Y .
The conditions of this theorem are fairly extreme, in that we require every single triangle
to be balanced. What if we only know that most triangles are balanced? It turns out that
the conditions of the theorem can be relaxed in a very natural way, allowing us to prove
statements like the following one. We phrase it so that the wording remains completely
parallel to that of the Balance Theorem.
Claim: If at least 99.9% of all triangles in a labeled complete graph are balanced,
then either
(a) there is a set consisting of at least 90% of the nodes in which at least 90%
of all pairs are friends, or else
(b) the nodes can be divided into two groups, X and Y , such that
(i) at least 90% of the pairs in X like each other,
(ii) at least 90% of the pairs in Y like each other, and
(iii) at least 90% of the pairs with one end in X and the other end in Y are
enemies.
This is a true statement, though the choice of numbers is very specific. Here is a more general
statement that includes both the Balance Theorem and the preceding claim as special cases.
Claim: Let " be any number such that 0 " < 1 8
, and define � = 3 p
". If at least
1 � " of all triangles in a labeled complete graph are balanced, then either
(a) there is a set consisting of at least 1�� of the nodes in which at least 1�� of all pairs are friends, or else
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE145
(b) the nodes can be divided into two groups, X and Y , such that
(i) at least 1 � � of the pairs in X like each other, (ii) at least 1 � � of the pairs in Y like each other, and (iii) at least 1�� of the pairs with one end in X and the other end in Y are
enemies.
Notice that the Balance Theorem is the case in which " = 0, and the other claim above is
the case in which " = .001 (since in this latter case, � = 3 p
" = .1).
We now prove this last claim. The proof is self-contained, but it is most easily read
with some prior experience in what is sometimes called the analysis of “permutations and
combinations” — counting the number of ways to choose particular subsets of larger sets.
The proof loosely follows the style of the proof we used for the Balance Theorem: we
will define the two sets X and Y to be the friends and enemies, respectively, of a designated
node A. Things are trickier here, however, because not all choices of A will give us the
structure we need — in particular, if a node is personally involved in too many unbalanced
triangles, then splitting the graph into its friends and enemies may give a very disordered
structure. Consequently, the proof consists of two steps. We first find a “good” node that
is not involved in too many unbalanced triangles. We then show that if we divide the graph
into the friends and enemies of this good node, we have the desired properties.
Warm-Up: Counting Edges and Triangles. Before launching into the proof itself,
let’s consider some basic counting questions that will show up as ingredients in the proof.
Recall that we have a complete graph, with an (undirected) edge joining each pair of nodes.
If N is the number of nodes in the graph, how many edges are there? We can count this
quantity as follows. There are N possible ways to choose one of the two endpoints, and then
N �1 possible ways to choose a di↵erent node as the other endpoint, for a total of N (N �1) possible ways to choose the two endpoints in succession. If we write down a list of all these
possible pairs of endpoints, then an edge with endpoints A and B will appear twice on the
list: once as AB and once as BA. In general, each edge will appear twice on the list, and so
the total number of edges is N (N � 1)/2. A very similar argument lets us count the total number of triangles in the graph. Specif-
ically, there are N ways to pick the first corner, then N �1 ways to pick a di↵erent node as the second corner, and then N � 2 ways to pick a third corner di↵erent from the first two. This yields a total of N (N � 1)(N � 2) sequences of three corners. If we write down this list of N (N �1)(N �2) sequences, then a triangle with corners A, B, and C will appear six times: as ABC, ACB, BAC, BCA, CAB, and CBA. In general, each triangle will appear
six times in this list, and so the total number of triangles is
N (N � 1)(N � 2) 6
.
146 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
The First Step: Finding a “Good” Node. Now let’s move on to the first step of the
proof, which is to find a node that isn’t involved in too many unbalanced triangles.
Since we are assuming that at most an " fraction of triangles are unbalanced, and the
total number of triangles in the graph is N (N �1)(N �2)/6, it follows that the total number of unbalanced triangles is at most "N (N � 1)(N � 2)/6. Suppose we define the weight of a node to be the number of unbalanced triangles that it is a part of; thus, a node of low weight
will be precisely what we’re seeking — a node that is in relatively few unbalanced triangles.
One way to count the total weight of all nodes would be to list — for each node —
the unbalanced triangles that it belongs to, and then look at the length of all these lists
combined. In these combined lists, each triangle will appear three times — once in the
list for each of its corners — and so the total weight of all nodes is exactly three times
the number of unbalanced triangles. As a result, the total weight of all nodes is at most
3"N (N � 1)(N � 2)/6 = "N (N � 1)(N � 2)/2. There are N nodes, so the average weight of a node is at most "(N � 1)(N � 2)/2. It’s
not possible for all nodes to have weights that are strictly above the average, so there is at
least one node whose weight is equal to the average or below it. Let’s pick one such node and
call it A. This will be our “good” node: a node whose weight is at most "(N �1)(N �2)/2.2
Since (N � 1)(N � 2) < N 2, this good node is in at most "N 2/2 triangles, and because the algebra is a bit simpler with this slightly larger quantity, we will use it in the rest of the
analysis.
The Second Step: Splitting the Graph According to the Good Node. By analogy
with the proof of the Balance Theorem, we divide the graph into two sets: a set X consisting
of A and all its friends, and a set Y consisting of all the enemies of A, as illustrated in
Figure 5.17. Now, using the definition of unbalanced triangles, and the fact that node A is
not involved in too many of them, we can argue that there are relatively few negative edges
inside each of X and Y , and relatively few positive edges between them. Specifically, this
works as follows.
• Each negative edge connecting two nodes in X creates a distinct unbalanced triangle involving node A. Since there are at most "N 2/2 unbalanced triangles involving A,
there are at most "N 2/2 negative edges inside X.
• A closely analogous argument applies to Y : Each negative edge connecting two nodes in Y creates a distinct unbalanced triangle involving node A, and so there are at most
"N 2/2 negative edges inside Y .
2This is a very common trick in counting arguments, referred to as the pigeonhole principle: to compute the average value of a set of objects, and then argue that there must be at least one node that is equal to the average or below. (Also, of course, there must be at least at least one object that is equal to the average or above, although this observation isn’t useful for our purposes here.)
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE147
A
B
C E
D
+
+
-
-
friends of A enemies of Aa good node A
mainly negative
mainly
positive
mainly
positive
Figure 5.17: The characterization of approximately balanced complete graphs follows from an analysis similar to the proof of the original Balance Theorem. However, we have to be more careful in dividing the graph by first finding a “good” node that isn’t involved in too many unbalanced triangles.
• And finally, an analogous argument applies to edges with one end in X and the other end in Y . Each such edge that is positive creates a distinct unbalanced triangle in-
volving A, and so there are at most "N 2/2 positive edges with one end in X and the
other end in Y .
We now consider several possible cases, depending on the sizes of the sets X and Y . Essen-
tially, if either of X or Y consists of almost the entire graph, then we show that alternative
(a) in the claim holds. Otherwise, if each of X and Y contain a non-negligible number of
nodes, then we show that alternative (b) in the claim holds. We’re also going to assume, to
make the calculations simpler, that N is even and that the quantity �N is a whole number,
although this is not in fact necessary for the proof.
To start, let x be the number of nodes in X and y be the number of nodes in Y . Suppose
first that x � (1��)N . Since " < 1 8
and � = 3 p
", it follows that � < 1 2
, and so x > 1 2
N . Now,
recall our earlier counting argument that gave a formula for the number of edges in a complete
graph, in terms of its number of nodes. In this case, X has x nodes, so it has x(x � 1)/2 edges. Since x > 1
2
N , this number of edges is at least ( 1 2
N + 1)( 1 2
N )/2 � ( 1 2
N )2/2 = N 2/8.
There are at most "N 2/2 negative edges inside X, and so the fraction of negative edges
inside X is at most "N 2/2
N 2/8 = 4" = 4�3 < �,
148 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
where we use the facts that " = �3 and � < 1 2
. We thus conclude that if X contains at least
(1 � �)N nodes, then it is a set containing at least a 1 � � fraction of the nodes in which at least 1 � � of all pairs are friends, satisfying part (a) in the conclusion of the claim.
The same argument can be applied if Y contains at least (1 � �)N nodes. Thus we are left with the case in which both X and Y contain strictly fewer than (1 � �)N , and in this case we will show that part (b) in the conclusion of the claim holds. First, of all the edges
with one end in X and the other in Y , what fraction are positive? The total number of edges
with one end in X and the other end in Y can be counted as follows: there are x ways to
choose the end in X, and then y ways to choose the end in Y , for a total of xy such edges.
Now, since each of x and y are less than (1 � �)N , and they add up to N , this product xy is at least (�N )(1 � �)N = �(1 � �)N 2 � �N 2/2, where the last inequality follows from the fact that � < 1
2
. There are at most "N 2/2 positive edges with one end in X and the other
in Y , so as a fraction of the total this is at most
"N 2/2
�N 2/2 =
"
� = �2 < �.
Finally, what fraction of edges inside each of X and Y are negative? Let’s calculate
this for X; the argument for Y is exactly the same. There are x(x � 1)/2 edges inside X in total, and since we’re in the case where x > �N , this total number of edges is at least
(�N + 1)(�N )/2 � (�N )2/2 = �2N 2/2. There are at most "N 2/2 negative edges inside X, so as a fraction of the total this is at most
"N 2/2
�2N 2/2 =
"
�2 = �.
Thus, the division of nodes into sets X and Y satisfies all the requirements in conclusion (b)
of the claim, and so the proof is complete.
As a final comment on the claim and its proof, one might feel that the di↵erence between
1 � " in the assumption of the claim and 1 � 3 p
" is a bit excessive: as we saw above, when
" = .001, this means we need to assume that 99.9% of all triangles are balanced in order to
get sets with a 90% density of edges having the correct sign. But in fact, it is possible to
construct examples showing that this relationship between " and � is in fact essentially the
best one can do. In short, the claim provides the kind of approximate version of the Balance
Theorem that we wanted at a qualitative level, but we need to assume a fairly small fraction
of unbalanced triangles in order to be able to start drawing strong conclusions.
5.6 Exercises
1. Suppose that a team of anthropologists is studying a set of three small villages that
neighbor one another. Each village has 30 people, consisting of 2-3 extended families.
5.6. EXERCISES 149
Everyone in each village knows all the people in their own village, as well as the people
in the other villages.
When the anthropologists build the social network on the people in all three villages
taken together, they find that each person is friends with all the other people in their
own village, and enemies with everyone in the two other villages. This gives them a
network on 90 people (i.e., 30 in each village), with positive and negative signs on its
edges.
According to the definitions in this chapter, is this network on 90 people balanced?
Give a brief explanation for your answer.
2. Consider the network shown in Figure 5.18: there is an edge between each pair of
nodes, with five of the edges corresponding to positive relationships, and the other five
of the edges corresponding to negative relationships.
C
A
ED
B
+ +
+
+
+
-
-
-
--
Figure 5.18: A network with five positive edges and five negative edges.
Each edge in this network participates in three triangles: one formed by each of the
additional nodes who is not already an endpoint of the edge. (For example, the A-B
edge participates in a triangle on A, B, and C, a triangle on A, B, and D, and a
triangle on A, B, and E. We can list triangles for the other edges in a similar way.)
For each edge, how many of the triangles it participates in are balanced, and how many
are unbalanced. (Notice that because of the symmetry of the network, the answer will
be the same for each positive edge, and also for each negative edge; so it is enough to
consider this for one of the positive edges and one of the negative edges.)
3. When we think about structural balance, we can ask what happens when a new node
150 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
tries to join a network in which there is existing friendship and hostility. In Fig-
ures 5.19–5.22, each pair of nodes is either friendly or hostile, as indicated by the + or
� label on each edge.
A
B C
+ +
+
Figure 5.19: A 3-node social network in which all pairs of nodes know each other, and all pairs of nodes are friendly toward each other.
A
B C
+ +
+
D
++ +
(a) D joins the network by becom- ing friends with all nodes.
A
B C
+ +
+
D
-- -
(b) D joins the network by becom- ing enemies with all nodes.
Figure 5.20: There are two distinct ways in which node D can join the social network from Figure 5.19 without becoming involved in any unbalanced triangles.
First, consider the 3-node social network in Figure 5.19, in which all pairs of nodes
know each other, and all pairs of nodes are friendly toward each other. Now, a fourth
node D wants to join this network, and establish either positive or negative relations
with each existing node A, B, and C. It wants to do this in such a way that it
doesn’t become involved in any unbalanced triangles. (I.e. so that after adding D and
the labeled edges from D, there are no unbalanced triangles that contain D.) Is this
possible?
In fact, in this example, there are two ways for D to accomplish this, as indicated in
Figure 5.20. First, D can become friends with all existing nodes; in this way, all the
5.6. EXERCISES 151
triangles containing it have three positive edges, and so are balanced. Alternately, it
can become enemies with all existing nodes; in this way, each triangle containing it has
exactly one positive edge, and again these triangles would be balanced.
So for this network, it was possible for D to join without becoming involved in any
unbalanced triangles. However, the same is not necessarily possible for other networks.
We now consider this kind of question for some other networks.
A
B C
- -
-
Figure 5.21: All three nodes are mutual enemies.
(a) Consider the 3-node social network in Figure 5.21, in which all pairs of nodes
know each other, and each pair is either friendly or hostile as indicated by the
+ or � label on each edge. A fourth node D wants to join this network, and establish either positive or negative relations with each existing node A, B, and
C. Can node D do this in such a way that it doesn’t become involved in any
unbalanced triangles?
• If there is a way for D to do this, say how many di↵erent such ways there are, and give an explanation. (That is, how many di↵erent possible labelings
of the edges out of D have the property that all triangles containing D are
balanced?)
• If there is no such way for D to do this, give an explanation why not. (In this and the subsequent questions, it possible to work out an answer by rea-
soning about the new node’s options without having to check all possibilities.)
(b) Same question, but for a di↵erent network. Consider the 3-node social network in
Figure 5.22, in which all pairs of nodes know each other, and each pair is either
friendly or hostile as indicated by the + or � label on each edge. A fourth node D wants to join this network, and establish either positive or negative relations
with each existing node A, B, and C. Can node D do this in such a way that it
doesn’t become involved in any unbalanced triangles?
• If there is a way for D to do this, say how many di↵erent such ways there are, and give an explanation. (That is, how many di↵erent possible labelings
152 CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
A
B C
+ +
-
Figure 5.22: Node A is friends with nodes B and C, who are enemies with each other.
of the edges out of D have the property that all triangles containing D are
balanced?)
• If there is no such way for D to do this, give an explanation why not.
(c) Using what you’ve worked out in Questions 2 and 3, consider the following ques-
tion. Take any labeled complete graph — on any number of nodes — that is not
balanced; i.e. it contains at least one unbalanced triangle. (Recall that a labeled
complete graph is a graph in which there is an edge between each pair of nodes,
and each edge is labeled with either + or �.) A new node X wants to join this network, by attaching to each node using a positive or negative edge. When, if
ever, is it possible for X to do this in such a way that it does not become involved
in any unbalanced triangles? Give an explanation for your answer. (Hint: Think
about any unbalanced triangle in the network, and how X must attach to the nodes
in it.)
4. Together with some anthropologists, you’re studying a sparsely populated region of a
rain forest, where 50 farmers live along a 50-mile-long stretch of river. Each farmer
lives on a tract of land that occupies a 1-mile stretch of the river bank, so their tracts
exactly divide up the 50 miles of river bank that they collectively cover. (The numbers
are chosen to be simple and to make the story easy to describe.)
The farmers all know each other, and after interviewing them, you’ve discovered that
each farmer is friends with all the other farmers that live at most 20 miles from him or
her, and is enemies with all the farmers that live more than 20 miles from him or her.
You build the signed complete graph corresponding to this social network, and you
wonder whether it satisfies the Structural Balance property. This is the question: is
the network structurally balanced or not? Provide an explanation for your answer.
Part II
Game Theory
153
Chapter 6
Games
In the opening chapter of the book, we emphasized that the “connectedness” of a complex
social, natural, or technological system really means two things: first, an underlying structure
of interconnecting links; and second, an interdependence in the behaviors of the individuals
who inhabit the system, so that the outcome for any one depends at least implicitly on the
combined behaviors of all. The first issue – network structure – was addressed in the first part
of the book using graph theory. In this second part of the book, we study interconnectedness
at the level of behavior, developing basic models for this in the language of game theory.
Game theory is designed to address situations in which the outcome of a person’s decision
depends not just on how they choose among several options, but also on the choices made
by the people they are interacting with. Game-theoretic ideas arise in many contexts. Some
contexts are literally games; for example, choosing how to target a soccer penalty kick and
choosing how to defend against it can be modeled using game theory. Other settings are
not usually called games, but can be analyzed with the same tools. Examples include the
pricing of a new product when other firms have similar new products; deciding how to bid in
an auction; choosing a route on the Internet or through a transportation network; deciding
whether to adopt an aggressive or a passive stance in international relations; or choosing
whether to use performance-enhancing drugs in a professional sport. In these examples,
each decision-maker’s outcome depends on the decisions made by others. This introduces a
strategic element that game theory is designed to analyze.
As we will see later in Chapter 7, game-theoretic ideas are also relevant to settings where
no one is overtly making decisions. Evolutionary biology provides perhaps the most striking
example. A basic principle is that mutations are more likely to succeed in a population
when they improve the fitness of the organisms that carry the mutation. But often, this
fitness cannot be assessed in isolation; rather, it depends on what all the other (non-mutant)
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
155
156 CHAPTER 6. GAMES
organisms are doing, and how the mutant’s behavior interacts with the non-mutants’ be-
haviors. In such situations, reasoning about the success or failure of the mutation involves
game-theoretic definitions, and in fact very closely resembles the process of reasoning about
decisions that intelligent actors make. Similar kinds of reasoning have been applied to the
success or failure of new cultural practices and conventions — it depends on the existing
patterns of behavior into which they are introduced. This indicates that the ideas of game
theory are broader than just a model of how people reason about their interactions with oth-
ers; game theory more generally addresses the question of which behaviors tend to sustain
themselves when carried out in a larger population.
Game-theoretic ideas will appear in many places throughout the book. Chapters 8 and
9 describe two initial and fundamental applications: to network tra�c, where travel time
depends on the routing decisions of others; and to auctions, where the success of a bidder
depends on how the other bidders behave. There will be many further examples later in the
book, including the ways in which prices are set in markets and the ways in which people
choose to adopt new ideas in situations where adoption decisions are a↵ected by what others
are doing.
As a first step, then, we begin with a discussion of the basic ideas behind game theory.
For now, this will involve descriptions of situations in which people interact with one an-
other, initially without an accompanying graph structure. Once these ideas are in place, we
will bring graphs back into the picture in subsequent chapters, and begin to consider how
structure and behavior can be studied simultaneously.
6.1 What is a Game?
Game theory is concerned with situations in which decision-makers interact with one another,
and in which the happiness of each participant with the outcome depends not just on his
or her own decisions but on the decisions made by everyone. To help make the definitions
concrete, it’s useful to start with an example.
A First Example. Suppose that you’re a college student, and you have two large pieces of
work due the next day: an exam, and a presentation. You need to decide whether to study
for the exam, or to prepare for the presentation. For simplicity, and to make the example as
clean as possible, we’ll impose a few assumptions. First, we’ll assume you can either study
for the exam or prepare for the presentation, but not both. Second, we’ll assume you have an
accurate estimate of the expected grade you’ll get under the outcomes of di↵erent decisions.
The outcome of the exam is easy to predict: if you study, then your expected grade is a
92, while if you don’t study, then your expected grade is an 80.
The presentation is a bit more complicated to think about. For the presentation, you’re
6.1. WHAT IS A GAME? 157
doing it jointly with a partner. If both you and your partner prepare for the presentation,
then the presentation will go extremely well, and your expected joint grade is a 100. If just
one of you prepares (and the other doesn’t), you’ll get an expected joint grade of 92; and if
neither of you prepares, your expected joint grade is 84.
The challenge in reasoning about this is that your partner also has the same exam the
next day, and we’ll assume that he has the same expected outcome for it: 92 if he studies,
and 80 if he doesn’t. He also has to choose between studying for the exam and preparing for
the presentation. We’ll assume that neither of you is able to contact the other, so you can’t
jointly discuss what to do; each of you needs to make a decision independently, knowing that
the other will also be making a decision.
Both of you are interested in maximizing the average grade you get, and we can use the
discussion above to work out how this average grade is determined by the way the two of
you invest your e↵orts:
• If both of you prepare for the presentation, you’ll both get 100 on the presentation and 80 on the exam, for an average of 90.
• If both of you study for the exam, you’ll both get 92 on the exam and 84 on the presentation, for an average of 88.
• If one of you studies for the exam while the other prepares for the presentation, the result is as follows.
– The one who prepares for the presentation gets a 92 on the presentation but only
an 80 on the exam, for an average of 86.
– On the other hand, the one who studies for the exam still gets a 92 on the
presentation — since it’s a joint grade, this person benefits from the fact that one
of the two of you prepared for it. This person also get a 92 on the exam, through
studying, and so gets an average of 92.
There’s a simple tabular way to summarize all these outcomes, as follows. We represent
your two choices — to prepare for the presentation, or to study for the exam — as the rows
of a 2 ⇥ 2 table. We represent your partner’s two choices as the columns. So each box in this table represents a decision by each of you. In each box, we record the average grade
you each receive: first yours, then your partner’s. Writing all this down, we have the table
shown in Figure 6.1.
This describes the set-up of the situation; now you need to figure out what to do: prepare
for the presentation, or study for the exam? Clearly, your average grade depends not just on
which of these two options you choose, but also on what your partner decides. Therefore, as
part of your decision, you have to reason about what your partner is likely to do. Thinking
158 CHAPTER 6. GAMES
You
Your Partner Presentation Exam
Presentation 90, 90 86, 92 Exam 92, 86 88, 88
Figure 6.1: Exam or Presentation?
about the strategic consequences of your own actions, where you need to consider the e↵ect
of decisions by others, is precisely the kind of reasoning that game theory is designed to
facilitate. So before moving on to the actual outcome of this exam-or-presentation scenario,
it is useful to introduce some of the basic definitions of game theory, and then continue the
discussion in this language.
Basic Ingredients of a Game. The situation we’ve just described is an example of a
game. For our purposes, a game is any situation with the following three aspects.
(i) There is a set of participants, whom we call the players. In our example, you and your
partner are the two players.
(ii) Each player has a set of options for how to behave; we will refer to these as the player’s
possible strategies. In the example, you and your partner each have two possible
strategies: to prepare for the presentation, or to study for the exam.
(iii) For each choice of strategies, each player receives a payo↵ that can depend on the
strategies selected by everyone. The payo↵s will generally be numbers, with each
player preferring larger payo↵s to smaller payo↵s. In our current example, the payo↵
to each player is the average grade he or she gets on the exam and the presentation.
We will generally write the payo↵s in a payo↵ matrix as in Figure 6.1.
Our interest is in reasoning about how players will behave in a given game. For now we
focus on games with only two players, but the ideas apply equally well to games with any
number of players. Also, we will focus on simple, one-shot games: games in which the
players simultaneously and independently choose their actions, and they do so only once. In
Section 6.10 at the end of this chapter, we discuss how to reinterpret the theory to deal with
dynamic games, in which actions can be played sequentially over time.
6.2 Reasoning about Behavior in a Game
Once we write down the description of a game, consisting of the players, the strategies, and
the payo↵s, we can ask how the players are likely to behave — that is, how they will go
about selecting strategies.
6.2. REASONING ABOUT BEHAVIOR IN A GAME 159
Underlying Assumptions. In order to make this question tractable, we will make a
few assumptions. First, we assume everything that a player cares about is summarized
in the player’s payo↵s. In the Exam-or-Presentation Game described in Section 6.1, this
means that the two players are solely concerned with maximizing their own average grade.
However, nothing in the framework of game theory requires that players care only about
personal rewards. For example, a player who is altruistic may care about both his or her
own benefits, and the other player’s benefit. If so, then the payo↵s should reflect this; once
the payo↵s have been defined, they should constitute a complete description of each player’s
happiness with each of the possible outcomes of the game.
We also assume that each player knows everything about the structure of the game. To
begin with, this means that each player knows his or her own list of possible strategies. It
seems reasonable in many settings to assume that each player also knows who the other
player is (in a two-player game), the strategies available to this other player, and what his
or her payo↵ will be for any choice of strategies. In the Exam-or-Presentation Game, this
corresponds to the assumption that you realize you and your partner are each faced with the
choice of studying for the exam or preparing for the presentation, and you have an accurate
estimate of the expected outcome under di↵erent courses of action. There is considerable
research on how to analyze games in which the players have much less knowledge about the
underlying structure, and in fact John Harsanyi shared the 1994 Nobel Prize in Economics
for his work on games with incomplete information [208].
Finally, we suppose that each individual chooses a strategy to maximize her own payo↵,
given her beliefs about the strategy used by the other player. This model of individual
behavior, which is usually called rationality, actually combines two ideas. The first idea is
that each player wants to maximize her own payo↵. Since the individual’s payo↵ is defined to
be whatever the individual cares about, this hypothesis seems reasonable. The second idea
is that each player actually succeeds in selecting the optimal strategy. In simple settings,
and for games played by experienced players, this too seems reasonable. In complex games,
or for games played by inexperienced players, it is surely less reasonable. It is interesting
to consider players who make mistakes and learn from the play of the game. There is an
extensive literature which analyzes problems of this sort [175], but we will not consider these
issues here.
Reasoning about Behavior in the Exam-or-Presentation Game. Let’s go back to
the Exam-or-Presentation Game and ask how we should expect you and your partner — the
two players in the game — to behave.
We first focus on this from your point of view. (The reasoning for your partner will be
symmetric, since the game looks the same from his point of view.) It would be easier to
decide what to do if you could predict what your partner would do, but to begin with, let’s
160 CHAPTER 6. GAMES
consider what you should do for each possible choice of strategy by your partner.
• First, if you knew your partner was going to study for the exam, then you would get a payo↵ of 88 by also studying, and a payo↵ of only 86 by preparing for the presentation.
So in this case, you should study for the exam.
• On the other hand, if you knew that your partner was going to prepare for the pre- sentation, then you’d get a payo↵ of 90 by also preparing for the presentation, but a
payo↵ of 92 by studying for the exam. So in this case too, you should study for the
exam.
This approach of considering each of your partner’s options separately turns out to be a very
useful way of analyzing the present situation: it reveals that no matter what your partner
does, you should study for the exam.
When a player has a strategy that is strictly better than all other options regardless of
what the other player does, we will refer to it as a strictly dominant strategy. When a player
has a strictly dominant strategy, we should expect that they will definitely play it. In the
Exam-or-Presentation Game, studying for the exam is also a strictly dominant strategy for
your partner (by the same reasoning), and so we should expect that the outcome will be for
both of you to study, each getting an average grade of 88.
So this game has a very clean analysis, and it’s easy to see how to end up with a prediction
for the outcome. Despite this, there’s something striking about the conclusion. If you and
your partner could somehow agree that you would both prepare for the presentation, you
would each get an average grade of 90 — in other words, you would each be better o↵.
But despite the fact that you both understand this, this payo↵ of 90 cannot be achieved by
rational play. The reasoning above makes it clear why not: even if you were to personally
commit to preparing for the presentation — hoping to achieve the outcome where you both
get 90 — and even if your partner knew you were doing this, your partner would still have
an incentive to study for the exam so as to achieve a still-higher payo↵ of 92 for himself.
This result depends on our assumption that the payo↵s truly reflect everything each
player values in the outcome — in this case, that you and your partner only care about
maximizing your own average grade. If, for example, you cared about the grade that your
partner received as well, then the payo↵s in this game would look di↵erent, and the outcome
could be di↵erent. Similarly, if you cared about the fact that your partner will be angry at
you for not preparing for the joint presentation, then this too should be incorporated into
the payo↵s, again potentially a↵ecting the results. But with the payo↵s as they are, we are
left with the interesting situation where there is an outcome that is better for both of you —
an average grade of 90 each — and yet it cannot be achieved by rational play of the game.
6.2. REASONING ABOUT BEHAVIOR IN A GAME 161
A Related Story: The Prisoner’s Dilemma. The outcome of the Exam-or-Presentation
Game is closely related to one of the most famous examples in the development of game the-
ory, the Prisoner’s Dilemma. Here is how this example works.
Suppose that two suspects have been apprehended by the police and are being interro-
gated in separate rooms. The police strongly suspect that these two individuals are respon-
sible for a robbery, but there is not enough evidence to convict either of them of the robbery.
However, they both resisted arrest and can be charged with that lesser crime, which would
carry a one-year sentence. Each of the suspects is told the following story. “If you confess,
and your partner doesn’t confess, then you will be released and your partner will be charged
with the crime. Your confession will be su�cient to convict him of the robbery and he will
be sent to prison for 10 years. If you both confess, then we don’t need either of you to testify
against the other, and you will both be convicted of the robbery. (Although in this case
your sentence will be less — 4 years only — because of your guilty plea.) Finally, if neither
of you confesses, then we can’t convict either of you of the robbery, so we will charge each
of you with resisting arrest. Your partner is being o↵ered the same deal. Do you want to
confess?”
To formalize this story as a game we need to identify the players, the possible strategies,
and the payo↵s. The two suspects are the players, and each has to choose between two possi-
ble strategies — Confess (C) or Not-Confess (N C). Finally, the payo↵s can be summarized
from the story above as in Figure 6.2. (Note that the payo↵s are all 0 or less, since there are
no good outcomes for the suspects, only di↵erent gradations of bad outcomes.)
Suspect 1
Suspect 2 N C C
N C �1,�1 �10, 0 C 0,�10 �4,�4
Figure 6.2: Prisoner’s Dilemma
As in the Exam-or-Presentation Game, we can consider how one of the suspects — say
Suspect 1 — should reason about his options.
• If Suspect 2 were going to confess, then Suspect 1 would receive a payo↵ of �4 by confessing and a payo↵ of �10 by not confessing. So in this case, Suspect 1 should confess.
• If Suspect 2 were not going to confess, then Suspect 1 would receive a payo↵ of 0 by confessing and a payo↵ of �1 by not confessing. So in this case too, Suspect 1 should confess.
So confessing is a strictly dominant strategy — it is the best choice regardless of what the
other player chooses. As a result, we should expect both suspects to confess, each getting a
162 CHAPTER 6. GAMES
payo↵ of �4. We therefore have the same striking phenomenon as in the Exam-or-Presentation Game:
there is an outcome that the suspects know to be better for both of them — in which they
both choose not to confess — but under rational play of the game there is no way for them
to achieve this outcome. Instead, they end up with an outcome that is worse for both of
them. And here too, it is important that the payo↵s reflect everything about the outcome of
the game; if, for example, the suspects could credibly threaten each other with retribution
for confessing, thereby making confessing a less desirable option, then this would a↵ect the
payo↵s and potentially the outcome.
Interpretations of the Prisoner’s Dilemma. The Prisoner’s Dilemma has been the
subject of a huge amount of literature since its introduction in the early 1950s [343, 346],
since it serves as a highly streamlined depiction of the di�culty in establishing cooperation
in the face of individual self-interest. While no model this simple can precisely capture
complex scenarios in the real world, the Prisoner’s Dilemma has been used as an interpretive
framework for many di↵erent real-world situations.
For example, the use of performance-enhancing drugs in professional sports has been
modeled as a case of the Prisoner’s Dilemma game [210, 367]. Here the athletes are the
players, and the two possible strategies are to use performance-enhancing drugs or not. If
you use drugs while your opponent doesn’t, you’ll get an advantage in the competition, but
you’ll su↵er long-term harm (and may get caught). If we consider a sport where it is di�cult
to detect the use of such drugs, and we assume athletes in such a sport view the downside as a
smaller factor than the benefits in competition, we can capture the situation with numerical
payo↵s that might look as follows. (The numbers are arbitrary here; we are only interested
in their relative sizes.)
Athlete 1
Athlete 2 Don’t Use Drugs Use Drugs
Don’t Use Drugs 3, 3 1, 4 Use Drugs 4, 1 2, 2
Figure 6.3: Performance-Enhancing Drugs
Here, the best outcome (with a payo↵ of 4) is to use drugs when your opponent doesn’t,
since then you maximize your chances of winning. However, the payo↵ to both using drugs
(2) is worse than the payo↵ to both not using drugs (3), since in both cases you’re evenly
matched, but in the former case you’re also causing harm to yourself. We can now see that
using drugs is a strictly dominant strategy, and so we have a situation where the players use
drugs even though they understand that there’s a better outcome for both of them.
More generally, situations of this type are often referred to as arms races, in which
6.3. BEST RESPONSES AND DOMINANT STRATEGIES 163
two competitors use an increasingly dangerous arsenal of weapons simply to remain evenly
matched. In the example above, the performance-enhancing drugs play the role of the
weapons, but the Prisoner’s Dilemma has also been used to interpret literal arms races
between opposing nations, where the weapons correspond to the nations’ military arsenals.
To wrap up our discussion of the Prisoner’s Dilemma, we should note that it only arises
when the payo↵s are aligned in a certain way — as we will see in the remainder of the chapter,
there are many situations where the structure of the game and the resulting behavior looks
very di↵erent. Indeed, even simple changes to a game can change it from an instance of
the Prisoner’s Dilemma to something more benign. For example, returning to the Exam-
or-Presentation Game, suppose that we keep everything the same as before, except that we
make the exam much easier, so that you’ll get a 100 on it if you study, and a 96 if you don’t.
Then we can check that the payo↵ matrix now becomes
You
Your Partner Presentation Exam
Presentation 98, 98 94, 96 Exam 96, 94 92, 92
Figure 6.4: Exam-or-Presentation Game with an easier exam.
Furthermore, we can check that with these new payo↵s, preparing for the presentation
now becomes a strictly dominant strategy; so we can expect that both players will play this
strategy, and both will benefit from this decision. The downsides of the previous scenario
no longer appear: like other dangerous phenomena, the Prisoner’s Dilemma only manifests
itself when the conditions are right.
6.3 Best Responses and Dominant Strategies
In reasoning about the games in the previous section, we used two fundamental concepts
that will be central to our discussion of game theory. As such, it is useful to define them
carefully here, and then to look further at some of their implications.
The first concept is the idea of a best response: it is the best choice of one player, given a
belief about what the other player will do. For instance, in the Exam-or-Presentation Game,
we determined your best choice in response to each possible choice of your partner.
We can make this precise with a bit of notation, as follows. If S is a strategy chosen by
Player 1, and T is a strategy chosen by Player 2, then there is an entry in the payo↵ matrix
corresponding to the pair of chosen strategies (S, T ). We will write P 1
(S, T ) to denote the
payo↵ to Player 1 as a result of this pair of strategies, and P 2
(S, T ) to denote the payo↵ to
Player 2 as a result of this pair of strategies. Now, we say that a strategy S for Player 1 is
a best response to a strategy T for Player 2 if S produces at least as good a payo↵ as any
164 CHAPTER 6. GAMES
other strategy paired with T :
P 1
(S, T ) � P 1
(S0, T )
for all other strategies S0 of Player 1. Naturally, there is a completely symmetric definition
for Player 2, which we won’t write down here. (In what follows, we’ll present the definitions
from Player 1’s point of view, but there are direct analogues for Player 2 in each case.)
Notice that this definition allows for multiple di↵erent strategies of Player 1 to be tied as
the best response to strategy T . This can make it di�cult to predict which of these multiple
di↵erent strategies Player 1 will use. We can emphasize that one choice is uniquely the best
by saying that a strategy S of Player 1 is a strict best response to a strategy T for Player 2
if S produces a strictly higher payo↵ than any other strategy paired with T :
P 1
(S, T ) > P 1
(S0, T )
for all other strategies S0 of Player 1. When a player has a strict best response to T , this is
clearly the strategy she should play when faced with T .
The second concept, which was central to our analysis in the previous section, is that of
a strictly dominant strategy. We can formulate its definition in terms of best responses as
follows.
• We say that a dominant strategy for Player 1 is a strategy that is a best response to every strategy of Player 2.
• We say that a strictly dominant strategy for Player 1 is a strategy that is a strict best response to every strategy of Player 2.
In the previous section, we made the observation that if a player has a strictly dominant
strategy, then we can expect him or her to use it. The notion of a dominant strategy is
slightly weaker, since it can be tied as the best option against some opposing strategies. As
a result, a player could potentially have multiple dominant strategies, in which case it may
not be obvious which one should be played.
The analysis of the Prisoner’s Dilemma was facilitated by the fact that both players had
strictly dominant strategies, and so it was easy to reason about what was likely to happen.
But most settings won’t be this clear-cut, and we now begin to look at games which lack
strictly dominant strategies.
A Game in Which Only One Player Has a Strictly Dominant Strategy. As a first
step, let’s consider a setting in which one player has a strictly dominant strategy and the
other one doesn’t. As a concrete example, we consider the following story.
Suppose there are two firms that are each planning to produce and market a new product;
these two products will directly compete with each other. Let’s imagine that the population
6.3. BEST RESPONSES AND DOMINANT STRATEGIES 165
of consumers can be cleanly divided into two market segments: people who would only buy
a low-priced version of the product, and people who would only buy an upscale version.
Let’s also assume that the profit any firm makes on a sale of either a low price or an upscale
product is the same. So to keep track of profits it’s good enough to keep track of sales. Each
firm wants to maximize its profit, or equivalently its sales, and in order to do this it has to
decide whether its new product will be low-priced or upscale.
So this game has two players — Firm 1 and Firm 2 — and each has two possible strategies:
to produce a low-priced product or an upscale one. To determine the payo↵s, here is how
the firms expect the sales to work out.
• First, people who would prefer a low-priced version account for 60% of the population, and people who would prefer an upscale version account for 40% of the population.
• Firm 1 is the much more popular brand, and so when the two firms directly compete in a market segment, Firm 1 gets 80% of the sales and Firm 2 gets 20% of the sales.
(If a firm is the only one to produce a product for a given market segment, it gets all
the sales.)
Based on this, we can determine payo↵s for di↵erent choices of strategies as follows.
• If the two firms market to di↵erent market segments, they each get all the sales in that segment. So the one that targets the low-priced segment gets a payo↵ .60 and the one
that targets the upscale segment gets .40.
• If both firms target the low-priced segment, then Firm 1 gets 80% of it, for a payo↵ of .48, and Firm 2 gets 20% of it, for a payo↵ of .12.
• Analogously, if both firms target the upscale segment, then Firm 1 gets a payo↵ of (.8)(.4) = .32 and Firm 2 gets a payo↵ of (.2)(.4) = .08.
This can be summarized in the following payo↵ matrix.
Firm 1
Firm 2 Low-Priced Upscale
Low-Priced .48, .12 .60, .40 Upscale .40, .60 .32, .08
Figure 6.5: Marketing Strategy
Notice that in this game, Firm 1 has a strictly dominant strategy: for Firm 1, Low-Priced
is a strict best response to each strategy of Firm 2. On the other hand, Firm 2 does not
have a dominant strategy: Low-Priced is its best response when Firm 1 plays Upscale, and
Upscale is its best response when Firm 1 plays Low-Priced.
166 CHAPTER 6. GAMES
Still, it is not hard to make a prediction about the outcome of this game. Since Firm 1 has
a strictly dominant strategy in Low-Priced, we can expect it will play it. Now, what should
Firm 2 do? If Firm 2 knows Firm 1’s payo↵s, and knows that Firm 1 wants to maximize
profits, then Firm 2 can confidently predict that Firm 1 will play Low-Priced. Then, since
Upscale is the strict best response by Firm 2 to Low-Priced, we can predict that Firm 2 will
play Upscale. So our overall prediction of play in this marketing game is Low-Priced by Firm
1 and Upscale by Firm 2, resulting in payo↵s of .60 and .40 respectively.
Note that although we’re describing the reasoning in two steps — first the strictly dom-
inant strategy of Firm 1, and then the best response of Firm 2 — this is still in the context
of a game where the players move simultaneously: both firms are developing their marketing
strategies concurrently and in secret. It is simply that the reasoning about strategies natu-
rally follows this two-step logic, resulting in a prediction about how the simultaneous play
will occur. It’s also interesting to note the intuitive message of this prediction. Firm 1 is
so strong that it can proceed without regard to Firm 2’s decision; given this, Firm 2’s best
strategy is to stay safely out of the way of Firm 1.
Finally, we should also note how the Marketing Strategy Game makes use of the knowl-
edge we assume players have about the game being played and about each other. In particu-
lar, we assume that each player knows the entire payo↵ matrix. And in reasoning about this
specific game, it is important that Firm 2 knows that Firm 1 wants to maximize profits, and
that Firm 2 knows that Firm 1 knows its own profits. In general, we will assume that the
players have common knowledge of the game: they know the structure of the game, they know
that each of them know the structure of the game, they know that each of them know that
each of them know, and so on. While we will not need the full technical content of common
knowledge in anything we do here, it is an underlying assumption and a topic of research in
the game theory literature [28]. As mentioned earlier, it is still possible to analyze games in
situations where common knowledge does not hold, but the analysis becomes more complex
[208]. It’s also worth noting that the assumption of common knowledge is a bit stronger than
we need for reasoning about simple games such as the Prisoner’s Dilemma, where strictly
dominant strategies for each player imply a particular course of action regardless of what
the other player is doing.
6.4 Nash Equilibrium
When neither player in a two-player game has a strictly dominant strategy, we need some
other way of predicting what is likely to happen. In this section, we develop methods for
doing this; the result will be a useful framework for analyzing games in general.
6.4. NASH EQUILIBRIUM 167
An Example: A Three-Client Game. To frame the question, it helps to think about a
simple example of a game that lacks strictly dominant strategies. Like our previous example,
it will be a marketing game played between two firms; however, it has a slightly more intricate
set-up. Suppose there are two firms that each hope to do business with one of three large
clients, A, B, and C. Each firm has three possible strategies: whether to approach A, B, or
C. The results of their two decisions will work out as follows.
• If the two firms approach the same client, then the client will give half its business to each.
• Firm 1 is too small to attract business on its own, so if it approaches one client while Firm 2 approaches a di↵erent one, then Firm 1 gets a payo↵ of 0.
• If Firm 2 approaches client B or C on its own, it will get their full business. However, A is a larger client, and will only do business with the firms if both approach A.
• Because A is a larger client, doing business with it is worth 8 (and hence 4 to each firm if it’s split), while doing business with B or C is worth 2 (and hence 1 to each firm if
it’s split).
From this description, we can work out the following payo↵ matrix.
Firm 1
Firm 2 A B C
A 4, 4 0, 2 0, 2 B 0, 0 1, 1 0, 2 C 0, 0 0, 2 1, 1
Figure 6.6: Three-Client Game
If we study how the payo↵s in this game work, we see that neither firm has a dominant
strategy. Indeed, each strategy by each firm is a strict best response to some strategy by the
other firm. For Firm 1, A is a strict best response to strategy A by Firm 2, B is a strict best
response to B, and C is a strict best response to C. For Firm 2, A is a strict best response
to strategy A by Firm 1, C is a strict best response to B, and B is a strict best response to
C. So how should we reason about the outcome of play in this game?
Defining Nash Equilibrium. In 1950, John Nash proposed a simple but powerful prin-
ciple for reasoning about behavior in general games [313, 314], and its underlying premise
is the following: even when there are no dominant strategies, we should expect players to
use strategies that are best responses to each other. More precisely, suppose that Player 1
chooses a strategy S and Player 2 chooses a strategy T . We say that this pair of strategies
168 CHAPTER 6. GAMES
(S, T ) is a Nash equilibrium if S is a best response to T , and T is a best response to S.
This is not a concept that can be derived purely from rationality on the part of the players;
instead, it is an equilibrium concept. The idea is that if the players choose strategies that
are best responses to each other, then no player has an incentive to deviate to an alternative
strategy — so the system is in a kind of equilibrium state, with no force pushing it toward a
di↵erent outcome. Nash shared the 1994 Nobel Prize in Economics for his development and
analysis of this idea.
To understand the idea of Nash equilibrium, we should first ask why a pair of strategies
that are not best responses to each other would not constitute an equilibrium. The answer is
that the players cannot both believe that these strategies will be actually used in the game,
as they know that at least one player would have an incentive to deviate to another strategy.
So Nash equilibrium can be thought of as an equilibrium in beliefs. If each player believes
that the other player will actually play a strategy that is part of a Nash equilibrium, then
she is willing to play her part of the Nash equilibrium.
Let’s consider the Three-Client Game from the perspective of Nash equilibrium. If Firm
1 chooses A and Firm 2 chooses A, then we can check that Firm 1 is playing a best response
to Firm 2’s strategy, and Firm 2 is playing a best response to Firm 1’s strategy. Hence, the
pair of strategies (A, A) forms a Nash equilibrium. Moreover, we can check that this is the
only Nash equilibrium. No other pair of strategies are best responses to each other.1
This discussion also suggests two ways to find Nash equilibria. The first is to simply
check all pairs of strategies, and ask for each one of them whether the individual strategies
are best responses to each other. The second is to compute each player’s best response(s) to
each strategy of the other player, and then find strategies that are mutual best responses.
6.5 Multiple Equilibria: Coordination Games
For a game with a single Nash equilibrium, such as the Three-Client Game in the previ-
ous section, it seems reasonable to predict that the players will play the strategies in this
equilibrium: under any other play of the game, at least one player will not be using a best
response to what the other is doing. Some natural games, however, can have more than one
Nash equilibrium, and in this case it becomes di�cult to predict how rational players will
actually behave in the game. We consider some fundamental examples of this problem here.
A Coordination Game. A simple but central example is the following Coordination
Game, which we can motivate through the following story. Suppose you and a partner are
1In this discussion, each player only has three available strategies: A, B, or C. Later in this we will introduce the possibility of more complex strategies in which players can randomize over their available options. With this more complex formulation of possible strategies, we will find additional equilibria for the Three-Client Game.
6.5. MULTIPLE EQUILIBRIA: COORDINATION GAMES 169
each preparing slides for a joint project presentation; you can’t reach your partner by phone,
and need to start working on the slides now. You have to decide whether to prepare your
half of the slides in PowerPoint or in Apple’s Keynote software. Either would be fine, but
it will be much easier to merge your slides together with your partner’s if you use the same
software.
So we have a game in which you and your partner are the two players, choosing Power-
Point or choosing Keynote form the two strategies, and the payo↵s are as shown in Figure 6.7.
You
Your Partner PowerPoint Keynote
PowerPoint 1, 1 0, 0 Keynote 0, 0 1, 1
Figure 6.7: Coordination Game
This is called a Coordination Game because the two players’ shared goal is really to
coordinate on the same strategy. There are many settings in which coordination games
arise. For example, two manufacturing companies that work together extensively need to
decide whether to configure their machinery in metric units of measurement or English units
of measurement; two platoons in the same army need to decide whether to attack an enemy’s
left flank or right flank; two people trying to find each other in a crowded mall need to decide
whether to wait at the north end of the mall or at the south end. In each case, either choice
can be fine, provided that both participants make the same choice.
The underlying di�culty is that the game has two Nash equilibria — i.e., (Power-
Point,PowerPoint) and (Keynote,Keynote) in our example from Figure 6.7. If the players
fail to coordinate on one of the Nash equilibria, perhaps because one player expects Power-
Point to be played and the other expects Keynote, then they receive low payo↵s. So what
do the players do?
This remains a subject of considerable discussion and research, but some proposals have
received attention in the literature. Thomas Schelling [364] introduced the idea of a focal
point as a way to resolve this di�culty. He noted that in some games there are natural
reasons (possibly outside the payo↵ structure of the game) that cause the players to focus
on one of the Nash equilibria. For example, suppose two drivers are approaching each other
at night on an undivided country road. Each driver has to decide whether to move over to
the left or the right. If the drivers coordinate — making the same choice of side — then they
pass each other, but if they fail to coordinate, then they get a severely low payo↵ due to the
resulting collision. Fortunately, social convention can help the drivers decide what to do in
this case: if this game is being played in the U.S., convention strongly suggests that they
should move to the right, while if the game is being played in England, convention strongly
suggests that they should move to the left. In other words, social conventions, while often
170 CHAPTER 6. GAMES
arbitrary, can sometimes be useful in helping people coordinate among multiple equilibria.
Variants on the Basic Coordination Game. One can enrich the structure of our basic
Coordination Game to capture a number of related issues surrounding the problem of mul-
tiple equilibria. To take a simple extension of our previous example, suppose that both you
and your project partner each prefer Keynote to PowerPoint. You still want to coordinate,
but you now view the two alternatives as unequal. This gives us the payo↵ matrix for an
Unbalanced Coordination Game, shown in Figure 6.8.
You
Your Partner PowerPoint Keynote
PowerPoint 1, 1 0, 0 Keynote 0, 0 2, 2
Figure 6.8: Unbalanced Coordination Game
Notice that (PowerPoint,PowerPoint) and (Keynote,Keynote) are still both Nash equi-
libria for this game, despite the fact that one of them gives higher payo↵s to both players.
(The point is that if you believe your partner will choose PowerPoint, you still should choose
PowerPoint as well.) Here, Schelling’s theory of focal points suggests that we can use a
feature intrinsic to the game — rather than an arbitrary social convention — to make a
prediction about which equilibrium will be chosen by the players. That is, we can predict
that when the players have to choose, they will select strategies so as to reach the equilib-
rium that gives higher payo↵s to both of them. (To take another example, consider the two
people trying to meet at a crowded mall. If the north end of the mall has a bookstore they
both like, while the south end consists of a loading dock, the natural focal point would be
the equilibrium in which they both choose the north end.)
Things get more complicated if you and your partner don’t agree on which software you
prefer, as shown in the payo↵ matrix of Figure 6.9.
You
Your Partner PowerPoint Keynote
PowerPoint 1, 2 0, 0 Keynote 0, 0 2, 1
Figure 6.9: Battle of the Sexes
In this case, the two equilibria still correspond to the two di↵erent ways of coordinating,
but your payo↵ is higher in the (Keynote,Keynote) equilibrium, while your partner’s payo↵
is higher in the (PowerPoint,PowerPoint) equilibrium. This game is traditionally called the
Battle of the Sexes, because of the following motivating story. A husband and wife want to
see a movie together, and they need to choose between a romantic comedy and an action
6.5. MULTIPLE EQUILIBRIA: COORDINATION GAMES 171
movie. They want to coordinate on their choice, but the (Romance,Romance) equilibrium
gives a higher payo↵ to one of them while the (Action,Action) equilibrium gives a higher
payo↵ to the other.
In Battle of the Sexes, it can be hard to predict the equilibrium that will be played using
either the payo↵ structure or some purely external social convention. Rather, it helps to know
something about conventions that exist between the two players themselves, suggesting how
they resolve disagreements when they prefer di↵erent ways of coordinating.
It’s worth mentioning one final variation on the basic Coordination Game, which has
attracted attention in recent years. This is the Stag Hunt Game [374]; the name is motivated
by the following story from writings of Rousseau. Suppose that two people are out hunting;
if they work together, they can catch a stag (which would be the highest-payo↵ outcome),
but on their own each can catch a hare. The tricky part is that if one hunter tries to catch
a stag on his own, he will get nothing, while the other one can still catch a hare. Thus, the
hunters are the two players, their strategies are Hunt Stag and Hunt Hare, and the payo↵s
are as shown in Figure 6.10.
Hunter 1
Hunter 2 Hunt Stag Hunt Hare
Hunt Stag 4, 4 0, 3 Hunt Hare 3, 0 3, 3
Figure 6.10: Stag Hunt
This is quite similar to the Unbalanced Coordination Game, except that if the two players
miscoordinate, the one who was trying for the higher-payo↵ outcome gets penalized more
than the one who was trying for the lower-payo↵ outcome. (In fact, the one trying for the
lower-payo↵ outcome doesn’t get penalized at all.) As a result, the challenge in reasoning
about which equilibrium will be chosen is based on the trade-o↵ between the high payo↵ of
one and the low downside of miscoordination from the other.
It has been argued that the Stag Hunt Game captures some of the intuitive challenges
that are also raised by the Prisoner’s Dilemma. The structures are clearly di↵erent, since
the Prisoner’s Dilemma has strictly dominant strategies; both, however, have the property
that players can benefit if they cooperate with each other, but risk su↵ering if they try
cooperating while their partner doesn’t. Another way to see some of the similarities between
the two games is to notice that if we go back to the original Exam-or-Presentation Game and
make one small change, then we end up changing it from an instance of Prisoner’s Dilemma
to something closely resembling Stag Hunt. Specifically, suppose that we keep the grade
outcomes the same as in Section 6.1, except that we require both you and your partner to
prepare for the presentation in order to have any chance of a better grade. That is, if you
both prepare, you both get a 100 on the presentation, but if at most one of you prepares, you
172 CHAPTER 6. GAMES
both get the base grade of 84. With this change, the payo↵s for the Exam-or-Presentation
Game become what is shown in Figure 6.11.
You
Your Partner Presentation Exam
Presentation 90, 90 82, 88 Exam 88, 82 88, 88
Figure 6.11: Exam-or-Presentation Game (Stag Hunt version)
We now have a structure that closely resembles the Stag Hunt Game: coordinating on
(Presentation,Presentation) or (Exam,Exam) are both equilibria, but if you attempt to go
for the higher-payo↵ equilibrium, you risk getting a low grade if your partner opts to study
for the exam.
6.6 Multiple Equilibria: The Hawk-Dove Game
Multiple Nash equilibria also arise in a di↵erent but equally fundamental kind of game, in
which the players engage in a kind of “anti-coordination” activity. Probably the most basic
form of such a game is the Hawk-Dove Game, which is motivated by the following story.
Suppose two animals are engaged in a contest to decide how a piece of food will be
divided between them. Each animal can choose to behave aggressively (the Hawk strategy)
or passively (the Dove strategy). If the two animals both behave passively, they divide the
food evenly, and each get a payo↵ of 3. If one behaves aggressively while the other behaves
passively, then the aggressor gets most of the food, obtaining a payo↵ of 5, while the passive
one only gets a payo↵ of 1. But if both animals behave aggressively, then they destroy the
food (and possibly injure each other), each getting a payo↵ of 0. Thus we have the payo↵
matrix in Figure 6.12.
Animal 1
Animal 2 D H
D 3, 3 1, 5 H 5, 1 0, 0
Figure 6.12: Hawk-Dove Game
This game has two Nash equilibria: (D, H) and (H, D). Without knowing more about the
animals we cannot predict which of these equilibria will be played. So as in the coordination
games we looked at earlier, the concept of Nash equilibrium helps to narrow down the set of
reasonable predictions, but it does not provide a unique prediction.
The Hawk-Dove game has been studied in many contexts. For example, suppose we sub-
stitute two countries for the two animals, and suppose that the countries are simultaneously
6.7. MIXED STRATEGIES 173
choosing whether to be aggressive or passive in their foreign policy. Each country hopes
to gain through being aggressive, but if both act aggressively they risk actually going to
war, which would be disastrous for both. So in equilibrium, we can expect that one will
be aggressive and one will be passive, but we can’t predict who will follow which strategy.
Again we would need to know more about the countries to predict which equilibrium will be
played.
Hawk-Dove is another example of a game that can arise from a small change to the payo↵s
in the Exam-or-Presentation Game. Let’s again recall the set-up from the opening section,
and now vary things so that if neither you nor your partner prepares for the presentation,
you will get a very low joint grade of 60. (If one or both of you prepare, the grades for the
presentation are the same as before.) If we compute the average grades you get for di↵erent
choices of strategies in this version of the game, we have the payo↵s in Figure 6.13.
You
Your Partner Presentation Exam
Presentation 90, 90 86, 92 Exam 92, 86 76, 76
Figure 6.13: Exam or Presentation? (Hawk-Dove version)
In this version of the game, there are two equilibria: (Presentation, Exam) and (Exam,
Presentation). Essentially, one of you must behave passively and prepare for the presentation,
while the other achieves the higher payo↵ by studying for the exam. If you both try to avoid
the role of the passive player, you end up with very low payo↵s, but we cannot predict from
the structure of the game alone who will play this passive role.
The Hawk-Dove game is also known by a number of other names in the game theory
literature. For example, it is frequently referred to as the game of Chicken, to evoke the
image of two teenagers racing their cars toward each other, daring each other to be the one
to swerve out of the way. The two strategies here are Swerve and Don’t Swerve: the one who
swerves first su↵ers humiliation from his friends, but if neither swerves, then both su↵er an
actual collision.
6.7 Mixed Strategies
In the previous two sections, we have been discussing games whose conceptual complexity
comes from the existence of multiple equilibria. However, there are also games which have
no Nash equilibria at all. For such games, we will make predictions about players’ behavior
by enlarging the set of strategies to include the possibility of randomization; once players
are allowed to behave randomly, one of John Nash’s main results establishes that equilibria
always exist [313, 314].
174 CHAPTER 6. GAMES
Probably the simplest class of games to expose this phenomenon are what might be called
“attack-defense” games. In such games, one player behaves as the attacker, while the other
behaves as the defender. The attacker can use one of two strategies — let’s call them A
and B — while the defender’s two strategies are “defend against A” or “defend against B.”
If the defender defends against the attack the attacker is using, then the defender gets the
higher payo↵; but if the defender defends against the wrong attack, then the attacker gets
the higher payo↵.
Matching Pennies. A simple attack-defense game is called Matching Pennies, and is
based on a game in which two people each hold a penny, and simultaneously choose whether
to show heads (H) or tails (T ) on their penny. Player 1 loses his penny to player 2 if they
match, and wins player 2’s penny if they don’t match. This produces a payo↵ matrix as
shown in Figure 6.14.
Player 1
Player 2 H T
H �1, +1 +1,�1 T +1,�1 �1, +1
Figure 6.14: Matching Pennies
Matching pennies is a simple example of a large class of interesting games with the
property that the payo↵s of the players sum to zero in every outcome. Such games are called
zero-sum games, and many attack-defense games — and more generally, games where the
players’ interests are in direct conflict — have this structure. Games like Matching Pennies
have in fact been used as metaphorical descriptions of decisions made in combat; for example,
the Allied landing in Europe on June 6, 1944 — one of the pivotal moments in World War
II — involved a decision by the Allies whether to cross the English Channel at Normandy or
at Calais, and a corresponding decision by the German army whether to mass its defensive
forces at Normandy or Calais. This has an attack-defense structure that closely resembles
the Matching Pennies game [123].
The first thing to notice about Matching Pennies is that there is no pair of strategies
that are best responses to each other. To see this, observe that for any pair of strategies,
one of the players gets a payo↵ of �1, and this player would improve his or her payo↵ to +1 by switching strategies. So for any pair of strategies, one of the players wants to switch
what they’re doing.2
2Incidentally, although it’s not crucial for the discussion here, it’s interesting to note that the Three- Client Game used as an example in Section 6.4 can be viewed intuitively as a kind of hybrid of the Matching Pennies Game and the Stag Hunt Game. If we look just at how the two players evaluate the options of approaching Clients B and C, we have Matching Pennies: Firm 1 wants to match, while Firm 2 wants to not match. However, if they coordinate on approaching Client A, then they both get even higher payo↵s –
6.7. MIXED STRATEGIES 175
This means that if we treat each player as simply having the two strategies H or T , then
there is no Nash equilibrium for this game. This is not so surprising if we consider how
Matching Pennies works. A pair of strategies, one for each player, forms a Nash equilibrium
if even given knowledge of each other’s strategies, neither player would have an incentive to
switch to an alternate strategy. But in Matching Pennies, if Player 1 knows that Player 2 is
going to play a particular choice of H or T , then Player 1 can exploit this by choosing the
opposite and receiving a payo↵ of +1. Analogous reasoning holds for Player 2.
When we think intuitively about how games of this type are played in real life, we see
that players generally try to make it di�cult for their opponents to predict what they will
play. This suggests that in our modeling of a game like Matching Pennies, we shouldn’t treat
the strategies as simply H or T , but as ways of randomizing one’s behavior between H and
T . We now see how to build this into a model for the play of this kind of game.
Mixed Strategies. The simplest way to introduce randomized behavior is to say that
each player is not actually choosing H or T directly, but rather is choosing a probability with
which she will play H. So in this model, the possible strategies for Player 1 are numbers
p between 0 and 1; a given number p means that Player 1 is committing to play H with
probability p, and T with probability 1 � p. Similarly, the possible strategies for Player 2 are numbers q between 0 and 1, representing the probability that Player 2 will play H.
Since a game consists of a set of players, strategies, and payo↵s, we should notice that
by allowing randomization, we have actually changed the game. It no longer consists of
two strategies by each player, but instead a set of strategies corresponding to the interval
of numbers between 0 and 1. We will refer to these as mixed strategies, since they involve
“mixing” between the options H and T . Notice that the set of mixed strategies still includes
the original two options of committing to definitely play H or T ; these two choices correspond
to selecting probabilities of 1 or 0 respectively, and we will refer to them as the two pure
strategies in the game. To make things more informal notationally, we will sometimes refer
to the choice of p = 1 by Player 1 equivalently as the “pure strategy H”, and similarly for
p = 0 and q = 1 or 0.
Payo↵s from Mixed Strategies. With this new set of strategies, we also need to deter-
mine the new set of payo↵s. The subtlety in defining payo↵s is that they are now random
quantities: each player will get +1 with some probability, and will get �1 with the remain- ing probability. When payo↵s were numbers it was obvious how to rank them: bigger was
better. Now that payo↵s are random, it is not immediately obvious how to rank them: we
want a principled way to say that one random outcome is better than another.
To think about this issue, let’s start by considering Matching Pennies from Player 1’s
analogously to the two hunters coordinating to hunt stag.
176 CHAPTER 6. GAMES
point of view, and focus first on how she evaluates her two pure strategies of definitely playing
H or definitely playing T . Suppose that Player 2 chooses the strategy q; that is, he commits
to playing H with probability q and T with probability 1�q. Then if Player 1 chooses pure strategy H, she receives a payo↵ of �1 with probability q (since the two pennies match with probability q, in which event she loses), and she receives a payo↵ of +1 with probability 1�q (since the two pennies don’t match with probability 1�q). Alternatively, if Player 1 chooses pure strategy T , she receives +1 with probability q, and �1 with probability (1�q). So even if Player 1 uses a pure strategy, her payo↵s can still be random due to the randomization
employed by Player 2. How should we decide which of H or T is more appealing to Player
1 in this case?
In order to rank random payo↵s numerically, we will attach a number to each distribution
that represents how attractive this distribution is to the player. Once we have done this, we
can then rank outcomes according to their associated number. The number we will use for
this purpose is the expected value of the payo↵. So for example, if Player 1 chooses the pure
strategy H while Player 2 chooses a probability of q, as above, then the expected payo↵ to
Player 1 is
(�1)(q) + (1)(1 � q) = 1 � 2q.
Similarly, if Player 1 chooses the pure strategy T while Player 2 chooses a probability of q,
then the expected payo↵ to Player 1 is
(1)(q) + (�1)(1 � q) = 2q � 1.
We will assume players are seeking to maximize the expected payo↵ they get from a choice
of mixed strategies. Although the expectation is a natural quantity, it is a subtle question
whether maximizing expectation is a reasonable modeling assumption about the behavior
of players. By now, however, there is a well-established foundation for the assumption that
players rank distributions over payo↵s according to their expected values [288, 363, 398], and
so we will follow it here.
We have now defined the mixed-strategy version of the Matching Pennies game: strategies
are probabilities of playing H, and payo↵s are the expectations of the payo↵s from the four
pure outcomes (H, H), (H, T ), (T, H), and (T, T ). We can now ask whether there is a Nash
equilibrium for this richer version of the game.
Equilibrium with Mixed Strategies. We define a Nash equilibrium for the mixed-
strategy version just as we did for the pure-strategy version: it is a pair of strategies (now
probabilities) so that each is a best response to the other.
First, let’s observe that no pure strategy can be part of a Nash equilibrium. This is
equivalent to the reasoning we did at the outset of this section. Suppose, for example, that
the pure strategy H (i.e. probability p = 1) by Player 1 were part of a Nash equilibrium.
6.7. MIXED STRATEGIES 177
Then Player 2’s unique best response would be the pure strategy H as well (since Player 2
gets +1 whenever he matches). But H by Player 1 is not a best response to H by Player
2, so in fact this couldn’t be a Nash equilibrium. Analogous reasoning applies to the other
possible pure strategies by the two players. So we reach the natural conclusion that in any
Nash equilibrium, both players must be using probabilities that are strictly between 0 and
1.
Next, let’s ask what Player 1’s best response should be to the strategy q used by Player
2. Above, we determined that the expected payo↵ to Player 1 from the pure strategy H in
this case is
1 � 2q,
while the expected payo↵ to Player 1 from the pure strategy T is
2q � 1.
Now, here’s the key point: if 1 � 2q 6= 2q � 1, then one of the pure strategies H or T is in fact the unique best response by Player 1 to a play of q by Player 2. This is simply
because one of 1 � 2q or 2q � 1 is larger in this case, and so there is no point for Player 1 to put any probability on her weaker pure strategy. But we already established that pure
strategies cannot be part of any Nash equilibrium for Matching Pennies, and because pure
strategies are the best responses whenever 1 � 2q 6= 2q � 1, probabilities that make these two expectations unequal cannot be part of a Nash equilibrium either.
So we’ve concluded that in any Nash equilibrium for the mixed-strategy version of Match-
ing Pennies, we must have
1 � 2q = 2q � 1,
or in other words, q = 1/2. The situation is symmetric when we consider things from Player
2’s point of view, and evaluate the payo↵s from a play of probability p by Player 1. We
conclude from this that in any Nash equilibrium, we must also have p = 1/2.
Thus, the pair of strategies p = 1/2 and q = 1/2 is the only possibility for a Nash
equilibrium. We can check that this pair of strategies in fact do form best responses to each
other. As a result, this is the unique Nash equilibrium for the mixed-strategy version of
Matching Pennies.
Interpreting the Mixed-Strategy Equilibrium for Matching Pennies. Having de-
rived the Nash equilibrium for this game, it’s useful to think about what it means, and how
we can apply this reasoning to games in general.
First, let’s picture a concrete setting in which two people actually sit down to play
Matching Pennies, and each of them actually commits to behaving randomly according to
probabilities p and q respectively. If Player 1 believes that Player 2 will play H strictly more
than half the time, then she should definitely play T — in which case Player 2 should not
178 CHAPTER 6. GAMES
be playing H more than half the time. The symmetric reasoning applies if Player 1 believes
that Player 2 will play T strictly more than half the time. In neither case would we have
a Nash equilibrium. So the point is that the choice of q = 1/2 by Player 2 makes Player 1
indi↵erent between playing H or T : the strategy q = 1/2 is e↵ectively “non-exploitable” by
Player 1. This was in fact our original intuition for introducing randomization: each player
wants their behavior to be unpredictable to the other, so that their behavior can’t be taken
advantage of. We should note that the fact that both probabilities turned out to be 1/2 is
a result of the highly symmetric structure of Matching Pennies; as we will see in subsequent
examples in the next section, when the payo↵s are less symmetric, the Nash equilibrium can
consist of unequal probabilities.
This notion of indi↵erence is a general principle behind the computation of mixed-strategy
equilibria in two-player, two-strategy games when there are no equilibria involving pure
strategies: each player should randomize so as to make the other player indi↵erent between
their two alternatives. This way, neither player’s behavior can be exploited by a pure strategy,
and the two choices of probabilities are best responses to each other. And although we won’t
pursue the details of it here, a generalization of this principle applies to games with any
finite number of players and any finite number of strategies: Nash’s main mathematical
result accompanying his definition of equilibrium was to prove that every such game has at
least one mixed-strategy equilibrium [313, 314].
It’s also worth thinking about how to interpret mixed-strategy equilibria in real-world
situations. There are in fact several possible interpretations that are appropriate in di↵erent
situations:
• Sometimes, particularly when the participants are genuinely playing a sport or game, the players may be actively randomizing their actions [107, 337, 405]: a tennis player
may be randomly deciding whether to serve the ball up the center or out to the side of
the court; a card-player may be randomly deciding whether to blu↵ or not; two children
may be randomizing among rock, paper, and scissors in the perennial elementary-school
contest of the same name. We will look at examples of this in the next section.
• Sometimes the mixed strategies are better viewed as proportions within a population. Suppose for example that two species of animals, in the process of foraging for food,
regularly engage in one-on-one attack-defense games with the structure of Matching
Pennies. Here, a single member of the first species always plays the role of attacker,
and a single member of the second species always plays the role of defender.
Let’s suppose that each individual animal is genetically hard-wired to always play H
or always play T ; and suppose further that the population of each species consists half
of animals hard-wired to play H, and half of animals hard-wired to play T . Then with
this population mixture, H-animals in each species do exactly as well on average, over
6.8. MIXED STRATEGIES: EXAMPLES AND EMPIRICAL ANALYSIS 179
many random interactions, as T -animals. Hence the population as a whole is in a kind
of mixed equilibrium, even though each individual is playing a pure strategy. This
story suggests an important link with evolutionary biology, which has in fact been
developed through a long line of research [375, 376]; this topic will be our focus in
Chapter 7.
• Maybe the most subtle interpretation is based on recalling, from Section 6.4, that Nash equilibrium is often best thought of as an equilibrium in beliefs. If each player believes
that her partner will play according to a particular Nash equilibrium, then she too
will want to play according to it. In the case of Matching Pennies, with its unique
mixed equilibrium, this means that it is enough for you to expect that when you meet
an arbitrary person, they will play their side of Matching Pennies with a probability
of 1/2. In this case, playing a probability of 1/2 makes sense for you too, and hence
this choice of probabilities is self-reinforcing — it is in equilibrium — across the entire
population.
6.8 Mixed Strategies: Examples and Empirical Anal- ysis
Because mixed-strategy equilibrium is a subtle concept, it’s useful to think about it through
further examples. We will focus on two main examples, both drawn from the realm of sports,
and both with attack-defense structures. The first is stylized and partly metaphorical, while
the second represents a striking empirical test of whether people in high-stakes situations
actually follow the predictions of mixed-strategy equilibrium. We conclude the section with
a general discussion of how to identify all the equilibria of a two-player, two-strategy game.
The Run-Pass Game. First, let’s consider a streamlined version of the problem faced by
two American football teams as they plan their next play in a football game. The o↵ense
can choose either to run or to pass, and the defense can choose either to defend against the
run or to defend against the pass. Here is how the payo↵s work.
• If the defense correctly matches the o↵ense’s play, then the o↵ense gains 0 yards.
• If the o↵ense runs while the defense defends against the pass, the o↵ense gains 5 yards.
• If the o↵ense passes while the defense defends against the run, the o↵ense gains 10 yards.
Hence we have the payo↵ matrix shown in Figure 6.15.
(If you don’t know the rules of American football, you can follow the discussion simply
by taking the payo↵ matrix as self-contained. Intuitively, the point is simply that we have
180 CHAPTER 6. GAMES
O↵ense
Defense Defend Pass Defend Run
Pass 0, 0 10,�10 Run 5,�5 0, 0
Figure 6.15: Run-Pass Game
an attack-defense game with two players named “o↵ense” and “defense” respectively, and
where the attacker has a stronger option (pass) and a weaker option (run).)
Just as in Matching Pennies, it’s easy to check that there is no Nash equilibrium where
either player uses a pure strategy: both have to make their behavior unpredictable by ran-
domizing. So let’s work out a mixed-strategy equilibrium for this game: let p be the prob-
ability that the o↵ense passes, and let q be the probability that the defense defends against
the pass. (We know from Nash’s result that at least one mixed-strategy equilibrium must
exist, but not what the actual values of p and q should be.)
We use the principle that a mixed equilibrium arises when the probabilities used by each
player makes his opponent indi↵erent between his two options.
• First, suppose the defense chooses a probability of q for defending against the pass. Then the expected payo↵ to the o↵ense from passing is
(0)(q) + (10)(1 � q) = 10 � 10q,
while the expected payo↵ to the o↵ense from running is
(5)(q) + (0)(1 � q) = 5q.
To make the o↵ense indi↵erent between its two strategies, we need to set 10�10q = 5q, and hence q = 2/3.
• Next, suppose the o↵ense chooses a probability of p for passing. Then the expected payo↵ to the defense from defending against the pass is
(0)(p) + (�5)(1 � p) = 5p � 5,
with the expected payo↵ to the defense from defending against the run is
(�10)(p) + (0)(1 � p) = �10p.
To make the defense indi↵erent between its two strategies, we need to set 5p�5 = �10p, and hence p = 1/3.
Thus, the only possible probability values that can appear in a mixed-strategy equilibrium
are p = 1/3 for the o↵ense, and q = 2/3 for the defense, and this in fact forms an equilibrium.
6.8. MIXED STRATEGIES: EXAMPLES AND EMPIRICAL ANALYSIS 181
Notice also that the expected payo↵ to the o↵ense with these probabilities is 10/3, and the
corresponding expected payo↵ to the defense is �10/3. Also, in contrast to Matching Pennies, notice that because of the asymmetric structure of the payo↵s here, the probabilities that
appear in the mixed-strategy equilibrium are unbalanced as well.
Strategic Interpretation of the Run-Pass Game. There are several things to notice
about this equilibrium. First, the strategic implications of the equilibrium probabilities are
intriguing and a bit subtle. Specifically, although passing is the o↵ense’s more powerful
weapon, it uses it less than half the time: it places only probability p = 1/3 on passing. This
initially seems counter-intuitive: why not spend more time using your more powerful option?
But the calculation that gave us the equilibrium probabilities also supplies the answer to
this question. If the o↵ense placed any higher probability on passing, then the defense’s
best response would be to always defend against the pass, and the o↵ense would actually do
worse in expectation.
We can see how this works by trying a larger value for p, like p = 1/2. In this case, the
defense will always defend against the pass, and so the o↵ense’s expected payo↵ will be 5/2,
since it gains 5 half the time and 0 the other half the time:
(1/2)(0) + (1/2)(5) = 5/2.
Above, we saw that with the equilibrium probabilities, the o↵ense has an expected payo↵
of 10/3 > 5/2. Moreover, because p = 1/3 makes the defense indi↵erent between its two
strategies, an o↵ense that uses p = 1/3 is guaranteed to get 10/3 > 5/2 no matter what the
defense does.
One way to think about the real power of passing as a strategy is to notice that in
equilibrium, the defense is defending against the pass 2/3 of the time, even though the
o↵ense is using it only 1/3 of the time. So somehow the threat of passing is helping the
o↵ense, even though it uses it relatively rarely.
This example clearly over-simplifies the strategic issues at work in American football:
there are many more than just two strategies, and teams are concerned with more than just
their yardage on the very next play. Nevertheless, this type of analysis has been applied
quantitatively to statistics from American football, verifying some of the main qualitative
conclusions at a broad level — that teams generally run more than they pass, and that the
expected yardage gained per play from running is close to the expected yardage gained per
play from passing for most teams [82, 84, 355].
The Penalty-Kick Game. The complexity of American football makes it hard to cast it
truly accurately as a two-person, two-strategy game. We now focus on a di↵erent setting,
also from professional sports, where such a formalization can be done much more exactly —
the modeling of penalty kicks in soccer as a two-player game.
182 CHAPTER 6. GAMES
In 2002, Ignacio Palacios-Huerta undertook a large study of penalty kicks from the per-
spective of game theory [337], and we focus on his analysis here. As he observed, penalty
kicks capture the ingredients of two-player, two-strategy games remarkably faithfully. The
kicker can aim the ball to the left or the right of the goal, and the goalie can dive to either
the left or right as well.3 The ball moves to the goal fast enough that the decisions of the
kicker and goalie are e↵ectively being made simultaneously; and based on these decisions the
kicker is likely to score or not. Indeed, the structure of the game is very much like Matching
Pennies: if the goalie dives in the direction where the ball is aimed, he has a good chance of
blocking it; if the goalie dives in the wrong direction, it is very likely to go in the goal.
Based on an analysis of roughly 1400 penalty kicks in professional soccer, Palacios-Huerta
determined the empirical probability of scoring for each of the four basic outcomes: whether
the kicker aims left or right, and whether the goalie dives left or right. This led to a payo↵
matrix as shown in Figure 6.16.
Kicker
Goalie L R
L 0.58,�0.58 0.95,�0.95 R 0.93,�0.93 0.70,�0.70
Figure 6.16: The Penalty-Kick Games (from empirical data [337]).
There are a few contrasts to note in relation to the basic Matching Pennies Game. First,
a kicker has a reasonably good chance of scoring even when the goalie dives in the correct
direction (although a correct choice by the goalie still greatly reduces this probability).
Second, kickers are generally right-footed, and so their chance of scoring is not completely
symmetric between aiming left and aiming right.4
Despite these caveats, the basic premise of Matching Pennies is still present here: there
is no equilibrium in pure strategies, and so we need to consider how players should random-
ize their behavior in playing this game. Using the principle of indi↵erence as in previous
examples, we see that if q is the probability that a goalie chooses L, we need to set q so as
to make the kicker indi↵erent between his two options:
(.58)(q) + (.95)(1 � q) = (.93)(q) + (.70)(1 � q).
Solving for q, we get q = .42. We can do the analogous calculation to obtain the value of p
that makes the goalie indi↵erent, obtaining p = .39.
The striking punchline to this study is that in the dataset of real penalty kicks, the
goalies dive left a .42 fraction of the time (matching the prediction to two decimal places),
3Kicks up the center, and decisions by the goalie to remain in the center, are very rare, and can be ignored in a simple version of the analysis.
4For purposes of the analysis, we take all the left-footed kickers in the data and apply a left-right reflection to all their actions, so that R always denotes the “natural side” for each kicker.
6.8. MIXED STRATEGIES: EXAMPLES AND EMPIRICAL ANALYSIS 183
and the kickers aim left a .40 fraction of the time (coming within .01 of the prediction). It is
particularly nice to find the theory’s predictions borne out in a setting such as professional
soccer, since the two-player game under study is being played by experts, and the outcome
is important enough to the participants that they are investing significant attention to their
choice of strategies.
Finding all Nash Equilibria. To conclude our discussion of mixed-strategy equilibria, we
consider the general question of how to find all Nash equilibria of a two-player, two-strategy
game.
First, it is important to note that a game may have both pure-strategy and mixed-
strategy equilibria. As a result, one should first check all four pure outcomes (given by
pairs of pure strategies) to see which, if any, form equilibria. Then, to check whether there
are any mixed-strategy equilibria, we need to see whether there are mixing probabilities p
and q that are best responses to each other. If there is a mixed-strategy equilibrium, then
we can determine Player 2’s strategy (q) from the requirement that Player 1 randomizes.
Player 1 will only randomize if his pure strategies have equal expected payo↵. This equality
of expected payo↵s for Player 1 gives us one equation which we can solve to determine q.
The same process gives an equation to solve for determining Player 2’s strategy p. If both
of the obtained values p and q are strictly between 0 and 1, and are thus legitimate mixed
strategies, then we have a mixed-strategy equilibrium.
Thus far, our examples of mixed-strategy equilibria have been restricted to games with
an attack-defense structure, and so we have not seen an example exhibiting both pure and
mixed equilibria. However, it is not hard to find such examples: in particular, Coordination
and Hawk-Dove games with two pure equilibria will have a third mixed equilibrium in which
each player randomizes. As an example, let’s consider the Unbalanced Coordination Game
from Section 6.5:
You
Your Partner PowerPoint Keynote
PowerPoint 1, 1 0, 0 Keynote 0, 0 2, 2
Figure 6.17: Unbalanced Coordination Game
Suppose that you place a probability of p strictly between 0 and 1 on PowerPoint, and
your partner places a probability of q strictly between 0 and 1 on PowerPoint. Then you’ll
be indi↵erent between PowerPoint and Keynote if
(1)(q) + (0)(1 � q) = (0)(q) + (2)(1 � q),
or in other words, if q = 2/3. Since the situation is symmetric from your partner’s point
of view, we also get p = 2/3. Thus, in addition to the two pure equilibria, we also get an
184 CHAPTER 6. GAMES
equilibrium in which each of you chooses PowerPoint with probability 2/3. Note that unlike
the two pure equilibria, this mixed equilibrium comes with a positive probability that the
two of you will miscoordinate; but this is still an equilibrium, since if you truly believe that
your partner is choosing PowerPoint with probability 2/3 and Keynote with probability 1/3,
then you’ll be indi↵erent between the two options, and will get the same expected payo↵
however you choose.
6.9 Pareto-Optimality and Social Optimality
In a Nash equilibrium, each player’s strategy is a best response to the other player’s strategies.
In other words, the players are optimizing individually. But this doesn’t mean that, as a
group, the players will necessarily reach an outcome that is in any sense good. The Exam-or-
Presentation Game from the opening section, and related games like the Prisoner’s Dilemma,
serve as examples of this. (We redraw the payo↵ matrix for the basic Exam-or-Presentation
Game in Figure 6.18.)
You
Your Partner Presentation Exam
Presentation 90, 90 86, 92 Exam 92, 86 88, 88
Figure 6.18: Exam or Presentation?
It is interesting to classify outcomes in a game not just by their strategic or equilibrium
properties, but also by whether they are “good for society.” In order to reason about this
latter issue, we first need a way of making it precise. There are two useful candidates for
such a definition, as we now discuss.
Pareto-Optimality. The first definition is Pareto-optimality, named after the Italian
economist Vilfredo Pareto who worked in the late 1800’s and early 1900’s.
A choice of strategies — one by each player — is Pareto-optimal if there is no
other choice of strategies in which all players receive payo↵s at least as high, and
at least one player receives a strictly higher payo↵.
To see the intuitive appeal of Pareto-optimality, let’s consider a choice of strategies that is
not Pareto-optimal. In this case, there’s an alternate choice of strategies that makes at least
one player better o↵ without harming any player. In basically any reasonable sense, this
alternate choice is superior to what’s currently being played. If the players could jointly
agree on what to do, and make this agreement binding, then surely they would prefer to
move to this superior choice of strategies.
6.9. PARETO-OPTIMALITY AND SOCIAL OPTIMALITY 185
The motivation here relies crucially on the idea that the players can construct a binding
agreement to actually play the superior choice of strategies: if this alternate choice is not a
Nash equilibrium, then absent a binding agreement, at least one player would want to switch
to a di↵erent strategy. As an illustration of why this is a crucial point, consider the outcomes
in the Exam-or-Presentation Game. The outcome in which you and your partner both study
for the exam is not Pareto-optimal, since the outcome in which you both prepare for the
presentation is strictly better for both of you. This is the central di�culty at the heart of
this example, now phrased in terms of Pareto-optimality. It shows that even though you
and your partner realize there is a superior solution, there is no way to maintain it without
a binding agreement between the two of you.
In this example, the two outcomes in which exactly one of you prepares for the pre-
sentation are also Pareto-optimal. In this case, although one of you is doing badly, there
is no alternate choice of strategies in which everyone is doing at least as well. So in fact,
the Exam-or-Presentation Game – and the Prisoner’s Dilemma — are examples of games in
which the only outcome that is not Pareto-optimal is the one corresponding to the unique
Nash equilibrium.
Social Optimality. A stronger condition that is even simpler to state is social optimality.
A choice of strategies — one by each player — is a social welfare maximizer (or
socially optimal) if it maximizes the sum of the players’ payo↵s.
In the Exam-or-Presentation Game, the social optimum is achieved by the outcome in which
both you and your partner prepare for the presentation, which produces a combined payo↵
of 90 + 90 = 180. Of course, this definition is only appropriate to the extent that it makes
sense to add the payo↵s of di↵erent players together — it’s not always clear that we can
meaningfully combine my satisfaction with an outcome and your satisfaction by simply
adding them up.
Outcomes that are socially optimal must also be Pareto-optimal: if such an outcome
weren’t Pareto-optimal, there would be a di↵erent outcome in which all payo↵s were at least
as large, and one was larger — and this would be an outcome with a larger sum of payo↵s.
On the other hand, a Pareto-optimal outcome need not be socially optimal. For example,
the Exam-or-Presentation Game has three outcomes that are Pareto-optimal, but only one
of these is the social optimum.
Finally, of course, it’s not the case that Nash equilibria are at odds with the goal of social
optimality in every game. For example, in the version of the Exam-or-Presentation Game
with an easier exam, yielding the payo↵ matrix that we saw earlier in Figure 6.4, the unique
Nash equilibrium is also the unique social optimum.
186 CHAPTER 6. GAMES
6.10 Advanced Material: Dominated Strategies and Dynamic Games
In this final section, we consider two further issues that arise in the analysis of games. First,
we study the role of dominated strategies in reasoning about behavior in a game, and find
that the analysis of dominated strategies can provide a way to make predictions about play
based on rationality, even when no player has a dominant strategy. Second, we discuss how
to reinterpret the strategies and payo↵s in a game to deal with situations in which play
actually occurs sequentially through time.
Before doing this, however, we begin with a formal definition for games that have more
than two players.
A. Multi-Player Games
A (multi-player) game consists, as in the two-player case, of a set of players, a set of strategies
for each player, and a payo↵ to each player for each possible outcome.
Specifically, suppose that a game has n players named 1, 2, . . . , n. Each player has a set
of possible strategies. An outcome (or joint strategy) of the game is a choice of a strategy for
each player. Finally, each player i has a payo↵ function P i
that maps outcomes of the game
to a numerical payo↵ for i: that is, for each outcome consisting of strategies (S 1
, S 2
, . . . , S n
),
there is a payo↵ P i
(S 1
, S 2
, . . . , S n
) to player i.
Now, we can say that a strategy S i
is a best response by Player i to a choice of strategies
(S 1
, S 2
, . . . , S i�1, Si+1, . . . , Sn) by all the other players if
P i
(S 1
, S 2
, . . . , S i�1, Si, Si+1, . . . , Sn) � Pi(S1, S2, . . . , Si�1, S0
i
, S i+1
, . . . , S n
)
for all other possible strategies S0 i
available to player i.
Finally, an outcome consisting of strategies (S 1
, S 2
, . . . , S n
) is a Nash equilibrium if each
strategy it contains is a best response to all the others.
B. Dominated Strategies and their Role in Strategic Reasoning
In Sections 6.2 and 6.3, we discussed (strictly) dominant strategies — strategies that are a
(strict) best response to every possible choice of strategies by the other players. Clearly if a
player has a strictly dominant strategy then this is the strategy she should employ. But we
also saw that even for two-player, two-strategy games, it is common to have no dominant
strategies. This holds even more strongly for larger games: although dominant and strictly
dominant strategies can exist in games with many players and many strategies, they are
rare.
6.10. ADVANCED MATERIAL: DOMINATED STRATEGIES AND DYNAMIC GAMES187
A B C D E F
Figure 6.19: In the Facility Location Game, each player has strictly dominated strategies but no dominant strategy.
However, even if a player does not have a dominant strategy, she may still have strategies
that are dominated by other strategies. In this section, we consider the role that such
dominated strategies play in reasoning about behavior in games.
We begin with a formal definition: a strategy is strictly dominated if there is some other
strategy available to the same player that produces a strictly higher payo↵ in response to
every choice of strategies by the other players. In the notation we’ve just developed, strategy
S i
for player i is strictly dominated if there is another strategy S0 i
for player i such that
P i
(S 1
, S 2
, . . . , S i�1, S
0 i
, S i+1
, . . . , S n
) > P i
(S 1
, S 2
, . . . , S i�1, Si, Si+1, . . . , Sn)
for all choices of strategies (S 1
, S 2
, . . . , S i�1, Si+1, . . . , Sn) by the other players.
Now, in the two-player, two-strategy games we’ve been considering thus far, a strategy is
strictly dominated precisely when the other strategy available to the same player is strictly
dominant. In this context, it wouldn’t make sense to study strictly dominated strategies
as a separate concept on their own. However, if a player has many strategies, then it’s
possible for a strategy to be strictly dominated without any strategy being dominant. In
such cases, we will find that strictly dominated strategies can play a very useful role in
reasoning about play in a game. In particular, we will see that there are cases in which
there are no dominant strategies, but where the outcome of the game can still be uniquely
predicted using the structure of the dominated strategies. In this way, reasoning based on
dominated strategies forms an intriguing intermediate approach between dominant strategies
and Nash equilibrium: on the one hand, it can be more powerful than reasoning based solely
on dominant strategies; but on the other hand, it still relies only on the premise that players
seek to maximize payo↵s, and doesn’t require the introduction of an equilibrium notion.
To see how this works, it’s useful to introduce the approach in the context of a basic
example.
Example: The Facility Location Game. Our example is a game in which two firms
compete through their choice of locations. Suppose that two firms are each planning to open
a store in one of six towns located along six consecutive exits on a highway. We can represent
the arrangement of these towns using a six-node graph as in Figure 6.19.
188 CHAPTER 6. GAMES
Now, based on leasing agreements, Firm 1 has the option of opening its store in any of
towns A, C, or E, while Firm 2 has the option of opening its store in any of towns B, D,
or F . These decisions will be executed simultaneously. Once the two stores are opened,
customers from the towns will go to the store that is closer to them. So for example, if Firm
1 open its store in town C and Firm 2 opens its store in town B, then the store in town B will
attract customers from A and B, while the store in town C will attract customers from C,
D, E, and F . If we assume that the towns contain an equal number of customers, and that
payo↵s are directly proportional to the number of customers, this would result in a payo↵
of 4 for Firm 1 and 2 for Firm 2, since Firm 1 claims customers from 4 towns while Firm 2
claims customers from the remaining 2 towns. Reasoning in this way about the number of
towns claimed by each store, based on proximity to their locations, we get the payo↵ matrix
shown in Figure 6.20.
Firm 1
Firm 2 B D F
A 1, 5 2, 4 3, 3 C 4, 2 3, 3 4, 2 E 3, 3 2, 4 5, 1
Figure 6.20: Facility Location Game
We refer to this as a Facility Location Game. The competitive location of facilities is a
topic that has been the subject of considerable study in operations research and other areas
[135]. Moreover, closely related models have been used when the entities being “located” are
not stores along a one-dimensional highway but the positions of political candidates along
a one-dimensional ideological spectrum — here too, choosing a certain position relative to
one’s electoral opponent can attract certain voters while alienating others [350]. We will
return to issues related to political competition, though in a slightly di↵erent direction, in
Chapter 23.
We can verify that neither player has a dominant strategy in this game: for example,
if Firm 1 locates at node A, then the strict best response of Firm 2 is B, while if Firm 1
locates at node E, then the strict best response of Firm 2 is D. The situation is symmetric
if we interchange the roles of the two firms (and read the graph from the other direction).
Dominated Strategies in the Facility Location Game. We can make progress in
reasoning about the behavior of the two players in the Facility Location Game by thinking
about their dominated strategies. First, notice that A is a strictly dominated strategy for
Firm 1: in any situation where Firm 1 has the option of choosing A, it would receive a
strictly higher payo↵ by choosing C. Similarly, F is a strictly dominated strategy for Firm
2: in any situation where Firm 1 has the option of choosing F , it would receive a strictly
higher payo↵ by choosing D.
6.10. ADVANCED MATERIAL: DOMINATED STRATEGIES AND DYNAMIC GAMES189
It is never in a player’s interest to use a strictly dominated strategy, since it should
always be replaced by a strategy that does better. So Firm 1 isn’t going to use strategy A.
Moreover, since Firm 2 knows the structure of the game, including Firm 1’s payo↵s, Firm
2 knows that Firm 1 won’t use strategy A. It can be e↵ectively eliminated from the game.
The same reasoning shows that F can be eliminated from the game.
We now have a smaller instance of the Facility Location Game, involving only the four
nodes B, C, D, and E, and the payo↵ matrix shown in Figure 6.21.
Firm 1
Firm 2 B D
C 4, 2 3, 3 E 3, 3 2, 4
Figure 6.21: Smaller Facility Location Game
Now something interesting happens. The strategies B and E weren’t previously strictly
dominated: they were useful in case the other player used A or F respectively. But with
A and F eliminated, the strategies B and E now are strictly dominated — so by the same
reasoning, both players know they won’t be used, and so we can eliminate them from the
game. This gives us the even smaller game shown in Figure 6.22.
Firm 1
Firm 2 D
C 3, 3
Figure 6.22: Even smaller Facility Location Game
At this point, there is a very clear prediction for the play of the game: Firm 1 will play
C, and Firm 2 will play D. And the reasoning that led to this is clear: after repeatedly
removing strategies that were (or became) strictly dominated, we were left with only a single
plausible option for each player.
The process that led us to this reduced game is called the iterative deletion of strictly
dominated strategies, and we will shortly describe it in its full generality. Before doing this,
however, it’s worth making some observations about the example of the Facility Location
Game.
First, the pair of strategies (C, D) is indeed the unique Nash equilibrium in the game,
and when we discuss the iterated deletion of strictly dominated strategies in general, we will
see that it is an e↵ective way to search for Nash equilibria. But beyond this, it is also an
e↵ective way to justify the Nash equilibria that one finds. When we first introduced Nash
equilibrium, we observed that it couldn’t be derived purely from an assumption of rationality
on the part of the players; rather, we had to assume further that play of the game would
be found at an equilibrium from which neither player had an incentive to deviate. On the
190 CHAPTER 6. GAMES
other hand, when a unique Nash equilibrium emerges from the iterated deletion of strictly
dominated strategies, it is in fact a prediction made purely based on the assumptions of the
players’ rationality and their knowledge of the game, since all the steps that led to it were
based simply on removing strategies that were strictly inferior to others from the perspective
of payo↵-maximization.
A final observation is that iterated deletion can in principle be carried out for a very
large number of steps, and the Facility Location Game illustrates this. Suppose that instead
of a path of length six, we had a path of length 1000, with the options for the two firms
still strictly alternating along this path (constituting 500 possible strategies for each player).
Then it would be still be the case that only the outer two nodes would be strictly dominated;
after their removal, we’d have a path of length 998 in which the two new outer nodes had
now become strictly dominated. We can continue removing nodes in this way, and after 499
steps of such reasoning, we’ll have a game in which only the 500th and 501st nodes have
survived as strategies. This is the unique Nash equilibrium for the game, and this unique
prediction can be justified by a very long sequence of deletions of dominated strategies.
It’s also interesting how this prediction is intuitively natural, and one that is often seen in
real life: two competing stores staking out positions next to each other near the center of the
population, or two political candidates gravitating toward the ideological middle ground as
they compete for voters in a general election. In each case, this move toward the center is the
unique way to maximize the territory that you can claim at the expense of your competitor.
Iterated Deletion of Dominated Strategies: The General Principle. In general,
for a game with an arbitrary number of players, the process of iterated deletion of strictly
dominated strategies proceeds as follows.
• We start with any n-player game, find all the strictly dominated strategies, and delete them.
• We then consider the reduced game in which these strategies have been removed. In this reduced game there may be strategies that are now strictly dominated, despite not
having been strictly dominated in the full game. We find these strategies and delete
them.
• We continue this process, repeatedly finding and removing strictly dominated strategies until none can be found.
An important general fact is that the set of Nash equilibria of the original game coincides
with the set of Nash equilibria for the final reduced game, consisting only of strategies
that survive iterated deletion. To prove this fact, it is enough to show that the set of
Nash equilibria does not change when we perform one round of deleting strictly dominated
6.10. ADVANCED MATERIAL: DOMINATED STRATEGIES AND DYNAMIC GAMES191
strategies; if this is true, then we have established that the Nash equilibria continue to remain
unchanged through an arbitrary finite sequence of deletions.
To prove that the set of Nash equilibria remains the same through one round of deletion,
we need to show two things. First, any Nash equilibrium of the original game is a Nash
equilibrium of the reduced game. To see this, note that otherwise there would be a Nash
equilibrium of the original game involving a strategy S that was deleted. But in this case,
S is strictly dominated by some other strategy S0. Hence S cannot be part of a Nash
equilibrium of the original game: it is not a best response to the strategies of the other
players, since the strategy S0 that dominates it is a better response. This establishes that
no Nash equilibrium of the original game can be removed by the deletion process. Second,
we need to show that any Nash equilibrium of the reduced game is also a Nash equilibrium
of the original game. In order for this not to be the case, there would have to be a Nash
equilibrium E = (S 1
, S 2
, . . . , S n
) of the reduced game, and a strategy S0 i
that was deleted
from the original game, such that player i has an incentive to deviate from its strategy S i
in
E to the strategy S0 i
. But strategy S0 i
was deleted because it was strictly dominated by at
least one other strategy; we can therefore find a strategy S00 i
that strictly dominated it and
was not deleted. Then player i also has an incentive to deviate from S i
to S00 i
, and S00 i
is still
present in the reduced game, contradicting our assumption that E is a Nash equilibrium of
the reduced game.
This establishes that the game we end up with, after iterated deletion of strictly domi-
nated strategies, still has all the Nash equilibria of the original game. Hence, this process
can be a powerful way to restrict the search for Nash equilibria. Moreover, although we
described the process as operating in rounds, with all currently strictly dominated strategies
being removed in each round, this is not essential. One can show that eliminating strictly
dominated strategies in any order will result in the same set of surviving strategies.
Weakly Dominated Strategies. It is also natural to ask about notions that are slightly
weaker than our definition of strictly dominated strategies. One fundamental definition in
this spirit is that of a weakly dominated strategy. We say that a strategy is weakly dominated
if there is another strategy that does at least as well no matter what the other players do,
and does strictly better against some joint strategy of the other players. In our notation
from earlier, we say that a strategy S i
for player i is weakly dominated if there is another
strategy S0 i
for player i such that
P i
(S 1
, S 2
, . . . , S i�1, S
0 i
, S i+1
, . . . , S n
) � P i
(S 1
, S 2
, . . . , S i�1, Si, Si+1, . . . , Sn)
for all choices of strategies (S 1
, S 2
, . . . , S i�1, Si+1, . . . , Sn) by the other players, and
P i
(S 1
, S 2
, . . . , S i�1, S
0 i
, S i+1
, . . . , S n
) > P i
(S 1
, S 2
, . . . , S i�1, Si, Si+1, . . . , Sn)
for at least one choice of strategies (S 1
, S 2
, . . . , S i�1, Si+1, . . . , Sn) by the other players.
192 CHAPTER 6. GAMES
For strictly dominated strategies, the argument for deleting them was compelling: they
are never best responses. For weakly dominated strategies, the issue is more subtle. Such
strategies could be best responses to some joint strategy by the other players. So a rational
player could play a weakly dominated strategy, and in fact Nash equilibria can involve weakly
dominated strategies.
There are simple examples that make this clear even in two-player, two-strategy games.
Consider for example a version of the Stag Hunt Game in which the payo↵ from successfully
catching a stag is the same as the payo↵ from catching a hare:
Hunter 1
Hunter 2 Hunt Stag Hunt Hare
Hunt Stag 3, 3 0, 3 Hunt Hare 3, 0 3, 3
Figure 6.23: Stag Hunt: A version with a weakly dominated strategy
In this case, Hunt Stag is a weakly dominated strategy, since each player always does
at least as well, and sometimes strictly better, by playing Hunt Hare. Nevertheless, the
outcome in which both players choose Hunt Stag is a Nash equilibrium, since each is playing
a best response to the other’s strategy. Thus, deleting weakly dominated strategies is not in
general a safe thing to do, if one wants to preserve the essential structure of the game: such
deletion operations can destroy Nash equilibria.
Of course, it might seem reasonable to suppose that a player should not play an equilib-
rium involving a weakly dominated strategy (such as (Hunt Stag, Hunt Stag)) if he had any
uncertainty about what the other players would do — after all, why not use an alternate
strategy that is at least as good in every eventuality? But Nash equilibrium does not take
into account this idea of uncertainty about the behavior of others, and hence has no way to
rule out such outcomes. In the next chapter, we will discuss an alternate equilibrium concept
known as evolutionary stability that in fact does eliminate weakly dominated strategies in
a principled way. The relationship between Nash equilibrium, evolutionary stability and
weakly dominated strategies is considered in the exercises at the end of Chapter 7.
C. Dynamic Games
Our focus in this chapter has been on games in which all players choose their strategies
simultaneously, and then receive payo↵s based on this joint decision. Of course, actual
simultaneity is not crucial for the model, but it has been central to our discussions so far
that each player is choosing a strategy without knowledge of the actual choices made by the
other players.
Many games, however, are played over time: some player or set of players moves first,
other players observe the choice(s) made, and then they respond, perhaps according to a
6.10. ADVANCED MATERIAL: DOMINATED STRATEGIES AND DYNAMIC GAMES193
8
4
12
6
6
12
4
2
Player 1
Player 2
A B
A B A B
Figure 6.24: A simple game in extensive form.
predetermined order of governing who moves when. Such games are called dynamic games,
and there are many basic examples: board games and card games in which players alternate
turns; negotiations, which usually involve a sequence of o↵ers and counter-o↵ers; and bidding
in an auction or pricing competing goods where participants must make decisions over time.
Here we’ll discuss an adaptation of the theory of games that incorporates this dynamic
aspect.
Normal and Extensive Forms of a Game. To begin with, specifying a dynamic game
is going to require a new kind of notation. Thus far, we’ve worked with something called
the normal-form representation of a game; this specifies the list of players, their possible
strategies, and the payo↵s arising from every possible (simultaneous) choice of strategies by
the players. (For two-player games, the payo↵ matrices we’ve seen in this chapter encode
the normal-form representation of a game in a compact way.)
To describe a dynamic game, we’re going to need a richer representation; we need to be
able to specify who moves when, what each player knows at any opportunity they have to
move, what they can do when it is their turn to move, and what the payo↵s are at the end
of the game. We refer to this specification as the extensive-form representation of the game.
194 CHAPTER 6. GAMES
Let’s start with a very simple example of a dynamic game, so that we can discuss what
its extensive-form representation looks like. As we’ll see, this game is simple enough that it
avoids some of the subtleties that arise in the analysis of dynamic games, but it is useful as
a first illustration, and we’ll proceed to a more complex second example afterward.
In our first example, we imagine that there are two firms — Firm 1 and Firm 2 — each
of whom is trying to decide whether to focus its advertising and marketing on two possible
regions, named A or B. Firm 1 gets to choose first. If Firm 2 follows Firm 1 into the same
region, then Firm 1’s “first-mover advantage” gives it 2/3 of the profit obtainable from the
market in that region, while Firm 2 will only get 1/3. If Firm 2 moves into the other region,
then each firm gets all the profit obtainable in their respective region. Finally, Region A has
twice as large a market as Region B: the total profit obtainable in region A is equal to 12,
while in Region B, it’s 6.
We write the extensive-form representation as a “game tree,” depicted in Figure 6.24.
This tree is designed to be read downward from the top. The top node represents Firm 1’s
initial move, and the two edges descending from this node represent its two options A or B.
Based on which branch is taken, this leads to a node representing Firm 2’s subsequent move.
Firm 2 can then also choose option A or B, again represented by edges descending from the
node. This leads to a terminal node representing the end of play in the game; each terminal
node is labeled with the payo↵s to the two players.
Thus, a specific play — determined by a sequence of choices by Firm 1 and Firm 2 —
corresponds to a path from the top node in the tree down to some terminal node. First
Firm 1 chooses A or B, then Firm 2 chooses A or B, and then the two players receive their
payo↵s. In a more general model of dynamic games, each node could contain an annotation
saying what information about the previous moves is known to the player currently making
a move; however, for our purposes here, we will focus on the case in which each player knows
the complete history of past moves when they go to make their current move.
Reasoning about Behavior in a Dynamic Game. As with simultaneous-move games,
we’d like to make predictions for what players will do in dynamic games. One way is to
reason from the game tree. In our current example, we can start by considering how Firm
2 will behave after each of the two possible opening moves by Firm 1. If Firm 1 chooses
A, then Firm 2 maximizes its payo↵ by choosing B, while if Firm 1 chooses B, then Firm
2 maximizes its payo↵ by choosing A. Now let’s consider Firm 1’s opening move, given
what we’ve just concluded about Firm 2’s subsequent behavior. If Firm 1 chooses A, then
it expects Firm 2 to choose B, yielding a payo↵ of 12 for Firm 1. If Firm 1 chooses B, then
it expects Firm 2 to choose A, yielding a payo↵ of 6 for Firm 1. Since we expect the firms
to try maximizing their payo↵s, we predict that Firm 1 should choose A, after which Firm
2 should choose B.
6.10. ADVANCED MATERIAL: DOMINATED STRATEGIES AND DYNAMIC GAMES195
This is a useful way to analyze dynamic games. We start one step above the terminal
nodes, where the last player to move has complete control over the outcome of the payo↵s.
This lets us predict what the last player will do in all cases. Having established this, we
then move one more level up the game tree, using these predictions to reason about what
the player one move earlier will do. We continue in this way up the tree, eventually making
predictions for play all the way up to the top node.
A di↵erent style of analysis exploits an interesting connection between normal and exten-
sive forms, allowing us to write a normal-form representation for a dynamic game as follows.
Suppose that, before the game is played, each player makes up a plan for how to play the
entire game, covering every possible eventuality. This plan will serve as the player’s strategy.
One way to think about such strategies, and a useful way to be sure that they include a
complete description of every possibility, is to imagine that each player has to provide all of
the information needed to write a computer program which will actually play the game in
their place.
For the game in Figure 6.24, Firm 1 only has two possible strategies: A or B. Since Firm
2 moves after observing what Firm 1 did, and Firm 2 has two possible choices for each of the
two options by Firm 1, Firm 2 has four possible plans for playing the game. They can be
written as contingencies, specifying what Firm 2 will do in response to each possible move
by Firm 1:
(A if A, A if B), (A if A, B if B), (B if A, A if B), and (B if A, B if B),
or in abbreviated form as
(AA, AB), (AA, BB), (BA, AB), and (BA, BB),
If each player chooses a complete plan for playing the game as its strategy, then we can
determine the payo↵s directly from this pair of chosen strategies via a payo↵ matrix.
Firm 1
Firm 2 AA, AB AA, BB BA, AB BA, BB
A 8, 4 8, 4 12, 6 12, 6 B 6, 12 4, 2 6, 12 4, 2
Figure 6.25: Conversion to normal form.
Because the plans describe everything about how a player will behave, we have managed
to describe this dynamic game in normal form: each player chooses a strategy (consisting of a
complete plan) in advance, and from this joint choice of strategies, we can determine payo↵s.
We will see later that there are some important subtleties in using this interpretation of the
underlying dynamic game, and in particular the translation from extensive to normal form
will sometimes not preserve the full structure implicit in the game. But the translation is a
196 CHAPTER 6. GAMES
useful tool for analysis, and the subtle lack of fidelity that can arise in the translation is in
itself a revealing notion to develop and explore.
With this in mind, we first finish our simple example, where the translation will work
perfectly, and then move on to a second example where the complications begin to arise.
For the normal-form payo↵ matrix corresponding to our first example, the payo↵ matrix
has eight cells, while the extensive-form representation only has four terminal nodes with
payo↵s. This occurs because each terminal node can be reached with two di↵erent pairs of
strategies, with each pair forming a cell of the payo↵ matrix. Both pairs of strategies dictate
the same actions in the path of the game tree which actually occurs, but describe di↵erent
hypothetical actions in other unrealized paths. For example, the payo↵s in the entries for
(A, (AA, AB)) and for (A, (AA, BB)) are the same because both strategy combinations lead
to the same terminal node. In both cases Firm 2 chooses A in response to what Firm 1
actually does; Firm 2’s plan for what to do in the event Firm 1 chose B is not realized by
the actual play.
Now, using the normal-form representation, we can quickly see that for Firm 1, strategy
A is strictly dominant. Firm 2 does not have a strictly dominant strategy, but it should
play a best response to Firm 1, which would be either (BA, AB) or (BA, BB). Notice that
this prediction of play by Firm 1 and Firm 2 based on the normal-form representation is the
same as our prediction based on direct analysis of the game tree, where we reasoned upward
from the terminal nodes: Firm 1 will play A, and in response Firm 2 will play B.
A More Complex Example: The Market Entry Game. In our first dynamic game,
reasoning based on the extensive and normal form representations led to essentially identical
conclusions. As games get larger, extensive forms are representationally more streamlined
than normal forms for dynamic games, but if this were the only distinction, it would be
hard to argue that dynamic games truly add much to the overall theory of games. In fact,
however, the dynamic aspect leads to new subtleties, and this can be exposed by considering
a case in which the translation from extensive form to normal form ends up obscuring some
of the structure that is implicit in the dynamic game.
For this, we consider a second example of a dynamic game, also played between two
competing firms. We call this the Market Entry Game, and it’s motivated by the following
scenario. Consider a region where Firm 2 is currently the only serious participant in a given
line of business, and Firm 1 is considering whether to enter the market.
• The first move in this game is made by Firm 1, who must decide whether to stay out of the market or enter it.
• If Firm 1 chooses to stay out, then the game ends, with Firm 1 getting a payo↵ of 0 and Firm 2 keeping the payo↵ from the entire market.
6.10. ADVANCED MATERIAL: DOMINATED STRATEGIES AND DYNAMIC GAMES197
-1 -1
1 1
Player 1
Stay Out Enter
Retaliate Cooperate 0 2
Player 2
Figure 6.26: Extensive-form representation of the Market Entry Game.
• If Firm 1 chooses to enter, then the game continues to a second move by Firm 2, who must choose whether to cooperate and divide the market evenly with Firm 1, or
retaliate and engage in a price war.
– If Firm 2 cooperates, then each firm gets a payo↵ corresponding to half the market.
– If Firm 2 retaliates, then each firm gets a negative payo↵.
Choosing numerical payo↵s to fill in this story, we can write the extensive-form representation
of the Market Entry Game as in Figure 6.26.
Subtle Distinctions Between Extensive and Normal Form Representations. Let’s
take the two ways we developed to analyze our previous dynamic game and apply them here.
First, we can work our way up the game tree starting at the terminal nodes, as follows. If
Firm 1 chooses to enter the market, then Firm 2 achieves a higher payo↵ by cooperating
than by retaliating, so we should predict cooperation in the event the game reaches this
point. Given this, when Firm 1 goes to make its first move, it can expect a payo↵ of 0 by
staying out, and a payo↵ of 1 by entering, so it should choose to enter the market. We can
therefore predict that Firm 1 will enter the market, and then Firm 2 will cooperate.
198 CHAPTER 6. GAMES
Now let’s consider the normal-form representation. Firm 1’s possible plans for playing
the game are just to choose Stay Out (S) or Enter (E). Firm 2’s possible plans are to choose
retaliation in the event of entry, or cooperation in the event of entry. We’ll denote these two
plans by R and C respectively. This gives us the payo↵ matrix in Figure 6.27.
Firm 1
Firm 2 R C
S 0, 2 0, 2 E �1,�1 1, 1
Figure 6.27: Normal Form of the Market Entry Game
Here’s the surprise: when we look at this game in normal form, we discover two distinct
(pure-strategy) Nash equilibria: (E, C), and (S, R). The first of these corresponds to the
prediction for play that we obtained by analyzing the extensive-form representation. What
does the second one correspond to?
To answer this, it helps to recall our view of the normal-form representation as capturing
the idea that each player commits in advance to a computer program that will play the game
in its place. Viewed this way, the equilibrium (S, R) corresponds to an outcome in which
Firm 2 commits in advance to a computer program that will automatically retaliate in the
event that Firm 1 enters the market. Firm 1, meanwhile, commits to a program that stays
out of the market. Given this pair of choices, neither firm has an incentive to change the
computer program they’re using: for example, if Firm 1 were to switch to a program that
entered the market, it would trigger retaliation by the program that Firm 2 is using.
This contrast between the prediction from the extensive and normal forms highlights some
important points. First, it shows that the premise behind our translation from extensive to
normal form — that each player commits ahead of time to a complete plan for playing the
game — is not really equivalent to our initial premise in defining dynamic games — namely,
that each player makes an optimal decision at each intermediate point in the game, based
on what has already happened up to that point. Firm 2’s decision to retaliate on entry
highlights this clearly. If Firm 2 can truly pre-commit to this plan, then the equilibrium
(S, R) makes sense, since Firm 1 will not want to provoke the retaliation that is encoded in
Firm 2’s plan. But if we take the dynamic game as originally defined in extensive form, then
pre-commitment to a plan is not part of the model: rather, Firm 2 only gets to evaluate its
decision to cooperate or retaliate once Firm 1 has already entered the market, and at that
point its payo↵ is better if it cooperates. Given this, Firm 1 can predict that it is safe to
enter.
In game theory, the standard model for dynamic games in extensive form assumes that
players will seek to maximize their payo↵ at any intermediate stage of play that can be
reached in the game. In this interpretation, there is a unique prediction for play in our
6.10. ADVANCED MATERIAL: DOMINATED STRATEGIES AND DYNAMIC GAMES199
Market Entry Game, corresponding to the equilibrium (E, C) in normal form. However, the
issues surrounding the other equilibrium (S, R) are not simply notational or representational;
they are deeper than this. For any given scenario, it is really a question of what we believe
is being modeled by the underlying dynamic game in extensive form. It is a question of
whether we are in a setting where a player can irrevocably pre-commit to a certain plan, to
the extent that other players will believe the commitment as a credible threat — or not.
Further, the Market Entry Game shows how the ability to commit to a particular course
of action — when possible — can in fact be a valuable thing for an individual player, even if
that course of action would be bad for everyone if it were actually carried out. In particular,
if Firm 2 could make Firm 1 believe that it really would retaliate in the event of entry, then
Firm 1 would choose to stay out, resulting in a higher payo↵ for Firm 2. In practice, this
suggests particular courses of action that Firm 2 could take before the game even starts. For
example, suppose that before Firm 1 had decided whether to enter the market, Firm 2 were
to publically advertise an o↵er to beat any competitor’s price by 10%. This would be a safe
thing to do as long as Firm 2 is the only serious participant in the market, but it becomes
dangerous to both firms if Firm 1 actually enters. The fact that the plan has been publically
announced means that it would be very costly (reputationally, and possibly legally) for Firm
2 to back away from it. In this way, the announcement can serve as a way of switching the
underlying model from one in which Firm 2’s threat to retaliate is not credible to one in
which Firm 2 can actually pre-commit to a plan for retaliation.
Relationship to Weakly Dominated Strategies. In discussing these distinctions, it is
also interesting to note the role that weakly dominated strategies play here. Notice that in the
normal-form representation in Figure 6.27, the strategy R for Firm 2 is weakly dominated,
and for a simple reason: it yields the same payo↵ if Firm 1 chooses S (since then Firm
2 doesn’t actually get to move), and it yields a lower payo↵ if Firm 1 chooses E. So
our translation from extensive form to normal form for dynamic games provides another
reason to be careful about predictions of play in a normal-form game that rely on weakly
dominated strategies: if the structure in fact arises from a dynamic game in extensive form,
then information about the dynamic game that is lost in the translation to normal form
could potentially be su�cient to eliminate such equilibria.
However, we can’t simply fix up the translation by eliminating weakly dominated strate-
gies. We saw earlier that iterated deletion of strictly dominated strategies can be done in any
order: all orders yield the same final result. But this is not true for the iterated deletion of
weakly dominated strategies. To see this, suppose we vary the Market Entry Game slightly
so that the payo↵ from the joint strategy (E, C) is (0, 0). (In this version, both firms know
they will fail to gain a positive payo↵ even if Firm 2 cooperates on entry, although they still
don’t do as badly as when Firm 2 retaliates.) R is a weakly dominated strategy as before,
200 CHAPTER 6. GAMES
but now so is E. (E and S produce the same payo↵ for Firm 1 when Firm 2 chooses C, and
S produces a strictly higher payo↵ when Firm 2 chooses R.)
In this version of the game, there are now three (pure-strategy) Nash equilibria: (S, C),
(E, C), and (S, R). If we first eliminate the weakly dominated strategy R, then we are left
with (S, C) and (E, C) as equilibria. Alternately, if we first eliminate the weakly dominated
strategy E, then we are left with (S, C) and (S, R) as equilibria. In both cases, no further
elimination of weakly dominated strategies is possible, so the order of deletion a↵ects the
final set of equilibria. We can ask which of these equilibria actually make sense as predictions
of play in this game. If this normal form actually arose from the dynamic version of the
Market Entry Game, then C is still the only reasonable strategy for Firm 2, while Firm 1
could now play either S or E.
Final Comments. The analysis framework we developed for most of this chapter is based
on games in normal form. One approach to analyzing dynamic games in extensive form is
to first find all Nash equilibria of the translation to normal form, treating each of these as a
candidate prediction of play in the dynamic game, and then go back to the extensive-form
version to see which of these make sense as actual predictions.
There is an alternate theory that works directly with the extensive-form representation.
The simplest technique used in this theory is the style of analysis we employed to analyze an
extensive-form representation from the terminal nodes upward. But there are more complex
components to the theory as well, allowing for richer structure such as the possibility that
players at any given point have only partial information about the history of play up to that
point. While we will not go further into this theory here, it is developed in a number of
books on game theory and microeconomic theory [263, 288, 336, 398].
6.11 Exercises
1. Say whether the following claim is true or false, and provide a brief (1-3 sentence)
explanation for your answer.
Claim: If player A in a two-person game has a dominant strategy s A
, then
there is a pure strategy Nash equilibrium in which player A plays s A
and
player B plays a best response to s A
.
2. Consider the following statement:
In a Nash equilibrium of a two-player game each player is playing an optimal
strategy, so the two player’s strategies are social-welfare maximizing.
6.11. EXERCISES 201
Is this statement correct or incorrect? If you think it is correct, give a brief (1-3
sentence) explanation for why. If you think it is incorrect, give an example of a game
discussed in Chapter 6 that shows it to be incorrect (you do not need to spell out all
the details of the game, provided you make it clear what you are referring to), together
with a brief (1-3 sentence) explanation.
3. Find all pure strategy Nash equilibria in the game below. In the payo↵ matrix below
the rows correspond to player A’s strategies and the columns correspond to player B’s
strategies. The first entry in each box is player A’s payo↵ and the second entry is
player B’s payo↵.
Player A
Player B L R
U 1, 2 3, 2 D 2, 4 0, 2
4. Consider the two-player game with players, strategies and payo↵s described in the
following game matrix.
Player A
Player B L M R
t 0, 3 6, 2 1, 1 m 2, 3 0, 1 7, 0 b 5, 3 4, 2 3, 1
Figure 6.28: Payo↵ Matrix
(a) Does either player have a dominant strategy? Explain briefly (1-3 sentences).
(b) Find all pure strategy Nash equilibria for this game.
5. Consider the following two-player game in which each player has three strategies.
Player A
Player B L M R
U 1, 1 2, 3 1, 6 M 3, 4 5, 5 2, 2 D 1, 10 4, 7 0, 4
Find all the (pure strategy) Nash equilibria for this game.
6. In this question we will consider several two-player games. In each payo↵ matrix below
the rows correspond to player A’s strategies and the columns correspond to player B’s
202 CHAPTER 6. GAMES
strategies. The first entry in each box is player A’s payo↵ and the second entry is
player B’s payo↵.
(a) Find all pure (non-randomized) strategy Nash equilibria for the game described
by the payo↵ matrix below.
Player A
Player B L R
U 2, 15 4, 20 D 6, 6 10, 8
(b) Find all pure (non-randomized) strategy Nash equilibria for the game described
by the payo↵ matrix below.
Player A
Player B L R
U 3, 5 4, 3 D 2, 1 1, 6
(c) Find all Nash equilibria for the game described by the payo↵ matrix below.
Player A
Player B L R
U 1, 1 4, 2 D 3, 3 2, 2
[Hint: This game has a both pure strategy equilibria and a mixed strategy equilibrium.
To find the mixed strategy equilibrium let the probability that player A uses strategy
U be p and the probability that player B uses strategy L be q. As we learned in our
analysis of matching pennies, if a player uses a mixed strategy (one that is not really
just some pure strategy played with probability one) then the player must be indi↵erent
between two pure strategies. That is the strategies must have equal expected payo↵s.
So, for example, if p is not 0 or 1 then it must be the case that q+4(1�q) = 3q+2(1�q) as these are the expected payo↵s to player A from U and D when player B uses
probability q.]
7. In this question we will consider several two-player games. In each payo↵ matrix below
the rows correspond to player A’s strategies and the columns correspond to player B’s
strategies. The first entry in each box is player A’s payo↵ and the second entry is
player B’s payo↵.
(a) Find all Nash equilibria for the game described by the payo↵ matrix below.
6.11. EXERCISES 203
Player A
Player B L R
U 1, 1 3, 2 D 0, 3 4, 4
Player A
Player B L R
U 5, 6 0, 10 D 4, 4 2, 2
(b) Find all Nash equilibria for the game described by the payo↵ matrix below (include
an explanation for your answer).
[Hint: This game has a mixed strategy equilibrium. To find the equilibrium let the
probability that player A uses strategy U be p and the probability that player B uses
strategy L be q. As we learned in our analysis of matching pennies, if a player uses a
mixed strategy (one that is not really just some pure strategy played with probability
one) then the player must be indi↵erent between two pure strategies. That is, the
strategies must have equal expected payo↵s. So, for example, if p is not 0 or 1 then it
must be the case that 5q + 0(1 � q) = 4q + 2(1 � q) as these are the expected payo↵s to player A from U and D when player B uses probability q.]
8. Consider the two-player game described by the payo↵ matrix below.
Player A
Player B L R
U 1, 1 0, 0 D 0, 0 4, 4
(a) Find all pure-strategy Nash equilibria for this game.
(b) This game also has a mixed-strategy Nash equilibrium; find the probabilities the
players use in this equilibrium, together with an explanation for your answer.
(c) Keeping in mind Schelling’s focal point idea from Chapter 6, what equilibrium do
you think is the best prediction of how the game will be played? Explain.
9. For each of the following two player games find all Nash equilibria. In each payo↵
matrix below the rows correspond to player A’s strategies and the columns correspond
to player B’s strategies. The first entry in each box is player A’s payo↵ and the second
entry is player B’s payo↵.
204 CHAPTER 6. GAMES
(a)
Player A
Player B
L R
U 8, 4 5, 5
D 3, 3 4, 8
(b)
Player A
Player B
L R
U 0, 0 �1, 1 D �1, 1 2,�2
10. In the payo↵ matrix below the rows correspond to player A’s strategies and the columns
correspond to player B’s strategies. The first entry in each box is player A’s payo↵ and
the second entry is player B’s payo↵.
Player A
Player B L R
U 3, 3 1, 2 D 2, 1 3, 0
(a) Find all pure strategy Nash equilibria of this game.
(b) Notice from the payo↵ matrix above that Player A’s payo↵ from the pair of
strategies (U, L) is 3. Can you change player A’s payo↵ from this pair of strate-
gies to some non-negative number in such a way that the resulting game has no
pure-strategy Nash equilibrium? Give a brief (1-3 sentence) explanation for your
answer.
(Note that in answering this question, you should only change Player A’s payo↵
for this one pair of strategies (U, L). In particular, leave the rest of the structure
of the game unchanged: the players, their strategies, the payo↵ from strategies
other than (U, L), and B’s payo↵ from (U, L).)
(c) Now let’s go back to the original payo↵ matrix from part (a) and ask an analogous
question about player B. So we’re back to the payo↵ matrix in which players A
and B each get a payo↵ of 3 from the pair of strategies (U, L).
Can you change player B’s payo↵ from the pair of strategies (U, L) to some non-
negative number in such a way that the resulting game has no pure-strategy Nash
equilibrium? Give a brief (1-3 sentence) explanation for your answer.
(Again, in answering this question, you should only change Player B’s payo↵ for
this one pair of strategies (U, L). In particular, leave the rest of the structure of
the game unchanged: the players, their strategies, the payo↵ from strategies other
than (U, L), and A’s payo↵ from (U, L).)
6.11. EXERCISES 205
11. In the text we’ve discussed dominant strategies and noted that if a player has a domi-
nant strategy we would expect it to be used. The opposite of a dominant strategy is
a strategy that is dominated. The definition of dominated is:
A strategy s⇤ i
is dominated if player i has another strategy s0 i
with the property that
player i’s payo↵ is greater from s0 i
than from s⇤ i
no matter what the other players in
the game do.
We do not expect a player to use a strategy that is dominated and this can help in
finding Nash equilibria. Here is an example of this idea. In this game, M is a dominated
strategy (it is dominated by R) and player B will not use it.
Player A
Player B L M R
U 2, 4 2, 1 3, 2 D 1, 2 3, 3 2, 4
So in analyzing the game we can delete M and look at the remaining game
Player A
Player B L R
U 2, 4 3, 2 D 1, 2 2, 4
Now player A has a dominant strategy (U) and it is easy to see that the Nash equilib-
rium of the 2-by-2 game is (U,L). You can check the original game to see that (U,L)
is a Nash equilibrium. Of course, using this procedure requires that we know that a
dominated strategy cannot be used in Nash equilibrium.5
Consider any two player game which has at least one (pure strategy) Nash equilibrium.
Explain why the strategies used in an equilibrium of this game will not be dominated
strategies.
12. In Chapter 6 we discussed dominant strategies and noted that if a player has a dominant
strategy we would expect it to be used. The opposite of a dominant strategy is a
strategy that is dominated. There are several possible notions of what it means for a
strategy to be dominated. In this problem we will focus on weak domination.
A strategy s⇤ i
is weakly dominated if player i has another strategy s0 i
with the property
that:
5This is actually true for any number of players. It would also help to know that if we iteratively remove dominated strategies (in any order) and analyze the reduced games we still find the Nash equilibria of the original game. This is also true, but it is a bit more complicated.
206 CHAPTER 6. GAMES
(a) No matter what the other player does, player i’s payo↵ from s0 i
is at least as large
as the payo↵ from s⇤ i
, and
(b) There is some strategy for the other player so that player i’s payo↵ from s0 i
is
strictly greater than the payo↵ from s⇤ i
.
(a) It seems unlikely that a player would use a weakly dominated strategy, but these
strategies can occur in a Nash equilibrium. Find all pure (non-randomized) Nash
equilibria for the game below. Do any of them use weakly dominated strategies?
Player A
Player B L R
U 1, 1 1, 1 D 0, 0 2, 1
(b) One way to reason about the weakly dominated strategies that you should have
found in answering the question above is to consider the following sequential game.
Suppose that the players actually move sequentially, but the player to move second
does not know what the player moving first chose. Player A moves first, and if he
chooses U, then player B’s choice does not matter. E↵ectively the game is over if A
chooses U as no matter what B does the payo↵ is (1, 1). If player A chooses D, then
player B’s move matters, and the payo↵ is (0, 0) if B chooses L or (2, 1) if B chooses R.
[Note that as B does not observe A’s move the simultaneous move game with payo↵
matrix above is equivalent to this sequential move game.]
In this game how would you expect the players to behave? Explain your reasoning.
[The players are not allowed to change the game. They play it once just as it is given
above. You may reason from the payo↵ matrix or the story behind the game, but if
you use the story remember that B does not observe A’s move until after the game is
over.]
13. Here we consider a game with three players, named 1, 2 and 3. To define the game
we need to specify the sets of strategies available to each player; also, when each of
the three players chooses a strategy, this gives a triple of strategies, and we need to
specify the payo↵ each player receives from any possible triple of strategies played.
Let’s suppose that player 1’s strategy set is {U, D}, players 2’s strategy set is {L, R} and player 3’s strategy set is {l, r}.
One way to specify the payo↵s would be to write down every possible triple of strategies,
and the payo↵s for each. A di↵erent but equivalent way to interpret triples of strategies,
which makes it easier to specify the payo↵s, is to imagine that player 3 chooses which
of two distinct two-player games players 1 and 2 will play. If 3 chooses l then the payo↵
matrix is
6.11. EXERCISES 207
Payo↵ Matrix l:
Player A
Player B L R
U 4, 4, 4 0, 0, 1 D 0, 2, 1 2, 1, 0
where the first entry in each cell is the payo↵ to player 1, the second entry is the payo↵
to player 2 and the third entry is the payo↵ to player 3.
If 3 chooses r then the payo↵ matrix is
Payo↵ Matrix r:
Player A
Player B L R
U 2, 0, 0 1, 1, 1 D 1, 1, 1 2, 2, 2
So, for example, if player 1 chooses U , player 2 chooses R and player 3 chooses r the
payo↵s are 1 for each player.
(a) First suppose the players all move simultaneously. That is, players 1 and 2 do not
observe which game player 3 has selected until after they each chose a strategy. Find
all of the (pure strategy) Nash equilibria for this game.
(b) Now suppose that player 3 gets to move first and that players 1 and 2 observe player
3’s move before they decide how to play. That is, if player 3 chooses the strategy r
then players 1 and 2 play the game defined by payo↵ matrix r and they both know that
they are playing this game. Similarly, if player 3 chooses the strategy l then players
1 and 2 play the game defined by payo↵ matrix l and they both know that they are
playing this game.
Let’s also suppose that if players 1 and 2 play the game defined by payo↵ matrix r
they play a (pure strategy) Nash equilibrium for that game; and similarly, if players
1 and 2 play the game defined by payo↵ matrix l they play a (pure strategy) Nash
equilibrium for that game. Finally, let’s suppose that player 3 understands that this is
how players 1 and 2 will behave.
What do you expect player 3 to do and why? What triple of strategies would you
expect to see played? Is this list of strategies a Nash equilibrium of the simultaneous
move game between the three players?
14. Consider the two-player game with players, strategies and payo↵s described in the
following game matrix.
208 CHAPTER 6. GAMES
Player 1
Player 2 L R
U 1, 1 4, 0 D 4, 0 3, 3
(a) Find all of the Nash equilibria of this game.
(b) In the mixed strategy equilibrium you found in part (a), you should notice that
player 1 plays strategy U more often than strategy D. One of your friends remarks
that your answer to part (a) must be wrong because clearly for player 1 strategy D is
a more attractive strategy than strategy U. Both U and D give player 1 a payo↵ of 4
on the o↵-diagonal elements of the payo↵ matrix, but D gives player 1 a payo↵ of 3 on
the diagonal while U only gives player 1 a payo↵ of 1 on the diagonal. Explain what
is wrong with this reasoning.
15. Two identical firms — let’s call them firm 1 and firm 2 — must decide simultaneously
and independently whether to enter a new market and what product to produce if they
do enter the market. Each firm, if it enters, can develop and produce either product
A or product B. If both firms enter and produce product A they each lose ten million
dollars. If both firms enter and both produce product B, they each make a profit of
five million dollars. If both enter and one produces A while the other produces B, then
they each make a profit of ten million dollars. Any firm that does not enter makes
a profit of zero. Finally, if one firm does not enter and the other firm produces A it
makes a profit of fifteen million dollars, while if the single entering firm produces B it
makes a profit of thirty million dollars.
You are the manager of firm 1 and you have to choose a strategy for your firm.
(a) Set this situation up as a game with two players, firms 1 and 2, and three strategies
for each firm: produce A, produce B or do not enter.
(b) One of your employees argues that you should enter the market (although he is not
sure what product you should produce) because no matter what firm 2 does, entering
and producing product B is better than not entering. Evaluate this argument.
(c) Another employee agrees with the person in part (b) and argues that as strategy A
could result in a loss (if the other firm also produces A) you should enter and produce
B. If both firms reason this way, and thus enter and produce product B, will their play
of the game form a Nash equilibrium? Explain.
(d) Find all the pure strategy Nash equilibria of this game.
(e) Another employee of your firm suggests merging the two firms and deciding co-
operatively on strategies so as to maximize the sum of profits. Ignoring whether this
merger would be allowed by the regulators do you think its a good idea? Explain.
Chapter 7
Evolutionary Game Theory
In Chapter 6, we developed the basic ideas of game theory, in which individual players
make decisions, and the payo↵ to each player depends on the decisions made by all. As we
saw there, a key question in game theory is to reason about the behavior we should expect
to see when players take part in a given game.
The discussion in Chapter 6 was based on considering how players simultaneously reason
about what the other players may do. In this chapter, on the other hand, we explore the
notion of evolutionary game theory, which shows that the basic ideas of game theory can be
applied even to situations in which no individual is overtly reasoning, or even making explicit
decisions. Rather, game-theoretic analysis will be applied to settings in which individuals can
exhibit di↵erent forms of behavior (including those that may not be the result of conscious
choices), and we will consider which forms of behavior have the ability to persist in the
population, and which forms of behavior have a tendency to be driven out by others.
As its name suggests, this approach has been applied most widely in the area of evolu-
tionary biology, the domain in which the idea was first articulated by John Maynard Smith
and G. R. Price [375, 376]. Evolutionary biology is based on the idea that an organism’s
genes largely determine its observable characteristics, and hence its fitness in a given envi-
ronment. Organisms that are more fit will tend to produce more o↵spring, causing genes
that provide greater fitness to increase their representation in the population. In this way,
fitter genes tend to win over time, because they provide higher rates of reproduction.
The key insight of evolutionary game theory is that many behaviors involve the interaction
of multiple organisms in a population, and the success of any one of these organisms depends
on how its behavior interacts with that of others. So the fitness of an individual organism
can’t be measured in isolation; rather it has to be evaluated in the context of the full
population in which it lives. This opens the door to a natural game-theoretic analogy:
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
209
210 CHAPTER 7. EVOLUTIONARY GAME THEORY
an organism’s genetically-determined characteristics and behaviors are like its strategy in a
game, its fitness is like its payo↵, and this payo↵ depends on the strategies (characteristics) of
the organisms with which it interacts. Written this way, it is hard to tell in advance whether
this will turn out to be a superficial analogy or a deep one, but in fact the connections turn
out to run very deeply: game-theoretic ideas like equilibrium will prove to be a useful way
to make predictions about the results of evolution on a population.
7.1 Fitness as a Result of Interaction
To make this concrete, we now describe a first simple example of how game-theoretic ideas
can be applied in evolutionary settings. This example will be designed for ease of explanation
rather than perfect fidelity to the underlying biology; but after this we will discuss examples
where the phenomenon at the heart of the example has been empirically observed in a variety
of natural settings.
For the example, let’s consider a particular species of beetle, and suppose that each
beetle’s fitness in a given environment is determined largely by the extent to which it can
find food and use the nutrients from the food e↵ectively. Now, suppose a particular mutation
is introduced into the population, causing beetles with the mutation to grow a significantly
larger body size. Thus, we now have two distinct kinds of beetles in the population — small
ones and large ones. It is actually di�cult for the large beetles to maintain the metabolic
requirements of their larger body size — it requires diverting more nutrients from the food
they eat — and so this has a negative e↵ect on fitness.
If this were the full story, we’d conclude that the large-body-size mutation is fitness-
decreasing, and so it will likely be driven out of the population over time, through multiple
generations. But in fact, there’s more to the story, as we’ll now see.
Interaction Among Organisms. The beetles in this population compete with each other
for food – when they come upon a food source, there’s crowding among the beetles as they
each try to get as much of the food as they can. And, not surprisingly, the beetles with large
body sizes are more e↵ective at claiming an above-average share of the food.
Let’s assume for simplicity that food competition in this population involves two beetles
interacting with each other at any given point in time. (This will make the ideas easier
to describe, but the principles we develop can also be applied to interactions among many
individuals simultaneously.) When two beetles compete for some food, we have the following
possible outcomes.
• When beetles of the same size compete, they get equal shares of the food.
• When a large beetle competes with a small beetle, the large beetle gets the majority of the food.
7.2. EVOLUTIONARILY STABLE STRATEGIES 211
• In all cases, large beetles experience less of a fitness benefit from a given quantity of food, since some of it is diverted into maintaining their expensive metabolism.
Thus, the fitness that each beetle gets from a given food-related interaction can be
thought of as a numerical payo↵ in a two-player game between a first beetle and a second
beetle, as follows. The first beetle plays one of the two strategies Small or Large, depending
on its body size, and the second beetle plays one of these two strategies as well. Based on
the two strategies used, the payo↵s to the beetles are described by Figure 7.1.
Beetle 1
Beetle 2 Small Large
Small 5, 5 1, 8 Large 8, 1 3, 3
Figure 7.1: The Body-Size Game
Notice how the numerical payo↵s satisfy the principles just outlined: when two small
beetles meet, they share the fitness from the food source equally; large beetles do well at
the expense of small beetles; but large beetles cannot extract the full amount of fitness from
the food source. (In this payo↵ matrix, the reduced fitness when two large beetles meet is
particularly pronounced, since a large beetle has to expend extra energy in competing with
another large beetle.)
This payo↵ matrix is a nice way to summarize what happens when two beetles meet,
but compared with the game in Chapter 6, there’s something fundamentally di↵erent in
what’s being described here. The beetles in this game aren’t asking themselves, “What do
I want my body size to be in this interaction?” Rather, each is genetically hard-wired to
play one of these two strategies through its whole lifetime. Given this important di↵erence,
the idea of choosing strategies — which was central to our formulation of game theory —
is missing from the biological side of the analogy. As a result, in place of the idea of Nash
equilibrium — which was based fundamentally on the relative benefit of changing one’s own
personal strategy — we will need to think about strategy changes that operate over longer
time scales, taking place as shifts in a population under evolutionary forces. We develop the
fundamental definitions for this in the next section.
7.2 Evolutionarily Stable Strategies
In Chapter 6, the notion of Nash equilibrium was central in reasoning about the outcome
of a game. In a Nash equilibrium for a two-player game, neither player has an incentive to
deviate from the strategy they are currently using — the equilibrium is a choice of strategies
that tends to persist once the players are using it. The analogous notion for evolutionary
212 CHAPTER 7. EVOLUTIONARY GAME THEORY
settings will be that of an evolutionarily stable strategy — a genetically-determined strategy
that tends to persist once it is prevalent in a population.
We formulate this as follows. Suppose, in our example, that each beetles is repeatedly
paired o↵ with other beetles in food competitions over the course of its lifetime. We will
assume the population is large enough that no two particular beetles have a significant
probability of interacting with each other repeatedly. A beetle’s overall fitness will be equal
to the average fitness it experiences from each of its many pairwise interactions with others,
and this overall fitness determines its reproductive success — the number of o↵spring that
carry its genes (and hence its strategy) into the next generation.
In this setting, we say that a given strategy is evolutionarily stable if, when the whole
population is using this strategy, any small group of invaders using a di↵erent strategy
will eventually die o↵ over multiple generations. (We can think of these invaders either
as migrants who move to join the population, or as mutants who were born with the new
behavior directly into the population.) We capture this idea in terms of numerical payo↵s
by saying that when the whole population is using a strategy S, then a small group of
invaders using any alternate strategy T should have strictly lower fitness than the users
of the majority strategy S. Since fitness translates into reproductive success, evolutionary
principles posit that strictly lower fitness is the condition that causes a sub-population (like
the users of strategy T ) to shrink over time, through multiple generations, and eventually
die o↵ with high probability.
More formally, we will phrase the basic definitions as follows.
• We say the fitness of an organism in a population is the expected payo↵ it receives from an interaction with a random member of the population.
• We say that a strategy T invades a strategy S at level x, for some small positive number x, if an x fraction of the underlying population uses T and a 1 � x fraction of the underlying population uses S.
• Finally, we say that a strategy S is evolutionarily stable if there is a (small) positive number y such that when any other strategy T invades S at any level x < y, the fitness
of an organism playing S is strictly greater than the fitness of an organism playing T .
Evolutionarily Stable Strategies in our First Example. Let’s see what happens when
we apply this definition to our example involving beetles competing for food. We will first
check whether the strategy Small is evolutionarily stable, and then we will do the same for
the strategy Large.
Following the definition, let’s suppose that for some small positive number x, a 1 � x fraction of the population uses Small and an x fraction of the population uses Large. (This
7.2. EVOLUTIONARILY STABLE STRATEGIES 213
is what the picture would look like just after a small invader population of large beetles
arrives.)
• What is the expected payo↵ to a small beetle in a random interaction in this popula- tion? With probability 1 � x, it meets another small beetle, receiving a payo↵ of 5, while with probability x, it meets a large beetle, receiving a payo↵ of 1. Therefore its
expected payo↵ is
5(1 � x) + 1 · x = 5 � 4x.
• What is the expected payo↵ to a large beetle in a random interaction in this population? With probability 1 � x, it meets a small beetle, receiving a payo↵ of 8, while with probability x, it meets another large beetle, receiving a payo↵ of 3. Therefore its
expected payo↵ is
8(1 � x) + 3 · x = 8 � 5x.
It’s easy to check that for small enough values of x (and even for reasonably large ones
in this case), the expected fitness of large beetles in this population exceeds the expected
fitness of small beetles. Therefore Small is not evolutionarily stable.
Now let’s check whether Large is evolutionarily stable. For this, we suppose that for some
very small positive number x, a 1�x fraction of the population uses Large and an x fraction of the population uses Small.
• What is the expected payo↵ to a large beetle in a random interaction in this population? With probability 1�x, it meets another large beetle, receiving a payo↵ of 3, while with probability x, it meets a small beetle, receiving a payo↵ of 8. Therefore its expected
payo↵ is
3(1 � x) + 8 · x = 3 + 5x.
• What is the expected payo↵ to a small beetle in a random interaction in this popu- lation? With probability 1 � x, it meets a large beetle, receiving a payo↵ of 1, while with probability x, it meets another small beetle, receiving a payo↵ of 5. Therefore its
expected payo↵ is
(1 � x) + 5 · x = 1 + 4x.
In this case, the expected fitness of large beetles in this population exceeds the expected
fitness of small beetles, and so Large is evolutionarily stable.
214 CHAPTER 7. EVOLUTIONARY GAME THEORY
Interpreting the Evolutionarily Stable Strategy in our Example. Intuitively, this
analysis can be summarized by saying that if a few large beetles are introduced into a
population consisting of small beetles, then the large beetles do extremely well — since
they rarely meet each other, they get most of the food in almost every competition they
experience. As a result, the population of small beetles cannot drive out the large ones, and
so Small is not evolutionarily stable.
On the other hand, in a population of large beetles, a few small beetles will do very badly,
losing almost every competition for food. As a result, the population of large beetles resists
the invasion of small beetles, and so Large is evolutionarily stable.
Therefore, if we know that the large-body-size mutation is possible, we should expect to
see populations of large beetles in the wild, rather than populations of small ones. In this
way, our notion of evolutionary stability has predicted a strategy for the population — as we
predicted outcomes for games among rational players in Chapter 6, but by di↵erent means.
What’s striking about this particular predicted outcome, though, is the fact that the
fitness of each organism in a population of small beetles is 5, which is larger than the fitness
of each organism in a population of large beetles. In fact, the game between small and large
beetles has precisely the structure of a Prisoner’s Dilemma game; the motivating scenario
based on competition for food makes it clear that the beetles are engaged in an arms race,
like the game from Chapter 6 in which two competing athletes need to decide whether to use
performance-enhancing drugs. There it was a dominant strategy to use drugs, even though
both athletes understand that they are better o↵ in an outcome where neither of them uses
drugs — it’s simply that this mutually better joint outcome is not sustainable. In the present
case, the beetles individually don’t understand anything, nor could they change their body
sizes even if they wanted to. Nevertheless, evolutionary forces over multiple generations are
achieving a completely analogous e↵ect, as the large beetles benefit at the expense of the
small ones. Later in this chapter, we will see that this similarity in the conclusions of two
di↵erent styles of analysis is in fact part of a broader principle.
Here is a di↵erent way to summarize the striking feature of our example: Starting from
a population of small beetles, evolution by natural selection is causing the fitness of the
organisms to decrease over time. This might seem troubling initially, since we think of
natural selection as being fitness-increasing. But in fact, it’s not hard to reconcile what’s
happening with this general principle of natural selection. Natural selection increases the
fitness of individual organisms in a fixed environment — if the environment changes to
become more hostile to the organisms, then clearly this could cause their fitness to go down.
This is what is happening to the population of beetles. Each beetle’s environment includes
all the other beetles, since these other beetles determine its success in food competitions;
therefore the increasing fraction of large beetles can be viewed, in a sense, as a shift to an
environment that is more hostile for everyone.
7.2. EVOLUTIONARILY STABLE STRATEGIES 215
Empirical Evidence for Evolutionary Arms Races. Biologists have o↵ered recent evi-
dence for the presence of evolutionary games in nature with the Prisoner’s-Dilemma structure
we’ve just seen. It is very di�cult to truly determine payo↵s in any real-world setting, and so
all of these studies are the subject of ongoing investigation and debate. For our purposes in
this discussion, they are perhaps most usefully phrased as deliberately streamlined examples,
illustrating how game-theoretic reasoning can help provide qualitative insight into di↵erent
forms of biological interaction.
It has been argued that the heights of trees can obey Prisoner’s-Dilemma payo↵s [156,
226]. If two neighboring trees both grow short, then they share the sunlight equally. They
also share the sunlight equally if they both grow tall, but in this case their payo↵s are each
lower because they have to invest a lot of resources in achieving the additional height. The
trouble is that if one tree is short while its neighbor is tall, then the tall tree gets most of
the sunlight. As a result, we can easily end up with payo↵s just like the Body-Size Game
among beetles, with the trees’ evolutionary strategies Short and Tall serving as analogues to
the beetles’ strategies Small and Large. Of course, the real situation is more complex than
this, since genetic variation among trees can lead to a wide range of di↵erent heights and
hence a range of di↵erent strategies (rather than just two strategies labeled Short and Tall).
Within this continuum, Prisoner’s-Dilemma payo↵s can only apply to a certain range of tree
heights: there is some height beyond which further height-increasing mutations no longer
provide the same payo↵ structure, because the additional sunlight is more than o↵set by the
fitness downside of sustaining an enormous height.
Similar kinds of competition take place in the root systems of plants [181]. Suppose you
grow two soybean plants at opposite ends of a large pot of soil; then their root systems
will each fill out the available soil and intermingle with each other as they try to claim as
many resources as they can. In doing so, they divide the resources in the soil equally. Now,
suppose that instead you partition the same quantity of soil using a wall down the middle, so
that the two plants are on opposite sides of the wall. Then each still gets half the resources
present in the soil, but each invests less of its energy in producing roots and consequently
has greater reproductive success through seed production.
This observation has implications for the following simplified evolutionary game involving
root systems. Imagine that instead of a wall, we had two kinds of root-development strategies
available to soybean plants: Conserve, where a plant’s roots only grow into its own share of
the soil, and Explore, where the roots grow everywhere they can reach. Then we again have
the scenario and payo↵s from the Body-Size Game, with the same conclusion: all plants are
better o↵ in a population where everyone plays Conserve, but only Explore is evolutionarily
stable.
As a third example, there was recent excitement over the discovery that virus populations
can also play an evolutionary version of the Prisoner’s Dilemma [326, 392]. Turner and Chao
216 CHAPTER 7. EVOLUTIONARY GAME THEORY
studied a virus called Phage �6, which infects bacteria and manufactures products needed
for its own replication. A mutational variant of this virus called Phage �H2 is also able to
replicate in bacterial hosts, though less e↵ectively on its own. However, �H2 is able to take
advantage of chemical products produced by �6, which gives �H2 a fitness advantage when
it is in the presence of �6. This turns out to yield the structure of the Prisoner’s Dilemma:
viruses have the two evolutionary strategies �6 and �H2; viruses in a pure �6 population all
do better than viruses in a pure �H2 population; and regardless of what the other viruses are
doing, you (as a virus) are better o↵ playing �H2. Thus only �H2 is evolutionarily stable.
The virus system under study was so simple that Turner and Chao were able to infer an
actual payo↵ matrix based on measuring the relative rates at which the two viral variants
were able to replicate under di↵erent conditions. Using an estimation procedure derived
from these measurements, they obtained the payo↵s in Figure 7.2. The payo↵s are re-scaled
so that the upper-left box has the value 1.00, 1.00.1
Virus 1
Virus 2 �6 �H2
�6 1.00, 1.00 0.65, 1.99 �H2 1.99, 0.65 0.83, 0.83
Figure 7.2: The Virus Game
Whereas our earlier examples had an underlying story very much like the use of performance-
enhancing drugs, this game among phages is actually reminiscent of a di↵erent story that
also motivates the Prisoner’s Dilemma payo↵ structure: the scenario behind the Exam-or-
Presentation game with which we began Chapter 6. There, two college students would both
be better o↵ if they jointly prepared for a presentation, but the payo↵s led them to each think
selfishly and study for an exam instead. What the Virus Game here shows is that shirking
a shared responsibility isn’t just something that rational decision-makers do; evolutionary
forces can induce viruses to play this strategy as well.
7.3 A General Description of Evolutionarily Stable Strate- gies
The connections between evolutionary games and games played by rational participants are
suggestive enough that it makes sense to understand how the relationship works in general.
We will focus here, as we have thus far, on two-player two-strategy games. We will also
1It should be noted that even in a system this simple, there are many other biological factors at work, and hence this payo↵ matrix is still just an approximation to the performance of �6 and �H2 populations under real experimental and natural conditions. Other factors appear to a↵ect these populations, including the density of the population and the potential presence of additional mutant forms of the virus [393].
7.3. A GENERAL DESCRIPTION OF EVOLUTIONARILY STABLE STRATEGIES 217
restrict our attention to symmetric games, as in the previous sections of this chapter, where
the roles of the two players are interchangeable.
The payo↵ matrix for a completely general two-player, two-strategy game that is sym-
metric can be written as in Figure 7.3.
Organism 1
Organism 2 S T
S a, a b, c T c, b d, d
Figure 7.3: General Symmetric Game
Let’s check how to write the condition that S is evolutionarily stable in terms of the four
variables a, b, c, and d. As before, we start by supposing that for some very small positive
number x, a 1�x fraction of the population uses S and an x fraction of the population uses T .
• What is the expected payo↵ to an organism playing S in a random interaction in this population? With probability 1�x, it meets another player of S, receiving a payo↵ of a, while with probability x, it meets a player of T , receiving a payo↵ of b. Therefore
its expected payo↵ is
a(1 � x) + bx.
• What is the expected payo↵ to an organism playing T in a random interaction in this population? With probability 1�x, it meets a player of S, receiving a payo↵ of c, while with probability x, it meets another player of T , receiving a payo↵ of d. Therefore its
expected payo↵ is
c(1 � x) + dx.
Therefore, S is evolutionarily stable if for all su�ciently small values of x > 0, the
inequality
a(1 � x) + bx > c(1 � x) + dx
holds. As x goes to 0, the left-hand side becomes a and the right-hand side becomes c.
Hence, if a > c, then the left-hand side is larger once x is su�ciently small, while if a < c
then the left-hand side is smaller once x is su�ciently small. Finally, if a = c, then the
left-hand side is larger precisely when b > d. Therefore, we have a simple way to express the
condition that S is evolutionarily stable:
In a two-player, two-strategy, symmetric game, S is evolutionarily stable precisely
when either (i) a > c, or (ii) a = c and b > d.
218 CHAPTER 7. EVOLUTIONARY GAME THEORY
It is easy to see the intuition behind our calculations that translates into this condition,
as follows.
• First, in order for S to be evolutionarily stable, the payo↵ to using strategy S against S must be at least as large as the payo↵ to using strategy T against S. Otherwise, an
invader who uses T would have a higher fitness than the rest of population, and the
fraction of the population who are invaders would have a good probability of growing
over time.
• Second, if S and T are equally good responses to S, then in order for S to be evolu- tionarily stable, players of S must do better in their interactions with T than players
of T do with each other. Otherwise, players of T would do as well as against the S
part of the population as players of S, and at least as well against the T part of the
population, so their overall fitness would be at least as good as the fitness of players
of S.
7.4 Relationship Between Evolutionary and Nash Equi- libria
Using our general way of characterizing evolutionarily stable strategies, we can now under-
stand how they relate to Nash equilibria. If we go back to the General Symmetric Game
from the previous section, we can write down the condition for (S, S) (i.e. the choice of S
by both players) to be a Nash equilibrium. (S, S) is a Nash equilibrium when S is a best
response to the choice of S by the other player: this translates into the simple condition
a � c.
If we compare this to the condition for S to be evolutionarily stable,
(i) a > c, or (ii) a = c and b > d,
we immediately get the conclusion that
If strategy S is evolutionarily stable, then (S, S) is a Nash equilibrium.
We can also see that the other direction does not hold: it is possible to have a game
where (S, S) is a Nash equilibrium, but S is not evolutionarily stable. The di↵erence in the
two conditions above tells us how to construct such a game: we should have a = c and b < d.
To get a sense for where such a game might come from, let’s recall the Stag Hunt Game
from Chapter 6. Here, each player can hunt stag or hunt hare; hunting hare successfully just
requires your own e↵ort, while hunting the more valuable stag requires that you both do so.
This produces payo↵s as shown in Figure 7.4.
7.4. RELATIONSHIP BETWEEN EVOLUTIONARY AND NASH EQUILIBRIA 219
Hunter 1
Hunter 2 Hunt Stag Hunt Hare
Hunt Stag 4, 4 0, 3 Hunt Hare 3, 0 3, 3
Figure 7.4: Stag Hunt
In this game, as written, Hunt Stag and Hunt Hare are both evolutionarily stable, as we
can check from the conditions on a, b, c, and d. (To check the condition for Hunt Hare, we
simply need to interchange the rows and columns of the payo↵ matrix, to put Hunt Hare in
the first row and first column.)
However, suppose we make up a modification of the Stag Hunt Game, by shifting the
payo↵s as follows. In this new version, when the players mis-coordinate, so that one hunts
stag while the other hunts hare, then the hare-hunter gets an extra benefit due to the lack
of competition for hare. In this way, we get a payo↵ matrix as in Figure 7.5.
Hunter 1
Hunter 2 Hunt Stag Hunt Hare
Hunt Stag 4, 4 0, 4 Hunt Hare 4, 0 3, 3
Figure 7.5: Stag Hunt: A version with added benefit from hunting hare alone
In this case, the choice of strategies (Hunt Stag, Hunt Stag) is still a Nash equilibrium:
if each player expects the other to hunt stag, then hunting stag is a best response. But
Hunt Stag is not an evolutionarily stable strategy for this version of the game, because (in
the notation from our General Symmetric Game) we have a = c and b < d. Informally,
the problem is that a hare-hunter and a stag-hunter do equally well when each is paired
with a stag-hunter; but hare-hunters do better than stag-hunters when each is paired with
a hare-hunter.
There is also a relationship between evolutionarily stable strategies and the concept of a
strict Nash equilibrium. We say that a choice of strategies is a strict Nash equilibrium if each
player is using the unique best response to what the other player is doing. We can check
that for symmetric two-player, two-strategy games, the condition for (S, S) to be a strict
Nash equilibrium is that a > c. So we see that in fact these di↵erent notions of equilibrium
naturally refine each other. The concept of an evolutionarily stable strategy can be viewed as
a refinement of the concept of a Nash equilibrium: the set of evolutionarily stable strategies
S is a subset of the set of strategies S for which (S, S) is a Nash equilibrium. Similarly, the
concept of a strict Nash equilibrium (when the players use the same strategy) is a refinement
of evolutionary stability: if (S, S) is a strict Nash equilibrium, then S is evolutionarily stable.
220 CHAPTER 7. EVOLUTIONARY GAME THEORY
It is intriguing that, despite the extremely close similarities between the conclusions
of evolutionary stability and Nash equilibrium, they are built on very di↵erent underlying
stories. In a Nash equilibrium, we consider players choosing mutual best responses to each
other’s strategy. This equilibrium concept places great demands on the ability of the players
to choose optimally and to coordinate on strategies that are best responses to each other.
Evolutionary stability, on the other hand, supposes no intelligence or coordination on the part
of the players. Instead, strategies are viewed as being hard-wired into the players, perhaps
because their behavior is encoded in their genes. According to this concept, strategies which
are more successful in producing o↵spring are selected for.
Although this evolutionary approach to analyzing games originated in biology, it can be
applied in many other contexts. For example, suppose a large group of people are being
matched repeatedly over time to play the General Symmetric Game from Figure 7.3. Now
the payo↵s should be interpreted as reflecting the welfare of the players, and not their
number of o↵spring. If any player can look back at how others have played and can observe
their payo↵s, then imitation of the strategies that have been most successful may induce
an evolutionary dynamic. Alternatively, if a player can observe his own past successes and
failures then his learning may induce an evolutionary dynamic. In either case, strategies that
have done relatively well in the past will tend to be used by more people in the future. This
can lead to the same behavior that underlies the concept of evolutionarily stable strategies,
and hence can promote the play of such strategies.
7.5 Evolutionarily Stable Mixed Strategies
As a further step in developing an evolutionary theory of games, we now consider how to
handle cases in which no strategy is evolutionarily stable.
In fact, it is not hard to see how this can happen, even in two-player games that have
pure-strategy Nash equilibria.2 Perhaps the most natural example is the Hawk-Dove Game
from Chapter 6, and we use this to introduce the basic ideas of this section. Recall that in
the Hawk-Dove Game, two animals compete for a piece of food; an animal that plays the
strategy Hawk (H) behaves aggressively, while an animal that plays the strategy Dove (D)
behaves passively. If one animal is aggressive while the other is passive, then the aggressive
animal benefits by getting most of the food; but if both animals are aggressive, then they
risk destroying the food and injuring each other. This leads to a payo↵ matrix as shown in
Figure 7.6.
In Chapter 6, we considered this game in contexts where the two players were making
choices about how to behave. Now let’s consider the same game in a setting where each
2Recall that a player is using a pure strategy if she always plays a particular one of the strategies in the game, as opposed to a mixed strategy in which she chooses at random from among several possible strategies.
7.5. EVOLUTIONARILY STABLE MIXED STRATEGIES 221
Animal 1
Animal 2 D H
D 3, 3 1, 5 H 5, 1 0, 0
Figure 7.6: Hawk-Dove Game
animal is genetically hard-wired to play a particular strategy. How does it look from this
perspective, when we consider evolutionary stability?
Neither D nor H is a best response to itself, and so using the general principles from
the last two sections, we see that neither is evolutionarily stable. Intuitively, a hawk will do
very well in a population consisting of doves — but in a population of all hawks, a dove will
actually do better by staying out of the way while the hawks fight with each other.
As a two-player game in which players are actually choosing strategies, the Hawk-Dove
Game has two pure Nash equilibria: (D, H) and (H, D). But this doesn’t directly help us
identify an evolutionarily stable strategy, since thus far our definition of evolutionary stability
has been restricted to populations in which (almost) all members play the same pure strategy.
To reason about what will happen in the Hawk-Dove Game under evolutionary forces, we
need to generalize the notion of evolutionary stability by allowing some notion of “mixing”
between strategies.
Defining Mixed Strategies in Evolutionary Game Theory. There are at least two
natural ways to introduce the idea of mixing into the evolutionary framework. First, it
could be that each individual is hard-wired to play a pure strategy, but some portion of
the population plays one strategy while the rest of the population plays another. If the
fitness of individuals in each part of the population is the same, and if invaders eventually
die o↵, then this could be considered to exhibit a kind of evolutionary stability. Second,
it could be that each individual is hard-wired to play a particular mixed strategy — that
is, they are genetically configured to choose randomly from among certain options with
certain probabilities. If invaders using any other mixed strategy eventually die o↵, then this
too could be considered a kind of evolutionary stability. We will see that for our purposes
here, these two concepts are actually equivalent to each other, and we will focus initially
on the second idea, in which individuals use mixed strategies. Essentially, we will find that
in situations like the Hawk-Dove game, the individuals or the population as a whole must
display a mixture of the two behaviors in order to have any chance of being stable against
invasion by other forms of behavior.
The definition of an evolutionarily stable mixed strategy is in fact completely parallel
to the definition of evolutionary stability we have seen thus far — it is simply that we now
greatly enlarge the set of possible strategies, so that each strategy corresponds to a particular
222 CHAPTER 7. EVOLUTIONARY GAME THEORY
randomized choice over pure strategies.
Specifically, let’s consider the General Symmetric Game from Figure 7.3. A mixed strat-
egy here corresponds to a probability p between 0 and 1, indicating that the organism plays S
with probability p and plays T with probability 1�p. As in our discussion of mixed strategies from Chapter 6, this includes the possibility of playing the pure strategies S or T by simply
setting p = 1 or p = 0. When Organism 1 uses the mixed strategy p and Organism 2 uses
the mixed strategy q, the expected payo↵ to Organism 1 can be computed as follows. There
is a probability pq of an (X, X) pairing, yielding a for the first player; there is a probability
p(1�q) of an (X, Y ) pairing, yielding b for the first player; there is a probability (1�p)q of a (Y, X) pairing, yielding c for the first player; and there is a probability (1 � p)(1 � q) of a (Y, Y ) pairing, yielding d for the first player. So the expected payo↵ for this first player is
V (p, q) = pqa + p(1 � q)b + (1 � p)qc + (1 � p)(1 � q)d.
As before, the fitness of an organism is its expected payo↵ in an interaction with a random
member of the population. We can now give the precise definition of an evolutionarily stable
mixed strategy.
In the General Symmetric Game, p is an evolutionarily stable mixed strategy if
there is a (small) positive number y such that when any other mixed strategy q
invades p at any level x < y, the fitness of an organism playing p is strictly
greater than the fitness of an organism playing q.
This is just like our previous definition of evolutionarily stable (pure) strategies, except
that we allow the strategy to be mixed, and we allow the invaders to use a mixed strategy.
An evolutionarily stable mixed strategy with p = 1 or p = 0 is evolutionarily stable under
our original definition for pure strategies as well. However, note the subtle point that even
if S were an evolutionarily stable strategy under our previous definition, it is not necessarily
an evolutionarily stable mixed strategy under this new definition with p = 1. The problem
is that it is possible to construct games in which no pure strategy can successfully invade a
population playing S, but a mixed strategy can. As a result, it will be important to be clear
in any discussion of evolutionary stability on what kinds of behavior an invader can employ.
Directly from the definition, we can write the condition for p to be an evolutionarily
stable mixed strategy as follows: for some y and any x < y, the following inequality holds
for all mixed strategies q 6= p:
(1 � x)V (p, p) + xV (p, q) > (1 � x)V (q, p) + xV (q, q). (7.1)
This inequality also makes it clear that there is a relationship between mixed Nash
equilibria and evolutionarily stable mixed strategies, and this relationship parallels the one
we saw earlier for pure strategies. In particular, if p is an evolutionarily stable mixed strategy,
7.5. EVOLUTIONARILY STABLE MIXED STRATEGIES 223
then we must have V (p, p) � V (q, p), and so p is a best response to p. As a result, the pair of strategies (p, p) is a mixed Nash equilibrium. However, because of the strict inequality
in Equation (7.1), it is possible for (p, p) to be a mixed Nash equilibrium without p being
evolutionarily stable. So again, evolutionary stability serves as a refinement of the concept
of mixed Nash equilibrium.
Evolutionarily Stable Mixed Strategies in the Hawk-Dove Game. Now let’s see
how to apply these ideas to the Hawk-Dove Game. First, since any evolutionarily stable
mixed strategy must correspond to a mixed Nash equilibrium of the game, this gives us a
way to search for possible evolutionarily stable strategies: we first work out the mixed Nash
equilibria for the Hawk-Dove, and then we check if they are evolutionarily stable.
As we saw in Chapter 6, in order for (p, p) to be a mixed Nash equilibrium, it must make
the two players indi↵erent between their two pure strategies. When the other player is using
the strategy p, the expected payo↵ from playing D is 3p+(1�p) = 1+2p, while the expected payo↵ from playing H is 5p. Setting these two quantities equal (to capture the indi↵erence
between the two strategies), we get p = 1/3. So (1/3, 1/3) is a mixed Nash equilibrium. In
this case, both pure strategies, as well as any mixture between them, produce an expected
payo↵ of 5/3 when played against the strategy p = 1/3.
Now, to see whether p = 1/3 is evolutionarily stable, we must check Inequality (7.1)
when some other mixed strategy q invades at a small level x. Here is a first observation
that makes evaluating this inequality a bit easier. Since (p, p) is a mixed Nash equilibrium
that uses both pure strategies, we have just argued that all mixed strategies q have the same
payo↵ when played against p. As a result, we have V (p, p) = V (q, p) for all q. Subtracting
these terms from the left and right of Inequality (7.1), and then dividing by x, we get the
following inequality to check:
V (p, q) > V (q, q). (7.2)
The point is that since (p, p) is a mixed equilibrium, the strategy p can’t be a strict best
response to itself — all other mixed strategies are just as good against it. Therefore, in order
for p to be evolutionarily stable, it must be a strictly better response to every other mixed
strategy q than q is to itself. That is what will cause it to have higher fitness when q invades.
In fact, it is true that V (p, q) > V (q, q) for all mixed strategies q 6= p, and we can check this as follows. Using the fact that p = 1/3, we have
V (p, q) = 1/3 · q · 3 + 1/3(1 � q) · 1 + 2/3 · q · 5 = 4q + 1/3
while
V (q, q) = q2 · 3 + q(1 � q) · 1 + (1 � q) · q · 5 = 6q � 3q2.
Now we have
V (p, q) � V (q, q) = 3q2 � 2q + 1/3 = 1
3 (9q2 � 6q + 1) =
1
3 (3q � 1)2.
224 CHAPTER 7. EVOLUTIONARY GAME THEORY
This last way of writing V (p, q)�V (q, q) shows that it is a perfect square, and so it is positive whenever q 6= 1/3. This is just what we want for showing that V (p, q) > V (q, q) whenever q 6= p, and so it follows that p is indeed an evolutionarily stable mixed strategy.
Interpretations of Evolutionarily Stable Mixed Strategies. The kind of mixed equi-
librium that we see here in the Hawk-Dove Game is typical of biological situations in which
organisms must break the symmetry between two distinct behaviors, when consistently
adopting just one of these behaviors is evolutionarily unsustainable.
We can interpret the result of this example in two possible ways. First, all participants
in the population may actually be mixing over the two possible pure strategies with the
given probability. In this case, all members of the population are genetically the same, but
whenever two of them are matched up to play, any combination of D and H could potentially
be played. We know the empirical frequency with which any pair of strategies will be played,
but not what any two animals will actually do. A second interpretation is that the mixture
is taking place at the population level: it could be that 1/3 of the animals are hard-wired
to always play D, and 2/3 are hard-wired to always play H. In this case, no individual is
actually mixing, but as long as it is not possible to tell in advance which animal will play D
and which will play H, the interaction of two randomly selected animals results in the same
frequency of outcomes that we see when each animal is actually mixing. Notice also that
in this case, the fitnesses of both kinds of animals are the same, since both D and H are
best responses to the mixed strategy p = 1/3. Thus, these two di↵erent interpretation of the
evolutionarily stable mixed strategy lead to the same calculations, and the same observed
behavior in the population.
There are a number of other settings in which this type of mixing between pure strategies
has been discussed in biology. A common scenario is that there is an undesirable, fitness-
lowering role in a population of organisms — but if some organisms don’t choose to play this
role, then everyone su↵ers considerably. For example, let’s think back to the Virus Game in
Figure 7.2 and suppose (purely hypothetically, for the sake of this example) that the payo↵
when both viruses use the strategy �H2 were (0.50, 0.50), as shown in Figure 7.7.
Virus 1
Virus 2 �6 �H2
�6 1.00, 1.00 0.65, 1.99 �H2 1.99, 0.65 0.50, 0.50
Figure 7.7: The Virus Game: Hypothetical payo↵s with stronger fitness penalties to �H2.
In this event, rather than having a Prisoner’s Dilemma type of payo↵ structure, we’d have
a Hawk-Dove payo↵ structure: having both viruses play �H2 is su�ciently bad that one of
them needs to play the role of �6. The two pure equilibria of the resulting two-player game
7.6. EXERCISES 225
— viewed as a game among rational players, rather than a biological interaction — would be
(�6,�H2) and (�H2,�6). In a virus population we’d expect to find an evolutionarily stable
mixed strategy in which both kinds of virus behavior were observed.
This example, like the examples from our earlier discussion of the Hawk-Dove Game
in Section 6.6, suggests the delicate boundary that exists between Prisoner’s Dilemma and
Hawk-Dove. In both cases, a player can choose to be “helpful” to the other player or “selfish”.
In Prisoner’s Dilemma, however, the payo↵ penalties from selfishness are mild enough that
selfishness by both players is the unique equilibrium — while in Hawk-Dove, selfishness is
su�ciently harmful that at least one player should try to avoid it.
There has been research into how this boundary between the two games manifests itself
in other biological settings as well. One example is the implicit game played by female
lions in defending their territory [218, 327]. When two female lions encounter an attacker
on the edge of their territory, each can choose to play the strategy Confront, in which she
confronts the attacker, or Lag, in which she lags behind and tries to let the other lion confront
the attacker first. If you’re one of the lions, and your fellow defender chooses the strategy
Confront, then you get a higher payo↵ by choosing Lag, since you’re less likely to get injured.
What’s harder to determine in empirical studies is what a lion’s best response should be to a
play of Lag by her partner. Choosing Confront risks injury, but joining your partner in Lag
risks a successful assault on the territory by the attacker. Understanding which is the best
response is important for understanding whether this game is more like Prisoner’s Dilemma
or Hawk-Dove, and what the evolutionary consequences might be for the observed behavior
within a lion population.
In this, as in many examples from evolutionary game theory, it is beyond the power of
current empirical studies to work out detailed fitness values for particular strategies. How-
ever, even in situations where exact payo↵s are not known, the evolutionary framework can
provide an illuminating perspective on the interactions between di↵erent forms of behav-
ior in an underlying population, and how these interactions shape the composition of the
population.
7.6 Exercises
1. In the payo↵ matrix below the rows correspond to player A’s strategies and the columns
correspond to player B’s strategies. The first entry in each box is player A’s payo↵ and
the second entry is player B’s payo↵.
Player A
Player B x y
x 2, 2 0, 0 y 0, 0 1, 1
226 CHAPTER 7. EVOLUTIONARY GAME THEORY
(a) Find all pure strategy Nash equilibria.
(b) Find all Evolutionarily Stable strategies. Give a brief explanation for your answer.
(c) Briefly explain how the sets of predicted outcomes relate to each other.
2. In the payo↵ matrix below the rows correspond to player A’s strategies and the columns
correspond to player B’s strategies. The first entry in each box is player A’s payo↵ and
the second entry is player B’s payo↵.
Player A
Player B x y
x 4, 4 3, 5 y 5, 3 5, 5
(a) Find all pure strategy Nash equilibria.
(b) Find all Evolutionarily Stable strategies. Give a brief explanation for your answer.
(c) Briefly explain how the answers in parts (2a) and (2b) relate to each other.
3. In this problem we will consider the relationship between Nash equilibria and evolu-
tionarily stable strategies for games with a strictly dominant strategy. First, let’s define
what we mean by strictly dominant. In a two-player game, strategy, X is said to be a
strictly dominant strategy for a player i if, no matter what strategy the other player j
uses, player i’s payo↵ from using strategy X is strictly greater than his payo↵ from any
other strategy. Consider the following game in which a, b, c, and d are non-negative
numbers.
Player A
Player B X Y
X a, a b, c Y c, b d, d
Suppose that strategy X is a strictly dominant strategy for each player, i.e. a > c and
b > d.
(a) Find all of the pure strategy Nash equilibria of this game.
(b) Find all of the evolutionarily stable strategies of this game.
(c) How would your answers to parts (a) and (b) change if we change the assumption
on payo↵s to: a > c and b = d?
7.6. EXERCISES 227
Player A
Player B X Y
X 1, 1 2, x Y x, 2 3, 3
4. Consider following the two-player, symmetric game where x can be 0, 1, or 2.
(a) For each of the possible values of x find all (pure strategy) Nash equilibria and all
evolutionarily stable strategies.
(b) Your answers to part (a) should suggest that the di↵erence between the predictions
of evolutionary stability and Nash equilibrium arises when a Nash equilibrium uses a
weakly dominated strategy. We say that a strategy s⇤ i
is weakly dominated if player i
has another strategy s0 i
with the property that:
(a) No matter what the other player does, player i’s payo↵ from s0 i
is at least as large
as the payo↵ from s⇤ i
, and
(b) There is some strategy for the other player so that player i’s payo↵ from s0 i
is
strictly greater than the payo↵ from s⇤ i
.
Now, consider the following claim that makes a connection between evolutionarily
stable strategies and weakly dominated strategies.
Claim: Suppose that in the game below, (X, X) is a Nash equilibrium and
that strategy X is weakly dominated. Then X is not an evolutionarily stable
strategy.
Player A
Player B X Y
X a, a b, c Y c, b d, d
Explain why this claim is true. (You do not need to write a formal proof; a careful
explanation is fine.)
228 CHAPTER 7. EVOLUTIONARY GAME THEORY
Chapter 8
Modeling Network Tra�c using Game Theory
Among the initial examples in our discussion of game theory in Chapter 6, we noted
that traveling through a transportation network, or sending packets through the Internet,
involves fundamentally game-theoretic reasoning: rather than simply choosing a route in
isolation, individuals need to evaluate routes in the presence of the congestion resulting from
the decisions made by themselves and everyone else. In this chapter, we develop models for
network tra�c using the game-theoretic ideas we’ve developed thus far. In the process of
doing this, we will discover a rather unexpected result — known as Braess’s Paradox [76]
— which shows that adding capacity to a network can sometimes actually slow down the
tra�c.
8.1 Tra�c at Equilibrium
Let’s begin by developing a model of a transportation network and how it responds to tra�c
congestion; with this in place, we can then introduce the game-theoretic aspects of the
problem.
We represent a transportation network by a directed graph: we consider the edges to be
highways, and the nodes to be exits where you can get on or o↵ a particular highway. There
are two particular nodes, which we’ll call A and B, and we’ll assume everyone wants to drive
from A to B. For example, we can imagine that A is an exit in the suburbs, B is an exit
downtown, and we’re looking at a large collection of morning commuters. Finally, each edge
has a designated travel time that depends on the amount of tra�c it contains.
To make this concrete, consider the graph in Figure 8.1. The label on each edge gives the
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
229
230 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
BA
C
D
x/100
45
45
x/100
Figure 8.1: A highway network, with each edge labeled by its travel time (in minutes) when there are x cars using it. When 4000 cars need to get from A to B, they divide evenly over the two routes at equilibrium, and the travel time is 65 minutes.
travel time (in minutes) when there are x cars using the edge. In this simplified example,
the A-D and C-B edges are insensitive to congestion: each takes 45 minutes to traverse
regardless of the number of cars traveling on them. On the other hand, the A-C and D-B
edges are highly sensitive to congestion: for each one, it takes x/100 minutes to traverse
when there are x cars using the edge.1
Now, suppose that 4000 cars want to get from A to B as part of the morning commute.
There are two possible routes that each car can choose: the upper route through C, or the
lower route through D. For example, if each car takes the upper route (through C), then
the total travel time for everyone is 85 minutes, since 4000/100 + 45 = 85. The same is true
if everyone takes the lower route. On the other hand, if the cars divide up evenly between
the two routes, so that each carries 2000 cars, then the total travel time for people on both
routes is 2000/100 + 45 = 65.
Equilibrium tra�c. So what do we expect will happen? The tra�c model we’ve described
is really a game in which the players correspond to the drivers, and each player’s possible
strategies consist of the possible routes from A to B. In our example, this means that each
player only has two strategies; but in larger networks, there could be many strategies for
each player. The payo↵ for a player is the negative of his or her travel time (we use the
negative since large travel times are bad).
1The travel times here are simplified to make the reasoning clearer: in any real application, each road would have both some minimum travel time, and some sensitivity to the number of cars x that are using it. However, the analysis here adapts directly to more intricate functions specifying the travel times on edges.
8.2. BRAESS’S PARADOX 231
This all fits very naturally into the framework we’ve been using. One thing to notice,
of course, is that up to now we have focused primarily on games with two players, whereas
the current game will generally have an enormous number of players (4000 in our example).
But this poses no direct problem for applying any of the ideas we’ve developed. A game can
have any number of players, each of whom can have any number of available strategies, and
the payo↵ to each player depends on the strategies chosen by all. A Nash equilibrium is still
a list of strategies, one for each player, so that each player’s strategy is a best response to
all the others. The notions of dominant strategies, mixed strategies and Nash equilibrium
with mixed strategies all have direct parallels with their definitions for two-player games.
In this tra�c game, there is generally not a dominant strategy; for example, in Figure 8.1
either route has the potential to be the best choice for a player if all the other players are
using the other route. The game does have Nash equilibria, however: as we will see next,
any list of strategies in which the drivers balance themselves evenly between the two routes
(2000 on each) is a Nash equilibrium, and these are the only Nash equilibria.
Why does equal balance yield a Nash equilibrium, and why do all Nash equilibria have
equal balance? To answer the first question, we just observe that with an even balance
between the two routes, no driver has an incentive to switch over to the other route. For the
second question, consider a list of strategies in which x drivers use the upper route and the
remaining 4000�x drivers use the lower route. Then if x is not equal to 2000, the two routes will have unequal travel times, and any driver on the slower route would have an incentive to
switch to the faster one. Hence any list of strategies in which x is not equal to 2000 cannot
be a Nash equilibrium; and any list of strategies in which x = 2000 is a Nash equilibrium.
8.2 Braess’s Paradox
In Figure 8.1, everything works out very cleanly: self-interested behavior by all drivers causes
them — at equilibrium — to balance perfectly between the available routes. But with only a
small change to the network, we can quickly find ourselves in truly counterintuitive territory.
The change is as follows: suppose that the city government decides to build a new, very
fast highway from C to D, as indicated in Figure 8.2. To keep things simple, we’ll model its
travel time as 0, regardless of the number of cars on it, although the resulting e↵ect would
happen even with more realistic (but small) travel times. It would stand to reason that
people’s travel time from A to B ought to get better after this edge from C to D is added.
Does it?
Here’s the surprise: there is a unique Nash equilibrium in this new highway network,
but it leads to a worse travel time for everyone. At equilibrium, every driver uses the
route through both C and D; and as a result, the travel time for every driver is 80 (since
4000/100 + 0 + 4000/100 = 80). To see why this is an equilibrium, note that no driver can
232 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
BA
C
D
x/100
45
45
x/100
0
Figure 8.2: The highway network from the previous figure, after a very fast edge has been added from C to D. Although the highway system has been “upgraded,” the travel time at equilibrium is now 80 minutes, since all cars use the route through C and D.
benefit by changing their route: with tra�c snaking through C and D the way it is, any
other route would now take 85 minutes. And to see why it’s the only equilibrium, you can
check that the creation of the edge from C to D has in fact made the route through C and
D a dominant strategy for all drivers: regardless of the current tra�c pattern, you gain by
switching your route to go through C and D.
In other words, once the fast highway from C to D is built, the route through C and
D acts like a “vortex” that draws all drivers into it — to the detriment of all. In the new
network there is no way, given individually self-interested behavior by the drivers, to get
back to the even-balance solution that was better for everyone.
This phenomenon — that adding resources to a transportation network can sometimes
hurt performance at equilibrium — was first articulated by Dietrich Braess in 1968 [76],
and it has become known as Braess’s Paradox. Like many counterintuitive anomalies, it
needs the right combination of conditions to actually pop up in real life; but it has been
observed empirically in real transportation networks — including in Seoul, Korea, where the
destruction of a six-lane highway to build a public park actually improved travel time into
and out of the city (even though tra�c volume stayed roughly the same before and after the
change) [37].
Some reflections on Braess’s paradox. Having now seen how Braess’s paradox works,
we can also appreciate that there is actually nothing really “paradoxical” about it. There are
many settings in which adding a new strategy to a game makes things worse for everyone.
For example, the Prisoner’s Dilemma from Chapter 6 can be used to illustrate this point: if
8.3. ADVANCED MATERIAL: THE SOCIAL COST OF TRAFFIC AT EQUILIBRIUM233
the only strategy for each player were Not-Confess (an admittedly very simple game), then
both players would be better o↵ compared with the game where Confess is added as an
option. (Indeed, that’s why the police o↵er Confess as an option in the first place.)
Still, it’s reasonable to view the analogous phenomenon at the heart of the Braess Paradox
as more paradoxical, at an intuitive level. We all have an informal sense that “upgrading”
a network has to be a good thing, and so it is surprising when it turns out to make things
worse.
The example in this section is actually the starting point for a large body of work on
game-theoretic analysis of network tra�c. For example, we could ask how bad Braess’s
Paradox can be for networks in general: how much larger can the equilibrium travel time be
after the addition of an edge, relative to what it was before? Suppose in particular that we
allow the graph to be arbitrary, and we assume that the travel time on each edge depends in
a linear way on the number of cars traversing it — that is, all travel times across edges have
the form ax + b, where each of a and b is either 0 or a positive number. In this case, elegant
results of Tim Roughgarden and Éva Tardos can be used to show that if we add edges to
a network with an equilibrium pattern of tra�c, there is always an equilibrium in the new
network whose travel time is no more than 4/3 times as large [18, 353]. Moreover, 4/3 is the
factor increase that we’d get in the example from Figures 8.1 and 8.2, if we replace the two
travel times of 45 with 40. (In that case, the travel time at equilibrium would jump from 60
to 80 when we add the edge from C to D.) So the Roughgarden-Tardos result shows that
this simple example is as bad as the Braess Paradox can get, in a quantitative sense, when
edges respond linearly to tra�c. (When edges can respond non-linearly, things can be much
worse.)
There are many other types of questions that can be pursued as well. For example, we
could think about ways of designing networks to prevent bad equilibria from arising, or to
avoid bad equilibria through the judicious use of tolls on certain parts of the network. Many
of these extensions, as well as others, are discussed by Tim Roughgarden in his book on
game-theoretic models of network tra�c [352].
8.3 Advanced Material: The Social Cost of Tra�c at Equilibrium
The Braess Paradox is one aspect of a larger phenomenon, which is that network tra�c at
equilibrium may not be socially optimal. In this section, we try to quantify how far from
optimal tra�c can be at equilibrium.
We would like our analysis to apply to any network, and so we introduce the following
general definitions. The network can be any directed graph. There is a set of drivers, and
di↵erent drivers may have di↵erent starting points and destinations. Now, each edge e has
234 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
BA
C
D T
AC
CD
AD
CB
DB
(x)=x (x)=5
(x)=0
T T
T
T (x)=5 (x)=x
(a) Travel times written as explicit functions of x.
BA
C
D
x
5
5
x
0
(b) Travel times written as annotations on the edges.
Figure 8.3: A network annotated with the travel-time function on each edge.
a travel-time function T e
(x), which gives the time it takes all drivers to cross the edge when
there are x drivers using it. These travel times are simply the functions that we drew as labels
inside the edges in the figures in Section 8.1. We will assume that all travel-time functions
are linear in the amount of tra�c, so that T e
(x) = a e
x + b e
for some choice of numbers a e
and
b e
that are either positive or zero. For example, in Figure 8.3 we draw another network on
which Braess’s Paradox arises, with the travel-time functions scaled down to involve smaller
numbers. The version of the drawing in Figure 8.3(a) has the travel-time functions explicitly
written out, while the version of the drawing in Figure 8.3(b) has the travel-time functions
written as labels inside the edges.
Finally, we say that a tra�c pattern is simply a choice of a path by each driver, and the
social cost of a given tra�c pattern is the sum of the travel times incurred by all drivers when
they use this tra�c pattern. For example, Figure 8.4 shows two di↵erent tra�c patterns on
the network from Figure 8.3, when there are four drivers, each with starting node A and
destination node B. The first of these tra�c patterns, in Figure 8.4(a), achieves the minimum
possible social cost — each driver requires 7 units of time to get to their destination, and
so the social cost is 28. We will refer to such a tra�c pattern, which achieves the minimum
possible social cost, as socially optimal. (There are other tra�c patterns on this network that
also achieve a social cost of 28; that is, there are multiple tra�c patterns for this network
that are socially optimal.) Note that socially optimal tra�c patterns are simply the social
welfare maximizers of this tra�c game, since the sum of the drivers’ payo↵s is the negative
of the social cost. The second tra�c pattern, Figure 8.4(b), is the unique Nash equilibrium,
and it has a larger social cost of 32.
The main two questions we consider in the remainder of this chapter are the following.
First, in any network (with linear travel-time functions), is there always an equilibrium tra�c
8.3. ADVANCED MATERIAL: THE SOCIAL COST OF TRAFFIC AT EQUILIBRIUM235
BA
C
D
x
5
5
x
0
(a) The social optimum.
BA
C
D
x
5
5
x
0
(b) The Nash equilibrium.
Figure 8.4: A version of Braess’s Paradox: In the socially optimal tra�c pattern (on the left), the social cost is 28, while in the unique Nash equilibrium (on the right), the social cost is 32.
pattern? We have seen examples in Chapter 6 of games where equilibria do not exist using
pure strategies, and it is not a priori clear that they should always exist for the tra�c game
we’ve defined here. However, we will find in fact that equilibria always do exist. The second
main question is whether there always exists an equilibrium tra�c pattern whose social cost
is not much more than the social optimum. We will find that this is in fact the case: we
will show a result due to Roughgarden and Tardos that there is always an equilibrium whose
social cost is at most twice that of the optimum [353].2
A. How to Find a Tra�c Pattern at Equilibrium
We will prove that an equilibrium exists by analyzing the following procedure that explicitly
searches for one. The procedure starts from any tra�c pattern. If it is an equilibrium, we
are done. Otherwise, there is at least one driver whose best response, given what everyone
else is doing, is some alternate path providing a strictly lower travel time. We pick one such
driver and have him switch to this alternate path. We now have a new tra�c pattern and
we again check whether it is an equilibrium — if it isn’t, then we have some driver switch
to his best response, and we continue in this fashion.
This procedure is called best-response dynamics, since it dynamically reconfigures the
2In fact, stronger results of Roughgarden and Tardos, supplemented by subsequent results of Anshelevich et al. [18], establish that in fact every equilibrium tra�c pattern has social cost at most 4/3 times the optimum. (One can show that this implies their result on the Braess Paradox cited in the previous section — that with linear travel times, adding edges can’t make things worse by a factor of more than 4/3.) However, since it is harder to prove the bound of 4/3, we limit ourselves here to proving the easier but weaker factor of 2 between the social optimum and some equilibrium tra�c pattern.
236 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
players’ strategies by constantly having some player perform his or her best response to the
current situation. If the procedure ever stops, in a state where everyone is in fact playing
their best response to the current situation, then we have an equilibrium. So the key is to
show that in any instance of our tra�c game, best-response dynamics must eventually stop
at an equilibrium.
But why should it? Certainly for games that lack an equilibrium, best-response dynamics
will run forever: for example, in the Matching Pennies game from Chapter 6, when only pure
strategies are allowed, best-response dynamics will simply consist of the two players endlessly
switching their strategies between H and T . It seems plausible that for some network, this
could happen in the tra�c game as well: one at a time, drivers shift their routes to ones
that are better for them, thus increasing the delay for another driver who then switches and
continues the cascade.
In fact, however, this cannot happen in the tra�c game. We now show that best-response
dynamics must always terminate in an equilibrium, thus proving not only that equilibria exist
but also that they can be reached by a simple process in which drivers constantly update
what they’re doing according to best responses.
Analyzing Best-Response Dynamics Via Potential Energy. How should we go about
proving that best-response dynamics must come to a halt? When you have a process that
runs according to some set of instructions like, “Do the following ten things and then stop,”
it’s generally obvious that it will eventually come to an end: the process essentially comes
with its own guarantee of termination. But we have a process that runs according to a
di↵erent kind of rule, one that says, “Keep doing something until a particular condition
happens to hold.” In this case, there is no a priori reason to believe it will ever stop.
In such cases, a useful analysis technique is to define some kind of progress measure that
tracks the process as it operates, and to show that eventually enough “progress” will be
made that the process must stop. For the tra�c game, it’s natural to think of the social
cost of the current tra�c pattern as a possible progress measure, but in fact the social cost
is not so useful for this purpose. Some best-response updates by drivers can make the social
cost better (for example, if a driver leaves a congested road for a relatively empty one),
but others can make it worse (as in the sequence of best-response updates that shifts the
tra�c pattern from the social optimum to the inferior equilibrium in the Braess Paradox).
So in general, as best-response dynamics runs, the social cost of the current tra�c pattern
can oscillate between going up and going down, and it’s not clear how this is related to our
progress toward an equilibrium.
Instead, we’re going to define an alternate quantity that initially seems a bit mysterious.
However, we will see that it has the property that it strictly decreases with each best-response
update, and so it can be used to track the progress of best-response dynamics [303]. We will
8.3. ADVANCED MATERIAL: THE SOCIAL COST OF TRAFFIC AT EQUILIBRIUM237
refer to this quantity as the potential energy of a tra�c pattern.
The potential energy of a tra�c pattern is defined edge-by-edge, as follows. If an edge e
currently has x drivers on it, then we define the potential energy of this edge to be
Energy(e) = T e
(1) + T e
(2) + · · · + T e
(x).
If an edge has no drivers on it, its potential energy is defined to be 0. The potential energy
of a tra�c pattern is then simply the sum of the potential energies of all the edges, with
their current number of drivers in this tra�c pattern. In Figure 8.5, we show the potential
energy of each edge for the five tra�c patterns that best-response dynamics produces as it
moves from the social optimum to the unique equilibrium in the Braess-Paradox network
from Figure 8.4.
Notice that the potential energy of an edge e with x drivers is not the total travel time
experienced by the drivers that cross it. Since there are x drivers each experiencing a travel
time of T e
(x), their total travel time is xT e
(x), which is a di↵erent number. The potential
energy, instead, is a sort of “cumulative” quantity in which we imagine drivers crossing the
edge one by one, and each driver only “feels” the delay caused by himself and the drivers
crossing the edge in front of him.
Of course, the potential energy is only useful for our purpose if it lets us analyze the
progress of best-response dynamics. We show how to do this next.
Proving that Best-Response Dynamics Comes to an End. Our main claim is the
following: each step of best-response dynamics causes the potential energy of the current
tra�c pattern to strictly decrease. Proving this will be enough to show that best-response
dynamics must come to an end, for the following reason. The potential energy can only take
a finite number of possible values — one for each possible tra�c pattern. If it is strictly
decreasing with each step of best-response dynamics, this means that it is “consuming” this
finite supply of possible values, since it can never revisit a value once it drops below it. So
best-response dynamics must come to a stop by the time the potential energy reaches its
minimum possible value (if not sooner). And once best-response dynamics comes to a stop,
we must be at an equilibrium — for otherwise, the dynamics would have a way to continue.
Thus, showing that the potential energy strictly decreases in every step of best-response
dynamics is enough to show the existence of an equilibrium tra�c pattern.
As an example, let’s return to the sequence of best-response steps from Figure 8.5. Al-
though the social cost is rising through the five tra�c patterns (increasing from 28 to 32),
the potential energy decreases strictly in each step (in the sequence 26, 24, 23, 21, 20). In
fact, it is easy to track the change in potential energy through this sequence as follows. From
one tra�c pattern to the next, the only change is that one driver abandons his current path
and switches to a new one. Suppose we really view this switch as a two-step process: first the
238 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
BA
C
D
x
5
5
x
0
energy = 1+2 energy = 5+5
energy = 5+5 energy = 1+2
(a) The initial tra�c pattern. (Potential energy is 26.)
BA
C
D
x
5
5
x
0
energy = 1+2 energy = 5
energy = 5+5 energy = 1+2+3
(b) After one step of best-response dynamics. (Po- tential energy is 24.)
BA
C
D
x
5
5
x
0
energy = 1+2 energy = 0
energy = 5+5 energy = 1+2+3+4
(c) After two steps. (Potential energy is 23.)
BA
C
D
x
5
5
x
0
energy = 1+2+3 energy = 0
energy = 5 energy = 1+2+3+4
(d) After three steps. (Potential energy is 21.)
BA
C
D
x
5
5
x
0
energy = 1+2+3+4 energy = 0
energy = 0 energy = 1+2+3+4
(e) After four steps: Equilibrium is reached. (Potential energy is 20.)
Figure 8.5: We can track the progress of best-response dynamics in the tra�c game by watching how the potential energy changes.
8.3. ADVANCED MATERIAL: THE SOCIAL COST OF TRAFFIC AT EQUILIBRIUM239
BA
C
D
x
5
5
x
0
energy = 1+2 energy = 5+5
energy = 5+5 energy = 1+2
(a) The potential energy of a tra�c pattern not in equilibrium.
BA
C
D
x
5
5
x
0
energy = 1 energy = 5
energy = 5+5 energy = 1+2
(b) Potential energy is released when a driver aban- dons their current path.
BA
C
D
x
5
5
x
0
energy = 1+2 energy = 5
energy = 5+5 energy = 1+2+3
(c) Potential energy is put back into the system when the driver chooses a new path.
Figure 8.6: When a driver abandons one path in favor of another, the change in potential energy is exactly the improvement in the driver’s travel time.
drivers abandons his current path, temporarily leaving the system; then, the driver returns
to the system by adopting a new path. This first step releases potential energy as the driver
leaves the system, and the second step adds potential energy as he re-joins. What’s the net
change?
For example, the transition from Figure 8.5(a) to 8.5(b) occurs because one driver aban-
dons the upper path and adopts the zigzag path. As shown in Figure 8.6, abandoning the
upper path releases 2 + 5 = 7 units of potential energy, while adopting the zigzag path puts
2 + 0 + 3 units of potential energy back into the system. The resulting change is a decrease
of 2.
Notice that the decrease of 7 is simply the travel time the driver was experiencing on
the path he abandoned, and the subsequent increase of 5 is the travel time the driver now
experiences on the path he has adopted. This relationship is in fact true for any network and
240 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
any best response by a driver, and it holds for a simple reason. Specifically, the potential
energy of edge e with x drivers is
T e
(1) + T e
(2) + · · · + T e
(x � 1) + T e
(x),
and when one of these drivers leaves it drops to
T e
(1) + T e
(2) + · · · + T e
(x � 1).
Hence the change in potential energy on edge e is T e
(x), exactly the travel time that the
driver was experiencing on e. Summing this over all edges used by the driver, we see that
the potential energy released when a driver abandons his current path is exactly equal to the
travel time the driver was experiencing. By the same reasoning, when a driver adopts a new
path, the potential energy on each edge e he joins increases from
T e
(1) + T e
(2) + · · · + T e
(x)
to
T e
(1) + T e
(2) + · · · + T e
(x) + T e
(x + 1),
and the increase of T e
(x + 1) is exactly the new travel time the driver experiences on this
edge. Hence, the potential energy added to the system when a driver adopts a new path is
exactly equal to the travel time the driver now experiences.
It follows when a driver switches paths, the net change in potential energy is simply his
new travel time minus his old travel time. But in best-response dynamics, a driver only
changes paths when it causes his travel time to decrease — so the change in potential energy
is negative for any best-response move. This establishes what we wanted to show: that the
potential energy in the system strictly decreases throughout best-response dynamics. As
argued above, since the potential energy cannot decrease forever, best-response dynamics
must therefore eventually come to an end, at a tra�c pattern in equilibrium.
B. Comparing Equilibrium Tra�c to the Social Optimum
Having shown that an equilibrium tra�c pattern always exists, we now consider how its
travel time compares to that of a socially optimal tra�c pattern. We will see that the
potential energy we’ve defined is very useful for making this comparison. The basic idea is
to establish a relationship between the potential energy of an edge and the total travel time
of all drivers crossing the edge. Once we do this, we will sum these two quantities over all
the edges to compare travel times at equiibrium and at social optimality.
8.3. ADVANCED MATERIAL: THE SOCIAL COST OF TRAFFIC AT EQUILIBRIUM241
......
Te(1)
Te(x)
Te(3)
Te(2)
Figure 8.7: The potential energy is the area under the shaded rectangles; it is always at least half the total travel time, which is the area inside the enclosing rectangle.
Relating Potential Energy to Travel Time for a Single Edge. We denote the po-
tential energy of an edge by Energy(e), and we recall that when there are x drivers, this
potential energy is defined by
Energy(e) = T e
(1) + T e
(2) + · · · + T e
(x).
On the other hand, each of the x drivers experiences a travel time of T e
(x), and so the total
travel time experienced by all drivers on the edge is
Total-Travel-Time(e) = xT e
(x).
For purposes of comparison with the potential energy, it is useful to write this as follows:
Total-Travel-Time(e) = T e
(x) + T e
(x) + · · · + T e
(x) | {z }
x terms
.
Since the potential energy and the total travel time each have x terms, but the terms in the
latter expression are at least as large as the terms in the former, we have
Energy(e) Total-Travel-Time(e).
Figure 8.7 shows how the potential energy and the total travel time compare when T e
is a linear function: the total travel time is the shaded area under the horizontal line with
242 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
y-value T e
(x), while the potential energy is the total area under all the unit-width rectangles
of heights T e
(1), T e
(2), . . . , T e
(x). As this figure makes clear geometrically, since T e
is a linear
function, we have
T e
(1) + T e
(2) + · · · + T e
(x) � 1
2 xT
e
(x).
Alternately, we can see this by a bit of simple algebra, recalling that T e
(x) = a e
x + b e
:
T e
(1) + T e
(2) + · · · + T e
(x) = a e
(1 + 2 + · · · + x) + b e
x
= a
e
x(x + 1)
2 + b
e
x
= x
✓ a
e
(x + 1)
2 + b
e
◆
� 1
2 x(a
e
x + b e
)
= 1
2 xT
e
(x).
In terms of energies and total travel times, this says
Energy(e) � 1
2 · Total-Travel-Time(e).
So the conclusion is that the potential energy of an edge is never far from the total travel
time: it is sandwiched between the total travel time and half the total travel time.
Relating the Travel Time at Equilibrium and Social Optimality. We now use this
relationship between potential energy and total travel to relate the equilibrium and socially
optimal tra�c patterns.
Let Z be a tra�c pattern; we define Energy(Z) to be the total potential energy of all
edges when drivers follow the tra�c pattern Z. We write Social-Cost(Z) to denote the
social cost of the tra�c pattern; recall that this is the sum of the travel times experienced by
all drivers. Equivalently, summing the social cost edge-by-edge, Social-Cost(Z) is the sum
of the total travel times on all the edges. So applying our relationships between potential
energy and travel time on an edge-by-edge basis, we see that the same relationships govern
the potential energy and social cost of a tra�c pattern:
1
2 · Social-Cost(Z) Energy(Z) Social-Cost(Z).
Now, suppose that we start from a socially optimal tra�c pattern Z, and we then allow
best-response dynamics to run until they stop at an equilibrium tra�c pattern Z0. The social
cost may start increasing as we run best-response dynamics, but the potential energy can
only go down — and since the social cost can never be more than twice the potential energy,
this shrinking potential energy keeps the social cost from ever getting more than twice as
8.4. EXERCISES 243
high as where it started. This shows that the social cost of the equilibrium we reach is at
most twice the cost of the social optimum we started with — hence there is an equilibrium
with at most twice the socially optimal cost, as we wanted to show.
Let’s write this argument out in terms of the inequalities on energies and social costs.
First, we saw in the previous section that the potential energy decreases as best-response
dynamics moves from Z to Z0, and so
Energy(Z0) Energy(Z).
Second, the quantitative relationships between energies and social cost say that
Social-Cost(Z0) 2 · Energy(Z0)
and
Energy(Z) Social-Cost(Z).
Now we just chain these inequalities together, concluding that
Social-Cost(Z0) 2 · Energy(Z0) 2 · Energy(Z) 2 · Social-Cost(Z).
Note that this really is the same argument that we made in words in the previous paragraph:
the potential energy decreases during best-response dynamics, and this decrease prevents the
social cost from every increasing by more than a factor of two.
Thus, tracking potential energy is not only useful for showing that best-response dynamics
must reach an equilibrium; by relating this potential energy to the social cost, we can use it
to put a bound on the social cost of the equilibrium that is reached.
8.4 Exercises
1. There are 1000 cars which must travel from town A to town B. There are two possible
routes that each car can take: the upper route through town C or the lower route
through town D. Let x be the number of cars traveling on the edge AC and let y be
the number of cars traveling on the edge DB. The directed graph in Figure 8.8 indicates
that travel time per car on edge AC is x/100 if x cars use edge AC, and similarly the
travel time per car on edge DB is y/100 if y cars use edge DB. The travel time per
car on each of edges CB and AD is 12 regardless of the number of cars on these edges.
Each driver wants to select a route to minimize his travel time. The drivers make
simultaneous choices.
(a) Find Nash equilibrium values of x and y.
(b) Now the government builds a new (one-way) road from town C to town D. The
new road adds the path ACDB to the network. This new road from C to D has a travel
244 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
BA
C
D
x/100 12
12 y/100
Figure 8.8: Tra�c Network.
time of 0 per car regardless of the number of cars that use it. Find a Nash equilibrium
for the game played on the new network. What are the equilibrium values of x and y?
What happens to total cost-of-travel (the sum of total travel times for the 1000 cars)
as a result of the availability of the new road?
(c) Suppose now that conditions on edges CB and AD are improved so that the travel
times on each edge are reduced to 5. The road from C to D that was constructed in
part (b) is still available. Find a Nash equilibrium for the game played on the network
with the smaller travel times for CB and AD. What are the equilibrium values of x and
y? What is the total cost-of-travel? What would happen to the total cost-of-travel if
the government closed the road from C to D?
2. There are two cities A and B joined by two routes. There are 80 travelers who begin
in city A and must travel to city B. There are two routes between A and B. Route
I begins with a highway leaving city A, this highway takes one hour of travel time
regardless of how many travelers use it, and ends with a local street leading into city
B. This local street near city B requires a travel time in minutes equal to 10 plus the
number of travelers who use the street. Route II begins with a local street leaving city
A, which requires a travel time in minutes equal to 10 plus the number of travelers
who use this street, and ends with a highway into city B which requires one hour of
travel time regardless of the number of travelers who use this highway.
(a) Draw the network described above and label the edges with the travel time needed
to move along the edge. Let x be the number of travelers who use Route I. The network
should be a directed graph as all roads are one-way.
(b) Travelers simultaneously chose which route to use. Find the Nash equilibrium
value of x.
(c) Now the government builds a new (two-way) road connecting the nodes where local
streets and highways meet. This adds two new routes. One new route consists of the
8.4. EXERCISES 245
BA
C
D
x/100
3.1
3.1
y/100
Figure 8.9: Tra�c Network
local street leaving city A (on Route II), the new road and the local street into city B
(on Route I). The second new route consists of the highway leaving city A (on Route
I), the new road and the highway leading into city B (on Route II). The new road is
very short and takes no travel time. Find the new Nash equilibrium. (Hint: There is
an equilibrium in which no one chooses to use the second new route described above.)
(d) What happens to total travel time as a result of the availability of the new road?
(e) If you can assign travelers to routes, then in fact it’s possible to reduce total travel
time relative to what it was before the new road was built. That is, the total travel
time of the population can be reduced (below that in the original Nash equilibrium
from part (b)) by assigning travelers to routes. There are many assignments of routes
that will accomplish this. Find one. Explain why your reassignment reduces total
travel time. (Hint: Remember that travel on the new road can go in either direction.
You do not need find the total travel time minimizing assignment of travelers. One
approach to this question is to start with the Nash equilibrium from part (b) and look
for a way to assign some travelers to di↵erent routes so as to reduce total travel time.)
3. There are 300 cars which must travel from city A to city B. There are two possible
routes that each car can take. The upper route through city C or the lower route
through city D. Let x be the number of cars traveling on the edge AC and let y be the
number of cars traveling on the edge DB. The directed graph in Figure 8.9 indicates
that total travel time per car along the upper route is (x/100) + 3.1 if x cars use
the upper route, and similarly the total travel time per car along the lower route is
3.1 + (y/100) if y cars take the lower route. Each driver wants to select a route to
246 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
minimize his total travel time. The drivers make simultaneous choices.
(a) Find Nash equilibrium values of x and y.
(b) Now the government builds a new (one-way) road from city A to city B. The new
route has a travel time of 5 per car regardless of the number of cars that use it. Draw
the new network and label the edges with the cost-of-travel needed to move along the
edge. The network should be a directed graph as all roads are one-way. Find a Nash
equilibrium for the game played on the new network. What happens to total cost-of-
travel (the sum of total travel times for the 300 cars) as a result of the availability of
the new road?
(c) Now the government closes the direct route between city A and city B and builds
a new one-way road which links city C to city D. This new road between C and D
is very short and has a travel time of 0 regardless of the number of cars that use it.
Draw the new network and label the edges with the cost-of-travel needed to move along
the edge. The network should be a directed graph as all roads are one-way. Find a
Nash equilibrium for the game played on the new network. What happens to total
cost-of-travel as a result of the availability of the new road?
(d) The government is unhappy with the outcome in part (c) and decides to reopen the
road directly linking city A and city B (the road that was built in part (b) and closed in
part (c)). The route between C and D that was constructed in part (c) remains open.
This road still has a travel time of 5 per car regardless of the number of cars that use
it. Draw the new network and label the edges with the cost-of-travel needed to move
along the edge. The network should be a directed graph as all roads are one-way. Find
a Nash equilibrium for the game played on the new network. What happens to total
cost-of-travel as a result of re-opening the direct route between A and B?
4. There are two cities, A and B, joined by two routes, I and II. All roads are one-way
roads. There are 100 travelers who begin in city A and must travel to city B. Route I
links city A to city B through city C. This route begins with a road linking city A to
city C which has a cost-of-travel for each traveler equal to 0.5 + x/200, where x is the
number of travelers on this road. Route I ends with a highway from city C to city B
which has a cost-of-travel for each traveler of 1 regardless of the number of travelers
who use it. Route II links city A to city B through city D. This route begins with
a highway linking city A to city D which has a cost-of-travel for each traveler of 1
regardless of the number of travelers who use it. Route II ends with a road linking city
D to city B which has a cost-of-travel for each traveler equal to 0.5 + y/200, where y
is the number of travelers on this road.
These costs of travel are the value that travelers put on the time lost due to travel plus
the cost of gasoline for the trip. Currently there are no tolls on these roads. So the
8.4. EXERCISES 247
government collects no revenue from travel on them.
(a) Draw the network described above and label the edges with the cost-of-travel
needed to move along the edge. The network should be a directed graph as all roads
are one-way.
(b) Travelers simultaneously chose which route to use. Find Nash equilibrium values
of x and y.
(c) Now the government builds a new (one-way) road from city C to city D. The new
road is very short and has 0 cost-of-travel. Find a Nash equilibrium for the game
played on the new network.
(d) What happens to total cost-of-travel as a result of the availability of the new road?
(e) The government is unhappy with the outcome in part (c) and decides to impose a
toll on users of the road from city A to city C and to simultaneously subsidize users of
the highway from city A to city D. They charge a toll of 0.125 to each user, and thus
increase the cost-of-travel by this amount, for users of the road from city A to city C.
They also subsidize travel, and thus reduce the cost-of-travel by this amount, for each
user of the highway from city A to city D by 0.125. Find a new Nash equilibrium.
[If you are curious about how a subsidy could work you can think of it as a negative
toll. In this economy all tolls are collected electronically, much as New York State
attempts to do with its E-ZPass system. So a subsidy just reduces the total amount
that highway users owe.]
(f) As you will observe in solving part (e) the toll and subsidy in part (e) were designed
so that there is a Nash equilibrium in which the amount the government collects from
the toll just equals the amount it loses on the subsidy. So the government is breaking
even on this policy. What happens to total cost-of-travel between parts (c) and (e)?
Can you explain why this occurs? Can you think of any break-even tolls and subsidies
that could be placed on the roads from city C to city B, and from city D to city B,
that would lower the total cost-of-travel even more?
248 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY
Chapter 9
Auctions
In Chapter 8, we considered a first extended application of game-theoretic ideas, in our
analysis of tra�c flow through a network. Here we consider a second major application —
the behavior of buyers and sellers in an auction.
An auction is a kind of economic activity that has been brought into many people’s
everyday lives by the Internet, through sites such as eBay. But auctions also have a long
history that spans many di↵erent domains. For example, the U.S. government uses auctions
to sell Treasury bills and timber and oil leases; Christie’s and Sotheby’s use them to sell art;
and Morrell & Co. and the Chicago Wine Company use them to sell wine.
Auctions will also play an important and recurring role in the book, since the simplified
form of buyer-seller interaction they embody is closely related to more complex forms of
economic interaction as well. In particular, when we think in the next part of the book
about markets in which multiple buyers and sellers are connected by an underlying network
structure, we’ll use ideas initially developed in this chapter for understanding simpler auction
formats. Similarly, in Chapter 15, we’ll study a more complex kind of auction in the context
of a Web search application, analyzing the ways in which search companies like Google,
Yahoo!, and Microsoft use an auction format to sell advertising rights for keywords.
9.1 Types of Auctions
In this chapter we focus on di↵erent simple types of auctions, and how they promote di↵erent
kinds of behavior among bidders. We’ll consider the case of a seller auctioning one item to
a set of buyers. We could symmetrically think of a situation in which a buyer is trying to
purchase a single item, and runs an auction among a set of multiple sellers, each of whom is
able to provide the item. Such procurement auctions are frequently run by governments to
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
249
250 CHAPTER 9. AUCTIONS
purchase goods. But here we’ll focus on the case in which the seller runs the auction.
There are many di↵erent ways of defining auctions that are much more complex than
what we consider here. The subsequent chapters will generalize our analysis to the case in
which there are multiple goods being sold, and the buyers assign di↵erent values to these
goods. Other variations, which fall outside the scope of the book, include auctions in which
goods are sold sequentially over time. These more complex variations can also be analyzed
using extensions of the ideas we’ll talk about here, and there is a large literature in economics
that considers auctions at this broad level of generality [256, 292].
The underlying assumption we make when modeling auctions is that each bidder has an
intrinsic value for the item being auctioned; she is willing to purchase the item for a price
up to this value, but not for any higher price. We will also refer to this intrinsic value as the
bidder’s true value for the item. There are four main types of auctions when a single item is
being sold (and many variants of these types).
1. Ascending-bid auctions, also called English auctions. These auctions are carried out
interactively in real time, with bidders present either physically or electronically. The
seller gradually raises the price, bidders drop out until finally only one bidder remains,
and that bidder wins the object at this final price. Oral auctions in which bidders
shout out prices, or submit them electronically, are forms of ascending-bid auctions.
2. Descending-bid auctions, also called Dutch auctions. This is also an interactive auction
format, in which the seller gradually lowers the price from some high initial value until
the first moment when some bidder accepts and pays the current price. These auctions
are called Dutch auctions because flowers have long been sold in the Netherlands using
this procedure.
3. First-price sealed-bid auctions. In this kind of auction, bidders submit simultaneous
“sealed bids” to the seller. The terminology comes from the original format for such
auctions, in which bids were written down and provided in sealed envelopes to the
seller, who would then open them all together. The highest bidder wins the object and
pays the value of her bid.
4. Second-price sealed-bid auctions, also called Vickrey auctions. Bidders submit simul-
taneous sealed bids to the sellers; the highest bidder wins the object and pays the
value of the second-highest bid. These auctions are called Vickrey auctions in honor
of William Vickrey, who wrote the first game-theoretic analysis of auctions (including
the second-price auction [400]). Vickery won the Nobel Memorial Prize in Economics
in 1996 for this body of work.
9.2. WHEN ARE AUCTIONS APPROPRIATE? 251
9.2 When are Auctions Appropriate?
Auctions are generally used by sellers in situations where they do not have a good estimate
of the buyers’ true values for an item, and where buyers do not know each other’s values. In
this case, as we will see, some of the main auction formats can be used to elicit bids from
buyers that reveal these values.
Known Values. To motivate the setting in which buyers’ true values are unknown, let’s
start by considering the case in which the seller and buyers know each other’s values for an
item, and argue that an auction is unnecessary in this scenario. In particular, suppose that
a seller is trying to sell an item that he values at x, and suppose that the maximum value
held by a potential buyer of the item is some larger number y. In this case, we say there is a
surplus of y � x that can be generated by the sale of the item: it can go from someone who values it less (x) to someone who values it more (y).
If the seller knows the true values that the potential buyers assign to the item, then he
can simply announce that the item is for sale at a fixed price just below y, and that he will
not accept any lower price. In this case, the buyer with value y will buy the item, and the
full value of the surplus will go to the seller. In other words, the seller has no need for an
auction in this case: he gets as much as he could reasonably expect just by announcing the
right price.
Notice that there is an asymmetry in the formulation of this example: we gave the seller
the ability to commit to the mechanism that was used for selling the object. This ability
of the seller to “tie his hands” by committing to a fixed price is in fact very valuable to
him: assuming the buyers believe this commitment, the item is sold for a price just below
y, and the seller makes all the surplus. In contrast, consider what would happen if we gave
the buyer with maximum value y the ability to commit to the mechanism. In this case,
this buyer could announce that she is willing to purchase the item for a price just above
the larger of x and the values held by all other buyers. With this announcement, the seller
would still be willing to sell — since the price would be above x — but now at least some
of the surplus would go to the buyer. As with the seller’s commitment, this commitment by
the buyer also requires knowledge of everyone else’s values.
These examples show how commitment to a mechanism can shift the power in the trans-
action in favor of the seller or the buyer. One can also imagine more complex scenarios in
which the seller and buyers know each other’s values, but neither has the power to unilater-
ally commit to a mechanism. In this case, one may see some kind of bargaining take place
over the price; we discuss the topic of bargaining further in Chapter 12. As we will discover
in the current chapter, the issue of commitment is also crucial in the context of auctions —
specifically, it is important that a seller be able to reliably commit in advance to a given
auction format.
252 CHAPTER 9. AUCTIONS
Unknown Values. Thus far we’ve been discussing how sellers and buyers might interact
when everyone knows each other’s true values for the item. Beginning in the next section,
we’ll see how auctions come into play when the participants do not know each other’s values.
For most of this chapter we will restrict our attention to the case in which the buyers
have independent, private values for the item. That is, each buyer knows how much she
values the item, she does not know how much others value it, and her value for it does not
depend on others’ values. For example, the buyers could be interested in consuming the
item, with their values reflecting how much they each would enjoy it.
Later we will also consider the polar opposite of this setting — the case of common
values. Suppose that an item is being auctioned, and instead of consuming the item, each
buyer plans to resell the item if she gets it. In this case (assuming the buyers will do a
comparably good job of reselling it), the item has an unknown but common value regardless
of who acquires it: it is equal to how much revenue this future reselling of the item will
generate. Buyers’ estimates of this revenue may di↵er if they have some private information
about the common value, and so their valuations of the item may di↵er. In this setting, the
value each buyer assigns to the object would be a↵ected by knowledge of the other buyers’
valuations, since the buyers could use this knowledge to further refine their estimates of the
common value.
9.3 Relationships between Di↵erent Auction Formats
Our main goal will be to consider how bidders behave in di↵erent types of auctions. We begin
in this section with some simple, informal observations that relate behavior in interactive
auctions (ascending-bid and descending-bid auctions, which play out in real time) with
behavior in sealed-bid auctions. These observations can be made mathematically rigorous,
but for the discussion here we will stick to an informal description.
Descending-Bid and First-Price Auctions. First, consider a descending-bid auction.
Here, as the seller is lowering the price from its high initial starting point, no bidder says
anything until finally someone actually accepts the bid and pays the current price. Bidders
therefore learn nothing while the auction is running, other than the fact that no one has
yet accepted the current price. For each bidder i, there’s a first price b i
at which she’ll be
willing to break the silence and accept the item at price b i
. So with this view, the process
is equivalent to a sealed-bid first-price auction: this price b i
plays the role of bidder i’s bid;
the item goes to the bidder with the highest bid value; and this bidder pays the value of her
bid in exchange for the item.
9.3. RELATIONSHIPS BETWEEN DIFFERENT AUCTION FORMATS 253
Ascending-Bid and Second-Price Auctions. Now let’s think about an ascending-bid
auction, in which bidders gradually drop out as the seller steadily raises the price. The
winner of the auction is the last bidder remaining, and she pays the price at which the
second-to-last bidder drops out.1
Suppose that you’re a bidder in such an auction; let’s consider how long you should stay
in the auction before dropping out. First, does it ever make sense to stay in the auction after
the price reaches your true value? No: by staying in, you either lose and get nothing, or else
you win and have to pay more than your value for the item. Second, does it ever make sense
to drop out before the price reaches your true value for the item? Again, no: if you drop
out early (before your true value is reached), then you get nothing, when by staying in you
might win the item at a price below your true value.
So this informal argument indicates that you should stay in an ascending-bid auction up
to the exact moment at which the price reaches your true value. If we think of each bidder
i’s “drop-out price” as her bid b i
, this says that people should use their true values as their
bids.
Moreover, with this definition of bids, the rule for determining the outcome of an ascending-
bid auction can be reformulated as follows. The person with the highest bid is the one who
stays in the longest, thus winning the item, and she pays the price at which the second-to-
last person dropped out — in other words, she pays the bid of this second-to-last person.
Thus, the item goes to the highest bidder at a price equal to the second-highest bid. This
is precisely the rule used in the sealed-bid second-price auction, with the di↵erence being
that the ascending-bid auction involves real-time interaction between the buyers and seller,
while the sealed-bid version takes place purely through sealed bids that the seller opens and
evaluates. But the close similarity in rules helps to motivate the initially counter-intuitive
pricing rule for the second-price auction: it can be viewed as a simulation, using sealed
bids, of an ascending-bid auction. Moreover, the fact that bidders want to remain in an
ascending-bid auction up to exactly the point at which their true value is reached provides
the intuition for what will be our main result in the next section: after formulating the
sealed-bid second-price auction in terms of game theory, we will find that bidding one’s true
value is a dominant strategy.
1It’s conceptually simplest to think of three things happening simultaneously at the end of an ascending- bid auction: (i) the second-to-last bidder drops out; (ii) the last remaining bidder sees that she is alone and stops agreeing to any higher prices; and (iii) the seller awards the item to this last remaining bidder at the current price. Of course, in practice we might well expect that there is some very small increment by which the bid is raised in each step, and that the last remaining bidder actually wins only after one more raising of the bid by this tiny increment. But keeping track of this small increment makes for a more cumbersome analysis without changing the underlying ideas, and so we will assume that the auction ends at precisely the moment when the second-highest bidder drops out.
254 CHAPTER 9. AUCTIONS
Comparing Auction Formats. In the next two sections we will consider the two main
formats for sealed-bid auctions in more detail. Before doing this, it’s worth making two
points. First, the discussion in this section shows that when we analyze bidder behav-
ior in sealed-bid auctions, we’re also learning about their interactive analogues — with
the descending-bid auction as the analogue of the sealed-bid first-price auction, and the
ascending-bid auction as the analogue of the sealed-bid second-price auction.
Second, a purely superficial comparison of the first-price and second-price sealed-bid
auctions might suggest that the seller would get more money for the item if he ran a first-
price auction: after all, he’ll get paid the highest bid rather than the second-highest bid. It
may seem strange that in a second-price auction, the seller is intentionally undercharging
the bidders. But such reasoning ignores one of the main messages from our study of game
theory — that when you make up rules to govern people’s behavior, you have to assume
that they’ll adapt their behavior in light of the rules. Here, the point is that bidders in a
first-price auction will tend to bid lower than they do in a second-price auction, and in fact
this lowering of bids will tend to o↵set what would otherwise look like a di↵erence in the
size of the winning bid. This consideration will come up as a central issue at various points
later in the chapter.
9.4 Second-Price Auctions
The sealed-bid second-price auction is particularly interesting, and there are a number of
examples of it in widespread use. The auction form used on eBay is essentially a second-price
auction. The pricing mechanism that search engines use to sell keyword-based advertising is
a generalization of the second-price auction, as we will see in Chapter 15. One of the most
important results in auction theory is the fact we mentioned toward the end of the previous
section: with independent, private values, bidding your true value is a dominant strategy in
a second price sealed-bid auction. That is, the best choice of bid is exactly what the object
is worth to you.
Formulating the Second-Price Auction as a Game. To see why this is true, we
set things up using the language of game theory, defining the auction in terms of players,
strategies, and payo↵s. The bidders will correspond to the players. Let v i
be bidder i’s true
value for the object. Bidder i’s strategy is an amount b i
to bid as a function of her true
value v i
. In a second-price sealed-bid auction, the payo↵ to bidder i with value v i
and bid b i
is defined as follows.
If b i
is not the winning bid, then the payo↵ to i is 0. If b i
is the winning bid, and
some other b j
is the second-place bid, then the payo↵ to i is v i
� b j
.
9.4. SECOND-PRICE AUCTIONS 255
Alternate bid bi''
Truthful bid bi = vi
Alternate bid bi'
Raised bid affects outcome only if highest other bid bj is in between.
If so, i wins but pays more than value.
Lowered bid affects outcome only if highest other bid bk is in between.
If so, i loses when it was possible to win with non-negative payoff
Figure 9.1: If bidder i deviates from a truthful bid in a second-price auction, the payo↵ is only a↵ected if the change in bid changes the win/loss outcome.
To make this completely well-defined, we need to handle the possibility of ties: what do
we do if two people submit the same bid, and it’s tied for the largest? One way to handle this
is to assume that there is a fixed ordering on the bidders that is agreed on in advance, and if
a set of bidders ties for the numerically largest bid, then the winning bid is the one submitted
by the bidder in this set that comes first in this order. Our formulation of the payo↵s works
with this more refined definition of “winning bid” and “second-place bid.” (And note that
in the case of a tie, the winning bidder receives the item but pays the full value of her own
bid, for a payo↵ of zero, since in the event of a tie the first-place and second-place bids are
equal.)
There is one further point worth noting about our formulation of auctions in the language
of game theory. When we defined games in Chapter 6, we assumed that each player knew the
payo↵s of all players in the game. Here this isn’t the case, since the bidders don’t know each
other’s values, and so strictly speaking we need to use a slight generalization of the notions
256 CHAPTER 9. AUCTIONS
from Chapter 6 to handle this lack of knowledge. For our analysis here, however, since we
are focusing on dominant strategies in which a player has an optimal strategy regardless of
the other players’ behavior, we will be able to disregard this subtlety.
Truthful Bidding in Second-Price Auctions. The precise statement of our claim about
second-price auctions is as follows.
Claim: In a sealed-bid second-price auction, it is a dominant strategy for each
bidder i to choose a bid b i
= v i
.
To prove this claim, we need to show that if bidder i bids b i
= v i
, then no deviation from
this bid would improve her payo↵, regardless of what strategy everyone else is using. There
are two cases to consider: deviations in which i raises her bid, and deviations in which i
lowers her bid. The key point in both cases is that the value of i’s bid only a↵ects whether
i wins or loses, but never a↵ects how much i pays in the event that she wins — the amount
paid is determined entirely by the other bids, and in particular by the largest among the
other bids. Since all other bids remain the same when i changes her bid, a change to i’s bid
only a↵ects her payo↵ if it changes her win/loss outcome. This argument is summarized in
Figure 9.1.
With this in mind, let’s consider the two cases. First, suppose that instead of bidding
v i
, bidder i chooses a bid b0 i
> v i
. This only a↵ects bidder i’s payo↵ if i would lose with bid
v i
but would win with bid b0 i
. In order for this to happen, the highest other bid b j
must be
between b i
and b0 i
. In this case, the payo↵ to i from deviating would be at most v i
� b j
0, and so this deviation to bid b0
i
does not improve i’s payo↵.
Next, suppose that instead of bidding v i
, bidder i chooses a bid b00 i
< v i
. This only a↵ects
bidder i’s payo↵ if i would win with bid v i
but would lose with bid b00 i
. So before deviating,
v i
was the winning bid, and the second-place bid b k
was between v i
and b00 i
. In this case, i’s
payo↵ before deviating was v i
� b k
� 0, and after deviating it is 0 (since i loses), so again this deviation does not improve i’s payo↵.
This completes the argument that truthful bidding is a dominant strategy in a sealed-
bid second-price auction. The heart of the argument is the fact noted at the outset: in a
second-price auction, your bid determines whether you win or lose, but not how much you
pay in the event that you win. Therefore, you need to evaluate changes to your bid in light
of this. This also further highlights the parallels to the ascending-bid auction. There too,
the analogue of your bid — i.e. the point up to which you’re willing to stay in the auction
— determines whether you’ll stay in long enough to win; but the amount you pay in the
event that you win is determined by the point at which the second-place bidder drops out.
The fact that truthfulness is a dominant strategy also makes second-price auctions con-
ceptually very clean. Because truthful bidding is a dominant strategy, it is the best thing
to do regardless of what the other bidders are doing. So in a second-price auction, it makes
9.5. FIRST-PRICE AUCTIONS AND OTHER FORMATS 257
sense to bid your true value even if other bidders are overbidding, underbidding, colluding,
or behaving in other unpredictable ways. In other words, truthful bidding is a good idea even
if the competing bidders in the auction don’t know that they ought to be bidding truthfully
as well.
We now turn to first-price auctions, where we’ll find that the situation is much more
complex. In particular, each bidder now has to reason about the behavior of her competitors
in order to arrive at an optimal choice for her own bid.
9.5 First-Price Auctions and Other Formats
In a sealed-bid first-price auction, the value of your bid not only a↵ects whether you win but
also how much you pay. As a result, most of the reasoning from the previous section has to
be redone, and the conclusions are now di↵erent.
To begin with, we can set up the first-price auction as a game in essentially the same
way that we did for second-price auctions. As before, bidders are players, and each bidder’s
strategy is an amount to bid as a function of her true value. The payo↵ to bidder i with
value v i
and bid b i
is simply the following.
If b i
is not the winning bid, then the payo↵ to i is 0. If b i
is the winning bid,
then the payo↵ to i is v i
� b i
.
The first thing we notice is that bidding your true value is no longer a dominant strategy.
By bidding your true value, you would get a payo↵ of 0 if you lose (as usual), and you would
also get a payo↵ of 0 if you win, since you’d pay exactly what it was worth to you.
As a result, the optimal way to bid in a first-price auction is to “shade” your bid slightly
downward, so that if you win you will get a positive payo↵. Determining how much to shade
your bid involves balancing a trade-o↵ between two opposing forces. If you bid too close to
your true value, then your payo↵ won’t be very large in the event that you win. But if you
bid too far below your true value, so as to increase your payo↵ in the event of winning, then
you reduce your chance of being the high bid and hence your chance of winning at all.
Finding the optimal trade-o↵ between these two factors is a complex problem that de-
pends on knowledge of the other bidders and their distribution of possible values. For
example, it is intuitively natural that your bid should be higher — i.e. shaded less, closer
to your true value — in a first-price auction with many competing bidders than in a first-
price auction with only a few competing bidders (keeping other properties of the bidders the
same). This is simply because with a large pool of other bidders, the highest competing bid
is likely to be larger, and hence you need to bid higher to get above this and be the highest
bid. We will discuss how to determine the optimal bid for a first-price auction in Section 9.7.
258 CHAPTER 9. AUCTIONS
All-pay auctions. There are other sealed-bid auction formats that arise in di↵erent set-
tings. One that initially seems counter-intuitive in its formulation is the all-pay auction:
each bidder submits a bid; the highest bidder receives the item; and all bidders pay their
bids, regardless of whether they win or lose. That is, the payo↵s are now as follows.
If b i
is not the winning bid, then the payo↵ to i is �b i
. If b i
is the winning bid,
then the payo↵ to i is v i
� b i
.
Games with this type of payo↵ arise in a number of situations, usually where the notion
of “bidding” is implicit. Political lobbying can be modeled in this way: each side must
spend money on lobbying, but only the successful side receives anything of value for this
expenditure. While it is not true that the side spending more on lobbying always wins, there
is a clear analogy between the amount spent on lobbying and a bid, with all parties paying
their bid regardless of whether they win or lose. One can picture similar considerations
arising in settings such as design competitions, where competing architectural firms spend
money on preliminary designs to try to win a contract from a client. This money must be
spent before the client makes a decision.
The determination of an optimal bid in an all-pay auction shares a number of qualitative
features with the reasoning in a first-price auction: in general you want to bid below your true
value, and you must balance the trade-o↵ between bidding high (increasing your probability
of winning) and bidding low (decreasing your expenditure if you lose and increasing your
payo↵ if you win). In general, the fact that everyone must pay in this auction format means
that bids will typically be shaded much lower than in a first-price auction. The framework
we develop for determining optimal bids in first-price auctions will also apply to all-pay
auctions, as we will see in Section 9.7.
9.6 Common Values and The Winner’s Curse
Thus far, we have assumed that bidders’ values for the item being auctioned are independent:
each bidder knows her own value for the item, and is not concerned with how much it is worth
to anyone else. This makes sense in a lot of situations, but it clearly doesn’t apply to a setting
in which the bidders intend to resell the object. In this case, there is a common eventual
value for the object — the amount it will generate on resale — but it is not necessarily
known. Each bidder i may have some private information about the common value, leading
to an estimate v i
of this value. Individual bidder estimates will typically be slightly wrong,
and they will also typically not be independent of each other. One possible model for such
estimates is to suppose that the true value is v, and that each bidder i’s estimate v i
is defined
by v i
= v + x i
, where x i
is a random number with a mean of 0, representing the error in i’s
estimate.
9.6. COMMON VALUES AND THE WINNER’S CURSE 259
Auctions with common values introduce new sources of complexity. To see this, let’s
start by supposing that an item with a common value is sold using a second-price auction.
Is it still a dominant strategy for bidder i to bid v i
? In fact, it’s not. To get a sense for why
this is, we can use the model with random errors v + x i
. Suppose there are many bidders,
and that each bids her estimate of the true value. Then from the result of the auction, the
winning bidder not only receives the object, she also learns something about her estimate
of the common value — that it was the highest of all the estimates. So in particular, her
estimate is more likely to be an over-estimate of the common value than an under-estimate.
Moreover, with many bidders, the second-place bid — which is what she paid — is also likely
to be an over-estimate. As a result she will likely lose money on the resale relative to what
she paid.
This is known as the winner’s curse, and it is a phenomenon that has a rich history in
the study of auctions. Richard Thaler’s review of this history [387] notes that the winner’s
curse appears to have been first articulated by researchers in the petroleum industry [95].
In this domain, firms bid on oil-drilling rights for tracts of land that have a common value,
equal to the value of the oil contained in the tract. The winner’s curse has also been studied
in the context of competitive contract o↵ers to baseball free agents [98] — with the unknown
common value corresponding to the future performance of the baseball player being courted.2
Rational bidders should take the winner’s curse into account in deciding on their bids: a
bidder should bid her best estimate of the value of the object conditional on both her private
estimate v i
and on winning the object at her bid. That is, it must be the case that at an
optimal bid, it is better to win the object than not to win it. This means in a common-value
auction, bidders will shade their bids downward even when the second-price format is used;
with the first-price format, bids will be reduced even further. Determining the optimal bid
is fairly complex, and we will not pursue the details of it here. It is also worth noting that
in practice, the winner’s curse can lead to outright losses on the part of the winning bidder
[387], since in a large pool of bidders, anyone who in fact makes an error and overbids is
more likely to be the winner of the auction.
2In these cases as well as others, one could argue that the model of common values is not entirely accurate. One oil company could in principle be more successful than another at extracting oil from a tract of land; and a baseball free agent may flourish if he joins one team but fail if he joins another. But common values are a reasonable approximation to both settings, as to any case where the purpose of bidding is to obtain an item that has some intrinsic but unknown future value. Moreover, the reasoning behind the winner’s curse arises even when the item being auctioned has related but non-identical values to the di↵erent bidders.
260 CHAPTER 9. AUCTIONS
9.7 Advanced Material: Bidding Strategies in First- Price and All-Pay Auctions
In the previous two sections we o↵ered some intuition about the way to bid in first-price
auctions and in all-pay auctions, but we did not derive optimal bids. We now develop models
of bidder behavior under which we can derive equilibrium bidding strategies in these auctions.
We then explore how optimal behavior varies depending on the number of bidders and on
the distribution of values. Finally, we analyze how much revenue the seller can expect to
obtain from various auctions. The analysis in this section will use elementary calculus and
probability theory.
A. Equilibrium Bidding in First-Price Auctions
As the basis for the model, we want to capture a setting in which bidders know how many
competitors they have, and they have partial information about their competitors’ values
for the item. However, they do not know their competitors’ values exactly.
Let’s start with a simple case first, and then move on to a more general formulation.
In the simple case, suppose that there are two bidders, each with a private value that is
independently and uniformly distributed between 0 and 1.3 This information is common
knowledge among the two bidders. A strategy for a bidder is a function s(v) = b that maps
her true value v to a non-negative bid b. We will make the following simple assumptions
about the strategies the bidders are using:
(i) s(·) is a strictly increasing, di↵erentiable function; so in particular, if two bidders have di↵erent values, then they will produce di↵erent bids.
(ii) s(v) v for all v: bidders can shade their bids down, but they will never bid above their true values. Notice that since bids are always non-negative, this also means that
s(0) = 0.
These two assumptions permit a wide range of strategies. For example, the strategy of
bidding your true value is represented by the function s(v) = v, while the strategy of shading
your bid downward to by a factor of c < 1 times your true value is represented by s(v) = cv.
More complex strategies such as s(v) = v2 are also allowed, although we will see that in
first-price auctions they are not optimal.
The two assumptions help us narrow the search for equilibrium strategies. The second
of our assumptions only rules out strategies (based on overbidding) that are non-optimal.
3The fact that the 0 and 1 are the lowest and highest possible values is not crucial; by shifting and re-scaling these quantities, we could equally well consider values that are uniformly distributed between any other pair of endpoints.
9.7. ADVANCED MATERIAL: BIDDING STRATEGIES IN FIRST-PRICE AND ALL-PAY AUCTIONS261
The first assumption restricts the scope of possible equilibrium strategies, but it makes the
analysis easier while still allowing us to study the important issues.
Finally, since the two bidders are identical in all ways except the actual value they draw
from the distribution, we will narrow the search for equilibria in one further way: we will
consider the case in which the two bidders follow the same strategy s(·).
Equilibrium with two bidders: The Revelation Principle. Let’s consider what such
an equilibrium strategy should look like. First, assumption (i) says that the bidder with the
higher value will also produce the higher bid. If bidder i has a value of v i
, the probability that
this is higher than the value of i’s competitor in the interval [0, 1] is exactly v i
. Therefore,
i will win the auction with probability v i
. If i does win, i receives a payo↵ of v i
� s(v i
).
Putting all this together, we see that i’s expected payo↵ is
g(v i
) = v i
(v i
� s(v i
)). (9.1)
Now, what does it mean for s(·) to be an equilibrium strategy? It means that for each bidder i, there is no incentive for i to deviate from strategy s(·) if i’s competitor is also using strategy s(·). It’s not immediately clear how to analyze deviations to an arbitrary strategy satisfying assumptions (i) and (ii) above. Fortunately, there is an elegant device that lets
us reason about deviations as follows: rather then actually switching to a di↵erent strategy,
bidder i can implement her deviation by keeping the strategy s(·) but supplying a di↵erent “true value” to it.
Here is how this works. First, if i’s competitor is also using strategy s(·), then i should never announce a bid above s(1), since i can win with bid s(1) and get a higher payo↵
with bid s(1) than with any bid b > s(1). So in any possible deviation by i, the bid she will
actually report will lie between s(0) = 0 and s(1). Therefore, for the purposes of the auction,
she can simulate her deviation to an alternate strategy by first pretending that her true value
is v0 i
rather than v i
, and then applying the existing function s(·) to v0 i
instead of v i
. This is
a special case of a much broader idea known as the Revelation Principle [124, 207, 310]; for
our purposes, we can think of it as saying that deviations in the bidding strategy function
can instead be viewed as deviations in the “true value” that bidder i supplies to her current
strategy s(·). With this in mind, we can write the condition that i does not want to deviate from
strategy s(·) as follows: v
i
(v i
� s(v i
)) � v(v i
� s(v)) (9.2)
for all possible alternate “true values” v between 0 and 1 that bidder i might want to supply
to the function s(·). Is there a function that satisfies this property? In fact, it is not hard to check that
s(v) = v/2 satisfies it. To see why, notice that with this choice of s(·), the left-hand
262 CHAPTER 9. AUCTIONS
side of Inequality (9.2) becomes v i
(v i
� v i
/2) = v2 i
/2 while the right-hand side becomes
v(v i
�v/2) = vv i
�v2/2. Collecting all the terms on the left, the inequality becomes simply
1
2 (v2 � 2vv
i
+ v2 i
) � 0,
which holds because the left-hand side is the square 1 2
(v � v i
)2.
Thus, the conclusion in this case is quite simple to state. If two bidders know they are
competing against each other, and know that each has a private value drawn uniformly at
random from the interval [0, 1], then it is an equilibrium for each to shade their bid down by
a factor of 2. Bidding half your true value is optimal behavior if the other bidder is doing
this as well.
Note that unlike the case of the second-price auction, we have not identified a dominant
strategy, only an equilibrium. In solving for a bidder’s optimal strategy we used each bidder’s
expectation about her competitor’s bidding strategy. In an equilibrium, these expectations
are correct. But if other bidders for some reason use non-equilibrium strategies, then any
bidder should optimally respond and potentially also play some other bidding strategy.
Deriving the two-bidder equilibrium. In our discussion of the equilibrium s(v) = v/2,
we initially conjectured the form of the function s(·), and then checked that it satisfied Inequality (9.2). But this approach does not suggest how to discover a function s(·) to use as a conjecture.
An alternate approach is to derive s(·) directly by reasoning about the condition in Inequality (9.2). Here is how we can do this. In order for s(·) to satisfy Inequality (9.2), it must have the property that for any true value v
i
, the expected payo↵ function g(v) =
v(v i
� s(v)) is maximized by setting v = v i
. Therefore, v i
should satisfy g0(v i
) = 0, where g0
is the first derivative of g(·) with respect to v. Since
g0(v) = v i
� s(v) � vs0(v)
by the Product Rule for derivatives, we see that s(·) must solve the di↵erential equation
s0(v i
) = 1 � s(v
i
)
v i
for all v i
in the interval [0, 1]. This di↵erential equation is solved by the function s(v i
) = v i
/2.
Equilibrium with Many Bidders. Now let’s suppose that there are n bidders, where n
can be larger than two. To start with, we’ll continue to assume that each bidder i draws her
true value v i
independently and uniformly at random from the interval between 0 and 1.
Much of the reasoning for the case of two bidders still works here, although the basic
formula for the expected payo↵ changes. Specifically, assumption (i) still implies that the
9.7. ADVANCED MATERIAL: BIDDING STRATEGIES IN FIRST-PRICE AND ALL-PAY AUCTIONS263
bidder with the highest true value will produce the highest bid and hence win the auction. For
a given bidder i with true value v i
, what is the probability that her bid is the highest? This
requires each other bidder to have a value below v i
; since the values are chosen independently,
this event has a probability of vn�1 i
. Therefore, bidder i’s expected payo↵ is
G(v i
) = vn�1 i
(v i
� s(v i
)). (9.3)
The condition for s(·) to be an equilibrium strategy remains the same as it was in the case of two bidders. Using the Revelation Principle, we view a deviation from the bidding
strategy as supplying a “fake” value v to the function s(·); given this, we require that the true value v
i
produces an expected payo↵ at least as high as the payo↵ from any deviation:
vn�1 i
(v i
� s(v i
)) � vn�1(v i
� s(v)) (9.4)
for all v between 0 and 1.
From this, we can derive the form of the bidding function s(·) using the di↵erential- equation approach that worked for two bidders. The expected payo↵ function G(v) =
vn�1(v i
� s(v)) must be maximized by setting v = v i
. Setting the derivative G0(v i
) = 0
and applying the Product Rule to di↵erentiate G, we get
(n � 1)vn�2v i
� (n � 1)vn�2s(v i
) � vn�1 i
s0(v i
) = 0
for all v i
between 0 and 1. Dividing through by (n � 1)vn�2 and solving for s0(v i
), we get
the equivalent but typographically simpler equation
s0(v i
) = (n � 1) ✓
1 � s(v
i
)
v i
◆ (9.5)
for all v i
between 0 and 1. This di↵erential equation is solved by the function
s(v i
) =
✓ n � 1
n
◆ v
i
.
So if each bidder shades her bid down by a factor of (n � 1)/n, then this is optimal behavior given what everyone else is doing. Notice that when n = 2 this is our two-bidder
strategy. The form of this strategy highlights an important principle that we discussed in
Section 9.5 about strategic bidding in first-price auctions: as the number of bidders increases,
you generally have to bid more “aggressively,” shading your bid down less, in order to win.
For the simple case of values drawn independently from the uniform distribution, our analysis
here quantifies exactly how this increased aggressiveness should depend on the number of
bidders n.
264 CHAPTER 9. AUCTIONS
General Distributions. In addition to considering larger numbers of bidders, we can
also relax the assumption that bidders’ values are drawn from the uniform distribution on
an interval.
Suppose that each bidder has her value drawn from a probability distribution over the
non-negative real numbers. We can represent the probability distribution by its cumulative
distribution function F (·): for any x, the value F (x) is the probability that a number drawn from the distribution is at most x. We will assume that F is a di↵erentiable function.
Most of the earlier analysis continues to hold at a general level. The probability that a
bidder i with true value v i
wins the auction is the probability that no other bidder has a
larger value, so it is equal to F (v i
)n�1. Therefore, the expected payo↵ to v i
is
F (v i
)n�1(v i
� s(v i
)).
Then, the requirement that bidder i does not want to deviate from this strategy becomes
F (v i
)n�1(v i
� s(v i
)) � F (v)n�1(v i
� s(v)) (9.6)
for all v between 0 and 1.
Finally, this equilibrium condition can be used to write a di↵erential equation just as
before, using the fact that the function of v on the right-hand side of Inequality (9.6) should
be maximized when v = v i
. We apply the Product Rule, and also the Chain Rule for
derivatives, keeping in mind that the derivative of the cumulative distribution function F (·) is the probability density function f (·) for the distribution. Proceeding by analogy with the analysis for the uniform distribution, we get the di↵erential equation
s0(v i
) = (n � 1) ✓
f (v i
)v i
� f (v i
)s(v i
)
F (v i
)
◆ . (9.7)
Notice that for the uniform distribution on the interval [0, 1], the cumulative distribution
function is F (v) = v and the density is f (v) = 1, which applied to Equation (9.7) gives us
back Equation (9.5).
Finding an explicit solution to Equation (9.7) isn’t possible unless we have an explicit form
for the distribution of values, but it provides a framework for taking arbitrary distributions
and solving for equilibrium bidding strategies.
B. Seller Revenue
Now that we’ve analyzed bidding strategies for first-price auctions, we can return to an issue
that came up at the end of Section 9.3: how to compare the revenue a seller should expect
to make in first-price and second-price auctions.
There are two competing forces at work here. On the one hand, in a second-price auction,
the seller explicitly commits to collecting less money, since he only charges the second-highest
9.7. ADVANCED MATERIAL: BIDDING STRATEGIES IN FIRST-PRICE AND ALL-PAY AUCTIONS265
bid. On the other hand, in a first-price auction, the bidders reduce their bids, which also
reduces what the seller can collect.
To understand how these opposing factors trade o↵ against each other, suppose we have
n bidders with values drawn independently from the uniform distribution on the interval
[0, 1]. Since the seller’s revenue will be based on the values of the highest and second-highest
bids, which in turn depend on the highest and second-highest values, we need to know the
expectations of these quantities.4 Computing these expectations is complicated, but the
form of the answer is very simple. Here is the basic statement:
Suppose n numbers are drawn independently from the uniform distribution on the
interval [0, 1] and then sorted from smallest to largest. The expected value of the
number in the kth position on this list is k
n + 1 .
Now, if the seller runs a second-price auction, and the bidders follow their dominant
strategies and bid truthfully, the seller’s expected revenue will be the expectation of the
second-highest value. Since this will be the value in position n � 1 in the sorted order of the n random values from smallest to largest, the expected value is (n � 1)/(n + 1), by the formula just described. On the other hand, if the seller runs a first-price auction, then in
equilibrium we expect the winning bidder to submit a bid that is (n � 1)/n times her true value. Her true value has an expectation of n/(n + 1) (since it is the largest of n numbers
drawn independently from the unit interval), and so the seller’s expected revenue is
✓ n � 1
n
◆ ✓ n
n + 1
◆ =
n � 1 n + 1
.
The two auctions provide exactly the same expected revenue to the seller!
Revenue Equivalence. As far as seller revenue is concerned, this calculation is in a sense
the tip of the iceberg: it is a reflection of a much broader and deeper principle known in
the auction literature as revenue equivalence [256, 288, 311]. Roughly speaking, revenue
equivalence asserts that a seller’s revenue will be the same across a broad class of auctions
and arbitrary independent distributions of bidder values, when bidders follow equilibrium
strategies. A formalization and proof of the revenue equivalence principle can be found in
[256].
From the discussion here, it is easy to see how the ability to commit to a selling mechanism
is valuable for a seller. Consider, for example, a seller using a second-price auction. If the
bidders bid truthfully and the seller does not sell the object as promised, then the seller
knows the bidders’ values and can bargain with them from this advantaged position. At
worst, the seller should be able to sell the object to the bidder with the highest value at a
4In the language of probability theory, these are known as the expectations of the order statistics.
266 CHAPTER 9. AUCTIONS
price equal to the second highest value. (The bidder with the highest value knows that if
she turns down the trade at this price, then the bidder with the second-highest value will
take it.) But the seller may be able to do better than this in the negotiation, and so overall
the bidders lose relative to the originally promised second-price auction. If bidders suspect
that this scenario may occur with some probability, then they may no longer find it optimal
to bid truthfully in the auction, and so it is not clear what the seller receives.
Reserve Prices. In our discussion of how a seller should choose an auction format, we
have implicitly assumed that the seller must sell the object. Let’s briefly consider how the
seller’s expected revenue changes if he has the option of holding onto the item and choosing
not to sell it. To be able to reason about the seller’s payo↵ in the event that this happens,
let’s assume that the seller values the item at u � 0, which is thus the payo↵ he gets from keeping the item rather than selling it.
It’s clear that if u > 0, then the seller should not use a simple first-price or second-price
auction. In either case, the winning bid might be less than u, and the seller would not want
to sell the object. If the seller refuses to sell after having specified a first-price or second-price
auction, then we are back in the case of a seller who might break his initial commitment to
the format.
Instead, it is better for the seller to announce a reserve price of r before running the
auction. With a reserve price, the item is sold to the highest bidder if the highest bid is
above r; otherwise, the item is not sold. In a first-price auction with a reserve price, the
winning bidder (if there is one) still pays her bid. In a second-price auction with a reserve
price, the winning bidder (if there is one) pays the maximum of the second-place bid and
the reserve price r. As we will see, it is in fact useful for the seller to declare a reserve price
even if his value for the item is u = 0.
Let’s consider how to reason about the optimal value for the reserve price in the case
of a second-price auction. First, it is not hard to go back over the argument that truthful
bidding is a dominant strategy in second-price auctions and check that it still holds in the
presence of a reserve price. Essentially, it is as if the seller were another “simulated” bidder
who always bids r; and since truthful bidding is optimal regardless of how other bidders
behave, the presence of this additional simulated bidder has no e↵ect on how any of the real
bidders should behave.
Now, what value should the seller choose for the reserve price? If the item is worth u to
the seller, then clearly he should set r � u. But in fact the reserve price that maximizes the seller’s expected revenue is strictly greater than u. To see why this is true, let’s first consider
a very simple case: a second-price auction with a single bidder, whose value is uniformly
distributed on [0, 1], and a seller whose value for the item is u = 0. With only one bidder,
the second-price auction with no reserve price will sell the item to the bidder at a price of
9.7. ADVANCED MATERIAL: BIDDING STRATEGIES IN FIRST-PRICE AND ALL-PAY AUCTIONS267
0. On the other hand, suppose the seller sets a reserve price of r > 0. In this case, with
probability 1 � r, the bidder’s value is above r, and the object will be sold to the bidder at a price of r. With probability r, the bidder’s value is below r, and so the seller keeps the
item, receiving a payo↵ of u = 0. Therefore, the seller’s expected revenue is r(1 � r), and this is maximized at r = 1/2. If the seller’s value u is greater than zero, then his expected
payo↵ is r(1 � r) + ru (since he receives a payo↵ of u when the item is not sold), and this is maximized by setting r = (1 + u)/2. So with a single bidder, the optimal reserve price
is halfway between the value of the object to the seller and the maximum possible bidder
value. With more intricate analyses, one can similarly determine the optimal reserve price
for a second-price auction with multiple bidders, as well as for a first-price auction with
equilibrium bidding strategies of the form we derived earlier.
C. Equilibrium Bidding in All-Pay Auctions
The style of analysis we’ve been using for first-price auctions can be adapted without much
di�culty to other formats as well. Here we will show how this works for the analysis of
all-pay auctions: recall from Section 9.5 that this is an auction format — designed to model
activities such as lobbying — where the highest bidder wins the item but everyone pays their
bid.
We will keep the general framework we used for first-price auctions earlier in this section,
with n bidders, each with a value drawn independently and uniformly at random from
between 0 and 1. As before, we want to find a function s(·) mapping values to bids, so that using s(·) is optimal if all other bidders are using it.
With an all-pay auction, the expected payo↵ for bidder i has a negative term if i does
not win. The formula is now
vn�1 i
(v i
� s(v i
)) + (1 � vn�1 i
)(�s(v i
)),
where the first term corresponds to the payo↵ in the event that i wins, and the second term
corresponds to the payo↵ in the event that i loses. As before, we can think of a deviation
from this bidding strategy as supplying a fake value v to the function s(·); so if s(·) is an equilibrium choice of strategies by the bidders, then
vn�1 i
(v i
� s(v i
)) + (1 � vn�1 i
)(�s(v i
)) � vn�1(v i
� s(v)) + (1 � vn�1)(�s(v)) (9.8)
for all v in the interval [0, 1].
Notice that the expected payo↵ consists of a fixed cost s(v) that is paid regardless of the
win/loss outcome, plus a value of v i
in the event that i wins. Canceling the common terms
in Inequality (9.8), we can rewrite it as
vn i
� s(v i
) � vn�1v i
� s(v). (9.9)
268 CHAPTER 9. AUCTIONS
for all v in the interval [0, 1]. Now, writing the right-hand side as a function g(v) = vn�1v i
� s(v), we can view Inequality (9.9) as requiring that v = v
i
maximizes the function g(·). The resulting equation g0(v
i
) = 0 then gives us a di↵erential equation that specifies s(·) quite simply:
s0(v i
) = (n � 1)vn�1 i
,
and hence s(v) =
✓ n � 1
n
◆ vn
i
Since v i
< 1, raising it to the nth power (as specified by the function s(·)) reduces it exponentially in the number of bidders. This shows that bidders will shade their bids
downward significantly as the number of bidders in an all-pay auction increases.
We can also work out the seller’s expected revenue. The seller collects money from
everyone in an all-pay auction; on the other hand, the bidders all submit low bids. The
expected value of a single bidder’s contribution to seller revenue is simply
Z 1
0
s(v) dv =
✓ n � 1
n
◆ Z 1
0
vn dv =
✓ n � 1
n
◆ ✓ 1
n + 1
◆ .
Since the seller collects this much in expectation from each bidder, the seller’s overall ex-
pected revenue is
n
✓ n � 1
n
◆ ✓ 1
n + 1
◆ =
n � 1 n + 1
.
This is exactly the same as the seller’s expected revenue in the first-price and second-price
auctions with the same assumptions about bidder values. Again, this is a reflection of the
much broader revenue equivalence principle [256, 288], which includes all-pay auctions in the
general set of auction formats it covers.
9.8 Exercises
1. In this question we will consider an auction in which there is one seller who wants to
sell one unit of a good and a group of bidders who are each interested in purchasing
the good. The seller will run a sealed-bid, second-price auction. Your firm will bid in
the auction, but it does not know for sure how many other bidders will participate in
the auction. There will be either two or three other bidders in addition to your firm.
All bidders have independent, private values for the good. Your firm’s value for the
good is c. What bid should your firm submit, and how does it depend on the number
of other bidders who show up? Give a brief (1-3 sentence) explanation for your answer.
2. In this problem we will ask how the number of bidders in a second-price, sealed-bid
auction a↵ects how much the seller can expect to receive for his object. Assume that
there are two bidders who have independent, private values v i
which are either 1 or 3.
9.8. EXERCISES 269
For each bidder, the probabilities of 1 and 3 are both 1/2. (If there is a tie at a bid of
x for the highest bid the winner is selected at random from among the highest bidders
and the price is x.)
(a) Show that the seller’s expected revenue is 6/4.
(b) Now let’s suppose that there are three bidders who have independent, private
values v i
which are either 1 or 3. For each bidder, the probabilities of 1 and 3 are both
1/2. What is the seller’s expected revenue in this case?
(c) Briefly explain why changing the number of bidders a↵ects the seller’s expected
revenue.
3. In this problem we will ask how much a seller can expect to receive for his object in
a second-price, sealed-bid auction. Assume that all bidders have independent, private
values v i
which are either 0 or 1. The probability of 0 and 1 are both 1/2.
(a) Suppose there are two bidders. Then there are four possible pairs of their values
(v 1
, v 2
): (0, 0), (1, 0), (0, 1), and (1, 1). Each pair of values has probability 1/4. Show
that the seller’s expected revenue is 1/4. (Assume that if there is a tie at a bid of x
for the highest bid the winner is selected at random from among the highest bidders
and the price is x.)
(b) What is the seller’s expected revenue if there are three bidders?
(c) This suggests a conjecture that as the number of bidders increases the seller’s
expected revenue also increases. In the example we are considering the seller’s expected
revenue actually converges to 1 as the number of bidders grows. Explain why this
should occur. You do not need to write a proof; an intuitive explanation is fine.
4. A seller will run a second-price, sealed-bid auction for an object. There are two bidders,
a and b, who have independent, private values v i
which are either 0 or 1. For both
bidders the probabilities of v i
= 0 and v i
= 1 are each 1/2. Both bidders understand
the auction, but bidder b sometimes makes a mistake about his value for the object.
Half of the time his value is 1 and he is aware that it is 1; the other half of the time
his value is 0 but occasionally he mistakenly believes that his value is 1. Let’s suppose
that when b’s value is 0 he acts as if it is 1 with probability 1 2
and as if it is 0 with
probability 1 2
. So in e↵ect bidder b sees value 0 with probability 1 4
and value 1 with
probability 3 4
. Bidder a never makes mistakes about his value for the object, but he
is aware of the mistakes that bidder b makes. Both bidders bid optimally given their
perceptions of the value of the object. Assume that if there is a tie at a bid of x for
the highest bid the winner is selected at random from among the highest bidders and
the price is x.
270 CHAPTER 9. AUCTIONS
(a) Is bidding his true value still a dominant strategy for bidder a? Explain briefly
(b) What is the seller’s expected revenue? Explain briefly.
5. Consider a second-price, sealed-bid auction with one seller who has one unit of the
object which he values at s and two buyers 1, 2 who have values of v 1
and v 2
for the
object. The values s, v 1
, v 2
are all independent, private values. Suppose that both
buyers know that the seller will submit his own sealed bid of s, but they do not know
the value of s. Is it optimal for the buyers to bid truthfully; that is should they each
bid their true value? Give an explanation for your answer.
6. In this question we will consider the e↵ect of collusion between bidders in a second-
price, sealed-bid auction. There is one seller who will sell one object using a second-
price sealed-bid auction. The bidders have independent, private values drawn from a
distribution on [0, 1]. If a bidder with value v gets the object at price p, his payo↵ is
v�p; if a bidder does not get the object his payo↵ is 0. We will consider the possibility of collusion between two bidders who know each others’ value for the object. Suppose
that the objective of these two colluding bidders is to choose their two bids as to
maximize the sum of their payo↵s. The bidders can submit any bids they like as long
as the bids are in [0, 1].
(a) Let’s first consider the case in which there are only two bidders. What two bids
should they submit? Explain.
(b) Now suppose that there is a third bidder who is not part of the collusion. Does the
existence of this bidder change the optimal bids for the two bidders who are colluding?
Explain.
7. A seller announces that he will sell a case of rare wine using a sealed-bid, second-price
auction. A group of I individuals plan to bid on this case of wine. Each bidder is
interested in the wine for his or her personal consumption; the bidders’ consumption
values for the wine may di↵er, but they don’t plan to resell the wine. So we will view
their values for the wine as independent, private values (as in Chapter 9). You are one
of these bidders; in particular, you are bidder number i and your value for the wine is
v i
.
How should you bid in each of the following situations? In each case, provide an
explanation for your answer; a formal proof is not necessary.
(a) You know that a group of the bidders will collude on bids. This group will chose
one bidder to submit a “real bid” of v and the others will all submit bids of 0. You are
not a member of this collusive group and you cannot collude with any other bidder.
9.8. EXERCISES 271
(b) You, and all of the other bidders, have just learned that this seller will collect
bids, but won’t actually sell the wine according to the rules of a second-price auction.
Instead, after collecting the bids the seller will tell all of the bidders that some other
fictional bidder actually submitted the highest bid and so won the auction. This bidder,
of course, doesn’t exist so the seller will still have the wine after the auction is over.
The seller plans to privately contact the highest actual bidder and tell him or her that
the fictional high bidder defaulted (he didn’t buy the wine after all) and that this
bidder can buy the wine for the price he or she bid in the auction. You cannot collude
with any bidder. [You do not need to derive an optimal bidding strategy. It is enough
to explain whether your bid would di↵er from your value and if so in what direction.]
8. In this problem we will ask how irrational behavior on the part of one bidder a↵ects
optimal behavior for the other bidders in an auction. In this auction the seller has one
unit of the good which will be sold using a second-price, sealed-bid auction. Assume
that there are three bidders who have independent, private values for the good, v 1
, v 2
v 3
, which are uniformly distributed on the interval [0, 1].
(a) Suppose first that all bidders behave rationally; that is they submit optimal bids.
Which bidder (in terms of values) wins the auction and how much does this bidder pay
(again in terms of the bidder’s values)?
(b) Suppose now that bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v 3
+ 1)/2. All other bidders know that bidder
3 is irrational in this way, although they do not know bidder 3’s actual value for the
object. How does this a↵ect the behavior of the other bidders?
(c) What e↵ect does bidder 3’s irrational behavior have on the expected payo↵s of
bidder 1? Here the expectation is over the values of v 2
and v 3
which bidder 1 does
not know. You do not need to provide an explicit solution or write a proof for your
answer; an intuitive explanation of the e↵ect is fine. [Remember a bidder’s payo↵ is
the bidder’s value for the object minus the price, if the bidder wins the auction; or 0,
if the bidder does not win the auction.]
9. In this problem we will ask how much a seller can expect to receive for his object
in a second-price, sealed-bid auction. Assume that there are two bidders who have
independent, private values v i
which are either 1 or 2. For each bidder, the probabilities
of v i
= 1 and v i
= 2 are each 1/2. Assume that if there is a tie at a bid of x for the
highest bid the winner is selected at random from among the highest bidders and the
price is x. We also assume that the value of the object to the seller is 0.
(a) Show that the seller’s expected revenue is 5/4.
272 CHAPTER 9. AUCTIONS
(b) Now let’s suppose that the seller sets a reserve price of R with 1 < R < 2: that
is, the object is sold to the highest bidder if her bid is at least R, and the price
this bidder pays is the maximum of the second highest bid and R. If no bid is
at least R, then the object is not sold, and the seller receives 0 revenue. Suppose
that all bidders know R. What is the seller’s expected revenue as a function of
R?
(c) Using the previous part, show that a seller who wants to maximize expected
revenue would never set a reserve price, R, that is more than 1 and less than 1.5.
10. In this problem we will examine a second-price, sealed-bid auction. Assume that there
are two bidders who have independent, private values v i
which are either 1 or 7. For
each bidder, the probabilities of v i
= 1 and v i
= 7 are each 1/2. So there are four
possible pairs of the bidders’ values (v 1
, v 2
): (1, 1), (1, 7), (7, 1), and (7, 7). Each pair
of values has probability 1/4.
Assume that if there is a tie at a bid of x for the highest bid the winner is selected at
random from among the highest bidders and the price is x.
(a) For each pair of values, what bid will each bidder submit, what price will the
winning bidder pay, and how much profit (the di↵erence between the winning bidder’s
value and price he pays) will the winning bidder earn?
(b) Now let’s examine how much revenue the seller can expect to earn and how much
profit the bidders can expect to make in the second price auction. Both revenue
and profit depend on the values, so let’s calculate the average of each of these numbers
across all four of the possible pairs of values. [Note that in doing this we are computing
each bidder’s expected profit before the bidder knows his value for the object.] What is
the seller’s expected revenue in the second price auction? What is the expected profit
for each bidder?
(c) The seller now decides to charge an entry fee of 1. Any bidder who wants to
participate in the auction must pay this fee to the seller before bidding begins and, in
fact, this fee is imposed before each bidder knows his or her own value for the object.
The bidders know only the distribution of values and that anyone who pays the fee
will be allowed to participate in a second price auction for the object. This adds a
new first stage to the game in which bidders decide simultaneously whether to pay the
fee and enter the auction, or to not pay the fee and stay out of the auction. This first
stage is then followed by a second stage in which anyone who pays the fee participates
in the auction. We will assume that after the first stage is over both potential bidders
learn their own value for the object (but not the other potential bidder’s value for the
9.8. EXERCISES 273
object) and that they both learn whether or not the other potential bidder decided to
enter the auction.
Let’s assume that any potential bidder who does not participate in the auction has a
profit of 0, if no one chooses to participate then the seller keeps the object and does not
run an auction, if only one bidder chooses to participate in the auction then the seller
runs a second price auction with only this one bidder (and treats the second highest
bid as 0), and finally if both bidders participate the second price auction is the one
you solved in part (a).
Is there an equilibrium in which each bidder pays the fee and participates in the
auction? Give an explanation for your answer.
11. In this question we will examine a second-price, sealed-bid auction for a single item.
We’ll consider a case in which true values for the item may di↵er across bidders, and it
requires extensive research by a bidder to determine her own true value for an item —
maybe this is because the bidder needs to determine her ability to extract value from
the item after purchasing it (and this ability may di↵er from bidder to bidder).
There are three bidders. Bidders 1 and 2 have values v 1
and v 2
, each of which is a ran-
dom number independently and uniformly distributed on the interval [0, 1]. Through
having performed the requisite level of research, bidders 1 and 2 know their own values
for the item, v 1
and v 2
, respectively, but they do not know each other’s value for item.
Bidder 3 has not performed enough research to know his own true value for the item.
He does know that he and bidder 2 are extremely similar, and therefore that his true
value v 3
is exactly equal to the true value v 2
of bidder 2. The problem is that bidder
3 does not know this value v 2
(nor does he know v 1
).
(a) How should bidder 1 bid in this auction? How should bidder 2 bid?
(b) How should bidder 3 behave in this auction? Provide an explanation for your
answer; a formal proof is not necessary.
274 CHAPTER 9. AUCTIONS
Part III
Markets and Strategic Interaction in Networks
275
Chapter 10
Matching Markets
We have now seen a number of ways of thinking both about network structure and
about the behavior of agents as they interact with each other. A few of our examples have
brought these together directly — such as the issue of tra�c in a network, including Braess’s
Paradox — and in the next few chapters we explore this convergence of network structure
and strategic interaction more fully, and in a range of di↵erent settings.
First, we think about markets as a prime example of network-structured interaction
between many agents. When we consider markets creating opportunities for interaction
among buyers and sellers, there is an implicit network encoding the access between these
buyers and sellers. In fact, there are a number of ways of using networks to model interactions
among market participants, and we will discuss several of these models. Later, in Chapter 12
on network exchange theory, we will discuss how market-style interactions become a metaphor
for the broad notion of social exchange, in which the social dynamics within a group can be
modeled by the power imbalances of the interactions within the group’s social network.
10.1 Bipartite Graphs and Perfect Matchings
Matching markets form the first class of models we consider, as the focus of the current
chapter. Matching markets have a long history of study in economics, operations research,
and other areas because they embody, in a very clean and stylized way, a number of basic
principles: the way in which people may have di↵erent preferences for di↵erent kinds of
goods, the way in which prices can decentralize the allocation of goods to people, and the
way in which such prices can in fact lead to allocations that are socially optimal.
We will introduce these various ingredients gradually, by progressing through a succession
of increasingly rich models. We begin with a setting in which goods will be allocated to people
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
277
278 CHAPTER 10. MATCHING MARKETS
Room1
Room2
Room3
Room4
Room5
Vikram
Wendy
Xin
Yoram
Zoe
(a) Bipartite Graph
Room1
Room2
Room3
Room4
Room5
Vikram
Wendy
Xin
Yoram
Zoe
(b) A Perfect Matching
Figure 10.1: (a) An example of a bipartite graph. (b) A perfect matching in this graph, indicated via the dark edges.
based on preferences, and these preferences will be expressed in network form, but there is
no explicit buying, selling, or price-setting. This first setting will also be a crucial component
of the more complex ones that follow.
Bipartite Graphs. The model we start with is called the bipartite matching problem, and
we can motivate it via the following scenario. Suppose that the administrators of a college
dormitory are assigning rooms to returning students for a new academic year; each room
is designed for a single student, and each student is asked to list several acceptable options
for the room they’d like to get. Students can have di↵erent preferences over rooms; some
people might want larger rooms, quieter rooms, sunnier rooms, and so forth — and so the
lists provided by the students may overlap in complex ways.
We can model the lists provided by the students using a graph, as follows. There is a
node for each student, a node for each room, and an edge connecting a student to a room if
the student has listed the room as an acceptable option. Figure 10.1(a) shows an example
with five students and five rooms (indicating, for instance, that the student named Vikram
has listed each of Rooms 1, 2, and 3 as acceptable options, while the student named Wendy
only listed Room 1).
This type of graph is bipartite, an important property that we saw earlier, in a di↵erent
10.1. BIPARTITE GRAPHS AND PERFECT MATCHINGS 279
context, in talking about a�liation networks in Chapter 4. In a bipartite graph the nodes
are divided into two categories, and each edge connects a node in one category to a node
in the other category. In this case, the two categories are students and rooms. Just as
bipartite graphs were useful in Chapter 4 to represent the participation of people in di↵erent
activities, here they are useful for modeling situations in which individuals or objects of one
type are being assigned or matched up with individuals or objects of another type. As in
Chapter 4, we will generally draw bipartite graphs as in Figure 10.1(a), with the two di↵erent
categories of nodes drawn as two parallel vertical columns, and the edges crossing between
the two columns.
Perfect Matchings. Let’s return to the task that the college dorm administrators were
trying to solve: assigning each student a room that they’d be happy to accept. This task has
a natural interpretation in terms of the graph we’ve just drawn: since the edges represent
acceptable options for students, we want to assign a distinct room to each student, so that
each student is assigned a room to which he or she is connected by an edge. Figure 10.1(b)
shows such an assignment, with the darkened edges indicating who gets which room.
We will refer to such an assignment as a perfect matching:
When there are an equal number of nodes on each side of a bipartite graph, a
perfect matching is an assignment of nodes on the left to nodes on the right, in
such a way that
(i) each node is connected by an edge to the node it is assigned to, and
(ii) no two nodes on the left are assigned to the same node on the right.
As this picture makes clear, there is an equivalent way to think about perfect matchings in
terms of the edges that form the assignment: a perfect matching can also be viewed as a
choice of edges in the bipartite graph so that each node is the endpoint of exactly one of the
chosen edges.
Constricted Sets. If a bipartite graph has a perfect matching, it’s easy to demonstrate
this: you just indicate the edges that form the perfect matching. But what if a bipartite
graph has no perfect matching? What could you show someone to convince them that there
isn’t one?
At first glance, this is not clear; one naturally worries that the only way to convince
someone that there is no perfect matching is to plow through all the possibilities and show
that no pairing works. But in fact there is a clean way to demonstrate that no perfect
matching exists, based on the idea illustrated in Figure 10.2. First, Figure 10.2(a) shows
a bipartite graph that contains no perfect matching. In Figure 10.2(b) we show a succinct
reason why there is no perfect matching in this graph: the set consisting of Vikram, Wendy,
280 CHAPTER 10. MATCHING MARKETS
Room1
Room2
Room3
Room4
Room5
Vikram
Wendy
Xin
Yoram
Zoe
(a) Bipartite graph with no perfect matching
Room1
Room2
Room3
Room4
Room5
Vikram
Wendy
Xin
Yoram
Zoe
(b) A constricted set demonstrating there is no perfect matching
Figure 10.2: (a) A bipartite graph with no perfect matching. (b) A constricted set demon- strating there is no perfect matching.
and Xin, taken together, has collectively provided only two options for rooms that would
be acceptable to any of them. With three people and only two acceptable rooms, there is
clearly no way to construct a perfect matching — one of these three people would have to
get an option they didn’t want in any assignment of rooms.
We call the set of three students in this example a constricted set, since their edges to
the other side of the bipartite graph “constrict” the formation of a perfect matching. This
example points to a general phenomenon, which we can make precise by defining in general
what it means for a set to be constricted, as follows. First, for any set of nodes S on the
right-hand side of a bipartite graph, we say that a node on the left-hand side is a neighbor
of S if it has an edge to some node in S. We define the neighbor set of S, denoted N (S), to
be the collection of all neighbors of S. Finally, we say that a set S on the right-hand side is
constricted if S is strictly larger than N (S) — that is, S contains strictly more nodes than
N (S) does.
Any time there’s a constricted set S in a bipartite graph, it immediately shows that there
can be no perfect matching: each node in S would have to be matched to a di↵erent node
in N (S), but there are more nodes in S than there are in N (S), so this is not possible.
10.1. BIPARTITE GRAPHS AND PERFECT MATCHINGS 281
Room1
Room2
Room3
Xin
Yoram
Zoe
12, 2, 4
8, 7, 6
7, 5, 2
Valuations
(a) A set of valuations
Room1
Room2
Room3
Xin
Yoram
Zoe
12, 2, 4
8, 7, 6
7, 5, 2
Valuations
(b) An optimal assignment
Figure 10.3: (a) A set of valuations. Each person’s valuations for the objects appears as a list next to them. (b) An optimal assignment with respect to these valuations.
So it’s fairly easy to see that constricted sets form one kind of obstacle to the presence of
perfect matchings. What’s also true, though far from obvious, is that constricted sets are in
fact the only kind of obstacle. This is the crux of the following fact, known as the Matching
Theorem.
Matching Theorem: If a bipartite graph (with equal numbers of nodes on the left
and right) has no perfect matching, then it must contain a constricted set.
The Matching Theorem was independently discovered by Denes König in 1931 and Phillip
Hall in 1935 [280]. Without the theorem, one might have imagined that a bipartite graph
could fail to have a perfect matching for all sorts of reasons, some of them perhaps even too
complicated to explain; but what the theorem says is that the simple notion of a constricted
set is in fact the only obstacle to having a perfect matching. For our purposes in this chapter,
we will only need to use the fact that the Matching Theorem is true, without having to go
into the details of its proof. However, its proof is elegant as well, and we describe a proof of
the theorem in Section 10.6 at the end of this chapter.
One way to think about the Matching Theorem, using our example of students and
rooms, is as follows. After the students submit their lists of acceptable rooms, it’s easy for
the dormitory administrators to explain to the students what happened, regardless of the
outcome. Either they can announce the perfect matching giving the assignment of students
to rooms, or they can explain that no assignment is possible by indicating a set of students
who collectively gave too small a set of acceptable options. This latter case is a constricted
set.
282 CHAPTER 10. MATCHING MARKETS
Room1
Room2
Room3
Xin
Yoram
Zoe
(a) A bipartite graph
Room1
Room2
Room3
Xin
Yoram
Zoe
1, 1, 0
1, 0, 0
0, 1, 1
(b) A set of valuations encoding the search for a perfect matching
Figure 10.4: (a) A bipartite graph in which we want to search for a perfect matching. (b) A corresponding set of valuations for the same nodes so that finding the optimal assignment lets us determine whether there is a perfect matching in the original graph.
10.2 Valuations and Optimal Assignments
The problem of bipartite matching from the previous section illustrates some aspects of a
market in a very simple form: individuals express preferences in the form of acceptable op-
tions; a perfect matching then solves the problem of allocating objects to individuals accord-
ing to these preferences; and if there is no perfect matching, it is because of a “constriction”
in the system that blocks it.
We now want to extend this model to introduce some additional features. First, rather
than expressing preferences simply as binary “acceptable-or-not” choices, we allow each
individual to express how much they’d like each object, in numerical form. In our example
of students and dorm rooms from Section 10.1, suppose that rather than specifying a list
of acceptable rooms, each student provides a numerical score for each room, indicating how
happy they’d be with it. We will refer to these numbers as the students’ valuations for the
respective rooms. Figure 10.3(a) shows an example of this with three students and three
rooms; for instance, Xin’s valuations for Rooms 1, 2, and 3 are 12, 2, and 4 respectively
(while Yoram’s valuations for Rooms 1, 2, and 3 are 8, 7, and 6 respectively). Notice that
students may disagree on which rooms are better, and by how much.
We can define valuations whenever we have a collection of individuals evaluating a col-
lection of objects. And using these valuations, we can evaluate the quality of an assignment
of objects to individuals, as follows: it is the sum of each individual’s valuation for what
10.2. VALUATIONS AND OPTIMAL ASSIGNMENTS 283
they get.1 Thus, for example, the quality of the assignment illustrated in Figure 10.3(b) is
12 + 6 + 5 = 23.
If the dorm administrators had accurate data on each student’s valuations for each room,
then a reasonable way to assign rooms to students would be to choose the assignment of
maximum possible quality. We will refer to this as the optimal assignment, since it maximizes
the total happiness of everyone for what they get. You can check that the assignment in
Figure 10.3(b) is in fact the optimal assignment for this set of valuations. Of course, while
the optimal assignment maximizes total happiness, it does not necessarily give everyone their
favorite item; for example, in Figure 10.3(b), all the students think Room 1 is the best, but
it can only go to one of them.
In a very concrete sense, the problem of finding an optimal assignment also forms a
natural generalization of the bipartite matching problem from Section 10.1. Specifically,
it contains the bipartite matching problem as a special case. Here is why. Suppose, as in
Section 10.1, that there are an equal number of students and rooms, and each student simply
submits a list of acceptable rooms without providing a numerical valuation; this gives us a
bipartite graph as in Figure 10.4(a). We would like to know if this bipartite graph contains
a perfect matching, and we can express precisely this question in the language of valuations
and optimal assignments as follows. We give each student a valuation of 1 for each room
they included on their acceptable list, and a valuation of 0 for each room they omitted from
their list. Applying this translation to the graph in Figure 10.4(a), for example, we get the
valuations shown in Figure 10.4(b). Now, there is a perfect matching precisely when we can
find an assignment that gives each student a room that he or she values at 1 rather than 0
— that is, precisely when the optimal assignment has a total valuation equal to the number
of students. This simple translation shows how the problem of bipartite matching is implicit
in the broader problem of finding an optimal assignment.
While the definition of an optimal assignment is quite natural and general, it is far
from obvious whether there is a comparably natural way to find or characterize the optimal
assignment for a given set of valuations. This is in fact a bit subtle; we will describe a way
to determine an optimal assignment, in the context of a broader market interpretation of
this problem, in the two next sections.
1Of course, this notion of the quality of an assignment is appropriate only if adding individual’s valuations makes sense. We can interpret individual valuations here as the maximum amount the individuals are willing to pay for items, so the sum of their valuations for the items they are assigned is just the maximum amount the group would be willing to pay in total for the assignment. The issue of adding individuals’ payo↵s was also discussed in Chapter 6, where we defined social optimality using the sum of payo↵s in a game.
284 CHAPTER 10. MATCHING MARKETS
10.3 Prices and the Market-Clearing Property
Thus far, we have been using the metaphor of a central “administrator” who determines
a perfect matching, or an optimal assignment, by collecting data from everyone and then
performing a centralized computation. And while there are clearly instances of market-like
activity that function this way (such as our example of students and dorm rooms), a more
standard picture of a market involves much less central coordination, with individuals making
decisions based on prices and their own valuations.
Capturing this latter idea brings us to the crucial step in our formulation of matching
markets: understanding the way in which prices can serve to decentralize the market. We
will see that if we replace the role of the central administrator by a particular scheme for
pricing items, then allowing individuals to follow their own self-interest based on valuations
and prices can still produce optimal assignments.
To describe this, let’s change the housing metaphor slightly, from students and dorm
rooms to one where the role of prices is more natural. Suppose that we have a collection
of sellers, each with a house for sale, and an equal-sized collection of buyers, each of whom
wants a house. By analogy with the previous section, each buyer has a valuation for each
house, and as before, two di↵erent buyers may have very di↵erent valuations for the same
houses. The valuation that a buyer j has for the house held by seller i will be denoted v ij
,
with the subscripts i and j indicating that the valuation depends on both the identity of
the seller i and the buyer j. We also assume that each valuation is a non-negative whole
number (0, 1, 2, . . .). We assume that sellers have a valuation of 0 for each house; they care
only about receiving payment from buyers, which we define next.2
Prices and Payo↵s. Suppose that each seller i puts his house up for sale, o↵ering to sell
it for a price p i
� 0. If a buyer j buys the house from seller i at this price, we will say that the buyer’s payo↵ is her valuation for this house, minus the amount of money she had to pay:
v ij
� p i
. So given a set of prices, if buyer j wants to maximize her payo↵, she will buy from
the seller i for which this quantity v ij
�p i
is maximized — with the following caveats. First,
if this quantity is maximized in a tie between several sellers, then the buyer can maximize
her payo↵ by choosing any one of them. Second, if her payo↵ v ij
� p i
is negative for every
choice of seller i, then the buyer would prefer not to buy any house: we assume she can
obtain a payo↵ of 0 by simply not transacting.
We will call the seller or sellers that maximize the payo↵ for buyer j the preferred sellers
2Our assumption that sellers all have valuations of 0 for their houses is done for the sake of simplicity; if we wanted, we could directly adapt the arguments here to the case in which “zero” is really some minimum base level, and all other valuations and prices represent amounts above this base level. It is also not hard to adapt our analysis to the case in which sellers each might have di↵erent valuations for their houses. Since none of these more general models add much to the underlying set of ideas, we will stick with the simple assumption that sellers have valuations of 0 for houses.
10.3. PRICES AND THE MARKET-CLEARING PROPERTY 285
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
Sellers Buyers Valuations
(a) Buyer Valuations
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
5
2
0
Prices Sellers Buyers Valuations
(b) Market-Clearing Prices
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
2
1
0
Prices Sellers Buyers Valuations
(c) Prices that Don’t Clear the Market
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
3
1
0
Prices Sellers Buyers Valuations
(d) Market-Clearing Prices (Tie-Breaking Required)
Figure 10.5: (a) Three sellers (a, b, and c) and three buyers (x, y, and z). For each buyer node, the valuations for the houses of the respective sellers appear in a list next to the node. (b) Each buyer creates a link to her preferred seller. The resulting set of edges is the preferred-seller graph for this set of prices. (c) The preferred-seller graph for prices 2, 1, 0. (d) The preferred-seller graph for prices 3, 1, 0.
of buyer j, provided the payo↵ from these sellers is not negative. We say that buyer j has
no preferred seller if the payo↵s v ij
� p i
are negative for all choices of i.
In Figures 10.5(b)-10.5(d), we show the results of three di↵erent sets of prices for the
same set of buyer valuations. Note how the sets of preferred sellers for each buyer change
depending on what the prices are. So for example, in Figure 10.5(b), buyer x would receive a
payo↵ of 12�5 = 7 if she buys from a, a payo↵ of 4�2 = 2 if she buys from b, and 2�0 = 2 if she buys from c. This is why a is her unique preferred seller. We can similarly determine
the payo↵s for buyers y (3, 5, and 6) and z (2, 3, and 2) for transacting with sellers a, b, and
c respectively.
286 CHAPTER 10. MATCHING MARKETS
Market-Clearing Prices. Figure 10.5(b) has the particularly nice property that if each
buyer simply claims the house that she likes best, each buyer ends up with a di↵erent house:
somehow the prices have perfectly resolved the contention for houses. And this happens
despite the fact that each of the three buyers value the house of seller a the highest; it is the
high price of 5 that dissuades buyers y and z from pursuing this house.
We will call such a set of prices market-clearing, since they cause each house to get bought
by a di↵erent buyer. In contrast, Figure 10.5(c) shows an example of prices that are not
market-clearing, since buyers x and z both want the house o↵ered by seller a — so in this
case, when each buyer pursues the house that maximizes their payo↵, the contention for
houses is not resolved. (Notice that although each of a, b, and c is a preferred seller for y,
since they all give y equal payo↵s, this does not help with the contention between x and z.)
Figure 10.5(d) illustrates one further subtlety in the notion of market-clearing prices.
Here, if the buyers coordinate so that each chooses the appropriate preferred seller, then
each buyer gets a di↵erent house. (This requires that y take c’s house and z take b’s house.)
Since it is possible to eliminate contention using preferred sellers, we will say that this set
of prices is market-clearing as well, even though a bit of coordination is required due to ties
in the maximum payo↵s. In some cases, ties like this may be inevitable: for example, if
all buyers have the same valuations for everything, then no choice of prices will break this
symmetry.
Given the possibility of ties, we will think about market-clearing prices more generally
as follows. For a set of prices, we define the preferred-seller graph on buyers and sellers by
simply constructing an edge between each buyer and her preferred seller or sellers. (There
will be no edge out of a buyer if she has no preferred seller.) So in fact, Figures 10.5(b)-
10.5(d) are just drawings of preferred-seller graphs for each of the three indicated sets of
prices. Now we simply say: a set of prices is market-clearing if the resulting preferred-seller
graph has a perfect matching.
Properties of Market-Clearing Prices. In a way, market-clearing prices feel a bit too
good to be true: if sellers set prices the right way, then self-interest runs its course and
(potentially with a bit of coordination over tie-breaking) all the buyers get out of each
other’s way and claim di↵erent houses. We’ve seen that such prices can be achieved in one
very small example; but in fact, something much more general is true:
Existence of Market-Clearing Prices: For any set of buyer valuations, there exists
a set of market-clearing prices.
So market-clearing prices are not just a fortuitous outcome in certain cases; they are always
present. This is far from obvious, and we will turn shortly to a method for constructing
market-clearing prices that, in the process, proves they always exist.
10.3. PRICES AND THE MARKET-CLEARING PROPERTY 287
Before doing this, we consider another natural question: the relationship between market-
clearing prices and social welfare. Just because market-clearing prices resolve the contention
among buyers, causing them to get di↵erent houses, does this mean that the total valuation
of the resulting assignment will be good? In fact, there is something very strong that can
be said here as well: market-clearing prices (for this buyer-seller matching problem) always
provide socially optimal outcomes:
Optimality of Market-Clearing Prices: For any set of market-clearing prices, a
perfect matching in the resulting preferred-seller graph has the maximum total
valuation of any assignment of sellers to buyers.
Compared with the previous claim on the existence of market-clearing prices, this fact about
optimality can be justified by a much shorter, if somewhat subtle, argument.
The argument is as follows. Consider a set of market-clearing prices, and let M be
a perfect matching in the preferred-seller graph. Now, consider the total payo↵ of this
matching, defined simply as the sum of each buyer’s payo↵ for what she gets. Since each
buyer is grabbing a house that maximizes her payo↵ individually, M has the maximum total
payo↵ of any assignment of houses to buyers. Now how does total payo↵ relate to total
valuation, which is what we’re hoping that M maximizes? If buyer j chooses house i, then
her valuation is v ij
and her payo↵ is v ij
� p i
. Thus, the total payo↵ to all buyers is simply
the total valuation, minus the sum of all prices:
Total Payo↵ of M = Total Valuation of M � Sum of all prices.
But the sum of all prices is something that doesn’t depend on which matching we choose
(it’s just the sum of everything the sellers are asking for, regardless of how they get paired up
with buyers). So a matching M that maximizes the total payo↵ is also one that maximizes
the total valuation. This completes the argument.
There is another important way of thinking about the optimality of market-clearing
prices, which turns out to be essentially equivalent to the formulation we’ve just described.
Suppose that instead of thinking about the total valuation of the matching, we think about
the total of the payo↵s received by all participants in the market — both the sellers and
the buyers. For a buyer, her payo↵ is defined as above: it is her valuation for the house she
gets minus the price she pays. A seller’s payo↵ is simply the amount of money he receives in
payment for his house. Therefore, in any matching, the total of the payo↵s to all the sellers
is simply equal to the sum of the prices (since they all get paid, and it doesn’t matter which
buyer pays which seller). Above, we just argued that the total of the payo↵s to all the buyers
is equal to the total valuation of the matching M , minus the sum of all prices. Therefore,
the total of the payo↵s to all participants — both the sellers and the buyers — is exactly
equal to the total valuation of the matching M ; the point is that the prices detract from
288 CHAPTER 10. MATCHING MARKETS
the total buyer payo↵ by exactly the amount that they contribute to the total seller payo↵,
and hence the sum of the prices cancels out completely from this calculation. Therefore,
to maximize the total payo↵s to all participants, we want prices and a matching that lead
to the maximum total valuation, and this is achieved by using market-clearing prices and a
perfect matching in the resulting preferred-seller graph. We can summarize this as follows.
Optimality of Market-Clearing Prices (equivalent version): A set of market-
clearing prices, and a perfect matching in the resulting preferred-seller graph,
produces the maximum possible sum of payo↵s to all sellers and buyers.
10.4 Constructing a Set of Market-Clearing Prices
Now let’s turn to the harder challenge: understanding why market-clearing prices must
always exist. We’re going to do this by taking an arbitrary set of buyer valuations, and
describing a procedure that arrives at market-clearing prices. The procedure will in fact
be a kind of auction — not a single-item auction of the type we discussed in Chapter 9,
but a more general kind taking into account the fact that there are multiple things being
auctioned, and multiple buyers with di↵erent valuations. This particular auction procedure
was described by the economists Demange, Gale, and Sotomayor in 1986 [129], but it’s
actually equivalent to a construction of market-clearing prices discovered by the Hungarian
mathematician Egerváry seventy years earlier, in 1916 [280].
Here’s how the auction works. Initially all sellers set their prices to 0. Buyers react by
choosing their preferred seller(s), and we look at the resulting preferred-seller graph. If this
graph has a perfect matching we’re done. Otherwise — and this is the key point — there is
a constricted set of buyers S. Consider the set of neighbors N (S), which is a set of sellers.
The buyers in S only want what the sellers in N (S) have to sell, but there are fewer sellers
in N (S) than there are buyers in S. So the sellers in N (S) are in “high demand” — too
many buyers are interested in them. They respond by each raising their prices by one unit,
and the auction then continues.
There’s one more ingredient, which is a reduction operation on the prices. It will be
useful to have our prices scaled so that the smallest one is 0. Thus, if we ever reach a point
where all prices are strictly greater than 0 — suppose the smallest price has value p > 0 —
then we reduce the prices by subtracting p from each one. This drops the lowest price to 0,
and shifts all other prices by the same relative amount.
A general round of the auction looks like what we’ve just described.
(i) At the start of each round, there is a current set of prices, with the smallest
one equal to 0.
(ii) We construct the preferred-seller graph and check whether there is a perfect
matching.
10.4. CONSTRUCTING A SET OF MARKET-CLEARING PRICES 289
(iii) If there is, we’re done: the current prices are market-clearing.
(iv) If not, we find a constricted set of buyers S and their neighbors N (S).
(v) Each seller in N (S) (simultaneously) raises his price by one unit.
(vi) If necessary, we reduce the prices — the same amount is subtracted from
each price so that the smallest price becomes zero.
(vii) We now begin the next round of the auction, using these new prices.
The full-page Figure 10.6 shows what happens when we apply the auction procedure to the
example from Figure 10.5.
The example in Figure 10.6 illustrates two aspects of this auction that should be empha-
sized. First, in any round where the set of “over-demanded” sellers N (S) consists of more
than one individual, all the sellers in this set raise their prices simultaneously. For example,
in the third round in Figure 10.6, the set N (S) consists of both a and b, and so they both
raise their prices so as to produce the prices used for the start of the fourth round. Second,
while the auction procedure shown in Figure 10.6 produces the market-clearing prices shown
in Figure 10.5(d), we know from Figure 10.5(b) that there can be other market-clearing
prices for the same set of buyer valuations.
Showing that the Auction Must Come to an End. Here is a key property of the
auction procedure we’ve defined: the only way it can come to end is if it reaches a set of
market-clearing prices; otherwise, the rounds continue. So if we can show that the auction
must come to an end for any set of buyer valuations — i.e. that the rounds cannot go on
forever — then we’ve shown that market-clearing prices always exist.
It’s not immediately clear, however, why the auction must always come to an end. Con-
sider, for example, the sequence of steps the auction follows in Figure 10.6: prices change,
di↵erent constricted sets form at di↵erent points in time, and eventually the auction stops
with a set of market-clearing prices. But why should this happen in general? Why couldn’t
there be a set of valuations that cause the prices to constantly shift around so that some set
of buyers is always constricted, and the auction never stops?
In fact, the prices can’t shift forever without stopping; the auction must always come to
an end. The way we’re going to show this is by identifying a precise sense in which a certain
kind of “potential energy” is draining out of the auction as it runs; since the auction starts
with only a bounded supply of this potential energy at the beginning, it must eventually run
out.
Here is how we define this notion of potential energy precisely. For any current set of
prices, define the potential of a buyer to be the maximum payo↵ she can currently get from
any seller. This is the buyer’s potential payo↵; the buyer will actually get this payo↵ if the
current prices are market-clearing prices. We also define the potential of a seller to be the
290 CHAPTER 10. MATCHING MARKETS
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
0
0
0
Prices Sellers Buyers Valuations
(a) Start of first round
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
1
0
0
Prices Sellers Buyers Valuations
(b) Start of second round
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
2
0
0
Prices Sellers Buyers Valuations
(c) Start of third round
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
3
1
0
Prices Sellers Buyers Valuations
(d) Start of fourth round
Figure 10.6: The auction procedure applied to the example from Figure 10.5. Each separate picture shows steps (i) and (ii) of successive rounds, in which the preferred-seller graph for that round is constructed.
(a) In the first round, all prices start at 0. The set of all buyers forms a constricted set S, with N (S) equal to the seller a. So a raises his price by one unit and the auction continues to the second round.
(b) In the second round, the set of buyers consisting of x and z forms a constricted set S, with N (S) again equal to the seller a. Seller a again raises his price by one unit and the auction continues to the third round. (Notice that in this round, we could alternately have identified the set of all buyers as a di↵erent constricted set S, in which case N (S) would have been the set of sellers a and b. There is no problem with this — it just means that there can be multiple options for how to run the auction procedure in certain rounds, with any of these options leading to market-clearing prices when the auction comes to an end.)
(c) In the third round, the set of all buyers forms a constricted set S, with N (S) equal to the set of two sellers a and b. So a and b simultaneously raise their prices by one unit each, and the auction continues to the fourth round.
(d) In the fourth round, when we build the preferred-seller graph, we find it contains a perfect matching. Hence, the current prices are market-clearing and the auction comes to an end.
10.5. HOW DOES THIS RELATE TO SINGLE-ITEM AUCTIONS? 291
current price he is charging. This is the seller’s potential payo↵; the seller will actually get
this payo↵ if the current prices are market-clearing prices. Finally, we define the potential
energy of the auction to be the sum of the potential of all participants, both buyers and
sellers.
How does the potential energy of the auction behave as we run it? It begins with all
sellers having potential 0, and each buyer having a potential equal to her maximum valuation
for any house — so the potential energy of the auction at the start is some whole number
P 0
� 0. Also, notice that at the start of each round of the auction, everyone has potential at least 0. The sellers always have potential at least 0 since the prices are always at least 0.
Because of the price-reduction step in every round, the lowest price is always 0, and therefore
each buyer is always doing at least as well as the option of buying a 0-cost item, which gives
a payo↵ of at least 0. (This also means that each buyer has at least one preferred seller at
the start of each round.) Finally, since the potentials of the sellers and buyers are all at least
0 at the start of each round, so is the potential energy of the auction.
Now, the potential only changes when the prices change, and this only happens in steps
(v) and (vi). Notice that the reduction of prices, as defined above, does not change the
potential energy of the auction: if we subtract p from each price, then the potential of each
seller drops by p, but the potential of each buyer goes up by p — it all cancels out. Finally,
what happens to the potential energy of the auction in step (v), when the sellers in N (S)
all raise their prices by one unit? Each of these sellers’ potentials goes up by one unit. But
the potential of each buyer in S goes down by one unit, since all their preferred houses just
got more expensive. Since S has strictly more nodes than N (S) does, this means that the
potential energy of the auction goes down by at least one unit more than it goes up, so it
strictly decreases by at least one unit.
So what we’ve shown is that in each step that the auction runs, the potential energy of
the auction decreases by at least one unit. It starts at some fixed value P 0
, and it can’t drop
below 0, so the auction must come to an end within P 0
steps — and when it comes to an
end, we have our market-clearing prices.
10.5 How Does this Relate to Single-Item Auctions?
We talked in Chapter 9 about single-item auctions, and we’ve now seen a more complex
type of auction based on bipartite graphs. It makes sense to ask how these di↵erent kinds
of auctions relate to each other. In fact, there is a very natural way to view the single-item
auction — both the outcome and the procedure itself — as a special case of the bipartite
graph auction we’ve just defined. We can do this as follows.
Suppose we have a set of n buyers and a single seller auctioning an item; let buyer j
have valuation v j
for the item. To map this to our model based on perfect matchings, we
292 CHAPTER 10. MATCHING MARKETS
a
b
c
x
y
z
3, 0, 0
2, 0, 0
1, 0, 0
0
0
0
Prices Sellers Buyers Valuations
(a) Start of the Auction
a
b
c
x
y
z
3, 0, 0
2, 0, 0
1, 0, 0
2
0
0
Prices Sellers Buyers Valuations
(b) End of the Auction
Figure 10.7: A single-item auction can be represented by the bipartite graph model: the item is represented by one seller node, and then there are additional seller nodes for which all buyers have 0 valuation. (a) The start of the bipartite graph auction. (b) The end of the bipartite graph auction, when buyer x gets the item at the valuation of buyer y.
need an equal number of buyers and sellers, but this is easily dealt with: we create n � 1 “fake” additional sellers (who conceptually represent n � 1 di↵erent ways to fail to acquire the item), and we give buyer j a valuation of 0 for the item o↵ered by each of these fake
sellers. With the real seller labeled 1, this means we have v 1j
= v j
, the valuation of buyer j
for the real item; and v ij
= 0 for larger values of i.
Now we have a genuine instance of our bipartite graph model: from a perfect matching
of buyers to sellers, we can see which buyer ends up paired with the real seller (this is the
buyer who gets the item), and from a set of market-clearing prices, we will see what the real
item sells for.
Moreover, the price-raising procedure to produce market-clearing prices — based on
finding constricted sets — has a natural meaning here as well. The execution of the procedure
on a simple example is shown in Figure 10.7. Initially, all buyers will identify the real seller
as their preferred seller (assuming that they all have positive valuations for the item). The
first constricted set S we find is the set of all buyers, and N (S) is just the single real seller.
Thus, the seller raises his price by one unit. This continues as long as at least two buyers
have the real seller as their unique preferred seller: they form a constricted set S with N (S)
equal to the real seller, and this seller raises his price by a unit. The prices of the fake items
remain fixed at 0 throughout the auction. Finally, when all but one buyer has identified other
sellers as preferred sellers, the graph has a perfect matching. This happens at precisely the
moment that the buyer with the second-highest valuation drops out — in other words, the
buyer with the highest valuation gets the item, and pays the second-highest valuation. So
the bipartite graph procedure precisely implements an ascending bid (English) auction.
10.6. ADVANCED MATERIAL: A PROOF OF THE MATCHING THEOREM 293
10.6 Advanced Material: A Proof of the Matching Theorem
The discussion in this chapter provides a complete proof that market-clearing prices exist,
omitting the details in only one place: we deferred the proof of the Matching Theorem
in Section 10.1, since it was enough for our purposes to use it in a “black-box” fashion.
However, the standard proof of the Matching Theorem in fact provides important insights
into the structure of bipartite graphs, and so we present it here.
Recall that the statement is the following.
Claim: If a bipartite graph (with equal numbers of nodes on the left and right)
has no perfect matching, then it must contain a constricted set.
The tricky part of proving this is to come up with some means of identifying a constricted
set in a bipartite graph, knowing only that it contains no perfect matching. Our general
plan for doing this will be as follows. We will take a bipartite graph, with equal numbers of
nodes on the left and right but no perfect matching, and consider a matching that includes
as many nodes as possible — we will call this a maximum matching. We then try to enlarge
it, by searching for a way to include one more node from each side in the matching. This
will fail (since the matching is already as large as possible for the given graph), and we show
that when this search for a larger matching fails, it produces a constricted set.
Naturally, there is a lot that needs to be fleshed out in this strategy, and the first step is
to consider how one goes about “enlarging” a matching in a bipartite graph. This will turn
out to be the key issue in the whole proof.
Alternating and Augmenting Paths. With this in mind, let’s forget about constricted
sets for a little while and simply think about matchings and how they can be enlarged. As a
first example, consider the bipartite graph in Figure 10.8(a), with a matching indicated using
bold edges. (We’ll say that the matching edges are the edges used in a given matching, and
the non-matching edges are the other, unused ones.) The matching shown in Figure 10.8(a)
is not a maximum matching — we could clearly pair up W with A and X with B to get a
larger one, as in Figure 10.8(c).
For examples this small, it’s easy just to look at the picture of the graph and see how
to find a larger matching. But for bipartite graphs that are more complicated, it’s useful
to have a more principled way of growing a smaller matching into a larger one. Here’s how
we can do this in Figure 10.8(a). We start at node W , looking for a matching that would
include it while still also including everyone who’s currently matched. It’s not immediately
obvious whether we can pair up W with A, since A is already matched to X. So we try
“undoing” the pairing of A and X, which would let W and A be matched. This in turn frees
up X, which can then be matched with B, and we’ve enlarged the matching.
294 CHAPTER 10. MATCHING MARKETS
WA
B X
(a) A matching that is not of max- imum size
WA
B X
(b) An augmenting path
WA
B X
(c) A larger (perfect) matching
Figure 10.8: (a) A matching that does not have maximum size. (b) What a matching does not have maximum size, we can try to find an augmenting path that connects unmatched nodes on opposite sides while alternating between non-matching and matching edges. (c) If we then swap the edges on this path — taking out the matching edges on the path and replacing them with the non-matching edges — then we obtain a larger matching.
This process is depicted in Figure 10.8(b). We followed a zigzag path through the bipartite
graph, alternately adding unused edges to the matching while removing currently used edges
from the matching: in particular, we added the edges A-W and B-X to the matching, while
removing the edge A-X. It was also important that the path was simple — it did not repeat
any nodes. We will call a simple path that alternates between non-matching and matching
edges in this way an alternating path.
This example illustrates a principle that’s true in general. In any bipartite graph with a
matching, if we can find an alternating path that begins and ends at an unmatched node,
then we can swap the roles of all edges on this path: each non-matching edge gets put into
the matching, and each edge on the path that is currently in the matching gets removed from
it. In this way, all the nodes in the path become paired up: we have managed to include the
two formerly unmatched endpoints in the matching, thereby enlarging it. We can summarize
this as follows:
Claim: In a bipartite graph with a matching, if there is an alternating path whose
endpoints are unmatched nodes, then the matching can be enlarged.
In view of this, we call an alternating path with unmatched endpoints an augmenting path,
since it gives us a way to augment the matching.
Augmenting paths can get much longer than what we see in Figure 10.8. For example, in
Figure 10.9, we show an augmenting path that includes a total of eight nodes, and succeeds
in including its two endpoints W and D in the matching. It’s also the case that augmenting
paths can be harder to find than these two simple examples might suggest. In each of these
examples, looking for the augmenting path never involves any real choices — we just keep
10.6. ADVANCED MATERIAL: A PROOF OF THE MATCHING THEOREM 295
WA
B X
C
D
Y
Z
(a) A matching that is not of max- imum size
WA
B X
C
D
Y
Z
(b) An augmenting path
WA
B X
C
D
Y
Z
(c) A larger (perfect) matching
Figure 10.9: The principle used in Figure 10.8 can be applied to larger bipartite graphs as well, sometimes producing long augmenting paths.
following the only edge available at each step. But in more complicated bipartite graphs,
there can be lots of dead-ends in the search for an augmenting path. Consider, for example,
the graph with a matching in Figure 10.10(a). There is in fact an augmenting path that
succeeds in including W and D in the matching, but even on this relatively small example
one needs to check a bit carefully to find it. Moreover, there are other alternating paths
starting from W — such as W -A-X and W -B-Y -C-Z — that don’t make it to the other
unmatched node D, as well as paths from W to D — such as W -B-Z-C-Y -D — that are
not alternating.
Searching for an Augmenting Path. Fortunately, however, there is a natural procedure
we can use to search for an augmenting path in a bipartite graph with a matching. It works
by simply adapting the breadth-first search (BFS) procedure to include the requirement of
alternation — as a result, we will refer to this new procedure as alternating BFS.
Here is how it works. We start at any unmatched node on the right. Then, as in
traditional BFS, we explore the rest of the graph layer by layer, adding new nodes to the
next layer when they are connected by an edge to a node in the current layer. Because
the graph is bipartite, these layers will alternate between consisting of nodes on the left
296 CHAPTER 10. MATCHING MARKETS
WA
B X
C
D
Y
Z
(a) A matching that is not of max- imum size
WA
B X
C
D
Y
Z
(b) An augmenting path
WA
B X
C
D
Y
Z
(c) A larger (perfect) matching
Figure 10.10: In more complex graphs, finding an augmenting path can require a more careful search, in which choices lead to “dead ends” while others connect two unmatched nodes.
and nodes on the right. Now, here is the di↵erence from traditional BFS: because we are
searching specifically for an augmenting path, we want the paths that move downward layer-
by-layer to all be alternating. Thus, when we build a new layer of nodes from the left-hand
side, we should only use non-matching edges to discover new nodes; and when we build a
new layer of nodes from the right-hand side, we should only use matching edges to discover
new nodes.
Figure 10.11 shows how this works on the example from Figure 10.10(a). Starting at W
(which we’ll think of as layer 0), we build the first layer by following non-matching edges to
A and B. We then build the second layer by only following matching edges, which leads us
to nodes X and Y . Following non-matching edges from this layer to new nodes not already
discovered, we get a third layer consisting of C and D; and finally, taking the matching edge
from C brings us to Z in the fourth layer. Notice that in this process, we never used the
edge B-Z: we couldn’t use it out of B in the first layer, because we were only allowed to
follow matching edges at that point; and we couldn’t use it out of Z in the fourth layer,
because by then B had already been discovered.
Now, the crucial thing to observe is that if this alternating BFS procedure ever produces
10.6. ADVANCED MATERIAL: A PROOF OF THE MATCHING THEOREM 297
W
A B
X
C D
Y
Z
B-Z edge not part of search
Figure 10.11: In an alternating breadth-first search, one constructs layers that alternately use non-matching and matching edges; if an unmatched node is ever reached, this results in an augmenting path.
a layer containing an unmatched node from the left-hand side of the graph, we have found
an augmenting path (and can thus enlarge the matching). We simply move downward in
a path from the unmatched node in layer 0 to the unmatched node from the left-hand
side, proceeding one layer at a time. The edges on this path will alternate between being
non-matching and matching, and so this will be an augmenting path.
Augmenting Paths and Constricted Sets. This gives us a systematic procedure to
search for an augmenting path. However, it leaves a basic question unresolved: if this search
procedure fails to find an augmenting path, can we necessarily conclude that there is no
perfect matching? This is certainly not a priori clear: why couldn’t it be that there is a
perfect matching hidden somewhere in the graph, and we just need a more powerful way to
find it? But in fact, alternating BFS is all that we need: what we’ll show now is that when
298 CHAPTER 10. MATCHING MARKETS
W
Layer 1
Layer 2
Layer 3
Layer 4
Layer 0
}
}
equal numbers of nodes
equal numbers of nodes
Figure 10.12: A schematic view of alternating breadth-first search, which produces pairs of layers of equal size.
alternating BFS fails to find an augmenting path, we can in fact extract from this failed
search a constricted set that proves there is no perfect matching.
Here is how. Consider any bipartite graph, and suppose we are currently looking at a
matching in it that is not perfect. Suppose further that we perform an alternating BFS
from an unmatched node W on the right-hand side, and we fail to reach any unmatched
node on the left-hand side. The resulting set of layers at the end of the search will look
schematically like what’s depicted in Figure 10.12. More concretely, Figure 10.13(a) shows
a specific example of a graph with no perfect matching, and Figure 10.13(b) shows a set of
layers from a failed alternating BFS on this example.
Let’s make some observations about the structure after a failed search.
1. First, the even-numbered layers consist of nodes from the right-hand side, while the
odd-numbered layers consist of nodes from the left-hand side.
10.6. ADVANCED MATERIAL: A PROOF OF THE MATCHING THEOREM 299
WA
B X
C
D
Y
Z
(a) A maximum matching that is not perfect
W
A B
X Y
(b) A failed search for an aug- menting path
WA
B X
C
D
Y
Z
(c) The resulting constricted set
Figure 10.13: (a) A matching that has maximum size, but is not perfect. (b) For such a matching, the search for an augment path using alternating breadth-first search will fail. (c) The failure of this search exposes a constricted set: the set of nodes belonging to the even layers.
2. Moreover, each odd layer contains exactly the same number of nodes as the subsequent
even layer. This is because we never reach an unmatched node in an odd layer: so in
every odd layer, the nodes are all connected by their matching edges to distinct nodes
in the next layer, as illustrated in Figure 10.12.
3. So not counting node W in layer 0, there are exactly the same number of nodes in even
layers (numbered 2 and higher) as there are in odd layers. Counting the one extra
node in layer 0, there are strictly more nodes in even layers overall than there are in
odd layers.
4. Finally, every node in an even layer has all of its neighbors in the graph present in some
layer. This is because each even-layer node other than W has its matched partner just
above it in the previous layer; and if any of its other neighbors were not already present
in a higher layer, they would be added to the next layer down, when we’re allowed to
explore using non-matching edges.
(Notice that it’s not necessarily true that every node in an odd layer has all of its
300 CHAPTER 10. MATCHING MARKETS
neighbors in the graph present in some layer. For example, in Figure 10.13(b), node
B’s neighbor Z is not present in any layer. This is because we were not allowed to add
Z when we got to B in the search, since we could only follow the matching edge out
of B.)
Putting these observations together, we discover the following fact: the set of nodes in all
even layers, at the end of a failed alternating BFS, forms a constricted set. This is simply
because it’s a set of nodes S on the right-hand side whose set of neighbors — because they’re
contained among the nodes in the odd layers — is strictly smaller than S is. Figures 10.13(b)
and 10.13(c) show how this works in one specific example.
This completes our plan — to extract a constricted set from the failure of alternating
BFS. Here is one way to summarize the conclusion.
Claim: Consider any bipartite graph with a matching, and let W be any un-
matched node on the right-hand side. Then either there is an augmenting path
beginning at W , or there is a constricted set containing W .
The Matching Theorem. The fact we’ve just discovered is the crucial step in proving
the Matching Theorem; from here it’s easy, as follows.
Consider a bipartite graph with an equal number of nodes on the left and right, and
suppose it has no perfect matching. Let’s take a maximum matching in it — one that
includes as many edges as possible. Since this matching is not perfect, and since there are
an equal number of nodes on the two sides of the bipartite graph, there must be a node W
on the right-hand side that is unmatched. We know there cannot be an augmenting path
containing W , since then we’d be able to enlarge the matching — and that isn’t possible
since we chose a matching of maximum size. Now, by our previous claim, since there is
no augmenting path beginning at W , there must be a constricted set containing W . Since
we’ve deduced the existence of a constricted set from the fact that the graph has no perfect
matching, this completes the proof of the Matching Theorem.
Computing a Perfect Matching. One final dividend from this analysis is that we actu-
ally have a reasonably e�cient method to determine whether a graph has a perfect matching
— enormously more e�cient than the brute-force approach of trying all ways to pair up the
nodes on the left and right.
The method works as follows. Given a bipartite graph with an equal number of nodes
on the left and right, we will progress through a sequence of matchings, and each matching
in the sequence will be one edge larger than the previous one. We can start from the empty
matching — the trivial one in which no nodes at all are paired. Now in general, we look at
our current matching and find an unmatched node W . We use alternating BFS to search for
10.6. ADVANCED MATERIAL: A PROOF OF THE MATCHING THEOREM 301
WA
B X
C
D
Y
Z
Figure 10.14: If the alternating breadth-first search fails from any node on the right-hand side, this is enough to expose a constricted set and hence prove there is no perfect matching. However, it is still possible that an alternating breadth-first search could still succeed from some other node. (In this case, the search from W would fail, but the search from Y would succeed.)
an augmenting path beginning at W . If we find one, we use this augmenting path to enlarge
the matching, and we continue with this new matching. If we don’t find one, we can stop
with a constricted set that proves the graph has no perfect matching.
Since the matchings get larger in every step while the process is running, the number of
matchings we pass through can be at most the number of nodes on each side of the graph.
By then, we will either have reached a perfect matching, or stopped earlier with a constricted
set.
An interesting question is the following: when the procedure stops with a constricted
set, are we guaranteed to have a maximum matching? As we’ve described the procedure so
far, the answer is no. Consider for example Figure 10.14. If we try to find an augmenting
path starting at W , then we will fail (producing the constricted set consisting of W and X).
This is indeed enough to prove there is no perfect matching. However, it does not mean that
the current matching has maximum size: if we instead had searched for an augmenting path
starting from Y , we would have succeeded, producing the path Y -B-Z-D. In other words,
if we’re looking for a maximum matching and not just a perfect matching, it can matter
302 CHAPTER 10. MATCHING MARKETS
where we start our search for an augmenting path; certain parts of the graph can become
“wedged,” while other still contain the potential for enlarging the matching.
However, there is a variation on our procedure that is guaranteed to produce a maximum
matching. We won’t go through all the details of this (see e.g. [260] for more), but the
idea is as follows. By revisiting the analysis we’ve used thus far, and adapting it a little
bit, one can show that if there is no augmenting path beginning at any node on the right-
hand side, then in fact the current matching has maximum size. This shows that if, as we
progress through larger and larger matchings, we always search for an augmenting path from
every node on the right-hand side, then either one of these searches will succeed, or else we
can conclude that the current matching has maximum size. And while this sounds like an
expensive thing to do — having to search separately from each node on the right — in fact
it can be done e�ciently by making all the unmatched nodes on the right constitute layer 0
in the alternating BFS, and otherwise running it as before. Then if an unmatched node on
the left is ever reached in some layer, we can follow the path from the appropriate node in
layer 0 down to it, producing an augmenting path.
A lot of work has gone into the design of e�cient methods for finding maximum matchings
in bipartite graphs, and there are a number of further improvements possible, including
versions of alternating BFS that try to find many augmenting paths simultaneously, thereby
cutting down the number of intermediate matchings one must pass through on the way to
the maximum. Determining how e�ciently maximum matchings can be found remains an
open area of research.
10.7 Exercises
1. Suppose we have a set of 2 sellers labeled a and b, and a set of 2 buyers labeled x and
y. Each seller is o↵ering a distinct house for sale, and the valuations of the buyers for
the houses are as follows.
Buyer Value for a’s house
Value for b’s house
x 2 4 y 3 6
Suppose that a charges a price of 0 for his house, and b charges a price of 1 for his
house. Is this set of prices market-clearing? Give a brief (1-3 sentence) explanation; as
part of your answer, say what the preferred-seller graph is with this given set of prices,
and use this in your explanation.
2. Suppose we have a set of 3 sellers labeled a, b, and c, and a set of 3 buyers labeled
x, y, and z. Each seller is o↵ering a distinct house for sale, and the valuations of the
buyers for the houses are as follows.
10.7. EXERCISES 303
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 5 7 1 y 2 3 1 z 5 4 4
Suppose that sellers a and b each charge 2, and seller c charges 1. Is this set of prices
market-clearing? Give a brief explanation.
3. Suppose we have a set of 3 sellers labeled a, b, and c, and a set of 3 buyers labeled
x, y, and z. Each seller is o↵ering a distinct house for sale, and the valuations of the
buyers for the houses are as follows.
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 2 4 6 y 3 5 1 z 4 7 5
Suppose that sellers a and c each charge 1, and seller b charges 3. Is this set of prices
market-clearing? Give a brief explanation.
4. Suppose we have a set of 3 sellers labeled a, b, and c, and a set of 3 buyers labeled
x, y, and z. Each seller is o↵ering a distinct house for sale, and the valuations of the
buyers for the houses are as follows.
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 12 9 8 y 10 3 6 z 8 6 5
Suppose that a charges a price of 3 for his house, b charges a price of 1 for his house,
and c charges a price of 0. Is this set of prices market-clearing? If so, explain which
buyer you would expect to get which house; if not, say which seller or sellers should
raise their price(s) in the next round of the bipartite-graph auction procedure from
Chapter 10.
5. Suppose we have a set of 3 sellers labeled a, b, and c, and a set of 3 buyers labeled
x, y, and z. Each seller is o↵ering a distinct house for sale, and the valuations of the
buyers for the houses are as follows.
304 CHAPTER 10. MATCHING MARKETS
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 7 7 4 y 7 6 3 z 5 4 3
Suppose that a charges a price of 4 for his house, b charges a price of 3 for his house,
and c charges a price of 1. Is this set of prices market-clearing? Give an explanation
for your answer, using the relevant definitions from Chapter 10.
6. Suppose we have a set of 3 sellers labeled a, b, and c, and a set of 3 buyers labeled
x, y, and z. Each seller is o↵ering a distinct house for sale, and the valuations of the
buyers for the houses are as follows.
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 6 3 2 y 10 5 4 z 7 8 6
Suppose that a charges a price of 4 for his house, b charges a price of 1 for his house,
and c charges a price of 0. Is this set of prices market-clearing? If so, explain which
buyer you would expect to get which house; if not, say which seller or sellers should
raise their price(s) in the next round of the bipartite-graph auction procedure from
Chapter 10.
7. Suppose we have a set of 3 sellers labeled a, b, and c, and a set of 3 buyers labeled
x, y, and z. Each seller is o↵ering a distinct house for sale, and the valuations of the
buyers for the houses are as follows.
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 6 8 7 y 5 6 6 z 3 6 5
Suppose that a charges a price of 2 for his house, b charges a price of 5 for his house,
and c charges a price of 4. Is this set of prices market-clearing? If so, explain which
buyer you would expect to get which house; if not, say which seller or sellers should
raise their price(s) in the next round of the bipartite-graph auction procedure from
Chapter 10.
10.7. EXERCISES 305
8. Suppose we have a set of 2 sellers labeled a and b, and a set of 2 buyers labeled x and
y. Each seller is o↵ering a distinct house for sale, and the valuations of the buyers for
the houses are as follows.
Buyer Value for a’s house
Value for b’s house
x 7 5 y 4 1
Describe what happens if we run the bipartite graph auction procedure to determine
market-clearing prices, by saying what the prices are at the end of each round of the
auction, including what the final market-clearing prices are when the auction comes to
an end.
9. Suppose we have a set of 3 sellers labeled a, b, and c, and a set of 3 buyers labeled
x, y, and z. Each seller is o↵ering a distinct house for sale, and the valuations of the
buyers for the houses are as follows.
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 3 6 4 y 2 8 1 z 1 2 3
Describe what happens if we run the bipartite graph auction procedure from Chap-
ter 10, by saying what the prices are at the end of each round of the auction, including
what the final market-clearing prices are when the auction comes to an end.
(Note: In some rounds, you may notice that there are multiple choices for the con-
stricted set of buyers A. Under the rules of the auction, you can choose any such
constricted set. It’s interesting to consider — though not necessary for this question
— how the eventual set of market-clearing prices depends on how one chooses among
the possible constricted sets.)
10. Suppose we have a set of 3 sellers labeled a, b, and c, and a set of 3 buyers labeled
x, y, and z. Each seller is o↵ering a distinct house for sale, and the valuations of the
buyers for the houses are as follows.
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 9 7 4 y 5 9 7 z 11 10 8
306 CHAPTER 10. MATCHING MARKETS
Describe what happens if we run the bipartite graph auction procedure from Chap-
ter 10, by saying what the prices are at the end of each round of the auction, including
what the final market-clearing prices are when the auction comes to an end.
(Note: In some rounds, you may notice that there are multiple choices for the con-
stricted set of buyers A. Under the rules of the auction, you can choose any such
constricted set. It’s interesting to consider — though not necessary for this question
— how the eventual set of market-clearing prices depends on how one chooses among
the possible constricted sets.)
x
y
a
b
z
c
Figure 10.15: The map for a parking-space market. (Image from Google Maps, http://maps.google.com/)
11. Figure 10.15 shows a map of part of the Back Bay section of Boston. Suppose that the
dark circles labeled x, y, and z represent people living in apartments in Back Bay who
want to rent parking spaces by the month for parking their cars. (Due to the density
of buildings, these parking spaces may be a short walk from where they live, rather
than right at their apartment.) The dark circles labeled a, b, and c represent parking
10.7. EXERCISES 307
spaces available for rent.
Let’s define the distance between a person and a parking space to be the number
of blocks they’d have to walk from their apartment to the parking space. Thus, for
example, z is at a distance of 2 from space c, while y is at a distance of 5 from c and x
is at a distance of 6 from c. (We’ll ignore the fact that the block between Gloucester
and Hereford is a bit shorter than the others; all blocks will be treated as the same in
counting distance.)
Suppose that a person has a valuation for a potential parking space equal to
8 � (their distance to the parking space).
(Notice that this formula gives higher valuations to closer parking spaces.) In terms of
these valuations, we’d like to think about prices that could be charged for the parking
spaces.
(a) Describe how you would set up this question as a matching market in the style of
Chapter 10. Say who the sellers and buyers would be in your set-up, as well as the
valuation each buyer has for the item o↵ered by each seller.
(b) Describe what happens if we run the bipartite graph auction procedure from
Chapter 10 on the matching market you set up in (a), by saying what the prices
are at the end of each round of the auction, including what the final market-clearing
prices are when the auction comes to an end.
(Note: In some rounds, you may notice that there are multiple choices for the con-
stricted set of buyers. Under the rules of the auction, you can choose any such con-
stricted set. It’s interesting to consider — though not necessary for this question —
how the eventual set of market-clearing prices depends on how one chooses among the
possible constricted sets.)
(c) At a more informal level, how do the prices you determined for the parking spaces
in (b) relate to these spaces’ intuitive “attractiveness” to the people in apartments x,
y, and z? Explain.
12. Suppose we have a set of 2 sellers labeled a and b, and a set of 2 buyers labeled x and
y. Each seller is o↵ering a distinct house for sale, and the valuations of the buyers for
the houses are as follows.
Buyer Value for a’s house
Value for b’s house
x 4 1 y 3 2
308 CHAPTER 10. MATCHING MARKETS
In general, there will be multiple sets of market-clearing prices for a given set of sellers,
buyers, and valuations: any set of prices that produces a preferred-seller graph with a
perfect matching is market-clearing.
As a way of exploring this issue in the context of the example above, give three di↵erent
sets of market-clearing prices for this matching market. The prices should be whole
numbers (i.e. they should be numbers from 0, 1, 2, 3, 4, 5, 6, . . .). (Note that for two sets
of market-clearing prices to be di↵erent, it is enough that they not consist of exactly
the same set of numbers.) Explain your answer.
13. Suppose you want to design an auction for the following type of situation: you have
two identical copies of a valuable object, and there are four potential buyers for the
object. Each potential buyer i wants at most one copy, and has a value v i
for either
copy.
You decide to design the auction by analogy with the way in which we derived the
single-item ascending-bid (English) auction from the general procedure for matching
markets. In the present case, as there, you want to create a bipartite graph that
encodes the situation, and then see what prices the bipartite graph auction procedure
comes up with.
(a) Describe how this construction would work using an example with four potential
buyers. In creating your example, first choose specific valuations for the potential
buyers, and then show how the auction proceeds and what the market-clearing prices
are.
(b) In the case of the single-item auction, the bipartite graph procedure yielded the
simple rule from the ascending-bid (English) auction: sell to the highest bidder at the
second-highest price. Describe in comparably simple terms what the rule is for the
current case of two identical items (i.e. your description should not involve the terms
“bipartite”, “graph,” or “matching”).
14. In Chapter 10, we discussed the notion of social-welfare maximization for matching
markets: finding a matching M that maximizes the sum of buyers’ valuations for what
they get, over all possible perfect matchings. We can call such a matching social-
welfare-maximizing. However, the sum of buyers’ valuations is not the only quantity
one might want to maximize; another natural goal might be to make sure that no
individual buyer gets a valuation that is too small.
With this in mind, let’s define the baseline of a perfect matching M to be the minimum
valuation that any buyer has for the item they get in M . We could then seek a perfect
matching M whose baseline is as large as possible, over all possible perfect matchings.
We will call such a matching baseline-maximizing.
10.7. EXERCISES 309
For example, in the following set of valuations,
Buyer Value for a’s house
Value for b’s house
Value for c’s house
x 9 7 4 y 5 9 7 z 11 10 8
the matching M consisting of the pairs a-x, b-y, and c-z has a baseline of 8 (this is
the valuation of z for what she gets, which is lower than the valuations of x and y for
what they get), while the matching M 0 consisting of the pairs b-x, c-y, and a-z has a
baseline of 7. In fact the first of these example matchings, M , is baseline-maximizing
for this sample set of valuations.
Now, finding a perfect matching that is baseline-maximizing is grounded in a kind of
“egalitarian” motivation — no one should be left too badly o↵. This may sometimes
be at odds with the goal of social-welfare maximization. We now explore this tension
further.
(a) Give an example of equal-sized sets of sellers and buyers, with valuations on the
buyers, so that there is no perfect matching that is both social-welfare-maximizing and
baseline-maximizing. (In other words, in your example, social-welfare maximization
and baseline maximization should only occur with di↵erent matchings.)
(b) It is also natural to ask whether a baseline-maximizing matching can always be
supported by market-clearing prices. Here is a precise way to ask the question.
For any equal-sized sets of sellers and buyers, with valuations on the buyers,
is there always a set of market-clearing prices so that the resulting preferred-
seller graph contains a baseline-maximizing perfect matching M ?
Give a yes/no answer to this question, together with a justification of your answer. (If
you answer “yes,” you should explain why there must always exist such a set of market-
clearing prices; if you answer “no,” you should explain why there can be examples in
which a baseline-maximizing matching cannot be found in the preferred-seller graph
resulting from market-clearing prices.)
15. Consider again the set-up for the bipartite graph auction, with an equal number of
buyers and sellers, and with each buyer having a valuation for the object being sold
by each seller. Suppose that we have an instance of this problem in which there is a
particular seller i who is the favorite: every buyer j has a higher valuation for seller
i’s object than for the object being sold by any other seller k. (In notation, we have
v ij
> v kj
for all choices of j and k 6= i.)
310 CHAPTER 10. MATCHING MARKETS
Consider a set of market-clearing prices in this situation. Must it be the case that the
price charged by seller i is at least as high as the price charged by any other seller?
Give an explanation for your answer.
Chapter 11
Network Models of Markets with Intermediaries
11.1 Price-Setting in Markets
In Chapter 10 we developed an analysis of trade and prices on a bipartite graph consisting of
buyers, sellers, and the edges connecting them. Most importantly, we showed that market-
clearing prices exist, and that trade at these prices results in maximal total valuation among
the buyers and sellers; and we found a procedure that allowed us to construct market-
clearing prices. This analysis shows in a striking way how prices have the power to direct
the allocation of goods in a desirable way. What it doesn’t do is provide a clear picture of
where prices in real markets tend to come from. That is, who sets the prices in real markets,
and why do they choose the particular prices they do?
Auctions, which we discussed in Chapter 9, provide a concrete example of price determi-
nation in a controlled setting. In our discussion of auctions, we found that if a seller with
a single object runs a second-price sealed-bid auction — or equivalently an ascending-bid
auction — then buyers bid their true values for the seller’s object. In that discussion, the
buyers were choosing prices (via their bids) in a procedure selected by the seller. We could
also consider a procurement auction in which the roles of buyers and sellers are reversed,
with a single buyer interested in purchasing an object from one of several sellers. Here, our
auction results imply that if the buyer runs a second-price sealed-bid auction (buying from
the lowest bidder at the second-lowest price), or equivalently a descending-o↵er auction, then
the sellers will o↵er to sell at their true costs. In this case, the sellers are choosing prices
(their o↵ers) in a procedure selected by the buyer.
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
311
312 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
But who sets prices, and who trades with whom, if there are many buyers and many
sellers? To get a feel for what happens, let’s look first at how trade takes place in an actual
market.
Trade with Intermediaries. In a wide range of markets, individual buyers and sellers do
not interact directly with each other, but instead trade through intermediaries — brokers,
market-markers, or middlemen who set the prices. This is true in settings that range from
the trade of agricultural goods in developing countries to the trade of assets in financial
markets.
To get a sense for how markets with intermediaries typically work, let’s focus on the latter
example, and consider how buyers and sellers interact in the stock market. In the U.S., buyers
and sellers trade over a billion shares of stock daily. But there is no one market for trade in
stocks in the U.S. Instead, trade occurs on multiple exchanges such as the New York Stock
Exchange (NYSE) or the NASDAQ-OMX, as well as on alternative trading systems such as
those run by Direct Edge, Goldman Sachs, or Investment Technologies Group (ITG), which
arrange trades in stocks for their clients. These markets operate in various ways: some (such
as NYSE or NASDAQ-OMX) determine prices that look very much like our market-clearing
prices from Chapter 10, while others (like Direct Edge, Goldman, or ITG) simply match
up orders to buy and sell stocks at prices determined in other markets. Some have people
(called specialists in the NYSE) directly involved in setting prices, while others are purely
electronic markets with prices set by algorithms; some trade continuously throughout the
day, while others trade less frequently as they wait for batches of buy and sell orders to
arrive; some allow anyone at least indirect access to the market, while others restrict the
group of buyers and sellers that they will deal with (often to large institutional traders).
Many of these markets create something called an order book for each stock that they
trade. An order book is simply a list of the orders that buyers and sellers have submitted
for that stock. A trader might for instance submit an order to sell 100 shares if the price is
at $5 or more per share; another trader might submit an order to sell 100 shares if the price
is at $5.50 or more per share. Two other traders might submit orders to buy 100 shares if
the price is no more than $4 per share, and to buy 100 shares if the price is no more than
$3.50 per share. Orders of this type are called limit orders, since they are commitments to
buy or sell only once the price reaches some limit set by the trader. If these were the only
orders that existed, then the order book for this stock would look like Figure 11.1(a).
The highest outstanding o↵er to buy the stock is referred to as the current bid for the
stock, while the lowest outstanding o↵er to sell it is referred to as the ask. If the market
uses a specialist then this person knows the book of orders, and may choose to submit his
or her own better o↵er to buy or sell out of inventory of the stock, which becomes the bid or
ask respectively. For example, if Figure 11.1(a) describes the order book, then the specialist
11.1. PRICE-SETTING IN MARKETS 313
$3.50 $4.00
$5.50
BID
ASK$5.00
(a)
$3.50 $4.00
$5.50
BID
ASK
(b)
Figure 11.1: (a) A book of limit orders for a stock with a bid of $4 and an ask of $5. (b) A book of limit orders for a stock with a bid of $4 and an ask of $5.50.
may choose to display a bid of $4.25, based on his or her own better o↵er, and an ask of
$5.00. These are the prices displayed to the trading public.
Retail traders (small traders who buy and sell stocks using their own assets) most often
do not submit limit orders; instead they typically submit orders to buy or sell at the existing
quotes — the current bid and ask. This type of order to trade immediately at market prices
is called a market order. For example, if a trader submits a market order to buy 100 shares
of a stock with the order book described by Figure 11.1(a), then the seller whose limit order
was placed at $5.00 sells 100 shares and the buyer who submitted the market order buys 100
shares at $5.00. (Note that the seller can be either a member of the public or the specialist.)
The new order book would then be as displayed in Figure 11.1(b), and new ask would be
$5.50. This process continues throughout the trading day with new limit orders, specialist
o↵ers, and market orders arriving over time, and transactions being performed.
Of course, orders to buy or sell are not always for 100 shares, and in fact order sizes vary
greatly. For example, if the order book is as depicted in Figure 11.1(a), and a market order
to buy 200 shares arrives then both sellers on the book sell at their ask prices. The buyer
will buy 100 shares at $5.00 and 100 shares at $5.50. We can think of this order as “walking
up the book,” since executing it exposes multiple orders at di↵erent prices.
Large mutual funds such as Fidelity or Vanguard, and other institutional traders such
as banks, pension funds, insurance companies and hedge funds, buy and sell a very large
number of shares each day. They don’t really want to trade many small lots of shares with
retail traders and, as in our 200-share example, walk up or down the book. They also don’t
want to submit a single large limit order to the market, as then other market participants
314 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
will know their trading desires, and can take advantage of them.1 Instead of submitting a
single large market or limit order these traders use a variety of orders and trading venues.
They typically split their order into many pieces and trade these pieces over the trading day,
or over several days, in order to minimize the impact of their trading desire on the price.
One way in which large traders hide their trading desires is to submit pieces of it to many
di↵erent trading systems to which they have access. One particularly interesting group of
trading systems are called dark pools. Examples of these alternative trading systems are
Goldman Sachs’s Sigma-X and the systems run by ITG. Access to these systems is limited
and orders submitted to these systems are not displayed to the public. Instead these systems
simply match orders submitted by their clients at prices established in the public market,
and charge their clients a fee for the service. This is a relatively new, but growing segment
of the market; in April 2009, for example, approximately 9% of the trade in U.S. equities
was done on dark pools.
As you might imagine, the actual structure of the stock market is very complex and
rapidly evolving. There are many trading systems, many types of orders that buyers and
sellers can use, and a wide variety of market participants. The questions of how prices evolve
over time and how they relate to the underlying fundamental value of the assets being traded
are also important, and we have ignored these issues so far. We will discuss some aspects
of the evolution of prices and their relation to the underlying values in Chapter 22; more
detailed analyses of the stock market are carried out in a number of books [206, 209, 332].
The collection of di↵erent trading venues for stocks ultimately results in a variety of
markets with restricted participation. So when we take into account the full set of trading
options for all market participants — both large and small — we see a network structure
emerge, connecting buyers and sellers to di↵erent possible intermediaries. A fundamental
question is how to reason about trade when there are multiple markets connected by a
network in this way. In the next section, we develop a network model for trade which
abstracts away the specific details of the stock market, focusing on the general issue of how
the underlying structure constrains who can trade with whom, and how prices are set by
market participants.
1A large order to buy, for example, may provide information to other market participants suggesting that the stock is currently undervalued and its price is likely to increase. These other market participants may then jump into the market, perhaps getting ahead of the execution of some part of the large order, and drive the price up quickly. This would harm the trader who submitted the large order as he may then have to pay more than expected for his order. This is related to broader questions about the role of information in markets, a topic we discuss in Chapter 22.
11.2. A MODEL OF TRADE ON NETWORKS 315
11.2 A Model of Trade on Networks
Our network model will be based on three fundamental principles that we saw in discussing
the stock market: Individual buyers and sellers often trade through intermediaries, not all
buyers and sellers have access to the same intermediaries, and not all buyers and sellers trade
at the same price. Rather, the prices that each buyer and seller commands are determined
in part by the range of alternatives that their respective network positions provide.
Before specifying the model, let’s first look at another example of trade, in a very di↵erent
setting, that exhibits these properties. This is the market for agricultural goods between
local producers and consumers in a developing country. In many cases there are middlemen,
or traders, who buy from farmers and then resell to consumers. Given the often poor
transportation networks, the perishability of the products and limited access to capital by
farmers, individual farmers can sell only to a limited number of intermediaries [46, 153].
Similarly, consumers can buy from only a limited number of intermediaries. A developing
country may have many such partially overlapping local markets existing alongside modern,
more global markets.
We can use a graph to describe the trading opportunities available to sellers, buyers,
and middlemen (traders). Figure 11.2 depicts a simple example of such a trading network,
superimposed on its geographic setting. Here we’ve labeled seller nodes with S, buyer nodes
with B, and trader nodes with T; and we’ve placed an edge between any two agents who can
trade with each other. Notice in this example that the seller and buyer on the right-hand
margin of the picture only have access to the trader on their side of the river. The buyer
at the top of the figure has access to both traders — perhaps he has a boat. You might
imagine that the extra trading opportunities available to this buyer, and the similar extra
trading opportunities available to the seller on the west bank of the river, would result in
better prices for them. We will see that this is exactly what happens in the trading outcomes
determined by our model in networks of this type.
Network Structure. We now describe a simple model of trade on a network which is
general enough to incorporate important features of the trading and price-setting process
for commodities as varied as financial assets traded in developed countries and agricultural
goods in developing countries [63].
For the simplest form of the model, we don’t try to address the issue of multiple goods
for sale, or multiple possible quantities; instead, we assume there is a single type of good
that comes in indivisible units. Each seller i initially holds one unit of the good which he
values at v i
; he is willing to sell it at any price that is at least v i
. Each buyer j values one
copy of the good at v j
, and will try to obtain a copy of the good if she can do it by paying
no more than v j
. No individual wants more than one copy of the good, so additional copies
are valued at 0. All buyers, sellers, and traders are assumed to know these valuations. As a
316 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
T
T
S
B
S
B
S
B
Figure 11.2: Trading networks for agricultural markets can be based on geographic con- straints, giving certain buyers (nodes labeled B) and sellers (nodes labeled S) greater access to traders (nodes labeled T).
result, this model is best thought of as describing interaction between individuals who have
a history of trade with each other, and hence know each other’s willingness to pay for goods.
Trade takes place on a network that represents who can trade with whom. As in the
example depicted in Figure 11.2 the nodes consist of buyers, sellers, and traders, with each
edge representing an opportunity for trade. Since we are assuming that the traders act as
intermediaries for the possible seller-buyer transactions, we require that each edge connects
a buyer or seller to a trader. In Figure 11.3, we depict the same graph from Figure 11.2,
redrawn to emphasize these features of the network model. (In all of our figures depicting
trading networks we will use the following conventions. Sellers are represented by circles
on the left, buyers are represented by circles on the right, and traders are represented by
squares in the middle. The value that each seller and buyer places on a copy of the good is
written next to the respective node that represents them.)
Beyond the fact that we now have intermediaries, there are a few other di↵erences between
this model and our model of matching markets from Chapter 10. First, we are assuming
11.2. A MODEL OF TRADE ON NETWORKS 317
S1
S2
S3
T1
T2
B1
B2
B3
sellers traders buyers
0
0
0
1
1
1
vi vj
Figure 11.3: A standardized view of the trading network from Figure 11.2: Sellers are represented by circles on the left, buyers are represented by circles on the right, and traders are represented by squares in the middle. The value that each seller and buyer places on a copy of the good is written next to the respective node that represents them.
that buyers have the same valuation for all copies of a good, whereas in matching markets
we allowed buyers to have di↵erent valuations for the goods o↵ered by di↵erent sellers. The
model in this chapter can be extended to allow for valuations that vary across di↵erent copies
of the good; things become more complicated, but the basic structure of the model and its
conclusions remain largely the same. A second di↵erence is that the network here is fixed and
externally imposed by constraints such as geography (in agricultural markets) or eligibility
to participate (in di↵erent financial markets). In matching markets, we began the chapter
with fixed graphs such as this, but then focused the core of the analysis on preferred-seller
graphs that were determined not by external forces but by the preferences of buyers with
respect to an evolving set of prices.
Prices and the Flow of Goods. The flow of goods from sellers to buyers is determined
by a game in which traders first set prices, and then sellers and buyers react to these prices.
Specifically, each trader t o↵ers a bid price to each seller i that he is connected to; we
will denote this bid price by b ti
. (The notation indicates that this is a price for a transaction
between t and i). This bid price is an o↵er by t to buy i’s copy of the good at a value of b ti
.
Similarly, each trader t o↵ers an ask price to each buyer j that he is connected to. This ask
price, denoted a tj
, is an o↵er by t to sell a copy of the good to buyer j at a value of a tj
. In
Figure 11.4(a), we show an example of bid and ask prices on the graph from Figure 11.3.
318 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
S1
S2
S3
T1
T2
B1
B2
B3
sellers traders buyers
0
0
0
1
1
1
0.2 0.8
0.2 0.8
0.3 0.7
0 1
(a) Prices
S1
S2
S3
T1
T2
B1
B2
B3
sellers traders buyers
0
0
0
1
1
1
0.2 0.8
0.2 0.8
0.3 0.7
0 1
(b) Flow of goods
Figure 11.4: (a) Each trader posts bid prices to the sellers he is connected to, and ask prices to the buyers he is connected to. (b) This in turn determines a flow of goods, as sellers and buyers each choose the o↵er that is most favorable to them.
Once traders announce prices, each seller and buyer chooses at most one trader to deal
with — each seller sells his copy of the good to the trader he selects (or keeps his copy of
the good if he chooses not to sell it), and each buyer purchases a copy of the good from
the trader she selects (or receives no copy of the good if she does not select a trader). This
determines a flow of goods from sellers, through traders, to buyers; Figure 11.4(b) depicts
such a flow of goods, with the sellers’ and buyers’ choices of traders indicated by the edges
with arrows on them.
Because each seller has only one copy of the good, and each buyer only wants one copy,
at most one copy of the good moves along any edge in the network. On the other hand,
there is no limit on the number of copies of the good that can pass through a single trader
node. Note that a trader can only sell as many goods to buyers as he receives from sellers;
we will include in the model a large penalty imposed on a trader who defaults on an o↵er to
sell to a buyer as a result of not having enough goods on hand. Due to this, there are strong
incentives for a trader not to produce bid and ask prices that cause more buyers than sellers
to accept his o↵ers. There are also incentives for a trader not to be caught in the reverse
di�culty, with more sellers than buyers accepting his o↵ers — in this case, he ends up with
excess inventory that he cannot sell. We will see that neither of these will happen in the
solutions we consider; traders will choose bid and ask prices such that the number of goods
they receive from sellers is equal to the number of goods they pass on to buyers.
Finally, notice something else about the flow of goods in this example: seller S3 accepts
the bid even though it is equal to his value, and likewise buyer B3 accepts the ask even
11.2. A MODEL OF TRADE ON NETWORKS 319
though it is equal to his value. In fact, each of S3 and B3 is indi↵erent between accepting and
rejecting the o↵er. Our assumption in this model is that when a seller or buyer is indi↵erent
between accepting or rejecting, then we (as the modelers) can choose either alternative as the
outcome that actually happens. Finding a way to handle indi↵erence is an important aspect
in most market models, since transactions will typically take place right at the boundary
of an individual’s willingness to trade. This is similar to the tie-breaking issue inherent in
the formulation of market-clearing prices in Chapter 10 as well. An alternate way to handle
indi↵erence in the present case is to assume a miniscule positive amount of payo↵ (e.g. a
penny) that is required for an agent to be willing to trade, in which case we would see bid
and ask values like 0.01 and 0.99. While this makes the tie-breaking decision more explicit,
the model becomes much messier and ultimately harder to reason about. As a result, we
will stick to the approach where we allow trades at zero payo↵, with ties broken as needed;
in doing so, we will remember that this is essentially a formal way to represent the idea of
a price or a profit margin being driven to (almost) zero. And if it makes things simpler to
think about, whenever you see an indi↵erent seller or buyer choosing to transact or not, you
can imagine the price being shifted either upward or downward by 0.01 to account for this
decision by the seller or buyer.
Payo↵s. Recall that specifying a game requires a description of the strategies and the
payo↵s. We have already discussed the strategies: a trader’s strategy is a choice of bid and
ask prices to propose to each neighboring seller and buyer; a seller or buyer’s strategy is a
choice of a neighboring trader to deal with, or the decision not to take part in a transaction.
The payo↵s follow naturally from the discussion thus far.
• A trader’s payo↵ is the profit he makes from all his transactions: it is the sum of the ask prices of his accepted o↵ers to buyers, minus the sum of the bid prices of his accepted
o↵ers to sellers. (As discussed above, we also subtract a large penalty if the trader has
more accepted asks than bids, but the e↵ect of this is primarily to ensure that traders
will never expose themselves to this situation in the solutions we consider.)
• For a seller i, the payo↵ from selecting trader t is b ti
, while the payo↵ from selecting
no trader is v i
. In the former case, the seller receives b ti
units of money, while in the
latter he keeps his copy of the good, which he values at v i
. (We will consider only cases
in which all the seller v i
’s are 0.)
• For each buyer j, the payo↵ from selecting trader t is v j
� a tj
, while the payo↵ from
selecting no trader is 0. In the former case, the buyer receives the good but gives up
a tj
units of money.
So for example, with prices and the flow of goods as in Figure 11.4(b), the payo↵ to the
first trader is (0.8�0.2) = 0.6 while the payo↵ to the second trader is (0.7+1�0.3�0) = 1.4.
320 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
S1
S2
S3
T1
T2
B1
B2
B3
sellers traders buyers
0
0
0
1
1
1
0 1
0.4 0.6
0.3 0.7
0 1
Figure 11.5: Relative to the choice of strategies in Figure 11.4(b), trader T 1 has a way to improve his payo↵ by undercutting T 2 and performing the transaction that moves S2’s copy of the good to B2.
The payo↵s to the three sellers are 0.2, 0.3, and 0 respectively, while the payo↵s to the three
buyers are 1 � 0.8 = 0.2, 1 � 0.7 = 0.3, and 1 � 1 = 0 respectively. The game we’ve defined here has a further important feature, which forms a contrast with
other games we have discussed earlier. In earlier games, all players moved (i.e. executed
their chosen strategies) simultaneously, while in this game the moves happen in two stages.
In the first stage, all the traders simultaneously choose bid and ask prices. In the second
stage, all the sellers and buyers then simultaneously choose traders to deal with. For us,
this two-stage structure will not make things too complicated, particularly since the second
stage is extremely simple: the best response for each seller and buyer is always simply to
choose the trader with the best o↵er, and so we can essentially view the sellers and buyers
as “drones” who are hard-wired to follow this rule. Still, we will have to take the two-stage
structure into account when we consider the equilibria for this game, which we do next.
Best Responses and Equilibrium. Let’s think about the strategies that the two traders
have chosen in Figure 11.4(b). The upper trader T 1 is making several bad decisions. First,
because of the o↵ers he is making to seller S2 and buyer B2, he is losing out on this deal to
the lower trader T 2. If for example he were to raise his bid to seller S2 to 0.4, and lower his
ask to buyer B2 to 0.6, then he’d take the trade away from trader T 2: seller S2 and buyer
B2 would both choose him, and he’d make a profit of 0.2.
Second, and even more simply, there is no reason for trader T 1 not to lower his bid to seller
11.2. A MODEL OF TRADE ON NETWORKS 321
S1, and raise his ask to buyer B1. Even with worse o↵ers, S1 and B1 will still want to deal
with T 1, since they have no other options aside from choosing not to transact. Given this,
T 1 will make more money with a lower bid to S1 and a higher ask to B1. Figure 11.5 shows
the results of a deviation by the upper trader that takes both of these points into account;
his payo↵ has now increased to (1 + 0.6 � 0 � 0.4) = 1.2. Note that seller S1 and buyer B1 are now indi↵erent between performing the transaction or not, and as discussed earlier, we
give ourselves (as the modelers) the power to break ties in determining the equilibrium for
such situations.
This discussion motivates the equilibrium concept we will use for this game, which is
a generalization of Nash equilibrium. As in the standard notion of Nash equilibrium from
Chapter 6, it will be based on a set of strategies such that each player is choosing a best
response to what all the other players are doing. However, the definition also needs to take
the two-stage structure of the game into account.
To do this, we first think about the problem faced by the buyers and sellers in the second
stage, after traders have already posted prices. Here, we have a standard game among the
buyers and sellers, and each of them chooses a strategy that is a best response to what all
other players are doing. Next, we think about the problem faced by the traders, in deciding
what prices to post in the first stage. Here, each trader chooses a strategy that is a best
response both to the strategies the sellers and buyers will use (what bids and asks they will
accept) and the strategies the other traders use (what bids and asks they post). So everyone
is employing a best response just as in any Nash equilibrium. The one di↵erence here is
that since the sellers and buyers move second they are required to choose optimally given
whatever prices the traders have posted, and the traders know this. This equilibrium is
called a subgame perfect Nash equilibrium; in this chapter, we will simply refer to it as an
equilibrium.2
The two-stage nature of our game here is particularly easy to think about, since the
behavior of sellers and buyers is very simple. Thus, for purposes of reasoning about equilibria,
we can mainly think about the strategies of the traders in the first stage, just as in a
simultaneous-move game, knowing that sellers and buyers will simply choose the best o↵ers
(possibly with tie-breaking) once the traders post prices.
In the next section, we’ll work out the set of possible equilibria for the trading network
in Figures 11.3–11.5, by first dissecting the network into simpler “building blocks.” In
particular, these building blocks will correspond to two of the basic structures contained
within the network in Figures 11.3–11.5: buyers and sellers who are monopolized by having
2The word “subgame” refers to the fact that once traders post prices, the buyers and sellers are faced with a new free-standing game in the second stage. The word “perfect” refers to the requirement that in the subgame, the players who have choices remaining are required to behave optimally given the choices that have already been made. This concept is considered at a general level, although without this particular terminology, in the discussion of games with sequential moves in Section 6.10.
322 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
S1 T1 B1
sellers traders buyers
0 1
0 1
Figure 11.6: A simple example of a trading network in which the trader has a monopoly and extracts all of the surplus from trade.
only a single trader they can deal with, and buyers and sellers who benefit from perfect
competition between multiple traders. In the process, we’ll see that network structure and
access to alternatives can significantly a↵ect the power of participants in the market.
11.3 Equilibria in Trading Networks
We now discuss the process of analyzing equilibria in trading networks. We begin with
simple network structures and build up to the example from the previous section. Following
our plan, we’ll begin by considering simple networks corresponding to monopoly and perfect
competition.
Monopoly. Buyers and sellers are subject to monopoly in our model when they have access
to only a single trader. Perhaps the simplest example of this is depicted in Figure 11.6. Here
we have one seller who values the good at 0, one trader, and one buyer who values the good
at 1.
In this trading network the trader is in a monopoly position relative to both the seller
and the buyer (there is only one trader available to each of them). The only equilibrium is
for the trader to set a bid of 0 to the seller and an ask of 1 to the buyer; the seller and buyer
will accept these prices and so the good will flow from the seller to the trader and then on to
the buyer. Note that we are using the indi↵erence of the seller and buyer as in the example
from the previous section: since the seller and buyer are indi↵erent between engaging in a
transaction or not, we as the modelers are choosing the outcome and having them perform
the transaction.
To see why this is the only equilibrium, we simply notice that with any other bid and ask
between 0 and 1, the trader could slightly lower the bid or raise the ask, thereby performing
the transaction at a higher profit.
11.3. EQUILIBRIA IN TRADING NETWORKS 323
S1
T1
T2
B1
sellers traders buyers
0 1
x x
x x
Figure 11.7: A trading network in which there is perfect competition between the two traders, T1 and T2. The equilibrium has a common bid and ask of x, where x can be any real number between 0 and 1.
Perfect Competition. Now let’s look at a basic example showing perfect competition
between two traders, as depicted in Figure 11.7.
In Figure 11.7 there is competition between traders T 1 and T 2 to buy the copy of the
good from S1 and sell it to B1. To help in thinking about what forms an equilibrium, let’s
first think about things that are out of equilibrium, in a manner similar to what we saw in
Figure 11.5. In particular, suppose trader T 1 is performing the trade and making a positive
profit: suppose his bid to the seller is some number b, and his ask to the buyer is a number
a > b. Since T 2 is not performing the trade, he currently has a payo↵ of zero. But then
it must be that T 2’s current strategy is not a best response to what T 1 is doing: T 2 could
instead o↵er a bid slightly above b and an ask slightly below a, thereby taking the trade
away from T 1 and receiving a positive payo↵ instead of zero.
So it follows that whichever trader is performing the trade at equilibrium must have a
payo↵ of 0: he must be o↵ering the same value x as his bid and ask. Suppose that trader T 1
is performing the trade. Notice that this equilibrium involves indi↵erence on his part: he is
indi↵erent between performing the trade at zero profit and not performing the trade. As in
the earlier case of indi↵erence by sellers and buyers, we assume that we (the modelers) can
choose an outcome in this case, and we will assume that the transaction is performed. Here
too, we could handle indi↵erence by assuming a minimum increment of money (e.g. 0.01),
and having the transaction take place with a bid and ask of x�0.01 and x respectively, but again handling indi↵erence via zero payo↵s (and keeping in mind that they are designed to
model profits that come arbitrarily close to 0) makes the analysis simpler without a↵ecting
324 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
the outcome.
Next we want to argue that the trader not performing the trade at equilibrium (T 2 in
this case) must also have bid and ask values of x. First, notice that in equilibrium, we cannot
have a trader buy the good from the seller without also selling it to the buyer; therefore,
T 2 must be o↵ering a bid b x, (or else the seller would sell to T 2) and an ask a � x. (or else the buyer would buy from T 2). But if this bid and ask were not the same — that is, if
a > b — then T 1 could lower his bid or raise his ask so that they still lie strictly between a
and b. In that case T 1 could perform the trade while making a positive profit, and hence his
current strategy of bidding and asking x would not be a best response to what T 2 is doing.
So the equilibrium occurs at a common bid and ask of x. What can we say about the
value of x? It clearly has to be between 0 and 1: otherwise either the seller wants to sell but
the buyer wouldn’t want to buy, or conversely the buyer wants to buy but the seller wouldn’t
want to sell. In fact this is all that we can say about x. Any equilibrium consists of a common
bid and ask by each trader, and a flow of goods from the seller to the buyer through one of
the traders. A key feature of the equilibrium is that the seller sells to the same trader that
the buyer buys from: this is another kind of coordination in the face of indi↵erence that is
reminiscent of the tie-breaking issues in market-clearing prices from Chapter 10. It is also
interesting that while traders make no profit in any equilibrium, the choice of equilibrium
— captured in the value of x — determines which of the seller or buyer receives a higher
payo↵. It ranges from the extreme cases of x = 0 (where the buyer consumes all the available
payo↵) and x = 1 (where the seller consumes it) to the intermediate value of x = 1 2
(where
the seller and buyer receive equal payo↵s). In the end, the choice of equilibrium reflects
something about the relative power of the seller and buyer that can only be inferred by
looking outside the formulation of the trading game — the game itself can determine only
the range of possible equilibria.
The Network from Section 11.2. Using the networks in Figures 11.6 and 11.7 as build-
ing blocks, it is not hard to work out the equilibria in the example from Section 11.2. This
is illustrated in Figure 11.8. Sellers S1 and S3, and buyers B1 and B3, are monopolized by
their respective traders, and so in any equilibrium these traders will drive the bids and asks
all the way to 0 and 1 respectively.
Seller S2 and buyer B2, on the other hand, benefit from perfect competition between
the two traders. Here the argument follows what we used in analyzing the simpler network
in Figure 11.7: the trader performing the transaction must have bid and ask values equal to
the same number x (for some real number x between 0 and 1), or else the other trader could
take the trade away from him; and given this, the other trader must also have bid and ask
values equal to x.
These types of reasoning are useful in analyzing other more complex networks as well.
11.3. EQUILIBRIA IN TRADING NETWORKS 325
S1
S2
S3
T1
T2
B1
B2
B3
sellers traders buyers
0
0
0
1
1
1
0 1
x x
x x
0 1
Figure 11.8: The equilibria for the trading network from Section 11.2. This network can be analyzed using the ideas from the simpler networks representing monopoly and perfect competition.
When you see a seller or buyer connected to only a single trader, they will receive zero payo↵
in any equilibrium, since the trader will drive the bid or ask to as extreme a value as possible.
On the other hand, when two traders both connect the same seller and buyer, then neither
can make a positive profit in conveying a good from this seller to this buyer: if one trader
performed the trade at a positive profit, the other could undercut them.
We now consider an example illustrating how the network structure can also produce
more complex e↵ects that are not explained by these two principles.
Implicit Perfect Competition. In our examples so far, when a trader makes no profit
from a transaction, it is always because there is another trader who can precisely replicate
the transaction — i.e., a trader who is connected to the same seller and buyer. However, it
turns out that traders can make zero profit for reasons based more on the global structure
of the network, rather than on direct competition with any one trader.
The network in Figure 11.9 illustrates how this can arise. In this trading network there
is no direct competition for any one “trade route” from a seller to a buyer. However, in any
equilibrium, all bid and ask prices take on some common value x between 0 and 1, and the
goods flow from the sellers to the buyers. So all traders again make zero profit.
It is easy to see that this is an equilibrium: we can simply check that each trader is using
a best response to all the other traders’ strategies. It takes a bit more work to verify that in
every equilibrium, all bid and ask prices are the same value x: this is most easily done by
326 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
T1
T2
T3
T4
S1
S2
B1
B2
0
0
1
1
x
x
x
x
x
x
x
x
Figure 11.9: A form of implicit perfect competition: all bid/ask spreads will be zero in equilibrium, even though no trader directly “competes” with any other trader for the same buyer-seller pair.
checking alternatives in which some trader posts a bid that is less than the corresponding
ask, and identifying a deviation that arises.
11.4 Further Equilibrium Phenomena: Auctions and Ripple E↵ects
The network model we’ve been considering is expressive enough that it can represent a
diverse set of other phenomena. Here we consider two distinct examples: the first showing
how the second-price auction for a single item arises from a trading network equilibrium,
and the second exploring how small changes to a network can produce e↵ects that ripple to
other nodes.
Second-price auctions. Figure 11.10 shows how we can represent the structure of a
single-item auction using a trading network. Suppose there is a single individual S1 with
an item to sell, and four potential buyers who value the item at values w, x, y, z, listed in
descending order w > x > y > z. We use four buyers in this example but our analysis would
work for an arbitrary number of buyers.
In keeping with our model in which trading happens through intermediaries, we assume
11.4. FURTHER EQUILIBRIUM PHENOMENA: AUCTIONS AND RIPPLE EFFECTS327
S1
B1
B2
B3
B4
T1
T2
T3
T4
0
w
x
y
z
(a) A single-item auction
S1
B1
B2
B3
B4
T1
T2
T3
T4
0
w
x
y
z
x
x
y
z
x
y
z
w
(b) Equilibrium for the auction
Figure 11.10: (a) A single-item auction can be represented using a trading network. (b) Equilibrium prices and flow of goods. The resulting equilibrium implements the second- price rule from Chapter 9.
that each buyer is represented by a distinct trader — essentially, someone who serves as
the buyer’s “proxy” for the transaction. This gives us a trading network as depicted in
Figure 11.10(a).
Now let’s consider a possible equilibrium for this network. Trader T 1 has the ability to
outbid all the other traders, since he has ability to sell to his buyer for an ask up to w. In
equilibrium, he will outbid them by the minimum he needs to in order to make the trade,
which he can do by o↵ering x to outbid T 2. Here we use indi↵erence to assume that the sale
at x will go to T 1 rather than T 2, that buyer B1 will buy from T 1 at a price of w, and that
buyers B2 through B4 will choose not to buy the good from their respective traders.
We therefore get the equilibrium depicted in Figure 11.10(b). Notice how this equilibrium
has exactly the form of a second-price auction, in which the item goes to the highest bidder,
with the seller receiving the second-highest valuation in payment.3 What’s interesting is
3With a little more work, we can describe the full set of equilibria for this network, and show that the second-price rule is unique over equilibria that avoid a certain “pathological” structure, as follows. In any equilibrium, the good flows from the seller to buyer B1, and each trader o↵ers to sell the good to his monopolized buyer for the buyer’s value. In the equilibrium we consider, T 1 and T 2 both bid x. However, there are other bids that can be part of an equilibrium: Essentially, as long as one of traders T 2, T 3, or T 4 bids between x and w, and T 1 matches this bid, we have an equilibrium. If the highest bid among T 2, T 3, and T 4 is strictly greater than x, then we have a situation in which this high-bidding trader among T 2 through T 4 has a crossing pair of bid and ask values: his bid is higher than his corresponding ask. This is an equilibrium, since T 1 still makes the trade, the trader with the crossing bid/ask pair doesn’t lose money, and no one has an incentive to deviate. However, it is a pathological kind of equilibrium since there is a trader who is o↵ering to buy for more than he is o↵ering to sell [63]. Thus, if we consider only equilibria
328 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
S1
S2
S3
T1
T2
B1
B2
B4
0
0
0
1
2
3
0 1
0 x
x
0
B3
4
3
4
(a) Equilibrium before new link.
S1
S2
S3
T1
T2
B1
B2
B4
0
0
0
1
2
3
0 1
z y
y
0 B3
4
3
4
z
(b) Equilibrium after new link.
Figure 11.11: (a) Equilibrium before the new S2-T 2 link is added. (b) When the S2-T 2 edge is added, a number of changes take place in the equilibrium. Among these changes is the fact that buyer B1 no longer gets a copy of the good, and B3 gets one instead.
that the second-price rule wasn’t in any sense “built in” to the formulation of the auction;
it emerged naturally as an equilibrium in our network representation.
Ripple E↵ects from Changes to a Network. Our network model also allows us to
explore how small changes to the network structure can a↵ect the payo↵s of nodes that are
not directly involved in the change. This suggests a way of reasoning about how “shocks”
to highly interconnected trading networks can ripple to more distant parts of the network.
This is a very general issue; we consider it here in a specific example where we can see
concretely how such e↵ects can arise. Consider the pair of networks in Figure 11.11: the
second network is obtained from the first simply by adding a link from S2 to T 2. We first
work out the equilibria for each of these networks, and then consider how they di↵er.
In Figure 11.11(a), all sellers and buyers are monopolized except for B2, so their payo↵s
will all be zero; we use indi↵erence to assume that B3 will not buy the good but B1 and B4
will. (As in all our examples, we can view this as modeling the fact that T 2 can charge a
price very slightly above 3 to B3, dissuading B3 from purchasing.) The one part that needs
additional analysis is the pair of asks charged to B2. In equilibrium, these must be the same
(otherwise, the trader making the sale could slightly raise his ask), and this common value x
can be anything between 0 and 2. We exploit indi↵erence here to assume that B2 buys from
without such crossing pairs of bid/ask values, then T 2 bids the “second-price value” x, and this is what the good sells for. So the second-price rule is unique over equiibria without crossing pairs.
11.4. FURTHER EQUILIBRIUM PHENOMENA: AUCTIONS AND RIPPLE EFFECTS329
trader T 1. Note that there cannot be an equilibrium in which buyer B2 buys from trader
T 2, since B2 can pay only 2 while trader T 2 can sell the unit of the good he is able to buy
at a price of 4.
In Figure 11.11(b), once the edge from S2 to T 2 has been added, we need to work out the
equilibrium bids and asks charged to S2 and B2, and the flow of goods. Reasoning about
these bids and asks requires a bit more work than we’ve seen in previous examples, so we
build up to it in a sequence of steps.
• The two bids to S2 must be the same as each other, since otherwise the trader getting the good could slightly lower his bid; for a similar reason, the two asks to B2 must
also be the same as each other. Let’s call the common bid z and the common ask y.
• We can next determine how the seller-trader transactions work out in equilibrium. S2 will sell to T 2 rather than T 1 in equilibrium: if S2 were selling to T 1, and T 1 were
receiving a positive payo↵ from this transaction, then S2 would be selling for at most
2. In this case, T 2 could slightly outbid T 1 and sell S2’s copy of the good to B3. So
in equilibrium, T 2 buys two copies of the good, while T 1 buys only one.
• Now let’s figure out the possible values for the ask y. The ask y must be at least 1: otherwise, one of the traders is selling to B2 for a low price, and the trader performing
this sale has an alternate trader whom he monopolizes, and from whom he would get
a higher payo↵. This can’t happen in equilibrium, so y is at least 1.
Also, the ask y cannot be above 2 in equilibrium: in this case, B2 would not buy, and
so T 1 could perform a payo↵-improving change in strategy by lowering his ask to B2,
thereby getting B2 to buy from T 1 for a price between 1 and 2.
• Next, we determine how the trader-buyer transactions work out in equilibrium. We’ve already concluded that T 2 is buying two copies of the good, and so he maximizes his
payo↵ by selling them to B3 and B4. Therefore T 2 is not selling to B2 in equilibrium.
Since, the ask y is at least 1, trader T 1 will buy from S1 and sell to B2.
• Finally, what do we know about the value of z? It has to be at least 1, or else T 1 could outbid T 2 for S2’s copy of the good, and receive a positive payo↵ by selling it to B1.
It also has to be at most 3, or else T 2 would prefer not to buy it from S2.
This sums up the analysis: in equilibrium, the bid z can be anything between 1 and 3, the
ask y can be anything between 1 and 2, and the goods flow from S1 through T 1 to B2, and
from S2 and S3 through T 2 to B3 and B4. Notice that this flow of goods maximizes the
total valuation of the buyers who obtain the good, given the constraints on trade imposed
by the network. We will see later in this chapter that any equilibrium has this e�ciency
property.
330 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
Let’s consider at a high level what’s going on in this pair of examples. In Figure 11.11(a),
trader T 2 has access to a set of buyers who want copies of the good very badly (they value
it highly), but his access to sellers is very limited. T 1 on the other hand is able to use all
his available trading opportunities. The market in this respect has a “bottleneck” that is
restricting the flow of goods.
Once S2 and T 2 form a link, creating the network in Figure 11.11(b), a number of things
change. First, and most noticeably, buyer B3 now gets a copy of the good while B1 doesn’t.
Essentially, the bottleneck in the market has been broken open, so the high-value buyers can
now obtain the good at the expense of the low-value buyers. From B1’s perspective, this
is a “non-local” e↵ect: a link formed between two nodes, neither of which are neighbors of
hers, has caused her to no longer be able to obtain the good.
There are other changes as well. Seller S2 is now in a much more powerful position, and
will command a significantly higher price (since y is at least 1 in any equilibrium). Moreover,
the range of possible equilibrium asks to B2 has been reduced from the interval [0, 2] to the
interval [1, 2]. So in particular, if we were previously in an equilibrium where the ask to B2
was a value x < 1, then this equilibrium gets disrupted and replaced by one in which the ask
is a higher number y � 1. This indicates a subtle way in which B2 was implicitly benefitting from the weak position of the sellers, which has been now strengthened by the creation of
the edge between S2 and T 2.
This is a simple example, but it already illustrates some of the complexities that can
arise when the structure of a trading network changes to alleviate (or create) bottlenecks for
the flow of goods. With more work, one can create examples where the e↵ects of changes in
the network ripple much farther through the structure.
This style of reasoning also points to a line of questions in which we view the network
as something malleable, and partially under the control of the market participants. For
example, how much should S2 and T 2 be willing to spend to create a link between each other,
shifting the network structure from the one in Figure 11.11(a) to the one in Figure 11.11(b)?
More generally, how should di↵erent nodes evaluate the trade-o↵s between investing resources
to create and maintain links, and the benefits they get in terms of increased payo↵s? This
is a question that has been considered in other models of trading networks [150, 261], and it
is part of a much broader research activity that investigates the formation of networks as a
game-theoretic activity under a variety of di↵erent kinds of payo↵s [19, 39, 121, 152, 227, 385].
11.5 Social Welfare in Trading Networks
When we’ve looked at games in earlier settings, we’ve considered not just equilibrium solu-
tions, but also the question of whether these solutions are socially optimal. That is, do they
maximize social welfare, the sum of the payo↵s of all players?
11.5. SOCIAL WELFARE IN TRADING NETWORKS 331
In the context of our game, each good that moves from a seller i to a buyer j contributes
v j
�v i
to the social welfare. This is how much more j values the good than i, and the money
that is spent in moving the good from i to j is simply transferred from one player to another,
creating a net e↵ect of zero to the total payo↵. In more detail, if the good moves through
trader t, who o↵ers a bid of b ti
to i and an ask of a tj
to j, then the sum of the payo↵s of i
and j, plus the portion of t’s payo↵s arising from this transaction, is equal to
(b ti
� v i
) + (a tj
� b ti
) + (v j
� a tj
) = v j
� v i
.
Thus the social welfare is simply the sum of v j
�v i
over all goods that move from a seller
i to a buyer j. This makes sense, since it reflects how much happier, in total, the new owners
of the goods are compared to the original owners of the goods. The maximum value of this
quantity over all possible flows of goods — i.e., the socially optimal value — depends not just
on the valuations of the sellers and buyers, but also on the network structure. Networks that
are more richly connected can potentially allow a flow of goods achieving a higher social
welfare than networks that are more sparsely connected, with bottlenecks that prevent a
desirable flow of goods.
For example, let’s go back to the pair of networks in Figure 11.11. In each case, the
equilibrium yields a flow of goods that achieves the social optimum. In Figure 11.11(a), the
best possible value of the social welfare is 1 + 2 + 4 = 7, since there is no way to use the
network to get copies of the goods to both B3 and B4. However, when the single edge from
S2 to T 2 is added, it suddenly becomes possible for both of these buyers to receive copies
of the good, and so the value of the social welfare increases to 2 + 3 + 4 = 9. This provides
a simple illustration of how a more richly connected network structure can enable greater
social welfare from trade.
In our discussion of social optimality we count the gains to traders as part of the social
welfare (since they are part of the society of players, along with sellers and buyers). In the
next section, we will consider how the total payo↵s are divided between sellers, buyers, and
traders, and how it depends on the network structure.
Equilibria and Social Welfare. In both the networks in Figure 11.11, the flow of goods
achieving maximum social welfare can be achieved by an equilibrium.
In fact, this holds for all the examples we’ve seen thus far, and it’s a fact that’s true in
general: it can be shown that in every trading network, there is always at least one equilib-
rium, and every equilibrium produces a flow of goods that achieves the social optimum [63].
While we won’t go into the details of the proof here, it is similar in structure to the Existence
and Optimality of Market-Clearing Prices that we discussed in the previous chapter. There
too, without intermediaries present, we were able to show that prices achieving a certain
type of equilibrium always exist (in that case the market-clearing property), and that all
such prices produce an allocation that maximize social welfare.
332 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
S1
S2
S3
B1
B2
B3
T3
T2
T1
T4
T5
0
0
0
1
1
1
0 x
x
x
x
x
x
x
0
1
Figure 11.12: Whether a trader can make a profit may depend on the choice of equilibrium. In this trading network, when x = 1, traders T 1 and T 5 make a profit, while when x = 0, only trader T 3 makes a profit.
11.6 Trader Profits
We now consider the question of how the social welfare in an equilibrium is divided up as
payo↵s among the sellers, buyers, and traders. In particular, the examples we’ve studied
so far suggest the informal principle that as the network becomes more richly connected,
individual traders have less and less power, and their payo↵s go down. Understanding this
more precisely points to a basic question that can be expressed in terms of our network
model: what is the structural basis of perfect competition?
Our examples suggest that in order to make a profit (i.e. a positive payo↵), a trader
must in some way be “essential” to the functioning of the trading network. Certainly, if there
is another trader who can replicate his function completely, then he cannot make a profit;
nor can he make a profit in more complex settings like the implicit perfect competition in
Figure 11.9. In fact, it will turn out that a version of this “essentiality” principle is true;
but it is a bit more subtle than it might initially appear. To motivate it, we start with two
illuminating examples.
First, whether a trader makes a profit can depend on the equilibrium: in some networks,
it can be possible for a trader to make a profit in some equilibria but not others. Figure 11.12
shows how this can occur. Any choice of x between 0 and 1 will result in an equilibrium,
with the traders T 2 and T 4 who are left out of trades serving to “lock” the value of x in
place. However, when x = 1, traders T 1 and T 5 make a profit, while when x = 0, only
trader T 3 makes a profit. Furthermore, while every equilibrium produces a flow of goods to
all three sellers, for a social welfare of 3, the amount of this social welfare that goes to the
11.6. TRADER PROFITS 333
T1
T2
S1
S2
S3
B1
B2
(a) A network in which trader T1 is essential.
T1
T2
S1
S2
S3
B1
B2
0
0
0
x
x
y
y
(b) An equilibrium where T1 trades one good.
T1
T2
S1
S2
S3
B1
B2
0
0
0
x
x
y
y
(c) An equilibrium where T1 trades two goods.
Figure 11.13: Despite a form of monopoly power in this network, neither trader can make a profit in any equilibrium: we must have x = y = 0.
buyers and sellers — rather than the traders — varies between 1 and 2 as x ranges from 1
to 0.
The second example, in Figure 11.13, is even more counter-intuitive. Here, traders T 1
and T 2 both have monopoly power over their respective sellers, and yet their profits are
zero in every equilibrium. We can verify this fact as follows. First, we notice that any
equilibrium must look like one of the solutions in Figure 11.13(b) or Figure 11.13(c). The
sellers are monopolized and will get bids of 0. For each buyer, the two asks must be the
same, since otherwise the trader making the sale could slightly raise his ask. Now, finally,
notice that if the common ask to either buyer were positive, then the trader left out of the
trade on the higher one has a profitable deviation by slightly undercutting this ask.
Therefore, in this example, all bids and asks equal 0 in any equilibrium, and so neither
trader profits. This happens despite the monopoly power of the traders — and moreover
T 1 fails to make a profit despite the fact that T 2 can only perform one trade on his own.
We can interpret this as a situation in which a small trader competes with a larger trader
across his full set of potential buyers, despite having access to an insu�cient set of sellers to
actually perform all the available trades — a situation in which “the threat is stronger than
its execution.” While this fits naturally within the scope of the model, examples such as
this one also suggest natural extensions to the model, in which each trader has an intrinsic
limit on the number of trades he can perform, and this a↵ects the behavior of competing
traders. While the possibility of such intrinsic limits haven’t played a role in our earlier
examples, Figure 11.13 suggests that allowing for such limits could change the outcome in
certain settings.
With these examples in mind, let’s return to the question of when, for a given trader T
in a network, there exists an equilibrium in which T receives a positive payo↵. It turns out
that there exists such an equilibrium precisely when T has an edge e to a seller or buyer
334 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
such that deleting e would change the value of the social optimum. In such a situation, we
say that e is an essential edge from T to the other node. The proof of this statement is
somewhat involved, and we refer the reader to [63] for the details. Figures 11.6 and 11.8 are
examples of trading networks in which each trader has an essential edge and thus makes a
profit in equilibrium, while Figures 11.7 and 11.9 are examples in which no trader has an
essential edge and no trader makes a profit.
This essential-edge condition is a stronger form of monopoly power than we saw in Fig-
ure 11.13(a). There, although deleting the node T 1 would change the value of the social
optimum, there is no single edge whose deletion would reduce the value of the social opti-
mum below 2; rather, after the removal of any one edge, there would still be a flow of goods
to both buyers. This is the crux of why T 1 is not able to make a profit in Figure 11.13(a),
despite his powerful position.
The example in Figure 11.12 also shows that this condition only implies a profit in some
equilibrium, as opposed to every equilibrium. In Figure 11.12, the available profit essentially
“slides” smoothly from one trader to another as we vary the value of x in the equilibrium.
11.7 Reflections on Trade with Intermediaries
In closing, it is useful to reflect on how our analysis of trade on networks relates to the
motivating examples from the beginning of this chapter: trade in the stock market and the
trade of agricultural goods in developing countries. The network model we analyzed in this
chapter is an abstraction that captures some essential features of these real markets, and
misses other features. Our trade model reflects the constraint that trade takes place through
intermediaries and that there is di↵erential access to these intermediaries. Equilibria in our
trading networks reflect the fact that buyers and sellers in intermediated markets, such as
the stock market, face a bid-ask spread. In our model, as in actual intermediated markets,
the size of this spread, and how much profit intermediaries make, depends on the amount of
competition between intermediaries for the trade flow.
However, there are other interesting aspects of trade in intermediated markets that are
not captured by our simple network model. In particular, we do not ask where buyers’ and
sellers’ values come from; nor do we ask about how they might use information revealed by
bids, asks, or trades to update these values. We discuss the role of beliefs and information
in the stock market in Chapter 22.
11.8 Exercises
1. Consider a trading network with intermediaries in which there is one seller S, two
buyers B1, B2 and two traders (intermediaries) T 1, T 2. The seller is allowed to trade
11.8. EXERCISES 335
with either trader. The buyers can each trade with only one of the traders: buyer
B1 can only trade with trader T 1; and buyer B2 can only trade with trader T 2. The
seller has one unit of the object and values it at 0; the buyers are not endowed with
the object. Buyer B1 values a unit at 1 and buyer B2 values a unit at 2.
(a) Draw the trading network, with the traders as squares, the buyers and the seller as
circles, and with edges connecting nodes who are able to trade with each other. Label
each node as S, B1, B2, T 1 or T 2.
(b) Suppose the traders o↵er prices as follows.
• Trader T 1 o↵ers a bid price of 1 3
to S and an ask price of 1 to B1.
• Trader T 2 o↵ers a bid price of 2 3
to S and an ask price of 2 to B2.
Does this set of prices form a Nash equilibrium? If you think the answer is yes, give a
brief (1-3 sentence) explanation why. If you think the answer is no, describe a way in
which one of the traders could changes its prices so as to increase its profit.
2. Consider a trading network in which there are two buyers (B1 and B2), two sellers (S1
and S2) and one trader (T1). All of the buyers and the sellers are allowed to trade with
the trader. The sellers each have one unit of the object and value it at 0; the buyers
are not endowed with the object, but they each want one unit; buyer B1 attaches a
value of 1 to one unit, while buyer B2 attaches a value of 2 to one unit.
(a) Draw the trading network, with the trader as a square, the buyers and the sellers
as circles, and edges representing pairs of people who are able to trade with each other.
Label the nodes as T1, B1, B2, S1, and S2. Find Nash equilibrium bid and ask prices.
(You do not need to provide an explanation for your answer.)
(b) Suppose now that we add a second trader (T2) who can trade with each seller and
each buyer. In the new network is there a Nash equilibrium in which each trader’s bid
price to each seller is 1; each trader’s ask price to buyer B1 is 1; each trader’s ask price
to buyer B2 is 2; one unit of the good flows from S1 to B1 through trader T1; and,
one unit of the good flows from S2 to B2 through trader T2? Draw the new trading
network and give a brief (1-3 sentence) explanation for your answer.
3. Consider a trading network with intermediaries in which there are two sellers S1, S2,
three buyers B1, B2, B3, and two traders (intermediaries) T 1, T 2. Each seller can
trade with either trader. Buyer B1 can only trade with trader T 1. Buyer B2 can trade
with either trader. Buyer B3 can only trade with trader T 2. The sellers each have
one unit of the object and value it at 0; the buyers are not endowed with the object.
Buyer B1 values a unit at 1, buyer B2 values a unit at 2 and buyer B3 values a unit
at 3.
336 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
(a) Draw the trading network, with the traders as squares, the buyers and the seller as
circles, and with edges connecting nodes who are able to trade with each other. Label
each node as S1, S2, B1, B2, B3, T 1 or T 2.
(b) Suppose the prices and the flow of goods are as follows.
• Trader T 1 o↵ers a bid price of 1 to each seller, an ask price of 1 to B1, and an ask price of 2 to B2.
• Trader T 2 o↵ers a bid price of 1 to each seller, an ask price of 2 to B2, and an ask price of 3 to B3.
• One unit of the good flows from seller S1 to buyer B2 through trader T 1 and one unit of the good flows from seller S2 to buyer B3 through trader T 2.
(If it is useful, it is okay to write these prices and this flow of goods on the picture you
drew for part (a), provided the picture itself is still clear.) Do these prices and this
flow of goods form a Nash equilibrium? If you think the answer is yes, give a brief (1-3
sentence) explanation why. If you think the answer is no, describe a way in which one
of the traders could changes its prices so as to increase its profit.
4. Consider a trading network with intermediaries in which there is one buyer, one seller
and two traders (intermediaries). The buyer and the seller each are allowed to trade
with either trader. The seller has one unit of the object and values it at 0; the buyer
is not endowed with the object but attaches a value of 1 to one unit of it.
Draw the trading network, with traders as squares, the buyer and the seller as circles,
and edges representing pairs of people who are able to transact directly. Then describe
what the possible Nash equilibrium outcomes are, together with an explanation for
your answer.
5. Consider a trading network with intermediaries in which there is one seller S, two
buyers B1, B2 and two traders (intermediaries) T 1, T 2. The seller is allowed to trade
with either trader. The buyers can each trade with only one of the traders: buyer
B1 can only trade with trader T 1; and buyer B2 can only trade with trader T 2. The
seller has one unit of the object and values it at 0; the buyers are not endowed with
the object. Buyer B1 values a unit at 3 and buyer B2 values a unit at 1.
(a) Draw the trading network, with the traders as squares, the buyers and the seller as
circles, and with edges connecting nodes who are able to trade with each other. Label
each node as S, B1, B2, T 1 or T 2.
(b) Find Nash equilibrium bid and ask prices for this trading network. How much
profit do the traders make?
11.8. EXERCISES 337
(c) Suppose now that we add edges representing the idea that each buyer can trade
with each trader. Find a Nash equilibrium in this new trading game. What happens
to trader profits? Why?
6. Consider a trading network with intermediaries in which there are three sellers S1, S2, S3,
two buyers B1, B2, and two traders (intermediaries) T 1, T 2. Sellers S1 and S2 can
trade only with trader T 1; and, seller S3 can trade only with trader T 2. The buyers
can each trade with only one of the traders: buyer B1 can only trade with trader T 1;
and buyer B2 can only trade with trader T 2. The sellers each have one unit of the
object and value it at 0. The buyers are not endowed with the object and they each
value a unit at 1.
(a) Draw the trading network, with the traders as squares, the buyers and sellers as
circles, and with edges connecting nodes who are able to trade with each other. Label
each node as S1, S2, S3, B1, B2, T 1 or T 2.
(b) Describe what the possible Nash equilibria are, including both prices and the flow
of goods. Give an explanation for your answer.
(c) Suppose now that we add an edge between buyer B2 and trader T 1. We want to
examine whether this new edge changes the outcome in the game. To do this, take
the equilibrium from your answer to (b), keep the prices and the flow of goods on the
edges from (b) the same as before, and then suppose that the ask price on the new
B2-T 1 edge is 1, and that no good flows on this new edge. Do these prices and this
overall flow of goods still form an equilibrium? If you think that the answer is yes,
give a brief (1-3 sentence) explanation why. If you think the answer is no, describe a
way in which one of the participants in the game would deviate.
7. Consider a trading network in which there are two buyers (B1 and B2), two sellers (S1
and S2), and two traders (T1 and T2). The sellers each have one unit of the object
and value it at 0; the buyers are not endowed with the object, but they each want one
unit and attach a value of 1 to one unit. Seller S1 and Buyer B1 can trade only with
trader T1; seller S2 and Buyer B2 can each trade with either trader.
(a) Draw the trading network, with the traders as squares, the buyers and the sellers
as circles, and edges representing pairs of people who are able to trade with each other.
Label the nodes as T1, T2, B1, B2, S1, and S2.
(b) Consider the following prices and flow of goods:
• T1’s bid price to Seller S1 is 0, his bid price to Seller S2 is 1/2, his ask price to Buyer B1 is 1, and his ask price to Buyer B2 is 1/2.
338 CHAPTER 11. NETWORK MODELS OF MARKETS WITH INTERMEDIARIES
• T2’s bid price to Seller S2 is 1/2 and his ask price to Buyer B2 is 1/2.
• One unit of the good flows from Seller S1 to Buyer B1 through Trader T1; and, one unit of the good flows from Seller S2 to Buyer B2 through trader T2.
Do these prices and this flow of goods describe an equilibrium of the trading game? If
you think that the answer is No, then briefly describe how someone should deviate. If
you think that the answer is Yes, then briefly explain (1-3 sentences) why the answer
is Yes.
(c) Suppose now that we add a third trader (T3) who can trade with Seller S1 and
Buyer B1. This trader cannot trade with the other seller or buyer, and the rest of the
trading network remains unchanged. Consider the following prices and flow of goods:
• The prices on the old edges are unchanged from those in part (b).
• The prices on the new edges are: a bid of 1/2 to Seller S1 by Trader T3 and an ask of 1/2 to Buyer B1 by Trader T3.
• The flow of goods is the same as in (b).
Do these prices and this flow of goods describe an equilibrium of the trading game? If
you think that the answer is No, then briefly describe how someone should deviate. If
you think that the answer is Yes, then briefly explain (1-3 sentences) why the answer
is Yes.
Chapter 12
Bargaining and Power in Networks
In our analysis of economic transactions on networks, particularly the model in Chapter
11, we considered how a node’s position in a network a↵ects its power in the market. In
some cases, we were able to come up with precise predictions about prices and power, but
in others the analysis left open a range of possibilities. For example, in the case of perfect
competition between traders, we could conclude that the traders would make no profit, but
it was not possible to say whether the resulting situation would favor particular buyers or
sellers — di↵erent divisions of the available surplus were possible. This is an instance of
a broader phenomenon that we discussed earlier, in Chapter 6: when there are multiple
equilibria, some of which favor one player and some of which favor another, we may need to
look for additional sources of information to predict how things will turn out.
In this chapter, we formulate a perspective on power in networks that can help us further
refine our predictions for the outcomes of di↵erent participants. This perspective arises
dominantly from research in sociology, and it addresses not just economic transactions, but
also a range of social interactions more generally that are mediated by networks. We will
develop a set of formal principles that aim to capture some subtle distinctions in how a
node’s network position a↵ects its power. The goal will be to create a succinct mathematical
framework enabling predictions of which nodes have power, and how much power they have,
for arbitrary networks.
12.1 Power in Social Networks
The notion of power is a central issue in sociology, and it has been studied in many forms. Like
many related notions, a fundamental question is the extent to which power is a property of
individuals (i.e. someone is particularly powerful because of their own exceptional attributes)
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
339
340 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
and the extent to which it is a property of network structure (i.e. someone is particularly
powerful because they hold a pivotal position in the underlying social structure).
The goal here is to understand power not just as a property of agents in economic settings,
or in legal or political settings, but in social interaction more generally — in the roles people
play in groups of friends, in communities, or in organizations. A particular focus is on the
way in which power is manifested between pairs of people linked by edges in a larger social
network. Indeed, as Richard Emerson has observed in his fundamental work on this subject,
power is not so much a property of an individual as it is a property of a relation between
two individuals — it makes more sense to study the conditions under which one person has
power over another, rather than simply asserting that a particular person is “powerful” [148].
A common theme in this line of work is to view a social relation between two individuals
as producing value for both of them. We will be deliberately vague in specifying what this
value is, since it clearly depends on the type of social relation we are discussing, but the
idea adapts naturally to many contexts. In an economic setting, it could be the revenue that
two people can produce by working together; in a political setting, it could be the ability of
each person in the relationship to do useful favors for the other; in the context of friendship,
it could be the social or psychological value that the two people derive from being friends
with one another. In any of these examples, the value may be divided equally or unequally
between the two parties. For example, one of the two parties in the relationship may get
more benefit from it than the other — they may get more than half the profits in a joint
business relationship, or in the context of a friendship they may be the center of attention,
or get their way more often in the case of disagreements. The way in which the value in the
relationship is divided between the two parties can be viewed as a kind of social exchange,
and power then corresponds to the imbalance in this division — with the powerful party in
the relationship getting the majority of the value.
Now, in some cases this imbalance in a relationship may be almost entirely the result of
the personalities of the two people involved. But in other cases, it may also be a function
of the larger social network in which the two people are embedded — one person may be
more powerful in a relationship because they occupy a more dominant position in the social
network, with greater access to social opportunities outside this single relationship. In this
latter case, the imbalance in the relationship may be rooted in considerations of network
structure, and transcend the individual characteristics of the two people involved. The
ways in which social imbalances and power can be partly rooted in the structure of the
social network has motivated the growth of a research area in sociology known as network
exchange theory [417].
An Example of a Powerful Network Position. It is useful to discuss this in the
context of a simple example. Consider the group of five friends depicted in Figure 12.1,
12.1. POWER IN SOCIAL NETWORKS 341
A B C
D
E
Figure 12.1: A social network on five people, with node B occupying an intuitively powerful position.
with strong friendships indicated by the social network links. Intuitively, node B appears
to hold a powerful position in the network, and in particular to be powerful relative to two
of her three neighbors, A and C. What general principle or principles should lead us to
this conclusion? Here are several proposals, which we state informally here but make more
precise in what follows.
(i) Dependence. Recalling that social relations confer value, nodes A and C are completely
dependent on B as a source of such value; B on the other hand, has multiple sources.
(ii) Exclusion. Related to (i), B has the ability to exclude A and C. In particular, suppose
each person were to choose a “best friend” in the group; then B has the unilateral
power to choose one of A and C, excluding the other. (However, B does not have the
analogous power over D.)
(iii) Satiation. A somewhat di↵erent basis for B’s power might be implicit in the psy-
chological principle of satiation: having diminishing rewards for increased amounts of
something. Again, viewing social relations as conferring value, B will acquire value
at a greater rate than the other members of the group; having thus become satiated,
B may be interested in maintaining these social relations only if she can receive an
unequal share of their value.
(iv) Betweenness. If we believe that the value generated in social relations flows not just
across single edges but more generally along paths, then we are led to consider notions
such as betweenness. Betweenness was considered extensively in Section 3.6; for our
342 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
purposes here, it is enough to think informally of a node as having high betweenness if
it lies on paths (and particularly short paths) between many pairs of other nodes. In
our example, B has high betweenness because she is the unique access point between
multiple di↵erent pairs of nodes in the network, and this potentially confers power.
More generally, betweenness is one example of a centrality measure that tries to find
the “central” points in a network. We saw in our discussion of structural holes in
Section 3.5 that evaluating a node’s power in terms of its role as an access point
between di↵erent parts of the network makes sense in contexts where we are concerned
about issues like the flow of information. Here, however, where we are concerned about
power arising from the asymmetries in pairwise relations, we will see some concrete
cases where a simple application of ideas about centrality can in fact be misleading.
12.2 Experimental Studies of Power and Exchange
While all of these principles are presumably at work in many situations, it is di�cult to
make precise or quantify their e↵ects in most real-world settings. As a result, researchers have
turned to laboratory experiments in which they ask test subjects to take part in stylized forms
of social exchange under controlled conditions. This style of research grew into an active
experimental program carried out by a number of research groups in network exchange theory
[417]. The basic idea underlying the experiments is to take the notion of “social value” and
represent it under laboratory conditions using a concrete economic framework, of the type
that we have seen in Chapters 10 and 11. In these experiments, the value that relationships
produce is represented by an amount of money that the participants in a relationship get
to share. This does not mean, however, that an individual necessarily cares only about the
amount of money that he receives. As we shall see, it’s clear from the results that even
subjects in the experiments may also care about other aspects of the relationship, such as
the fairness of the sharing.
While the details vary across experiments, here is the set-up for a typical one. Roughly,
people are placed at the nodes of a small graph representing a social network; a fixed sum
of money is placed on each edge of a graph; and nodes joined by an edge negotiate over how
the money placed between them should be divided up. The final, crucial part of the set-up
is that each node can take part in a division with only one neighbor, and so is faced with
the choice not just of how large a share to seek, but also with whom. The experiment is run
over multiple periods to allow for repeated interaction by the participants, and we study the
divisions of money after many rounds.
Here are the mechanics in more detail.
1. A small graph (such as the one in Figure 12.1) is chosen, and a distinct volunteer
test subject is chosen to represent each node. Each person, representing a node, sits
12.2. EXPERIMENTAL STUDIES OF POWER AND EXCHANGE 343
at a computer and can exchange instant messages with the people representing the
neighboring nodes.
2. The value in each social relation is made concrete by placing a resource pool on each
edge — let’s imagine this as a fixed sum of money, say $1, which can be divided between
the two endpoints of the edge. We will refer to a division of this money between the
endpoints as an exchange. Whether this division ends up equal or unequal will be
taken as a sign of the asymmetric amounts of power in the relationship that the edge
represents.
3. Each node is given a limit on the number of neighbors with whom she can perform
an exchange. The most common variant is to impose the extreme restriction that
each node can be involved in a successful exchange with only one of her neighbors;
this is called the 1-exchange rule. Thus, for example, in Figure 12.1, node B can
ultimately make money from an exchange with only one of her three neighbors. Given
this restriction, the set of exchanges that take place in a given round of the experiment
can be viewed as a matching in the graph: a set of edges that have no endpoints in
common. However, it will not necessarily be a perfect matching, since some nodes may
not take part in any exchange. For example, in the graph in Figure 12.1, the exchanges
will definitely not form a perfect matching, since there are an odd number of nodes.
4. Here is how the money on each edge is divided. A given node takes part in simultaneous
sessions of instant messaging separately with each of her neighbors in the network. In
each, she engages in relatively free-form negotiation, proposing splits of the money on
the edge, and potentially reaching an agreement on a proposed split. These negotiations
must be concluded by a fixed time limit; and to enforce the 1-exchange rule defined
above, as soon as a node reaches an agreement with one neighbor, her negotiations
with all other neighbors are immediately terminated.
5. Finally, the experiment is run for multiple rounds. The graph and the assignment
of subjects to nodes as described in point (1) are kept fixed across rounds. In each
round, new money is placed on each edge as in point (2), each node can take part in
an exchange as in point (3), and the money is divided as in point (4). The experiment
is run for multiple rounds to allow for repeated interactions among the nodes, and we
study the exchange values that occur after many rounds.
Thus the general notion of “social value” on edges is implemented using a specific eco-
nomic metaphor: the value is represented using money, and people are negotiating explicitly
over how to divide it up. We will mainly focus on the 1-exchange rule, unless noted other-
wise. We can view the 1-exchange rule as encoding the notion of choosing “best friends”,
which we discussed earlier when we talked about exclusion. That is, the 1-exchange rule
344 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
A B
(a) 2-Node Path
A B C
(b) 3-Node Path
A B C D
(c) 4-Node Path
A B C D E
(d) 5-Node Path
Figure 12.2: Paths of lengths 2, 3, 4, and 5 form instructive examples of di↵erent phenomena in exchange networks.
models a setting in which the nodes are trying to form partnerships: each node wants to be
in a partnership; and subject to this, the node wants to get a reasonable share of the value
implicit in this partnership. Later in this chapter we will see that varying the number of
successful exchanges in which a node can participate has e↵ects on which nodes hold power,
often in interesting ways.
There are many variations in the precise way these experiments are implemented. One
particularly interesting dimension is the amount of information provided to the participants
about the exchanges made by other participants. This has ranged in experiments from a
high-information version — in which each person sees not just what is happening on their
edges, but also what is happening on every edge in the network, in real-time — to a low-
information version — in which each person is told only what is happening on the edges
she is directly involved in; for example, she may have no idea how many other potential
partners each of her neighbors has. An interesting finding from this body of work is that
the experimental results do not change much with the amount of information available [389];
this suggests a certain robustness to the results, and also allows us to draw some conclusions
about the kinds of reasoning that participants are engaging in as they take part in these
experiments.
12.3 Results of Network Exchange Experiments
Let’s start by discussing what happens when one runs this type of experiment on some
simple graphs using human test subjects. Since the results are intuitively reasonable and
fairly robust, we’ll then consider — in increasing levels of detail — what sorts of principles
can be inferred about power in these types of exchange situations.
Figure 12.2 depicts four basic networks that have been used in experiments. Notice
that these are just paths of lengths 2, 3, 4, and 5. Despite their simplicity, however, each
12.3. RESULTS OF NETWORK EXCHANGE EXPERIMENTS 345
introduces novel issues, and we will discuss them in order.
The 2-Node Path. The 2-node path is as simple as it gets: two people are given a
fixed amount of time in which to agree on a way to split $1. Yet even this simple setting
introduces a lot of conceptual complexity — a large amount of work in game theory has
been devoted precisely to the problem of reasoning about outcomes when two parties with
oppositely aligned interests sit down to negotiate. As we will discuss more fully later in this
chapter, most of the standard theoretical treatments predict a 1 2
- 1 2
split. This seems to be
a reasonable prediction, and it is indeed approximately what happens in network exchange
experiments on a 2-node graph.
The 3-Node Path. On a 3-node path with nodes labeled A, B, and C in order, node B
intuitively has power over both A and C. For example, as B negotiates with A, she has the
ability to fall back on her alternative with C, while A has no other alternatives. The same
reasoning applies to B’s negotiations with C.
Moreover, at least one of A or C must be excluded from an exchange in each round. In
experiments, one finds that subjects who are excluded tend to ask for less in the next round
in the hope of becoming included. Thus, the repeated exclusion of A and C tends to drive
down what they ask for, and one finds in practice that B indeed receives the overwhelming
majority of the money in her exchanges (roughly 5/6 in one recent set of experiments [281]).
An interesting variation on this experiment is to modify the 1-exchange rule to allow B
to take part in two exchanges in each round. One now finds that B negotiates on roughly
equal footing with both A and C. This is consistent with the notions of dependence and
exclusion discussed earlier: in order for B to achieve half the value from each exchange in
each round, she needs A and C as much as they need her.
This result for the version in which B is allowed two exchanges is less consistent with
satiation, however: if B were becoming satiated by money twice as quickly as A and C, one
could expect to start seeing an e↵ect in which A and C need to o↵er unequal splits to B in
order to keep B interested. But this is not what actually happens.
The 4-Node Path. The 4-node path is already significantly more subtle than the previous
two examples. There is an outcome in which all nodes take part in an exchange — A
exchanges with B and C exchanges with D — but there is also an outcome in which B and
C exchange with each other while excluding A and D.
Thus, B should have some amount of power over A, but it is a weaker kind of power
than in the 3-node path. In the 3-node path, B could exclude A and seek an exchange with
C, who has no other options. In the 4-node path, on the other hand, if B excludes A, then
B herself pays a price by having to seek an exchange with C, who already has an attractive
346 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
A B
C
D
Figure 12.3: An exchange network with a weak power advantage for node B.
alternative in D. In other words, B’s threat to exclude A is a costly one to actually execute.
Experiments bear out this notion of weak power: in A-B exchanges, B gets roughly between
7/12 and 2/3 of the money, but not more [281, 373].
The 5-Node Path. Paths of length 5 introduce a further subtlety: node C, which intu-
itively occupies the “central” position in the network, is in fact weak when the 1-exchange
rule is used. This is because C’s only opportunities for exchange are with B and D, and
each of these nodes have very attractive alternatives in A and E respectively. Thus, C can
be excluded from exchange almost as easily as A and E can. Put succinctly, C’s partners
for negotiation all have access to very weak nodes as alternatives, and this makes C weak as
well.
In experiments, one finds that C does slightly better than A and E do, but only slightly.
Thus, the 5-node path shows that simple centrality notions like betweenness can be mislead-
ing measures of power in some kinds of exchange networks.
Note that the weakness of C really does depend on the fact that the 1-exchange rule
is used. Suppose, for example, that we instead allowed A, C, and E to take part in one
exchange each, but allowed B and D to take part in two exchanges each. Then suddenly
each of B and D need C to make full use of their exchange opportunities, and C is now the
node with the ability to exclude some of his exchange partners.
Other Networks. Many other networks have been studied experimentally. In a number
of cases, the outcomes can be understood by combining ideas from the four basic networks
in Figure 12.2.
For example, the graph in Figure 12.1 has been extensively studied by network exchange
theorists. Since B has the ability to exclude both A and C, she tends to achieve highly favor-
able exchanges with them. Given these two alternatives, B and D almost never exchange;
as a result, D doesn’t have a realistic second option besides E, and hence D and E tend to
12.3. RESULTS OF NETWORK EXCHANGE EXPERIMENTS 347
A
B C
Figure 12.4: An exchange network in which negotiations never stabilize.
exchange on roughly equal footing. All these observations are borne out by the experimental
results.
Another interesting example that has been extensively studied is the “stem graph” shown
in Figure 12.3. Here, C and D typically exchange with each other, while B exchanges with
A, obtaining favorable terms. The position of node B in this network is conceptually similar
to the position of B in the 4-node-path: in the stem graph, B has a power advantage in her
dealings with A, but it is a weak power advantage, since to exclude A she has to exchange
with C or D, who each have exchange options in each other. Experiments have shown that
node B in the stem graph makes slightly more money than node B in the 4-node path, and
there is an intuitive, if somewhat subtle, reason for this: B’s threat over A in the 4-node
path is to negotiate with the comparably powerful node C, while B’s threat in the stem
graph is to negotiate with people who are slightly weaker.
An Unstable Network. A common theme in all the networks we have discussed thus far
is that the negotiations among participants tend to wrap up reliably by the time limit, with
fairly consistent outcomes. But there exist pathological networks in which negotiations tend
to drag out until the very end, with unpredictable individual outcomes for the participants.
To see why this might happen, we consider the simplest of these pathological examples,
depicted in Figure 12.4: three nodes each connected to each other. It is not hard to see what
happens when an exchange experiment is run on the triangle. Only one exchange can be
completed among the three nodes; so as time is running out, two of the nodes — say, A and
B — will be wrapping up negotiations, while the third node (C in this case) is completely
left out and stands to get nothing. This means that C will be willing to break into the A-B
negotiations up to the very end, o↵ering an exchange to either of these nodes in which they
get almost everything as long as C can get a small amount. If this happens — say that C
breaks up the A-B negotiations by o↵ering highly favorable terms to A — then there will be
a di↵erent node left out (B in this case), who will in turn be willing to o↵er highly favorable
terms to get back in on an exchange.
This process, by itself, would cycle indefinitely — with some node always left out and
348 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
A
B
C
D
sells to A for x: payoff of x
receives good of value 1: payoff of 1-x
Figure 12.5: An exchange network built from the 4-node path can also be viewed as a buyer-seller network with 2 sellers and 2 buyers.
trying anything to get back in — and it is only brought to a halt by the arbitrary arrival of
the time limit. Under these conditions, you have nodes “playing for the last shot,” with the
outcome for any one node correspondingly hard to predict.
Again, this is not an issue that comes up in any of the earlier examples we discussed;
what’s di↵erent in the triangle network is that no matter what tentative exchanges are being
planned, excluded nodes have a natural way to “break in” to the negotiations. This prevents
the outcome from ever stabilizing across the network. It is also worth noting that the mere
presence of a triangle in a larger network does not necessarily cause problems: for example,
the stem graph in Figure 12.3 contains a triangle, but the exchange possibilities provided by
the additional node A allows for robust outcomes in which A exchanges with B and C and D
exchange with each other. The problem with a “free-standing” triangle as in Figure 12.4 is
fundamentally di↵erent: here, there is always a node who is left out, and yet has the ability
to do something about it.
12.4 A Connection to Buyer-Seller Networks
When we discussed matching markets in Chapter 10, we considered bipartite graphs con-
sisting of buyers and sellers. Here, on the other hand, we have been talking about graphs in
which the participants all play the same role (there is no division into buyers and sellers);
and rather than conducting trade, they negotiate over the division of money on the edges.
Despite these surface-level di↵erences, there is a close connection between the two set-
tings. To see this connection, let’s consider the 4-node path as an example. Suppose we
12.5. MODELING TWO-PERSON INTERACTION: THE NASH BARGAINING SOLUTION349
declare nodes A and C to be buyers, and nodes B and D to be sellers. We give one unit of a
good to each of B and D, and one unit of money to each of A and C; we assume that A and
C each have a valuation of 1 for one copy of the good, and that B and D have no valuation
for the good. We now consider the prices at which sales of the good will take place.
It takes a bit of thought, but this is completely equivalent to the exchange network
experiment on the length-4 path, as indicated in Figure 12.5. For example, if B sells to A
for a price of x, then B gets a payo↵ of x (from the x units of money), and A gets a payo↵
of 1 � x (from the 1 unit of value for the good, minus the x units of money he has to pay). Thus, the negotiation between A and B over a price x in the buyer-seller network is just like
the negotiation between A and B over the division of $1 into x and 1 � x in an exchange network. Furthermore, the 1-exchange rule corresponds to the requirement that each seller
can only sell a single unit of the good, and each buyer only wants one unit of the good.
One can perform a comparable translation for all the graphs in Figures 12.1 and 12.2.
However, it is important to note two caveats about this general observation on the relation-
ship between exchange networks and buyer-seller networks. First, the translation is only
possible for graphs that are bipartite (as all the graphs in Figures 12.1 and 12.2 are), even if
they are not drawn with the nodes in two parallel columns. The triangle graph in Figure 12.4
is not bipartite, and although we can still talk about the exchange network experiment, it is
not possible to label the nodes as buyers and sellers in such a way that all edges join a seller
to a buyer. We can make one node a seller, and another node a buyer, but then we have
no options for what to label the third node. Similarly, the stem graph in Figure 12.3 is not
bipartite, and so the analogy to buyer-seller networks cannot be applied there either.
A second caveat is that, for bipartite graphs, the two formulations are equivalent only
at a mathematical level. It is not at all clear that human subjects placed in a buyer-seller
experiment would behave in the same way as human subjects in a network exchange exper-
iment, even on the very same graph. Indeed, there is recent empirical evidence suggesting
that one may in fact see di↵erent outcomes from these two ways of describing the same
process to test subjects [397].
12.5 Modeling Two-Person Interaction: The Nash Bar- gaining Solution
Thus far, we have seen a range of networks on which exchange experiments have been
carried out, and we have developed some of the informal reasons why the outcomes turn
out the way they do. We’d now like to develop a more mathematical framework allowing
us to express predictions about what will happen when network exchange takes place in an
arbitrary network. Among the phenomena we’d like to be able to explain are the distinctions
between equal and asymmetric division of value across an edge; between strong power (when
350 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
A B
outside option
x
outside option
y
Figure 12.6: Two nodes bargaining with outside options.
imbalances go to extremes) and weak power (as in the four-node path, when the imbalance
remains moderate); and between networks where outcomes stabilize and networks (like the
triangle in Figure 12.4) where they don’t.
In fact, we will be able to achieve this goal to a surprising extent, capturing each of
these phenomena in a model based on simple principles. We begin the formulation of this
model here and in the next section by developing two important ingredients, each based on a
di↵erent type of two-person interaction. The first ingredient — the Nash bargaining solution
— has a more mathematical flavor, while the second — the ultimatum game — is based
primarily on human-subject experiments.
The Nash Bargaining Solution. Let’s start with a simple formulation of two-person
bargaining. Suppose, as in network exchange on a 2-node path, that two people A and B
are negotiating over how to split $1 between them. Now, however, we extend the story to
assume that A also has an outside option of x, and B has an outside option of y. By this we
mean that if A doesn’t like his share of the $1 arising from the negotiations with B, he can
leave and take x instead. This will presumably happen, for instance, if A is going to get less
than x from the negotiations. Similarly, B has the option of abandoning the negotiation at
any time and taking her outside option of y. Notice that if x + y > 1, then no agreement
between A and B is possible, since they can’t divide a dollar so that one gets at least x and
the other gets at least y. Consequently, we will assume x + y 1 when we consider this type of situation.
Given these conditions, A requires at least x from the negotiations over splitting the
dollar, and B requires at least y. Consequently, the negotiation is really over how to split
the surplus s = 1 � x � y (which is at least 0, given our assumption that x + y 1 in the previous paragraph). A natural prediction is that if the two people A and B have equal
bargaining power, then they will agree on the division that splits this surplus evenly: A gets
x + 1 2
s, and B gets y + 1 2
s. This is the prediction of a number of general theories including
12.5. MODELING TWO-PERSON INTERACTION: THE NASH BARGAINING SOLUTION351
the Nash bargaining solution [312], and we will use this as our term for the outcome:
Nash Bargaining Solution: When A and B negotiate over splitting a dollar, with
an outside option of x for A and an outside option of y for B (and x + y 1), the Nash bargaining outcome is
• x + 1
2 s =
x + 1 � y 2
to A, and
• y + 1
2 s =
y + 1 � x 2
to B.
In the literature on network exchange theory, this division is sometimes referred as the
equidependent outcome [120], since each person is depending equally on the other for conces-
sions to make the negotiation work. At a high level, the formulation of the Nash bargaining
solution emphasizes an important point about the process of negotiation in general: trying
to ensure that you have as strong an outside option as possible, before the negotiations even
begin, can be very important for achieving a favorable outcome. For most of this chapter,
it is enough to take the Nash bargaining solution as a self-contained principle, supported by
the results of experiments. In the final section of the chapter, however, we ask whether it
can be derived from more fundamental models of behavior. We show there that in fact it
can — it arises naturally as an equilibrium when we formulate the process of bargaining as
a game.
Experiments on Status E↵ects. When we think about bargaining in the context of
experiments with human subjects, we of course need to consider the assumption that the
two people have equal bargaining power. While in our models we will make use of this
assumption, it is interesting to think about how external information could a↵ect relative
bargaining power in settings such as these.
The e↵ects of perceived social status on bargaining power have been explored experimen-
tally by sociologists. In these experiments, two people are asked to divide money in situations
where they are led to believe that one is “higher-status” and the other is “lower-status.” For
example, in a recent set of these experiments, pairs of people A and B, each female college
sophomores, negotiated in the presence of outside options using instant messaging. How-
ever, each was given false information about the other: A was told that B was a high-school
student with low grades, while B was told that A was a graduate student with very high
grades [390]. Thus, A believed B to be low-status, while B believed A to be high-status.
The results of these experiments illustrate interesting ways in which beliefs about di↵er-
ential status can lead to deviations from theoretical predictions in bargaining. First, each
subject had to communicate information about their own outside options to their partners
as part of the negotiation (this information was not provided by the experimenters). It
was found that people tended to inflate the size of their outside option when they believed
352 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
their negotiating partner was lower-status; and they tended to reduce the size of their out-
side option when they believed their negotiating partner was higher-status. Compounding
this e↵ect, people tended to partially discount a negotiating partner’s claims about outside
options when they believed this partner to be lower-status. (In other words, lower-status
people tended to underreport the value of their outside options, and even these underre-
ported values were discounted by their partners.) Overall, for these and other reasons, the
subject who was believed to be higher-status by her partner tended to achieve significantly
better bargaining outcomes than the theoretical predictions.
Naturally, these status e↵ects are interesting additional factors to incorporate into models
of exchange. For developing the most basic family of models, however, we will focus on the
case of interaction in the absence of additional status e↵ects, using the Nash bargaining
outcome as a building block.
12.6 Modeling Two-Person Interaction: The Ultima- tum Game
The Nash bargaining outcome provides us with a way of reasoning about two people whose
power di↵erences arise through di↵erences in their outside options. In principle, this applies
even to situations with extreme power imbalances. For example, in network exchange on a
3-node path, we saw that the center node holds all the power, since it can exclude either
of the two other nodes. But in exchange experiments on this network, the center is not
generally able to drive its partners’ shares all the way down to 0; rather one sees splits like 5
6
- 1 6
.
What causes the negotiations to “pull back” from a completely unbalanced outcome?
This is in fact a recurring e↵ect in exchange experiments: human subjects placed in bargain-
ing situations with strong power imbalances will systematically deviate from the extreme
predictions of simple theoretical models. One of the most basic experimental frameworks for
exploring this e↵ect is called the Ultimatum Game [203, 386], and it works as follows.
Like the bargaining framework discussed in the previous section, the Ultimatum Game
also involves two people dividing a dollar, but following a very di↵erent procedure than what
we saw before:
(i) Person A is given a dollar and told to propose a division of it to person B. That is, A
should propose how much he keeps for himself, and how much he gives to B.
(ii) Person B is then given the option of approving or rejecting the proposed division.
(iii) If B approves, each person keeps the proposed amount. If B rejects, then each person
gets nothing.
12.6. MODELING TWO-PERSON INTERACTION: THE ULTIMATUM GAME 353
Moreover, let’s assume that A and B are communicating by instant messaging from di↵erent
rooms; they are told at the outset that they have never met each other before, and quite
possibly will never meet again. For all intents and purposes, this is a one-shot interaction.
Suppose first that both people are strictly interested in maximizing the amount of money
they walk away with; how should they behave? This is not hard to work out. First, let’s
consider how B should behave. If A proposes a division that gives any positive amount to B,
then B’s choice is between getting this positive amount of money (by accepting) and getting
nothing (by rejecting). Hence, B should accept any positive o↵er.
Given that this is how B is going to behave, how should A behave? Since B will accept
any positive o↵er, A should pick the division that gives B something and otherwise maximizes
A’s own earnings. Thus, A should propose $.99 for himself and $.01 for B, knowing that B
will accept this. A could alternately propose $1.00 for himself and $.00 for B, gambling that
B — who would then be indi↵erent between accepting and rejecting — would still accept.
But for this discussion we’ll stick with the division that gives B a penny.
This, then, is a prediction of how purely money-maximizing individuals would behave in
a situation of extreme power imbalance: the one holding all the power (A) will o↵er as little
as possible, and the one with essentially no power will accept anything o↵ered. Intuition —
and, as we will see next, experimental results — suggests that this is not how human beings
will typically behave.
The Results of Experiments on the Ultimatum Game. In 1982, Güth, Schmit-
tberger, and Schwarze [203] performed a series of influential experiments in which they
studied how people would actually play the Ultimatum Game. They found that people play-
ing the role of A tended to o↵er fairly balanced divisions of the money — on average, about
a third of the total, with a significant number of people playing A in fact o↵ering an even
split. Moreover, they found that very unbalanced o↵ers were often rejected by the person
playing the role of B.
A large amount of follow-up work has shown these finding to be highly robust [386], even
when relatively large amounts of money are at stake. The experiment has also been carried
out in a number of di↵erent countries, and there are interesting cultural variations, although
again the tendency toward relatively balanced divisions is consistent [93].
Relatively balanced o↵ers in the Ultimatum Game, and rejections of positive amounts
of money — can these observations be reconciled with the game-theoretic framework we’ve
used in previous chapters? There are in fact a number of ways to do so. Perhaps the most
natural is to keep in mind one of the basic principles we discussed when defining payo↵s in
game-theoretic situations: a player’s payo↵ should reflect his or her complete evaluation of a
given outcome. So when a player B evaluates an outcome in which she walks away with only
10% of the total, one interpretation is that there is a significant negative emotional payo↵ to
354 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
A B C
1/2 1/2 0
(a) Not a stable outcome
A B C
0 1 0
(b) A stable outcome
A B C D
1/2 1/2 1/4 3/4
(c) Not a stable outcome
A B C D
1/2 1/2 1/2 1/2
(d) A stable outcome
A B C D
1/3 2/3 2/3 1/3
(e) A stable outcome
Figure 12.7: Some examples of stable and unstable outcomes of network exchange on the 3-node path and the 4-node path. The darkened edges constitute matchings showing who exchanges with whom, and the numbers above the nodes represent the values.
being treated unfairly, and hence when we consider B’s complete evaluation of the options,
B finds a a greater overall benefit to rejecting the low o↵er and feeling good about it than
accepting the low o↵er and feeling cheated. Moreover, since people playing the role of A
understand that this is the likely evaluation that their partner B will bring to the situation,
they tend to o↵er relatively balanced divisions to avoid rejection, because rejection results
in A getting nothing as well.
It remains true that if you find yourself playing the role of A in an instance of the
Ultimatum Game where player B is a money-maximizing robot, you should o↵er as little as
possible. What the line of experiments on this topic have shown is simply that real people’s
payo↵s are not well modeled by strict money-maximization. Even a robot will reject low
o↵ers if you instruct it to care about feeling cheated.
All these observations are useful when we think about network exchange experiments
where there are strong power imbalances between adjacent nodes — in these situations, we
should expect to see wide asymmetries in the division of resources, but not necessarily as
wide as the basic models might predict.
12.7. MODELING NETWORK EXCHANGE: STABLE OUTCOMES 355
12.7 Modeling Network Exchange: Stable Outcomes
Having now built up some principles — both theoretical and empirical — that govern two-
person interactions, we apply these to build a model that can approximately predict the
outcomes of network exchange on arbitrary graphs.
Outcomes. Let’s begin by making precise what we mean by an outcome. We say that an
outcome of network exchange on a given graph consists of two things:
(i) A matching on the set of nodes, specifying who exchanges with whom. Recall that a
matching, as discussed in Chapter 10, is a set of edges so that each node is the endpoint
of at most one of them — this corresponds to the 1-exchange rule, in which each node
can complete at most one exchange, and some nodes may be left out.
(ii) A number associated with each node, called its value, indicating how much this node
gets from its exchange. If two nodes are matched in the outcome, then the sum of
their values should equal 1, indicating that they split the one unit of money in some
fashion between them. If a node is not part of any matching in the outcome, then its
value should equal 0, indicating that it does not take part in an exchange.
Figure 12.7 depicts examples of outcomes on the 3-node and 4-node paths.
Stable Outcomes. For any network, there is almost always a wide range of possible
outcomes. Our goal is to identify the outcome or outcomes that we should expect in a
network when an exchange experiment is actually performed.
A basic property we’d expect an outcome to have is stability: no node X can propose an
o↵er to some other node Y that makes both X and Y better o↵ — thus “stealing” node Y
away from an existing agreement. For example, consider Figure 12.7(a). In addition to C
feeling left out by the outcome, there is something that C can do to improve the situation:
for example, C can o↵er 2/3 to B (keeping 1/3 for himself), if B will break her agreement
with A and exchange with C instead. This o↵er from C to B would make B better o↵ (as
she would get 2/3 instead of her current 1/2) and it would also make C better o↵ (as he
would get 1/3 instead of 0). There is nothing to prevent this from happening, so the current
situation is unstable. (Although we’ve described this trade as having been initiated by C, it
could equally well be initiated by B, in an attempt to improve on her current value of 1/2.)
Compare this to the situation in Figure 12.7(b). Here too C is doing badly, but now
there is nothing he can do to remedy the situation. B is already getting 1 — the most she
possibly can — and so there is nothing that C can o↵er to B to break the current A-B
exchange. The situation, even though it is bad for some parties, is stable.
356 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
We can make this idea precise for any network, defining an instability in an outcome to
be a situation where two nodes have both the opportunity and the incentive to disrupt the
existing pattern of exchanges. Specifically, we have the following definition.
Instability: Given an outcome consisting of a matching and values for the nodes,
an instability in this outcome is an edge not in the matching, joining two nodes
X and Y , such that the sum of X’s value and Y ’s value is less than 1.
Notice how this captures the kind of situation we’re been discussing: in an instability, the two
nodes X and Y have the opportunity to disrupt the status quo (because they’re connected
by an edge, and hence allowed to exchange), and they also have the incentive — since the
sum of their values is less than 1, they can find a way to divide the dollar between them and
each end up better than they currently are doing.
In the example we discussed from Figure 12.7(a), the instability is the edge connecting
B and C — the sum of the values is 1/2, and so both B and C can end up better o↵ by
exchanging with each other. On the other hand, Figure 12.7(b) has no instabilities; there are
no inherent stresses that could disrupt the status quo in Figure 12.7(b). Thus, we introduce
a further definition, which we call stability.
Stability: An outcome of network exchange is stable if and only if it contains no
instabilities.
Given the inherent fragility of outcomes with instabilities, we expect to see stable outcomes
in practice, and for networks that have stable outcomes, we in fact typically do see results
that are close to stable outcomes.
Figures 12.7(c)–12.7(e) provide some further opportunities to test these definitions on
examples. There is an instability in Figure 12.7(c), since nodes B and C are connected
by an edge and collectively making less than the one unit of money they could split by
exchanging with each other. On the other hand, the outcomes in Figures 12.7(d) and 12.7(e)
are both stable, since on the one edge not in the matching, the two nodes are collectively
making at least one unit of money from the current situation.
Applications of Stable Outcomes. In addition to being intuitively natural, the notion
of a stable outcome helps to explain some of the general principles observed in network
exchange experiments.
First, stable outcomes are good at approximately capturing what’s going on in situations
with extreme power imbalances. If we think a bit about Figures 12.7(a) and 12.7(b), we
can convince ourselves that the only stable outcomes on the 3-node path are those in which
B exchanges with one of A or C and gets the full one unit of value for herself. Indeed, if
B got anything less than one unit, the unmatched edge would form an instability. Hence,
12.7. MODELING NETWORK EXCHANGE: STABLE OUTCOMES 357
stability shows why B occupies the dominant position in this network. In fact, with a bit of
analysis, we can see that on the 5-node path from Figure 12.2(d), the only stable outcomes
give values of 1 to the “o↵-center” nodes B and D. So stable outcomes are also able to pick
up on the subtlety that the central node C on the 5-node path is in fact very weak.
Now we know that in fact human subjects on the 3-node path or 5-node paths will not
push things all the way to 0-1 outcomes; rather, the powerful nodes tend to get amounts
more like 5 6
. But our discussion surrounding the Ultimatum Game shows that this is, in a
sense, the most extreme kind of outcome that we’ll see from real people. Since the notion
of stability isn’t designed to avoid extremes, we’ll view this mismatch between theory and
experiment as something that is relatively easy to explain and account for: when we see
strong-power outcomes in practice like 1 6
- 5 6
, we can think of this as being as close to 0-1 as
human players will get.1
Our current framework is also good at identifying situations where there is no stable
outcome. In particular, recall the pathological behavior of network exchange on the triangle
network in Figure 12.4, which never settles down to a predictable result. We can now explain
what’s going on by observing that there is no stable outcome for the triangle network. To
see why, notice first that in any outcome, some node will be unmatched and get a value of
0. Let’s suppose this is node C – due to the symmetry of the situation, it doesn’t matter
which node we choose for this argument. This unmatched node C has edges to both other
nodes, and no matter how these other two nodes divide the money, at least one of them (say,
B) will get less than 1. But now the edge connecting B and C is an instability, since they
collectively are getting less than 1 and yet have the ability to perform an exchange.
The fact that there is no stable outcome provides us with a way to think about the
dynamics of negotiation on the triangle — no matter what tentative agreement is reached,
the system necessarily contains internal stress that will disrupt it.
Limitations of Stable Outcomes. The explanatory power of stability also has significant
limitations, however. One source of limitations lies in the fact that it allows outcomes to go
to extremes that people will not actually follow in real life. But as we’ve already observed,
this di�culty is something where the theory is approximately accurate, and the discrepancies
can be recognized and dealt with relatively easily.
A more fundamental di�culty with the notion of a stable outcome is that it is too
ambiguous in situations where there is a weak power imbalance between individuals. For
example, let’s go back to Figures 12.7(d) and 12.7(e). Both of these represent stable outcomes
on the 4-node path, but the first of these gives equal values to all the nodes, despite the
power advantages of the middle nodes. In fact, there is a large range of possible stable
1In fact, one can extend the theory of stability fairly easily to be able to handle this e↵ect explicitly. For the discussion here, however, we’ll stick with the simpler version that allows things to go to extremes.
358 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
A B C D
1/2 1/2 1/2 1/2
outside option 0
outside option 1/2
outside option 1/2
outside option 0
(a) Not a balanced outcome
A B C D
1/3 2/3 2/3 1/3
outside option 0
outside option 1/3
outside option 1/3
outside option 0
(b) A balanced outcome
A B C D
1/4 3/4 3/4 1/4
outside option 0
outside option 1/4
outside option 1/4
outside option 0
(c) Not a balanced outcome
Figure 12.8: The di↵erence between balanced and unbalanced outcomes.
outcomes on the 4-node path: when the matching consists of the two outer edges, then any
way of dividing the value on these edges so that B and C cumulatively get at least 1 will be
stable.
To summarize, while stability is an important concept for reasoning about outcomes of
exchange, it is too weak in networks that exhibit subtle power di↵erences. On these networks,
it is not restrictive enough, since it permits too many outcomes that don’t actually occur.
Is there a way to strengthen the notion of stability so as to focus on the outcomes that are
most typical in real life? There is, and this will be the focus of the next section.
12.8. MODELING NETWORK EXCHANGE: BALANCED OUTCOMES 359
12.8 Modeling Network Exchange: Balanced Outcomes
In cases where there are many possible stable outcomes for a given network, we will show in
this section how to select a particularly natural set of outcomes that we call balanced.
The idea behind balanced outcomes is perhaps best illustrated by considering the four-
node path. In particular, Figure 12.7(d) is a stable outcome, but it doesn’t correspond to
what one sees in real experiments. Moreover, there is something clearly “not right” about
it: nodes B and C are being severely out-negotiated. Despite the fact that each of them has
an alternate option, they are splitting the money evenly with A and D respectively, even
though A and D have nowhere else to go.
We can think about this issue by noticing that network exchange can be viewed as a type
of bargaining in which the “outside options” — in the sense of the Nash bargaining solution
from Section 12.5 — are provided by the other nodes in the network. Figure 12.8(a) depicts
this for the all- 1 2
outcome we’ve been considering. Given the values for each node, we observe
that B in e↵ect has an outside option of 1 2
, since she can o↵er 1 2
to C (or an amount very
slightly higher than 1 2
) and steal C away from his current agreement with D. For the same
reason, C also has an outside option of 1 2
, by considering what he would need to o↵er B to
steal her away from her current agreement with A. On the other hand, the network with its
current node values provides A and D with outside options of 0 — they have no alternatives
to their current agreements.
Defining Balanced Outcomes. The discussion above suggests a useful way to view the
problem with the all- 1 2
outcome: the exchanges that are happening do not represent the Nash
bargaining outcomes with respect to the nodes’ outside options. And it is in this context
that the outcome in Figure 12.8(b) starts to look particularly natural. With these values, B
has an outside option of 1 3
, since to steal C away from his current partnership B would need
to o↵er C a value of 2 3
, keeping 1 3
for herself. Thus, B’s 2 3
- 1 3
split with A represents the Nash
bargaining solution for B and A with outside options provided by the values in the rest of
the network. The same reasoning holds for the C-D exchange. Hence, this set of values on
the 4-node path has an elegant self-supporting property: each exchange represents the Nash
bargaining outcome, given the exchanges and values elsewhere in the network.
We can define this notion of balance in general for any network, as follows [120, 349].
First, for any outcome in a network, we can identify each node’s best outside option just as
we did in the 4-node path: it is the most money the node can make by stealing a neighbor
away from his or her current partnership. Now we define a balanced outcome as follows.
Balanced Outcome: An outcome (consisting of a matching and node values) is
balanced if, for each edge in the matching, the split of the money represents
the Nash bargaining outcome for the two nodes involved, given the best outside
360 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
A B
C
D
1/4 3/4
1/2
1/2
outside option
0
outside option
1/2 outside option
1/4
outside option
1/4
Figure 12.9: A balanced outcome on the stem graph.
options for each node provided by the values in the rest of the network.
Notice how this type of outcome really is “balanced” between di↵erent extremes. On
the one hand, it prevents B and C from getting too little, as in Figure 12.8(a). But it also
prevents B and C from getting too much — for example, the outcome in Figure 12.8(c) is
not balanced either, because B and C are each getting more than their share under the Nash
bargaining outcome.
Notice also that all of the outcomes in Figure 12.8 are stable. So in this example it’s
reasonable to think of balance as a refinement of stability. In fact, for any network, every
balanced outcome is stable. In a balanced outcome each node in the matching gets at least
its best outside option, which is the most the node could get on any unused edge. So no two
nodes have an incentive to disrupt a balanced outcome by using a currently unused edge,
and therefore the outcome is stable. But balance is more restrictive than stability, in that
there can be many stable outcomes that are not balanced.
Applications and Interpretations of Balanced Outcomes. In addition to its elegant
definition, the balanced outcome corresponds approximately to the results of experiments
with human subjects. We have seen this already for the 4-node path. The results for the
stem graph provide another basic example.
Figure 12.9 shows the unique balanced outcome for the stem graph: C and D exchange on
even terms, providing B with an outside option of 1 2
, and hence leading to a Nash bargaining
outcome of 1 4
- 3 4
between A and B. The balanced outcome thus captures not just weak power
advantages, but also subtle di↵erences in these advantages across networks — in this case,
the idea that B’s advantage in the stem graph is slightly greater than in the 4-node path.
12.9. ADVANCED MATERIAL: A GAME-THEORETIC APPROACH TO BARGAINING361
Given the delicate self-reference in the definition of a balanced outcome — its values are
defined by determining outside options in terms of the values themselves — it is natural to
ask whether balanced outcomes even exist for all networks. Of course, since any balanced
outcome is stable, a balanced outcome can only exist when a stable outcome exists, and
we know from the previous section that for certain graphs (such as the triangle) there is
no stable outcome. But it can be shown that in any network with a stable outcome, there
is also a balanced outcome, and there are also methods to compute the set of all balanced
outcomes for a given network [31, 242, 254, 349, 378].
In fact, the concepts of stability and balance from this section and the previous one can
be framed in terms of ideas from an area known as cooperative game theory, which studies
how a collection of players will divide up the value arising from a collective activity (such as
network exchange in the present case). In this framework, stability can be formulated using
a central notion in cooperative game theory known as the core solution, and balance can be
formulated as a combination of the core solution and a second notion known as the kernel
solution [234, 289, 349].
Finally, we note that balance is one of several definitions proposed for refining stable
outcomes to produce reasonable alignment with experiments. There are competing theories,
including one called equiresistance, that achieve similar results [373]. It remains an open
research question to understand how closely the predictions of all these theories match up
with human-subject experiments when we move to significantly larger and more complex
networks.
12.9 Advanced Material: A Game-Theoretic Approach to Bargaining
In Section 12.5 we considered a basic setting in which two people, each with outside options,
bargain over a shared resource. We argued that the Nash bargaining solution provides a
natural prediction for how the surplus available in the bargaining will be divided. When
John Nash originally formulated this notion, he motivated it by first writing down a set of
axioms he believed the outcome of any bargaining solution should satisfy, and then showing
that these axioms characterize his bargaining solution [312]. But one can also ask whether
the same solution can be motivated through a model that takes into account the strategic
behavior of the people performing the bargaining — that is, whether we can formulate a game
that captures the essentials of bargaining as an activity, and in which the Nash bargaining
outcome emerges as an equilibrium. This was done in the 1980s by Binmore, Rubinstein,
and Wolinsky [60], using a game-theoretic formulation of bargaining due to Rubinstein [356].
Here we describe how this strategic approach to the Nash bargaining solution works; it
is based on the notion of a dynamic game as formulated in Section 6.10. In our formulation
362 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
of bargaining, we will use the basic set-up from Section 12.5. There are two individuals A
and B who negotiate over how to split $1 between them. Person A has an outside option of
x and person B has an outside option of y. We assume that x + y < 1 as otherwise there is
no way to split the $1 that would be beneficial to both people.
Formulating Bargaining as a Dynamic Game. The first step is to formulate bar-
gaining as a game. To do this, we imagine a stylized picture for how two people A and
B might negotiate over the division of a dollar, as suggested by the following hypothetical
conversation (in which A presumably has the stronger outside option):
A: I’ll give you 30% of the dollar.
B: No, I want 40%.
A: How about 34%?
B: I’ll take 36%.
A: Agreed.
To capture the intuition suggested by this conversation, we define a dynamic bargaining
game that proceeds over a sequence of periods that can continue indefinitely.
• In the first period, A proposes a split of the dollar in which he gets a 1
and B gets b 1
.
(The subscript “1” indicates that this is the split proposed in the first period.) We will
denote this split by (a 1
, b 1
).
• B can then either accept A’s proposal or reject it. If B accepts, the game ends and each player gets their respective portion. Otherwise, the game continues to period 2.
• In the second period, B proposes a split (a 2
, b 2
) in which she gets b 2
and A gets a 2
.
Now A can either accept or reject; again, the game ends if A accepts, and it continues
if A rejects.
• The periods continue indefinitely in this fashion, with A proposing a split in each odd- numbered period, and B proposing a split in each even-number period. Any accepted
o↵er ends the game immediately.
The conversation between A and B above fits the structure of this game, if we rewrite it
using our notation as follows.
(Period 1) A: (.70, 30)? B: Reject.
(Period 2) B: (.60, 40)? A: Reject.
12.9. ADVANCED MATERIAL: A GAME-THEORETIC APPROACH TO BARGAINING363
(Period 3) A: (.66, 34)? B: Reject.
(Period 4) B: (.64, 36)? A: Accept.
There is one more important part to the game, which models the idea that the two parties
experience some pressure to actually reach a deal. At the end of each round, and before the
next round begins, there is a fixed probability p > 0 that negotiations abruptly break down.
In the event of such a breakdown, there will be no further periods, and the players will be
forced to take their respective outside options.
This describes the full game: it proceeds through a sequence of alternating o↵ers, and it
continues until someone accepts an o↵er or negotiations break down. At its conclusion, each
player receives a payo↵ — either the accepted split of the dollar, or the outside options in
the event of a breakdown.
The possibility of a breakdown in negotiations means that if B decides to reject the
proposed split in the first period, for example, she is risking the possibility that there won’t
be a second round, and she will have to fall back to her outside option. Each player has
to take this risk into account each time they reject an o↵er. This breakdown probability
is necessary for the results we derive on bargaining, and we can view it as reflecting the
idea that each player believes there is some chance the game will end before they reach an
agreement. Perhaps the other player will give up on the negotiation or will abruptly be
drawn away by some unexpected better opportunity that comes along, or perhaps there is
simply some outside reason that the game itself suddenly ends.
Analyzing the Game: An Overview. The game we have just defined is a dynamic
game in the sense of Section 6.10, but with two di↵erences worth noting. The first di↵erence
is that each time a player makes a proposal, the set of available strategies is infinite rather
than finite: he or she can propose to keep a portion of the dollar equal to any real number
between 0 and 1. For our purposes, this di↵erence ends up being relatively minor, and it
doesn’t cause any trouble in the analysis. The second di↵erence is more significant. In
Section 6.10, we considered finite-horizon games that ran for at most a finite number of
periods, whereas here we have an infinite-horizon game in which the sequence of periods
can in principle go on forever. This poses a problem for the style of analysis we used in
Section 6.10, where we reasoned from the final period of the game (with just a single move
left to make) backward to the beginning. Here there is no final period, so we will need a
di↵erent way to analyze the game.
Despite this, the type of reasoning that we employed in Section 6.10 will help us to solve
this game. The equilibrium we will look for is a subgame perfect equilibrium — a notion that
we also saw in Chapter 11 associated with the trading game in which traders post prices,
and buyers and sellers subsequently react. A subgame perfect equilibrium is simply a Nash
364 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
equilibrium with the property that the strategies, beginning from any intermediate point in
the game, still form a Nash equilibrium for the play proceeding from that point onward.
Our main result is twofold. First, the bargaining game has a subgame perfect equilibrium
with a simple structure in which A’s initial o↵er is accepted. Second, for this equilibrium,
we can work out the values in the initial split (a 1
, b 1
) that is proposed and accepted. These
quantities a 1
and b 1
depend on the underlying value of the breakdown probability p, and as
p goes to 0, the split (a 1
, b 1
) converges to the Nash bargaining outcome. So the point is that
when two strategic bargainers interact through negotiations that are unlikely to break down
quickly, the Nash bargaining solution is a good approximate prediction for the outcome.
It is also worth considering how our formulation of bargaining here relates to the ex-
perimental work in network exchange theory from earlier in this chapter. There are a few
di↵erences. First, of course, the experiments discussed earlier involve multiple interlinked
negotiations that take place concurrently — one negotiation for each edge in a network. It
is an interesting but largely open question to adapt the kind of bargaining game formulated
here to a setting where negotiations take place simultaneously across all the edges of a net-
work. But beyond this consideration, there are still di↵erences between our game-theoretic
model here and the exchange-theory experiments even when we look just at a single edge
of the network. First, the experiments generally allowed for free-form discussion between
the two endpoints of an edge, whereas we have a specified a fixed format in which the two
bargainers take turns proposing splits, beginning with A. The fact that A gets to move
first in our game gives him some advantage, but in the case we are mainly interested in
for our results — as the breakdown probability p becomes small — this advantage becomes
negligible. Second, the experiments generally imposed a fixed time limit to ensure that ne-
gotiations would eventually end, while we are using a breakdown probability that applies to
each round. It is not clear exactly how these two sources of time pressure in a negotiation
relate to each other, since even with a fixed time limit, the fact that nodes may have multiple
network neighbors in the exchange-theory experiments makes it hard to reason about how
long the negotiation on any particular edge is likely to last.
A First Step: Analyzing a Two-Period Version of Bargaining. Because of the
complexity introduced by the infinite nature of the game, it is useful to get some initial
insight by first analyzing a finite version of it.
In particular, let’s take our earlier version of the game and assume that it ends for sure
at the end of the second period. (As before, it may also end with probability p at the end
of the first period.) Since this is now a game with a finite number of periods, we can solve
it backward through time as follows.
• First, A will accept B’s proposal (a 2
, b 2
) in period two provided that a 2
is at least
as large as A’s outside option x. (Since negotiations are guaranteed to end after this
12.9. ADVANCED MATERIAL: A GAME-THEORETIC APPROACH TO BARGAINING365
round, A is simply choosing at this point between a 2
and x.)
• Given this, there is no reason for B to o↵er A more than x, so B’s period-two proposal will be (x, 1 � x). Since we have assumed x + y < 1, we have 1 � x > y, and so B prefers this split to the outcome in which negotiations end and B gets only y.2
• Now, when B considers whether to accept or reject A’s o↵er in the first round, she should compare it to the expected payo↵ she’d get by rejecting it and allowing the game
to continue. If she rejects the o↵er, then with probability p, negotiations break down
immediately and she gets y. Otherwise, the game continues to its second and final
round, where we’ve already concluded that B will get 1 � x. Therefore, B’s expected payo↵ if she rejects the o↵er is
py + (1 � p)(1 � x).
Let’s call this quantity z; our conclusion is that in the first round, B will accept any
o↵er of at least z.
• Finally, we need to determine what A will propose in the first round. There is no point in A’s o↵ering to B anything more generous than (1 � z, z), since B will accept this, so the question is simply whether A prefers this split to his outside option x. In fact,
he does: since y < 1 � x, and z is a weighted average of y and 1 � x, it follows that z < 1 � x, and so 1 � z > x.
Therefore, A will propose (1 � z, z) in the first round, and it will be immediately accepted.
This describes the complete solution to the two-period bargaining game, and it’s inter-
esting to consider how the outcome for each player depends on the value of the breakdown
probability p. When p is close to one, so that negotiations are very likely to break down in
the first round, B’s payo↵ z = py + (1 � p)(1 � x) is very close to her back-up option y; correspondingly, A gets almost all the surplus. On the other hand, when p is close to zero,
so that negotiations are very likely to continue to the second round, B’s payo↵ is very close
to 1 � x, and so A is driven down to almost his back-up option. This makes sense intuitively. When p is close to one, A has most of the leverage in
the negotiations, since his o↵er is probably the only one that will get made. When p is
close to zero, B has most of the leverage in the negotiations, since she will probably get
to make the final o↵er, and can therefore safely ignore an undesirable initial o↵er from A.
Notice also that when p is exactly equal to 1 2
, the payo↵s correspond to the Nash bargaining
2We will exploit indi↵erence, as in many of our previous models, to assume that A accepts the proposed split (x, 1�x) rather than letting negotiations end. Alternately, as usual, we could imagine that B proposes an amount very slightly above x to A, to make sure A accepts.
366 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
outcome: each player gets an amount halfway between their backup option and their backup
option plus the full surplus. So this in fact provides us with a first way to obtain the Nash
bargaining solution from a two-player game: when the players take part in a two-round
negotiation that ends with probability 1 2
after the first round. As a reasonable model of
bargaining, however, this structure is a bit artificial: why only two rounds, and moreover,
why a breakdown probability of exactly 1 2
? It feels more reasonable to consider negotiations
that are allowed to go on for a long time, with the small underlying breakdown probability
imposing a mild form of pressure to reach an agreement. This is the infinite-horizon version
that we formulated initially, and which we will analyze next.
Back to the Infinite-Horizon Bargaining Game. One way to build up to the analysis
of the infinite-horizon game would be to consider finite-horizon bargaining games that are
allowed to last for a larger and larger number of rounds, and try to argue that these eventually
approximate the infinite-horizon version. Finite-horizon games of even length give B the last
o↵er, while those of odd length give A the last o↵er; but as the length increases, the chance
that the last round is ever reached will go down. It is possible to carry out this analysis, but
in fact it’s easier to use what we learned in the two-round version of the game to directly
conjecture the structure of an equilibrium for the infinite-horizon game.
In particular, we saw in the analysis of the two-round bargaining game that o↵ers are
not rejected in equilibrium. There are two reasons for this. First, both players stand to
gain from splitting the surplus 1 � x � y in some fashion, and delaying by rejecting o↵ers makes it possible that negotiations will break down and this surplus will be lost. Second,
each player can reason about the minimum amount that the other is willing to accept, and
so he or she can o↵er exactly this amount when given the opportunity to make an o↵er.
At a general level, these considerations still apply to the infinite-horizon game, and so it is
natural to conjecture there is an equilibrium in which A’s initial o↵er is accepted. We will
search for such an equilibrium — and in fact, more strongly for an equilibrium where from
any intermediate point in the game, the next o↵er to be made would be accepted.
There is another issue to consider: there is at least one sense in which the finite-horizon
bargaining games actually have a more complicated structure than the infinite-horizon game.
For a finite-horizon bargaining game, the reasoning in each period is slightly di↵erent — you
have to evaluate the expected payo↵ a bit di↵erently with 10 rounds left to go than you do
with 9 rounds or 8 rounds left to go. This means that the splits being proposed will also
change slightly in value as the time until the end of the game changes. The infinite-horizon
game, on the other hand, is fundamentally di↵erent: after a back-and-forth pair of o↵ers by
A and B, there is another copy of exactly the same infinite-horizon game left to be played.
The structure and payo↵s in the game don’t change over time. Of course, the players do
observe o↵ers being made and rejected if the game actually continues past the first period,
12.9. ADVANCED MATERIAL: A GAME-THEORETIC APPROACH TO BARGAINING367
and they could condition their behavior on this history of o↵ers. But given the stationary
nature of the game’s structure over time, it’s natural to look for an equilibrium among the
set of stationary strategies: those in which each of A and B plans to propose the same split
in every period in which they are scheduled to propose, and each of A and B also has a fixed
amount that they require in order to accept a proposal. An equilibrium that uses stationary
strategies will be called a stationary equilibrium.
Analyzing the Game: A Stationary Equilibrium. A nice feature of stationary strate-
gies is that they’re very easy to describe and work with. Although the game is complex, any
pair of stationary strategies for A and B can be represented by just a few numbers, namely:
• The split (a 1
, b 1
) that A will o↵er whenever he is scheduled to propose a split;
• the split (a 2
, b 2
) that B will o↵er whenever she is scheduled to propose a split; and
• reservation amounts a and b, constituting the minimum o↵ers that A and B respectively will accept from the other.
Moreover, since the o↵ers constitute proposed splits of one dollar, the two parts of each split
sum to 1; so we have b 1
= 1 � a 1
and a 2
= 1 � b 2
.
Our plan is to write down a set of equations on the values describing the stationary
strategies, such that any pair of stationary strategies satisfying these equations constitute
an equilibrium. We will then solve these equations, obtaining a stationary equilibrium, and
show that as the breakdown probability p converges to 0, the payo↵s to A and B converge
to the Nash bargaining outcome.
The equations are as follows. First, as in the two-period version of the game, A will o↵er
B the least he can in order to get B to accept his o↵er, so we set
b 1
= b. (12.1)
Similarly, B will o↵er the least she can in order to get A to accept her o↵er, so
a 2
= a. (12.2)
Again following the reasoning from the two-period version, B will set her reservation amount
b right at the level where she is indi↵erent between accepting A’s o↵er and rejecting it. If
she accepts, she gets b 1
; if she rejects, she gets the expected payo↵ that comes from allowing
the game to continue. We can determine this expected value as follows. With probability p,
the game ends right after her rejection, in which case she receives y. Otherwise, the game
continues with an o↵er by B to A, and this o↵er will be accepted since, by Equation (12.2),
we’ve set a 2
= a. In this case, B receives b 2
; so her overall expected payo↵ from allowing the
368 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
game to continue would be py + (1 � p)b 2
. For B to be indi↵erent between accepting and
rejecting, we need
b 1
= py + (1 � p)b 2
. (12.3)
Similar reasoning applies to A’s reservation amount: if he rejects an o↵er from B and allows
the game to continue, his expected payo↵ is px + (1 � p)a 1
, and so for him to be indi↵erent
between accepting and rejecting B’s o↵er we have
a 2
= px + (1 � p)a 1
. (12.4)
Following the reasoning above, we can check that these four equations (12.1)–(12.4) are
enough to ensure that the pair of stationary strategies forms an equilibrium.
Since b 1
= 1 � a 1
and a 2
= 1 � b 2
, this gives us two linear equation in two unknowns:
1 � a 1
= py + (1 � p)b 2
1 � b 2
= px + (1 � p)a 1
Solving these, we get
a 1
= (1 � p)x + 1 � y
2 � p
b 2
= (1 � p)y + 1 � x
2 � p .
In this equilibrium, A’s initial o↵er is accepted, so A gets a payo↵ of a 1
, and B gets a payo↵
of
b 1
= 1 � a 1
= y + (1 � p)(1 � x)
2 � p We can check how these values for a
1
and b 1
behave as a function of p. When p is close
to 1, they are approximately 1 � y and y respectively. A gets almost all the surplus and B gets very close to her outside option, because the negotiations are likely to break down after
the opening o↵er by A, and A is taking advantage of this fact.
More interestingly, as p converges to 0, so that the players can expect the negotiations
to continue for a long time, the opening o↵er is still accepted in this stationary equilibrium,
but the payo↵s are converging to
✓ x + 1 � y
2 , y + 1 � x
2
◆ ,
which are the values for the Nash bargaining solution. This completes the analysis, and
shows how the Nash bargaining outcome arises very naturally from a game-theoretic model
in which the two bargainers behave strategically, following a simple model of negotiations.
12.10. EXERCISES 369
12.10 Exercises
1. Suppose a network exchange theory experiment is run on the graph depicted in Fig-
ure 12.10 using the one-exchange rule. Say which node or nodes you would expect to
make the most money (i.e. receive the most favorable exchanges), and give a brief (1-3
sentence) explanation for your answer.
a b c
d
e
f g
Figure 12.10: A graph used for a network exchange theory experiment.
2. Suppose a network exchange theory experiment is run on the graph depicted in Fig-
ure 12.11 (i.e. a graph that is a 3-node path), using the one-exchange rule.
Now you, playing the role of a fourth node d, are told to attach by a single edge to one
of the nodes in the network. How should you attach to the network to put yourself
in as powerful a position as possible, where power will be determined by the result of
a network exchange theory experiment run on the resulting 4-node network? Give a
brief explanation for your answer.
a b c
Figure 12.11:
370 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
3. Suppose a network exchange theory experiment is run on the graph depicted in Fig-
ure 12.12 using the one-exchange rule with $10 placed on each edge.
a b c d e
Figure 12.12: A graph used for a network exchange theory experiment.
(a) Say which node or nodes you would expect to make the most money (i.e. receive
the most favorable exchanges), and give a brief (1-3 sentence) explanation for your
answer. You do not need to give actual numbers for the amounts of money the nodes
would receive.
(b) Now the experimenters vary the network: they introduce a sixth node f , which
is attached by a single edge to just the node c. A new person is brought in to play
the role of f , and a new round of experiments is performed with this new six-node
network.
Explain what you think will happen to the relative power of the participants, compared
to the situation in (a), and give a brief (1-3 sentence) explanation for your answer.
Again, you do not need to give actual numbers for the amounts of money the nodes
would receive.
4. Suppose a network exchange theory experiment is run on the graph depicted in Fig-
ure 12.13 using the one-exchange rule with $10 placed on each edge.
a b c d
Figure 12.13:
(a) Say which node or nodes you would expect to make the most money (i.e. receive
the most favorable exchanges), and give a brief (1-3 sentence) explanation for your
answer. You do not need to give actual numbers for the amounts of money the nodes
would receive.
(b) Now the experimenters vary the conditions slightly: instead of placing $10 on the
b-c edge, they place only $2. Otherwise, the experiment is run exactly as before.
12.10. EXERCISES 371
Explain what you think will happen to the relative power of each of the participants,
compared to the situation in (a), and give a brief (1-3 sentence) explanation for your
answer. Again, you do not need to give actual numbers for the amounts of money the
nodes would receive.
5. Suppose a network exchange theory experiment is run on the graph depicted in Fig-
ure 12.14 using the one-exchange rule with $10 placed on each edge.
a b c d
Figure 12.14: A 4-node path in a network exchange theory experiment.
(a) After running the experiment for a while, the experimenters vary the network:
they introduce two further nodes e and f , and bring in additional people to play the
roles of these nodes. Node e is attached by a single edge to node b, while node f is
attached by a single edge to node c.
A new round of experiments is performed with this new six-node network. Explain
what you think will happen to the relative power of the participants, compared to the
situation in the original four-node network. Give a brief (1-3 sentence) explanation for
your answer. You do not need to give actual numbers for the amounts of money the
nodes would receive.
(b) The experimenters now decide to vary the network again. They keep the same
set of nodes, but now they add an edge linking e directly to f . (The existing edges
continue to remain in place as well.)
A new round of experiments is performed with this modified six-node network. Explain
what you think will happen to the relative power of the participants, compared to the
situation on the previous six-node network in part (a). Give a brief (1-3 sentence)
explanation for your answer. You do not need to give actual numbers for the amounts
of money the nodes would receive.
6. (a) Suppose that two di↵erent network exchange theory experiments are run, using
the one-exchange rule — one on the 3-node path depicted in Figure 12.15, and the
other on the 4-node path depicted in Figure 12.15. In which set of experiments do you
expect node b to receive more money (i.e. receive more favorable exchanges)? Give
a brief (1-3 sentence) explanation for your answer. (You do not need to give actual
numbers for the amounts of money the nodes would receive.)
(b) Suppose a network exchange theory experiment is run on the graph depicted in
372 CHAPTER 12. BARGAINING AND POWER IN NETWORKS
a b c
(a) 3-Node Path
a b c d
(b) 4-Node Path
Figure 12.15: A 3-node path (right) and a 4-node path (left).
Figure 12.16 using the one-exchange rule. Say which node or nodes you would expect
to make the most money (i.e. receive the most favorable exchanges).
Also, do you think the advantage experienced by the most powerful nodes in Fig-
ure 12.16 will be more similar to the advantage experienced by node b on the 3-node
path from part (a), or more similar to the advantage experienced by node b on the
4-node path from part (a)?
Give a brief (1-3 sentence) explanation for your answers. (Again, you do not need to
give actual numbers for the amounts of money the nodes would receive.)
B
A
C D
E F
Figure 12.16: The graph for the network exchange theory experiment in part (b).
Part IV
Information Networks and the World Wide Web
373
Chapter 13
The Structure of the Web
Up to this point in the book, we’ve considered networks in which the basic units being
connected were people or other social entities, like firms or organizations. The links connect-
ing them have generally corresponded to opportunities for some kind of social or economic
interaction.
In the next several chapters, we consider a di↵erent type of network, in which the basic
units being connected are pieces of information, and links join pieces of information that are
related to each other in some fashion. We will call such a network an information network.
As we will see, the World Wide Web is arguably the most prominent current example of
such a network, and while the use of information networks has a long history, it was really
the growth of the Web that brought such networks to wide public awareness.
While there are basic di↵erences between information networks and the kinds of social
and economic networks that we’ve discussed earlier, many of the central ideas developed
earlier in the book will turn out to be fundamental here as well: we’ll be using the same
basic ideas from graph theory, including short paths and giant components; formulating
notions of power in terms of the underlying graph structure; and even drawing connections
to matching markets when we consider some of the ways in which search companies on the
Web have designed their businesses.
Because the Web plays such a central role in the modern version of this topic, we begin
with some context about the Web, and then look further back into the history of information
networks that led up to the Web.
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
375
376 CHAPTER 13. THE STRUCTURE OF THE WEB
I teach a class
on Networks.
Networks
Course:
We have a
class blog
Networks
Class Blog:
This blog post
is about
Microsoft
Microsoft
Home Page
Figure 13.1: A set of four Web pages.
13.1 The World Wide Web
If you’re reading this book, it’s likely that you use the Web on a daily basis. But since
the Web is so enmeshed in the broader information infrastructure of the world (including
the Internet, wireless communication systems, and the global media industry), it’s actually
useful to think a bit about what the Web is and how it came about, starting from first
principles.
At a basic level, the Web is an application developed to let people share information
over the Internet; it was created by Tim Berners-Lee during the period 1989-1991 [54, 55].
Although it is a simplification, we can view the original conception and design of the Web
as involving two central features. First, it provided a way for you to make documents easily
available to anyone on the Internet, in the form of Web pages that you could create and
store on a publically accessible part of your computer. Second, it provided a way for others
to easily access such Web pages, using a browser that could connect to the public spaces on
computers across the Internet and retrieve the Web pages stored there.
13.1. THE WORLD WIDE WEB 377
I teach a class
on Networks
Networks
Course:
We have a
class blog
Networks
Class Blog:
This blog post
is about
Microsoft
Microsoft
Home Page
Figure 13.2: Information on the Web is organized using a network metaphor: The links among Web pages turn the Web into a directed graph.
To a first approximation, this is still how we experience the Web today: as a sequence of
Web pages rendered inside a browser. For example, Figure 13.1 shows a set of four separate
Web pages: the home page of a college instructor who teaches a class on networks; the home
page of the networks class he teaches; the blog for the class, with a post about Microsoft
listed at the top; and the corporate home page for Microsoft. Because of the underlying
design, we can think of these pages both as part of a single coherent system (the Web), but
also as files that likely reside on four separate computers, controlled by several di↵erent and
completely independent organizations, and made publically accessible by a now-universal
consensus to participate in the protocols of the Web.
Hypertext. Beyond these basic features, there is a crucial design principle embedded in
the Web — the decision to organize the information using a network metaphor. This is what
turns the set of Web pages from Figure 13.1 into the “web” of Web pages in Figure 13.2: in
writing a Web page, you can annotate any portion of the document with a virtual link to
378 CHAPTER 13. THE STRUCTURE OF THE WEB
another Web page, allowing a reader to move directly from your page to this other one. The
set of pages on the Web thereby becomes a graph, and in fact a directed graph: the nodes
are the pages themselves, and the directed edges are the links that lead from one page to
another.
Much as we’re familiar with the idea of links among Web pages, we should appreciate
that the idea to organize Web pages as a network was both inspired and non-obvious. There
are many ways to arrange information — according to a classification system, like books in
a library; as a series of folders, like the files on your computer; even purely alphabetically,
like the terms in an index or the names in a phone directory. Each of these organizational
systems can make sense in di↵erent contexts, and any of them could in principle have been
used for the Web. But the use of a network structure truly brings forth the globalizing power
of the Web by allowing anyone authoring a Web page to highlight a relationship with any
other existing page, anywhere in the world.
The decision to use this network metaphor also didn’t arise out of thin air; it’s an applica-
tion of a computer-assisted style of authoring known as hypertext that had been explored and
refined since the middle of the twentieth century [316, 324]. The motivating idea behind hy-
pertext is to replace the traditional linear structure of text with a network structure, in which
any portion of the text can link directly to any other part — in this way, logical relationships
within the text that are traditionally implicit become first-class objects, foregrounded by the
use of explicit links. In its early years, hypertext was a cause passionately advocated by a
relatively small group of technologists; the Web subsequently brought hypertext to a global
audience, at a scale that no one could have anticipated.
13.2 Information Networks, Hypertext, and Associa- tive Memory
The hypertextual structure of the Web provides us with a familiar and important example
of an information network — nodes (Web pages in this case) containing information, with
explicit links encoding relationships between the nodes. But the notion of an information
network significantly predates the development of computer technology, and the creators of
hypertext were in their own right motivated by earlier networks that wove together large
amounts of information.
Intellectual Precursors of Hypertext. A first important intellectual precursor of hy-
pertext is the concept of citation among scholarly books and articles. When the author
or authors of a scholarly work wish to credit the source of an idea they are invoking, they
include a citation to the earlier paper that provides the source of this idea. For example,
Figure 13.3 shows the citations among a set of sociology papers that provided some of the
13.2. INFORMATION NETWORKS, HYPERTEXT, AND ASSOCIATIVE MEMORY379
Kossinets-
Watts 2006
Burt 2004
Travers-
Milgram 1969
Lazarsfeld-
Merton 1954
Cartwright-
Harary 1956
Rapoport
1953
Granovetter
1973
Burt 2000
Davis 1963
Feld 1981
Granovetter
1985
Coleman
1988
Milgram
1967
Figure 13.3: The network of citations among a set of research papers forms a directed graph that, like the Web, is a kind of information network. In contrast to the Web, however, the passage of time is much more evident in citation networks, since their links tend to point strictly backward in time.
380 CHAPTER 13. THE STRUCTURE OF THE WEB
key ideas in the first part of this book. (At the bottom of this figure are seminal papers on
— from left to right — triadic closure, the small-world phenomenon, structural balance, and
homophily.) We can see how work in this field — as in any academic discipline — builds
on earlier work, with the dependence represented by a citation structure. We can also see
how this citation structure naturally forms a directed graph, with nodes representing books
and articles, and directed edges representing citations from one work to another. The same
structure arises among patents, which provide citations to prior work and earlier inventions;
and among legal decisions, which provide citations to earlier decisions that are being used
as precedents, or are being distinguished from the present case. Of course, the example in
Figure 13.3 is a tiny piece of a much larger directed graph; for instance, Mark Granovetter’s
1973 paper on the strength of weak ties has been cited several thousand times in the aca-
demic literature, so in the full citation structure we should imagine thousands of arrows all
pointing to this single node.
One distinction between citation networks and the Web is that citations are governed
much more strongly by an underlying “arrow of time.” A book, article, patent, or legal
decision is written at a specific point in time, and the citations it contains — the edges
pointing outward to other nodes — are e↵ectively “frozen” at the point when it is written.
In other words, citations lead back into the past: if paper X cites paper Y , then we generally
won’t find a citation from Y back to X for the simple reason that Y was written at a time
before X existed. Of course, there are exceptions to this principle — two papers that
were written concurrently, with each citing the other; or a work that is revised to include
more recent citations — but this flow backward in time is a dominant pattern in citation
networks. On the Web, in contrast, while some pages are written once and then frozen
forever, a significant portion of them are evolving works in progress where the links are
updated over long periods of time. This means that while links are directed, there is no
strong sense of “flow” from the present into the past.
Citation networks are not the only earlier form of information network. The cross-
references within a printed encyclopedia or similar reference work form another important
example; one article will often include pointers to other related articles. An on-line reference
work like Wikipedia (even when viewed simply as a collection of linked articles, independent
of the fact that it exists on the Web) is structured in the same way. This organizing principle
is a clear precursor of hypertext, in that the cross-referencing links make relationships among
the articles explicit. It is possible to browse a printed or on-line encyclopedia through its
cross-references, pursuing serendipitious leads from one topic to another.
For example, Figure 13.4 shows the cross-references among Wikipedia articles on certain
topics in game theory, together with connections to related topics.1 We can see, for example,
1Since Wikipedia changes constantly, Figure 13.4 necessarily represents the state of the links among these articles only at the time of this writing. The need to stress this point reinforces the contrast with the “frozen” nature of the citations in a collection of papers such as those in Figure 13.3.
13.2. INFORMATION NETWORKS, HYPERTEXT, AND ASSOCIATIVE MEMORY381
Nash
Equilibrium
A Beautiful
Mind (film)
Apollo 13
(film)
Game
Theory
John Forbes
Nash
Ron Howard
NASA
Conspiracy
Theories
RAND
Figure 13.4: The cross-references among a set of articles in an encyclopedia form another kind of infor- mation network that can be represented as a directed graph. The figure shows the cross-references among a set of Wikipedia articles on topics in game theory, and their connections to related topics including popular culture and government agencies.
how it’s possible to get from the article on Nash Equilibrium to the article on NASA (the U.S.
National Aeronautics and Space Administration) by passing through articles on John Nash
(the creator of Nash equilibrium), A Beautiful Mind (a film about John Nash’s life), Ron
Howard (the director of A Beautiful Mind), Apollo 13 (another film directed by Ron Howard),
and finally on to the article about NASA (the U.S. government agency that managed the
real Apollo 13 space mission). In short: Nash equilibrium was created by someone whose
life was the subject of a movie by a director who also made a movie about NASA. Nor
is this the only short chain of articles from Nash equilibrium to NASA. Figure 13.4 also
contains a sequence of cross-references based on the fact that John Nash worked for a period
of time at RAND, and RAND is the subject of several conspiracy theories, as is NASA.
These short paths between seemingly distant concepts reflect an analogue, for information
382 CHAPTER 13. THE STRUCTURE OF THE WEB
networks, of the “six degrees of separation” phenomenon in social networks from Chapter 2,
where similarly short paths link apparently distant pairs of people.
Indeed, browsing through chains of cross-references is closely related to the stream-of-
consciousness way in which one mentally free-associates between di↵erent ideas. For example,
suppose you’ve just been reading about Nash equilibrium in a book, and while thinking about
it during a walk home your mind wanders, and you suddenly notice that you’ve shifted to
thinking about NASA. It may take a bit of reflection to figure out how this happened, and
to reconstruct a chain of free-association like the one pictured in Figure 13.4, carried out
entirely among the existing associations in your mind. This idea has been formalized in
another kind of information network: a semantic network, in which nodes literally represent
concepts, and edges represent some kind of logical or perceived relationship between the
concepts. Researchers have used techniques like word association studies (e.g. “Tell me
what you think of when I say the word ‘cold’ ”) as a way to probe the otherwise implicit
structure of semantic networks as they exist in people’s minds [381].
Vannevar Bush and the Memex. Thus, information networks date back into much
earlier periods in our history; for centuries, they were associated with libraries and scholarly
literature, rather than with computer technology and the Internet. The idea that they
could assume a strongly technological incarnation, in the form of something like the Web, is
generally credited to Vannevar Bush and his seminal 1945 article in the Atlantic Monthly,
entitled “As We May Think” [89]. Written at the end of World War II, it imagined with
eerie prescience the ways in which nascent computing and communication technology might
revolutionize the ways we store, exchange, and access information.
In particular, Bush observed that traditional methods for storing information in a book, a
library, or a computer memory are highly linear — they consist of a collection of items sorted
in some sequential order. Our conscious experience of thinking, on the other hand, exhibits
what might be called an associative memory, the kind that a semantic network represents
— you think of one thing; it reminds you of another; you see a novel connection; some
new insight is formed. Bush therefore called for the creation of information systems that
mimicked this style of memory; he imagined a hypothetical prototype called the Memex that
functioned very much like the Web, consisting of digitized versions of all human knowledge
connected by associative links, and he imagined a range of commercial applications and
knowledge-sharing activities that could take place around such a device. In this way, Bush’s
article foreshadowed not only the Web itself, but also many of the dominant metaphors that
are now used to think about the Web: the Web as universal encyclopedia; the Web as giant
socio-economic system; the Web as global brain.
The fact that Vannever Bush’s vision was so accurate is not in any sense coincidental;
Bush occupied a prominent position in the U.S. government’s scientific funding establish-
13.2. INFORMATION NETWORKS, HYPERTEXT, AND ASSOCIATIVE MEMORY383
ment, and his ideas about future directions had considerable reach. Indeed, the creators of
early hypertext systems explicitly invoked Bush’s ideas, as did Tim Berners-Lee when he set
out to develop the Web.
The Web and its Evolution. This brings us back to the 1990s, the first decade of the
Web, in which it grew rapidly from a modest research project to a vast new medium with
global reach. In the early phase of this period, the simple picture in Figure 13.2 captured the
Web’s essential nature: most pages were relatively static documents, and most links served
primarily navigational functions — to transport you from one page to another, according to
the relational premise of hypertext.
This is still a reasonable working approximation for large portions of the Web, but the
Web has also increasingly outgrown the simple model of documents connected by navigational
links, and it is important to understand how this has happened in order to be able to interpret
any analysis of the Web’s structure. In the earliest days of the Web, the computers hosting
the content played a relatively passive role: they mainly just served up pages in response
to requests for them. Now, on the other hand, the powerful computation available at the
other end of a link is often brought more directly into play: links now often trigger complex
programs on the computer hosting the page. Links with labels like “Add to Shopping Cart,”
“Submit my Query,” “Update my Calendar,” or “Upload my Image,” are not intended by
their authors primarily to transport you to a new page (though they may do that incidentally
as part of their function) — such links exist to activate computational transactions on the
machine that runs the site. Here’s an example to make this concrete. If we continued
following links from the Microsoft Home Page in the example from Figure 13.2, we could
imagine taking a next step to the on-line shopping site that Microsoft hosts for its products.
From this page, clicking on a link labeled “Buy Now” next to one of the featured products
would result in a charge to your credit card and the delivery of the product to your home
in the physical, o↵-line world. There would also be a new page providing a receipt, but the
purpose of this last “Buy Now” link was not primarily to transport you, hypertextually, to
a “receipt page”; rather, it was to perform the indicated transaction.
In view of these considerations, it is useful to think of a coarse division of links on the
Web into navigational and transactional, with the former serving the traditional hypertex-
tual functions of the Web and the latter primarily existing to perform transactions on the
computers hosting the content. This is not a perfect or clear-cut distinction, since many links
on the Web have both navigational and transactional functions, but it is a useful dichotomy
to keep in mind when evaluating the function of the Web’s pages and links.
While a lot of content on the Web now has a primarily transactional nature, this content
still remains largely linked together by a navigational “backbone” — it is reachable via
relatively stable Web pages connected to each other by more traditional navigational links.
384 CHAPTER 13. THE STRUCTURE OF THE WEB
This is the portion of the Web we will focus on in our analysis of its global structure. Sorting
out what should belong to this navigational backbone and what shouldn’t is ultimately a type
of judgment call, but fortunately there is a lot of experience in making and even codifying
such judgments. This is because distinguishing between navigational and transactional links
has long been essential to Web search engines, when they build their indexes of the available
content on the Web. It’s clearly not in a search engine’s interest to index, for the general
public, every receipt from an on-line purchase that every user of the Web has ever made, or
every query result for available airline flight times or product specifications that every Web
user ever has made. As a result, search engines have developed and refined automated rules
that try to assess whether the content they are collecting is relatively stable and intended for
public consumption, and they tend to collect content that is reachable via navigational links.
We will implicitly be following such working definitions when we talk about the structure of
the Web; and when we discuss empirical data on large sets of Web pages in Section 13.4, it
will be based on collections assembled by search engines according to such rules.
13.3 The Web as a Directed Graph
Viewing social and economic networks in terms of their graph structures provides significant
insights, and the same is true for information networks such as the Web. When we view the
Web as a graph, it allows us to better understand the logical relationships expressed by its
links; to break its structure into smaller, cohesive units; and — as we will see in Chapter 14
— to identify important pages as a step in organizing the results of Web searches.
To begin with, it is important to note two things. First, in discussing the graph structure
of the Web, we will be following the plan outlined at the end of Section 13.2 and focusing
on its navigational links. As we observed in that discussion, the navigational links still form
the bulk of the Web’s structural backbone, despite the increasing richness of Web content
as a whole.
Second, we need to appreciate that the fundamentally directed nature of the Web makes
it di↵erent from many of the networks we’ve considered thus far. Recall that in a directed
graph, the edges don’t simply connect pairs of nodes in a symmetric way — they point from
one node to another. This is clearly true on the Web: just because you write a blog post and
include a link to the Web page of a company or organization, there is no reason to believe
that they will necessarily reciprocate and include a link back to the blog post.
This distinction between directedness and undirectedness is an important aspect of the
di↵erence between social and information networks; an analogy here is to the di↵erence
between the global friendship network that we discussed in Chapter 2, showing who is friends
with whom, and the global name-recognition network, in which there is a link from person
A to person B if A has heard of B. This latter network is directed and in fact quite
13.3. THE WEB AS A DIRECTED GRAPH 385
I'm a student at Univ. of X
Univ. of X
Classes
Networks
Networks class blog
Blog post about
Company Z
Company Z's home page
Our Founders
Press Releases
Contact Us
Blog post about college rankings
I teach at Univ. of X
USNews College
Rankings
USNews Featured Colleges
I'm applying to college
My song lyrics
Figure 13.5: A directed graph formed by the links among a small set of Web pages.
asymmetric — famous celebrities are recognizable to millions of people, and in fact millions
closely track the goings-on in their lives, but one doesn’t expect that such celebrities are
in any sense aware of the names or identities of all these fans. In other words, the global
name-recognition network is structurally more similar to an information network like the
Web than it is to a traditional social network defined by friendship.
Paths and Strong Connectivity. The connectivity of undirected graphs was defined in
terms of paths: two nodes are linked by a path if we can follow a sequence of edges from
one to the other; a graph is connected if every pair of nodes is linked by a path; and we can
break up a disconnected graph into its connected components. Now that we’re dealing with
a directed graph, we’re going to try following the same general strategy for talking about
connectivity; but to do this, we first need to rework the definition of a path to take directions
386 CHAPTER 13. THE STRUCTURE OF THE WEB
into account, and this will necessarily make the subsequent definitions more subtle.
First, a path from a node A to a node B in a directed graph is a sequence of nodes,
beginning with A and ending with B, with the property that each consecutive pair of nodes
in the sequence is connected by an edge pointing in the forward direction. This “pointing in
the forward direction” condition makes the definition of a path in a directed graph di↵erent
from the corresponding definition for undirected graphs, where edges have no direction. On
the Web, this notion of following links only in the forward direction corresponds naturally to
the notion of viewing Web pages with a browser: we can follow a link when it’s emanating
from the page we’re on, but we aren’t in general aware of all the links that point to the page
we’re currently visiting.
We can try out this definition on the example in Figure 13.5, which shows the directed
graph formed by the links among a small set of Web pages; it depicts some of the people
and classes associated with the hypothetical University of X, which we imagine to have once
been a Featured College in a national magazine. By following a sequence of links in this
example (all in the forward direction), we can discover that there’s a path from the node
labeled Univ. of X to the node labeled US News College Rankings: we can follow a link
from Univ. of X to its Classes page, then to the home page of its class entitled Networks,
then to the Networks class blog, then to a class blog post about college rankings, and finally
via a link from this blog post to the page US News College Rankings. On the other hand,
there’s no path from the node labeled Company Z’s home page to the node labeled US News
College Rankings — there would be if we were allowed to follow directed edges in the reverse
direction, but following edges forward from Company Z’s home page, we can only reach Our
Founders, Press Releases, and Contact Us.
With the definition of a path in hand, we can adapt the notion of connectivity to the
setting of directed graphs. We say that a directed graph is strongly connected if there is a
path from every node to every other node. So for example, the directed graph of Web pages
in Figure 13.5 is not strongly connected, since as we’ve just observed, there are certain pairs
of nodes for which there’s no path from the first to the second.
Strongly Connected Components. When a directed graph is not strongly connected,
it’s important to be able to describe its reachability properties: identifying which nodes are
“reachable” from which others using paths. To define this notion precisely, it’s again useful
to draw an analogy to the simpler case of undirected graphs, and try to start from there.
For an undirected graph, its connected components serve as a very e↵ective summary of
reachability: if two nodes belong to the same component, then they can reach each other by
paths; and if two nodes belong to di↵erent components then they can’t.
But reachability in a directed graph is a harder thing to summarize. In a directed graph,
we can have pairs of nodes for which each can reach the other (like Univ. of X and US
13.3. THE WEB AS A DIRECTED GRAPH 387
I'm a student at Univ. of X
Company Z's home page
Our Founders
Press Releases
Contact Us
Univ. of X
Classes
Networks
Networks class blog
Blog post about college rankings
I teach at Univ. of X
USNews: College
Rankings
USNews: Featured Colleges
Blog post about
Company Z
I'm a applying to college
My song lyrics
Figure 13.6: A directed graph with its strongly connected components identified.
388 CHAPTER 13. THE STRUCTURE OF THE WEB
News College Rankings), pairs for which one can reach the other but not vice versa (like
US News College Rankings and Company Z’s home page), and pairs for which neither can
reach the other (like I’m a student at Univ. of X and I’m applying to college). Moreover, the
conceptual complexity of reachability in a directed graph corresponds to a kind of “visual”
complexity as well: whereas the components of an undirected graph naturally correspond
to separate chunks of the graph with no edges between them, a directed graph that is not
strongly connected does not break equally obviously into non-interacting pieces. How then
should we describe its reachability properties?
The key is to find the right notion of a “component” for directed graphs, and in fact one
can do this with a definition that strictly mirrors the formal definition of a component in an
undirected graph.
We say that a strongly connected component (SCC) in a directed graph is a
subset of the nodes such that: (i) every node in the subset has a path to every
other; and (ii) the subset is not part of some larger set with the property that
every node can reach every other.
As in the undirected case, part (i) of this definition says that all nodes within a strongly
connected component can reach each other, and part (ii) of this definition says that the
strongly connected components correspond as much as possible to separate “pieces,” not
smaller portions of larger pieces.
It helps to consider an example: in Figure 13.6 we show the strongly connected compo-
nents for the directed graph from Figure 13.5. Notice the role that part (ii) of the definition
plays in producing the separate pieces of the graph in this picture: the set of four nodes
consisting of Univ. of X, Classes, Networks, and I teach at Univ. of X collectively satisfy
part (i) of the definition, but they do not form a strongly connected component because they
belong to a larger set that also satisfies (i).
Looking at this picture, one can see how the SCCs serve as a compact summary of the
reachability properties of the directed graph. Given two nodes A and B, we can tell if there
is a path from A to B as follows. First, we find the SCCs containing A and B respectively. If
A and B belong to the same SCC, then they can each reach each other by paths. Otherwise,
viewing the SCCs themselves as larger “super-nodes”, we see if there is a way to walk from
the SCC of A to the SCC of B, following edges between SCCs in the forward direction. If
there is a way to do this, then this walk can be opened up into a path from A to B in the
graph; if there is no way to do this, then there is no path from A to B.
13.4 The Bow-Tie Structure of the Web
In 1999, after the Web had been growing for the better part of a decade, Andrei Broder and
his colleagues set out to build a global map of the Web, using strongly connected components
13.4. THE BOW-TIE STRUCTURE OF THE WEB 389
Figure 13.7: A schematic picture of the bow-structure of the Web (image from [80]). Al- though the numbers are now outdated, the structure has persisted.
as the basic building blocks [80]. For their raw data, they used the index of pages and links
from one of the largest commercial search engines at the time, AltaVista. Their influential
study has since been replicated on other, even larger snapshots of the Web, including an
early index of Google’s search engine [56] and large research collections of Web pages [133].
Similar analyses have been carried out for particular well-defined pieces of the Web, including
the links among articles on Wikipedia [83], and even for complex directed graph structures
arising in other domains, such as the network of interbank loans depicted in Figure 1.3 from
Chapter 1 [50]. In this way, although the actual snapshot of the Web used by Broder et al. in
their original study comes from an earlier time in the Web’s history, the mapping paradigm
they proposed continues to be a useful way of thinking about giant directed graphs in the
context of the Web and more generally.
390 CHAPTER 13. THE STRUCTURE OF THE WEB
A Giant Strongly Connected Component. A “map” of the Web clearly can’t resemble
a map of the physical world in any serious sense, given the scale and complexity of the network
being analyzed. Rather, what Broder et al. [80] wanted was something more conceptual —
an abstract map dividing the Web into a few large pieces, and showing in a stylized way how
these pieces fit together.
Their first finding was that the Web contains a giant strongly connected component.
Recall from our discussions in Chapter 2 that many naturally occuring undirected graphs
have a giant connected component — a single component containing a significant fraction
of all the nodes. The fact that the directed analogue of this phenomenon holds for the
Web is not hard to believe based on analogous thought experiments. Roughly, the point is
that a number of major search engines and other “starting page” sites have links leading to
directory-type pages from which you can, in turn, reach the home pages of major educational
institutions, large companies, and governmental agencies. From here one can reach most of
the pages within each of these large sites. Further, many of the pages within these sites
link back to the search engines and starting pages themselves. (The path from US News
College Rankings to a class blog and back in Figures 13.5 and 13.6 suggests a concrete
example for how this happens.) Thus, all these pages can mutually reach one another, and
hence all belong to the same strongly connected component. Given that this SCC contains
(at least) the home pages of many of the major commercial, governmental, and non-profit
organizations in the world, it is easy to believe that it is a giant SCC.
From here, we can invoke an argument — familiar from the undirected case as well —
that there is almost surely at most one giant SCC. For if there were two giant SCCs — call
them X and Y — all it would take is a single link from any node in X to any node Y , and
another link from any node in Y to any node in X, and X and Y would merge to become
part of a single SCC.
The Bow-Tie Structure. The second step in the analysis by Broder et al. [80] was to
position all the remaining SCCs in relation to the giant one. This involves classfying nodes
by their ability to reach and be reached from the giant SCC. The first two sets in this
classification are the following.
(1) IN: nodes that can reach the giant SCC but cannot be reached from it — i.e., nodes
that are “upstream” of it.
(2) OUT: nodes that can be reached from the giant SCC but cannot reach it — i.e., nodes
are “downstream” of it.
Figure 13.6 forms a useful example for trying out these definitions. Although the network in
Figure 13.6 is much too small for any of its SCCs to be considered “giant,” we can imagine its
largest SCC as the giant one and consider how the other nodes are positioned in relation to
13.4. THE BOW-TIE STRUCTURE OF THE WEB 391
it. In this case, the pages I’m a student at Univ. of X and I’m applying to college constitute
IN, and the pages Blog post about Company Z and the whole SCC involving Company Z
constitute OUT. And this is roughly what one intuitively expects to find in these sets: IN
contains pages that have not been “discovered” by members of the giant SCC, while OUT
contains pages that may receive links from the giant SCC, but which choose not to link back.
Figure 13.7 shows the original schematic image from Broder et al. [80], depicting the
relation of IN, OUT, and the giant SCC. Because of the visual e↵ect of IN and OUT as
large lobes hanging o↵ the central SCC, Broder et al. termed this the “bow-tie picture” of
the Web, with the giant SCC as the “knot” in the middle. The actual sizes of the di↵erent
pieces shown in the Figure come from the 1999 AltaVista data, and are long since obsolete
— the main point is that all three of these pieces are very large.
As Figure 13.7 also makes clear, there are pages that belong to none of IN, OUT, or the
giant SCC — that is, they can neither reach the giant SCC nor be reached from it. These
can be further classified as
(3) Tendrils: The “tendrils” of the bow-tie consist of (a) the nodes reachable from IN that
cannot reach the giant SCC, and (b) the nodes that can reach OUT but cannot be
reached from the giant SCC. For example, the page My song lyrics in Figure 13.6 is
an example of a tendril page, since it’s reachable from IN but has no path to the giant
SCC. It’s possible for a tendril node to satisfy both (a) and (b), in which case it’s
part of a “tube” that travels from IN to OUT without touching the giant SCC. (For
example, if the page My song lyrics happened to link to Blog post about Company Z in
Figure 13.6, it would be part of a tube.)
(4) Disconnected: Finally, there are nodes that would not have a path to the giant SCC
even if we completely ignored the directions of the edges. These belong to none of the
preceding categories.
Taken as a whole, then, the bow-tie picture of the Web provides a high-level view of
the Web’s structure, based on its reachability properties and how its strongly connected
components fit together. From it, we see that the Web contains a central “core” containing
many of its most prominent pages, with many other nodes that lie upstream, downstream,
or “o↵ to the side” relative to this core. It is also a highly dynamic picture: as people
create pages and links, the constituent pieces of the bow-tie are constantly shifting their
boundaries, with nodes entering (and also leaving) the giant SCC over time. But subsequent
studies suggest that the aggregate picture remains relatively stable over time, even as the
detailed structure changes continuously.
While the bow-tie picture gives us a global view of the Web, it doesn’t give us insight into
the more fine-grained patterns of connections within the constituent parts — connections
which could serve to highlight important Web pages or communities of thematically related
392 CHAPTER 13. THE STRUCTURE OF THE WEB
pages. Addressing these latter issues will require more detailed network analysis, which we
undertake in Chapter 14; as we will see, this requires us to think about what it means for
a Web page to occupy a “powerful” position, and it leads to methods that bear directly on
the design of Web search engines. More generally, network analysis of the Web forms one
ingredient in a broader emerging research agenda that aims to understand the structure,
behavior, and evolution of the Web as a phenomenon in itself [220].
13.5 The Emergence of Web 2.0
The increasing richness of Web content, which we’ve encountered through the distinction
between navigational and transactional links, fueled a series of further significant changes in
the Web during its second decade of existence, between 2000 and 2009. Three major forces
behind these changes were
(i) the growth of Web authoring styles that enabled many people to collectively create
and maintain shared content;
(ii) the movement of people’s personal on-line data (including e-mail, calendars, photos,
and videos) from their own computers to services o↵ered and hosted by large companies;
and
(iii) the growth of linking styles that emphasize on-line connections between people, not
just between documents.
Taken together, this set of changes altered user experience on the Web su�ciently that
technologists led by Tim O’Reilly and others began speaking in 2004 and 2005 about the
emergence of Web 2.0 [335]. While the term evokes images of a new software release, there
is agreement that Web 2.0 is principally “an attitude, not a technology” [125]. There has
never been perfect consensus on the meaning of the term, but it has generally connoted a
major next step in the evolution of the Web, driven by versions of principles (i), (ii), and
(iii) above (as well as others), and arising from a confluence of factors rather than any one
organization’s centralized decisions.
Indeed, there was an explosion of prominent new sites during the period 2004–2006 that
exemplified these three principles (i), (ii), and (iii), sometimes in combination. To name
just a few examples: Wikipedia grew rapidly during this period, as people embraced the
idea of collectively editing articles to create an open encyclopedia on the Web (principle (i));
Gmail and other on-line e-mail services encouraged individuals to let companies like Google
host their archives of e-mail (principle (ii)); MySpace and Facebook achieved widespread
adoption with a set of features that primarily emphasized the creation of on-line social
networks (principle (iii)).
13.5. THE EMERGENCE OF WEB 2.0 393
Many sites during this period combined versions of all three principles. For example, the
photo-sharing site Flickr and subsequently the video-sharing site YouTube provided users
with a centralized place to store their own photos and videos (principle (ii)), simultane-
ously enriched this content by allowing a large user community to tag and comment on it
(principle (i)), and allowed users to form social connections to others whose content they
followed (principle (iii)). The micro-blogging service Twitter extended principle (ii) further,
by creating an on-line forum for personal data (in the form of short real-time descriptions of
one’s experiences, thoughts, and questions) that would otherwise never have been recorded
at all. Because many people will all comment at roughly the same time on a current event
in the news, Twitter also creates collective summaries of worldwide reactions to such events
(principle (i)), and allows users to construct links by which they follow the writings of other
users (principle (iii)).
Even if some (or many) of these specific sites are replaced by others in the coming years,
the principles they embody have clearly brought about a lasting change in perspective on
Web content. These principles have also led to a point that we discussed early in Chapter 1:
designers of Web sites today need to think not just about organizing information, but also
about the social feedback e↵ects inherent in maintaining an audience of millions of users —
users who are able to interact directly not just with the site itself but with one another.
This helps to explain why many of the central concepts in this book relate to phenomena
that surround this current phase of the Web’s evolution. For example, many of the key
rallying cries that accompanied the emergence of Web 2.0 are in a sense shorthand for social
phenomena that we discuss in other chapters:
• “Software that gets better the more people use it.” A core principle of Web 2.0 is that on-line Web sites and services can become more appealing to users — and in fact, often
genuinely more valuable to them — as their audiences grow larger. When and how
this process takes place forms a central focus in chapters from the next two parts of
the book, particularly Chapters 16, 17, and 19.
• “The wisdom of crowds.” The collaborative authoring of an encyclopedia by millions on Wikipedia, the elevation of news content by group evaluation on Digg, the fact that
photos of breaking news now often appear on Flickr before they do in the mainstream
news, and many similar developments highlighted the ways in which the audience of a
Web 2.0 site — each contributing specific expertise and sometimes misinformation —
can produce a collective artifact of significant value. But the “wisdom of the crowds,”
as this process is now often called, is a subtle phenomenon that can fail as easily as
it can succeed. In Chapter 22 we discuss some of the basic work in the theory of
markets that helps explain how collective information residing in a large group can be
synthesized successfully; and in Chapter 16 we describe ways in which this process can
also lead to unexpected and sometimes undesirable outcomes.
394 CHAPTER 13. THE STRUCTURE OF THE WEB
• “The Long Tail.” With many people contributing content to a Web 2.0 site, the system will generally reach a balance between a small amount of hugely popular content and
a “long tail” of content with various levels of niche appeal. Such distributions of
popularity have important consequences, and will be the topic of Chapter 18.
In addition to the ideas suggested by such mantras, the premises underlying Web 2.0
appear in many other contexts in the book as well. The social-networking aspects of Web 2.0
sites provide rich data for large-studies of social network structure, as discussed in Chapter 2.
They o↵er a basis for empirical studies of the ideas of triadic closure and group a�liation from
Chapters 3 and 4, and have been used to evaluate the theories underlying the small-world
phenomenon in Chapter 20.
Moreover, many of the features that are common to Web 2.0 sites are designed to ex-
plicitly steer some of the underlying social feedback mechanisms in desirable directions. For
example, reputation systems and trust systems enable users to provide signals about the
behavior — and misbehavior — of other users. We discussed such systems in the context of
structural balance in Chapter 5, and will see their role in providing information essential to
the functioning of on-line markets in Chapter 22. Web 2.0 sites also make use of recommen-
dation systems, to guide users toward items that they may not know about. In addition to
serving as helpful features for a site’s users, such recommendation systems interact in com-
plex but important ways with distributions of popularity and the long tail of niche content,
as we will see in Chapter 18.
The development of the current generation of Web search engines, led by Google, is
sometimes seen as a crucial step in the pivot from the early days of the Web to the era of
Web 2.0. In the next two chapters we will discuss how thinking of the Web as a network
helped form the foundation for these search engines, and how models based on matching
markets helped turn search into a profitable business.
13.6 Exercises
1. Consider the set of 18 Web pages drawn in Figure 13.8, with links forming a directed
graph. Which nodes constitute the largest strongly connected component (SCC) in
this graph? Taking this as the giant SCC, which nodes then belong to the sets IN and
OUT defined in Section 13.4? Which nodes belong to the tendrils of the graph?
2. As new links are created and old ones are removed among an existing set of Web pages,
the pages move between di↵erent parts of the bow-tie structure.
(a) Name an edge you could add or delete from the graph in Figure 13.8 so as to
increase the size of the largest strongly connected component.
13.6. EXERCISES 395
7
6
8
11
2
12 13
14
9
3
4
5
10
15
16
17
1
18
Figure 13.8: A directed graph of Web pages.
(b) Name an edge you could add or delete from the graph in Figure 13.8 so as to
increase the size of the set IN.
(c) Name an edge you could add or delete from the graph in Figure 13.8 so as to
increase the size of the set OUT.
3. In Exercise 2, we considered how the consistuent parts of the bow-tie structure change
as edges are added to or removed from the graph. It’s also interesting to ask about
the magnitude of these changes.
(a) Describe an example of a graph where removing a single edge can reduce the size
of the largest strongly connected component by at least 1000 nodes. (Clearly you
shouldn’t attempt to draw the full graph; rather, you can describe it in words,
and also draw a schematic picture if it’s useful.)
(b) Describe an example of a graph where adding a single edge can reduce the size
of the set OUT by at least 1000 nodes. (Again, you should describe the graph
rather than actually drawing it.)
396 CHAPTER 13. THE STRUCTURE OF THE WEB
Chapter 14
Link Analysis and Web Search
14.1 Searching the Web: The Problem of Ranking
When you go to Google and type “Cornell,” the first result it shows you is www.cornell.edu,
the home page of Cornell University. It’s certainly hard to argue with this as a first choice,
but how did Google “know” that this was the best answer? Search engines determine how to
rank pages using automated methods that look at the Web itself, not some external source
of knowledge, so the conclusion is that there must be enough information intrinsic to the
Web and its structure to figure this out.
Before discussing some of the ideas behind the ranking of pages, let’s begin by considering
a few of the basic reasons why it’s a hard problem. First, search is a hard problem for com-
puters to solve in any setting, not just on the Web. Indeed, the field of information retrieval
[36, 360] has dealt with this problem for decades before the creation of the Web: automated
information retrieval systems starting in the 1960s were designed to search repositories of
newspaper articles, scientific papers, patents, legal abstracts, and other document collections
in reponse to keyword queries. Information retrieval systems have always had to deal with
the problem that keywords are a very limited way to express a complex information need;
in addition to the fact that a list of keywords is short and inexpressive, it su↵ers from the
problems of synonymy (multiple ways to say the same thing, so that your search for recipes
involving scallions fails because the recipe you wanted called them “green onions”) and pol-
ysemy (multiple meanings for the same term, so that your search for information about the
animal called a jaguar instead produces results primarily about automobiles, football players,
and an operating system for the Apple Macintosh.)
For a long time, up through the 1980s, information retrieval was the province of reference
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
397
398 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
librarians, patent attorneys, and other people whose jobs consisted of searching collections of
documents; such people were trained in how to formulate e↵ective queries, and the documents
they were searching tended to be written by professionals, using a controlled style and
vocabulary. With the arrival of the Web, where everyone is an author and everyone is a
searcher, the problems surrounding information retrieval exploded in scale and complexity.
To begin with, the diversity in authoring styles makes it much harder to rank documents
according to a common criterion: on a single topic, one can easily find pages written by
experts, novices, children, conspiracy theorists — and not necessarily be able to tell which is
which. Once upon a time, the fact that someone had the money and resources to produce a
professional-looking, typeset, bound document meant that they were very likely (even if not
always) someone who could be taken seriously. Today, anyone can create a Web page with
high production values.
There is a correspondingly rich diversity in the set of people issuing queries, and the
problem of multiple meanings becomes particularly severe. For example, when someone
issues the single-word query “Cornell,” a search engine doesn’t have very much to go on.
Did the searcher want information about the university? The university’s hockey team? The
Lab of Ornithology run by the university? Cornell College in Iowa? The Nobel-Prize-winning
physicist Eric Cornell? The same ranking of search results can’t be right for everyone.
These represent problems that were also present in traditional information retrieval sys-
tems, just taken to new extremes. But the Web also introduces new kinds of problems. One
is the dynamic and constantly-changing nature of Web content. On September 11, 2001,
many people ran to Google and typed “World Trade Center.” But there was a mismatch
between what people thought they could get from Google and what they really got: since
Google at the time was built on a model in which it periodically collected Web pages and
indexed them, the results were all based on pages that were gathered days or weeks earlier,
and so the top results were all descriptive pages about the building itself, not about what
had occurred that morning. In response to such events, Google and the other main search
engines built specialized “News Search” features, which collect articles more or less contin-
uously from a relatively fixed number of news sources, so as to be able to answer queries
about news stories minutes after they appear. Even today, such news search features are
only partly integrated into the core parts of the search engine interface, and emerging Web
sites such as Twitter continue to fill in the spaces that exist between static content and
real-time awareness.
More fundamental still, and at the heart of many of these issues, is the fact that the
Web has shifted much of the information retrieval question from a problem of scarcity to a
problem of abundance. The prototypical applications of information retrieval in the pre-Web
era had a “needle-in-a-haystack” flavor — for example, an intellectual-property attorney
might express the information need, “find me any patents that have dealt with the design
14.2. LINK ANALYSIS USING HUBS AND AUTHORITIES 399
of elevator speed regulators based on fuzzy-logic controllers.” Such issues still arise today,
but the hard part for most Web searches carried out by the general public is in a sense the
opposite: to filter, from among an enormous number of relevant documents, the few that
are most important. In other words, a search engine has no problem finding and indexing
literally millions of documents that are genuinely relevant to the one-word query “Cornell”;
the problem is that the human being performing the search is going to want to look at only
a few of these. Which few should the search engine recommend?
An understanding of the network structure of Web pages will be crucial for addressing
these questions, as we now discuss.
14.2 Link Analysis using Hubs and Authorities
So we’re back to our question from the beginning of the chapter: in response to the one-word
query “Cornell,” what are the clues that suggest Cornell’s home page, www.cornell.edu, is a
good answer?
Voting by In-Links. In fact, there is a natural way to address this, provided we start
from the right perspective. This perspective is to note that there is not really any way to
use features purely internal to the page www.cornell.edu to solve this problem: it does not
use the word “Cornell” more frequently or more prominently than thousands of other pages.
and so there is nothing on the page itself that makes it stand out. Rather, it stands out
because of features on other Web pages: when a page is relevant to the query “Cornell,”
very often www.cornell.edu is among the pages it links to.
This is the first part of the argument that links are essential to ranking: that we can use
them to assess the authority of a page on a topic, through the implicit endorsements that
other pages on the topic confer through their links to it. Of course, each individual link
may have many possible meanings: it may be o↵-topic; it may convey criticism rather than
endorsement; it may be a paid advertisement. It is hard for search engines to automatically
assess the intent of each link. But we hope that in aggregate, if a page receives many links
from other relevant pages, then it is receiving a kind of collective endorsement.
In the case of the query “Cornell,” we could operationalize this by first collecting a large
sample of pages that are relevant to the query — as determined by a classical, text-only,
information retrieval approach. We could then let pages in this sample “vote” through their
links: which page on the Web receives the greatest number of in-links from pages that are
relevant to Cornell? Even this simple measure of link-counting works quite well for queries
such as “Cornell,” where, ultimately, there is a single page that most people agree should be
ranked first.
400 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Wall St.
Journal
New York
Times
USA Today
Yahoo!
Amazon
2 votes
4 votes
3 votes
1 vote
3 votes
3 votes
SJ Merc
News 2 votes
Figure 14.1: Counting in-links to pages for the query “newspapers.”
A List-Finding Technique. It’s possible to make deeper use of the network structure
than just counting in-links, and this brings us to the second part of the argument that links
are essential. Consider, as a typical example, the one-word query “newspapers.” Unlike
the query “Cornell,” there is not necessarily a single, intuitively “best” answer here; there
are a number of prominent newspapers on the Web, and an ideal answer would consist of a
list of the most prominent among them. With the query “Cornell,” we discussed collecting
a sample of pages relevant to the query and then let them vote using their links. What
happens if we try this for the query “newspapers”?
What you will typically observe, if you try this experiment, is that you get high scores for a
mix of prominent newspapers (i.e. the results you’d want) along with pages that are going to
receive a lot of in-links no matter what the query is — pages like Yahoo!, Facebook, Amazon,
and others. In other words, to make up a very simple hyperlink structure for purposes of
14.2. LINK ANALYSIS USING HUBS AND AUTHORITIES 401
Wall St.
Journal
New York
Times
USA Today
Yahoo!
Amazon
11
7
3
6
3
3
5
2 votes
4 votes
3 votes
1 vote
3 votes
3 votes
8
SJ Merc
News
6
2 votes
Figure 14.2: Finding good lists for the query “newspapers”: each page’s value as a list is written as a number inside it.
this example, we’d see something like Figure 14.1: the unlabeled circles represent our sample
of pages relevant to the query “newspapers,” and among the four pages receiving the most
votes from them, two are newspapers (New York Times and USA Today) and two are not
(Yahoo! and Amazon). This example is designed to be small enough to try by hand; in
a real setting, of course there would be many plausible newspaper pages and many more
o↵-topic pages.
But votes are only a very simple kind of measure that we can get from the link structure
— there is much more to be discovered if we look more closely. To try getting more, we
ask a di↵erent question. In addition to the newspapers themselves, there is another kind of
useful answer to our query: pages that compile lists of resources relevant to the topic. Such
pages exist for most broad enough queries: for “newspapers,” they would correspond to lists
402 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Wall St.
Journal
New York
Times
USA Today
Yahoo!
Amazon
11
7
3
6
3
3
5
new score: 19
new score: 31
new score: 24
new score: 5
new score: 15
8
SJ Merc
News
6
new score: 19
new score: 12
Figure 14.3: Re-weighting votes for the query “newspapers”: each of the labeled page’s new score is equal to the sum of the values of all lists that point to it.
of links to on-line newspapers; for “Cornell,” one can find many alumni who maintain pages
with links to the University, its hockey team, its Medical School, its Art Museum, and so
forth. If we could find good list pages for newspapers, we would have another approach to
the problem of finding the newspapers themselves.
In fact, the example in Figure 14.1 suggests a useful technique for finding good lists. We
notice that among the pages casting votes, a few of them in fact voted for many of the pages
that received a lot of votes. It would be natural, therefore, to suspect that these pages have
some sense where the good answers are, and to score them highly as lists. Concretely, we
could say that a page’s value as a list is equal to the sum of the votes received by all pages
that it voted for. Figure 14.2 shows the result of applying this rule to the pages casting votes
in our example.
14.2. LINK ANALYSIS USING HUBS AND AUTHORITIES 403
The Principle of Repeated Improvement. If we believe that pages scoring well as lists
actually have a better sense for where the good results are, then we should weight their votes
more heavily. So, in particular, we could tabulate the votes again, but this time giving each
page’s vote a weight equal to its value as a list. Figure 14.3 shows what happens when we
do this on our example: now the other newspapers have surpassed the initially high-scoring
Yahoo! and Amazon, because these other newspapers were endorsed by pages that were
estimated to be good lists.
In fact, you can recognize the intuition behind this re-weighting of votes in the way we
evaluate endorsements in our everyday lives. Suppose you move to a new town and hear
restaurant recommendations from a lot of people. After discovering that certain restaurants
get mentioned by a lot of people, you realize that certain people in fact had mentioned most
of these highly-recommended restaurants when you asked them. These people play the role
of the high-value lists on the Web, and it’s only natural to go back and take more seriously
the more obscure restaurants that they recommended, since you now particularly trust their
judgment. This last step is exactly what we are doing in re-weighting the votes for Web
pages.
The final part of the argument for link analysis is then the following: Why stop here? If
we have better votes on the right-hand-side of the figure, we can use these to get still more
refined values for the quality of the lists on the left-hand-side of the figure. And with more
refined estimates for the high-value lists, we can re-weight the votes that we apply to the
right-hand-side once again. The process can go back and forth forever: it can be viewed
as a Principle of Repeated Improvement, in which each refinement to one side of the figure
enables a further refinement to the other.
Hubs and Authorities. This suggests a ranking procedure that we can try to make
precise, as follows [247]. First, we’ll call the kinds of pages we were originally seeking — the
prominent, highly endorsed answers to the queries — the authorities for the query. We’ll call
the high-value lists the hubs for the query. Now, for each page p, we’re trying to estimate
its value as a potential authority and as a potential hub, and so we assign it two numerical
scores: auth(p) and hub(p). Each of these starts out with a value equal to 1, indicating that
we’re initially agnostic as to which is the best in either of these categories.
Now, voting – in which we use the quality of the hubs to refine our estimates for the
quality of the authorities – is simply the following:
Authority Update Rule: For each page p, update auth(p) to be the sum of the
hub scores of all pages that point to it.
On the other hand, the list-finding technique – in which we use the quality of the authorities
to refine our estimates for the quality of the hubs, is the following:
404 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Wall St.
Journal
New York
Times
USA Today
Yahoo!
Amazon
11
7
3
6
3
3
5
normalized .152
normalized .248
normalized .192
normalized .040
normalized .120
8
SJ Merc
News
6
normalized .152
normalized .096
Figure 14.4: Re-weighting votes after normalizing for the query “newspapers.”
Hub Update Rule: For each page p, update hub(p) to be the sum of the authority
scores of all pages that it points to.
Notice how a single application of the Authority Update Rule (starting from a setting in
which all scores are initially 1) is simply the original casting of votes by in-links. A single
application of the Authority Update Rule followed by a single application the Hub Update
Rule produces the results of the original list-finding technique. In general, the Principle of
Repeated Improvement says that to obtain better estimates, we should simply apply these
rules in alternating fashion, as follows.
• We start with all hub scores and all authority scores equal to 1.
• We choose a number of steps k.
14.2. LINK ANALYSIS USING HUBS AND AUTHORITIES 405
Wall St.
Journal
New York
Times
USA Today
Yahoo!
Amazon
.321
.181
.015
.088
.003
.003
.123
limit .199...
limit .304...
limit .205...
limit .043...
limit .042...
.249
SJ Merc
News
.018
limit .199...
limit .008...
Figure 14.5: Limiting hub and authority values for the query “newspapers.”
• We then perform a sequence of k hub-authority updates. Each update works as follows:
– First apply the Authority Update Rule to the current set of scores.
– Then apply the Hub Update Rule to the resulting set of scores.
• At the end, the hub and authority scores may involve numbers that are very large. But we only care about their relative sizes, so we can normalize to make them smaller: we
divide down each authority score by the sum of all authority scores, and divide down
each hub score by the sum of all hub scores. (For example, Figure 14.4 shows the result
of normalizing the authority scores that we determined in Figure 14.3.)
What happens if we do this for larger and larger values of k? It turns out that the
normalized values actually converge to limits as k goes to infinity: in other words, the
406 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
results stabilize so that continued improvement leads to smaller and smaller changes in the
values we observe. We won’t prove this right now, but we provide a proof in Section 14.6 at
the end of this chapter. Moreover, the analysis in that section shows that something even
deeper is going on: except in a few rare cases (characterized by a certain kind of degenerate
property of the link structure), we reach the same limiting values no matter what we choose
as the initial hub and authority values, provided only that all of them are positive. In other
words, the limiting hub and authority values are a property purely of the link structure,
not of the initial estimates we use to start the process of computing them. (For the record,
the limiting values for our “newspapers” example are shown, to three decimal places, in
Figure 14.5.)
Ultimately, what these limiting values correspond to is a kind of equilibrium: their relative
sizes remain unchanged if we apply the Authority Update Rule or the Hub Update Rule. As
such, they reflect the balance between hubs and authorities that provided the initial intuition
for them: your authority score is proportional to the hub scores of the pages that point to
you, and your hub score is proportional to the authority scores of the pages you point to.
14.3 PageRank
The intuition behind hubs and authorities is based on the idea that pages play multiple
roles in the network, and in particular that pages can play a powerful endorsement role
without themselves being heavily endorsed. For queries with a commercial aspect — such
as our query for newspapers in the previous section, or searches for particular products to
purchase, or more generally searches that are designed to yield corporate pages of any type
— there is a natural basis for this intuition. Competing firms will not link to each other,
except in unusual circumstances, and so they can’t be viewed as directly endorsing each
other; rather, the only way to conceptually pull them together is through a set of hub pages
that link to all of them at once.
In other settings on the Web, however, endorsement is best viewed as passing directly
from one prominent page to another — in other words, a page is important if it is cited
by other important pages. This is often the dominant mode of endorsement, for example,
among academic or governmental pages, among bloggers, or among personal pages more
generally. It is also the dominant mode in the scientific literature. And it is this mode of
endorsement that forms the basis for the PageRank measure of importance [79].
As with hubs and authorities, the intuition behind PageRank starts with simple voting
based on in-links, and refines it using the Principle of Repeated Improvement. In particular,
the Principle is applied here by having nodes repeatedly pass endorsements across their
out-going links, with the weight of a node’s endorsement based on the current estimate of
its PageRank: nodes that are currently viewed as more important get to make stronger
14.3. PAGERANK 407
B
A
C
FD E
H
G
Figure 14.6: A collection of eight pages: A has the largest PageRank, followed by B and C (which collect endorsements from A).
endorsements.
The basic definition of PageRank. Intuitively, we can think of PageRank as a kind of
“fluid” that circulates through the network, passing from node to node across edges, and
pooling at the nodes that are the most important. Specifically, PageRank is computed as
follows.
• In a network with n nodes, we assign all nodes the same initial PageRank, set to be 1/n.
• We choose a number of steps k.
• We then perform a sequence of k updates to the PageRank values, using the following rule for each update:
Basic PageRank Update Rule: Each page divides its current PageRank equally
across its out-going links, and passes these equal shares to the pages it points
to. (If a page has no out-going links, it passes all its current PageRank to
itself.) Each page updates its new PageRank to be the sum of the shares it
receives.
408 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
B
A
C
FD E
H
G
4/13
2/13 2/13
1/13 1/13 1/13 1/13
1/13
Figure 14.7: Equilibrium PageRank values for the network of eight Web pages from Fig- ure 14.6.
Notice that the total PageRank in the network will remain constant as we apply these
steps: since each page takes its PageRank, divides it up, and passes it along links, PageRank
is never created nor destroyed, just moved around from one node to another. As a result,
we don’t need to do any normalizing of the numbers to prevent them from growing, the way
we had to with hub and authority scores.
As an example, let’s consider how this computation works on the collection of 8 Web
pages in Figure 14.6. All pages start out with a PageRank of 1/8, and their PageRank
values after the first two updates are given by the following table:
Step A B C D E F G H 1 1/2 1/16 1/16 1/16 1/16 1/16 1/16 1/8 2 3/16 1/4 1/4 1/32 1/32 1/32 1/32 1/16
For example, A gets a PageRank of 1/2 after the first update because it gets all of F ’s,
G’s, and H’s PageRank, and half each of D’s and E’s. On the other hand, B and C each
get half of A’s PageRank, so they only get 1/16 each in the first step. But once A acquires
a lot of PageRank, B and C benefit in the next step. This is in keeping with the principle of
repeated improvement: after the first update causes us to estimate that A is an important
page, we weigh its endorsement more highly in the next update.
14.3. PAGERANK 409
Equilibrium Values of PageRank. As with hub-authority computations, one can prove
that except in certain degenerate special cases the PageRank values of all nodes converge to
limiting values as the number of update steps k goes to infinity.
Because PageRank is conserved throughout the computation — with the total PageRank
in the network equal to one — the limit of the process has a simple interpretation. We can
think of the limiting PageRank values, one value for each node, as exhibiting the following
kind of equilibrium: if we take the limiting PageRank values and apply one step of the Basic
PageRank Update Rule, then the values at every node remain the same. In other words,
the limiting PageRank values regenerate themselves exactly when they are updated. This
description gives a simple way to check whether an assignment of numbers to a set of Web
pages forms such an equilibrium set of PageRank values: we check that they sum to 1, and
we check that when we apply the Basic PageRank Update Rule, we get the same values
back.
For example, on the network of Web pages from Figure 14.6, we can check that the
values shown in Figure 14.7 have the desired equilibrium property — assigning a PageRank
of 4/13 to page A, 2/13 to each of B and C, and 1/13 to the five other pages achieves this
equilibrium.
Now, depending on the network structure, the set of limiting values may not be the
only ones that exhibit this kind of equilibrium. However, one can show that if the network
is strongly connected — that is, each node can reach each other node by a directed path,
following the definition from Chapter 13 — then there is a unique set of equilibrium values,
and so whenever the limiting PageRank values exist, they are the only values that satisfy
this equilibrium.
Scaling the definition of PageRank. There is a di�culty with the basic definition
of PageRank, however: in many networks, the “wrong” nodes can end up with all the
PageRank. Fortunately, there is a simple and natural way to fix this problem. yielding the
actual definition of PageRank that is used in practice. Let’s first describe the problem and
then its solution.
To trigger the problem, suppose we take the network in Figure 14.6 and make a small
change, so that F and G now point to each other rather than pointing to A. The result is
shown in Figure 14.8. Clearly this ought to weaken A somewhat, but in fact a much more
extreme thing happens: PageRank that flows from C to F and G can never circulate back
into the rest of the network, and so the links out of C function as a kind of “slow leak” that
eventually causes all the PageRank to end up at F and G. We can indeed check that by
repeatedly running the Basic PageRank Update Rule, we converge to PageRank values of
1/2 for each of F and G, and 0 for all other nodes.
This is clearly not what we wanted, but it’s an inevitable consequence of the definition.
410 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
B
A
C
FD E
H
G
Figure 14.8: The same collection of eight pages, but F and G have changed their links to point to each other instead of to A. Without a smoothing e↵ect, all the PageRank would go to F and G.
And it becomes a problem in almost any real network to which PageRank is applied: as
long as there are small sets of nodes that can be reached from the rest of the graph, but
have no paths back, then PageRank will build up there.1 Fortunately, there is a simple and
natural way to modify the definition of PageRank to get around this problem, and it follows
from the “fluid” intuition for PageRank. Specifically, if we think about the (admittedly
simplistic) question of why all the water on earth doesn’t inexorably run downhill and reside
exclusively at the lowest points, it’s because there’s a counter-balancing process at work:
water also evaporates and gets rained back down at higher elevations.
We can use this idea here. We pick a scaling factor s that should be strictly between 0
and 1. We then replace the Basic PageRank Update Rule with the following:
Scaled PageRank Update Rule: First apply the Basic PageRank Update Rule.
Then scale down all PageRank values by a factor of s. This means that the total
PageRank in the network has shrunk from 1 to s. We divide the residual 1 � s units of PageRank equally over all nodes, giving (1 � s)/n to each.
1If we think back to the bow-tie structure of the Web from Chapter 13, there is a way to describe the problem in those terms as well: there are many “slow leaks” out of the giant SCC, and so in the limit, all nodes in the giant SCC will get PageRank values of 0; instead, all the PageRank will end up in the set OUT of downstream nodes.
14.3. PAGERANK 411
This rule also preserves the total PageRank in the network, since it is just based on redis-
tribution according to a di↵erent “water cycle” that evaporates 1 � s units of PageRank in each step and rains it down uniformly across all nodes.
The Limit of the Scaled PageRank Update Rule. One can show that repeated appli-
cation of the Scaled PageRank Update Rule converges to a set of limiting PageRank values
as the number of updates k goes to infinity. Moreover, for any network, these limiting values
form the unique equilibrium for the Scaled PageRank Update Rule: they are the unique set
of values that remain unchanged under the application of this update rule. Notice, of course,
that these values depend on our choice of the scaling factor s: we should really think of there
being a di↵erent update rule for each possible value of s.
This is the version of PageRank that is used in practice, with a scaling factor s that is
usually chosen to be between 0.8 and 0.9.2 The use of the scaling factor also turns out to
make the PageRank measure less sensitive to the addition or deletion of small numbers of
nodes or links [268, 422].
Random walks: An equivalent definition of PageRank. To conclude our discussion
in this section, we now describe an equivalent formulation of PageRank that looks quite
di↵erent on the surface, but in fact leads to exactly the same definition.
It works as follows. Consider someone who is randomly browsing a network of Web pages,
such as the one in Figure 14.6. They start by choosing a page at random, picking each page
with equal probability. They then follow links for a sequence of k steps: in each step, they
pick a random out-going link from their current page, and follow it to where it leads. (If their
current page has no out-going links, they just stay where they are.) Such an exploration of
nodes performed by randomly following links is called a random walk on the network. We
should stress that this is not meant to be an accurate model of an actual person exploring
the Web; rather, it is a thought experiment that leads to a particular definition.
In Section 14.6, we analyze this random walk and show the following fact:
Claim: The probability of being at a page X after k steps of this random walk is
precisely the PageRank of X after k applications of the Basic PageRank Update
Rule.
2As an aside about our earlier motivating example, one can check that using a value of s in this range doesn’t completely fix the problem with Figure 14.8: nodes F and G still get most (though no longer all) of the PageRank under the scaled update rule with such values of s. The problem is that an eight-node example is simply too small for the redistribution of the PageRank to truly o↵set the problem of a slow leak into a dead-end region of the network: on only eight nodes, a “slow leak” isn’t actually so slow. However, on large networks such as are used in real applications, the redistribution of PageRank works well to give nodes outside the giant strongly connected component of the network very small limiting PageRank values.
412 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Given that the two formulations of PageRank — based on repeated improvement and random
walks respectively — are equivalent, we do not strictly speaking gain anything at a formal
level by having this new definition. But the analysis in terms of random walks provides some
additional intuition for PageRank as a measure of importance: the PageRank of a page X
is the limiting probability that a random walk across hyperlinks will end up at X, as we run
the walk for larger and larger numbers of steps.
This equivalent definition using random walks also provides a new and sometimes useful
perspective for thinking about some of the issues that came up earlier in the section. For
example, the “leakage” of PageRank to nodes F and G in Figure 14.8 has a natural interpre-
tation in terms of the random walk on the network: in the limit, as the walk runs for more
and more steps, the probability of the walk reaching F or G is converging to 1; and once it
reaches either F or G, it is stuck at these two nodes forever. Thus, the limiting probabilities
of being at F and G are converging to 1/2 each, and the limiting probabilities are converging
to 0 for all other nodes.
We will also show in Section 14.6 how to formulate the Scaled PageRank Update Rule
in terms of random walks. Rather than simply following a random edge in each step, the
walker performs a “scaled” version of the walk as follows: With probability s, the walker
follows a random edge as before; and with probability 1 � s, the walker jumps to a random node anywhere in the network, choosing each node with equal probability.
14.4 Applying Link Analysis in Modern Web Search
The link analysis ideas described in Sections 14.2 and 14.3 have played an integral role
in the ranking functions of the current generation of Web search engines, including Google,
Yahoo!, Microsoft’s search engine Bing, and Ask. In the late 1990s, it was possible to produce
reasonable rankings using these link analysis methods almost directly on top of conventional
search techniques; but with the growth and enormously expanding diversity of Web content
since then, link analysis ideas have been extended and generalized considerably, so that they
are now used in a wide range of di↵erent ways inside the ranking functions of modern search
engines.
It is hard to say anything completely concrete about the current ranking functions of the
main search engines, given that they are constantly evolving in complexity, and given that
the search engine companies themselves are extremely secretive about what goes into their
ranking functions. (There are good reasons for this secrecy, as we will discuss later.) But
we can make general observations, coupled with sentiments that represent the conventional
wisdom of the search community. In particular, PageRank was one of the original and central
ingredients of Google, and it has always been a core component of its methodology. The
importance of PageRank as a feature in Google’s ranking function has long been claimed
14.4. APPLYING LINK ANALYSIS IN MODERN WEB SEARCH 413
to be declining over time, however. For example, in 2003 and 2004, a significant overhaul
of Google’s ranking function was generally believed to involve non-PageRank styles of link
analysis, including a method called Hilltop [58], developed by Krishna Bharat and George
Mihaila as an extension of the two-sided form of endorsement behind hubs and authorities.
Around a similar time period, the search engine Ask rebuilt its ranking function around hubs
and authorities, though its recent extensions have increasingly blended this in with many
other features as well.
Combining links, text, and usage data. While our emphasis on link analysis in this
chapter was meant to motivate the ideas in a clean setting, in practice one clearly needs to
closely integrate information from both network structure and textual content in order to
produce the highest-quality search results. One particularly e↵ective way to combine text
and links for ranking is through the analysis of anchor text, the highlighted bits of clickable
text that activate a hyperlink leading to another page [102]. Anchor text can be a highly
succinct and e↵ective description of the page residing at the other end of a link; for example,
if you read “I am a student at Cornell University” on someone’s Web page, it’s a good guess
that clicking on the highlighted link associated with the text “Cornell University” will take
you to a page that is in some way about Cornell.3
In fact, the link analysis methods we have been describing can be easily extended to
incorporate textual features such as anchor text. In particular, the basic forms of both hubs
and authorities and PageRank perform updates by simply adding up values across links.
But if certain links have highly relevant anchor text while others don’t, we can weight the
contributions of the relevant links more heavily than the others; for example, as we pass hub
or authority scores, or PageRank values, across a link, we can multiply them by a factor that
indicates the quality of the anchor text on the link [57, 102].
In addition to text and links, search engines use many other features as well. For example,
the way in which users choose to click or not click on a search result conveys a lot of
information: if among a search engine’s ranked results for the query “Cornell,” most users
skip the first result and click on the second, it suggests that the first two results should
potentially be reordered. There is ongoing research on methods for tuning search results
based on this type of feedback [228].
A moving target. A final important aspect of Web search serves to illustrate a basic
game-theoretic principle that we have encountered many times already — that you should
always expect the world to react to what you do. As search grew into the dominant means of
accessing information on the Web, it mattered to a lot of people whether they ranked highly
3Of course, not all anchor text is useful; consider the ubiquitous bit of Web page text, “For more informa- tion, click here.” Such examples make you realize that creating useful anchor text is an aspect of hypertext authoring style worth paying attention to.
414 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
in search engine results. For example, many small companies had business models that
increasingly depended on showing up among the first screen of Google’s results for common
queries ranging from “Caribbean vacations” to “vintage records.” An update to Google’s
ranking function that pushed them o↵ the first screen could spell financial ruin. Indeed,
search-industry publications began naming some of Google’s more significant updates to
its core ranking function in the alphabetic style usually reserved for hurricanes — and the
analogy was an apt one, since each of these updates was an unpredictable act of nature (in
this case, Google) that inflicted millions of dollars of economic damage.
With this in mind, people who depended on the success of their Web sites increasingly
began modifying their Web-page authoring styles to score highly in search engine rankings.
For people who had conceived of Web search as a kind of classical information retrieval
application, this was something novel. Back in the 1970s and 1980s, when people designed
information retrieval tools for scientific papers or newspaper articles, authors were not overtly
writing their papers or abstracts with these search tools in mind.4 From the relatively early
days of the Web, however, people have written Web pages with search engines quite explicitly
in mind. At first, this was often done using over-the-top tricks that aroused the ire of the
search industry; as the digital librarian Cli↵ Lynch noted at the time, “Web search is a new
kind of information retrieval application in that the documents are actively behaving badly.”
Over time though, the use of focused techniques to improve a page’s performance in
search engine rankings became regularized and accepted, and guidelines for designing these
techniques emerged; a fairly large industry known as search engine optimization (SEO) came
into being, consisting of search experts who advise companies on how to create pages and
sites that rank highly. And so to return to the game-theoretic view: the growth of SEO
followed naturally once search became such a widespread application on the Web; it simply
mattered too much to too many people that they be easily findable through search.
These developments have had several consequences. First, they mean that for search
engines, the “perfect” ranking function will always be a moving target: if a search engine
maintains the same method of ranking for too long, Web-page authors and their consultants
become too e↵ective at reverse-engineering the important features, and the search engine
is in e↵ect no longer in control of what ranks highly. Second, it means that search engines
are incredibly secretive about the internals of their ranking functions — not just to prevent
competing search engines from finding out what they’re doing, but also to prevent designers
of Web sites from finding out.
And finally, with so much money at stake, the search industry turned these developments
into a very successful business model based on advertising. Rather than simply showing
results computed by a ranking function, the search engine o↵ered additional slots on the
4One can argue, of course, that at a less overt level, the development of standard authoring styles in these domains has been motivated by the goal of making these kinds of documents easier to classify and organize.
14.5. APPLICATIONS BEYOND THE WEB 415
main results page through a market in which sites could pay for placement. Thus, when you
look at a search results page today, you see the results computed by the ranking function
alongside the paid results. We have just seen some of the ideas behind ranking functions;
the paid results, as we will see in the next chapter, are allocated using the kinds of matching
markets discussed in Chapter 10.
14.5 Applications beyond the Web
Link analysis techniques of the kind we’ve been discussing have been applied to a wide range
of other settings, both before and after their use on the Web. In essentially any domain
where information is connected by a network structure, it becomes natural to infer measures
of authority from the patterns of links.
Citation Analysis. As we discussed in Chapters 2 and 13, the study of citations among
scientific papers and journals has a long history that significantly predates the Web [145]. A
standard measure in this field is Garfield’s impact factor for a scientific journal [177], defined
to be the average number of citations received by a paper in the given journal over the
past two years. This type of voting by in-links can thus serve as a proxy for the collective
attention that the scientific community pays to papers published in the journal.
In the 1970s, Pinski and Narin extended the impact factor by taking into account the
idea that not all citations should be counted equally — rather, citations from journals that
are themselves high-impact should be viewed as more important [341]. This can be viewed
as a use of the principle of repeated improvement, in the context of the scientific literature,
just as we’ve seen it used for Web-page ranking. Pinski and Narin used this to formulate a
notion of influence weights for journals [180, 341] that is defined very similarly to the notion
of PageRank for Web pages.
Link Analysis of U.S. Supreme Court Citations. Recently, researchers have adapted
link analysis techniques from the Web to study the network of citations among legal decisions
by U.S. courts [166, 377]. Citations are crucial in legal writing, to ground a decision in
precedent and to explain the relation of a new decision to what has come before. Using link
analysis in this context can help in identifying cases that play especially important roles in
the overall citation structure.
In one example of this style of research, Fowler and Jeon applied hub and authority
measures to the set of all U.S. Supreme Court decisions, a collection of documents that spans
more than two centuries [166]. They found that the set of Supreme Court decisions with
high authority scores in the citation network align well with the more qualitative judgments
of legal experts about the Court’s most important decisions. This includes some cases that
416 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Figure 14.9: The rising and falling authority of key Fifth Amendment cases from the 20th
century illustrates some of the relationships among them. (Image from [166].)
acquired significant authority according to numerical measures shortly after they appeared,
but which took much longer to gain recognition from the legal community.
Supreme Court decisions also provide a rich setting for looking at how authority can
change over long time periods. For example, Fowler and Jeon analyzed the rising and falling
authority of some of the key Fifth Amendment cases from the 20th century, as illustrated
in Figure 14.9. In particular, Brown v. Mississippi — a 1936 case concerning confessions
obtained under torture — began rising rapidly in authority in the early 1960s as the Warren
Court forcefully took on a range of issues surrounding due process and self-incrimination.
This development ultimately led to the landmark case Miranda v. Arizona in 1966 — and
with this clear precedent established, the need for citations to Brown v. Mississippi quickly
declined as the authority of Miranda shot upward.
The analysis of Supreme Court citations also shows that significant decisions can vary
widely in the rate at which they acquire authority. For example, Figure 14.10 (also from
[166]) shows that Roe v. Wade — like Miranda — grew in authority very rapidly from the
time it was first issued. On the other hand, the equally consequential Brown v. Board of
Education only began acquiring significant authority in the citation network roughly a decade
after it was issued. Fowler and Jeon argue that this trajectory aligns with legal scholars’
14.6. ADVANCED MATERIAL: SPECTRAL ANALYSIS, RANDOM WALKS, AND WEB SEARCH417
Figure 14.10: Roe v. Wade and Brown v. Board of Education acquired authority at very di↵erent speeds. (Image from [166].)
views of the case, writing, “Judicial specialists often point towards the ruling issued in Brown
as an example of a precedent that was legally weak when first issued, and was strengthened
through the Civil Rights Act of 1964 and its application in subsequent civil rights cases”
[166].
This style of analysis thus shows how a strictly network-based analysis of a topic as
intricate as legal precedent can reveal subtleties that align well with the views of the scholarly
community. It also indicates some of the interesting e↵ects that emerge when one tries to
track the rising and falling pattern of authority in a complex domain — an activity that
stands to provide important insights in many other settings as well.
14.6 Advanced Material: Spectral Analysis, Random Walks, and Web Search
We now discuss how to analyze the methods for computing hub, authority, and PageRank
values. This will require some basic familiarity with matrices and vectors. Building on this,
we will show that the limiting values of these link-analysis measures can be interpreted as
coordinates in eigenvectors of certain matrices derived from the underlying networks. The
use of eigenvalues and eigenvectors to study the structure of networks is often referred to as
418 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
node 1 0 1 0 1
0 0 1 1
1 0 0 0
0 0 1 0
node 4node 3
node 2
Figure 14.11: The directed hyperlinks among Web pages can be represented using an adja- cency matrix M : the entry M
ij
is equal to 1 if there is a link from node i to node j, and M
ij
= 0 otherwise.
the spectral analysis of graphs, and we will see that this theory forms a natural language for
discussing the outcome of methods based on repeated improvement.
A. Spectral Analysis of Hubs and Authorities
Our first main goal will be to show why the hub-authority computation converges to limiting
values for the hub and authority scores, as claimed in Section 14.2. As a first important step
in this, we show how to write the Authority Update and Hub Update Rules from that section
as matrix-vector multiplications.
Adjacency Matrices and Hub/Authority Vectors. We will view a set of n pages as
a set of nodes in a directed graph. Given this set of nodes, labeled 1, 2, 3, . . . , n, let’s encode
the links among them in an n ⇥ n matrix M as follows: the entry in the ith row and jth
column of M , denoted M ij
, will be equal to 1 if there is a link from node i to node j, and
it will be equal to 0 otherwise. We will call this the adjancency matrix of the network.
Figure 14.11 shows an example of a directed graph and its adjacency matrix. Given a large
set of pages, we expect that most of them will have very few outlinks relative to the total
number of pages, and so this adjacency matrix will have most entries equal to 0. As a result,
the adjacency matrix is not necessarily a very e�cient way to represent the network, but as
we will see, it is conceptually very useful.
Now, since the hub and authority scores are lists of numbers — one associated with each
14.6. ADVANCED MATERIAL: SPECTRAL ANALYSIS, RANDOM WALKS, AND WEB SEARCH419
node 1 0 1 0 1
0 0 1 1
1 0 0 0
0 0 1 0
2
6
4
3
=
9
7
2
4
node 4node 3
node 2
Figure 14.12: By representing the link structure using an adjacency matrix, the Hub and Authority Update Rules become matrix-vector multiplication. In this example, we show how multiplication by a vector of authority scores produces a new vector of hub scores.
of the n nodes of the network — we can represent them simply as vectors in n dimensions,
where the ith coordinate gives the hub or authority score of node i. Specifically, we write h
for the vector of hub scores, with h i
equal to the hub score of node i, and we similarly write
a for the vector of authority scores.
Hub and Authority Update Rules as Matrix-Vector Multiplication. Let’s consider
the Hub Update Rule in terms of the notation we’ve just defined. For a node i, its hub score
h i
is updated to be the sum of a j
over all nodes j to which i has an edge. Note that these
nodes j are precisely the ones for which M ij
= 1. Thus we can write the update rule as
h i
M i1
a 1
+ M i2
a 2
+ · · · + M in
a n
, (14.1)
where we use the notation “ ” to mean that the quantity on the left-hand-side is updated to become the quantity on the right-hand-side. This is a correct way to write the update
rule, since the values M ij
as multipliers select out precisely the authority values that we wish
to sum.
But Equation (14.1) corresponds exactly to the definition of matrix-vector multiplication,
so we can write it in the following equivalent way:
h M a.
Figure 14.12 shows this for the example from Figure 14.11, with the authority scores (2, 6, 4, 3)
producing the hub scores (9, 7, 2, 4) via the Hub Update Rule. Indeed, this is an example
of a general principle: if you’re updating a collection of variables according to a rule that
420 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
selects out certain ones to add up, you can often write this update rule as a matrix-vector
multiplication for a suitably chosen matrix and vector.
Specifying the Authority Update Rule in this style is strictly analogous, except that the
scores flow in the other direction across the edges. That is, a i
is updated to be the sum of
h j
over all nodes j that have an edge to i, so
a i
M 1i
h 1
+ M 2i
h 2
+ · · · + M ni
h n
. (14.2)
This too corresponds to a matrix-vector multiplication, but using a matrix where the entries
have all been “reflected” so that the roles of rows and columns are interchanged. This can
be specified using the transpose of the matrix M , denoted M T , and defined by the property
that the (i, j) entry of M T is the (j, i) entry of M : that is, M T ij
= M ji
. Then Equation (14.2)
corresponds to the update rule
a M T h.
Unwinding the k-step hub-authority computation. So far we have discussed a single
application of each of the update rules. What happens when we perform the k-step hub-
authority computation for some large value of k?
We start with initial vectors of authority and hub scores that we denote ah0i and hh0i,
each of them equal to the vector all of whose coordinates are 1. Now, let ahki and hhki denote
the vectors of authority and hub scores after k applications of the Authority and then Hub
Update Rules in order, as in Section 14.2. If we simply follow the formulas above, we first
find that
ah1i = M T hh0i
and
hh1i = M ah1i = M M T hh0i.
That’s the result of the 1-step hub-authority computation. In the second step, we therefore
get
ah2i = M T hh1i = M T M M T hh0i
and
hh2i = M ah2i = M M T M M T hh0i = (M M T )2hh0i.
One more step makes the pattern clear:
ah3i = M T hh2i = M T M M T M M T hh0i = (M T M )2M T hh0i
and
hh3i = M ah3i = M M T M M T M M T hh0i = (M M T )3hh0i.
14.6. ADVANCED MATERIAL: SPECTRAL ANALYSIS, RANDOM WALKS, AND WEB SEARCH421
Proceeding for larger numbers of steps, then, we find that ahki and hhki are products of the
terms M and M T in alternating order, where the expression for ahki begins with M T and
the expression for hhki begins with M . We can write this much more compactly as
ahki = (M T M )k�1M T hh0i
and
hhki = (M M T )khh0i.
So that’s a direct picture of what’s happening in the k-step hub-authority computation:
the authority and hub vectors are the results of multiplying an initial vector by larger and
larger powers of M T M and M M T respectively. We now consider why this process converges
to stable values.
Thinking about multiplication in terms of eigenvectors. Let’s keep in mind that,
since the actual magnitude of the hub and authority values tend to grow with each update,
they will only converge when we take normalization into account. To put it another way, it
is the directions of the hub and authority vectors that are converging. Concretely, what we
will show is that there are constants c and d so that the sequences of vectors hhki
ck and
ahki
dk converge to limits as k goes to infinity.
We’ll talk first about the sequence of hub vectors, and then we’ll consider the authority
vectors largely by pursuing a direct analogy to the analysis of hub vectors. If
hhki
ck =
(M M T )khh0i
ck
is going to converge to a limit hh⇤i, what properties do we expect hh⇤i should have? Since the
direction is converging, we expect that at the limit, the direction of hh⇤i shouldn’t change
when it is multiplied by (M M T ), although its length might grow by a factor of c. That is,
we expect that hh⇤i will satisfy the equation
(M M T )hh⇤i = chh⇤i.
Any vector satisfying this property — that it doesn’t change direction when multiplied by a
given matrix — is called an eigenvector of the matrix, and the scaling constant c is called the
eigenvalue corresponding to the eigenvector. So we expect that hh⇤i should be an eigenvector
of the matrix M M T , with c a corresponding eigenvalue. We now prove that the sequence of
vectors hhki
ck indeed converges to an eigenvector of M M T .
To prove this, we use the following basic fact about matrices. We say that a square
matrix A is symmetric if it remains the same after transposing it: A ij
= A ji
for each choice
of i and j, or in other words A = AT . The fact we will use is the following [268]:
422 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Any symmetric matrix A with n rows and n columns has a set of n eigenvectors
that are all unit vectors and all mutually orthogonal — that is, they form a basis
for the space Rn.
Since M M T is symmetric, we can apply this fact to it. Let’s write the resulting mu-
tually orthogonal eigenvectors as z 1
, z 2
, . . . , z n
, with corresponding eigenvalues c 1
, c 2
, . . . , c n
respectively; and let’s order the eigenvalues so that |c 1
| � |c 2
| � · · · � |c n
|. Furthermore, to make things simpler in this explanation, let’s suppose that |c
1
| > |c 2
|. (This essentially always happens in link analysis applications; and below we explain the small changes that
need to be made in the discussion if this assumption does not hold.) Now, given any vector
x, a good way to think about the matrix-vector product (M M T )x is to first write x as a
linear combination of the vectors z 1
, . . . , z n
. That is, with x = p 1
z 1
+ p 2
z 2
+ · · · + p n
z n
for
coe�cients p 1
, . . . , p n
, we have
(M M T )x = (M M T )(p 1
z 1
+ p 2
z 2
+ · · · + p n
z n
)
= p 1
M M T z 1
+ p 2
M M T z 2
+ · · · + p n
M M T z n
= p 1
c 1
z 1
+ p 2
c 2
z 2
+ · · · + p n
c n
z n
,
where the third equality follows from the fact that each z i
is an eigenvector.
What this says is that z 1
, z 2
, . . . , z n
is a very useful set of coordinate axes for representing
x: multiplication by M M T consists simply of replacing each term p i
z i
in the representation
of x by c i
p i
z i
. We now see how this makes it easy to analyze multiplication by larger powers
of M M T , which will be the last step we need for showing convergence.
Convergence of the hub-authority computation. We’ve seen that when we take any
vector x and write it in the form p 1
z 1
+ · · · + p n
z n
, multiplication by M M T produces
c 1
p 1
z 1
+ · · · + c n
p n
z n
. When we multiply repeatedly by M M T , each successive multipli-
cation introduces an additional factor of c i
in front of the ith term. Therefore we have
(M M T )kx = ck 1
p 1
z 1
+ ck 2
p 2
z 2
+ · · · + ck n
p n
z n
.
Now let’s think of this in the context of the vectors of hub scores, where hhki = (M M )T hh0i.
Recall that hh0i is just the fixed starting vector in which each coordinate is equal to 1;
it can be represented in terms of the basis vectors z 1
, . . . , z n
as some linear combination
hh0i = q 1
z 1
+ q 2
z 2
· · · + q n
z n
. So
hhki = (M M T )khh0i = ck 1
q 1
z 1
+ ck 2
q 2
z 2
+ · · · + ck n
q n
z n
, (14.3)
and if we divide both sides by ck 1
, then we get
hhki
ck 1
= q 1
z 1
+
✓ c 2
c 1
◆ k
q 2
z 2
+ · · · + ✓
c n
c 1
◆ k
q n
z n
. (14.4)
14.6. ADVANCED MATERIAL: SPECTRAL ANALYSIS, RANDOM WALKS, AND WEB SEARCH423
Recalling our assumption that |c 1
| > |c 2
| (which we’ll relax shortly), we see that as k goes to infinity, every term on the right-hand side but the first is going to 0. As a result, the
sequence of vectors hhki
ck 1
is converging to the limit q 1
z 1
as k goes to infinity.
Wrapping up. We’re essentially done at this point; but to round out the picture of con-
vergence, we will show two important things. First, we need to make sure that the coe�cient
q 1
in the argument above is not zero, so as to be able to ensure so that the limit q 1
z 1
is in
fact a non-zero vector in the direction of z 1
. Second, we will find that in fact a limit in the
direction of z 1
is reached essentially regardless of our choice of starting hub scores hh0i: it is
in this sense that the limiting hub weights are really a function of the network structure, not
the starting estimates. We will show these two facts in reverse order, considering the second
point first.
To begin with, then, let’s suppose we began the computation of the hub vector from a
di↵erent starting point: rather than having hh0i be the vector with all coordinates equal to
1, we picked some other starting vector x. Let’s suppose only that x has a positive number
in each coordinate — we’ll call such a vector a positive vector. As we noted before, any
vector x can be written as x = p 1
z 1
+ · · · p n
z n
, for some choice of multipliers p 1
, . . . , p n
, and
so (M M T )kx = ck 1
p 1
z 1
+ · · · ck n
p n
z n
. Then hhki/ck 1
is converging to p 1
z 1
— in other words,
still converging to a vector in the direction of z 1
even with this new choice for the starting
vector hh0i = x.
Now, let’s show why q 1
and p 1
above are not zero (hence showing that the limits are
non-zero vectors). Given any vector x, there is an easy way to think about the value of p 1
in its representation as x = p 1
z 1
+ · · · + p n
z n
: we just compute the inner product of z 1
and
x. Indeed, since the vectors z 1
, . . . , z n
are all mutually orthogonal, we have
z 1
· x = z 1
· (p 1
z 1
+ · · · p n
z n
) = p 1
(z 1
· z 1
) + p 2
(z 1
· z 2
) + · · · + p n
(z 1
· z n
) = p 1
,
since all terms in the last sum are 0 except for p 1
(z 1
· z 1
) = p 1
. Since p 1
is just the inner
product of x and z 1
, we see that our sequence of hub vectors converges to a non-zero vector
in the direction of z 1
provided only that our starting hub vector hh0i = x is not orthogonal
to z 1
.
We now argue that no positive vector can be orthogonal to z 1
, which will conclude the
picture of convergence that we’ve been seeking to establish. The argument works via the
following steps.
1. It is not possible for every positive vector to be orthogonal to z 1
, and so there is some
positive vector x for which (M M T )kx/ck 1
converges to a non-zero vector p 1
z 1
.
2. Since the expressions for (M M T )kx/ck 1
only involve non-negative numbers, and their
values converge to p 1
z 1
, it must be that p 1
z 1
has only non-negative coordinates; and
p 1
z 1
must have at least one positive coordinate, since it is not equal to zero.
424 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
3. So if we consider the inner product of any positive vector with p 1
z 1
, the result must
be positive. Hence we conclude that no positive vector can be orthogonal to z 1
. This
establishes that in fact the sequence of hub vectors converges to a vector in the direction
of z 1
when we start from any positive vector (including the all-ones vector), which is
what we wanted to show.
This is pretty much the complete story, with the only loose end being our assumption
that |c 1
| > |c 2
|. Let’s now relax this assumption. In general, there may be ` > 1 eigenvalues that are tied for the largest absolute value: that is, we can have |c
1
| = · · · = |c `
|, and then eigenvalues c
`+1
, . . . , c n
are all smaller in absolute value. While we won’t go through all the
details here, it is not hard to show that all the eigenvalues of M M T are non-negative, so in
fact we have c 1
= · · · = c `
> c `+1
� · · · � c n
� 0. In this case, going back to Equations (14.3) and (14.4), we have
hhki
ck 1
= ck 1
q 1
z 1
+ · · · + ck n
q n
z n
ck 1
= q 1
z 1
+ · · · + q `
z `
+
✓ c `+1
c 1
◆ k
q `+1
z `+1
+ · · · + ✓
c n
c 1
◆ k
q n
z n
.
Terms ` + 1 through n of this sum go to zero, and so the sequence converges to q 1
z 1
+ · · · + q `
z `
. Thus, when c 1
= c 2
, we still have convergence, but the limit to which the sequence
converges might now depend on the choice of the initial vector hh0i (and particularly its inner
product with each of z 1
, . . . , z `
). We should emphasize, though, that in practice, with real
and su�ciently large hyperlink structures, one essentially always gets a matrix M with the
property that M M T has |c 1
| > |c 2
|. Finally, we observe that while this whole discussion has been in terms of the sequence
of hub vectors, it can be adapted directly to analyze the sequence of authority vectors as
well. For the authority vectors, we are looking at powers of (M T M ), and so the basic result
is that the vector of authority scores will converge to an eigenvector of the matrix M T M
associated with its largest eigenvalue.
B. Spectral Analysis of PageRank
The analysis we’ve just seen emphasizes how eigenvectors arise naturally as the limits of
repeated improvement. We now discuss how PageRank can be similarly analyzed using
matrix-vector multiplication and eigenvectors.
Recall that like hub and authority scores, the PageRank of a node is a numerical quantity
that is repeatedly refined using an update rule. Let’s start by thinking about the Basic
PageRank Update Rule from Section 14.3, and then move on to the scaled version. Under
the basic rule, each node takes its current PageRank and divides it equally over all the nodes
it points to. This suggests that the “flow” of PageRank specified by the update rule can be
naturally represented using a matrix N as depicted in Figure 14.13: we define N ij
to be the
share of i’s PageRank that j should get in one update step. This means that N ij
= 0 if i
14.6. ADVANCED MATERIAL: SPECTRAL ANALYSIS, RANDOM WALKS, AND WEB SEARCH425
node 1 0 1/2 0 1/2
0 0 1/2 1/2
1 0 0 0
0 0 1 0
node 4node 3
node 2
Figure 14.13: The flow of PageRank under the Basic PageRank Update Rule can be repre- sented using a matrix N derived from the adjacency matrix M : the entry N
ij
specifies the portion of i’s PageRank that should be passed to j in one update step.
doesn’t link to j, and otherwise N ij
is the reciprocal of the number of nodes that i points
to. In other words, when i links to j, then N ij
= 1/` i
, where ` i
is the number of links out of
i. (If i has no outgoing links, then we define N ii
= 1, in keeping with the rule that a node
with no outgoing links passes all its PageRank to itself.) In this way, N is similar in spirit
to the adjacency matrix M , but with a di↵erent definition when i links to j.
Now, let’s represent the PageRanks of all nodes using a vector r, where the coordinate
r i
is the PageRank of node i. Using this notation, we can write the Basic PageRank Update
Rule as
r i
N 1i
r 1
+ N 2i
r 2
+ · · · + N ni
r n
. (14.5)
This corresponds to multiplication by the transpose of the matrix, just as we saw for the
Authority Update Rule; thus, Equation (14.5) can be written as
r N T r. (14.6)
The Scaled PageRank Update Rule can be represented in essentially the same way, but
with a di↵erent matrix Ñ to represent the di↵erent flow of PageRank, as indicated in Fig-
ure 14.14. Recall that in the scaled version of the update rule, the updated PageRank is
scaled down by a factor of s, and the residual 1�s units are divided equally over all nodes. Thus, we can simply define Ñ
ij
to be sN ij
+ (1 � s)/n, and then the scaled update rule can be written as
r i
Ñ 1i
r 1
+ Ñ 2i
r 2
+ · · · + Ñ ni
r n
. (14.7)
426 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
node 1 .05 .45 .05 .45
.05 .05 .45 .45
.85 .05 .05 .05
.05 .05 .85 .05
node 4node 3
node 2
Figure 14.14: The flow of PageRank under the Scaled PageRank Update Rule can also be represented using a matrix derived from the adjacency matrix M (shown here with scaling factor s = 0.8). We denote this matrix by Ñ ; the entry Ñ
ij
specifies the portion of i’s PageRank that should be passed to j in one update step.
or equivalently
r Ñ T r. (14.8)
Repeated Improvement Using the Scaled PageRank Update Rule. As we apply
the scaled update rule repeatedly, starting from an initial PageRank vector rh0i, we produce
a sequence of vectors rh1i, rh2i, . . . where each is obtained from the previous via multiplication
by Ñ T . Thus, unwinding this process, we see that
rhki = (Ñ T )krh0i.
Moreover, since PageRank is conserved as it is updated — that is, the sum of the PageRanks
at all nodes remains constant through the application of the scaled update rule — we don’t
have to worry about normalizing these vectors as we proceed.
So by analogy with the limiting values of the hub-authority computation (but with the
added fact that normalization isn’t needed), one expects that if the Scaled PageRank Update
Rule converges to a limiting vector rh⇤i, this limit should satisfy Ñ T rh⇤i = rh⇤i — that is, we
should expect rh⇤i to be an eigenvector of Ñ T with corresponding eigenvalue 1. Such an rh⇤i
has the property that it will not change under further refinements by the Scaled PageRank
Update Rule.
In fact, all this turns out to be true: repeated application of the Scaled PageRank Update
Rule converges to precisely such an rh⇤i. To prove this, however, we can’t use the same
14.6. ADVANCED MATERIAL: SPECTRAL ANALYSIS, RANDOM WALKS, AND WEB SEARCH427
approach that we applied in the case of the hub-authority computation: there, the matrices
involved (M M T and M T M ) were symmetric, and so they had eigenvalues that were real
numbers and orthogonal eigenvectors that formed a basis. In general, for matrices such as
Ñ that are not symmetric, the eigenvalues can be complex numbers, and the eigenvectors
may have less clean relationships to one another.
Convergence of the Scaled PageRank Update Rule. Fortunately, for matrices such
as Ñ in which all entries are positive (i.e. Ñ ij
> 0 for all entries Ñ ij
), we can use a powerful
result known as Perron’s Theorem [268]. For our purposes, Perron’s Theorem says that any
matrix P in which all entries are positive has the following properties.
(i) P has a real eigenvalue c > 0 such that c > |c0| for all other eigenvalues c0.
(ii) There is an eigenvector y with positive real coordinates corresponding to the largest
eigenvalue c, and y is unique up to multiplication by a constant.
(iii) If the largest eigenvalue c is equal to 1, then for any starting vector x 6= 0 with non- negative coordinates, the sequence of vectors P kx converges to a vector in the direction
of y as k goes to infinity.
Interpreted in terms of the (scaled) version of PageRank, Perron’s Theorem tells us that
there is a unique vector y that remains fixed under the application of the scaled update rule,
and that repeated application of the update rule from any starting point will converge to y.
This vector y thus corresponds to the limiting PageRank values we have been seeking.
C. Formulation of PageRank Using Random Walks
To close this chapter, we consider how to formulate PageRank in terms of a random walk
on the nodes of the network, following the discussion at the end of Section 14.3.
First let’s make the description of the random walk precise. A walker chooses a starting
node at random, picking each node with equal probability. (When a random choice is made
with equal probability over the options, we will say it is made uniformly at random.) Then, in
each step, the walker follows an outgoing link selected uniformly at random from its current
node, and it moves to the node that this link points to. In this way, a random path through
the graph is constructed one node at a time.
Let’s ask the following question: if b 1
, b 2
, . . . , b n
denote the probabilities of the walk being
at nodes 1, 2, . . . , n respectively in a given step, what is the probability it will be at node i
in the next step? We can answer this by reasoning as follows.
1. For each node j that links to i, if we are given that the walk is currently at node j,
then there is a 1/` j
chance that it moves from j to i in the next step, where ` j
is the
number of links out of j.
428 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
2. The walk has to actually be at node j for this to happen, so node j contributes
b j
(1/` j
) = b j
/` j
to the probability of being at i in the next step.
3. Therefore, summing b j
/` j
over all nodes j that link to i gives the probability the walk
is at b i
in the next step.
So the overall probability that the walk is at i in the next step is the sum of b j
/` j
over
all nodes that link to i. We can use the matrix N defined in the analysis of PageRank to
write this update to the probability b i
as follows:
b i
N 1i
b 1
+ N 2i
b 2
+ · · · + N ni
b n
. (14.9)
If we represent the probabilities of being at di↵erent nodes using a vector b, where the
coordinate b i
is the probability of being at node i, then this update rule can be written using
matrix-vector multiplication by analogy with what we did in our earlier analyses:
b N T b. (14.10)
What we discover is that this is exactly the same as the Basic PageRank Update rule from
Equation (14.6). Since both PageRank values and random-walk probabilities start out the
same (they are initially 1/n for all nodes), and they then evolve according to exactly the same
rule, they remain the same forever. This justifies the claim that we made in Section 14.3:
Claim: The probability of being at a page X after k steps of this random walk is
precisely the PageRank of X after k applications of the Basic PageRank Update
Rule.
And this makes intuitive sense. Like PageRank, the probability of being at a given node in
a random walk is something that gets divided up evenly over all the outgoing links from a
given node, and then passed on to the nodes at the other ends of these links. In other words,
probability and PageRank both flow through the graph according to the same process.
A Scaled Version of the Random Walk. We can also formulate an interpretation of
the Scaled PageRank Update Rule in terms of random walks. As suggested at the end of
Section 14.3, this modified walk works as follows, for a number s > 0: With probability s,
the walk follows a random edge as before; and with probability 1 � s it jumps to a node chosen uniformly at random.
Again, let’s ask the following question: if b 1
, b 2
, . . . , b n
denote the probabilities of the
walk being at nodes 1, 2, . . . , n respectively in a given step, what is the probability it will be
at node i in the next step? The probability of being at node i will now be the sum of sb j
/` j
,
over all nodes j that link to i, plus (1 � s)/n. If we use the matrix Ñ from our analysis of the Scaled PageRank Update Rule, then we can write the probability update as
b i
Ñ 1i
b 1
+ Ñ 2i
b 2
+ · · · + Ñ ni
b n
. (14.11)
14.7. EXERCISES 429
or equivalently
b Ñ T b. (14.12)
This is the same as the update rule from Equation (14.8) for the scaled PageRank values.
The random-walk probabilities and the scaled PageRank values start at the same initial
values, and then evolve according to the same update, so they remain the same forever. This
argument shows the following
Claim: The probability of being at a page X after k steps of the scaled random
walk is precisely the PageRank of X after k applications of the Scaled PageRank
Update Rule.
It also establishes that as we let the number of these scaled random-walk steps go to infinity,
the limiting probability of being at a node X is equal to the limiting scaled PageRank value
of X.
14.7 Exercises
1. Show the values that you get if you run two rounds of computing hub and authority
values on the network of Web pages in Figure 14.15. (That is, the values computed by
the k-step hub-authority computation when we choose the number of steps k to be 2.)
Show the values both before and after the final normalization step, in which we divide
each authority score by the sum of all authority scores, and divide each hub score by
the sum of all hub scores. It’s fine to write the normalized scores as fractions rather
than decimals.)
A
B
C
D
E
Figure 14.15:
2. (a) Show the values that you get if you run two rounds of computing hub and authority
values on the network of Web pages in Figure 14.16. (That is, the values computed by
the k-step hub-authority computation when we choose the number of steps k to be 2.)
430 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
F
G
H
C
AD
BE
Figure 14.16: A network of Web pages.
Show the values both before and after the final normalization step, in which we divide
each authority score by the sum of all authority scores, and divide each hub score by
the sum of all hub scores. (We will call the scores obtained after this dividing-down
step the normalized scores. It’s fine to write the normalized scores as fractions rather
than decimals.)
F
G
H
C
AD
BE
Figure 14.17: A network of Web pages.
(b) Due to the symmetry of nodes A and B in part (a), you should have seen that
they get the same authority scores. Now let’s look at what happens to the scores when
14.7. EXERCISES 431
node E, which links to B, decides to link to C as well. This produces the new network
of Web pages shown in Figure 14.17.
Similarly to part (a), show the normalized hub and authority values that each node
gets when you run the 2-step hub-authority computation on the new network in Fig-
ure 14.17.
(c) In (b), which of nodes A or B now has the higher authority score? Give a brief
explanation in which you provide some intuition for why the di↵erence in authority
scores between A and B in (b) turned out the way it did.
3. In Chapter 14, we discussed the fact that designers of Web content often reason ex-
plicitly about how to create pages that will score highly on search engine rankings. In
a scaled-down setting, this question explores some reasoning in that style.
(a) Show the values that you get if you run two rounds of computing hub and authority
values on the network of Web pages in Figure 14.18. (That is, the values computed by
the k-step hub-authority computation when we choose the number of steps k to be 2.)
Show the values both before and after the final normalization step, in which we divide
each authority score by the sum of all authority scores, and divide each hub score by
the sum of all hub scores. (We will call the scores obtained after this dividing-down
step the normalized scores. It’s fine to write the normalized scores as fractions rather
than decimals.)
C
D
E
A
B
F
Figure 14.18:
(b) Now we come to the issue of creating pages so as to achieve large authority scores,
given an existing hyperlink structure.
432 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
In particular, suppose you wanted to create a new Web page X, and add it to the
network in Figure 14.18, so that it could achieve a (normalized) authority score that
is as large as possible. One thing you might try is to create a second page Y as well,
so that Y links to X and thus confers authority on it. In doing this, it’s natural to
wonder whether it helps or hurts X’s authority to have Y link to other nodes as well.
Specifically, suppose you add X and Y to the network in Figure 14.18. In order to add
X and Y to this network, one needs to specify what links they will have. Here are two
options; in the first option, Y links only to X, while in the second option, Y links to
other strong authorities in addition to X.
• Option 1: Add new nodes X and Y to Figure 14.18; create a single link from Y to X; create no links out of X.
• Option 2: Add new nodes X and Y to Figure 14.18; create links from Y to each of A, B, and X; create no links out of X.
For each of these two options, we’d like to know how X fares in terms of its authority
score. So, for each option, show the normalized authority values that each of A, B, and
X get when you run the 2-step hub-authority computation on the resulting network
(as in part (a)). (That is, you should perform the normalization step where you divide
each authority value down by the total.)
For which of Options 1 or 2 does page X get a higher authority score (taking normal-
ization into account)? Give a brief explanation in which you provide some intuition
for why this option gives X a higher score.
(c) Suppose instead of creating two pages, you create three pages X, Y , and Z, and
again try to strategically create links out of them so that X gets ranked as well as
possible.
Describe a strategy for adding three nodes X, Y , and Z to the network in Figure 14.18,
with choices of links out of each, so that when you run the 2-step hub-authority com-
putation (as in parts (a) and (b)), and then rank all pages by their authority score,
node X shows up in second place.
(Note that there’s no way to do this so that X shows up in first place, so second place
is the best one can hope for using only three nodes X, Y , and Z.)
4. Let’s consider the limiting values that result from the Basic PageRank Update Rule (i.e.
the version where we don’t introduce a scaling factor s). In Chapter 14, these limiting
values are described as capturing “a kind of equilibrium based on direct endorsement:
they are values that remain unchanged when everyone divides up their PageRank and
passes it forward across their out-going links.”
14.7. EXERCISES 433
This description gives a way to check whether an assignment of numbers to a set of
Web pages forms an equilibrium set of PageRank values: the numbers should add up
to 1, and they should remain unchanged when we apply the Basic PageRank Update
Rule. For example, this is illustrated in Chapter 14 via Figure 14.6: you can check
that if we assign a PageRank of 4/13 to page A, 2/13 to each of B and C, and 1/13
to the five other pages, then these numbers add up to 1 and they remain unchanged
when we apply the Basic PageRank Update Rule. Hence they form an equilibrium set
of PageRank values.
For each of the following two networks, use this approach to check whether the numbers
indicated in the figure form an equilibrium set of PageRank values. (In cases where
the numbers do not form an equilibrium set of PageRank values, you do not need to
give numbers that do; you simply need to explain why the given numbers do not.)
B
A
D
E
C
3/10
1/10 1/102/10
3/10
Figure 14.19: A network of Web pages.
(a) Does the assignment of numbers to the nodes in Figure 14.19 form an equilibrium
set of PageRank values for this network of Web pages? Give an explanation for your
answer.
(b) Does the assignment of numbers to the nodes in Figure 14.20 form an equilibrium
set of PageRank values for this network of Web pages? Give an explanation for your
answer.
5. Figure 14.21 depicts the links among 6 Web pages, and also a proposed PageRank
value for each one, expressed as a decimal next to the node.
Are these correct equilibrium values for the Basic PageRank Update Rule? Give a
brief (1-3 sentence) explanation for your answer.
434 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
B
A
C
D
E
G
1/4
1/8 1/8
1/8 1/8
1/4
Figure 14.20: A network of Web pages.
BA
C D E
F
.15 .15
.10 .20 .10
.30
Figure 14.21: A collection of 6 Web pages, with possible PageRank values.
6. One of the basic ideas behind the computation of hubs and authorities is to distinguish
between pages that have multiple reinforcing endorsements and those that simply have
high in-degree. (Recall that the in-degree of a node is the number of links coming into
it.)
Consider for example the graph shown in Figure 14.22. (Despite the fact that it has
two separate pieces, keep in mind that it is a single graph.) The contrast described
above can be seen by comparing node D to nodes B1, B2, and B3: whereas D has
many in-links from nodes that only point to D, nodes B1, B2, and B3 have fewer
in-links each, but from a mutually reinforcing set of nodes.
Let’s explore how this contrast plays out in the context of this stylized example.
14.7. EXERCISES 435
A1
A2
A3
B1
B2
B3
C1
C2
C3
C4
C5
D
Figure 14.22:
(a) Show the values you get from running the 2-step hub-authority computation from
the chapter on link analysis. (If you want, you can omit the final step in which the
values are normalized; i.e., you can just leave the values as large numbers.)
(b) Give formulas, in terms of k, for the values at each node that you get from running
the k-step hub-authority computation. (Again, if you want, you can omit the final
step in which the values are normalized, and give the formulas in terms of k without
normalization.)
(c) As k goes to infinity, what do the normalized values at each node converge to?
Give an explanation for your answer; this explanation does not have to constitute a
formal proof, but it should argue at least informally why the process is converging
to the values you claim. In addition to your explanation of what’s happening in
the computation, briefly discuss (in 1-2 sentences) how this relates to the intuition
suggested in the opening paragraph of this problem, about the di↵erence between
pages that have multiple reinforcing endorsements and those that simply have high
in-degree.
436 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Chapter 15
Sponsored Search Markets
15.1 Advertising Tied to Search Behavior
The problem of Web search, as traditionally formulated, has a very “pure” motivation: it
seeks to take the content people produce on the Web and find the pages that are most
relevant, useful, or authoritative for any given query. However, it soon became clear that
a lucrative market existed within this framework for combining search with advertising,
targeted to the queries that users were issuing.
The basic idea behind this is simple. Early Web advertising was sold on the basis of
“impressions,” by analogy with the print ads one sees in newspapers or magazines: a company
like Yahoo! would negotiate a rate with an advertiser, agreeing on a price for showing its
ad a fixed number of times. But if the ad you’re showing a user isn’t tied in some intrinsic
way to their behavior, then you’re missing one of the main benefits of the Internet as an
advertising venue, compared to print or TV. Suppose for example that you’re a very small
retailer who’s trying to sell a specialized product; say, for example, that you run a business
that sells calligraphy pens over the Web. Then paying to display ads to the full Internet-
using population seems like a very ine�cient way to find customers; instead, you might want
to work out an agreement with a search engine that said, “Show my ad to any user who
enters the query ‘calligraphy pens.’ ” After all, search engine queries are a potent way to
get users to express their intent — what it is that they’re interested in at the moment they
issue their query — and an ad that is based on the query is catching a user at precisely this
receptive moment.
Originally pioneered by the company Overture, this style of keyword-based advertising
has turned out to be an enormously successful way for search engines to make money. At
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
437
438 CHAPTER 15. SPONSORED SEARCH MARKETS
Figure 15.1: Search engines display paid advertisements (shown on the right-hand side of the page in this example) that match the query issued by a user. These appear alongside the results determined by the search engine’s own ranking method (shown on the left-hand side). An auction procedure determines the selection and ordering of the ads.
present it’s a business that generates tens of billions of dollars per year in revenue, and it is
responsible, for example, for nearly all of Google’s revenue. From our perspective, it’s also
a very nice blend of ideas that have come up earlier in this book: it creates markets out of
the information-seeking behavior of hundreds of millions of people traversing the Web; and
we will see shortly that it has surprisingly deep connections to the kinds of auctions and
matching markets that we discussed in Chapters 9 and 10.
Keyword-based ads show up on search engine results pages alongside the unpaid (“or-
ganic” or “algorithmic”) results. Figure 15.1 shows an example of how this currently looks
on Google for the query “Keuka Lake,” one of the Finger Lakes in upstate New York. The
algorithmic results generated by the search engine’s internal ranking procedure are on the
left, while the paid results (in this case for real estate and vacation rentals) are ordered on
the right. There can be multiple paid results for a single query term; this simply means that
the search engine has sold an ad on the query to multiple advertisers. Among the multiple
slots for displaying ads on a single page, the slots higher up on the page are more expensive,
since users click on these at a higher rate.
The search industry has developed certain conventions in the way it sells keyword-based
15.1. ADVERTISING TIED TO SEARCH BEHAVIOR 439
ads, and for thinking about this market it’s worth highlighting two of these at the outset.
Paying per click. First, ads such as those shown in Figure 15.1 are based on a cost-per-
click (CPC) model. This means that if you create an ad that will be shown every time a
user enters the query “Keuka Lake,” it will contain a link to your company’s Web site —
and you only pay when a user actually clicks on the ad. Clicking on an ad represents an
even stronger indication of intent than simply issuing a query; it corresponds to a user who
issued the query, read your ad, and is now visiting your site. As a result, the amount that
advertisers are willing to pay per click is often surprisingly high. For example, to occupy
the most prominent spot for “calligraphy pens” costs about $1.70 per click on Google as of
this writing; occupying the top spot for “Keuka Lake” costs about $1.50 per click. (For the
misspelling “calligaphy pens,” the cost is still about $0.60 per click — after all, advertisers
are still interested in potential customers even if their query contains a small but frequent
typo.)
For some queries, the cost per click can be positively stratospheric. Queries like “loan
consolidation,” “mortgage refinancing,” and “mesothelioma” often reach $50 per click or
more. One can take this as an advertiser’s estimate that it stands to gain an expected value
of $50 for every user who clicks through such an ad to its site.1
Setting prices through an auction. There is still the question of how a search engine
should set the prices per click for di↵erent queries. One possibility is simply to post prices, the
way that products in a store are sold. But with so many possible keywords and combinations
of keywords, each appealing to a relatively small number of potential advertisers, it would
essentially be hopeless for the search engine to maintain reasonable prices for each query in
the face of changing demand from advertisers.
Instead, search engines determine prices using an auction procedure, in which they solicit
bids from the advertisers. If there were a single slot in which an ad could be displayed, then
this would be just a single-item auction such as we saw in Chapter 9, and there we saw
that the sealed-bid second-price auction had many appealing features. The problem is more
complicated in the present case, however, since there are multiple slots for displaying ads,
and some are more valuable than others.
We will consider how to design an auction for this setting in several stages.
(1) First, if the search engine knew all the advertisers’ valuations for clicks, the situation
could be represented directly as a matching market in the style that we discussed in
1Naturally, you may be wondering at this point what mesothelioma is. As a quick check on Google reveals, it’s a rare form of lung cancer that is believed to be caused by exposure to asbestos in the workplace. So if you know enough to be querying this term, you may well have been diagnosed with mesothelioma, and are considering suing your employer. Most of the top ads for this query link to law firms.
440 CHAPTER 15. SPONSORED SEARCH MARKETS
Chapter 10 — essentially, the slots are the items being sold, and they’re being matched
with the advertisers as buyers.
(2) If we assume that the advertisers’ valuations are not known, however, then we need
to think about ways of encouraging truthful bidding, or to deal with the consequences
of untruthful bidding. This leads us directly to an interesting general question that
long predates the specific problem of keyword-based advertising: how do you design a
price-setting procedure for matching markets in which truthful bidding is a dominant
strategy for the buyers? We will resolve this question using an elegant procedure called
the Vickrey-Clarke-Groves (VCG) mechanism [112, 199, 400], which can be viewed as
a far-reaching generalization of the second-price rule for single-item auctions that we
discussed in Chapter 9.
(3) The VCG mechanism provides a natural way to set prices in matching markets, in-
cluding those arising from keyword-based advertising. For various reasons, however,
it is not the procedure that the search industry adopted. As a result, our third topic
will be an exploration of the auction procedure that is used to sell search advertising
in practice, the Generalized Second-Price Auction (GSP). We will see that although
GSP has a simple description, the bidding behavior it leads to is very complex, with
untruthful bidding and socially non-optimal outcomes. Trying to understand bidder
behavior under this auction turns out to be an interesting case study in the intricacies
of a complex auction procedure as it is implemented in a real application.
15.2 Advertising as a Matching Market
Clickthrough Rates and Revenues Per Click. To begin formulating a precise descrip-
tion of how search advertising is sold, let’s consider the set of available “slots” that the
search engine has for selling ads on a given query, like the three advertising slots shown in
Figure 15.1. The slots are numbered 1, 2, 3, . . . starting from the top of the page, and users
are more likely to click on the higher slots. We will assume that each slot has a specific
clickthrough rate associated with it — this is the number of clicks per hour that an ad placed
in that slot will receive.
In the models we discuss, we will make a few simplifying assumptions about the click-
through rates. First, we assume that advertisers know the clickthrough rates. Second, we
assume that the clickthrough rate depends only on the slot itself and not on the ad that is
placed there. Third, we assume that the clickthrough rate of a slot also doesn’t depend on
the ads that are in other slots. In practice, the first of these assumptions is not particularly
problematic since advertisers have a number of means (including tools provided by the search
engine itself) for estimating clickthrough rates. The second assumption is an important is-
15.2. ADVERTISING AS A MATCHING MARKET 441
a
b
c
x
y
z
3
2
1
slots advertisers revenues
per click
clickthrough
rates
10
5
2
Figure 15.2: In the basic set-up of a search engine’s market for advertising, there are a certain number of advertising slots to be sold to a population of potential advertisers. Each slot has a clickthrough rate: the number of clicks per hour it will receive, with higher slots generally getting higher clickthrough rates. Each advertisers has a revenue per click, the amount of money it expects to receive, on average, each time a user clicks on one of its ads and arrives at its site. We draw the advertisers in descending order of their revenue per click; for now, this is purely a pictorial convention, but in Section 15.2 we will show that the market in fact generally allocates slots to the advertisers in this order.
sue: a relevant, high-quality ad in a high slot will receive more clicks than an o↵-topic ad,
and in fact we will describe how to extend the basic models to deal with ad relevance and ad
quality at the end of the chapter. The third assumption — interaction among the di↵erent
ads being shown — is a more complex issue, and it is still not well understood even within
the search industry.
This is the full picture from the search engine’s side: the slots are the inventory that it is
trying to sell. Now, from the advertisers’ side, we assume that each advertiser has a revenue
per click: the expected amount of revenue it receives per user who clicks on the ad. Here too
we will assume that this value is intrinsic to the advertiser, and does not depend on what
was being shown on the page when the user clicked on the ad.
This is all the information we need to understand the market for a particular keyword:
the clickthrough rates of the slots, and the revenues per click of the advertisers. Figure 15.2
shows a small example with three slots and three advertisers: the slots have clickthrough
rates of 10, 5, and 2 respectively, while the advertisers have revenues per click of 3, 2, and 1
respectively.
442 CHAPTER 15. SPONSORED SEARCH MARKETS
a
b
c
x
y
z
30, 15, 6
20, 10, 4
10, 5, 2
slots advertisers valuations
(a) Advertisers’ valuations for the slots
a
b
c
x
y
z
30, 15, 6
20, 10, 4
10, 5, 2
slots advertisers valuationsprices
13
3
0
(b) Market-clearing prices for slots
Figure 15.3: The allocation of advertising slots to advertisers can be represented as a match- ing market, in which the slots are the items to be sold, and the advertisers are the buyers. An advertiser’s valuation for a slot is simply the product of its own revenue per click and the clickthrough rate of the slot; these can be used to determine market-clearing prices for the slots.
Constructing a Matching Market. We now show how to represent the market for a
particular keyword as a matching market of the type we studied in Chapter 10. To do this,
it is useful to first review the basic ingredients of matching market from Chapter 10.
• The participants in a matching market consist of a set of buyers and a set of sellers.
• Each buyer j has a valuation for the item o↵ered by each seller i. This valuation can depend on the identities of both the buyer and the seller, and we denote it v
ij
.
• The goal is to match up buyers with sellers, in such a way that no buyer purchases two di↵erent items, and the same item isn’t sold to two di↵erent buyers.
To cast the search engine’s advertising market for a particular keyword in this framework,
we use r i
to denote the clickthrough rate of slot i, and v j
to denote the revenue per click of
advertiser j. The benefit that advertiser j receives from being shown in slot i is then just
r i
v j
, the product of the number of clicks and the revenue per click.
In the language of matching markets, this is advertiser j’s valuation v ij
for slot i — that
is, the value it receives from acquiring slot i. So by declaring the slots to be the sellers,
the advertisers to be the buyers, and the buyers’ valuations to be v ij
= r i
v j
, the problem
of assigning slots to advertisers is precisely the problem of assigning sellers to buyers in a
matching market. In Figure 15.3(a), we show how this conversion is applied to the example in
Figure 15.2, yielding the buyer valuations shown. As this figure makes clear, the advertising
set-up produces a matching market with a special structure: since the valuations are obtained
15.2. ADVERTISING AS A MATCHING MARKET 443
by multiplying rates by revenues, we have a situation where all the buyers agree on their
preferences for the items being sold, and where in fact the valuations of one buyer simply
form a multiple of the valuations of any other buyer.
When we considered matching markets in Chapter 10, we focused on the special case
in which the number of sellers and the number of buyers were the same. This made the
discussion simpler in a number of respects; in particular, it meant that the buyers and sellers
could be perfectly matched, so that each item is sold, and each buyer purchases exactly one
item. We will make the analogous assumption here: with slots playing the role of sellers
and advertisers playing the role of buyers, we will focus on the case in which the numbers
of slots and advertisers are the same. But it is important to note that this assumption
is not at all essential, because for purposes of analysis we can always translate a scenario
with unequal numbers of slots and advertisers into an equivalent one with equal numbers, as
follows. If there are more advertisers than slots, we simply create additional “fictitious” slots
of clickthrough rate 0 (i.e. of valuation 0 to all buyers) until the number of slots is equal to
the number of advertisers. The advertisers who are matched with the slots of clickthrough
rate 0 are then simply the ones who don’t get assigned a (real) slot for advertising. Similarly,
if there are more slots than advertisers, we just create additional “fictitious” advertisers who
have a valuation of 0 for all slots.
Obtaining Market-Clearing Prices. With the connection to matching markets in place,
we can use the framework from Chapter 10 to determine market-clearing prices. Again, it
is worth reviewing this notion from Chapter 10 in a bit of detail as well, since we will be
using it heavily in what follows. Recall, roughly speaking, that a set of prices charged by
the sellers is market-clearing if, with these prices, each buyer prefers a di↵erent slot. More
precisely, the basic ingredients of market-clearing prices are as follows.
• Each seller i announces a price p i
for his item. (In our case, the items are the slots.)
• Each buyer j evaluates her payo↵ for choosing a particular seller i: it is equal to the valuation minus the price for this seller’s item, v
ij
� p i
.
• We then build a preferred-seller graph as in Figure 15.3(b) by linking each buyer to the seller or sellers from which she gets the highest payo↵.
• The prices are market-clearing if this graph has a perfect matching: in this case, we can assign distinct items to all the buyers in such a way that each buyer gets an item
that maximizes her payo↵.
In Chapter 10, we showed that market-clearing prices exist for every matching market, and
we gave a procedure to construct them. We also showed in Chapter 10 that the assignment
444 CHAPTER 15. SPONSORED SEARCH MARKETS
of buyers to sellers achieved by market-clearing prices always maximizes the buyers’ total
valuations for the items they get.
Returning to the specific context of advertising markets, market-clearing prices for the
search engine’s advertising slots have the desirable property that advertisers prefer di↵erent
slots, and the resulting assignment of advertisers to slots maximizes the total valuations of
each advertiser for what they get. (Again, see Figure 15.3(b).) In fact, it is not hard to work
out that when valuations have the special form that we see in advertising markets — each
consisting of a clickthrough rate times a revenue per click — then the maximum valuation
is always obtained by giving the slot with highest clickthrough rate to the advertiser with
maximum revenue per click, the slot with second highest rate to the advertiser with second
highest revenue per click, and so forth.
The connection with matching markets shows that we can in fact think about advertising
prices in the more general case where di↵erent advertisers can have arbitrary valuations for
slots — they need not be the product of a clickthrough rate and a revenue per click. This
allows advertisers, for example, to express how they feel about users who arrive via an ad
in the third slot compared with those who arrive via an ad in the first slot. (And indeed, it
is reasonable to believe that these two populations of users might have di↵erent behavioral
characteristics.)
Finally, however, this construction of prices can only be carried out by a search engine if
it actually knows the valuations of the advertisers. In the next section we consider how to
set prices in a setting where the search engine doesn’t know these valuations; it must rely
on advertisers to report them without being able to know whether this reporting is truthful.
15.3 Encouraging Truthful Bidding in Matching Mar- kets: The VCG Principle
What would be a good price-setting procedure when the search engine doesn’t know the
advertisers’ valuations? In the early days of the search industry, variants of the first-price
auction were used: advertisers were simply asked to report their revenues per click in the
form of bids, and then they were assigned slots in decreasing order of these bids. Recall from
Chapter 9 that when bidders are simply asked to report their values, they will generally
under-report, and this is what happened here. Bids were shaded downward, below their true
values; and beyond this, since the auctions were running continuously over time, advertisers
constantly adjusted their bids by small increments to experiment with the outcome and
to try slightly outbidding competitors. This resulted in a highly turbulent market and a
huge resource expenditure on the part of both the advertisers and the search engines, as the
constant price experimentation led to prices for all queries being updated essentially all the
time.
15.3. ENCOURAGING TRUTHFUL BIDDING IN MATCHING MARKETS: THE VCG PRINCIPLE445
In the case of a single-item auction, we saw in Chapter 9 that these problems are handled
by running a second-price auction, in which the single item is awarded to the highest bidder at
a price equal to the second-highest bid. As we showed there, truthful bidding is a dominant
strategy for second-price auctions — that is, it is at least as good as any other strategy,
regardless of what the other participants are doing. This dominant strategy result means
that second-price auctions avoid many of the pathologies associated with more complex
auctions.
But what is the analogue of the second-price auction for advertising markets with multiple
slots? Given the connections we’ve just seen to matching markets in the previous section,
this turns out to be a special case of an interesting and fundamental question: how can we
define a price-setting procedure for matching markets so that truthful reporting of valuations
is a dominant strategy for the buyers? Such a procedure would be a massive generalization
of the second-price auction, which — though already fairly subtle — only applies to the case
of single items.
The VCG Principle. Since a matching market contains many items, it is hard to directly
generalize the literal description of the second-price single-item auction, in which we assign
the item to the highest bidder at the second-highest price. However, by viewing the second-
price auction in a somewhat less obvious way, we get a principle that does generalize.
This view is the following. First, the second-price auction produces an allocation that
maximizes social welfare — the bidder who values the item the most gets it. Second, the
winner of the auction is charged an amount equal to the “harm” he causes the other bidders
by receiving the item. That is, suppose the bidders’ values for the item are v 1
, v 2
, v 3
, . . . , v n
in decreasing order. Then if bidder 1 were not present, the item would have gone to bidder 2,
who values it at v 2
. The other bidders still would not get the item, even if bidder 1 weren’t
there. Thus, bidders 2 through n collectively experience a harm of v 2
because bidder 1 is
there — since bidder 2 loses this much value, and bidders 3 through n are una↵ected. This
harm of v 2
is exactly what bidder 1 is charged. Indeed, the other bidders are also charged
an amount equal to the harm they cause to others — in this case, zero, since no bidder is
a↵ected by the presence of any of bidders 2 through n in the single-item auction.
Again, this is a non-obvious way to think about single-item auctions, but it is a principle
that turns out to encourage truthful reporting of values in much more general situations:
each individual is charged the harm they cause to the rest of the world. Or to put it
another way, each individual is charged a price equal to the total amount better o↵ everyone
else would be if this individual weren’t there. We will refer to this as the Vickrey-Clarke-
Groves (VCG) principle, after the work of Clarke and Groves, who generalized the central
idea behind Vickrey’s second-price auction for single items [112, 199, 400]. For matching
markets, we will describe an application of this principle due to Herman Leonard [270] and
446 CHAPTER 15. SPONSORED SEARCH MARKETS
Gabrielle Demange [128]; it develops a pricing mechanism in this context that causes buyers
to reveal their valuations truthfully.
Applying the VCG Principle to Matching Markets. In a matching market, we have
a set of buyers and a set of sellers — with equal numbers of each — and buyer j has a
valuation of v ij
for the item being sold by seller i.2 We are assuming here that each buyer
knows her own valuations, but that these valuations are not known to the other buyers or
to the sellers. Also, we assume that each buyer only cares which item she receives, not
about how the remaining goods are allocated to the other buyers. Thus, in the language of
auctions, the buyers have independent, private values.
Under the VCG principle, we first assign items to buyers so as to maximize total valuation.
Then, the price buyer j should pay for seller i’s item — in the event she receives it — is the
harm she causes to the remaining buyers through her acquisition of this item. This is equal
to the total boost in valuation everyone else would get if we computed the optimal matching
without buyer j present. To give a better sense of how this principle works for matching
markets, we first walk through how it would apply to the example in Figure 15.3. We then
define the VCG price-setting procedure in general, and in the next section we show that it
yields truth-telling as a dominant strategy — for each buyer, truth-telling is at least as good
as any other option, regardless of what the other buyers are doing.
In Figure 15.3, where the buyers are advertisers and the items are advertising slots,
suppose we assign items to maximize total valuation: item a to buyer x, item b to buyer
y, and item c to buyer z. What prices does the VCG principle dictate for each buyer? We
show the reasoning in Figure 15.4.
• First, in the optimal matching without buyer x present, buyer y gets item a and buyer z gets item b. This improves the respective valuations of y and z for their assigned
items by 20 � 10 = 10 and 5 � 2 = 3 respectively. The total harm caused by x is therefore 10 + 3 = 13, and so this is the price that x should pay.
• In the optimal matching without buyer y present, buyer x still gets a (so she is un- a↵ected), while buyer z gets item b, for an improved valuation of 3. The total harm
caused by y is 0 + 3 = 3, and so this is the price that y should pay.
• Finally, in the optimal matching without buyer z present, buyers x and y each get the same items they would have gotten had z been there. z causes no harm to the rest of
the world, and so her VCG price is 0.
With this example in mind, we now describe the VCG prices for a general matching
market. This follows exactly from the principle we’ve been discussing, but it requires a bit
2As always, we can handle unequal numbers of buyers and sellers by creating “fictitious” individuals and valuations of 0, as in Section 15.2.
15.3. ENCOURAGING TRUTHFUL BIDDING IN MATCHING MARKETS: THE VCG PRINCIPLE447
a
b
c
x
y
z
30, 15, 6
20, 10, 4
10, 5, 2
slots advertisers valuations
If x weren't there, y
would do better by
20-10=10, and z would
do better by 5-2=3,
for a total harm of 13.
(a) Determining how much better o↵ y and z would be if x were not present
a
b
c
x
y
z
30, 15, 6
20, 10, 4
10, 5, 2
slots advertisers valuations
If y weren't there, x
would be unaffected,
and z would do better
by 5-2=3, for a total
harm of 3.
(b) Determining how much better o↵ x and z would be if y were not present
Figure 15.4: The VCG price an individual buyer pays for an item can be determined by working out how much better o↵ all other buyers would be if this individual buyer were not present.
of notation due to the multiple items and valuations. First, let S denote the set of sellers and
B denote the set of buyers. Let V S B
denote the maximum total valuation over all possible
perfect matchings of sellers and buyers — this is simply the value of the socially optimal
outcome with all buyers and sellers present.
Now, let S �i denote the set of sellers with seller i removed, and let B�j denote the set of buyers with buyer j removed. So if we give item i to seller j, then the best total valuation
the rest of the buyers could get is V S�i B�j : this is the value of the optimal matching of sellers
and buyers when we’ve taken item i and buyer j out of consideration. On the other hand, if
buyer j simply didn’t exist, but item i were still an option for everyone else, then the best
total valuation the rest of the buyers could get is V S B�j. Thus, the total harm caused by
448 CHAPTER 15. SPONSORED SEARCH MARKETS
buyer j to the rest of the buyers is the di↵erence between how they’d do without j present
and how they do with j present; in other words, it is the di↵erence V S B�j �V
S�i B�j . This is the
VCG price p ij
that we charge to buyer j for item i, so we have the equation
p ij
= V S B�j � V
S�i B�j . (15.1)
The VCG Price-Setting Procedure. Using the ideas developed so far, we can now
define the complete VCG price-setting procedure for matching markets. We assume that
there is a single price-setting authority (an “auctioneer”) who can collect information from
buyers, assign items to them, and charge prices. Fortunately, this framework works very well
for selling advertising slots, where all the items (the slots) are under the control of a single
agent (the search engine).
The procedure is as follows:
1. Ask buyers to announce valuations for the items. (These announcements need not be
truthful.)
2. Choose a socially optimal assignment of items to buyers — that is, a perfect matching
that maximizes the total valuation of each buyer for what they get. This assignment
is based on the announced valuations (since that’s all we have access to.)
3. Charge each buyer the appropriate VCG price: that is, if buyer j receives item i
under the optimal matching, then charge buyer j a price p ij
determined according to
Equation (15.1).
Essentially, what the auctioneer has done is to define a game that the buyers play: they
must choose a strategy (a set of valuations to announce), and they receive a payo↵: their
valuation for the item they get, minus the price they pay. What turns out to be true, though
it is far from obvious, is that this game has been designed to make truth-telling — in which
a buyer announces her true valuations — a dominant strategy. We will prove this in the
next section; but before this, we make a few observations.
First, notice that there’s a crucial di↵erence between the VCG prices defined here, and
the market-clearing prices arising from the auction procedure in Chapter 10. The market-
clearing prices defined there were posted prices, in that the seller simply announced a price
and was willing to charge it to any buyer who was interested. The VCG prices here, on the
other hand, are personalized prices: they depend on both the item being sold and the buyer
it is being sold to, The VCG price p ij
paid by buyer j for item i might well di↵er, under
Equation (15.1), from the VCG price p ik
that buyer k would pay if it were assigned item i.3
3Despite this, there is are deep and subtle connections between the two kinds of prices; we explore this issue further in the final section of this chapter.
15.4. ANALYZING THE VCG PROCEDURE: TRUTH-TELLING AS A DOMINANT STRATEGY449
Another way to think about the relationship between the market-clearing prices from
Chapter 10 and the VCG prices here is to observe how each is designed to generalize a
di↵erent single-item auction format. The market-clearing prices in Chapter 10 were defined
by a significant generalization of the ascending (English) auction: prices were raised step-
by-step until each buyer favored a di↵erent item, and we saw in Section 10.5 that one could
encode the single-item ascending auction as a special case of the general construction of
market-clearing prices.
The VCG prices, on the other hand, are defined by an analogous and equally substantial
generalization of the sealed-bid second-price auction. At a qualitative level, we can see the
“harm-done-to-others” principle is behind both the second-price auction and the VCG prices,
but in fact we can also see fairly directly that the second-price auction is a special case of
the VCG procedure. Specifically, suppose there are n buyers who each want a single item,
and buyer i has valuation v i
for it, where the numbers v i
are sorted in descending order so
that v 1
is the largest. Let’s turn this into a matching market with n buyers and n sellers by
simply adding n�1 fictitious items; all buyers have valuation 0 for each fictitious item. Now, if everyone reports their values truthfully, then the VCG procedure will assign item 1 (the
real item — the only one with any value) to buyer 1 (who has the highest valuation), and
all the rest of the buyers would get fictitious items of zero value. What price should buyer
1 pay? According to Equation (15.1), she should pay V S B�1 � V
S�1 B�1 . The first term is buyer
2’s valuation, since with buyer 1 gone the socially optimal matching gives item 1 to buyer
2. The second term is 0, since with both buyer 1 and item 1 gone, there are no remaining
items of any value. Thus, buyer 1 pays buyer 2’s valuation, and so we have precisely the
pricing rule for second-price sealed-bid auctions.
15.4 Analyzing the VCG Procedure: Truth-Telling as a Dominant Strategy
We now show that the VCG procedure encourages truth-telling in a matching market. Con-
cretely, we will prove the following claim.
Claim: If items are assigned and prices computed according to the VCG procedure,
then truthfully announcing valuations is a dominant strategy for each buyer, and
the resulting assignment maximizes the total valuation of any perfect matching of
slots and advertisers.
The second part of this claim (that the total valuation is maximized) is easy to justify:
if buyers report their valuations truthfully, then the assignment of items is designed to
maximize the total valuation by definition.
The first part of the claim is the more subtle: Why is truth-telling a dominant strategy?
Suppose that buyer j announces her valuations truthfully, and in the matching we assign
450 CHAPTER 15. SPONSORED SEARCH MARKETS
i j
V S-i
B-j
(a) vij +V S�i B�j is the maximum valuation of any
matching.
h
j
V S-h
B-j
(b) vhj + V S�h B�j is the maximum valuation only
over matchings constrained to assign h to j.
Figure 15.5: The heart of the proof that the VCG procedure encourages truthful bidding comes down to a comparison of the value of two matchings.
her item i. Then her payo↵ is v ij
� p ij
. We want to show that buyer j has no incentive to
deviate from a truthful announcement.
If buyer j decides to lie about her valuations, then one of two things can happen: either
this lie a↵ects the item she gets, or it doesn’t. If buyer j lies but still gets the same item
i, then her payo↵ remains exactly the same, because the price p ij
is computed only using
announcements by buyers other than j. So if a deviation from truth-telling is going to be
beneficial for buyer j, it has to a↵ect the item she receives.
Suppose, therefore, that buyer j lies about her valuations and gets item h instead of item
i. In this case, her payo↵ would be v hj
� p hj
. Notice again that the price p hj
is determined
only by the announcements of buyers other than j. To show that there is no incentive to lie
and receive item h instead of i, we need to show that
v ij
� p ij
� v hj
� p hj
.
15.4. ANALYZING THE VCG PROCEDURE: TRUTH-TELLING AS A DOMINANT STRATEGY451
If we expand out the definitions of p ij
and p hj
using Equation (15.1), this is equivalent to
showing
v ij
� [V S B�j � V
S�i B�j ] � vhj � [V
S
B�j � V S�h B�j ].
Both sides of this inequality contain the term V S B�j, so we can add this to both sides; in this
way, the previous inequality is equivalent to showing
v ij
+ V S�i B�j � vhj + V
S�h B�j . (15.2)
We now argue why this last inequality holds. In fact, both the left-hand side and the
right-hand side describe the total valuation of di↵erent matchings, as shown in Figure 15.5.
The matching on the left-hand side is constructed by pairing j with the item i she would
get in an optimal matching, and then optimally matching the remaining buyers and items.
In other words, it is a matching that achieves the maximum total valuation over all possible
perfect matchings, so we can write the left-hand side as
v ij
+ V S�i B�j = V
S
B
. (15.3)
In contrast, the matching on the right-hand side of Inequality (15.2) is constructed by pairing
j with some other item h, and then optimally matching the remaining buyers and items. So
it is a matching that achieves the maximum total valuation only over those matchings that
pair j with h. Therefore,
v hj
+ V S�h B�j V
S
B
.
The left-hand side of Inequality (15.2), the maximum valuation with no restrictions on who
gets any slot, must be at least as large as the right-hand side, the maximum with a restriction.
And this is what we wanted to show.
Nothing in this argument depends on the decisions made by other buyers about what to
announce. For example, it doesn’t require them to announce their true values; the arguments
comparing di↵erent matchings can be applied to whatever valuations are announced by the
other buyers, with the same consequences. Thus we have shown that truthfully announcing
valuations is a dominant strategy in the VCG procedure.
To close this section, let’s go back to the specific case of keyword-based advertising, in
which the buyers correspond to advertisers and the items for sale correspond to advertising
slots. Our discussion so far has focused on finding and achieving an assignment of advertisers
to slots that maximizes the total valuation obtained by advertisers. But of course, this is
not what the search engine selling the advertising slots directly cares about. Instead it cares
about its revenue: the sum of the prices that it can charge for slots. It is not clear that
the VCG procedure is the best way to generate revenue for the search engine. Determining
which procedure will maximize seller revenue is a current topic of research. It could be
that the best a seller can do is to use some procedure that generates an optimal matching
452 CHAPTER 15. SPONSORED SEARCH MARKETS
— and potentially one that is better than VCG at converting more of the total valuation
into seller revenue. Or it could be that the seller is better o↵ using a procedure that does
not always yield an optimal matching. And it may be that some version of a revenue-
equivalence principle — such as we saw for single-item auctions in Chapter 9 — holds here
as well, showing that certain classes of auction provide equivalent amounts of revenue to the
seller when buyers behave strategically.
In the next sections, we sample the general flavor of some of these revenue issues by
considering the alternative to VCG that the search industry has adopted in practice — a
simple-to-describe auction called the Generalized Second Price auction that induces complex
bidding behavior.
15.5 The Generalized Second Price Auction
After some initial experiments with other formats, the main search engines have adopted a
procedure for selling advertising slots called the Generalized Second Price auction (GSP). At
some level, GSP — like VCG — is a generalization of the second-price auction for a single
item. However, as will see, GSP is a generalization only in a superficial sense, since it doesn’t
retain the nice properties of the second-price auction and VCG.
In the GSP procedure, each advertiser j announces a bid consisting of a single number
b j
— the price it is willing to pay per click. (This would correspond, for example, to the
$1.70 for “calligraphy pens” or $1.50 for “Keuka Lake” that we saw at the beginning of the
chapter.) As usual, it is up to the advertiser whether or not its bid is equal to its true
valuation per click v j
. Then, after each advertiser submits a bid, the GSP procedure awards
each slot i to the ith highest bidder, at a price per click equal to the (i + 1)st highest bid.
In other words, each advertiser who is shown on the results page is paying a price per click
equal to the bid of the advertiser just below them.
So GSP and VCG can be viewed in parallel terms, in that each asks for announced
valuations from the advertisers, and then each uses these announcements to determine an
assignment of slots to advertisers, as well as prices to charge. When there is a single slot,
both are equivalent to the second-price auction. But when there are multiple slots, their
rules for producing prices are di↵erent. VCG’s rule is given by Equation (15.1). GSP’s rule,
when the bids per click are b 1
, b 2
, b 3
, . . . in descending order, is to charge a cumulative price
of r i
b i+1
for slot i. This is because the ith highest bidder will get slot i at a price per click of
b i+1
; multiplying by the clickthrough rate of r i
gives a total price of r i
b i+1
for all the clicks
associated with slot i.
Analyzing GSP. GSP was originally developed at Google; once it had been in use for
a while in the search industry, researchers including Varian [399] and Edelman, Ostrovsky,
15.5. THE GENERALIZED SECOND PRICE AUCTION 453
and Schwarz [144] began working out some of its basic properties. Their analysis formulates
the problem as a game, using the definitions from Chapter 6. Each advertiser is a player, its
bid is its strategy, and its payo↵ is its revenue minus the price it pays. In this game, we will
consider Nash equilibria — we seek sets of bids so that, given these bids, no advertiser has
an incentive to change how it is behaving.4
First, we’ll see that GSP has a number of pathologies that VCG was designed to avoid:
truth-telling might not constitute a Nash equilibrium; there can in fact be multiple possible
equilibria; and some of these may produce assignments of advertisers to slots that do not
maximize total advertiser valuation. On the positive side, we show in the next section
that there is always at least one Nash equilibrium set of bids for GSP, and that among
the (possibly multiple) equilibria, there is always one that does maximize total advertiser
valuation. The analysis leading to these positive results about equilibria builds directly on
the market-clearing prices for the matching market of advertisers and slots, thus establishing
a connection between GSP and market-clearing prices.
Hence, while GSP possesses Nash equilibria, it lacks some of the main nice properties of
the VCG procedure from Sections 15.3 and 15.4. However, in keeping with our discussion
from the end of the last section, the search engines ultimately have an interest in choosing a
procedure that will maximize their revenue (given the behavior of the advertisers in response
to it). Viewed in this light, it is not clear that GSP is the wrong choice, though it is also far
from clear that it is the right choice. As mentioned at the end of Section 15.4, understanding
the revenue trade-o↵s among di↵erent procedures for selling keyword-based advertising is
largely an open question, and the subject of current research.
Truth-telling may not be an equilibrium. It is not hard to make an example to show
that truth-telling may not be an equilibrium when the GSP procedure is used. One example
of this is depicted in Figure 15.6:
• There are two slots for ads, with clickthrough rates of 10 and 4. In the figure, we also show a third fictitious slot of clickthrough rate 0, so as to equalize the number of
advertisers and slots.
• There are three advertisers x, y, and z, with values per click of 7, 6, and 1 respectively.
Now, if each advertiser bids its true valuation, then advertiser x gets the top slot at a
price per click of 6; since there are 10 clicks associated with this slot, x pays a cumulative
price of 6 · 10 = 60 for the slot. Advertiser x’s valuation for the top slot is 7 · 10 = 70, so its 4In order to analyze Nash equilibrium in the bidding game we will assume that each advertiser knows
the values of all other bidders. Otherwise, they do not know the payo↵s to all players in the bidding game and we could not use Nash equilibrium to analyze the game. The motivation for this assumption is that we envision a situation in which these bidders have been bidding against each other repeatedly and have learned each others’ willingnesses to pay for clicks.
454 CHAPTER 15. SPONSORED SEARCH MARKETS
a
b
c
x
y
z
7
6
1
slots advertisers revenues
per click
clickthrough
rates
10
4
0
Figure 15.6: An example of a set of advertisers and slots for which truthful bidding is not an equilibrium in the Generalized Second Price auction. Moreover, this example possesses multiple equilibria, some of which are not socially optimal.
payo↵ is 70�60 = 10. Now, if x were to lower its bid to 5, then it would get the second slot for a price per click of 1, implying a cumulative price of 4 for the slot. Since its valuation
for the second slot is 7 · 4 = 28, this is a payo↵ of 28�4 = 24, which is an improvement over the result of bidding truthfully.
Multiple and non-optimal equilibria. The example in Figure 15.6 turns out to illus-
trate some other complex properties of bidding behavior in GSP. In particular, there is more
than one equilibrium set of bids for this example, and among these equilibria are some that
produce a socially non-optimal assignment of advertisers to slots.
First, suppose that advertiser x bids 5, advertiser y bids 4, and advertiser z bids 2. With
a little e↵ort, we can check that this forms an equilibrium: checking the condition for z is
easy, and the main further things to observe are that x doesn’t want to lower its bid below
4 so as to move to the second slot, and y doesn’t want to raise its bid above 5 to get the
first slot. This is an equilibrium that produces a socially optimal allocation of advertisers to
slots, since x gets slot a, while y gets b and z gets c.
But one can also check that if advertiser x bids 3, advertiser y bids 5, and advertiser z
bids 1, then we also get a set of bids in Nash equilibrium. Again, the main thing to verify is
that x doesn’t want to raise its bid above y’s, and that y doesn’t want to lower its bid below
x’s. This equilibrium is not socially optimal, since it assigns y to the highest slot and x to
the second-highest.
There is much that is not understood in general about the structure of the sub-optimal
equilibria arising from GSP. For example, it is an interesting open question to try quantifying
15.5. THE GENERALIZED SECOND PRICE AUCTION 455
a
b
c
x
y
z
70, 28, 0
60, 24, 0
10, 4, 0
slots advertisers valuations
Figure 15.7: Representing the example in Figure 15.6 as a matching market, with advertiser valuations for the full set of clicks associated with each slot.
how far from social optimality a Nash equilibrium of GSP can be.
The Revenue of GSP and VCG. The existence of multiple equilibria also adds to the
di�culty in reasoning about the search engine revenue generated by GSP, since it depends
on which equilibrium (potentially from among many) is selected by the bidders. In the
example we’ve been working with, we’ll show that depending on which equilibrium of GSP
the advertisers actually use, the revenue to the search engine can be either higher or lower
than the revenue it would collect by charging the VCG prices.
Let’s start by determining the revenue to the search engine from the two GSP equilibria
that we worked out above.
• With bids of 5, 4, and 2, the 10 clicks in the top slot are sold for 4 per click, and the 4 clicks in the second slot are sold for 2 per click, for a total revenue to the search engine
of 48.
• On the other hand, with bids of 3, 5, and 1, the 10 clicks in the top slot are sold for 3 per click, and the 4 clicks in the second slot are sold for 1 per click, for a total revenue
to the search engine of 34.
Now, how do these compare with the revenue generated by the VCG procedure? To work
out the VCG prices, we first need to convert the example from Figure 15.6 into a matching
market, just as we did in Section 15.2: for each advertiser and each slot, we work out the
advertiser’s valuation for the full set of clicks associated with that slot. We show these
valuations in Figure 15.7.
456 CHAPTER 15. SPONSORED SEARCH MARKETS
a
b
c
x
y
z
70, 28, 0
60, 24, 0
10, 4, 0
slots advertisers valuationsprices
40
4
0
Figure 15.8: Determining market-clearing prices for the example in Figure 15.6, starting with its representation as a matching market.
The matching used by the VCG procedure is the one which maximizes the total valuation
of all advertisers for the slot they get; this is achieved by assigning slot a to x, slot b to y,
and slot c to z. Now, we work out a price to charge each advertiser for the full set of clicks
in the slot it gets, by determining the harm each advertiser causes to all others. The harm x
causes to y and z can be computed as follows: without x present, y would move up one slot,
obtaining an increased valuation of 60 � 24 = 36, and z would move up one slot, obtaining an increased valuation of 4 � 0 = 4. Therefore, x should pay 40 for the full set of clicks in the first slot. Similarly, without y present, z would get 4 instead of 0, so y should pay 4
for the set of clicks in the second slot. Finally, since z causes no harm to anyone, it pays 0.
Thus, the total revenue collected by the search engine is 44.
So we find that in this example, the answer to the question, “Does GSP or VCG provide
more revenue to the search engine?” is indeed that it depends on which equilibrium of GSP
the advertisers use. With the first equilibrium of GSP that we identified, the revenue is
48, while with the second, the revenue is 34. The revenue from the VCG mechanism is in
between these two values, at 44.
15.6 Equilibria of the Generalized Second Price Auc- tion
The examples in the previous section give a sense for some of the complex behavior of GSP.
Here, we show that there is nonetheless a natural connection between GSP and market-
clearing prices: from a set of market-clearing prices for the matching market of advertisers
15.6. EQUILIBRIA OF THE GENERALIZED SECOND PRICE AUCTION 457
and slots, one can always construct a set of bids in Nash equilibrium — and moreover one
that produces a socially optimal assignment of advertisers to slots. As a consequence, there
always exists a set of socially optimal equilibrium bids for the GSP procedure.
To give the basic idea for how to construct an equilibrium, we do it first on the example
from Figure 15.6. In fact, we’ve just seen two equilibria for this example in the previous
section, but the point here is to see how a socially optimal one can be easily constructed
by following a few simple principles, rather than by trial-and-error or guesswork. We’ll then
identify the principles from this example that carry over to construct equilibria in general.
An Equilibrium for Figure 15.6. The basic idea is to use market-clearing prices to guide
us to a set of bids that produce these prices. To construct market-clearing prices, we first
convert the example from Figure 15.6 into a matching market by determining advertisers’
valuations for each slot, as we did at the end of the previous section (in Figure 15.7). We
then determine market-clearing prices for this matching market, as shown in Figure 15.8.
These market-clearing prices are cumulative prices for each slot — single prices that cover
all the clicks associated with that slot. We can easily translate back to prices per click by
simply dividing by the clickthrough rate: this produces a price per click of 40/10 = 4 for the
first slot, and 4/4 = 1 for the second slot. It will turn out not to be important how we price
the fictitious third slot per click, but it is fine to give it a price of 0.
Next, we find bids that result in these prices per click. This is not hard to do: the prices
per click are 4 and 1 for the two slots, so these should be the bids of y and z respectively.
Then the bid of x can be anything as long as it’s more than 4. With these bids, x pays 4
per click for the first slot, y pays 1 per click for the second slot, and z pays 0 per click for
the (fake) third slot — and the allocation of advertisers to slots is socially optimal.
Having used the market-clearing prices to guide us toward a set of bids, we now use
the market-clearing property to verify that these bids form a Nash equilibrium. There are
several cases to consider, but the overall reasoning will form the general principles that
extend beyond just this example. First, let’s argue that x doesn’t want to lower its bid. If it
drops down to match y’s bid, then it can get the second slot at the price that y is currently
paying. Similarly, it could match z’s bid and get the third slot at the price that z is currently
paying. But since the prices are market-clearing, x doesn’t want to do either of these things.
For similar reasons, y doesn’t want to drop its bid to get the third slot at the price z is
currently paying.
Next, let’s argue that y doesn’t want to raise its bid. Indeed, suppose that it raised its
bid to get the first slot — to do this, it would need to match x’s current bid. But in this
case, x becomes the second-highest bidder, and so y would get the first slot at a price per
click equal to x’s current bid, which is above 4. Because the prices are market-clearing, y
doesn’t prefer the first slot to its current slot at a price per click of 4, so it certainly doesn’t
458 CHAPTER 15. SPONSORED SEARCH MARKETS
prefer the first slot to its current slot at a higher price per click. Thus, y doesn’t want to
raise its bid. Similar reasoning shows that z doesn’t want to raise its bid.
This concludes the analysis: no advertiser wants to raise or lower its current bid, and so
the set of bids in this example forms a Nash equilibrium.
It is not hard to carry out the construction and the reasoning used here in general; we
show how to do this next.
GSP always has a Nash equilibrium: The General Argument Now let’s consider
a general instance where we have a set of advertisers and a set of slots; by adding fake slots
of 0 value if necessary, we will assume that these two sets have the same size.
Let’s suppose that the advertisers are labeled 1, 2, . . . , n in decreasing order of their
valuations per click, and let’s suppose that the slots are labeled 1, 2, . . . , n in decreasing
order of their clickthrough rates. We first represent the set of advertisers and slots using a
matching market, and we consider any set of market-clearing prices for the slots, denoted
p 1
, p 2
, . . . , p n
in order. Again, these are prices for the full set of clicks in each slot; we
will consider the price per click of each slot below. In Section 15.2, we argued that since
a perfect matching in the resulting preferred-seller graph maximizes the total valuation of
each advertiser for the slot it gets, it follows that the advertiser with the highest valuation
per click gets the top slot, the advertiser with next-highest valuation gets the second slot,
and so forth, with advertiser i getting slot i.
We now show how to get this outcome from an equilibrium set of bids in GSP. Our plan is
first to construct a set of bids that produces this same set of market-clearing prices, together
with the same socially optimal matching of advertisers to slots. Then, we will show that
these bids form a Nash equilibrium.
Constructing the bids. For the first step, we start by considering the prices per click
that we get from the market-clearing prices: p⇤ j
= p j
/r j
. We start by arguing that these
prices per click decrease as we move down the slots: p⇤ 1
� p⇤ 2
� · · · � p⇤ n
. To see why this
is true, let’s compare two slots j and k, where j is numbered lower than k, and show that
p⇤ j
� p⇤ k
.
Since the prices are market-clearing, advertiser k is at least as happy with slot k as it
would be with slot j. In slot k, its total payo↵ is the product of its payo↵ per click, v k
�p⇤ k
,
times the clickthrough rate r k
. In slot j, its total payo↵ would be the product of its payo↵
per click there, v k
� p⇤ j
, times the clickthrough rate r j
. Now, the clickthrough rate is higher
in slot j, yet slot k is preferred; so it must be that the payo↵ per click is smaller in slot j.
That is, v k
�p⇤ j
is smaller than v k
�p⇤ k
, or equivalently, p⇤ j
� p⇤ k
. This inequality is precisely
the fact we were looking for.
Now that we have decreasing prices per click, we can construct the bids we’re looking
15.7. AD QUALITY 459
for. We simply have advertiser j place a bid of p⇤ j�1 for each j > 1, and we have advertiser 1
place any bid larger than p⇤ 1
. Notice that this is exactly what happened when we constructed
an equilibrium for the example in Figure 15.6. With these bids, we have all the desired
properties: for each j, advertiser j is assigned to slot j and pays a price per click of p⇤ j
.
Why do the bids form a Nash equilibrium? To show why these bids form a Nash
equilibrium, we adapt the principles that we used in analyzing the equilibrium for Figure 15.6.
We first argue that no advertiser will want to lower its bid, and then that no advertiser will
want to raise its bid either.
Consider an advertiser j, currently in slot j. If it were to lower its bid, the best it could
do is to pick some lower slot k, bid just under the current bid of advertiser k, and thereby
get slot k at the price that advertiser k is currently paying. But since the prices are market-
clearing, j is at least as happy with its current slot at its current price as it would be with
k’s current slot at k’s current price. So in fact, this shows that no advertiser will want to
lower its bid.
How about raising a bid? The best advertiser j could do here is to pick some higher slot
i, bid just above the current bid of advertiser i, and thereby get slot i. What price would j
pay for slot i if it did this? It’s forcing advertiser i one slot down, and so it would pay the
current bid of advertiser i. This is actually larger than what advertiser i is currently paying
for slot i: advertiser i is currently paying the bid of advertiser i + 1, which is lower. So the
upshot is that j would get slot i at a price higher than the current price of slot i. Since
the market-clearing condition says that j doesn’t even prefer slot i at the current price, it
certainly wouldn’t prefer it at a higher price. This shows that no advertiser wants to raise
its bid either, and so the set of bids indeed forms a Nash equilibrium.
15.7 Ad Quality
What we’ve discussed thus far forms part of the basic framework for thinking about search
advertising markets. Of course, there are numerous further issues that come up in the use
of this framework by the major search engines, and in this section and the next we briefly
discuss a few of these issues. We begin with the issue of ad quality.
The assumption of a fixed clickthrough rate. One of the assumptions we’ve made
throughout the analysis is that a fixed clickthrough rate r j
is associated with each slot j —
in other words, that the number of clicks this slot receives is independent of which ad you
place there. But in general this is not likely to be true: users will look at the thumbnail
description of an ad placed in a given slot (evaluating, for example, whether they recognize
the name of the company placing the ad), and this will a↵ect whether they click on the ad.
460 CHAPTER 15. SPONSORED SEARCH MARKETS
And this, in turn, a↵ects how much money the search engine makes, since it’s charging per
click, not per impression.
So from the search engine’s point of view, the worrisome scenario is that a low-quality
advertiser bids very highly, thus obtaining the first slot under GSP. Users are then not
interested in clicking through on this ad (maybe they don’t trust the company, or the ad is
only minimally relevant to the query term). As a result, it sits at the top of the list as the
high bidder, but the search engine makes almost no money from it because users rarely click
on the ad. If the search engine could somehow expel this ad and promote the higher-quality
ads, it could potentially make more money.
Again, our model as described can’t really address this, since it starts from the assumption
that an ad in position i will get clicks at rate r i
, regardless of which ad it is. This “pure”
version of GSP, using the model from Sections 15.5 and 15.6 is essentially what the company
Overture used at the time it was acquired by Yahoo!, and hence what Yahoo! used initially
as well. And indeed, it su↵ers from exactly this problem — advertisers can sometimes occupy
high slots without generating much money for the search engine.
The role of ad quality. When Google developed its system for advertising, it addressed
this problem as follows. For each ad submitted by an advertiser j, they determine an
estimated quality factor q j
. This is intended as a “fudge factor” on the clickthrough rate: if
advertiser j appears in slot i, then the clickthrough rate is estimated to be not r i
but the
product q j
r i
. The introduction of ad quality is simply a generalization of the model we’ve
been studying all along: in particular, if we assume that all factors q i
are equal to 1, then
we get back the model that we’ve been using thus far in the chapter.
From the perspective of our matching market formulation, it’s easy to incorporate these
quality factors: we simply change the valuation of advertiser j for slot i, from v ij
= r i
v j
to
v ij
= q j
r i
v j
. The rest of the analysis remains the same, using these new valuations.
Google has adapted the GSP procedure analogously. Rather than assigning advertisers to
slots in decreasing orders of their bids b j
, it assigns them in decreasing order of the product of
their bid and quality factor q j
b j
. This makes sense, since this is the ordering of advertisers by
expected revenue to the search engine. The payments change correspondingly. The previous
rule — paying the bid of the advertiser just below you — can, in retrospect, be interpreted
more generally as paying the minimum bid you would need in order to hold your current
position. This rule carries over to the version with quality factors: each advertiser pays the
minimum amount it would need to keep its current position, when ranked according to q j
b j
.
With these changes, it’s possible to go back and perform the analysis of GSP at this
more general level. Close analogues of all the previous findings still hold here; while the
introduction of quality factors makes the analysis a little bit more complicated, the basic
ideas remain largely the same [144, 399].
15.8. COMPLEX QUERIES AND INTERACTIONS AMONG KEYWORDS 461
The mysterious nature of ad quality. How is ad quality computed? To a significant
extent, it’s estimated by actually observing the clickthrough rate of the ad when shown on
search results pages — this makes sense, since the goal of the quality factor is to act as
a modifier on the clickthrough rate. But other factors are taken into account, including
the relevance of the ad text and the “landing page” that the ad links to. Just as with the
unpaid organic search engine results on the left-hand-side of the screen, search engines are
very secretive about how they compute ad quality, and will not reveal the function to the
advertisers who are bidding.
One consequence is that the introduction of ad quality factors makes the keyword-based
advertising market much more opaque to the advertisers. With pure GSP, the rules were
very simple: for a given set of bids, it was clear how the advertisers would be allocated to
slots. But since the ad quality factor is under the search engine’s control, it gives the search
engine nearly unlimited power to a↵ect the actual ordering of the advertisers for a given set
of bids.
How does the behavior of a matching market such as this one change when the precise
rules of the allocation procedure are being kept secret from the bidders? This is an issue
that is actively discussed in the search industry, and a topic for potential research.
15.8 Complex Queries and Interactions Among Key- words
At the outset, we observed that markets are being conducted simultaneously for millions of
query words and phrases. In our analysis, we’ve focused the model on what goes on in a
single one of these markets, for a single keyword; but in reality, of course, there are complex
interactions among the markets for di↵erent keywords.
In particular, consider the perspective of a company that’s trying to advertise a product
using keyword-based advertising; suppose, for example, that the company is selling ski va-
cation packages to Switzerland. There are a lot of di↵erent keywords and phrases on which
the company might want to place bids: “Switzerland,” “Swiss vacation,” “Swiss hotels,”
“Alps,” “ski vacation,” “European ski vacation,” and many others (including grammatical
permutations of these). With a fixed advertising budget, and some estimates about user
behavior and the behavior of other advertisers, how should the company go about dividing
its budget across di↵erent keywords? This is a challenging problem, and one that is the
subject of current research [357].
There’s an analogous problem from the search engine’s perspective. Suppose advertisers
have placed bids on many queries relevant to Swiss ski vacations, and then a user comes and
issues the query, “Zurich ski vacation trip December.” It’s quite likely that very few users
have ever issued this exact query before, and also very likely that no advertiser has placed
462 CHAPTER 15. SPONSORED SEARCH MARKETS
a bid on this exact phrase. If the rules of the keyword-based advertising market are defined
too strictly — that the search engine can only show ads for words or phrases that have been
explicitly bid on — then it seems as though both the search engine and the advertisers are
losing money: there clearly are advertisers who would be happy to be displayed for this
query.
The question of which ads to show, however, is quite a di�cult problem. A simple
rule, such as showing the advertisers that placed the maximum bid for any of the words
in the query, seems like a bad idea: probably there are advertisers who have placed very
high bids on “vacation” (e.g. companies that sell generic vacation packages) and “ski” (e.g.
companies that sell skis), and neither of these seems like the right match to the query. It
seems important to take into account the fact that the query, through its choice of terms, is
specifying something fairly narrow.
Furthermore, even if relevant advertisers can be identified, how much should they be
charged for a click, given that they never expressed a bid on exactly this query? The main
search engines tend to get agreements from advertisers that they’ll extrapolate from their
bids on certain queries to implied bids on more complex queries, such as in this example,
but working out the best way to do this is not fully understood. These issues are the subject
of active work at search engine companies, and again the subject of some very interesting
potential further research.
15.9 Advanced Material: VCG Prices and the Market- Clearing Property
At the end of Section 15.3, we noted some of the di↵erences between the two main ways we’ve
seen to assign prices to items in matching markets: the VCG prices defined in this chapter,
and the construction of market-clearing prices from Chapter 10. In particular, we observed
that the di↵erence reflected a contrast between personalized and posted prices. VCG prices
are selected only after a matching between buyers and sellers has been determined — the
matching that maximizes the total valuation of buyers for what they get. The VCG price
of an item thus makes use of information not just about the item itself, but also about who
is buying it in the matching. Market-clearing prices, in a sense, work the other way around:
the prices are chosen first, and they are posted prices that are o↵ered to any buyer who
is interested. The prices then cause certain buyers to select certain items, resulting in a
matching.5
Given these significant di↵erences, one might expect the prices to look di↵erent as well.
But a comparison of simple examples suggests that something intriguing might be going on.
5In the discussion that follows, we’ll refer to nodes on the left-hand side of the bipartite graph sometimes as “items” and sometimes as “sellers”; for our purposes here, we treat these as meaning the same thing.
15.9. ADVANCED MATERIAL: VCG PRICES AND THE MARKET-CLEARING PROPERTY463
a
b
c
x
y
z
12, 4, 2
8, 7, 6
7, 5, 2
3
1
0
Prices Sellers Buyers Valuations
Figure 15.9: A matching market, with valuations and market-clearing prices specified, and a perfect matching in the preferred-seller graph indicated by the bold edges.
Consider for instance the matching market shown in Figures 15.3 and 15.4. In Figure 15.3
we see a set of market-clearing prices constructed using the procedure from Chapter 10. In
Figure 15.4, we see that these same prices arise as the VCG prices too.
Nor is it the special structure of prices arising from clickthrough rates and revenues per
click that causes this. For instance, let’s go back to the example used in Figure 10.6 from
Chapter 10, which has valuations with a much more “scrambled” structure. We’ve re-drawn
the final preferred-seller graph arising from the auction procedure in Figure 15.9, with the
(unique) perfect matching in this graph indicated using bold edges. This is the matching
that maximizes the total valuation of buyers for the item they get, so we apply the definitions
from earlier in the current chapter to determine the VCG prices. For example, to determine
the price that should be charged for seller a’s item, we observe
• If neither a nor x were present, the maximum total valuation of a matching between the remaining sellers and buyers would be 11, by matching y to c and z to b.
• If x weren’t present but a were, then the maximum total valuation possible would be 14, by matching y to b and z to a.
• The di↵erence between these two quantities is the definition of the VCG price for item a; it is 14 � 11 = 3.
We could perform the corresponding analysis to get the VCG prices for items b and c, and
we’d see that the values are 1 and 0, respectively. In other words, we again find that the
VCG prices are also market-clearing prices.
464 CHAPTER 15. SPONSORED SEARCH MARKETS
In this section, we show that the relationship suggested by these examples holds in
general. Our main result is that despite their definition as personalized prices, VCG prices
are always market-clearing. That is, suppose we were to compute the VCG prices for a
given matching market, first determining a matching of maximum total valuation, and then
assigning each buyer the item they receive in this matching, with a price tailored for this
buyer-seller match. Then, however, suppose we go on to post the prices publicly: rather than
requiring buyers to follow the matching used in the VCG construction, we allow any buyer to
purchase any item at the indicated price. We will see that despite this greater freedom, each
buyer will in fact achieve the highest payo↵ by selecting the item she was assigned when the
VCG prices were constructed. This will establish that the prices are market-clearing under
the definition from Chapter 10.
First Steps Toward a Proof. Let’s think for a minute about how you might prove such
a fact, once you start to suspect from simple examples that it might be true. It’s tempting
to start with the very compact formula defining the VCG prices — Equation (15.1) — and
then somehow reason about this formula to show that it has the market-clearing property.
In fact, it’s tricky to make this approach work, and it’s useful to understand why. Recall
that Equation (15.1) says that if item i is assigned to buyer j in the optimal matching, then
we should charge a price of
V S B�j � V
S�i B�j ,
where V S B�j is the total valuation of an optimal matching with j removed, and V
S�i B�j is the
total valuation of an optimal matching with both i and j removed. Now, the term V S B�j is in
fact a sum of many smaller terms, each consisting of the valuation of a distinct buyer for the
item she is assigned in an optimal matching. V S�i B�j is similarly a sum of many terms. But
the key conceptual di�culty is the following: V S B�j and V
S�i B�j arise from di↵erent matchings
— potentially very di↵erent matchings — and so there is no direct way to compare the sums
that they represent and easily subtract the terms of one from the other.
To make progress, we need to actually understand how the matchings that define these
two terms V S B�j and V
S�i B�j relate to each other at a structural level. And to do this, we will
show that matchings achieving these respective quantities can in fact arise from a common
set of market-clearing prices: there is a single set of market-clearing prices on the set of
items S so that matchings achieving each of V S B�j and V
S�i B�j arise as perfect matchings in the
preferred-seller graphs of related but slightly di↵erent matching markets. This will enable
us to see how the two matchings relate to each other — and in particular how to build one
from the other — in a way that lets us subtract the relevant terms from each other and thus
analyze the right-hand side of Equation (15.1).
For all this to work, we need to first understand which set of market-clearing prices
actually correspond to the VCG prices. There are many possible sets of market-clearing
15.9. ADVANCED MATERIAL: VCG PRICES AND THE MARKET-CLEARING PROPERTY465
prices, but with some checking, we can see that in our examples, the VCG prices have
corresponded to prices that are as small as possible, subject to having the market-clearing
property. So let’s consider the following way to make this precise. Over all possible sets of
market-clearing prices, consider the ones that minimize the total sum of the prices. (For
example, in Figure 15.9, the total sum of prices is 3 + 1 + 0 = 4.) We will refer to such
prices as a set of minimum market-clearing prices. In principle, there could be multiple sets
of minimum market-clearing prices, but in fact we will see that there is only one such set,
and they form the VCG prices. This is the crux of the following result, proved by Leonard
[270] and Demange [128].
Claim: In any matching market, the VCG prices form the unique set of market-
clearing prices of minimum total sum.
This is the statement we will prove in this section.
The proof of this statement is quite elegant, but it is also arguably the most intricate piece
of analysis in the book; it has a level of complexity that involves bringing together several
non-trivial lines of argument as part of the overall proof. In approaching a proof with this
type of structure, it helps to proceed in two stages. First, we will outline a sequence of two
key facts that illuminate the structure of the underlying matchings. Each of these two facts
needs a proof, but we will first simply state the facts and show how the overall proof of the
claim follows directly from them. This provides a high-level overview of how the proof works,
in a way that is self-contained and contains the central ideas. After this, we will describe
how to prove the two facts themselves, which will fill in the remaining details of the proof.
Finally, here is one more point to note before beginning: As in a number of previous
places when we discussed matching markets, we will assume that all valuations are whole
numbers (0, 1, 2, . . .), and that all prices are whole numbers as well.
A. High-Level Overview of the Proof
Recall that our basic plan is to understand how matchings defining the quantities V S B�j and
V S�i B�j relate to each other, by showing how they arise from a common structure. To do this,
the first step is to show that the preferred-seller graph for the minimum market-clearing
prices contains not only the edges of a perfect matching, but also enough extra edges that
we can easily assemble other matchings once we begin removing buyers in the ways suggested
by the VCG formula.
The First Fact: The Preferred-Seller Graph for Minimum Market-Clearing Prices.
The first of our two facts talks about the structure of the preferred-seller graph in the case
when a set of market-clearing prices has minimum total sum. As a first step, let’s go back
466 CHAPTER 15. SPONSORED SEARCH MARKETS
i
h
i*
k
m
There is an alternating path, beginning with a non-matching edge,
from i to an item (i*) of price 0.
price 0
Figure 15.10: The key property of the preferred-seller graph for minimum market-clearing prices: for each item of price greater than 0, there is an alternating path, beginning with a non-matching edge, to an item of price 0.
to the initial example of market-clearing prices from Chapter 10, and in particular com-
pare preferred-seller graphs for two di↵erent sets of market-clearing prices on the same set
of valuations, shown in Figures 10.5(b) and 10.5(d). Notice that the prices in the first of
these, Figure 10.5(b), are larger and more “spread out,” while the prices in Figure 10.5(d)
in fact have minimum total sum. This corresponds to a di↵erence in the structures of the
preferred-seller graphs as well. The preferred-seller graph in Figure 10.5(b) is very sparse,
with just three separate edges that constitute a perfect matching. The preferred-seller graph
in Figure 10.5(d) is much denser: although it too contains only one perfect matching, it has
additional edges that seem to serve as supports, “anchoring” the matching in place.
We now show that this anchoring e↵ect is a general one: essentially, whenever a set of
market-clearing prices has minimum total sum, the preferred-seller graph must contain not
only a perfect matching, but also enough other edges to form a path linking each item to
an item of price 0. In fact, the paths we construct will be alternating paths in the sense
defined in Section 10.6: for a given perfect matching in the graph, the edges on the paths
will alternate between being part of the matching and not part of the matching. We will
refer to these two kinds of edges as matching edges and non-matching edges respectively.
Here is the exact statement of the first fact, shown schematically in Figure 15.10.
Fact 1: Consider the preferred-seller graph for a set of market-clearing prices
of minimum total sum, fix a particular perfect matching in this graph, and let
i be any item whose price is greater than 0. Then there is an alternating path,
15.9. ADVANCED MATERIAL: VCG PRICES AND THE MARKET-CLEARING PROPERTY467
wa
b x
c
d
y
z
Prices Sellers Buyers Valuations
4
3
1
0
7, 5, 4, 2
6, 5, 2, 1
5, 6, 2, 2
4, 4, 2, 1
Figure 15.11: A matching market with market-clearing prices of minimum total sum. Note how from each item, there is an alternating path, beginning with a non-matching edge, that leads to the item of zero price.
beginning with a non-matching edge, that connects i to some item of price 0.
For example, in Figure 15.9, with the matching indicated in bold, there is an alternating
path from b to y to c; this path begins with the non-matching b-y edge, and ends at the
zero-priced item c. Similarly, there is a longer alternating path from a through z, b, and
y, ending at c. Figure 15.11 shows a larger example, also with market-clearing prices of
minimum-total sum, and also with the matching indicated in bold; here too one can find
alternating paths from each of items a, b, and c down to the zero-priced item d.
Following our plan, we defer the proof of Fact 1 until later in the section. However, we can
give some intuition for the proof as follows. Roughly speaking, if there weren’t alternating
paths anchoring all the prices to items of price 0, then we could find a set of items that were
“floating” free of any relation to the zero-priced items. In this case, we could push the prices
of these free-floating items down slightly, while still preserving the market-clearing property.
This would yield a set of market-clearing prices of smaller total sum, contradicting our
assumption that we already have the minimum market-clearing prices. This contradiction
468 CHAPTER 15. SPONSORED SEARCH MARKETS
wa
b x
c
d
y
z
Sellers Buyers Valuations
7, 5, 4, 2
0, 0, 0, 0
5, 6, 2, 2
4, 4, 2, 1
Figure 15.12: If we start with the example in Figure 15.11 and zero out buyer x, the structure of the optimal matching changes significantly.
will show that the minimum market-clearing prices are all anchored via alternating paths to
zero-priced items.
The Second Fact: Zeroing Out a Buyer. Our second main fact will relate the minimum
market-clearing prices to a matching that achieves the value V S B�j, the first term on the right-
hand side of Equation (15.1).
To explain how this fact works, we start with a useful way to think about the quantity
V S B�j. Formally, V
S
B�j is the maximum total valuation of any matching in the market where
j has been removed, but where all items have been kept. But here’s a di↵erent, equivalent
way to define V S B�j. Suppose that we were to change j’s valuations for every item to 0; we’ll
call this the version of the matching market in which j has been zeroed out. To find an
optimal matching in this market with j zeroed out, we note that it doesn’t matter which
item j gets (since j values them all at zero); therefore, we can first optimally match all the
other buyers with items, and then give j whatever is left over. The value of the resulting
matching is V S B�j. In other words, V
S
B�j is the value of the optimal matching in the market
15.9. ADVANCED MATERIAL: VCG PRICES AND THE MARKET-CLEARING PROPERTY469
wa
b x
c
d
y
z
Prices Sellers Buyers Valuations
4
3
1
0
7, 5, 4, 2
0, 0, 0, 0
5, 6, 2, 2
4, 4, 2, 1
Figure 15.13: However, even after we zero out buyer x, the same set of prices remain market- clearing. This principle is true not just for this example, but in general.
where j is zeroed out: she is still present, but all her valuations are now equal to 0.
Now, an optimal matching in the market with j zeroed out may have a very di↵erent
structure than an optimal matching in the original market — di↵erent buyers may get
completely di↵erent items. For example, Figure 15.12 shows the unique optimal matching
in the market from Figure 15.11 after we zero out x: other than buyer y, who still gets
item b, the assignment of items to all other buyers has changed completely. This is another
reflection of the di�culty in reasoning about Equation (15.1): when we remove buyers or
items, the matchings can rearrange themselves in complex ways.
Despite this, there is an important connection between the original market and the zeroed-
out market: the minimum market-clearing prices for the original market are also market-
clearing for the zeroed-out market. We illustrate this for our example in Figure 15.13: keeping
the same prices that were used in Figure 15.11, we see that the preferred-seller graph still has
a perfect matching even after x has been zeroed out, and this means that the prices are still
market-clearing. Moreover, we can observe some additional features of this example. First,
x now receives an item of price 0. Second, consider the payo↵ of each other buyer, defined
470 CHAPTER 15. SPONSORED SEARCH MARKETS
as the valuation minus the price of the item she gets. For each other buyer, the payo↵ is the
same in Figures 15.11 and 15.13.
Our second fact shows that all these observations hold in general.
Fact 2: Consider any matching market with minimum market-clearing prices p,
and let j be any buyer.
(i) The prices p are also market-clearing for the market in which j is zeroed
out.
Moreover, for any perfect matching in the preferred-seller graph of the zeroed-out
market,
(ii) buyer j receives a zero-priced item; and
(iii) each buyer other than j obtains the same payo↵ that she did in the original
market.
Again, we defer the proof of Fact 2 to later in the section, but it is not hard to establish the
proof using Fact 1. Essentially, when we zero out j, we look at the item i that j formerly got
in the original market, before she was zeroed out. We follow the alternating path provided
by Fact 1 from i down to an item i⇤ of price 0. We then show that assigning item i⇤ to j, and
shifting the assignment to all other buyers using the edges on this alternating path, gives us
a perfect matching in the preferred-seller graph of the zeroed-out market at the same prices.
This shows that the same prices are in fact market-clearing for the zeroed-out market, and
will establish parts (ii) and (iii) of the claim as well.
Proving the Claim Using Facts 1 and 2. With Facts 1 and 2 in place, we can finish
the proof of our main claim, that the minimum market-clearing prices are defined by the
VCG formula.
To start, let’s review some notation. As before, let v ij
denote the valuation that a buyer
j has for an item i. Let p i
be the price charged for item i in our market-clearing prices, and
let P be the sum of the prices of all items. Suppose that buyer j is matched to item i in the
perfect matching in the preferred-seller graph. Buyer j receives a payo↵ of v ij
�p i
from this
item i; we will use z j
to denote this payo↵,
z j
= v ij
� p i
, (15.4)
and Z to denote the sum of the payo↵s of all buyers from the items they are matched with.
Next, let’s recall two basic observations that were made in earlier sections. First, each
buyer j achieves a payo↵ of v ij
� p i
from the item to which she is matched. As we noted in
15.9. ADVANCED MATERIAL: VCG PRICES AND THE MARKET-CLEARING PROPERTY471
Chapter 10, if we add these expressions up over all buyers, we get the following relationship
for the matching M of buyers to items:
Total Payo↵ of M = Total Valuation of M � Sum of all prices.
In our current notation, this is
Z = V S B
� P. (15.5)
Second, we argued in Section 15.4 that if i is matched to j in an optimal matching, then
v ij
+ V S�i B�j = V
S
B
. (15.6)
This is Equation (15.3) from Section 15.4, and it follows simply because one way to achieve
an optimal matching is to first pair i with j (obtaining a valuation of v ij
), and then optimally
match all the remaining buyers and items.
Finally, let’s consider this same formula
Total Payo↵ of M = Total Valuation of M � Sum of all prices,
for the market in which j has been zeroed out, using the same set of market-clearing prices
and a perfect matching in the preferred-seller graph that Fact 2 provides. The total valuation
of this matching is V S B�j, as we argued earlier. The prices haven’t changed, so their total
sum is still P . Finally, what’s the total payo↵? By part (ii) of Fact 2, the payo↵ for buyer
j has dropped from z j
, which it was in the original market, to 0. By part (iii) of Fact 2,
the payo↵ for every other buyer has remained the same. Therefore, the total payo↵ in the
zeroed-out market is Z � z i
. Putting all these together, we have the equation
Z � z i
= V S B�j � P. (15.7)
Since we now have equations that relate the two terms on the right-hand of Equation
(15.1) to a common set of quantities, we can finish the proof using a small amount of algebraic
manipulation. Let’s first subtract Equation (15.7) from Equation (15.5): this gives us
z i
= V S B
� V S B�j.
Next, let’s expand z i
using Equation (15.4) and expand V S B
using Equation (15.6). This
gives us
v ij
� p i
= v ij
+ V S�i B�j � V
S
B�j.
Canceling the common term of v ij
and negating everything, we get
p i
= V S B�j � V
S�i B�j ,
which is the VCG formula we were seeking. This shows that the market-clearing prices of
minimum total sum are defined by the VCG formula, and hence proves the claim.
472 CHAPTER 15. SPONSORED SEARCH MARKETS
i
h k
?
j
Figure 15.14: In order for a matching edge from a buyer k to an item h to leave the preferred- seller graph when the price of i is reduced by 1, it must be that k now strictly prefers i. In this case, k must have previously viewed i as comparable in payo↵ to h, resulting in a non-matching edge to i.
B. Details of the Proof
The discussion so far provides a complete proof, assuming that we take Facts 1 and 2 as
given. To finish the proof, therefore, we need to provide proofs of Facts 1 and 2. The crux
of this is proving Fact 1, which will consist of an analysis of alternating paths in the style of
Section 10.6. After this analysis, establishing Fact 2 is relatively quick using Fact 1.
A First Step Toward Fact 1. To prove Fact 1, we consider a set of minimum market-
clearing prices, and an item i whose price is greater than 0, and we try to construct an
alternating path (beginning with a non-matching edge) from i to some zero-priced item.
As a first step toward this, to convey the idea at the heart of the argument, let’s show
something simpler: that this item i, of price p i
> 0, is connected to at least one non-matching
edge (in addition to its matching edge to the buyer j that obtains it). Clearly it will be
necessary to establish the presence of such a non-matching edge in any case, if we want
ultimately to show that i has an alternating path all the way down to a zero-priced item.
So suppose, by way of contradiction, that i is not connected to a non-matching edge: its
only edge is the matching edge to buyer j. In this case, we claim that we can subtract 1
from the price p i
, and the resulting modified prices will still be market-clearing. This would
be a contradiction, since we assumed our market-clearing prices have minimum total sum.
Clearly if we subtract 1 from p i
, it is still non-negative, so we just need to show that
the preferred-seller graph still contains a perfect matching. In fact, we’ll show the stronger
fact that the preferred-seller graph still contains all the matching edges that it used to have.
Indeed, how could a matching edge leave the preferred-seller graph after the price reduction?
15.9. ADVANCED MATERIAL: VCG PRICES AND THE MARKET-CLEARING PROPERTY473
i
h k
m
X (1) No matching edges from a buyer in X to an item not in X.
(2) No non-matching edges from an item in X to a buyer
not in X.
Figure 15.15: Consider the set X of all nodes that can be reached from i using an alternating path that begins with a non-matching edge. As we argue in the text, if k is a buyer in X, then the item to which she is matched must also be in X. Also, if h is an item in X, then any buyer to which h is connected by a non-matching edge must also be in X. Here is an equivalent way to phrase this: there cannot be a matching edge connecting a buyer in X to an item not in X, or a non-matching edge connecting an item in X to a buyer not in X.
The only item that became more attractive was item i, so for a matching edge to leave the
preferred-seller graph, it must be that some buyer k other than j, who used to be matched to
an item h, drops its edge to h because it now strictly prefers i. This situation is pictured in
Figure 15.14. Now, since i’s price was only reduced by 1, and since all prices and valuations
are whole numbers, if k now strictly prefers i to h after the price reduction, it must have
formerly viewed them as tied. But this means that before the reduction in i’s price, k had a
preferred-seller edge to i. Since k was matched to h, this k-i edge would be a non-matching
edge in the preferred-seller graph, which is not possible since i’s only edge in the preferred-
seller graph was its matching edge to j. This completes the chain of conclusions we need:
no matching edge can leave the preferred-seller graph when i’s price is reduced by 1, so the
reduced prices are still market-clearing, and this contradicts the assumption that we had
474 CHAPTER 15. SPONSORED SEARCH MARKETS
f
e n
X If n was matched to e but now strictly prefers f, then:
(1) n must have had a non-matching edge to f,
(2) f must be in X, and (3) e must not be in X.
In this case, either the f-n edge or the e-n edge causes a contradiction.
......
Figure 15.16: We can reduce the prices of all items in X by 1 and still retain the market- clearing property: as we argue in the text, the only way this can fail is if some matching edge connects a buyer in X to an item not in X, or some non-matching edge connects an item in X to a buyer not in X. Either of these possibilities would contradict the facts in Figure 15.15.
minimum market-clearing prices.
A Proof of Fact 1. The argument above is the key to proving Fact 1; for the complete
proof, we need to move from simply showing the existence of a non-matching edge out of i
to a full alternating path, beginning with such an edge, all the way to a zero-priced item.
To do this, we start at the item i, and we consider the set X of all nodes in the bipartite
graph (both items and buyers) that can be reached from i, using an alternating path that
begins with a non-matching edge. Here are two simple observations about the set X.
(a) For any buyer k who is in X, the item h to which she is matched is also in X. Fig-
ure 15.15 helps make clear why this must be true. The alternating path that reached
k from i must have ended on a non-matching edge, so by adding the matching edge to
h to the end of this path, we see that h must also be in X.
15.9. ADVANCED MATERIAL: VCG PRICES AND THE MARKET-CLEARING PROPERTY475
(b) For any item h that is in X, and any buyer m connected to h by a non-matching edge
in the preferred-seller graph, the buyer m must also be in X. This is a direct companion
to the previous fact, and also illustrated by Figure‘15.15: the alternating path that
reached h from i must have ended on a matching edge, so by adding the non-matching
edge to m to the end of this path, we see that m must also be in X.
If this set X contains an item of price 0, we are done: we have the path we want. If
this set X doesn’t contains an item of price 0, then we complete the proof using the same
price-reduction idea we saw earlier, in our warm-up to the proof of Fact 1: in this case, we
will reduce the price of each item in X by 1, show that the resulting prices are still market-
clearing, and thereby contradict our assumption that we had the minimum market-clearing
prices. It will follow that X must contain a zero-priced item.
Here is the main thing we need to show.
Suppose we reduce the price of each item in X by 1. Then all matching edges
that were in the preferred-seller graph before the price reduction remain in the
preferred-seller graph after the price reduction.
The argument is essentially the same as the one we used earlier, when we were reducing the
price of just item i. We ask: how could a matching edge leave the preferred-seller graph after
the reduction? Figure 15.16 shows what must happen for this to be possible: a buyer n was
formerly matched to an item e, and now some other item f has strictly higher payo↵ after
the price reduction. Since all valuations, prices, and payo↵s are whole numbers, and no price
changed by more than 1, it must be that e and f used to be tied for the highest payo↵ to n
(so n had edges to both of them in the preferred-seller graph before the reduction) and f is
in the set X while e is not (so f had its price reduced while e’s price remained the same).
Now we get a contradiction to one of our basic observations (a) and (b) about the set X:
Since n was matched to e, and e is not in X, observation (a) says that n must not be in X;
but since n was not matched to f , and f is in X, observation (b) says that n must be in X.
This contradiction — n must both be in X and not be in X — shows that no matching edge
can leave the preferred-seller graph after the price reduction. And this in turn establishes
that the reduced prices are still market-clearing after the price-reduction, contradicting our
assumption that they were the minimum market-clearing prices.
This concludes the proof, and if we look back at how it worked, we can see that it bears
out our intuition for how the non-matching edges serve to anchor all the items via alternating
paths to the items of price 0. Specifically, if this anchoring did not happen, then there would
be a set X that was floating free of any connections to zero-priced items, and in this case
the prices of all items in X could be pushed further downward. This can’t happen if the
market-clearing prices are already as low as possible.
476 CHAPTER 15. SPONSORED SEARCH MARKETS
i
h
i*
j
k
m
When j is matched to i in the original
market, first find a path to a zero-
priced item i*.
Figure 15.17: The first step in analyzing the market with j zeroed out: find an alternating path from item i — to which buyer j was matched in the original market — to a zero-priced item i⇤.
A Proof of Fact 2. To prove Fact 2, we start with a matching market with minimum
market-clearing prices p, and we consider the preferred-seller graph for these prices. Now,
suppose that we zero out a buyer j, but keep the prices the same. The resulting preferred-
seller graph is now di↵erent, but we’d like to show that it still contains a perfect matching.
How does the preferred-seller graph change when we zero out j, keeping the prices fixed?
For buyers other than j, their edges remain the same, since they have the same valuations
and observe the same prices. For j, on the other hand, the zero-priced items are now the
only items that give her a non-negative payo↵, so her edges in the preferred-seller graph now
go to precisely this set of zero-priced items. Notice, for example, that this is what happens
to the preferred-seller graph as we move from Figure 15.11 to Figure 15.13: the zeroed-out
buyer x has its preferred-seller edge shift from item b to the zero-priced item d.
Because we know that the preferred-seller graph in the original market has the structure
guaranteed by Fact 1, we can view this change to the preferred-seller graph in the way
suggested by Figures 15.17 and 15.18. Before zeroing out j, when it is matched to some item
i, there is an alternating path in the preferred-seller graph, beginning with a non-matching
edge, from i to a zero-priced item i⇤. After zeroing out j, there is a preferred-seller edge
15.9. ADVANCED MATERIAL: VCG PRICES AND THE MARKET-CLEARING PROPERTY477
i
h
i*
j
k
m
In the zeroed-out market, j loses its
preferred-seller edge to i, but acquires
a preferred-seller edge to i*.
Figure 15.18: The second step in analyzing the market with j zeroed out: build the new preferred-seller graph by rewiring j’s preferred-seller edges to point to the zero-priced items.
from j directly to i⇤ (and to any other zero-priced items as well).
It is easy to see from this pair of pictures how to find a perfect matching in the preferred-
seller graph after this change to its structure. This is shown in Figure 15.19: for each buyer
other than j who is involved in the alternating path from i to i⇤, we simply shift her edge
“upward” along the alternating path. This makes room for j to match with i⇤, restoring the
perfect matching.
Since the preferred-seller graph has a perfect matching, this establishes that the prices
are still market-clearing for the zeroed-out market. We can also establish parts (ii) and (iii)
of Fact 2 directly from our construction. Part (ii) follows simply from the fact that j only
has edges to zero-priced items in the preferred-seller graph. For part (iii), note first of all
that it is a statement about the payo↵s that buyers receive. Even when there are potentially
multiple perfect matchings in a preferred-seller graph, any given buyer obtains the same
payo↵ in every one of these perfect matchings, since all of her edges in the preferred-seller
graph yield the same, maximum payo↵. As a result, it is enough to establish part (iii) for
the perfect matching we just constructed, and it will then apply to the payo↵ properties
of every perfect matching in the preferred-seller graph. So consider the matching we just
478 CHAPTER 15. SPONSORED SEARCH MARKETS
i
h
i*
j
k
m
We can still find a perfect matching in
this new preferred-seller graph. This
means that the same prices are also
market-clearing for the zeroed-out
market.
Figure 15.19: The third and final step in analyzing the market with j zeroed out: observe that the rewired preferred-seller graph still contains a perfect matching, in which j is now paired with i⇤.
constructed, and let k be any buyer other than j. Either k gets the same item she had in
the perfect matching for the original market, in which case she gets the same payo↵ — or
else k shifts from one item to another along the alternating path. In this latter case, since
k had edges to both of these items in the preferred-seller graph of the original market, she
receives the same payo↵ from each of them, and so again k’s payo↵ remains the same. This
completes the proof of Fact 2, and hence fills in the final details needed to complete the
proof of the overall claim.
15.10 Exercises
1. Suppose a search engine has two ad slots that it can sell. Slot a has a clickthrough
rate of 10 and slot b has a clickthrough rate of 5. There are three advertisers who are
interested in these slots. Advertiser x values clicks at 3 per click, advertiser y values
clicks at 2 per click, and advertiser z values clicks at 1 per click.
Compute the socially optimal allocation and the VCG prices for it. Give a brief
15.10. EXERCISES 479
explanation for your answer.
2. Suppose a search engine has three ad slots that it can sell. Slot a has a clickthrough
rate of 6, slot b has a clickthrough rate of 5 and slot c has a clickthrough rate of 1.
There are three advertisers who are interested in these slots. Advertiser x values clicks
at 4 per click, advertiser y values clicks at 2 per click, and advertiser z values clicks at
1 per click. Compute the socially optimal allocation and the VCG prices for it. Give
a brief explanation for your answer.
3. Suppose a search engine has three ad slots that it can sell. Slot a has a clickthrough
rate of 5, slot b has a clickthrough rate of 2, and slot c has a clickthrough rate of 1.
There are three advertisers who are interested in these slots. Advertiser x values clicks
at 3 per click, advertiser y values clicks at 2 per click, and advertiser z values clicks at
1 per click.
Compute the socially optimal allocation and the VCG prices for it. Give a brief
explanation for your answer.
4. Suppose a search engine has two ad slots that it can sell. Slot a has a clickthrough
rate of 4 and slot b has a clickthrough rate of 3. There are three advertisers who are
interested in these slots. Advertiser x values clicks at 4 per click, advertiser y values
clicks at 3 per click, and advertiser z values clicks at 1 per click.
(a) Suppose that the search engine runs the VCG Procedure to allocate slots. What
assignment of slots will occur and what prices will the advertisers pay? Give an expla-
nation for your answer.
(b) Now the search engine is considering the creation of a third ad slot which will have
a clickthrough rate of 2. Let’s call this new ad slot c. Suppose that search engine does
create this slot and again uses the VCG Procedure to allocate slots. What assignment
of slots will occur and what prices will the advertisers pay? Give an explanation for
your answer.
(c) What revenue will the search engine receive from the VCG Procedure in parts (a)
and (b)? If you were running the search engine, given this set of advertisers and slots,
and could choose whether to create slot c or not, what would you do? Why? (In
answering this question assume that you have to use the VCG Procedure to allocate
any slots you create.)
5. Suppose a search engine has two ad slots that it can sell. Slot a has a clickthrough
rate of 12 and slot b has a clickthrough rate of 5. There are two advertisers who are
480 CHAPTER 15. SPONSORED SEARCH MARKETS
interested in these slots. Advertiser x values clicks at 5 per click and advertiser y values
clicks at 4 per click.
(a) Compute the socially optimal allocation and the VCG prices for it.
(b) Suppose the search engine decides not to sell slot b. Instead, it sells only slot a
using a sealed-bid, second-price auction. What bids will the advertisers submit for slot
a, who will win, and what price will they pay?
(c) Which of these two possible procedures (a) and (b) generate the greater revenue
for the search engine? By how much?
(d) Now let’s see if the result in part (c) is general or not. That is, does it depend
on the clickthrough rates and values? Let there be two slots and two advertisers; let
the clickthrough rates be r a
for slot a and r b
for slot b, with r a
> r b
> 0; and let the
advertisers’ values be v x
and v y
, with v x
> v y
> 0. Can you determine which of the
two procedures generates the greater revenue for the search engine? Explain.
6. Chapter 15 discusses the relationship between the VCG Principle and second price
auctions. In particular, we saw that the VCG Principle is a generalization of the idea
behind second price auctions to a setting in which there is more than one object being
sold. In this problem we will explore this relationship in an example. Suppose that a
seller has one item, which we’ll call item x. There are three buyers, whom we’ll call a,
b, and c. The values that these buyers (a, b, and c) have for the item are 6, 3, and 1,
respectively.
(a) Suppose that the seller runs a second price auction for the item. Which buyer will
win the auction and how much will this buyer pay?
(b) Now let’s suppose that the seller uses the VCG procedure to allocate the item.
Remember that the first step in the running the VCG procedure when there are more
buyers than items is to create fictional items, which each buyer values at 0, so that the
number of items to be allocated is the same as the number of bidders. Let’s call these
additional (fictional) items y and z. Find the allocation that results from running the
VCG procedure. What are the prices charged to each buyer for the item that they
receive? Explain why the price that buyer a pays is the harm that he causes to the
remaining bidders by taking the item he is assigned.
Part V
Network Dynamics: Population Models
481
Chapter 16
Information Cascades
16.1 Following the Crowd
When people are connected by a network, it becomes possible for them to influence each
other’s behavior and decisions. In the next several chapters, we will explore how this ba-
sic principle gives rise to a range of social processes in which networks serve to aggregate
individual behavior and thus produce population-wide, collective outcomes.
There is a nearly limitless set of situations in which people are influenced by others:
in the opinions they hold, the products they buy, the political positions they support, the
activities they pursue, the technologies they use, and many other things. What we’d like to
do here is to go beyond this observation and consider some of the reasons why such influence
occurs. We’ll see that there are many settings in which it may in fact be rational for an
individual to imitate the choices of others even if the individual’s own information suggests
an alternative choice.
As a first example, suppose that you are choosing a restaurant in an unfamiliar town, and
based on your own research about restaurants you intend to go to restaurant A. However,
when you arrive you see that no one is eating in restaurant A while restaurant B next door
is nearly full. If you believe that other diners have tastes similar to yours, and that they too
have some information about where to eat, it may be rational to join the crowd at B rather
than to follow your own information. To see how this is possible, suppose that each diner
has obtained independent but imperfect information about which of the two restaurants is
better. Then if there are already many diners in restaurant B, the information that you
can infer from their choices may be more powerful than your own private information, in
which case it would in fact make sense for you to join them regardless of your own private
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
483
484 CHAPTER 16. INFORMATION CASCADES
information. In this case, we say that herding, or an information cascade, has occurred. This
terminology, as well as this example, comes from the work of Banerjee [40]; the concept was
also developed in other work around the same time by Bikhchandani, Hirshleifer, and Welch
[59, 412].
Roughly, then, an information cascade has the potential to occur when people make
decisions sequentially, with later people watching the actions of earlier people, and from
these actions inferring something about what the earlier people know. In our restaurant
example, when the first diners to arrive chose restaurant B, they conveyed information to
later diners about what they knew. A cascade then develops when people abandon their
own information in favor of inferences based on earlier people’s actions.
What is interesting here is that individuals in a cascade are imitating the behavior of
others, but it is not mindless imitation. Rather, it is the result of drawing rational inferences
from limited information. Of course, imitation may also occur due to social pressure to
conform, without any underlying informational cause, and it is not always easy to tell these
two phenomena apart. Consider for example the following experiment performed by Milgram,
Bickman, and Berkowitz in the 1960s [298]. The experimenters had groups of people ranging
in size from just one person to as many as fifteen people stand on a street corner and stare
up into the sky. They then observed how many passersby stopped and also looked up at
the sky. They found that with only one person looking up, very few passersby stopped. If
five people were staring up into the sky, then more passersby stopped, but most still ignored
them. Finally, with fifteen people looking up, they found that 45% of passersby stopped and
also stared up into the sky.
The experimenters interpreted this result as demonstrating a social force for conformity
that grows stronger as the group conforming to the activity becomes larger. But another
possible explanation — essentially, a possible mechanism giving rise to the conformity ob-
served in this kind of situation — is rooted in the idea of information cascades. It could be
that initially the passersby saw no reason to look up (they had no private or public infor-
mation that suggested it was necessary), but with more and more people looking up, future
passersby may have rationally decided that there was good reason to also look up (since
perhaps those looking up knew something that the passersby didn’t know).
Ultimately, information cascades may be at least part of the explanation for many types
of imitation in social settings. Fashions and fads, voting for popular candidates, the self-
reinforcing success of books placed highly on best-seller lists, the spread of a technological
choice by consumers and by firms, and the localized nature of crime and political movements
can all be seen as examples of herding, in which people make decisions based on inferences
from what earlier people have done.
16.2. A SIMPLE HERDING EXPERIMENT 485
Informational e↵ects vs. Direct-Benefit E↵ects. There is also a fundamentally dif-
ferent class of rational reasons why you might want to imitate what other people are doing.
You may want to copy the behavior of others if there is a direct benefit to you from aligning
your behavior with their behavior. For example, consider the first fax machines to be sold.
A fax machine is useless if no one else owns one, and so in evaluating whether to buy one,
it’s very important to know whether there are other people who own one as well — not just
because their purchase decisions convey information, but because they directly a↵ect the
fax machine’s value to you as a product. A similar argument can be made for computer
operating systems, social networking sites, and other kinds of technology where you directly
benefit from choosing an option that has a large user population.
This type of direct-benefit e↵ect is di↵erent from the informational e↵ects we discussed
previously: here, the actions of others are a↵ecting your payo↵s directly, rather than indi-
rectly by changing your information. Many decisions exhibit both information and direct-
benefit e↵ects — for example, in the technology-adoption decisions just discussed, you po-
tentially learn from others’ decisions in addition to benefitting from compatibility with them.
In some cases, the two e↵ects are even in conflict: if you have to wait in a long line to get
into a popular restaurant, you are choosing to let the informational benefits of imitating
others outweigh the direct inconvenience (from waiting) that this imitation causes you.
In this chapter, we develop some simple models of information cascades; in the next
chapter, we do this for direct-benefit e↵ects. One reason to develop minimal, stylized models
for these e↵ects is to see whether the stories we’ve been telling can have a simple basis —
and we will see that much of what we’ve been discussing at an informal level can indeed be
represented in very basic models of decision-making by individuals.
16.2 A Simple Herding Experiment
Before delving into the mathematical models for information cascades [40, 59, 412], we start
with a simple herding experiment created by Anderson and Holt [14, 15] to illustrate how
these models work.
The experiment is designed to capture situations with the basic ingredients from our
discussion in the previous section:
(a) There is a decision to be made — for example, whether to adopt a new technology,
wear a new style of clothing, eat in a new restaurant, or support a particular political
position.
(b) People make the decision sequentially, and each person can observe the choices made
by those who acted earlier.
(c) Each person has some private information that helps guide their decision.
486 CHAPTER 16. INFORMATION CASCADES
(d) A person can’t directly observe the private information that other people know, but
he or she can make inferences about this private information from what they do.
We imagine the experiment taking place in a classroom, with a large group of students
as participants. The experimenter puts an urn at the front of the room with three marbles
hidden in it; she announces that there is a 50% chance that the urn contains two red marbles
and one blue marble, and a 50% chance the urn contains two blue marbles and one red marble.
In the former case, we will say that it is a “majority-red” urn, and in the latter case, we will
say that it is a “majority-blue” urn.1
Now, one by one, each student comes to the front of the room and draws a marble from
the urn; he looks at the color and then places it back in the urn without showing it to the
rest of the class. The student then guesses whether the urn is majority-red or majority-
blue and publicly announces this guess to the class. (We assume that at the very end of
the experiment, each student who has guessed correctly receives a monetary reward, while
students who have guessed incorrectly receive nothing.) The public announcement is the
key part of the set-up: the students who have not yet had their turn don’t get to see which
colors the earlier students draw, but they do get to hear the guesses that are being made.
This parallels our original example with the two restaurants: one-by-one, each diner needs
to guess which is the better restaurant, and while they don’t get to see the reviews read by
the earlier diners, they do get to see which restaurant these earlier diners chose.
Let’s now consider what we should expect to happen when this experiment is performed.
We will assume that all the students reason correctly about what to do when it is their
turn to guess, using everything they have heard so far. We will keep the analysis of the
experiment informal, and later use a mathematical model to justify it more precisely.
We organize the discussion by considering what happens with each student in order.
Things are fairly straightforward for the first two students; they become interesting once we
reach the third student.
• The First Student. The first student should follow a simple decision rule for making a guess: if he sees a red marble, it is better to guess that the urn is majority-red; and if
he sees a blue marble, it is better to guess that the urn is majority-blue. (This is an
intuitively natural rule, and — as with the other conclusions we draw here — we will
justify it later mathematically using the model we develop in the subsequent sections.)
This means the first student’s guess conveys perfect information about what he has
seen.
1It’s important that the students believe this statement about probabilities. So you can imagine, if you like, that the experimenter has actually filled two urns with marbles. One has two red marbles and one blue marble, and the other urn contains two blue marbles and one red marble. One of these urns is selected at random, with equal probability on each urn, and this is the urn used in the experiment.
16.2. A SIMPLE HERDING EXPERIMENT 487
• The Second Student. If the second student sees the same color that the first student announced, then her choice is simple: she should guess this color as well.
Suppose she sees the opposite color — say that she sees red while the first guess was
blue. Since the first guess was exactly what the first student saw, the second student
can essentially reason as though she got to draw twice from the urn, seeing blue once
and red once. In this case, she is indi↵erent about which guess to make; we will assume
in this case that she breaks the tie by guessing the color she saw. Thus, whichever
color the second student draws, her guess too conveys perfect information about what
she has seen.
• The Third Student. Things start to get interesting here. If the first two students have guessed opposite colors, then the third student should just guess the color he sees,
since it will e↵ectively break the tie between the first two guesses.
But suppose the first two guesses have been the same — say they’ve both been blue —
and the third student draws red. Since we’ve decided that the first two guesses convey
perfect information, the third student can reason in this case as though he saw three
draws from the urn: two blue, and one red. Given this information, he should guess
that the urn is majority-blue, ignoring his own private information (which, taken by
itself, suggested that the urn is majority-red).
More generally, the point is that when the first two guesses are the same, the third
student should guess this color as well, regardless of which color he draws from the
urn. And the rest of class will only hear his guess; they don’t get to see which color
he’s drawn. In this case, an information cascade has begun. The third student makes
the same guess as the first two, regardless of which color he draws from the urn, and
hence regardless of his own private information.
• The Fourth Student and Onward. For purposes of this informal discussion, let’s consider just the “interesting” case above, in which the first two guesses were the same —
suppose they were both blue. In this case, we’ve argued that the third student will
also announce a guess of blue, regardless of what he actually saw.
Now consider the situation faced by the fourth student, getting ready to make a guess
having heard three guesses of “blue” in a row. She knows that the first two guesses
conveyed perfect information about what the first two students saw. She also knows
that, given this, the third student was going to guess “blue” no matter what he saw
— so his guess conveys no information.
As a result, the fourth student is in exactly the same situation — from the point of
view of making a decision — as the third student. Whatever color she draws, it will
488 CHAPTER 16. INFORMATION CASCADES
be outweighed by the two draws of blue by the first two students, and so she should
guess “blue” regardless of what she sees.
This will continue with all the subsequent students: if the first two guesses were “blue,”
then everyone in order will guess “blue” as well. (Of course, a completely symmetric
thing happens if the first two guesses are “red”.) An information cascade has taken
hold: no one is under the illusion that every single person is drawing a blue marble,
but once the first two guesses turn out “blue,” the future announced guesses become
worthless and so everyone’s best strategy is to rely on the limited genuine information
they have available.
In the next section, we’ll discuss a model of decision-making under uncertainty that
justifies the guesses made by the students. More generally, our discussion hasn’t considered
every possible eventuality (for example, what should you do if you’re the sixth student
and you’ve heard the guesses “blue, red, red, blue, blue”?), but our subsequent model will
actually predict an outcome for any sequence of guesses.
For now, though, let’s think about the particular scenario discussed here — the way in
which a cascade takes place as long as the first two guesses are the same. Although the setting
is very stylized, it teaches us a number of general principles about information cascades.
First, it shows how easily they can occur, given the right structural conditions. It also
shows how a bizarre pattern of decisions — each of a large group of students making exactly
the same guess — can take place even when all the decision-makers are being completely
rational.
Second, it shows that information cascades can lead to non-optimal outcomes. Suppose
for example that we have an urn that is majority-red. There is a 1 3
chance that the first
student draws a blue marble, and a 1 3
chance that the second student draws a blue marble;
since these draws are independent, there is a 1 3
· 1 3
= 1 9
chance that both do. In this case,
both of the first two guesses will be “blue”; so, as we have just argued, all subsequent guesses
will be “blue” — and all of these guesses will be wrong, since the urn is majority-red. This 1
9
chance of a population-wide error is not ameliorated by having many people participate,
since under rational decision-making, everyone will guess blue if the first two guesses are
blue, no matter how large the group is.
Third, this experiment illustrates that cascades — despite their potential to produce
long runs of conformity — can be fundamentally very fragile. Suppose, for example, that
in a class of 100 students, the first two guesses are “blue,” and all subsequent guesses are
proceeding — as predicted — to be “blue” as well. Now, suppose that students 50 and 51
both draw red marbles, and they each “cheat” by showing their marbles directly to the rest
of the class. In this case, the cascade has been broken: when student 52 gets up to make a
guess, she has four pieces of genuine information to go on: the colors observed by students
1, 2, 50, and 51. Since two of these colors are blue and two are red, she should make the
16.3. BAYES’ RULE: A MODEL OF DECISION-MAKING UNDER UNCERTAINTY489
A B
A B
Figure 16.1: Two events A and B in a sample space, and the joint event A \ B.
guess based on her own draw, which will break the tie.
The point is that everyone knew the initial run of 49 “blue” guesses had very little
information supporting it, and so it was easy for a fresh infusion of new information to
overturn it. This is the essential fragility of information cascades: even after they have
persisted for a long time, they can be overturned with comparatively little e↵ort.2
This style of experiment has generated a significant amount of subsequent research in
its own right, and understanding the extent to which human subjects follow the type of
behavior described above under real experimental conditions is a subtle issue [100, 223]. For
our purposes, however, the simple description of the experiment is intended to serve mainly
as a vivid illustration of some of the basic properties of information cascades in a controlled
setting. Having now developed some of these basic properties, we turn to the formulation
of a model that lets us reason precisely about the decision-making that takes place during a
cascade.
16.3 Bayes’ Rule: A Model of Decision-Making Under Uncertainty
If we want to build a mathematical model for how information cascades occur, it will nec-
essarily involve people asking themselves questions like, “What is the probability this is the
2It is important to note that not all imitative e↵ects are so easy to overturn. As we will see in the next chapter, for example, imitation based on direct-benefit e↵ects can be very di�cult to reverse once it is underway.
490 CHAPTER 16. INFORMATION CASCADES
better restaurant, given the reviews I’ve read and the crowds I see in each one?” Or, “What
is the probability this urn is majority-red, given the marble I just drew and the guesses I’ve
heard?” In other words, we need a way to determine probabilities of events given information
that is observed.
Conditional Probability and Bayes’ Rule. We will be computing the probability of
various events, and using these to reason about decision-making. In the context of the
experiment from Section 16.2, an event could be “The urn is majority-blue,” or “the first
student draws a blue marble.” Given any event A, we will denote its probability of occuring
by Pr [A]. Whether an event occurs or not is the result of certain random outcomes (which
urn was placed at the front of the room, which marble a particular student grabbed when
he reached in, and so forth). We therefore imagine a large sample space, in which each point
in the sample space consists of a particular realization for each of these random outcomes.
Given a sample space, events can be pictured graphically as in Figure 16.1: the unit-area
rectangle in the figure represents the sample space of all possible outcomes, and the event A
is then a region within this sample space — the set of all outcomes where event A occurs.
In the figure, the probability of A corresponds to the area of this region. The relationship
between two events can be illustrated graphically as well. In Figure 16.1 we see two events
A and B. The area where they overlap corresponds to the joint event when both A and B
occur. This event is the intersection of A and B, and it’s denoted by A \ B. If we think about the examples of questions at the start of this section, we see that it
is not enough to talk about the probability of an event A; rather, we need to consider the
probability of A, given that some other event B has occurred. For example, A may be the
event that the urn in the experiment from Section 16.2 is majority-blue, and B may be the
event that the ball you’ve drawn is blue. We will refer to this quantity as the conditional
probability of A given B, and denote it by Pr [A | B]. Again, the graphical depiction in Figure 16.1 is useful: to determine the conditional probability of A given B, we assume
that we are in the part of the sample space corresponding to B, and we want to know the
probability that we are also in A (that is, in A \ B). We can think of this as the fraction of the area of region B occupied by A \ B, and so we define
Pr [A | B] = Pr [A \ B]
Pr [B] . (16.1)
Similarly, the conditional probability of B given A is
Pr [B | A] = Pr [B \ A]
Pr [A] =
Pr [A \ B] Pr [A]
, (16.2)
where the second equality follows simply because A \ B and B \ A are the same set. Rewriting (16.1) and (16.2), we have
Pr [A | B] · Pr [B] = Pr [A \ B] = Pr [B | A] · Pr [A] , (16.3)
16.3. BAYES’ RULE: A MODEL OF DECISION-MAKING UNDER UNCERTAINTY491
and therefore, dividing through by Pr [B],
Pr [A | B] = Pr [A] · Pr [B | A]
Pr [B] . (16.4)
Equation (16.4) is called Bayes’ Rule. There is also a bit of extra useful terminology sur-
rounding Bayes’ Rule. When we want to make explicit that we’re interested in the e↵ect of
event B on the probability of an event A, we refer to Pr [A] as the prior probability of A,
since it reflects our understanding of the probability of A without knowing anything about
whether B has occurred. Correspondingly, we refer to Pr [A | B] as the posterior probability of A given B, since it reflects our new understanding of the probability of A now that we
know B has occurred. The e↵ect of knowing B is thus captured in the change from the prior
probability of A to the posterior probability of A, using Equation (16.4).
An Example of Bayes’ Rule. As noted above, we will be applying Bayes’ Rule in cases
where a decision-maker is assessing the probability that a particular choice is the best one,
given the event that he has received certain private information and/or observed certain
other decisions. To get used to Bayes’ Rule, we first work through a basic example that
illustrates how it is typically applied.
The example involves eyewitness testimony. Suppose that in some city 80% of taxi cabs
are black and the remaining 20% are yellow. A witness to a hit-and-run accident involving a
taxi states that the cab involved was yellow. Suppose that eyewitness testimony is imperfect
in the sense that witnesses sometimes misidentify the colors of cabs. In particular, let’s
suppose that if a taxi is yellow then a witness will claim it is yellow after the fact 80% of the
time; and if it is black, they will claim it is black 80% of the time.
Interpreting eyewitness testimony, therefore, is at some level a question of conditional
probability: what is the probability the cab is yellow (or black), given that the witness says
it is yellow? Introducing some notation, let true denote the true color of the cab, and let
report denote the reported color of the cab; let Y denote yellow and B denote black. We
are looking for the value of Pr [true = Y | report = Y ]. The data we have does not directly include the answer to this question, but we can
determine the answer using Bayes’ Rule. Applying Equation (16.4) with A equal to the
event true = Y and B equal to the event report = Y , we have
Pr [true = Y | report = Y ] = Pr [true = Y ] · Pr [report = Y | true = Y ] ·
Pr [report = Y ] . (16.5)
Now, we’ve been told that Pr [report = Y | true = Y ] is 0.8 (this is the accuracy of eyewitness testimony) and that Pr [true = Y ] is 0.2 (this is the frequency of yellow taxi cabs, and hence
provides the prior probability of the event true = Y ). We can also figure out the denominator
with a little work, as follows. There are two ways for a witness to report that a cab is yellow:
492 CHAPTER 16. INFORMATION CASCADES
one is for the cab to actually be yellow, and the other is for it to actually be black. The
probability of getting a report of yellow via the former option is
Pr [true = Y ] · Pr [report = Y | true = Y ] = 0.2 · 0.8 = 0.16,
and the probability of getting a report of yellow via the latter option is
Pr [true = B] · Pr [report = Y | true = B] = 0.8 · 0.2 = 0.16.
The probability of a report of yellow is the sum of these two probabilities,
Pr [report = Y ] = Pr [true = Y ] · Pr [report = Y | true = Y ] + Pr [true = B] · Pr [report = Y | true = B]
= 0.2 · 0.8 + 0.8 · 0.2 = 0.32.
We can now put everything together via Equation (16.5) so as to get
Pr [true = Y | report = Y ] = Pr [true = Y ] · Pr [report = Y | true = Y ]
Pr [report = Y ]
= 0.2 · 0.8
0.32 = 0.5.
So the conclusion is that if the witness says the cab was yellow, it is in fact equally
likely to have been yellow or black. Since the frequency of black and yellow cabs makes
black substantially more likely in the absence of any other information (0.8 versus 0.2), the
witness’s report had a substantial e↵ect on our beliefs about the color of the particular cab
involved. But the report should not lead us to believe that the cab was in fact more likely
to have been yellow than black.3
A second example: Spam filtering. As the example with taxi cabs illustrates, Bayes’
Rule is a fundamental way to make inferences from observations, and as such it is used in a
wide variety of settings. One application where it has been very influential is in e-mail spam
detection — automatically filtering unwanted e-mail out of a user’s incoming e-mail stream.
Bayes’ Rule was a crucial conceptual ingredient in the first generation of e-mail spam filters,
and it continues to form part of the foundation for many spam filters [187].
We can appreciate the connection between Bayes’ Rule and spam filtering through the
following example. Suppose that you receive a piece of e-mail whose subject line contains
3Kahneman and Tversky have run an experiment with a similar example which shows that people some- times do not make predictions according to Bayes’ Rule [231]. In their experiment, subjects place too much weight on their observations and too little weight on prior probabilities. The e↵ect of errors in predictions on actions, and the subsequent e↵ect on cascades is an interesting topic, but we will not address it here.
16.3. BAYES’ RULE: A MODEL OF DECISION-MAKING UNDER UNCERTAINTY493
the phrase “check this out” (a popular phrase among spammers). Based just on this (and
without looking at the sender or the message content), what is the chance the message is
spam?
This is already a question about conditional probability: we’re asking for the value of
Pr [message is spam | subject contains “check this out”] .
To make this equation and the ones that follow a bit simpler to read, let’s abbreviate message
is spam to just spam, and abbreviate subject contains “check this out” to just “check this
out”; so we want the value of
Pr [spam | “check this out”] .
To determine this value, we need to know some facts about your e-mail and the general
use of the phrase “check this out” in subject lines. Suppose that 40% of all your e-mail is
spam and the remaining 60% is e-mail you want to receive. Also, suppose that 1% of all
spam messages contain the phrase “check this out” in their subject lines, while 0.4% of all
non-spam messages contain this phrase. Writing these in terms of probabilities, it says that
Pr [spam] = 0.4; this is the prior probability that an incoming message is spam (without
conditioning on events based on the message itself). Also, we have
Pr [“check this out” | spam] = .01
and
Pr [“check this out” | not spam] = .004
We’re now in a situation completely analogous to the calculations involving eyewitness tes-
timony: we can use Bayes’ Rule to write
Pr [spam | “check this out”] = Pr [spam] · Pr [“check this out” | spam]
Pr [“check this out” ] .
Based on what we know, we can determine that the numerator is .4 · .01 = .004. For the denominator, as in the taxicab example, we note that there are two ways for a message
to contain “check this out” — either by being spam or by not being spam. As in that
calculation,
Pr [“check this out”] = Pr [spam] · Pr [“check this out” | spam] + Pr [not spam] · Pr [“check this out” | not spam]
= .4 · .01 + .6 · .004 = .0064.
Dividing numerator by denominator, we get our answer:
Pr [spam | “check this out”] = .004
.0064 =
5
8 = .625.
494 CHAPTER 16. INFORMATION CASCADES
In other words, although spam (in this example) forms less than half of your incoming e-mail,
a message whose subject line contains the phrase “check this out” is — in the absence of
any other information — more likely to be spam than not.
We can therefore view the presence of this phrase in the subject line as a weak “signal”
about the message, providing us with evidence about whether it’s spam. In practice, spam
filters built on Bayes’ Rule look for a wide range of di↵erent signals in each message — the
words in the message body, the words in the subject, properties of the sender (do you know
them? what kind of an e-mail address are they using?), properties of the mail program used
to compose the message, and other features. Each of these provides its own estimate for
whether the message is spam or not, and spam filters then combine these estimates to arrive
at an overall guess about whether the message is spam. For example, if we also knew that
the message above came from someone you send mail to every day, then presumably this
competing signal — strongly indicating that the message is not spam — should outweigh
the presence of the phrase “check this out” in the subject.
16.4 Bayes’ Rule in the Herding Experiment
Let’s now use Bayes’ Rule to justify the reasoning that the students used in the simple herding
experiment from Section 16.2. First, notice that each student’s decision is intrinsically based
on determining a conditional probability: each student is trying to estimate the conditional
probability that the urn is majority-blue or majority-red, given what she has seen and heard.
To maximize her chance of winning the monetary reward for guessing correctly, she should
guess majority-blue if
Pr [majority-blue | what she has seen and heard] > 1
2 and guess majority-red otherwise. If the two conditional probabilities are both exactly 0.5,
then it doesn’t matter what she guesses.
We know the following facts from the set-up of the experiment, before anyone has drawn
any marbles. First, the prior probabilities of majority-blue and majority-red are each 1 2
:
Pr [majority-blue] = Pr [majority-red] = 1
2 .
Also, based on the composition of the two kinds of urns,
Pr [blue | majority-blue] = Pr [red | majority-red] = 2
3 .
Now, following the scenario from Section 16.2, let’s suppose that the first student draws
a blue marble. He therefore wants to determine Pr [majority-blue | blue], and just as in the examples from Section 16.3, he can use Bayes’ Rule to calculate
Pr [majority-blue | blue] = Pr [majority-blue] · Pr [blue | majority-blue]
Pr [blue] . (16.6)
16.4. BAYES’ RULE IN THE HERDING EXPERIMENT 495
The numerator is 1 2
· 2 3
= 1 3
. For the denominator, we reason just as in Section 16.3 by noting
that there are two possible ways to get a blue marble — if the urn is majority-blue, or if it
is majority-red:
Pr [blue] = Pr [majority-blue] · Pr [blue | majority-blue] + Pr [majority-red] · Pr [blue | majority-red]
= 1
2 · 2
3 +
1
2 · 1
3 =
1
2 .
The answer Pr [blue] = 1 2
makes sense, given that the roles of blue and red in this experiment
are completely symmetric.
Dividing numerator by denominator, we get
Pr [majority-blue | blue] = 1/3
1/2 =
2
3 .
Since this conditional probability is greater than 1 2
, we get the intuitive result that the first
student should guess majority-blue when he sees a blue marble. Note that in addition to
providing the basis for the guess, Bayes’ Rule also provides a probability, namely 2 3
, that the
guess will be correct.
The calculation is very similar for the second student, and we skip this here to move
on to the calculation for the third student, where a cascade begins to form. Let’s suppose,
as in the scenario from Section 16.2, that the first two students have announced guesses of
blue, and the third student draws a red marble. As we discussed there, the first two guesses
convey genuine information, so the third student knows that there have been three draws
from the urn, consisting of the sequence of colors blue, blue, and red. What he wants to
know is
Pr [majority-blue | blue, blue, red]
so as to make a guess about the urn. Using Bayes’ Rule we get
Pr [majority-blue | blue, blue, red] = Pr [majority-blue] · Pr [blue, blue, red | majority-blue]
Pr [blue, blue, red] .
(16.7)
Since the draws from the urn are independent, the probability Pr [blue, blue, red | majority-blue] is determined by multiplying the probabilities of the three respective draws together:
Pr [blue, blue, red | majority-blue] = 2
3 · 2
3 · 1
3 =
4
27 .
To determine Pr [blue, blue, red], as usual we consider the two di↵erent ways this sequence
could have happened — if the urn is majority-blue, or if it is majority-red:
Pr [blue, blue, red] = Pr [majority-blue] · Pr [blue, blue, red | majority-blue] + Pr [majority-red] · Pr [blue, blue, red | majority-red]
= 1
2 · 2
3 · 2
3 · 1
3 +
1
2 · 1
3 · 1
3 · 2
3 =
6
54 =
1
9 .
496 CHAPTER 16. INFORMATION CASCADES
Plugging all this back into Equation (16.7), we get
Pr [majority-blue | blue, blue, red] = 4
27
· 1 2
1
9
= 2
3 .
Therefore, the third student should guess majority-blue (from which he will have a 2 3
chance
of being correct) — this confirms our intuitive observation in Section 16.2 that the student
should ignore what he sees (red) in favor of the two guesses he’s already heard (both blue).
Finally, once these three draws from the urn have taken place, all future students will have
the same information as the third student, and so they will all perform the same calculation,
resulting in an information cascade of blue guesses.
16.5 A Simple, General Cascade Model
Let’s return to the motivation for the herding experiment in Section 16.2: the experiment
served as a stylized metaphor for any situation in which people make decisions sequentially,
basing these decisions on a combination of their own private information and observations
of what earlier people have done. We now formulate a model that covers such situations in
general. We will see that Bayes’ Rule predicts in this general model that cascades will form,
with probability tending to 1 as the number of people goes to infinity.
Formulating the Model. Consider a group of people (numbered 1, 2, 3, . . .) who will
sequentially make decisions — that is, individual 1 will decide first, then individual 2 will
decide, and so on. We will describe the decision as a choice between accepting or rejecting
some option: this could be a decision about whether to adopt a new technology, wear a new
fashion, eat in a new restaurant, commit a crime, vote for a particular political candidate,
or choose one route to a common destination rather than an alternative route.
First model ingredient: States of the world. At the start of everything, before any indi-
vidual has made a decision, we assume that the world is randomly placed into one of two
possible states: it is either placed in a state in which the option is actually a good idea, or
a state in which the option is actually a bad idea. We imagine that the state of the world
is determined by some initial random event that the individuals can’t observe, but they will
try to use what they observe to make inferences about this state. For example, the world is
either in a state where the new restaurant is good or a state where it is bad; the individuals
in the model know that it was randomly placed in one of these two states, and they’re trying
to figure out which.
We write the two possible states of the world as G, representing the state where the
option is a good idea, and B, representing the state where the option is a bad idea. We
suppose that each individual knows the following fact: the initial random event that placed
16.5. A SIMPLE, GENERAL CASCADE MODEL 497
the world into state G or B placed it into state G with probability p, and into state B with
probability 1 � p. This will serve as the prior probabilities of G and B; in other words, Pr [G] = p, and hence Pr [B] = 1 � Pr [G] = 1 � p.
Second model ingredient: Payo↵s. Each individual receives a payo↵ based on her decision
to accept or reject the option. If the individual chooses to reject the option, she receives a
payo↵ of 0. The payo↵ for accepting depends on whether the option is a good idea or a bad
idea. Let’s suppose that if the option is a good idea, then the payo↵ obtained from accepting
it is a positive number v g
> 0. If the option is a bad idea, then the payo↵ is a negative
number v b
< 0. We will also assume that the expected payo↵ from accepting in the absence
of other information is equal to 0; in other words, v g
p + v b
(1 � p) = 0. That is, before an individual gets any additional information, the expected payo↵ from accepting is the same
as the payo↵ from rejecting.
Third model ingredient: Signals. In addition to the payo↵s, we also want to model the
e↵ect of private information. We assume that before any decisions are made, each individual
gets a private signal that provides information about whether accepting is a good idea or
a bad idea. The private signal is designed to model private information that the person
happens to know, beyond just the prior probability p that accepting the option is a good
idea.
The private signal does not convey perfect certainty about what to do (since we want
to model individual uncertainty even after the signal comes in), but it does convey useful
information. Specifically, there are two possible signals: a high signal (denoted H), suggesting
that accepting is a good idea; and a low signal (denoted L), suggesting that accepting
is a bad idea. We can make this precise by saying that if accepting is in fact a good
idea, then high signals are more frequent than low signals: Pr [H | G] = q > 1 2
, while
Pr [L | G] = 1 � q < 1 2
. Similarly, if accepting the option is a bad idea, then low signals are
more frequent: Pr [L | B] = q and Pr [H | B] = 1 � q, for this same value of q > 1 2
. This is
summed up in the table in Figure 16.2.
Signals
States B G
L q 1 � q H 1 � q q
Figure 16.2: The probability of receiving a low or high signal, as a function of the two possible states of the world (G or B).
Notice how the herding experiment from Section 16.2 fits the properties of this more
abstract model. The two possible states of the world are that the urn placed at the front
of the room was majority-blue, or that it was majority-red. We can think of “accepting” as
guessing “majority-blue”; this is a good idea (G) if the true urn really is majority-blue and
498 CHAPTER 16. INFORMATION CASCADES
a bad idea (B) otherwise. The prior probability of accepting being a good idea is p = 1 2
.
The private information in the experiment is the color of the ball the individual draws; it’s
a “high” signal if it is blue, and so Pr [H | G] = Pr [blue | majority-blue] = q = 2 3
.
Similarly, to return to the two-restaurant example from the opening section, “accepting”
could correspond to choosing the first restaurant A; it’s a good idea if restaurant A is actually
better than the second restaurant B. The private information could be a review that you read
of the first restaurant, with a high signal corresponding to a review comparing it favorably
to restaurant B next door. If choosing the first restaurant is actually good, there should be
a higher number of such reviews, so Pr [H | G] = q > 1 2
.
Individual decisions. We now want to model how people should make decisions about
accepting or rejecting. First, let’s consider how someone should do this based only on their
own private signal, and then consider the e↵ect of observing the earlier decisions of others.
Suppose that a person gets a high signal. This shifts their expected payo↵ from v g
Pr [G]+
v b
Pr [B] = 0 to v g
Pr [G | H] + v b
Pr [B | H]. To determine this new expected payo↵, we use Bayes’ Rule; the calculation is just like the ones from the previous sections.
Pr [G | H] = Pr [G] · Pr [H | G]
Pr [H]
= Pr [G] · Pr [H | G]
Pr [G] · Pr [H | G] + Pr [B] · Pr [H | B] =
pq
pq + (1 � p)(1 � q) > p,
where in the second line we compute the denominator Pr [H] as usual by expanding out the
two possible ways of getting a high signal (if the option is a good idea or a bad idea); and
the final inequality follows since pq + (1�p)(1�q) < pq + (1�p)q = q in the denominator. This result makes good sense: A high signal is more likely to occur if the option is good
than if it is bad, so if an individual observes a high signal they raise their estimate of the
probability that the option is good. As a result, the expected payo↵ shifts from 0 to a
positive number, and so they should accept the option.
A completely analogous calculation shows that if the individual receives a low signal,
they should reject the option.
Multiple signals. We know from the herding experiment that an important step in rea-
soning about how people make decisions in sequence is to understand how an individual
should use the evidence of multiple signals. Using Bayes’ Rule, it’s not hard to reason di-
rectly about an individual’s decision when they get a sequence S of independently generated
signals consisting of a high signals and b low signals, interleaved in some fashion. We do this
below, deriving the following facts:
16.5. A SIMPLE, GENERAL CASCADE MODEL 499
(i) the posterior probability Pr [G | S] is greater than the prior Pr [G] when a > b;
(ii) the posterior Pr [G | S] is less than the prior Pr [G] when a < b; and
(iii) the two probabilities Pr [G | S] and Pr [G] are equal when a = b.
As a result, individuals should accept the option when they get more high signals than low
signals, and reject it when they get more low signals than high signals; they are indi↵erent
when they get the same number of each. In other words, in this simple setting with a
sequence of signals, individuals can decide according to a majority-vote over the signals they
receive.
In the remainder of this section, we justify these facts (i)-(iii), using Bayes’ Rule and a
bit of algebra. In the next section, we then explore the consequences of this for sequential
decision-making in this model.
To apply Bayes’ Rule, we write
Pr [G | S] = Pr [G] · Pr [S | G]
Pr [S] , (16.8)
where S is a sequence with a high signals and b low signals. To compute Pr [S | G] in the numerator, we note that since the signals are generated independently, we can simply
multiply their probabilities: this gives us a factors of q and b factors of (1 � q), and so Pr [S | G] = qa(1 � q)b.
To compute Pr [S], we consider that S can arise if the option is a good idea or a bad
idea, so
Pr [S] = Pr [G] · Pr [S | G] + Pr [B] · Pr [S | B] = pqa(1 � q)b + (1 � p)(1 � q)aqb.
Plugging this back into Equation (16.8), we get
Pr [G | S] = pqa(1 � q)b
pqa(1 � q)b + (1 � p)(1 � q)aqb .
What we want to know is how this expression compares to p. One way to answer this question
is as follows. If we were to replace the second term in the denominator by (1�p)qa(1�q)b, then the denominator would become pqa(1�q)b + (1�p)qa(1�q)b = qa(1�q)b, and so the whole expression would become
pqa(1 � q)b
qa(1 � q)b = p.
So the question is: does this replacement make the denominator smaller or larger?
500 CHAPTER 16. INFORMATION CASCADES
(i) If a > b, then this replacement makes the denominator larger, since q > 1 2
and we now
have more factors of q and fewer factors of 1 � q. Since the denominator gets larger, the overall expression gets smaller as it is converted to a value of p, and therefore
Pr [G | S] > p = Pr [G].
(ii) If a < b, the argument is symmetric: this replacement makes the denominator smaller,
and hence the overall expression larger. So Pr [G | S] < p = Pr [G].
(ii) Finally, if a = b, then this replacement keeps the value of the denominator the same,
and so Pr [G | S] = p = Pr [G].
16.6 Sequential Decision-Making and Cascades
Let’s now consider what happens when individuals make decisions in sequence. As before, we
want to capture situations in which each person can see what earlier people do, but not what
they know. In our model, this means that when a given person decides whether to accept
or reject the option, they have access to their own private signal and also the accept/reject
decisions of all earlier people. Crucially, however, they do not see the actual private signals
of any of these earlier people.
The reasoning is now very similar to what we did for the sequence of students in the
herding experiment from Section 16.2. To start, let’s note the close parallels to our discussion
there.
• Person 1 will follow his own private signal, as we just saw in Section 16.5.
• Person 2 will know that person 1’s decision reveals their private signal, and so it’s as though person 2 gets two signals. If these signals are the same, person 2’s decision is
easy. If they are di↵erent, then as we saw at the end of Section 16.5, person 2 will be
indi↵erent between accepting and rejecting. Here we will assume she follows her own
private signal. Thus, either way, person 2 is following her own signal.
• As a result, person 3 knows that person 1 and person 2 both acted on their private signals, so it is as though person 3 has received three independent signals (the two he
infers, and his own). We know from the argument in Section 16.5 that person 3 will
follow the majority signal (high or low) in choosing whether to accept or reject.
This means that if person 1 and person 2 made opposite decisions (i.e. they received
opposite signals), then person 3 will use his own signal as the tie-breaker. Hence future
people will know that person 3’s decision was based on his own signal, and so they can
use this information in their own decisions.
16.6. SEQUENTIAL DECISION-MAKING AND CASCADES 501
0
1
2
3
-1
-2
-3
1 2 3 4 5 6 7 people
#acc - #rej
cascade starts above this line
cascade starts below this line
Figure 16.3: A cascade begins when the di↵erence between the number of acceptances and rejections reaches two.
On the other hand, if person 1 and person 2 made the same decision (i.e. had the
same signal), then person 3 will follow this regardless of what his own signal says.
Hence future people will know that person 3’s decision conveys no information about
his signal, and future people will all be in the same position as person 3. In this case, a
cascade has begun. That is, we are in a situation where no individual’s decision can be
influenced by his own signal. No matter what they see, every individual from person
3 on will make the same decision that 1 and 2 made.
Let’s now consider how this process unfolds through future people beyond person 3. In
particular, let’s consider the perspective of a person numbered N . Suppose that person
N knows that everyone before her has followed their own signal — that is, suppose the
accept/reject decisions of these earlier people exactly coincide with whether they received a
high or low signal, and person N knows this. There are several possible cases to consider.
• If the number of acceptances among the people before N is equal to the number of rejections, then N ’s signal will be the tie-breaker, and so N will follow her own signal.
• If the number of acceptances among the people before N di↵ers from the number of rejections by one, then either N ’s private signal will make her indi↵erent, or it will
reinforce the majority signal. Either way, N will follow her private signal (since we
assume a person follows their own signal in the case of indi↵erence).
502 CHAPTER 16. INFORMATION CASCADES
• If the number of acceptances among the people before N di↵ers from the number of rejections by two or more, then however N ’s private signal turns out, it won’t outweigh
this earlier majority. As a result, N will follow the earlier majority and ignore her own
signal.
Moreover, in this case, the people numbered N + 1, N + 2, and onward will know that
person N ignored her own signal (whereas we’ve assumed that all earlier people were
known to have followed their private signals). So they will each be in exactly the same
position as N . This means that each of them too will ignore their own signals and
follow the majority, and hence a cascade has begun.
We can therefore sum up the behavior of the decision-making process as follows. As long
as the number of acceptances di↵ers from the number of rejections by at most one, each
person in sequence is simply following their own private signal in deciding what to do. But
once the number of acceptances di↵ers from the number of rejections by two or more, a
cascade takes over, and everyone simply follows the majority decision forever. Figure 16.3
illustrates this for a sample outcome of the process, in which we plot the di↵erence between
the number of acceptances and the number of rejections over time as people make decisions.
This plot moves up or down by one each time a new decision is made, since either the number
of acceptances or the number of rejections grows by exactly one with each decision. Once
the di↵erence between the number of acceptances and the number of rejections escapes from
the narrow horizontal ribbon around zero — that is, once the plots moves at least two steps
away from the x-axis — a cascade begins and runs forever.
Finally, it is very hard for this di↵erence to remain in such a narrow interval (between
�1 and +1) forever. For example, during the period of time when people are following their own signals, if three people in a row ever happen to get the same signal, a cascade will
definitely have begun. (Note that a cascade can begin even without this happening, but you
can check that three matching signals in a row are always enough to make one begin.) Now,
let’s argue that the probability of finding three matching signals in a row is converging to
1 as the number of people N goes to infinity. To see why, suppose we divide the first N
people into blocks of three consecutive people each (people 1, 2, 3, people 4, 5, 6, people
7, 8, 9, and so on). Then the people in any one block will receive identical signals with
probability q3 + (1 � q)3. The probability that none of these blocks consists of identical signals is therefore (1 � q3 � (1 � q)3)N/3. As N goes to infinity this quantity goes to 0.
What this short argument shows is that, as the number of people goes to infinity, the
probability a cascade begins converges to 1. Thus, in the limit, a cascade takes place in this
model almost surely.
Having now completed the analysis, it is worth making a few final observations. First,
this is ultimately an extremely simple model of individual decision-making. In more general
versions, for example, it could well be the case that people don’t see all the decisions made
16.7. LESSONS FROM CASCADES 503
earlier but only some of them; that not all private signals convey equal information; or
that not everyone receives the same payo↵s [2, 38, 186]. Many of these more general variants
become much more complicated to analyze, and they can di↵er in their specifics (for example,
the condition for a cascade to begin is clearly not always as simple as having the number of
acceptances di↵er from the number of rejections by at least two). But the general conclusions
from these models tend to be qualitatively similar: when people can see what others do
but not what they know, there is an initial period when people rely on their own private
information, but as time goes on, the population can tip into a situation where people — still
behaving fully rationally — begin ignoring their own information and following the crowd.
We now turn to some of the general conclusions from these cascade models, relating
them to more qualitative principles that have been observed about information cascades in
di↵erent settings.
16.7 Lessons from Cascades
At the end of Section 16.2, we made some observations about the simple herding experiment
that are reinforced by the general model we’ve just analyzed.
(i) Cascades can be wrong. If, for example, accepting the option is in fact a bad idea
but the first two people happen to get high signals, a cascade of acceptances will start
immediately, even though it is the wrong choice for the population.
(ii) Cascades can be based on very little information. Since people ignore their private
information once a cascade starts, only the pre-cascade information influences the
behavior of the population. This means that if a cascade starts relatively quickly in a
large population, most of the private information that is collectively available to the
population (in the form of private signals to individuals) is not being used.
(iii) Cascades are fragile. The previous point, that cascades can be based on relatively
little information, makes them easy to start; but it can also make them easy to stop.
One manifestation of this is that people who receive slightly superior information can
overturn even long-lived cascades.
Suppose, for example, that a cascade of acceptances is underway in our model; this
means that the number of high signals exceeded the number of low signals by two at
the time the cascade began. Now suppose someone making a decision in the midst
of this cascade happens to receive two private signals. If they are both low signals,
then this person (taking into account the earlier signals he can infer) has now seen an
equal number of high and low signals. Since he is indi↵erent, our assumption is that
he will reject (since his own signal(s) were low), despite the long run of acceptances
504 CHAPTER 16. INFORMATION CASCADES
that preceded him. A single public signal can have the same e↵ect: if, in the midst of
a cascade, there is a public signal that everyone sees, then the next person to decide
in e↵ect receives two signals (the public one and her own private one), with similar
consequences.
More generally, perhaps the main lesson to be learned from studying cascades is to be
careful in drawing conclusions about the best course of action from the behavior of a crowd.
As we have just seen, the crowd can be wrong even if everyone is rational and everyone takes
the same action.
This forms an interesting contrast with an argument made by popular general-audience
books such as James Surowiecki’s The Wisdom of Crowds [383], that the aggregate behavior
of many people with limited information can sometimes produce very accurate results. In his
opening example, Surowiecki notes that if many people are guessing independently, then the
average of their guesses is often a surprisingly good estimate of whatever they are guessing
about (perhaps the number of jelly beans in a jar, or the weight of a bull at a fair). The
key to this argument of course is that the individuals each have private information (their
signals), and they guess independently, without knowing what the others have guessed. If
instead they guess sequentially, and can observe the earlier guesses of others, then we are
back in the cascade setting and there would be no reason to expect the average guess to be
good at all. Surowiecki also notes this possibility of cascades as a caution in following the
crowd.
These observations suggest how the possibility of cascades can a↵ect the behavior of
individuals or groups in a range of di↵erent situations. One setting that is susceptible to
cascades is the style of group decision-making in which a committee of people sit around a
table and discuss potential solutions to a problem. For example, consider a hiring committee
that needs to decide whether to make a job o↵er to candidate A or candidate B. In these
kinds of situations, a common strategy is to go around the table, asking people in sequence
to express their support for option A or option B. But if the participants assume that they
all have roughly equal insight into the problem, then a cascade can quickly develop: if a few
people initially favor A, others may be led to conclude that they should favor A, even if they
initially preferred B on their own. The cascade principles we’ve seen in this chapter suggest
that this may not just be an issue of social pressure to conform to the majority, but in fact a
rational approach to decision-making, in which you assume that the people speaking before
you have information about the problem that is comparable in quality to your own.
Such considerations suggest an inherent tension between getting a group of experts to
work together and build on each other’s ideas, on the one hand, and giving them the oppor-
tunity to form their own opinions, on the other. This in turn suggests strategies for balancing
this tension, potentially by forcing experts to reach partial decisions independently before
entering a phase of collaboration and consensus. It also suggests that if certain people are
16.8. EXERCISES 505
known to have particularly good information about a problem, it can matter whether they
weigh in earlier in the process or later.
Marketers also use the idea of cascades to attempt to get a buying cascade started for
a new product. If they can induce an initial set of people to adopt the new product, then
those who make purchasing decisions later on may also adopt the product even it is no
better than, or perhaps even worse than, competing products. This is most e↵ective if these
later consumers are able to observe the adoption decisions, but not how satisfied the early
customers actually were with their choice — this is consistent with the idea that cascades
arise naturally when people can see what others do but not what they know. If the payo↵s
(or statistics based on the payo↵s) from earlier consumers are available, this can help prevent
a cascade of bad choices.
16.8 Exercises
1. In this problem we will ask whether an information cascade can occur if each individual
sees only the action of his immediate neighbor rather than the actions of all those who
have chosen previously. Let’s keep the same setup as in the Information Cascades
chapter, except than when individual i chooses he observes only his own signal and the
action of individual i � 1.
(a) Briefly explain why the decision problems faced by individuals 1 and 2 are un-
changed by this modification to the information network.
(b) Individual 3 observes the action of individual 2, but not the action of individual
1. What can 3 infer about 2’s signal from 2’s action?
(c) Can 3 infer anything about 1’s signal from 2’s action? Explain.
(d) What should 3 do if he observes a high signal and he knows that 2 Accepted?
What if 3’s signal was low and 2 Accepted?
(e) Do you think that a cascade can form in this world? Explain why or why not. A
formal proof is not necessary, a brief argument is su�cient.
2. In this question we consider a variation on the model of information cascades from
Chapter 16. Suppose that there is a new technology which individuals sequentially
decide to adopt or reject. Let’s suppose that anyone who adopts the new technology
receives either a positive or a negative payo↵ from using the new technology. Unlike
the model used in Chapter 16, these payo↵s are random and have the property that
the average payo↵ is positive if the technology is Good, and negative if the technology
is Bad. Anyone who decides to reject the new technology always receives a payo↵ of
exactly zero.
506 CHAPTER 16. INFORMATION CASCADES
As in the model used in Chapter 16, each person receives a private signal about the
technology and observes the actions of all those who chose previously. However, unlike
the model used in Chapter 16, each person is also told the payo↵s received by each of
those who moved previously. [One interpretation of this is that a government agency
collects information about individuals’ experiences and distributes it for free as a public
service.]
(a) Suppose that the new technology is actually Bad. How does this new information
about payo↵s (the payo↵s received by each of those who moved previously) a↵ect the
potential for an information cascade of choices to adopt the new technology to form
and persist? [You do not need to write a proof. A brief argument is su�cient.]
(b) Suppose that the new technology is actually Good. Can an information cascade
of rejections of the new technology occur? Explain briefly.
3. In this problem we will consider the information cascades model from Chapter 16 with
specific values for the probabilities. Let’s suppose that the probability that Accept is
a good idea is p = 1/2; and the probability of a High signal if Good is true (as well as
the probability of a Low signal if Bad is true) is q = 3/4. Finally, let’s assume that
Good is actually true.
(a) What is the probability that the first person to decide will choose Accept; what’s
the probability that this person will choose Reject?
(b) What is the probability of observing each of the four possible pairs of choices by
the first two people: (A,A), (A,R), (R,A), and (R,R)? [A pair of choices such as (A,R)
means that the first person chose Accept and second person chose Reject.]
(c) What is the probability of an Accept or a Reject cascade emerging with the decision
by the third person to choose? Explain why a cascade emerges with this probability.
4. Let’s consider the model of information cascades. Assume that the probability that the
state is Good (G) is p = 1/2, and that the probability of a High signal given a Good
state is q = 2/3. (The probability of a Low signal given a Bad state is also q = 2/3.)
Remember that each person observes a signal and the choices (but not the signals) of
all those who chose before him. Each person chooses between Accept (A) and Reject
(R).
Suppose that you are the tenth person to make a choice and you have observed that
everyone before you chose R. That is, we are in an R-cascade.
(a) What is the probability that this is an incorrect cascade? [The probability that
the state is actually G given that we are in this R-cascade.]
16.8. EXERCISES 507
(b) Now let’s suppose that before you (person 10) receive your signal, you decide to
ask person 9 about the signal that they observed. Let’s suppose that person 9 observed
a High signal, that person 9 tells you that his signal was High, and that you know that
person 9 is telling the truth. After this, you receive your own signal. What decision
should you make, A or R, and how does it depend on which signal you receive?
(c) Now let’s consider person 11. Person 11 observes only his own signal and the choices
of those who decided before him (1 to 10). Person 11 knows that you have observed
both your signal and person 9’s signal. Person 11 cannot observe these signals; all
he knows is the choices that have been made. The first nine people have chosen R.
What should person 11 do if you choose R? What should he do if you choose A? Why?
Remember that person 11 observes a signal, and so his choice can depend on his signal
as well as the earlier choices.
5. Suppose you’re working at a company, and your boss asks you to explain what went
wrong in a recent hiring decision. The company decided to interview two candidates
for a single job. Let’s call the two candidates A and B. A hiring committee was formed
to attend the interviews and decide which of the two candidates to hire. Everyone on
the committee was interested in making the best possible hire, but after the interview
it was clear that members of the committee had di↵erent ideas about which of the two
candidates was the best choice. When the committee met to make the final decision
they decided to go around the room and ask each person on the committee to announce
which of the two candidates they believed to be the best choice for the company. In
fact, everyone on the committee said that candidate A seemed to be the best choice,
so the o↵er was made immediately to candidate A without additional discussion.
Now that candidate A has worked for the firm for a while it is clear that candidate B
would have been a better choice.
(a) Your boss has asked you to explain how the committee members could have
unanimously supported candidate A when she was reasonably certain that before
the committee meeting at least some of the members of the committee thought
that B was probably the best choice. What can you tell her?
(b) Can you suggest another procedure that the committee could have used that would
have revealed the initially di↵ering opinions about the candidates and which might
have resulted in the actually better choice of candidate B?
6. You have to make a choice between two alternatives. These alternatives might be,
for example, whether to believe a rumor or not, which of two competing products to
purchase, which of two competing political candidates to vote for, or which of two
508 CHAPTER 16. INFORMATION CASCADES
possible technologies to adopt for the new firm that you have just started. Unfortu-
nately, you do not know much about the potential benefits of choosing either of the
alternatives. We will represent this formally by saying that you believe that each of
the alternatives is equally likely to be the best choice. However, there are experts who
do have information about the benefit (to you) from each of the alternatives. Experts
are not perfect; they just know more than you do. We will represent this formally by
saying that each expert has some imperfect, private information about the benefit of
each alternative and we will assume that all experts are equally good in evaluating the
two alternatives.
The experts have made public recommendations about which of the alternatives is
best. (The experts cannot convey their information directly. It’s just too complex and
if they tried you would not know what to do with their statements anyhow.) Experts
make recommendations sequentially and each expert knows what all of those who have
announced their recommendation earlier have chosen. (This is a bit extreme, but the
opposite case in which they simultaneously choose is even more extreme.) You see the
recommendations of all of the experts, but you do not know the order in which they
made those recommendations. Finally, we will assume that experts are honest; that is,
they always make the recommendation that they believe is best (for you) given their
private information and any inferences that they can draw from the recommendations
of other experts.
(a) Suppose that the majority of the experts recommend alternative A. How confident
should you be that A is in fact the best choice? Should you be more confident that A
is the best choice if the fraction of experts who recommend A is even larger, i.e. not
just more than one-half, but close to one? Explain. [You can’t provide a numerical
answer to this question. Just discuss the inference that you can make from seeing
recommendations.]
(b) Suppose now that experts do not make public recommendations. Instead, you have
to hire an expert in order to obtain his or her recommendation. Let’s also suppose that
experts don’t talk to each other before you hire them; instead they privately obtain
information and update their opinion about the alternatives. You know experts are not
perfect so you plan to hire five of them in the hope that having more opinions is better.
Consider two procedures for how to organize the process of obtaining the experts’
recommendations. In procedure I, bring all of the experts together in a room and ask
them to announce their recommendations sequentially. In procedure II, privately ask
each expert to announce their recommendation. Which procedure will provide you
with the most information? Why?
Chapter 17
Network E↵ects
At the beginning of Chapter 16, we discussed two fundamentally di↵erent reasons why
individuals might imitate the behavior of others. One reason was based on informational
e↵ects: since the behavior of other people conveys information about what they know, observ-
ing this behavior and copying it (even against the evidence of one’s own private information)
can sometimes be a rational decision. This was our focus in Chapter 16. The other reason
was based on direct-benefit e↵ects, also called network e↵ects: for some kinds of decisions,
you incur an explicit benefit when you align your behavior with the behavior of others. This
is what we will consider in this chapter.
A natural setting where network e↵ects arise is in the adoption of technologies for which
interaction or compatibility with others is important. For example, when the fax machine
was first introduced as a product, its value to a potential consumer depended on how many
others were also using the same technology. The value of a social-networking or media-
sharing site exhibits the same properties: it’s valuable to the extent that other people are
using it as well. Similarly, a computer operating system can be more useful if many other
people are using it: even if the primary purpose of the operating system itself is not to
interact with others, an operating system with more users will tend to have a larger amount
of software written for it, and will use file formats (e.g. for documents, images, and movies)
that more people can easily read.
Network E↵ects as Externalities. The e↵ects we are describing here are called positive
externalities. An externality is any situation in which the welfare of an individual is a↵ected
by the actions of other individuals, without a mutually agreed-upon compensation. For
example, the benefit to you from a social networking site is directly related to the total
number of people who use the site. When someone else joins the site, they have increased your
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
509
510 CHAPTER 17. NETWORK EFFECTS
welfare even though no explicit compensation accounts for this. This is an externality, and
it is positive in the sense that your welfare increases. In this chapter, we will be considering
the consequences of positive externalities due to network e↵ects. In the settings we analyze
here, payo↵s depend on the number of others who use a good and not on the details of how
they are connected. In Chapter 19, we will look at the details of network connectivity and
ask how they a↵ect the positive externalities that result.
Notice that we have also seen examples of negative externalities earlier in the book —
these are cases where an externality causes a decrease in welfare. Tra�c congestion as dis-
cussed in Chapter 8 is an example in which your use of a (transportation or communication)
network decreases the payo↵ to other users of the network, again despite the lack of com-
pensation among the a↵ected parties. In the final section of this chapter, we will look at a
direct comparison of positive and negative externalities in more detail.
It’s important, also, to note that not everything is an externality — the key part is that
the e↵ect has to be uncompensated. For example, if you drink a can of Diet Coke then
there is one less can of Diet Coke for the rest of the world to consume, so you decrease the
welfare of others by your action. But in this case, in order to drink the can of Diet Coke
you have to pay for it, and if you pay what it costs to make another can of Diet Coke, then
you have exactly compensated the rest of the world for your action. That is, there is no
uncompensated e↵ect, and hence no externality. We explore the interaction of externalities
and compensation further when we discuss property rights in Chapter 24.
17.1 The Economy Without Network E↵ects
Our canonical setting in this chapter will be the market for a good: we will first consider
how the market functions when there is no network e↵ect — that is, when consumers do not
care how many other users of the good there are — and then we will see how things change
when a network e↵ect is present.
We want to analyze markets with a huge number of potential purchasers, each of whom
is small enough relative to the entire market that he or she can make individual decisions
without a↵ecting the aggregate behavior. For example, each individual considering the
purchase of a loaf of bread does so without worrying about whether her individual decision
— all else remaining the same — will a↵ect the price of bread. (Note that this is di↵erent
from worrying about whether decisions made by a large number of people will have an e↵ect,
which they certainly can.) Of course, in real markets the number of consumers is finite, and
each individual decision does have a very, very small e↵ect on the aggregate. But each
purchaser’s impact is so small relative to the market that we can model individuals as not
taking this into account when they make a decision.
Formally, we model the lack of individual e↵ects on the aggregate by representing the
17.1. THE ECONOMY WITHOUT NETWORK EFFECTS 511
consumers as the set of all real numbers in the interval strictly between 0 and 1. That is, each
consumer is named by a di↵erent real number, and the total mass of consumers is 1. This
naming of the consumers by real numbers will be notationally useful — for example, the set
of consumers with names between 0 and x < 1 represents an x fraction of the population. A
good way to think of this model of consumers is as a continuous approximation to a market
with a very large, but finite, number of consumers; the continuous model will be useful in
various places to avoid having to deal with the explicit e↵ect of any one individual on the
overall population.
Each consumer wants at most one unit of the good; each consumer has a personal intrinsic
interest in obtaining the good that can vary from one consumer to another. When there are
no network e↵ects at work, we model a consumer’s willingness to pay as being determined
entirely by this intrinsic interest. When there are network e↵ects, a consumer’s willingness
to pay is determined by two things:
• intrinsic interest; and
• the number of other people using the good — the larger the user population, the more she is willing to pay.
Our study of network e↵ects here can be viewed as an analysis of how things change once
this second factor comes into play.
To start understanding this issue, we first consider how a market works when there are
no network e↵ects.
Reservation Prices. With no network e↵ects, each consumer’s interest in the good is
specified by a single reservation price: the maximum amount she is willing to pay for one
unit of the good. We’ll assume that the individuals are arranged in the interval between 0
and 1 in order of decreasing reservation price, so that if consumer x has a higher reservation
price than consumer y, then x < y. Let r(x) denote the reservation price of consumer x. For
the analysis in this chapter, we will assume that this function r(·) is continuous, and that no two consumers have exactly the same reservation price — so the function r(·) is strictly decreasing as it ranges over the interval from 0 to 1.
Suppose that the market price for a unit of the good is p: everyone who wants to buy
the good can buy it at price p, and no units are o↵ered for sale at a price above or below
p. At price p, everyone whose reservation price is at least p will actually buy the good, and
everyone whose reservation price is below p will not buy it. Clearly at a price of r(0) or
more, no one will buy the good; and at a price of r(1) or less, everyone will buy the good.
So let’s consider the interesting region for the price p, when it lies strictly between r(1) and
r(0). In this region, there is some unique number x with the property that r(x) = p: as
512 CHAPTER 17. NETWORK EFFECTS
0 1
r(1)
r(0)
Price
Consumers
price p
x
y = r(x)
y = p
Figure 17.1: When there are no network e↵orts, the demand for a product at a fixed market price p can be found by locating the point where the curve y = r(x) intersects the horizontal line y = p.
Figure 17.1 illustrates, since r(·) is a continuous function that strictly decreases, it must cross the horizontal line y = p somewhere.
This means that all consumers between 0 and x buy the product, and all consumers
above x don’t — so an x fraction of the population buys the product. We can do this for
every price p: there is an x depending on p that specifies the fraction of the population that
will purchase at price p. This way of reading the relation between price and quantity (for
any price the quantity that will be demanded) is usually called the (market) demand for the
good, and it is a very useful way to think of the relation between the price and the number
of units purchased.1
The Equilibrium Quantity of the Good. Let’s suppose that this good can be produced
at a constant cost of p⇤ per unit, and that, as is the case for consumers, there are many
potential producers of the good so that none of them is large enough to be able to influence
the market price of the good. Then, in aggregate, the producers will be willing to supply
any amount of the good at a price of p⇤ per unit, and none of the good at any price below p⇤.
Moreover the assumption of a large number of potential producers who can create new copies
1In the language of microeconomics, the function r(·) describes the inverse demand function. The inverse of r(·), giving x in terms of p, is the demand function.
17.1. THE ECONOMY WITHOUT NETWORK EFFECTS 513
0 1
r(1)
r(0)
Price
Consumers
p*
y = r(x)
constant cost per unit p*
equilibrium quantity x*
Figure 17.2: When copies of a good can be produced at a constant cost p⇤ per unit, the equilibrium quantity consumed will be the number x⇤ for which r(x⇤) = p⇤.
of the good at a constant cost of p⇤ implies that the price cannot remain above p⇤, since any
profit to a producer would be driven to zero by competition from other producers. Thus, we
can assume a market price of p⇤, regardless of the number of units of the good produced.2
As above, cases in which p⇤ is above r(0) or below r(1) are not particularly interesting, since
then either everyone or no one buys the good. Therefore, we assume that r(0) > p⇤ > r(1).
To complete the picture of how the market operates without network e↵ects, we now
determine the supply of the good. Since p⇤ is between the highest and lowest reservation
prices, we can find a unique x⇤ between 0 and 1 so that r(x⇤) = p⇤. We call x⇤ the equilibrium
quantity of the good, given the reservation prices and the cost p⇤. Figure 17.2 revisits
Figure 17.1, including the cost p⇤ and the equilibrium quantity x⇤.
Notice the sense in which x⇤ represents an equilibrium in the population’s consumption
of the good. If less than an x⇤ fraction of the population purchased the good, there would
be consumers who have not purchased but who would have an incentive to do so, because
of reservation prices above p⇤. In other words, there would be “upward pressure” on the
consumption of the product, since there is a portion of the population that would not have
purchased but wished they had. On the other hand, if more than an x⇤ fraction of the
2Continuing with the microeconomic language, this is the long-run competitive supply for any good produced by a constant-cost industry.
514 CHAPTER 17. NETWORK EFFECTS
population purchased the good, there would be consumers who had purchased the good
but wished they had not, because of reservation prices below p⇤. In this case, we’d have
“downward pressure” on the consumption of the good.
One attractive feature of this equilibrium is that it is socially optimal (as defined in
Chapter 6). To see why, let’s consider the social welfare of the allocation, which we can
think of as the di↵erence between the total reservation prices of the consumers who receive
a copy of the good and the total cost of producing the corresponding quantity of the good.
Now, if society were going to produce enough of the good to give it to an x fraction of the
population, then social welfare would be maximized by giving it to all consumers between
0 and x, since they correspond to the x fraction of the population that values the good the
most. Which value of x would be the best choice? Since the contribution of a consumer x0
to the social welfare is the di↵erence r(x0) � p⇤, we can think of the social welfare, when consumers 0 through x get copies of the good, as the (signed) area between the curve y = r(x)
and the horizontal line y = p⇤. It’s signed in the sense that portions of the curve y = r(x)
that drop below y = p⇤ contribute negatively to the area. Given this, we’d want to choose x
so that we collect all the positive area between y = r(x) and y = p⇤, and none of the negative
area. This is achieved by choosing x to be the equilibrium x⇤. Hence the equilibrium quantity
x⇤ is socially optimal.
We now introduce network e↵ects; we’ll see that this causes several important features
of the market to change in fundamental ways.
17.2 The Economy with Network E↵ects
In this section, we discuss a model for network e↵ects in the market for a good. We will
follow a general approach suggested by Katz, Shapiro, and Varian [235, 368]; see also the
writings of Brian Arthur [25, 27] for influential early discussions of these ideas.
With network e↵ects, a potential purchaser takes into account both her own reservation
price and the total number of users of the good. A simple way to model this is to say that
there are now two functions at work: when a z fraction of the population is using the good,
the reservation price of consumer x is equal to r(x)f (z), where r(x) as before is the intrinsic
interest of consumer x in the good, and f (z) measures the benefit to each consumer from
having a z fraction of the population use the good. This new function f (z) is increasing
in z: it controls how much more valuable a product is when more people are using it. The
multiplicative form r(x)f (z) for reservation prices means that those who place a greater
intrinsic value on the good benefit more from an increase in the fraction of the population
using the good than do those who place a smaller intrinsic value on the good.
For now, in keeping with the motivation from communication technology and social
media, we will assume that f (0) = 0: if no one has purchased the good no one is willing to
17.2. THE ECONOMY WITH NETWORK EFFECTS 515
pay anything for the good. In Section 17.6 we will consider versions of the model where f (0)
is not 0. We will also assume that f is a continuous function. Finally, to make the discussion
a bit simpler, we will assume that r(1) = 0. This means that as we consider consumers x
tending to 1 (the part of the population least interested in purchasing), their willingness to
pay is converging to 0.3
Since a consumer’s willingness to pay depends on the fraction of the population using
the good, each consumer needs to predict what this fraction will be in order to evaluate
whether to purchase. Suppose that the price of the good is p⇤, and that consumer x expects
a z fraction of the population will use the good. Then x will want to purchase provided that
r(x)f (z) � p⇤. We begin by considering what happens in the case when all consumers make perfect
predictions about the number of users of the good; after this, we will then consider the
population-level dynamics that are caused by imperfect predictions.
Equilibria with Network E↵ects. What do we have in mind, in the context of the
current discussion, when we suppose that consumers’ predictions are perfect? We mean
that the consumers form a shared expectation that the fraction of the population using the
product is z, and if each of them then makes a purchasing decision based on this expectation,
then the fraction of people who actually purchase is in fact z. We call this a self-fulfilling
expectations equilibrium for the quantity of purchasers z: if everyone expects that a z fraction
of the population will purchase the product, then this expectation is in turn fulfilled by
people’s behavior.
Let’s consider what such an equilibrium value of z looks like, in terms of the price p⇤ > 0.
First of all, if everyone expects a z = 0 fraction of the population to purchase, then the
reservation price of each consumer x is r(x)f (0) = 0, which is below p⇤. Hence no one will
want to purchase, and the shared expectation of z = 0 has been fulfilled.
Now let’s consider a value of z strictly between 0 and 1. If exactly a z fraction of the
population purchases the good, which set of individuals does this correspond to? Clearly if
consumer x0 purchases the good and x < x0, then consumer x will as well. Therefore, the set
of purchasers will be precisely the set of consumers between 0 and z. What is the price p⇤
at which exactly these consumers want to purchase, and no one else? The lowest reservation
price in this set will belong to consumer z, who — because of the shared expectation that
a z fraction of the population will purchase — has a reservation price of r(z)f (z). In order
for exactly this set of consumers, and no one else, to purchase the good, we must have
p⇤ = r(z)f (z).
We can summarize this as follows:
3The assumption that r(1) = 0 isn’t necessary for our qualitative results, but it avoids various additional steps later on.
516 CHAPTER 17. NETWORK EFFECTS
0 1
Price
Consumers
p*
z' z''
Figure 17.3: Suppose there are network e↵ects and f (0) = 0, so that the good has no value to people when no one is using it. In this case, there can be multiple self-fulfilling expectations equilibria: at z = 0, and also at the points where the curve r(z)f (z) crosses the horizontal line at height p⇤.
If the price p⇤ > 0 together with the quantity z (strictly between 0 and 1) form a
self-fulfilling expectations equilibrium, then p⇤ = r(z)f (z).
This highlights a clear contrast with the model of the previous section, in which network
e↵ects were not present. There, we saw that in order to have more of the good sold, the price
has to be lowered — or equivalently, at high prices the number of units of the good that
can be sold is smaller. This follows directly from the fact that the equilibrium quantity x⇤
without network e↵ects is governed by p⇤ = r(x⇤), and r(x) is decreasing in x. The market
for a good with network e↵ects is more complicated, since the amount of the good demanded
by consumers depends on how much they expect to be demanded — this leads to the more
complex equation p⇤ = r(z)f (z) for the equilibrium quantity z. Under our assumption that
f (0) = 0, we’ve seen that one equilibrium with network e↵ects occurs at price p⇤ and z = 0:
Producers are willing to supply a zero quantity of the good, and since no one expects the
good to be used, none of it is demanded either.
A Concrete Example. To find whether other equilibria exist, we need to know the form
of the functions r(·) and f (·) in order to analyze the equation p⇤ = r(z)f (z). To show how this works, let’s consider a concrete example in which r(x) = 1 � x and f (z) = z. In this case, r(z)f (z) = z(1 � z), which has a parabolic shape as shown in Figure 17.3: it is 0 at z = 0 and z = 1, and it has a maximum at z = 1
2
, when it takes the value 1 4
. Of course, in
17.3. STABILITY, INSTABILITY, AND TIPPING POINTS 517
general the functions r(·) and f (·) need not look exactly like this example, but typically we expect to see something like the shape displayed in Figure 17.3.
Continuing with this concrete example, we can now work out the set of equilibria exactly.
If p⇤ > 1 4
, then there is no solution to p⇤ = r(z)f (z) = z(1 � z) (since the right-hand side has a maximum value of 1
4
, at z = 1 2
), and so the only equilibrium is when z = 0. This
corresponds to a good that is simply too expensive, and so the only equilibrium is when
everyone expects it not to be used.
On the other hand, when p⇤ is between 0 and 1 4
, there are two solutions to p⇤ = z(1�z): they are at points z0 and z00 where the horizontal line y = p⇤ slices through the parabola
defined by z(1 � z), as shown in Figure 17.3. Thus there are three possible equilibria in this case: when z is equal to any of 0, z0, or z00. For each of these three values of z, if
people expect exactly a z fraction of the population to buy the good, then precisely the top
z fraction of the population will do so.
There are two initial observations worth making from this example. First, the notion
of a self-fulfilling expectations equilibrium corresponds, in a general sense, to the e↵ects of
aggregate “consumer confidence.” If the population has no confidence in the success of the
good, then because of the network e↵ects, no one will want it, and this lack of confidence will
be borne out by the failure of people to purchase it. On the other hand — for the very same
good, at the same price — if the population is confident of its success, then it is possible
for a significant fraction of the population to decide to purchase it, thereby confirming its
success. The possibility of multiple equilibria in this way is characteristic of markets in which
network e↵ects are at work.
A second observation concerns the nature of consumer demand in this case. Compared
to the simple, decreasing curve in Figure 17.2, the curve in Figure 17.3 highlights the compli-
cated relationship between the price and the equilibrium quantity. In particular, as the price
p⇤ drops gradually below 1 4
, the “high” equilibrium z00 moves right (as in the simple model
without network e↵ects), but the “low” equilibrium z0 moves left, toward smaller fractions
of the population. To understand how these two equilibria relate to each other, we need to
consider an important qualitative contrast between them, which we formulate in the next
section.
17.3 Stability, Instability, and Tipping Points
Let’s continue with the example in Figure 17.3, and explore the properties of its equilibria.
To begin with, it’s useful to work through the details of why values of z other than 0, z0,
or z00 do not constitute equilibria. In particular, suppose that a z fraction of the population
were to purchase the good, where z is not one of these three equilibrium quantities.
• If z is between 0 and z0, then there is “downward pressure” on the consumption of the
518 CHAPTER 17. NETWORK EFFECTS
good: since r(z)f (z) < p⇤, the purchaser named z (and other purchasers just below z)
will value the good at less than p⇤, and hence will wish they hadn’t bought it. This
would push demand downward.
• If z is between z0 and z00, then there is “upward pressure” on the consumption of the good: since r(z)f (z) > p⇤, consumers with names slightly above z have not purchased
the good but will wish they had. This would drive demand upward.
• Finally, if z is above z00, then there is again downward pressure: since r(z)f (z) < p⇤, purchaser z and others just below will wish they hadn’t bought the good, pushing
demand down.
This set of three di↵erent possibilities for the non-equilibrium values of z has interesting
consequences for the equilibria z0 and z00. First, it shows that z00 has a strong stability
property. If slightly more than a z00 fraction buys the good, then the demand gets pushed back
toward z00; if slightly less than a z00 fraction buys the good, then the demand correspondingly
gets pushed up toward z00. So in the event of a “near miss” in the population’s expectations
around z00, we would expect the outcome to settle down to z00 anyway.
The situation looks di↵erent — and highly unstable — in the vicinity of the equilibrium
z0. If slightly more than a z0 fraction buys the good, then upward pressure drives the demand
away from z0 toward the higher equilibrium at z00. And if slightly less than a z0 fraction buys
the good, then downward pressure drives the demand away from z0 in the other direction,
down toward the equilibrium at 0. Thus, if exactly a z0 fraction of the population purchases
the good, then we are at equilibrium; but if the fraction is even slightly o↵ from this, the
system will tend to spiral up or spiral down to a significant extent.
Thus, z0 is not just an unstable equilibrium; it is really a critical point, or a tipping
point, in the success of the good. If the firm producing the good can get the population’s
expectations for the number of purchasers above z0, then they can use the upward pressure
of demand to get their market share to the stable equilibrium at z00. On the other hand,
if the population’s expectations are even slightly below z0, then the downward pressure will
tend to drive the market share to 0. The value z0 is the hump the firm must get over in order
to succeed.
This view of the equilibria suggests a way of thinking about the price p⇤. If the firm
were to price the good more cheaply — in other words, to lower the price p⇤ — then this
would have two beneficial e↵ects. Since the parabola in Figure 17.3 would now be sliced by
a lower horizontal line (reflecting the lower price), the low equilibrium z0 would move left;
this provides a critical point that is easier to get past. Moreover, the high equilibrium z00
would move right, so if the firm is able to get past the critical point, the eventual size of its
user population z00 would be even larger. Of course, if p⇤ is set below the cost of production
the firm loses money. But as part of a pricing strategy over time, in which early losses may
17.4. A DYNAMIC VIEW OF THE MARKET 519
Shared Expectation z
Outcome
z z = g(z)
Figure 17.4: From a model with network e↵ects, we can define a function ẑ = g(z): if everyone expects a z fraction of the population to purchase the good, then in fact a g(z) fraction will do so.
be o↵set by growth in the user population and later profits, this may be a viable strategy.
Many firms do this by o↵ering free trials for their products or by setting low introductory
prices.
17.4 A Dynamic View of the Market
There is another way to view this critical point idea that is particularly illuminating. We
have been focusing on an equilibrium in which consumers correctly predict the number of
actual users of the good. Let’s now ask what this would look like if consumers have common
beliefs about how many users there will be, but we allow for the possibility that these beliefs
are not correct.
This means that if everyone believes a z fraction of the population will use the product,
then consumer x — based on this belief — will want to purchase if r(x)f (z) � p⇤. Hence, if anyone at all wants to purchase, the set of people who will purchase will be between 0 and
ẑ, where ẑ solves the equation r(ẑ)f (z) = p⇤. Equivalently,
r(ẑ) = p⇤
f (z) , (17.1)
or, taking the inverse of the function r(·),
ẑ = r�1 ✓
p⇤
f (z)
◆ . (17.2)
520 CHAPTER 17. NETWORK EFFECTS
Shared Expectation z
Outcome
z
z' z''
z = z
z = g(z)
Figure 17.5: When r(x) = 1 � x and f (z) = z, we get the curve for g(z) shown in the plot: g(z) = 1 � p⇤/z if z � p⇤ and g(z) = 0 if z < p⇤. Where the curve ẑ = g(z) crosses the line ẑ = z, we have self-fulfilling expectations equilibria. When ẑ = g(z) lies below the line ẑ = z, we have downward pressure on the consumption of the good (indicated by the downward arrows); when ẑ = g(z) lies above the line ẑ = z, we have upward pressure on the consumption of the good (indicated by the upward arrows). This indicates visually why the equilibrium at z0 is unstable while the equilibrium at z00 is stable.
This provides a way of computing the outcome ẑ from the shared expectation z, but we
should keep in mind that we can only use this equation when there is in fact a value of ẑ
that solves Equation (17.1). Otherwise, the outcome is simply that no one purchases.
Since r(·) is a continuous function that decreases from r(0) down to r(1) = 0, such a solution will exist and be unique precisely when
p⇤
f (z) r(0). Therefore, in general, we can
define a function g(·) that gives the outcome ẑ in terms of the shared expectation z as follows. When the shared expectation is z � 0, the outcome is ẑ = g(z), where
• g(z) = r�1 ✓
p⇤
f (z)
◆ when the condition for a solution
p⇤
f (z) r(0) holds; and
g(z) = 0 otherwise.
Let’s try this on the example illustrated in Figure 17.3, where r(x) = 1�x and f (z) = z. In this case, r�1(x) turns out to be 1 � x. Also, z(0) = 1, so the condition for a solution p⇤
f (z) r(0) is just z � p⇤. Therefore, in this example
17.4. A DYNAMIC VIEW OF THE MARKET 521
Shared Expectation z
Outcome
z
z' z''
z = z
z = g(z)
Figure 17.6: The curve g(z), and its relation to the line ẑ = z, illustrates a pattern that we expect to see in settings more general than just the example used for Figure 17.5.
g(z) = 1 � p⇤
z when z � p⇤, and g(z) = 0 otherwise.
We can plot the function ẑ = g(z) as shown in Figure 17.4. Beyond the simple shape of the
curve, however, its relationship to the 45o line ẑ = z provides a striking visual summary of the
issues around equilibrium, stability, and instability that we’ve been discussing. Figure 17.5
illustrates this. To begin with, when the plots of the two functions ẑ = g(z) and ẑ = z cross,
we have a self-fulfilling expectations equilibrium: here g(z) = z, and so if everyone expects a
z fraction of the population to purchase, then in fact a z fraction will do so. When the curve
ẑ = g(z) lies below the line ẑ = z, we have downward pressure on the consumption of the
good: if people expect a z fraction of the population to use the good, then the outcome will
underperform these expectations, and we would expect a downward spiral in consumption.
And correspondingly, when the curve ẑ = g(z) lies above the line ẑ = z, we have upward
pressure on the consumption of the good.
This gives a pictorial interpretation of the stability properties of the equilibria. Based on
how the functions cross in the vicinity of the equilibrium z00, we see that it is stable: there
is upward pressure from below and downward pressure from above. On the other hand,
where the curves cross in the vicinity of the equilibrium z0, there is instability — downward
pressure from below and upward pressure from above, causing the equilibrium to quickly
unravel if it is perturbed in either direction.
The particular shape of the curve in Figure 17.5 depends on the functions we chose in our
522 CHAPTER 17. NETWORK EFFECTS
z = z
z = g(z)
Figure 17.7: A “zoomed-in” region of a curve ẑ = g(z) and its relation to the line ẑ = z.
example, but the intuition behind this picture is much more general than the example. With
network e↵ects in general, we would expect to see a relation between the expected number
of users and the actual number of purchasers that looks qualitatively like this curve, or more
generally like the smoother version in Figure 17.6. Where the curve ẑ = g(z) crosses the line
ẑ = z, we have equilibria that can be either stable or unstable depending on whether the
curve crosses from above or below the line.
The Dynamic Behavior of the Population. In the 1970s, Mark Granovetter and
Thomas Schelling used pictures like the ones in Figures 17.5 and 17.6 to model how a
population might react dynamically to a network e↵ect [192, 366]. Specifically, they were
interested in how the number of people participating in a given activity with network e↵ects
would tend to grow or shrink over time.
To motivate the kind of question they formulated, let’s imagine that instead of evaluating
the purchase of a discrete object like a fax machine, people in society are evaluating their
17.4. A DYNAMIC VIEW OF THE MARKET 523
z = z
z = g(z)
(z , z )0 0
(z , z )0 1
(z , z )1 1
Figure 17.8: The audience size changes dynamically as people react to the current audience size. This e↵ect can be tracked using the curve ẑ = g(z) and the line ẑ = z.
participation in a large social media site — something where you chat with friends, share
videos, or some similar activity. We are formulating the underlying story here in terms of
participation rather than purchasing because the dynamics of participation are more fluid
than the dynamics of purchasing: someone can change their mind about participation in a
social media site from one day to the next, whereas purchasing a physical good is a step that
isn’t as naturally undone.
Despite the change in the motivating story, the model remains exactly the same. Each
person x has an intrinsic interest in using the site, represented by a function r(x), and
the site is more attractive to people if it has more users, as governed by a function f (z).
Counterbalancing this, let’s suppose that there is a fixed level of e↵ort required to use the
site, which serves the role of a “price” p⇤ (except that the price may consist of the expenditure
of e↵ort rather than money). Thus, if person x expects a z fraction of the population to
want to participate, then x will participate if r(x)f (z) � p⇤. This is just the same as the
524 CHAPTER 17. NETWORK EFFECTS
z = z
z = g(z)
(z , z )0 0
(z , z )0 1
(z , z )1 1
(z , z )1 2
(z , z )2 2
Figure 17.9: Successive updates cause the audience size to converge to a stable equilibrium point (and to move away from the vicinities of unstable ones).
criterion we saw before.
Let’s suppose that time proceeds in a fixed set of periods (e.g. days, weeks, or months)
t = 0, 1, 2, . . .. At time t = 0, some initial fraction of the population z 0
is participating in the
site — let’s call this the initial audience size. Now, the audience size changes dynamically
over time as follows. In each period t, people evaluate whether to participate based on a
shared expectation that the audience size will be the same as what it was in the previous
period. In terms of our function g(·), which maps shared expectations to outcomes, this means that z
1
= g(z 0
), since everyone acts in period t = 1 on the expectation that the
audience size will be z 0
. After this, z 2
= g(z 1
), since in period t = 2 everyone will act based
on the expectation that the audience size is now z 1
; and more generally, we have z t
= g(z t�1)
for each t.
This is clearly a model in which the population is behaving in a myopic way — they
evaluate the benefits of participation as though the future will be the same as the present.
17.4. A DYNAMIC VIEW OF THE MARKET 525
However, it is an approximation that can be reasonable in settings where people have rela-
tively limited information, and where they are behaving according to simple rules. Moreover,
part of its value as an approximation in this case is that it produces dynamic behavior that
closely corresponds to our notions of equilibrium: if the population follows this model, then
it converges precisely to self-fulfilling expectations equilibria that are stable. We discuss the
reasons for this next.
Analyzing the Dynamics. The dynamic behavior of the population can be analyzed in a
way that is purely “pictorial” but nevertheless completely rigorous. Here is how this works,
using a zoomed-in region of the curve ẑ = g(z) in the vicinity of two equilibria as shown in
Figure 17.7.
We have an initial audience size z 0
, and we want to understand how the sequence of
audience sizes z 1
= g(z 0
), z 2
= g(z 1
), z 3
= g(z 2
), . . . behaves over time. We will do this by
tracking the points (z t
, z t
), as t ranges over t = 0, 1, 2, . . .; notice that all of these points lie
on the diagonal line ẑ = z. The basic way in which we move from one of these points to
the next one is shown in Figure 17.8. We start by locating the current audience size z 0
on
the line ẑ = z. Now, to determine z 1
, we simply move vertically until we reach the curve
ẑ = g(z), since this gives us the value of z 1
= g(z 0
). Then we again locate the audience size
z 1
on the line ẑ = z — this involves moving horizontally from the point (z 0
, z 1
= g(z 0
)) until
we reach the point (z 1
, z 1
). We have therefore gone from (z 0
, z 0
) to (z 1
, z 1
), following the
evolution of the audience in the first time period.
This is the basic operation: for each time period t, we can determine the new audience
size z t
from the current one z t�1 in the same way. We first move vertically from the point
(z t�1, zt�1) to the point (zt�1, zt) (which lies on the curve ẑ = g(z)); we then move horizontally
from the point (z t�1, zt) to the point (zt, zt).
Figure 17.9 now shows what happens as we track this sequence of points, following how
the audience size changes. When we’re following a part of the curve ẑ = g(z) that lies above
the diagonal line ẑ = z, the points move upward, converging to the nearest place where the
two functions cross, which will be at a stable equilibrium point. On the left- and right-hand
parts of the picture we show what happens to two other trajectories that start out from
points where the curve ẑ = g(z) lies below the diagonal line ẑ = z. Here, the sequence of
points that track the audience size will move downward, again converging to the first crossing
point it reaches; again, this will be a stable equilibrium. Notice that around the unstable
equilibrium point in the figure, this means that the trajectories of points move away from
it on either side — consistent with our view of unstable equilibria, there is no way to reach
this point unless you start right at it.
Thus, this simple dynamics for updating the audience size — although it is based on my-
opic behavior by the population — illustrates how stable and unstable equilibria govern the
526 CHAPTER 17. NETWORK EFFECTS
outcomes. Stable equilibria attract the population from both sides, while unstable equilibria
act like “branch points,” with the audience size flowing away from them on either side.
17.5 Industries with Network Goods
The discussion and models thus far provide some useful intuitions about how an industry
with network e↵ects might be expected to evolve over time. Let’s discuss what we might
learn at a qualitative level from these models. We’ll continue to use “audience size” for the
fraction of the population that purchases a product.
Let’s start with Figure 17.3, and suppose that a new product is introduced with a high
initial cost of production — in particular, suppose the horizontal line at height p⇤ is above
the top of the parabola. In this case the only equilibrium is at audience size z = 0. If over
time the cost of production falls then eventually a horizontal line drawn at p⇤ will intersect
the parabola in two points, much like we see in Figure 17.3, and there will be three possible
equilibria. But when p⇤ is large, near the top of the curve in Figure 17.3, it is likely that
none of the good will be sold: to have any sales occur consumers would have to expect an
audience size of at least z0, which will be large when p⇤ is large (near the fraction of the
population where the parabola reaches its peak). Given that none were sold previously —
when the cost was above the top of the curve — this seems an unlikely prediction. But as
the cost of production continues to fall, the critical point decreases (as z0 gets closer to 0),
and an audience size of at least z0 starts to seem more and more likely. Once consumers
expect the good to be viable, with an audience size of at least z0, the stable equilibrium is in
fact z00. So as costs decline we would expect to initially see no sales, and then once purchases
begin we would expect to see sales grow rapidly to the stable point.
Marketing a Product with Network E↵ects. How can a firm that wants to sell a
product with a network e↵ect use these insights to market its product? Suppose you are
running a firm that is producing a new product subject to network e↵ects; perhaps it’s a
new piece of software, communication technology, or social media. The marketing of this
product will not succeed unless you can get past the tipping point (at z0). Starting small
and hoping to grow slowly is unlikely to succeed, since unless your product is widely used it
has little value to any potential purchasers.
Thus, you somehow need to convince a large initial group to adopt your product before
others will be willing to buy it. How would you do this? One possibility is to set an initial
low, introductory price for the good, perhaps even o↵ering it for free. This price below the
cost of producing the good will result in early losses, but if the product catches on — if it
gets over the tipping point — then your firm can raise the price and perhaps make enough
profit to overcome the initial losses.
17.5. INDUSTRIES WITH NETWORK GOODS 527
Another alternative is to attempt to identify fashion leaders, those whose purchase or use
of the good will attract others to use it, and convince them to adopt the good. This strategy
also involves network e↵ects, but they are ones that cannot be studied at the population level.
Instead we would need to identify a network of connections between potential purchasers and
ask who influences whom in this network. We explore this idea in Chapter 19.
Social Optimality with Network E↵ects. We saw in Section 17.1 that for a market
with no network e↵ects, the equilibrium is socially optimal. That is, it maximizes the total
di↵erence between the reservation prices of the consumers who purchase the good and the
total cost of producing the good, over all possible allocations to people.
For goods with network e↵ects, however, the equilibria are typically not optimal. At a
high level, the reason is that each consumer’s choice a↵ects each other consumer’s payo↵,
and the consequences of this can be analyzed as follows. Suppose we are at an equilibrium
in which the audience size is z⇤. The consumer named z⇤ — the purchaser with the least
interest in the product — has a reservation price of r(z⇤)f (z⇤) = p⇤. Now, consider the set of
consumers with names above z⇤ and below z⇤+c for some small constant c > 0. None of these
consumers want to buy, since r(z)f (z⇤) < p for z in this range. But if they all did purchase
the good, then all the current purchasers would benefit: the value of the product to each
purchaser x < z⇤ would increase from r(x)f (z⇤) to r(x)f (z⇤ + c). The potential consumers
between z⇤ and z⇤ + c don’t take this e↵ect into account in their respective decisions about
purchasing the good.
It is easy to set up situations where this overall benefit to existing purchasers outweighs
the overall loss that consumers between z⇤ and z⇤ + c would experience from buying the
good. In such a case the equilibrium is not socially optimal, since society would be better
o↵ if these additional people bought the good. This example illustrates the more general
principle that for goods with network e↵ects, markets typically provide less of the good than
is socially optimal.
Network E↵ects and Competition. Finally, let’s ask what might happen if multiple
firms develop competing new products, each of which has its own network e↵ects. For
example, we could consider two competing social-networking sites that o↵er similar services,
or two technologies that do essentially the same thing, but where the value of each of these
technologies depends on how many people use it. There are a number of classic examples of
this from technology industries over the last several decades [27]. These include the rise of
Microsoft to dominate the market for personal-computer operating systems, and the triumph
of VHS over Betamax as the standard videotape format in the 1980s.
In such cases of product competition with network e↵ects, it is likely that one product
will dominate the market, as opposed to a scenario in which both products (or even more
528 CHAPTER 17. NETWORK EFFECTS
than two) flourish. The product that first gets over its own tipping point attracts many
consumers and this may make the competing product less attractive. Being the first to
reach this tipping point is very important — more important than being “best” in an abstract
sense. That is, suppose that if product A has audience size z, then consumer x values it
at r A
(x)f (z), while if product B has audience size z, then consumer x values it at a larger
amount r B
(x)f (z) > r A
(x)f (z). Let’s also suppose that each product can be produced at
the same price. Then it seems reasonable to say that product B is the better product. But
if product A is on the market first, and gets over its tipping point, then product B may not
be able to survive.4
These considerations help provide some intuition for how markets with strong network
e↵ects tend to behave. Writing in the Harvard Business Review in 1996, Brian Arthur sum-
marized the “hallmarks” of these markets in a way that reflects the discussion in the previous
paragraph: “market instability (the market tilts to favor a product that gets ahead), mul-
tiple potential outcomes ([e.g.,] under di↵erent events in history di↵erent operating systems
could have won), unpredictability, the ability to lock in a market, the possible predominance
of an inferior product, and fat profits for the winner” [27]. It is not the case that a given
market with network e↵ects will necessarily display all these characteristics, but they are
phenomena to watch for in this type of setting.
Of course, in our discussion of the dominance of product A over product B, we are
assuming that nothing else changes to shift the balance after A achieves dominance. If the
firm that makes product B improves its product su�ciently and markets it well, and if the
firm that makes product A doesn’t respond e↵ectively, then B may still overtake A and
become the dominant product.
17.6 Mixing Individual E↵ects with Population-Level E↵ects
Thus far we have been focusing on models of network e↵ects in which the product is useless
to consumers when it has an audience size of 0; this is captured by our assumption that
f (0) = 0. But of course one can also study more general kinds of network e↵ects, in which
a product has some value to a person even when he or she is the first purchaser, and its
value then increases as more people buy it. We can think of this as a model that mixes
individual e↵ects (a person’s value for the product on its own) with population-level e↵ects
(the increased value a person derives when the product has a large audience size). In such
a model, we would have f (0) > 0, and have f (z) increasing in z.
We won’t attempt to cover all the ways of fleshing out such a model; instead we develop
one general class of examples to illustrate how qualitatively new phenomena can arise when
4Exercises 3 and 4 at the end of this chapter o↵er simple models of this situation.
17.6. MIXING INDIVIDUAL EFFECTS WITH POPULATION-LEVEL EFFECTS 529
Shared Expectation z
Outcome
z
z = z
z = g(z)
Figure 17.10: When f (0) > 0, so that people have value for the product even when they are the only user, the curve ẑ = g(z) no longer passes through the point (0, 0), and so an audience size of 0 is no longer an equilibrium.
we mix individual and population-level e↵ects. In particular, we focus on a phenomenon
identified in this type of model by Mark Granovetter [192], and which corresponds to an
intuitively natural issue in the marketing of new products with network e↵ects.
A Concrete Model. For our example, let’s consider a function f (·) of the form f (z) = 1 + az2 for a constant parameter a. We’ll continue to use the simple example r(x) = 1 � x; so when the audience size is z, the value of the product to consumer x is
r(x)f (z) = (1 � x)(1 + az2).
Now let’s apply the analysis from Section 17.4 to this function to get the dynamic behavior
of the market. We will assume that the price p⇤ is strictly between 0 and 1. When everyone
expects an audience size of z, the fraction of people who actually use the product is ẑ = g(z),
where g(·) is defined as in Section 17.4:
530 CHAPTER 17. NETWORK EFFECTS
Shared Expectation z
Outcome
z
z = z
z = g(z)
(z , z )1 1
(z , z ) 0 0 = (0,0)
stable equilibrium (z*,z*)
stable equilibrium (z**,z**)
Figure 17.11: The audience grows dynamically from an initial size of zero to a relatively small stable equilibrium size of z⇤.
g(z) = r�1 ✓
p⇤
f (z)
◆ when the condition for a solution
p⇤
f (z) r(0) holds; and
g(z) = 0 otherwise.
As before, we have r�1(x) = 1 � x. Since in our case r(0) = 1, f (z) � 1, and p⇤ < 1, this means that the condition for a solution
p⇤
f (z) r(0) will always hold. Plugging this into the
formula for g(z), we get
g(z) = 1 � p⇤
1 + az2 .
When we plot this function ẑ = g(z) together with the 45o line ẑ = z, we get something
that looks like Figure 17.10.
Growing an Audience from Zero. In our earlier model with f (0) = 0, an audience size
of zero was a stable equilibrium: if everyone expected that no one would use the product,
then no one would. But when f (0) > 0, so that the product has value to people even when
17.6. MIXING INDIVIDUAL EFFECTS WITH POPULATION-LEVEL EFFECTS 531
Shared Expectation z
Outcome
z
z = z
z = g(z)
stable equilibrium
Figure 17.12: If the price is reduced slightly, the curve ẑ = g(z) shifts upward so that it no longer crosses the line ẑ = z in the vicinity of the point (z⇤, z⇤).
they are the only user, an audience size of zero is no longer an equilibrium (when p⇤ < 1):
even if everyone expects no one to use the product, some people will still purchase it.
As a result, it becomes natural to ask what happens when such a product starts at an
audience size of zero, and we then follow the dynamics that were defined in Section 17.4.
Figure 17.11 shows what happens when we do this: the sequence of audience sizes increases
from z 0
= 0 up to the first point (z⇤, z⇤) at which the curve ẑ = g(z) crosses the line ẑ = z.
This is the stable equilibrium that is reached when we run the dynamics of the market
starting from an audience size of 0.
Notice how the underlying story that we’re modeling with this process has no direct
analogue in the earlier model when f (0) = 0. There, because the product was useless if it
had an audience size of zero, a firm marketing the product needed alternate ways to get over
its tipping point at the low, unstable equilibrium in order to have any customers at all. But
when f (0) > 0, the audience can grow from zero up to some larger stable equilibrium z⇤
through the simple dynamics in Figure 17.11. In other words, we’re able to talk here about
532 CHAPTER 17. NETWORK EFFECTS
Shared Expectation z
Outcome
z
z = z
z = g(z)
(z , z )1 1
(z , z ) 0 0 = (0,0)
(z , z )2 2
(z , z )3 3
(z , z )4 4
(z , z )5 5
(z , z )6 6
stable equilibrium
Figure 17.13: The small reduction in price that shifted the curve ẑ = g(z) has a huge e↵ect on the equilibrium audience size that is reached starting from zero.
an audience that grows gradually and organically, starting from no users at all, rather than
one that needs to be pushed by other means over an initial tipping point.
Bottlenecks and Large Changes. The firm marketing the product in our example,
however, may well want more than what it gets in Figure 17.11. Although the audience
grows to some size z⇤ on its own, there is a much higher stable equilibrium, shown in
the figure at (z⇤⇤, z⇤⇤), that would be much more desirable if only it could be reached. But
starting from zero, the audience doesn’t reach this high equilibrium z⇤⇤, because it is blocked
by a “bottleneck” that stops it at z⇤.
Here is where we get to the surprising phenomenon at the heart of this example: small
changes in the properties of the market can cause enormous changes in the size of the
equilibrium audience that is reached, starting from zero [192]. Suppose that the firm is able
to lower the price p⇤ slightly, to some new value q⇤ < p⇤. Then we get a new function h(z)
17.7. ADVANCED MATERIAL: NEGATIVE EXTERNALITIES AND THE EL FAROL BAR PROBLEM533
mapping shared expectations to outcomes,
h(z) = 1 � q⇤
1 + az2 ,
which in turn defines a new dynamic process. As q⇤ is made smaller, the curve ẑ = h(z)
shifts upward until it no longer crosses the line ẑ = z at all in the vicinity of the point
(z⇤, z⇤); this is shown in Figure 17.12. However, h(·) still has a high stable equilibrium close to the high equilibrium (z⇤⇤, z⇤⇤) for the function g(·).
As soon as h(·) lifts enough that it no longer crosses ẑ = z near (z⇤, z⇤), the equilibrium audience size starting from zero changes abruptly and dramatically: it suddenly jumps from
a value near z⇤ to a much higher value near z⇤⇤. There is a natural reason for this: as shown
in Figure 17.13, the “bottleneck” at (z⇤, z⇤) has opened into a narrow passageway, and so
now the dynamics starting from the point (0, 0) can carry the audience all the way up to the
stable equilibrium near (z⇤⇤, z⇤⇤).5
This phenomenon shows how in models with network e↵ects, small changes in market
conditions can have strong, discontinuous e↵ects on the outcome. The contrast between
Figure 17.11 and Figure 17.13 relates to an important issue in the marketing of products
with network e↵ects. In Figures 17.11, the product reaches a small group of the most
enthusiastic consumers (the ones with the highest values for the product), but it fails to make
the leap from this group to the much broader set of people — mainstream, less enthusiastic
consumers — who could collectively push the audience size up to the higher equilibrium
at z⇤⇤. However, once the price is lowered very slightly, making the product slightly more
attractive to everyone, a passage is opened that enables the success of the product with
its most enthusiastic consumers to carry over to this larger mainstream set, driving the
equilibrium audience size up to a much larger value.
17.7 Advanced Material: Negative Externalities and The El Farol Bar Problem
In di↵erent contexts, we have now analyzed situations with both negative externalities (traf-
fic congestion and the Braess Paradox) and positive externalities (goods with network e↵ects,
in this chapter). The settings for these analyses each contained a number of details designed
to capture their respective contexts: in our discussion of negative externalities we had the
complexity of an underlying network through which the tra�c flowed; with positive exter-
nalities, we had a heterogeneous population with diverse reservation prices, all reacting to a
common market price.
5It is not hard to find specific numbers that cause this e↵ect to happen; for example, you can try f (z) = 1 + 4z2 and p⇤ = 0.93. In this case, the equilibrium audience size starting from zero is around 0.1. If we then lower the price slightly to q⇤ = 0.92, the equilibrium audience size starting from zero jumps to around 0.7.
534 CHAPTER 17. NETWORK EFFECTS
But even after eliminating all these details, reducing the problems to simpler forms, the
phenomena that surround negative and positive externalities remain quite di↵erent at a more
fundamental level. In this section we consider some of these contrasts using stylized, simple
examples that enable us to highlight the di↵erences more clearly. In the process, we will also
consider the question of how individuals coordinate on equilibrium behavior in each setting.
Simple Scenarios with Negative and Positive Externalities. As our simplified set-
ting for negative externalities, we use the widely-studied El Farol Bar problem created by
Brian Arthur [26]. The problem is named after a bar in Sante Fe that used to have live
music every Thursday evening. In the formulation of the problem, the bar has seating for
only 60 people, and so showing up for the music is enjoyable only when at most 60 people
do so. With more than 60 people in attendance, it becomes unpleasantly crowded, such that
it would be preferable to have stayed home. Now, unfortunately, there are 100 people each
week who are interested in going to the bar, and they all share the view that going is only
worthwhile when at most 60 people show up. How does each person reason in a given week
about whether to go or to stay home — knowing that everyone else is reasoning about this
decision as well?
The El Farol Bar problem describes a situation with a very simple negative externality:
the payo↵s to participating in the underlying activity (going to the bar) decrease as the
number of participants increases. And despite the appealingly simple statement of the
problem, it creates a situation in which the reasoning problem faced by the participants
is very complex. To illustrate where some of these complexities come from, it’s useful to
compare it to a simple analogous situation that contains a positive externality.
In this parallel scenario, let’s imagine a division of a large company consisting of 100
people, in which the management is encouraging the employees to use a particular corporate
social-networking site as part of their workflow. The management would like each employee
to create an account and maintain a presence on the site to facilitate increased interaction
across the division. The employees each believe that this would be worthwhile provided that
enough people in the division participate in the site; otherwise, the e↵ort required would
not be worth it. Thus, each employee wants to use the social-networking site if at least
60 other employees do so as well. (So counting the employee herself, this means the total
number of employees using the site should be strictly greater than 60 in order for it to
be worth the e↵ort.) This is closely analogous to the scenarios we’ve considered in earlier
sections of this chapter, concerning goods with network e↵ects — although here, instead
of each individual having a distinct reservation price, there is simply a common interest in
participating provided the audience size is large enough. (Also, the size of the population is
finite rather than infinite.) In a di↵erent way, the analogy to the El Farol Bar problem should
also be clear: in the scenario of the social-networking site we have a positive externality in
17.7. ADVANCED MATERIAL: NEGATIVE EXTERNALITIES AND THE EL FAROL BAR PROBLEM535
which a participation level above 60 is good, while in El Farol, we have a negative externality
in which a participation level above 60 is bad.
These two examples have been designed in such a way that one exhibits only negative
externalities, and the other exhibits only positive externalities. It’s important to keep in
mind, of course, that many real situations in fact display both kinds of externalities — some
level of participation by others is good, but too much is bad. For example, the El Farol Bar
might be most enjoyable if a reasonable crowd shows up, provided it does not exceed 60.
Similarly, an on-line social media site with limited infrastructure might be most enjoyable
if it has a reasonably large audience, but not so large that connecting to the Web site
becomes very slow due to the congestion. To keep the discussion here as clean as possible,
we are keeping the two kinds of externalities separate, but understanding how they work in
combination is an important topic of ongoing research [229]. We also consider a simple way
of combining the two e↵ects in Exercise 2 at the end of this chapter.
Basic Comparisons between the Two Scenarios. The contrasts between the two
scenarios — El Farol and corporate social-networking — translate into significant di↵erences
in how we should expect people to behave. Let’s first think about these di↵erences informally;
later we’ll carry out the analysis in more detail.
Reasoning about the social-networking scenario using what we’ve seen earlier in this
chapter, we find that there are two very natural equilibria the group of 100 people could
exhibit. If everyone participates, then everyone has an interest in participating; similarly, if
no one participates, then no one has an interest in participating. (There are also other more
complex equilibria, as we’ll see later, but these two all-or-nothing equilibria are by far the
two most natural.)
On the other hand, neither of these outcomes is an equilibrium in the El Farol Bar
problem. If everyone were to attend, then everyone would have an incentive to stay home;
and if no one were to attend, then everyone would have an incentive to attend. Instead, the
equilibria have a more complex structure in which individuals need to break the underlying
symmetry in such a way that some people attend and some stay home.
There is a second, essentially equivalent way to describe this contrast, using the idea
of shared expectations from earlier in the chapter. In the social-networking scenario, if the
individuals have a shared expectation that everyone will participate, then this expectation
is self-fulfilling: everyone will in fact participate. On the other hand, if they have a shared
expectation that no one will participate, this expectation too will be self-fulfilling. As we’ve
seen, understanding such self-fulfilling expectations is a key part of reasoning about situations
with positive externalities.
The negative externalities in the El Farol Bar problem, on the other hand, pose problems
for shared expectations. In particular, individuals cannot have a fixed, shared expectation
536 CHAPTER 17. NETWORK EFFECTS
of the audience size at El Farol that will be self-fulfilling. If everyone expects an audience
size of at most 60, then everyone will show up, thereby negating this prediction. Similarly,
if everyone expects an audience size above 60, then everyone will stay home, negating this
prediction too.6
These are fundamental contrasts: with positive externalities, there exist self-fulfilling
expectations and a natural set of outcomes to coordinate on; with negative externalities,
any shared expectation of a fixed audience size will be self-negating, and the individuals
must instead sort themselves out in much more complicated ways. Given this complexity,
the El Farol Bar problem has become a testing ground for a variety of models of individual
behavior. We now describe some of these models and styles of analysis in more detail.
Nash Equilibria in the El Farol Bar Problem. First, let’s consider how to model the
El Farol Bar problem as a game that will be played once by the set of 100 people. (We could
imagine that instead of the bar having music every Thursday, it is simply hosting a single
concert, and everyone needs to decide in advance whether to attend.) Each person has two
possible strategies, Go (to the bar) or Stay (home), and his payo↵s are as follows.
• If he chooses Stay, then he receives a payo↵ of 0 in all outcomes.
• If he chooses Go, then he receives a payo↵ of x > 0 when at most 60 people choose Go, and a payo↵ of �y < 0 when more than 60 people choose Go.
There are many di↵erent pure-strategy Nash equilibria for this game. Our discussion
above makes clear that there is no equilibrium in which all players use the same pure strategy,
but any outcome in which exactly 60 people choose Go and 40 people choose Stay is a pure-
strategy Nash equilibrium. Of course, it is far from clear how the group would settle on this
set of heterogeneous strategies, since at the outset of the game, they are all identical. We
will return to the question of how heterogeneous strategies might arise later in this section.
There is, however, an equilibrium in which all players behave symmetrically, and this
is through the use of mixed strategies, in which each player chooses Go with the same
probability p. In this case too, there are some subtleties. It would be natural to guess that
the shared probability p in this mixed-strategy equilibrium would be 0.6, but this is not
necessarily the case. Instead, p depends on the payo↵s x and �y: following the reasoning we saw in Chapter 6, we need to choose p so that each player is indi↵erent between choosing Go
and choosing Stay. This will ensure that no one has an incentive to deviate from randomizing
between the two alternatives.
6As Brian Arthur notes, this latter possibility is a reflection of the same phenomenon that the baseball player Yogi Berra invoked when he quipped about a popular restaurant, “Nobody goes there anymore; it’s too crowded” [26, 105].
17.7. ADVANCED MATERIAL: NEGATIVE EXTERNALITIES AND THE EL FAROL BAR PROBLEM537
Since the payo↵ for Stay is always 0, this means that we need to choose p so that the
expected payo↵ from Go is also 0. Therefore, we need to choose p so that the equation
x · Pr [at most 60 go] � y · Pr [more than 60 go] = 0 (17.3)
holds. Using the fact that
Pr [more than 60 go] = 1 � Pr [at most 60 go] ,
we can rearrange Equation (17.3) to get
Pr [at most 60 go] = y
x + y . (17.4)
So in order to have a mixed-strategy equilibrium, we must choose p so that Equation (17.4)
holds. When x = y, choosing p = 0.6 will work [212]. But suppose that x and y are di↵erent;
for example, perhaps the music at El Farol is pleasant, but the nights on which it is crowded
are truly unbearable, so that y is significantly larger than x. In this case, p must be chosen
so that the probability at most 60 people go is very high, and so p will be significantly
less than 0.6. Since the expected number of people attending is 100p, this means that in
expectation, significantly fewer than 60 people will be showing up. So with y > x, the bar
will be significantly underutilized in the mixed-strategy equilibrium, due to the shared fear
of overcrowding.
The existence of this mixed-strategy equilibrium is a useful counterpoint to our earlier
informal discussion about the di�culty of forming a shared expectation in the El Farol Bar
problem. It’s true that any shared expectation consisting of a fixed number representing the
audience size — the kind of shared expectation we used earlier in this chapter — will be
negated by what actually happens. But if we allow more complicated kinds of expectations,
then in fact there is a shared expectation that will be self-fulfilling — this is the expectation
that everyone plans to randomize their decision to attend the bar, choosing to go with the
probability p that makes Equation (17.4) come true.
Analogies with Equilibria in Related Games. To get some intuition about the equi-
libria we’ve found, it’s useful to compare them to the equilibria of some related games.
First, suppose we model the corporate social-networking scenario from earlier in this
section as a similar one-shot game: the two possible strategies are Join or Don’t Join (the
site); the payo↵ to Don’t Join is always 0, the payo↵ to Join is y when more than 60 people
join, and the payo↵ to Join is �x when at most 60 join. In this case, corresponding to what we saw in our informal discussion earlier, there are just two pure-strategy equilibria: one in
which everyone chooses Join, and one in which everyone choose Don’t Join. Interestingly,
the same mixed-strategy equilibrium that applied to the El Farol Bar problem also holds
here: if everyone chooses Join with a probability p for which
�x · Pr [at most 60 join] + y · Pr [more than 60 join] = 0 (17.5)
538 CHAPTER 17. NETWORK EFFECTS
then everyone is indi↵erent between joining and not joining, and so this is an equilibrium.
Since Equations (17.3) and (17.5) are equivalent (they are simply negations of each other),
we get the same value of p that we did in the El Farol Bar problem.
Since games with 100 players are inherently complex, it’s also instructive to ask what
the two-player versions of these games look like. Specifically, in the two-player version of
the El Farol Bar problem, each player wants to attend the bar as long as the other player
doesn’t; in the two-player social-networking game, each players wants to use the site as
long as the other one does. Scaled down to this size, each of these corresponds to one of
the fundamental games introduced in Chapter 6: the two-player El Farol Bar problem is a
Hawk-Dove game, in which the two players try to make their actions di↵erent, while the
two-player social-networking scenario is a Coordination game, in which the two players try
to make their actions the same.
Each of these games has pure-strategy equilibria, as well as a mixed-strategy equilibrium
in which the players randomize over their two available strategies. For example, in the
two-player version of the El Farol Bar problem, the payo↵ matrix is shown in Figure 17.14.
Player 1
Player 2 Stay Go
Stay 0, 0 0, x Go x, 0 �y,�y
Figure 17.14: Two-Player El Farol Problem
The two pure-strategy equilibria consist of one player choosing Go while the other chooses
Stay. For the mixed-strategy equilibrium, each player chooses Go with a probability p that
causes the expected payo↵ from Go to be equal to 0:
x(1 � p) � yp = 0,
and hence p = x/(x + y). By analogy with the multi-player version, p is not equal to 1/2
unless x = y. Also, just as there will be random fluctuations in the actual attendance at the
bar around the mean of 100p in the multi-player game, there is significant variation in how
many people choose Go in the two-player version as well. Specifically, with probability p2
both players choose Go, and with probability (1 � p)2 both choose Stay.
Repeated El Farol Problems. While the existence of a mixed-strategy equilibrium in
which all 100 people follow the same strategy is an important observation about the El
Farol Bar problem, it can’t be the whole story. It’s not clear why, or whether, a group of
people would actually arrive at this mixed-strategy equilibrium, or at any other particular
equilibrium or pattern of behavior from among the many that are possible. Once the group
is playing an equilibrium, no one has an incentive to deviate — that’s what it means for the
17.7. ADVANCED MATERIAL: NEGATIVE EXTERNALITIES AND THE EL FAROL BAR PROBLEM539
behaviors to be in equilibrium. But how do they coordinate on equilibrium behavior in the
first place?
To formulate models that address these questions, it is useful to think about a setting in
which the El Farol game is played repeatedly. That is, suppose that each Thursday night the
same 100 people must each decide whether to go to the bar or stay home, and each receives
a payo↵ of x (from going as part of a group of at most 60), �y (from going as part of a group of more than 60), or 0 (from staying home). Each person also knows the history of what has
happened on each prior Thursday, so they can use this information in making their decision
for the current Thursday. By reasoning about decisions over time in this repeated El Farol
game, we might hope to see how a pattern of behavior gradually emerges from rules that
take past experience into account.
There are a number of di↵erent formalisms that can be used to study the repeated El
Farol game. One approach is to view the full sequence of Thursdays as a dynamic game
of the type studied in Section 6.10, in which players choose sequences of strategies — in
this case, one for each Thursday — and correspondingly receive payo↵s over time. We
could consider Nash equilibria in this dynamic game and see whether the play of the game
eventually settles down to repeated play of some equilibrium of the one-shot El Farol game.
Essentially, we would be asking if equilibrium play by sophisticated players in the dynamic
game converges to something that looks simple. Although learning can go on during the play
of an equilibrium in a dynamic game [175], this approach can’t fully answer our underlying
question of how the individuals come to play a Nash equilibrium at all — since the learning
here would be taking place within a Nash equilibrium of the larger dynamic game.
An alternate approach is to ask what might happen if the players are potentially more
naive. A useful way to think about how players — sophisticated or naive — behave in a
repeated game is to decompose their choice of strategy into a forecasting rule and a choice
of action given the forecasting rule. A forecasting rule is any function that maps the past
history of play to a prediction about the actions that all other players will take in the
future. Forecasting rules can in principle be very complex. An individual could take all past
behavior into account in generating a forecast and he may also forecast that how others will
behave in the future depends on how he behaves now. Alternatively, a very naive player may
forecast that each other player will simply use a fixed action forever. From an individual’s
forecasting rule, we can then make a prediction about his behavior: we assume that each
individual behaves optimally given his forecasting rule. That is, he chooses an action that
maximizes his expected payo↵ given whatever he forecasts about the behavior of others.
For the repeated El Farol game, most attention has focused on forecasting rules that
work with audience sizes: a given forecasting rule is a function mapping the sequence of past
audience sizes to a prediction about the number of other people who will go to the bar on
the upcoming Thursday. (Thus, each forecasting rule produces a number between 0 and 99
540 CHAPTER 17. NETWORK EFFECTS
when given a history of past audience sizes.) This is a bit less expressive than a forecasting
rule that predicts, for each other person individually, whether they will go to the bar or stay
home, but it captures the main quantity of interest, which is the total number of people who
show up. For an individual using any such forecasting rule, his choice of action is easy to
describe: he goes to the bar if his forecasting rule produces a number that is at most 59,
and he stays home if it produces a number that is 60 or more.
In keeping with our informal discussion earlier in the section about self-fulfilling and self-
negating expectations, we observe first of all that if everyone uses the same forecasting rule
for the audience size, then everyone will make very bad predictions. In any given situation,
either this common forecasting rule will predict 59 or fewer others in attendance, or at
least 60 others in attendance. In the first case, everyone will show up, and in the second
case, everyone will stay home; in both cases, the forecasting rule is wrong. So to make any
progress, we will need for players to use a diversity of di↵erent forecasting rules.
A long line of research has considered how the group behaves when they use di↵erent
classes of forecasting rules; the goal is to understand whether the system converges to a
state in which, on any given Thursday, roughly 60% of the agents produce a forecast that
causes them to go the bar, and roughly 40% produce a forecast that causes them to stay
home (e.g. [26, 104, 167]). This investigation has been carried out both mathematically
and by computer simulation, with some of the analysis becoming quite complex. In general,
researchers have found that under a variety of conditions, the system converges to a state
where the average attendance varies around 60 — in other words, providing near-optimal
utilization of the bar over time.
While we won’t go into the details of this analysis, it is not hard to get some intuition for
why an average attendance of 60 arises very naturally when agents select from a diversity
of forecasting rules. To do this, we can analyze perhaps the simplest model of individual
forecasting, in which each person chooses a fixed prediction k for the number of others who
will show up, and uses this prediction every week. That is, he will ignore the past history
and always predict that k other people will be in attendance. Now, if each person picks their
fixed value of k uniformly at random from the 100 natural numbers between 0 and 99, what
is the expected audience size at the bar each week? The audience will consist of all people
whose forecasting rule is based on a value of k between 0 and 59, and the expected number
of such people is 60. Thus, with this very naive forecasting, we in fact get an expected
attendance of 60 each Thursday, as desired.
Of course, this analysis is based on people who make extremely naive forecasts, but it
shows how diversity in the set of forecasting rules can naturally lead to the right level of
attendance. One can also ask what happens when individuals select random forecasts from
a more complex space of possibilities, in which the prediction is based on the past several
audience sizes. Under fairly general assumptions, an average attendance of 60 continues to
17.8. EXERCISES 541
hold, although establishing this is significantly more complicated [104].
17.8 Exercises
1. Consider a product that has network e↵ects in the sense of our model from Chapter 17.
Consumers are named using real numbers between 0 and 1; the reservation price for
consumer x when a z fraction of the popuation uses the product is given by the formula
r(x)f (z), where r(x) = 1 � x and f (z) = z.
(a) Let’s suppose that this good is sold at cost 1/4 to any consumer who wants to
buy a unit. What are the possible equilibrium number of purchasers of the good?
(b) Suppose that the cost falls to 2/9 and that the good is sold at this cost to any
consumer who wants to buy a unit. What are the possible equilibrium number of
purchasers of the good?
(c) Briefly explain why the answers to parts (a) and (b) are qualitatively di↵erent.
(d) Which of the equilibria you found in parts (a) and (b) are stable? Explain your
answer.
2. In Chapter 17, we focused on goods with positive network e↵ects: ones for which addi-
tional users made the good more attractive for everyone. But we know from our earlier
discussion of Braess’s Paradox that network e↵ects can sometimes be negative: more
users can sometimes make an alternative less attractive, rather than more attractive.
Some goods actually have both e↵ects. That is, the good may become more attractive
as more people use it as long there aren’t too many users, and then once there are too
many users it becomes less attractive as more people use it. Think of a club in which
being a member is more desirable if there is a reasonable number of other members,
but once the number of members gets too large the club begins to seem crowded and
less attractive. Here we explore how our model of network e↵ects can incorporate such
a combination of e↵ects.
In keeping with the notation in Chapter 17, let’s assume that consumers are named
using real numbers between 0 and 1. Individual x has the reservation price r(x) = 1�x before we consider the network e↵ect. The network e↵ect is given by f (z) = z for
z 1/4 and by f (z) = (1/2)�z for z � 1/4. So the network benefit to being a user is maximized when the fraction of the population using the product is z = 1/4, once the
fraction is beyond 1/4 the benefit declines, and it becomes negative if more than 1/2 of
the population is using it. Suppose that the price of this good is p where 0 < p < 1/16.
(a) How many equilibria are there? Why? [You do not need to solve for the number(s)
of users; a graph and explanation is fine.]
542 CHAPTER 17. NETWORK EFFECTS
(b) Which equilibria are stable? Why?
(c) Consider an equilibrium in which someone is using the good. Is social welfare
maximized at this number of users, or would it go up if there were more users, or would
it go up if there were fewer users? Explain. [Again no calculations are necessary; a
careful explanation is su�cient.]
3. You have developed a new product which performs the same service as an established
product and your product is much better than the established product. Specifically, if
the number of users of the two products were the same, then each potential purchaser’s
reservation price for your product would be twice their reservation price for the ex-
isting product. The di�culty that you face is that these are products with network
e↵ects and no one wants to use more than one of the two products. Currently, every
potential purchaser is using the established product. Your cost of production and your
competitor’s costs of production are exactly the same and let’s suppose that they are
equal to the price at which your competitor’s product is sold.
If all of the potential purchasers switched to your product the maximum price that you
could charge (and still have all of them buy your product) would be twice the current
price. So clearly you could make a nice profit if you could attract these potential
purchasers. How would you attempt to convince users to switch to your product? You
do not need to construct a formal model of the situation described in this question. It
is su�cient to describe the strategies that you would try.
4. In the model of network e↵ects that we covered in Chapter 17 there was only one
product. Now let’s ask what might happen if there are two competing products which
both have network e↵ects. Assume that for each product:
(a) If no one is expected to use the product, then no one places a positive value on
the product.
(b) If one-half of the consumers are expected to use the product, then exactly one-half
of the consumers would buy the product.
(c) If all of the consumers are expected to use the product, then all consumers would
buy the product.
Using an analysis of network e↵ects, describe the possible equilibrium configurations of
numbers of consumers using each product and briefly discuss which of these equilibria
you would expect to be stable and which you would expect to be unstable. You do
not need to build a formal model to answer this question. Just describe in words what
might happen in this market.
Chapter 18
Power Laws and Rich-Get-Richer Phenomena
18.1 Popularity as a Network Phenomenon
For the past two chapters, we have been studying situations in which a person’s behavior
or decisions depend on the choices made by other people — either because the person’s
rewards are dependent on what other people do, or because the choices of other people
convey information that is useful in the decision-making process. We’ve seen that these
types of coupled decisions, where behavior is correlated across a population, can lead to
outcomes very di↵erent from what we find in cases where individuals make independent
decisions.
Here we apply this network approach to analyze the general notion of popularity. Pop-
ularity is a phenomenon characterized by extreme imbalances: while almost everyone goes
through life known only to people in their immediate social circles, a few people achieve
wider visibility, and a very, very few attain global name recognition. Analogous things could
be said of books, movies, or almost anything that commands an audience. How can we
quantify these imbalances? Why do they arise? Are they somehow intrinsic to the whole
idea of popularity?
We will see that some basic models of network behavior can provide significant insight
into these questions. To begin the discussion, we focus on the Web as a concrete domain
in which it is possible to measure popularity very accurately. While it may be di�cult to
estimate the number of people worldwide who have heard of famous individuals such as
Barack Obama or Bill Gates, it is easy to take a snapshot of the full Web and simply count
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
543
544 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
Figure 18.1: The density of values in the normal distribution.
the number of links to high-profile Web sites such as Google, Amazon, or Wikipedia. We
will refer to the full set of links pointing to a given Web page as the in-links to the page.
Thus, we will start by using the number of in-links to a Web page as a measure of the page’s
popularity; we will keep in mind, however, that this is just one example of a much broader
phenomenon.
Early in the Web’s history, people had already begun to ask a very basic version of the
page popularity question, phrased as follows:
As a function of k, what fraction of pages on the Web have k in-links?
Larger values of k indicate greater popularity, so this is precisely the question of how popu-
larity is distributed over the set of Web pages.
A Simple Hypothesis: The Normal Distribution. Before trying to resolve this ques-
tion, it’s useful to ask what we should expect the answer to be. A natural guess is the
normal, or Gaussian, distribution — the so-called “bell curve” — used widely throughout
probability and statistics. While we won’t need many details about the normal distribution
here, it’s worth recalling that it’s characterized by two quantities: a mean value, and a stan-
dard deviation around this mean. Figure 18.1 shows a plot of the density of values in the
normal distribution, scaled so that the mean is 0 and the standard deviation is 1. The basic
fact about normal distributions is that the probability of observing a value that exceeds the
mean by more than c times the standard deviation decreases exponentially in c.
The normal distribution is a natural guess in our case, since it is ubiquitous across the
natural sciences. A result from the early 1900s, the Central Limit Theorem, provides a
18.2. POWER LAWS 545
fundamental explanation for its appearance in so many settings: roughly speaking (and
suppressing the full details), the Central Limit Theorem says that if we take any sequence
of small independent random quantities, then in the limit their sum (or average) will be
distributed according to the normal distribution. In other words, any quantity that can
be viewed as the sum of many small independent random e↵ects will be well-approximated
by the normal distribution. Thus, for example, if one performs repeated measurements of
a fixed physical quantity, and if the variations in the measurements across trials are the
cumulative result of many independent sources of error in each trial, then the distribution
of measured values should be approximately normal.
How would we apply this in the case of Web pages? If we model the link structure of the
Web, for example, by assuming that each page decides independently at random whether to
link to any other given page, then the number of in-links to a given page is the sum of many
independent random quantities (i.e. the presence or absence of a link from each other page),
and hence we’d expect it to be normally distributed. In particular, this would suggest a
hypothesis for the answer to our original question: if we believe this model, then the number
of pages with k in-links should decrease exponentially in k, as k grows large.
18.2 Power Laws
When people measured the distribution of links on the Web, however, they found something
very di↵erent. In studies over many di↵erent Web snapshots, taken at di↵erent points in the
Web’s history, the recurring finding is that the fraction of Web pages that have k in-links is
approximately proportional to 1/k2 [80]. (More precisely, the exponent on k is generally a
number slightly larger than 2.)
Why is this so di↵erent from the normal distribution? The crucial point is that 1/k2
decreases much more slowly as k increases, so pages with very large numbers of in-links
are much more common than we’d expect with a normal distribution. For example, 1/k2
is only one in a million for k = 1000, while an exponentially decaying function like 2�k is
unimaginably tiny for k = 1000. A function that decreases as k to some fixed power, such as
1/k2 in the present case, is called a power law; when used to measure the fraction of items
having value k, it says, qualitatively, that it’s possible to see very large values of k.
This provides a quantitative form for one of the points we made initially: popularity
seems to exhibit extreme imbalances, with very large values likely to arise. And it accords
with our intuition about the Web, where there are certainly a reasonably large number of
extremely popular pages. One sees similar power laws arising in measures of popularity in
many other domains as well: for example, the fraction of telephone numbers that receive k
calls per day is roughly proportional to 1/k2; the fraction of books that are bought by k
people is roughly proportional to 1/k3; the fraction of scientific papers that receive k citations
546 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
Figure 18.2: A power law distribution (such as this one for the number of Web page in-links, from Broder et al. [80]) shows up as a straight line on a log-log plot.
in total is roughly proportional to 1/k3; and there are many related examples [10, 320].
Indeed, just as the normal distribution is widespread in a family of settings in the natural
sciences, power laws seem to dominate in cases where the quantity being measured can be
viewed as a type of popularity. Hence, if you are handed data of this sort — say, for example,
that someone gives you a table showing the number of monthly downloads for each song at
a large on-line music site that they’re hosting — one of the first things that’s worth doing
is to test whether it’s approximately a power law 1/kc for some c, and if so, to estimate the
exponent c.
There’s a simple method that provides at least a quick test for whether a dataset exhibits
a power-law distribution. Let f (k) be the fraction of items that have value k, and suppose you
want to know whether the equation f (k) = a/kc approximately holds, for some exponent
c and constant of proportionality a. Then, if we write this as f (k) = ak�c and take the
logarithms of both sides of this equation, we get
log f (k) = log a � c log k.
This says that if we have a power-law relationship, and we plot log f (k) as a function of log k,
then we should see a straight line: �c will be the slope, and log a will be the y-intercept. Such a “log-log” plot thus provides a quick way to see if one’s data exhibits an approximate
power-law: it is easy to see if one has an approximately straight line, and one can read o↵
the exponent from the slope. For example, Figure 18.2 does this for the fraction of Web
pages with k in-links [80].
But if we are going to accept that power laws are so widespread, we also need a simple
explanation for what is causing them: just as the Central Limit Theorem gave us a very
18.3. RICH-GET-RICHER MODELS 547
basic reason to expect the normal distribution, we’d like something comparable for power
laws. For example, it’s striking how closely the plot in Figure 18.2 follows a straight line
for much of the distribution, especially considering how many utterly uncontrollable factors
come into play in the formation of the Web’s link structure. What underlying process is
keeping the line so straight?
18.3 Rich-Get-Richer Models
Ideas from the analysis of information cascades and network e↵ects provide the basis for
a very natural mechanism to generate power laws. Just as normal distributions arise from
many independent random decisions averaging out, we will find that power laws arise from
the feedback introduced by correlated decisions across a population.
It is actually an open and very interesting research question to provide a fully satisfactory
model of power laws starting from simple models of individual decision-making (as we did
for information cascades). Instead, we will build our model not from the internals of each
person’s decision-making process, but from the observable consequences of decision-making
in the presence of cascades: we will assume simply that people have a tendency to copy the
decisions of people who act before them.
Based on this idea, here is a simple model for the creation of links among Web pages
[42, 265, 300, 340, 371].
(1) Pages are created in order, and named 1, 2, 3, . . . , N .
(2) When page j is created, it produces a link to an earlier Web page according to the
following probabilistic rule (which is controlled by a single number p between 0 and 1).
(a) With probability p, page j chooses a page i uniformly at random from among all
earlier pages, and creates a link to this page i.
(b) With probability 1�p, page j instead chooses a page i uniformly at random from among all earlier pages, and creates a link to the page that i points to.
(c) This describes the creation of a single link from page j; one can repeat this process
to create multiple, independently generated links from page j. (However, to keep
things simple, we will suppose that each page creates just one outbound link.)
Part (2b) of this process is the key: after finding a random earlier page i in the population,
the author of page j does not link to i, but instead copies the decision made by the author
of page i — linking to the same page that i did.
The main result about this model is that if we run it for many pages, the fraction of
pages with k in-links will be distributed approximately according to a power law 1/kc, where
the value of the exponent c depends on the choice of p [68]. This dependence goes in an
548 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
intuitively natural direction: as p gets smaller, so that copying becomes more frequent, the
exponent c gets smaller as well, making one more likely to see extremely popular pages.
Proving this result would require more intricate analysis than we’ll be able to do here,
but it is useful to work through some of the informal ideas behind this analysis. First of all,
the copying mechanism in (2b) is really an implementation of the following “rich-get-richer”
dynamics: when you copy the decision of a random earlier page, the probability that you
end up linking to some page ` is directly proportional to the total number of pages that
currently link to `. Thus, an equivalent way to write our copying process would have been
to phrase (2b) as
(2) ...
(b) With probability 1 � p, page j chooses a page ` with probability proportional to `’s current number of in-links, and creates a link to `.
Why do we call this a “rich-get-richer” rule? Because the probability that page ` experiences
an increase in popularity is directly proportional to `’s current popularity. This phenomenon
is also known as preferential attachment [42], in the sense that links are formed “preferen-
tially” to pages that already have high popularity. And the copying model provides an
operational story for why popularity should exhibit such rich-get-richer dynamics: essen-
tially, the more well-known someone is, the more likely you are to hear their name come up
in conversation, and hence the more likely you are to end up knowing about them as well.
The same holds for Web pages, the specific focus of our model here.
The remaining intuition behind the analysis runs as follows. With the rich-get-richer
dynamics in place, our model predicts that popularity should grow according to the same
rule that governs the growth of bacterial colonies and compound interest: a page’s popularity
grows at a rate proportional to its current value, and hence exponentially with time. A page
that gets a small lead over others will therefore tend to extend this lead; whereas the crux
of the Central Limit Theorem is that small independent random values tend to cancel each
other out, the rich-get-richer nature of copying actually amplifies the e↵ects of large values,
making them even larger. In Section 18.7 at the end of this chapter, we show how to
turn this reasoning into a calculation that produces the correct exponent on the power-law
distribution.
As with any simple model, the goal is not to capture all the reasons why people create
links on the Web, or in any other network, but to show that a simple and very natural
principle behind link creation leads directly to power laws — and hence, one should not find
them as surprising as they might first appear.
Indeed, rich-get-richer models can suggest a basis for power laws in a wide array of
settings, including some that have nothing at all to do with human decision-making. For
example, the populations of cities have been observed to follow a power law distribution: the
18.4. THE UNPREDICTABILITY OF RICH-GET-RICHER EFFECTS 549
fraction of cities with population k is roughly 1/kc for some constant c [371]. If we assume
that cities are formed at di↵erent times, and that, once formed, a city grows in proportion
to its current size simply as a result of people having children, then we have almost precisely
the same rich-get-richer model — and hence we should not be surprised to see the power law
that is in fact present in reality. To take a very di↵erent example, researchers in biology have
argued (though the data is still too sparse to be sure) that the number of copies of a gene
in a genome approximately follows a power-law distribution [99]. If we believe that gene
copies arise in large part through mutational events in which a random segment of DNA is
accidentally duplicated, then a gene which already has many copies is proportionally more
likely to be lying in a random stretch of DNA that gets copied — so “rich” genes (those with
many copies) get “richer,” and again we should not be surprised to see a power law.
A priori, finding similar laws governing Web page popularity, city populations, and gene
copies is quite mysterious; but if one views all these as outcomes of processes exhibiting rich-
get-richer e↵ects, then the picture starts to become clearer. Again, one must stress that these
are still simple models designed just to approximate what’s going on; and there are other
classes of simple models designed to capture power-law behavior that we have not discussed
here. For example, a parallel thread of research has argued how power laws can arise from
systems that are being optimized in the presence of constraints [96, 136, 151, 284, 300]. But
what all these simple models suggest is that when one sees a power law in data, the possible
reasons why it’s there can often be more important than the simple fact that it’s there.
18.4 The Unpredictability of Rich-Get-Richer E↵ects
Given the nature of the feedback e↵ects that produce power laws, it’s natural to suspect that
for a Web page, a book, a song, or any other object of popular attention, the initial stages
of its rise to popularity is a relatively fragile thing. Once any one of these items is well-
established, the rich-get-richer dynamics of popularity are likely to push it even higher; but
getting this rich-get-richer process ignited in the first place seems like a precarious process,
full of potential accidents and near-misses.
This sensitivity to unpredictable initial fluctuations is something that we saw in the
previous two chapters as well: information cascades can depend on the outcome of a small
number of initial decisions in the population, and a worse technology can win because it
reaches a certain critical audience size before its competitors do. The dynamics of popularity
suggest that random e↵ects early in the process should play a role here as well. For example,
if we could roll time back 15 years, and then run history forward again, would the Harry
Potter books again sell hundreds of millions of copies, or would they languish in obscurity
while some other works of children’s fiction achieved major success? One’s intuition suggests
the latter. More generally, if history were to be replayed multiple times, it seems likely that
550 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
there would be a power-law distribution of popularity each of these times, but it’s far from
clear that the most popular items would always be the same.
Thought experiments of this type are useful in considering the consequences of our mod-
els, but, needless to say, it’s di�cult to actually implement them as real experiments. Re-
cently, however, Salgankik, Dodds, and Watts performed an experiment that begins to pro-
vide some empirical support for this intuition [359]. They created a music download site,
populated with 48 obscure songs of varying quality written by actual performing groups.
Visitors to the site were presented with a list of the songs and given the opportunity to
listen to them. Each visitor was also shown a table listing the current “download count” for
each song — the number of times it had been downloaded from the site thus far. At the end
of a session, the visitor was given the opportunity to download copies of the songs that he
or she liked.
Now, unbeknownst to the visitors, upon arrival they were actually being assigned at
random to one of eight “parallel” copies of the site. The parallel copies started out identically,
with the same songs and with each song having a download count of zero. However, each
parallel copy then evolved di↵erently as users arrived. In a controlled, small-scale setting,
then, this experiment provided a way to observe what happens to the popularities of 48 songs
when you get to run history forward eight di↵erent times. And in fact, it was found that the
“market share” of the di↵erent songs varied considerably across the di↵erent parallel copies,
although the best songs never ended up at the bottom and the worst songs never ended up
at the top.
Salganik et al. also used this approach to show that, overall, feedback produced greater
inequality in outcomes. Specifically, they assigned some users to a ninth version of the site
in which no feedback about download counts was provided at all. In this version of the
site, there was no direct opportunity for users to contribute to rich-get-richer dynamics, and
indeed, there was significantly less variation in the market share of di↵erent songs.
There are clear implications for popularity in less controlled environments, parallel to
some of the conclusions we’ve drawn from our models — specifically, that the future success
of a book, movie, celebrity, or Web site is strongly influenced by these types of feedback
e↵ects, and hence may to some extent be inherently unpredictable.
Closer Relationships between Power Laws and Information Cascades? Consid-
erations of this sort suggest an important question for further research: understanding the
relationship between power laws and information cascades at a deeper level. When we looked
at information cascades, we saw how a population in which people were aware of earlier de-
cisions made between two alternatives (e.g. accepting an idea or rejecting it) could end up
in a cascade, even if each person is making an optimal decision given what they’ve observed.
Our copying model for power laws draws on the intuition behind this model, but it di↵ers in
18.5. THE LONG TAIL 551
several respects. First, a model for popularity should include choices among many possible
options (e.g. all possible Web pages), rather than just two options. Second, the copying
model involves a set of people who engage in very limited observation of the population:
when you create a new Web page, the model assumes you consult the decision of just one
other randomly selected person. And third, the copying model is based on the idea that
later people imitate the decisions of earlier people, but it doesn’t derive this imitation from
a more fundamental model of rational decision-making.
The first two of these di↵erences simply reflect the specifics of the problem being modeled
here — the way in which popularity evolves over time. But it would be very interesting to
overcome the third of these di↵erences, constructing a copying-style model to produce power
laws on top of a base model of individual decision-making. Such an approach could shed
further insight into the mechanisms behind rich-get-richer dynamics, and provide a picture of
popularity as arising from competing information cascades whose intensities vary according
to the power laws that we observe in real systems.
18.5 The Long Tail
The distribution of popularity can have important business consequences, particularly in the
media industry. In particular, let’s imagine a media company with a large inventory — a
giant retailer of books or music, for example — and consider the following question: are
most sales being generated by a small set of items that are enormously popular, or by a
much larger population of items that are each individually less popular? In the former case,
the company is basing its success on selling “hits” — a small number of blockbusters that
create huge revenues. In the latter case, the company is basing its success on a multitude of
“niche products,” each of which appeals to a small segment of the audience.
In a widely-read 2004 article entitled “The Long Tail,” Chris Anderson argued that
Internet-based distribution and other factors were driving the media and entertainment
industries toward a world in which the latter alternative would be dominant, with a “long
tail” of obscure products driving the bulk of audience interest [13]. As he wrote, “You can
find everything out there on the Long Tail. There’s the back catalog, older albums still
fondly remembered by longtime fans or rediscovered by new ones. There are live tracks,
B-sides, remixes, even (gasp) covers. There are niches by the thousands, genre within genre
within genre: Imagine an entire Tower Records devoted to ’80s hair bands or ambient dub.”
Although sales data indicates that the trends are in fact somewhat complex [146], this
tension between hits and niche products makes for a compelling organizing framework. It
also accords with the fundamental models of companies like Amazon or Netflix, where the
ability to carry huge inventories — without the restrictions imposed by physical stores —
makes it feasible to sell an astronomical diversity of products even when very few of them
552 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
k
j
sales volume
number
of books
There are j books that
have sold at least k
copies.
Figure 18.3: The distribution of popularity: how many items have sold at least k copies?
generate much volume on their own. And ultimately, quantifying the importance of the Long
Tail comes down to an analysis of power laws.
Visualizing the Long Tail. The first thing to notice, when we compare this discussion of
the Long Tail to our earlier analysis of power laws, is that in some sense we’re now viewing
things out the opposite end of the telescope. Initially, we started from a baseline in which we
expected to see Gaussian distributions and tight concentration around the average, and then
we observed that the number of highly popular items was much higher than this baseline
would suggest. Now, on the other hand, we’re starting from a very di↵erent default view of
the world — a sort of stereotype of the media business in which only blockbusters matter
— and we’re observing that the total sales volume of unpopular items, taken together, is
really very significant. In terms of the plot in Figure 18.2, this new view is focusing on the
upper-left part of the plot, whereas before we were primarily focused on the lower-right.
Once you recognize that this contrast is going on, it’s not hard to reconcile the two views
[4]. First, let’s modify our original definition of the popularity curve slightly, in a way that
doesn’t fundamentally change what we’re measuring. Rather than asking
As a function of k, what fraction of items have popularity exactly k?
let’s instead ask
As a function of k, what number of items have popularity at least k?
Notice that we’ve changed two things: “fraction” to “number” (a completely inconsequential
change), and “exactly k” to “at least k”. This second change modifies the function we’re
18.5. THE LONG TAIL 553
j
k
number of books
sales
volume
The j-th most popular
book has sold k
copies.
Figure 18.4: The distribution of popularity: how many copies of the jth most popular item have been sold?
considering; but while we won’t go through the derivation here, it’s possible to show that if
the original function was a power-law, then this new one is too. We show a schematic plot
of this new function in Figure 18.3; if we’re talking about the popularity of some item like
books, then a point (k, j) on this curve means, by definition, “There are j books that have
sold at least k copies.”
So far, this is still the conceptual view from the previous section: as we follow the x-axis
of the curve to the right, we’re essentially asking, “As you look at larger and larger sales
volumes, how few books do you find?” To capture the discussions of the Long Tail more
directly, we want to be asking the following question as we follow the x-axis to the right:
“As you look at less and less popular items, what sales volumes do you see?”
If we think about it, this simply involves switching the two axes. That is, suppose that
we plot exactly the same curve, but we interchange the roles of the x- and y-axes, as shown
in Figure 18.4. Interpreting this new curve literally from its definition, a point (j, k) on the
curve says, “The jth most popular book has sold k copies.” This is exactly what we want:
we order the books by “sales rank,” and then we look at the popularity of books as we move
out to larger and larger sales ranks — into the niche products.1 And the characteristic shape
of this curve, tailing o↵ slowly downward to the right, is the visual basis for the term “Long
Tail.”
One can now easily discuss trends in sales volume, and their consequences, in terms of
the curve in Figure 18.4. Essentially, the area under the curve from some point j outward is
1Our notion of “sales rank” simply reflects the sorted, decreasing order of all items by sales volume. When the term “sales rank” is used by on-line retailers such as Amazon, it tends to be a more complex measure that incorporates other factors as well.
554 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
the total volume of sales generated by all items of sales-rank j and higher; and so a concrete
version of the hits-vs.-niche question, for a particular set of products, is whether there is
significantly more area under the left part of this curve (hits) or the right (niche products).
And the debate over trends toward niche products becomes a question of whether this curve
is changing shape over time, adding more area under the right at the expense of the left.
It is worth noting that curves of the type in Figure 18.4 — with the axes ordered so that
the variable on the x-axis is rank rather than popularity — have a long history. They are
often called Zipf plots after the linguist George Kingsley Zipf, who produced such curves
for a number of human activities [423]. Most famously, he identified the empirical principle
known as Zipf’s Law, that the frequency of the jth most common word in English (or most
other widespread human languages) is proportional to 1/j. Thus, perhaps not surprisingly,
the debate within the media industry about curves like this echoes earlier fascination in other
areas.
18.6 The E↵ect of Search Tools and Recommendation Systems
We conclude by briefly discussing a further question that has been growing in importance as
people consider popularity and its distribution: are Internet search tools making the rich-
get-richer dynamics of popularity more extreme or less extreme? What is interesting is that
there are two compelling but juxtaposed sides to this question, and its ultimate resolution
will likely involve decisions about how individuals and corporations design and deploy future
generations of search tools.
On one side of this question, we’ve seen that a model in which people copy links from
uniformly random Web pages already gives an advantage to popular pages. But once people
are using search engines such as Google to find pages, then even the choice of what to copy
from becomes highly skewed: as we’ve seen, Google is using popularity measures to rank
Web pages, and the highly-ranked pages are in turn the main ones that users see in order
to formulate their own decisions about linking. A similar argument can be made for other
media in which a handful of the most popular items have the potential to crowd out all
others. In simple models, this kind of feedback can accentuate rich-get-richer dynamics,
producing even more inequality in popularity [103].
There are other forces at work, however. To begin with, users type a very wide range of
queries into Google, and so there isn’t a single list of “top pages on Google” — rather, by
getting results on relatively obscure queries, users are being led to pages that they are likely
never to have discovered through browsing alone. Search tools used in this style, targeted
more closely to users’ specific interests, can in fact provide ways around universally popular
pages, enabling people to find unpopular items more easily, and potentially counteracting
18.7. ADVANCED MATERIAL: ANALYSIS OF RICH-GET-RICHER PROCESSES 555
the rich-get-richer dynamics. Here too, simple mathematical models have demonstrated how
such e↵ects can work [165].
This latter view also forms an important part of Anderson’s Long-Tail argument: in
order to make money from a giant inventory of niche products, a company crucially needs
for its customers to be aware of these products, and to have some reasonable way to explore
them [13]. Viewed in this light, the types of recommendation systems that companies like
Amazon and Netflix have popularized can be seen as integral to their business strategies:
they are essentially search tools designed to expose people to items that may not be generally
popular, but which match user interests as inferred from their history of past purchases.
Ultimately, the design of search tools is an example of a kind of higher-order feedback
e↵ect: by causing people to process their available options in one way or another, we can
reduce rich-get-richer e↵ects, or amplify them, or potentially steer them in di↵erent direc-
tions altogether. These are among the subtle consequences that take place when we inject
sophisticated information systems into what is an already complex social system.
18.7 Advanced Material: Analysis of Rich-Get-Richer Processes
In Section 18.3, we described a simple model of a growing directed network based on copying
— or equivalently, based on rich-get-richer dynamics. We claimed there that the fraction of
nodes with k in-links is distributed approximately according to a power law 1/kc, where c
depends on the behavior of nodes in the model. Here we provide a heuristic argument that
analyzes the behavior of the model to indicate why the power law arises, and in fact goes
further to show how the power-law exponent c is related to more basic features of the model.
The analysis is based on the simple di↵erential equation governing exponential growth that
one sees in introductory calculus.
First, let’s reprise the description of the model from Section 18.3, as follows.
(1) Pages are created in order, and named 1, 2, 3, . . . , N .
(2) When page j is created, it produces a link to an earlier Web page according to the
following probabilistic rule (which is controlled by a single number p between 0 and 1).
(a) With probability p, page j chooses a page i uniformly at random from among all
earlier pages, and creates a link to this page i.
(b) With probability 1 � p, page j chooses a page ` with probability proportional to `’s current number of in-links, and creates a link to `.
(c) This describes the creation of a single link from page j; one can repeat this process
to create multiple, independently generated links from page j. (However, to keep
556 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
things simple, we will suppose that each page creates just one outbound link.)
Note that here we are using the rich-get-richer version of step (2b), rather the original
copying version. Recall that the two formulations are equivalent; but for our purposes, it
will be easier to analyze the phrasing of the model as we have it here.
At one level, we now have a purely probabilistic question: we have specified a randomized
process that runs for N steps (as the N pages are created one at a time), and we can simply
determine the expected number of pages with k in-links at the end of the process. (Or analyze
the distribution of this quantity.) While several groups of researchers have performed this
analysis [68], it is a bit more intricate than what we can feasibly cover here. Instead, we
describe how an approximation to the model allows for a much simpler calculation that gives
the correct value for the exponent c in the power law. This approximate analysis was the
first one to be carried out [42], and it thus provided heuristic evidence for the power-law
e↵ect that was then verified by more rigorous analysis of the full probabilistic model.
A deterministic approximation of the rich-get-richer process. Before describing the
approximation to the model, let’s discuss some simple properties of the original probabilistic
model itself. First, the number of in-links to a node j at a time step t � j is a random variable X
j
(t). Let’s observe two facts about X j
(t).
(a) The initial condition. Since node j starts with no in-links when it is first created at
time j, we know that X j
(j) = 0.
(b) The expected change to X j
over time. Node j gains an in-link in step t + 1 if and only
if the link from the newly created node t + 1 points to it. What is the probability that
this happens? With probability p, node t + 1 links to an earlier node chosen uniformly
at random, and with probability 1 � p it links to an earlier node with probability proportional to the node’s current number of in-links. In the former case, node t + 1
links to node j with probability 1/t. For the latter case, we observe that at the moment
node t + 1 is created, the total number of links in the network is t (one out of each
prior node), and of these, X j
(t) point to node j. Thus, in the latter case, node t + 1
links to node j with probability X j
(t)/t. Therefore, the overall probability that node
t + 1 links to node j is p
t +
(1 � p)X j
(t)
t .
The basic plan in building an approximation to the model is to analyze a di↵erent,
closely analogous, but simpler rich-get-richer process, in which it is correspondingly easier to
discover the power law. Again, this does not directly imply that the original model behaves
the same way, but the similarities between the two models o↵er evidence that can then be
verified by further analysis of the original model.
18.7. ADVANCED MATERIAL: ANALYSIS OF RICH-GET-RICHER PROCESSES 557
The central idea in formulating the simpler model is to make it deterministic — that is, a
model in which there are no probabilities, but in which everything instead evolves in a fixed
way over time, like an idealized physical system that behaves according to some “equations
of motion” starting from a set of initial conditions. To do this, we have time run continuously
from 0 to N (rather than in the discrete steps 1, 2, 3, . . .), and we approximate X j
(t) — the
number of in-links of node j — by a continuous function of time x j
(t). We characterize the
function x j
by two properties that seek to approximate the initial conditions and expected
change over time that we described above for X j
(t). The two properties of the function x j
are the following.
(a) The initial condition. Recall that X j
(j) = 0. We define x j
(j) = 0 as well.
(b) The growth equation. Recall that when node t + 1 arrives, the number of in-links to
node j increases with probability
p
t +
(1 � p)X j
(t)
t
In the determinstic approximation provided by the function x j
, we model this rate of
growth by the di↵erential equation
dx j
dt =
p
t +
(1 � p)x j
t . (18.1)
Using di↵erential equations, we have thus specified the behavior of x j
, our deterministic
approximation to the number of in-links to node j over time. Essentially, rather than dealing
with random variables X j
(t) that move in small probabilistic “jumps” at discrete points in
time, we get to work with a quantity x j
that grows completely smoothly over time, at a rate
tuned to match the expected changes in the corresponding random variables.
We now explore the consequences of the di↵erential equation defining x j
; this leads
quickly to the kind of power-law distribution we want.
Solving the deterministic approximation. We begin by solving the di↵erential equa-
tion (18.1) governing x j
. For notational simplicity, let’s write q = 1�p, so that the di↵erential equation becomes
dx j
dt =
p + qx j
t .
Dividing both sides by p + qx j
, we get
1
p + qx j
dx j
dt =
1
t .
Integrating both sides Z 1
p + qx j
dx j
dt dt =
Z 1
t dt,
558 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
we get
ln(p + qx j
) = q ln t + c
for a constant c. Exponentiating, and writing A = ec, we get
p + qx j
= Atq
and hence
x j
(t) = 1
q (Atq � p) . (18.2)
Now, we can determine the value of the constant A by using the initial condition x j
(j) = 0.
This condition gives us the equation
0 = x j
(j) = 1
q (Ajq � p) ,
and hence A = p/jq. Plugging this value for A into Equation (18.2), we get
x j
(t) = 1
q
✓ p
jq · tq � p
◆ =
p
q
✓ t
j
◆ q
� 1 �
. (18.3)
Identifying a power law in the deterministic approximation. Equation (18.3) is a
significant intermediate step in the analysis, since it gives us a closed-form expression for
how each x j
grows over time. Now we want to use this to ask the following question: For a
given value of k, and a time t, what fraction of all nodes have at least k in-links at time t?
Since x j
approximates the number of in-links of node j, the analogue to this question that
we consider in our simplified model is: For a given value of k, and a time t, what fraction of
all functions x j
satisfy x j
(t) � k? Using Equation (18.3), this corresponds to the inequality
x j
(t) = p
q
✓ t
j
◆ q
� 1 � � k,
or, re-writing this in terms of j,
j t q
p · k + 1
��1/q .
Out of all the functions x 1
, x 2
, . . . , x t
at time t, the fraction of values j that satisfy this is
simply 1
t · t
q
p · k + 1
��1/q =
q
p · k + 1
��1/q . (18.4)
We can already see the power law taking shape here: since p and q are constants, the
expression inside brackets on the right-hand-side is proportional to k, and so the fraction of
x j
that are at least k is proportional to k�1/q.
18.8. EXERCISES 559
For the final step, note that this has so far been about the fraction of nodes F (k) with at
least k in-links. But from this, we can directly approximate the fraction of nodes f (k) with
exactly k in-links simply by taking the derivative — in other words, approximating f (k) by
�dF/dk. Di↵erentiating the expression in Equation (18.4), we get
1
q
q
p
q
p · k + 1
��1�1/q .
In other words, the deterministic model predicts that the fraction of nodes with k in-links is
proportional to k�(1+1/q) — a power law with exponent
1 + 1
q = 1 +
1
1 � p .
Subsequent analysis of the original probabilistic model showed that, with high probability
over the random formation of links, the fraction of nodes with k in-links is indeed proportional
to k�(1+1/(1�p)) [68]. The heuristic argument supplied by the deterministic approximation to
the model thus provides a simple way to see where this power-law exponent 1 + 1/(1 � p) comes from.
The behavior of this exponent also makes sense intuitively as we vary p. When p is close
to 1, link formation is mainly based on uniform random choices, and so the role of rich-
get-richer dynamics is muted. Correspondingly, the power-law exponent tends to infinity,
showing that nodes with very large numbers of in-links become increasingly rare. On the
other hand, when p is close to 0, the growth of the network is strongly governed by rich-get-
richer behavior, and the power-law exponent decreases toward 2, allowing for many nodes
with very large numbers of in-links. The fact that 2 is a natural limit for the exponent as
rich-get-richer dynamics become stronger also provides a nice way to think about the fact
that many power-law exponents in real networks (such as for the number of in-links to a
Web page) tend to be slightly above 2.
A final appealing feature of this deterministic analysis is that it is very malleable — it
can be easily modified to cover extensions of the model, and this has been the subject of
considerable further research [10].
18.8 Exercises
1. Consider an on-line news site, such as cnn.com or nytimes.com, which consists of a
front page with links to many di↵erent articles. The people who operate such sites
generally track the popularity of the various articles that get posted, asking questions
like the ones that we’ve seen in this chapter: “As a function of k, what fraction of all
articles have been viewed by k people?” Let’s call this the popularity distribution of
the articles.
560 CHAPTER 18. POWER LAWS AND RICH-GET-RICHER PHENOMENA
Now suppose that the operators of such a news site are considering changing the front
page, so that next to each link is a counter showing how many people have clicked on
the link. (E.g., next to each link it will say something like, “30,480 people have viewed
this story,” with the number getting updated over time.)
First, what e↵ect do you think this change will have on the behavior of people using
the site? Second, do you expect that adding this feature will cause the popularity dis-
tribution of the articles to follow a power-law distribution more closely or less closely,
compared to the version of the site before these counters were added? Give an expla-
nation for your answer.
2. When we covered power laws in Chapter 18, we discussed a number of cases in which
power laws arise, generally reflecting some notion of “popularity” or a close analogue.
Consider, for example, the fraction of news articles each day that are read by k people:
if f (k) represents this fraction as a function of k, then f (k) approximately follows a
power-law distribution of the form f (k) ⇡ k�c for some exponent c.
Let’s think about this example in more detail, and in particular consider the following
questions. What mechanisms for providing news to the public will tend to accentuate
this power-law e↵ect, causing the most widely-read articles to be even more widely-
read? What mechanisms will tend to diminish the power-law e↵ect, more evenly bal-
ancing readership across more and less widely-read articles? Give an explanation for
your answer.
This is an open-ended question, in the sense that the range of possible correct answers
is quite broad; also, it is fine for your answer to be informally stated, provided the
explanation is clear.
3. Suppose that some researchers studying educational institutions decide to collect data
to address the following two questions.
(a) As a function of k, what fraction of Cornell classes have k students enrolled?
(b) As a function of k, what fraction of 3rd-grade elementary school classrooms in
New York State have k pupils?
Which one of these would you expect to more closely follow a power-law distribution
as a function of k? Give a brief explanation for your answer, using some of the ideas
about power-law distributions developed in Chapter 18.
Part VI
Network Dynamics: Structural Models
561
Chapter 19
Cascading Behavior in Networks
19.1 Di↵usion in Networks
A basic issue in the preceding several chapters has been the way in which an individual’s
choices depend on what other people do — this has informed our use of information cascades,
network e↵ects, and rich-get-richer dynamics to model the processes by which new ideas
and innovations are adopted by a population. When we perform this type of analysis,
the underlying social network can be considered at two conceptually very di↵erent levels
of resolution: one in which we view the network as a relatively amorphous population of
individuals, and look at e↵ects in aggregate; and another in which we move closer to the
fine structure of the network as a graph, and look at how individuals are influenced by their
particular network neighbors. Our focus in these past few chapters has been mainly on
the first of these levels of resolution, capturing choices in which each individual is at least
implicitly aware of the previous choices made by everyone else, and takes these into account.
In the next few chapters, we bring the analysis closer to the detailed network level.
What do we gain by considering this second level of resolution, oriented around network
structure? To begin with, we can address a number of phenomena that can’t be modeled
well at the level of homogeneous populations. Many of our interactions with the rest of the
world happen at a local, rather than a global, level — we often don’t care as much about
the full population’s decisions as about the decisions made by friends and colleagues. For
example, in a work setting we may choose technology to be compatible with the people we
directly collaborate with, rather than the universally most popular technology. Similarly,
we may adopt political views that are aligned with those of our friends, even if they are
nationally in the minority.
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
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564 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
In this way, considering individual choices with explicit network structure merges the
models of the past several chapters with a distinct line of thinking begun in Chapter 4,
when we examined how people link to others who are like them, and in turn can become
more similar to their neighbors over time. The framework in Chapter 4 dealt explicitly with
network connections, but did not explore the individual decision-making that leads people
to become similar to their neighbors: instead, a tendency toward favoring similarity was
invoked there as a basic assumption, rather than derived from more fundamental principles.
In contrast, the last several chapters have developed principles that show how, at an aggre-
gate population level, becoming similar to one’s neighbors can arise from the behavior of
individuals who are seeking to maximize their utility in given situations. We saw in fact that
there are two distinct kinds of reasons why imitating the behavior of others can be beneficial:
informational e↵ects, based on the fact that the choices made by others can provide indi-
rect information about what they know; and direct-benefit e↵ects, in which there are direct
payo↵s from copying the decisions of others — for example, payo↵s that arise from using
compatible technologies instead of incompatible ones.
We now connect these two approaches by exploring some of the decision-making principles
that can be used to model individual decision-making in a social network, leading people to
align their behaviors with those of their network neighbors.
The Di↵usion of Innovations. We will consider specifically how new behaviors, prac-
tices, opinions, conventions, and technologies spread from person to person through a social
network, as people influence their friends to adopt new ideas. Our understanding of how
this process works is built on a long history of empirical work in sociology known as the
di↵usion of innovations [115, 351, 382]. A number of now-classic studies done in the middle
of the 20th century established a basic research strategy for studying the spread of a new
technology or idea through a group of people, and analyzing the factors that facilitated or
impeded its progress.
Some of these early studies focused on cases in which the person-to-person influence was
due primarily to informational e↵ects: as people observed the decisions of their network
neighbors, it provided indirect information that led them to try the innovation as well. Two
of the most influential early pieces of research to capture such informational e↵ects were
Ryan and Gross’s study of the adoption of hybrid seed corn among farmers in Iowa [358]
and Coleman, Katz, and Menzel’s study of the adoption of tetracycline by physicians in the
United States [115]. In Ryan and Gross’s study, they interviewed farmers to determine how
and when they decided to begin using hybrid seed corn; they found that while most of the
farmers in their study first learned about hybrid seed corn from salesmen, most were first
convinced to try using it based on the experience of neighbors in their community. Coleman,
Katz, and Menzel went further when they studied the adoption of a new drug by doctors,
19.2. MODELING DIFFUSION THROUGH A NETWORK 565
mapping out the social connections among the doctors making decisions about adoption.
While these two studies clearly concerned very di↵erent communities and very di↵erent
innovations, they — like other important studies of that period — shared a number of basic
ingredients. In both cases, the novelty and initial lack of understanding of the innovation
made it risky to adopt, but it was ultimately highly beneficial; in both cases, the early
adopters had certain general characteristics, including higher socio-economic status and a
tendency to travel more widely; and in both cases, decisions about adoption were made in
the context of a social structure where people could observe what their neighbors, friends,
and colleagues were doing.
Other important studies in the di↵usion of innovations focused on settings in which
decisions about adoption were driven primarily by direct-benefit e↵ects rather than infor-
mational ones. A long line of di↵usion research on communication technologies has explored
such direct-benefit e↵ects; the spread of technologies such as the telephone, the fax machine,
and e-mail has depended on the incentives people have to communicate with friends who
have already adopted the technology [162, 285].
As studies of this type began proliferating, researchers started to identify some of the
common principles that applied across many di↵erent domains. In his influential book on
the di↵usion of innovations, Everett Rogers gathered together and articulated a number of
these principles [351], including a set of recurring reasons why an innovation can fail to
spread through a population, even when it is has significant relative advantage compared to
existing practices. In particular, the success of an innovation also depends on its complexity
for people to understand and implement; its observability, so that people can become aware
that others are using it; its trialability, so that people can mitigate its risks by adopting it
gradually and incrementally; and perhaps most crucially, its overall compatibility with the
social system that it is entering. Related to this, the principle of homophily that we have
encountered in earlier chapters can sometimes act as a barrier to di↵usion: since people
tend to interact with others who are like themselves, while new innovations tend to arrive
from “outside” the system, it can be di�cult for these innovations to make their way into a
tightly-knit social community.
With these considerations in mind, we now begin the process of formulating a model for
the spread of an innovation through a social network.
19.2 Modeling Di↵usion through a Network
We build our model for the di↵usion of a new behavior in terms of a more basic, underlying
model of individual decision-making: as individuals make decisions based on the choices
of their neighbors, a particular pattern of behavior can begin to spread across the links
of the network. To formulate such an individual-level model, it is possible to start either
566 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
from informational e↵ects [2, 38, 186] or direct-benefit e↵ects [62, 147, 308, 420]. In this
chapter, we will focus on the latter, beginning with a natural model of direct-benefit e↵ects
in networks due to Stephen Morris [308].
Network models based on direct-benefit e↵ects involve the following underlying consid-
eration: you have certain social network neighbors — friends, acquaintances, or colleagues
— and the benefits to you of adopting a new behavior increase as more and more of these
neighbors adopt it. In such a case, simple self-interest will dictate that you should adopt the
new behavior once a su�cient proportion of your neighbors have done so. For example, you
may find it easier to collaborate with co-workers if you are using compatible technologies;
similarly, you may find it easier to engage in social interaction — all else being equal — with
people whose beliefs and opinions are similar to yours.
A Networked Coordination Game. These ideas can be captured very naturally using
a coordination game, a concept we first encountered in Section 6.5. In an underlying social
network, we will study a situation in which each node has a choice between two possible
behaviors, labeled A and B. If nodes v and w are linked by an edge, then there is an
incentive for them to have their behaviors match. We represent this using a game in which
v and w are the players and A and B are the possible strategies. The payo↵s are defined as
follows:
• if v and w both adopt behavior A, they each get a payo↵ of a > 0;
• if they both adopt B, they each get a payo↵ of b > 0; and
• if they adopt opposite behaviors, they each get a payo↵ of 0.
We can write this in terms of a payo↵ matrix, as in Figure 19.1. Of course, it is easy to
imagine many more general models for coordination, but for now we are trying to keep things
as simple as possible.
v
w A B
A a, a 0, 0 B 0, 0 b, b
Figure 19.1: A-B Coordination Game
This describes what happens on a single edge of the network; but the point is that each
node v is playing a copy of this game with each of its neighbors, and its payo↵ is the sum of
its payo↵s in the games played on each edge. Hence v’s choice of strategy will be based on
the choices made by all of its neighbors, taken together.
19.2. MODELING DIFFUSION THROUGH A NETWORK 567
v
A
A
A
B
B
B
B pd neighbors
use A
(1-p)d neighbors
use B
Figure 19.2: v must choose between behavior A and behavior B, based on what its neighbors are doing.
The basic question faced by v will be the following: suppose that some of its neighbors
adopt A, and some adopt B; what should v do in order to maximize its payo↵? This clearly
depends on the relative number of neighbors doing each, and on the relation between the
payo↵ values a and b. With a little bit of algebra, we can make up a decision rule for v quite
easily, as follows. Suppose that a p fraction of v’s neighbors have behavior A, and a (1 � p) fraction have behavior B; that is, if v has d neighbors, then pd adopt A and (1 � p)d adopt B, as shown in Figure 19.2. So if v chooses A, it gets a payo↵ of pda, and if it chooses B, it
gets a payo↵ of (1 � p)db. Thus, A is the better choice if
pda � (1 � p)db,
or, rearranging terms, if
p � b
a + b .
We’ll use q to denote this expression on the right-hand side. This inequality describes a very
simple threshold rule: it says that if at least a q = b/(a + b) fraction of your neighbors follow
behavior A, then you should too. And it makes sense intuitively: when q is small, then
A is the much more enticing behavior, and it only takes a small fraction of your neighbors
engaging in A for you to do so as well. On the other hand, if q is large, then the opposite
holds: B is the attractive behavior, and you need a lot of your friends to engage in A before
you switch to A. There is a tie-breaking question when exactly a q fraction of a node’s
568 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
v
r
w
t
s
u
(a) The underlying network
v
r
w
t
s
u
(b) Two nodes are the initial adopters
v
r
w
t
s
u
(c) After one step, two more nodes have adopted
v
r
w
t
s
u
(d) After a second step, everyone has adopted
Figure 19.3: Starting with v and w as the initial adopters, and payo↵s a = 3 and b = 2, the new behavior A spreads to all nodes in two steps. Nodes adopting A in a given step are drawn with dark borders; nodes adopting B are drawn with light borders.
neighbors follow A; in this case, we will adopt the convention that the node chooses A rather
than B.
Notice that this is in fact a very simple — and in particular, myopic — model of individual
decision-making. Each node is optimally updating its decision based on the immediate
consideration of what its neighbors are currently doing, but it is an interesting research
question to think about richer models, in which nodes try to incorporate more long-range
considerations into their decisions about switching from B to A.
Cascading Behavior. In any network, there are two obvious equilibria to this network-
wide coordination game: one in which everyone adopts A, and another in which everyone
adopts B. Guided by di↵usion questions, we want to understand how easy it is, in a given
19.2. MODELING DIFFUSION THROUGH A NETWORK 569
situation, to “tip” the network from one of these equilibria to the other. We also want to
understand what other “intermediate” equilibria look like — states of coexistence where A
is adopted in some parts of the network and B is adopted in others.
Specifically, we consider the following type of situation. Suppose that everyone in the
network is initially using B as a default behavior. Then, a small set of “initial adopters”
all decide to use A. We will assume that the initial adopters have switched to A for some
reason outside the definition of the coordination game — they have somehow switched due
to a belief in A’s superiority, rather than by following payo↵s — but we’ll assume that all
other nodes continue to evaluate their payo↵s using the coordination game. Given the fact
that the initial adopters are now using A, some of their neighbors may decide to switch to
A as well, and then some of their neighbors might, and so forth, in a potentially cascading
fashion. When does this result in every node in the entire network eventually switching over
to A? And when this isn’t the result, what causes the spread of A to stop? Clearly the
answer will depend on the network structure, the choice of initial adopters, and the value of
the threshold q that nodes use for deciding whether to switch to A.
The above discussion describes the full model. An initial set of nodes adopts A while
everyone else adopts B. Time then runs forward in unit steps; in each step, each node uses
the threshold rule to decide whether to switch from B to A.1 The process stops either when
every node has switched to A, or when we reach a step where no node wants to switch, at
which point things have stabilized on coexistence between A and B.
Let’s consider an example of this process using the social network in Figure 19.3(a).
• Suppose that the coordination game is set up so that a = 3 and b = 2; that is, the payo↵ to nodes interacting using behavior A is 3/2 times what it is with behavior B.
Using the threshold formula, we see that nodes will switch from B to A if at least a
q = 2/(3 + 2) = 2/5 fraction of their neighbors are using A.
• Now, suppose that nodes v and w form the set of initial adopters of behavior A, while everyone else uses B. (See Figure 19.3(b), where dark circles denote nodes adopting
A and lighter circles denote nodes adopting B.) Then after one step, in which each
of the other nodes evaluates its behavior using the threshold rule, nodes r and t will
switch to A: for each of them, 2/3 > 2/5 of their neighbors are now using A. Nodes s
and u do not switch, on the other hand, because for each of them, only 1/3 < 2/5 of
their neighbors are using A.
1While we won’t go through the details here, it is not hard to show that no node that switches to A at some point during this process will ever switch back to B at a later point — so what we’re studying is indeed a strictly progressive sequence of switches from A to B. Informally, this fact is based on the observation that for any node that switches to A at some point in time, the number of neighbors of this node that follow A only continues to increase as time moves forward beyond this point — so if the threshold rule said to switch to A at some point in time, it will only say this more strongly at future times. This is the informal version of the argument, but it is not hard to turn this into a proof.
570 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
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Figure 19.4: A larger example.
• In the next step, however, nodes s and u each have 2/3 > 2/5 of their neighbors using A, and so they switch. The process now comes to an end, with everyone in the network
using A.
Notice how the process really is a chain reaction: nodes v and w aren’t able to get s and u
to switch by themselves, but once they’ve converted r and t, this provides enough leverage.
It’s also instructive to consider an example in which the adoption of A continues for a
while but then stops. Consider the social network in Figure 19.4, and again let’s suppose
that in the A-B coordination game, we have a = 3 and b = 2, leading to a threshold of
q = 2/5. If we start from nodes 7 and 8 as initial adopters (Figure 19.5(a)), then in the next
three steps we will first see (respectively) nodes 5 and 10 switch to A, then nodes 4 and 9,
and then node 6. At this point, no further nodes will be willing to switch, leading to the
outcome in Figure 19.5(b).
We’ll call this chain reaction of switches to A a cascade of adoptions of A, and we’d like
to distinguish between two fundamental possibilities: (i) that the cascade runs for a while
but stops while there are still nodes using B, or (ii) that there is a complete cascade, in
which every node in the network switches to A. We introduce the following terminology for
referring to the second possibility.
Consider a set of initial adopters who start with a new behavior A, while every
other node starts with behavior B. Nodes then repeatedly evaluate the decision to
switch from B to A using a threshold of q. If the resulting cascade of adoptions
19.2. MODELING DIFFUSION THROUGH A NETWORK 571
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(a) Two nodes are the initial adopters
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(b) The process ends after three steps
Figure 19.5: Starting with nodes 7 and 8 as the initial adopters, the new behavior A spreads to some but not all of the remaining nodes.
572 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
of A eventually causes every node to switch from B to A, then we say that the
set of initial adopters causes a complete cascade at threshold q.
Cascading Behavior and “Viral Marketing.” There are a few general observations to
note about the larger example in Figure 19.5. First, it nicely illustrates a point from the
opening section, that tightly-knit communities in the network can work to hinder the spread
of an innovation. Summarizing the process informally, A was able to spread to a set of nodes
where there was su�ciently dense internal connectivity, but it was never able to leap across
the “shores” in the network that separate nodes 8-10 from nodes 11-14, or that separate node
6 from node 2. As a result, we get coexistence between A and B, with boundaries in the
network where the two meet. One can see reflections of this in many instances of di↵usion
— for example, in di↵erent dominant political views between adjacent communities. Or, in
a more technological setting, consider the ways in which di↵erent social-networking sites are
dominated by di↵erent age groups and lifestyles — people will have an incentive to be on the
sites their friends are using, even when large parts of the rest of the world are using something
else. Similarly, certain industries heavily use Apple Macintosh computers despite the general
prevalence of Windows: if most of the people you directly interact with use Apple software,
it’s in your interest to do so as well, despite the increased di�culty of interoperating with
the rest of the world.
This discussion also suggests some of the strategies that might be useful if A and B
in Figure 19.5 were competing technologies, and the firm producing A wanted to push its
adoption past the point at which it has become stuck in Figure 19.5(b). Perhaps the most
direct way, when possible, would be for the maker of A to raise the quality of its product
slightly. For example, if we change the payo↵ a in the underlying coordination game from
a = 3 to a = 4, then resulting threshold for adopting A drops from q = 2/5 down to q = 1/3.
With this threshold, we could check that all nodes would eventually switch to A starting
from the situation in Figure 19.5(b). In other words, at this lower threshold, A would be able
to break into the other parts of the network that are currently resisting it. This captures an
interesting sense in which making an existing innovation slightly more attractive can greatly
increase its reach. It also shows that our discussion about the coexistence between A and B
along a natural boundary in the network depended not just on the network structure, but
also on the relative payo↵s of coordinating on A versus B.
When it’s not possible to raise the quality of A — in other words, when the marketer
of A can’t change the threshold — a di↵erent strategy for increasing the spread of A would
be to convince a small number of key people in the part of the network using B to switch
to A, choosing these people carefully so as to get the cascade going again. For example, in
Figure 19.5(b), we can check that if the marketer of A were to focus its e↵orts on convincing
node 12 or 13 to switch to A, then the cascading adoption of A would start up again,
19.3. CASCADES AND CLUSTERS 573
f
g
e h
j
k
i l
b
c
a d
Figure 19.6: A collection of four-node clusters, each of density 2/3.
eventually causing all of nodes 11–17 to switch. On the other hand, if the marketer of A
spent e↵ort getting node 11 or 14 to switch to A, then it would have no further consequences
on the rest of the network; all other nodes using B would still be below their threshold of
q = 2/5 for switching to A. This indicates that the question of how to choose the key nodes
to switch to a new product can be subtle, and based intrinsically on their position in the
underlying network. Such issues are important in discussions of “viral marketing” [230], and
have been analyzed in models of the type we are considering here [71, 132, 240, 309, 348].
Finally, it is useful to reflect on some of the contrasts between population-level network
e↵ects in technology adoption, as we formulated them in Chapter 17, and network-level cas-
cading adoption as illustrated here. In a population-level model, when everyone is evaluating
their adoption decisions based on the fraction of the entire population that is using a partic-
ular technology, it can be very hard for a new technology to get started, even when it is an
improvement on the status quo. In a network, however, where you only care about what your
immediate neighbors are doing, it’s possible for a small set of initial adopters to essentially
start a long fuse running that eventually spreads the innovation globally. This idea that a
new idea is initially propagated at a local level along social network links is something one
sees in many settings where an innovation gains eventual widespread acceptance.
19.3 Cascades and Clusters
We continue exploring some of the consequences of our simple model of cascading behavior
from the previous section: now that we’ve seen how cascades form, we look more deeply at
what makes them stop. Our specific goal will be to formalize something that is intuitively
apparent in Figure 19.5 — that the spread of a new behavior can stall when it tries to
break in to a tightly-knit community within the network. This will in fact provide a way
of formalizing a qualitative principle discussed earlier — that homophily can often serve
as a barrier to di↵usion, by making it hard for innovations to arrive from outside densely
connected communities.
574 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
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Figure 19.7: Two clusters of density 2/3 in the network from Figure 19.4.
As a first step, let’s think about how to make the idea of a “densely connected community”
precise, so that we can talk about it in the context of our model. A key property of such
communities is that when you belong to one, many of your friends also tend to belong. We
can take this as the basis of a concrete definition, as follows.
We say that a cluster of density p is a set of nodes such that each node in the set
has at least a p fraction of its network neighbors in the set.
For example, the set of nodes a, b, c, d forms a cluster of density 2/3 in the network in
Figure 19.6. The sets e, f, g, h and i, j, k, l each form clusters of density 2/3 as well.
As with any formal definition, it’s important to notice the ways in which it captures
our motivation as well as some of the ways in which it might not. Each node in a cluster
does have a prescribed fraction of its friends residing in the cluster as well, implying some
level of internal “cohesion.” On the other hand, our definition does not imply that any two
particular nodes in the same cluster necessarily have much in common. For example, in any
network, the set of all nodes is always a cluster of density 1 — after all, by definition, all
your network neighbors reside in the network. Also, if you have two clusters of density p,
then the union of these two clusters (i.e. the set of nodes that lie in at least one of them) is
also a cluster of density p. These observations are consistent with the notion that clusters
19.3. CASCADES AND CLUSTERS 575
in networks can exist simultaneously at many di↵erent scales.
The Relationship between Clusters and Cascades. The example in Figure 19.7 hints
at how the cluster structure of a network might tell us something about the success or failure
of a cascade. In this example, we see two communities, each of density 2/3, in the network
from Figure 19.4. These correspond precisely to the parts of the network that the cascading
behavior A was unable to break into, starting from nodes 7 and 8 as initial adopters. Could
this be a general principle?
In fact it is, at least within the context of the model we’ve developed. We now formulate
a result saying, essentially, that a cascade comes to a stop when it runs into a dense cluster;
and furthermore, that this is the only thing that causes cascades to stop [308]. In other
words, clusters are the natural obstacles to cascades. Here is the precise statement, phrased
in terms of the set of initial adopters and the remaining network — the portion of the network
consisting of all nodes other than these initial adopters.
Claim: Consider a set of initial adopters of behavior A, with a threshold of q for
nodes in the remaining network to adopt behavior A.
(i) If the remaining network contains a cluster of density greater than 1 � q, then the set of initial adopters will not cause a complete cascade.
(ii) Moreover, whenever a set of initial adopters does not cause a complete cas-
cade with threshold q, the remaining network must contain a cluster of den-
sity greater than 1 � q.
It is appealing how this result gives a precise characterization for the success or failure of a
cascade, in our simple model, using a natural feature of the network structure. Further, it
does so by concretely formalizing a sense in which tightly-knit communities block the spread
of cascades.
We now prove this result by separately establishing parts (i) and (ii). In going through
the proofs of the two parts, it’s useful to think about them both in general, and also in light
of the example in Figure 19.7, where clusters of density greater than 1 � 2 5
= 3 5
block the
spread of A at threshold 2 5
.
We begin with part (i).
Part (i): Clusters are Obstacles to Cascades. Consider an arbitrary network in which
behavior A is spreading with threshold q, starting from a set of initial adopters. Suppose
that the remaining network contains a cluster of density greater than 1 � q. We now argue that no node inside the cluster will ever adopt A.
Indeed, assume the opposite — that some node inside the cluster does eventually adopt
A — and consider the earliest time step t at which some node inside the cluster does so. Let
576 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
v
cluster
Figure 19.8: The spread of a new behavior, when nodes have threshold q, stops when it reaches a cluster of density greater than (1 � q).
v be the name of a node in the cluster that adopts A at time t. The situation is depicted
schematically in Figure 19.8 — essentially, we want to argue that, at the time v adopted,
it could not possibly have had enough neighbors using A to trigger its threshold rule. This
contradiction will show that v in fact could not have adopted.
Here is how we do this. At the time that v adopted A, its decision was based on the set
of nodes who had adopted A by the end of the previous time step, t � 1. Since no node in the cluster adopted before v did (that’s how we chose v), the only neighbors of v that were
using A at the time it decided to switch were outside the cluster. But since the cluster has
density greater than 1�q, more than a 1�q fraction of v’s neighbors are inside the cluster, and hence less than a q fraction of v’s neighbors are outside the cluster. Since these are the
only neighbors who could have been using A, and since the threshold rule requires at least a
q fraction of neighbors using v, this is a contradiction. Hence our original assumption, that
some node in the cluster adopted A at some point in time, must be false.
Having established that no node in the cluster ever adopts A, we are done, since this
shows that the set of initial adopters does not cause a complete cascade.
Part (ii): Clusters are the Only Obstacles to Cascades. We now establish part
(ii) of our claim, which says in e↵ect that not only are clusters a natural kind of obstacle
to cascades — they are in fact the only kind of obstacle. From a methodological point
of view (although all the details are di↵erent), this is reminiscent of a question we asked
with matching markets: having found that constricted sets are natural obstacles to perfect
matchings, we went on to find that they are in fact the only obstacle.
19.3. CASCADES AND CLUSTERS 577
w
initial adopters
nodes that eventually switch to A
nodes that don't
eventually switch to A
Figure 19.9: If the spread of A stops before filling out the whole network, the set of nodes that remain with B form a cluster of density greater than 1 � q.
To prove part (ii) we show that whenever a set of initial adopters fails to cause a complete
cascade with threshold q, there is a cluster in the remaining network of density greater than
(1�q). In fact, this is not di�cult: consider running the process by which A spreads, starting from the initial adopters, until it stops. It stops because there are still nodes using B, but
none of the nodes in this set want to switch, as illustrated in Figure 19.9.
Let S denote the set of nodes using B at the end of the process. We want to claim that
S is a cluster of density greater than 1 � q, which will finish the proof of part (ii). To see why this is true, consider any node w in this set S. Since w doesn’t want to switch to A,
it must be that the fraction of its neighbors using A is less than q — and hence that the
fraction of its neighbors using B is greater than 1 � q. But the only nodes using B in the whole network belong to the set S, so the fraction of w’s neighbors belonging to S is greater
than 1�q. Since this holds for all nodes in S, it follows that S is a cluster of density greater than 1 � q.
This wraps up our analysis of cascades and clusters; the punch-line is that in this model, a
set of initial adopters can cause a complete cascade at threshold q if and only if the remaining
network contains no cluster of density greater than (1 � q). So in this sense, cascades and clusters truly are natural opposites: clusters block the spread of cascades, and whenever a
cascade comes to a stop, there’s a cluster that can be used to explain why.
578 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
Figure 19.10: The years of first awareness and first adoption for hybrid seed corn in the Ryan-Gross study. (Image from [358].)
19.4 Di↵usion, Thresholds, and the Role of Weak Ties
One of the fundamental things we learn from studying di↵usion is that there is a crucial
di↵erence between learning about a new idea and actually deciding to adopt it. This contrast
was already important in the early days of di↵usion research. For example, Figure 19.10
comes from the original Ryan-Gross study of hybrid seed corn [358]; it shows a clear wave
of awareness of this innovation that significantly precedes the wave of adoptions.
Our models also illustrate this contrast. If we imagine that people first hear about an
innovation when any of their neighbors first adopts, then we see for example in Figure 19.5
that nodes 4 and 9 are aware of A as a new behavior right away, but it takes further time
for them to actually adopt it. In an even stronger direction, nodes 2 and 11-14 eventually
become aware of A but never adopt it.
Centola and Macy [101] and Siegel [369] make the interesting observation that threshold
models for di↵usion thus highlight an interesting subtlety in the strength-of-weak-ties theory
that we discussed in Chapter 3. Recall that the strength of weak ties is rooted in the idea
that weak social connections, to people we see infrequently, often form local bridges in a
social network. They therefore provide access to sources of information — things like new
job opportunities — that reside in parts of the network we otherwise wouldn’t have access
to. To take a canonical picture from Chapter 3, shown here in Figure 19.11, the u-w and v-w
edges span tightly-knit communities that wouldn’t otherwise be able to communicate, and
19.4. DIFFUSION, THRESHOLDS, AND THE ROLE OF WEAK TIES 579
w
x
v
u
Figure 19.11: The u-w and v-w edges are more likely to act as conduits for information than for high-threshold innovations.
thus we expect v, for example, to receive information from his edge to w that he wouldn’t
get from his other edges.
But things look very di↵erent if we consider the spread of a new behavior that requires
not just awareness, but an actual threshold for adoption. Suppose, for example, w and x in
Figure 19.11 are the initial adopters of a new behavior that is spreading with a threshold
of 1/2. Then we can check that everyone else in their tightly-knit six-node community will
adopt this behavior, but u and v will not. (Nor, therefore, will anyone else lying beyond
them in the network.)
This illustrates a natural double-edged aspect to bridges and local bridges in a social
network: they are powerful ways to convey awareness of new things, but they are weak at
transmitting behaviors that are in some way risky or costly to adopt — behaviors where you
need to see a higher threshold of neighbors doing it before you do it as well. In this sense,
nodes u and v in Figure 19.11 have strong informational advantages over other members of
their respective tightly-knit communities — they can learn from node w about a new behavior
currently spreading in w’s community — but for behaviors with higher thresholds they will
still want to align themselves with others in their own community. If we think about it, this
is actually remarkably consistent with the picture from Chapter 3, in which local bridges and
positions near structural holes can provide access to information that you’re not otherwise
580 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
learning about from your own cluster in the network: for behaviors that spread with high
thresholds, a local bridge may well connect you to someone whose network neighborhood
has caused them to settle on a di↵erent behavior than you have.
The trade-o↵s inherent in this picture have been used to motivate some of the reasons
why many social movements tend to build support locally and relatively slowly. Although
a world-spanning system of weak ties in the global friendship network is able to spread
awareness of a joke or an on-line video with remarkable speed, political mobilization moves
more sluggishly, needing to gain momentum within neighborhoods and small communities.
Thresholds provide a possible reason: social movements tend to be inherently risky under-
takings, and hence individuals tend to have higher thresholds for participating; under such
conditions, local bridges that connect very di↵erent parts of the network are less useful. Such
considerations provide a perspective on other well-known observations about social move-
ments in the di↵usion literature, such as Hedstrom’s findings that such movements often
spread geographically [215], and McAdam’s conclusion that strong ties, rather than weak
ties, played the more significant role in recruitment to student activism during Freedom
Summer in the 1960s [290, 291].
19.5 Extensions of the Basic Cascade Model
Our discussion thus far has shown how a very simple model of cascades in networks can
capture a number of qualitative observations about how new behaviors and innovations
di↵use. We now consider how the model can be extended and enriched, keeping its basic
points the same while hinting at additional subtleties.
Heterogeneous Thresholds. Thus far we have been keeping the underlying model of
individual behavior as simple as possible — everyone has the same payo↵s, and the same
intensity of interaction with their network neighbors. But we can easily make these assump-
tions more general while still preserving the structure of the model and the close connection
between cascades and clusters.
As the main generalization we consider, suppose that each person in the social network
values behaviors A and B di↵erently. Thus, for each node v, we define a payo↵ a v
— labeled
so that it is specific to v — that it receives when it coordinates with someone on behavior
A, and we define a payo↵ b v
that it receives when it coordinates with someone on behavior
B. When two nodes v and w interact across an edge in the network, they are thus playing
the coordination game in Figure 19.12.
Almost all of the previous analysis carries over with only small modifications; we now
briefly survey how these changes go. When we first defined the basic coordination game,
with all nodes agreeing on how to value A and B, we next asked how a given node v should
19.5. EXTENSIONS OF THE BASIC CASCADE MODEL 581
v
w A B
A a v
, a w
0, 0 B 0, 0 b
v
, b w
Figure 19.12: A-B Coordination Game
choose its behavior based on what its neighbors are doing. A similar question applies here,
leading to a similar calculation. If v has d neighbors, of whom a p fraction have behavior A
and a (1 � p) fraction have behavior B, then the payo↵ from choosing A is pda v
while the
payo↵ from choosing B is (1 � p)db v
. Thus A is the better choice if
p � b v
a v
+ b v
.
Using q v
to denote the right-hand side of this, we again have a very simple decision rule —
now, each node v has its own personal threshold q v
, and it chooses A if at least a q v
fraction
of its neighbors have done so. Moreover the variation in this set of heterogeneous node
thresholds has an intuitive meaning in terms of the variation in payo↵s: if a node values A
more highly relative to B, its threshold q v
is correspondingly lower.
The process now runs as before, starting from a set of initial adopters, with each node
evaluating its decision according to its own threshold rule in each time step, and switching
to A if its threshold is reached. Figure 19.13 shows an example of this process (where each
node’s threshold is drawn to the upper-right of the node itself).
A number of interesting general observations are suggested by what happens in Fig-
ure 19.13. First, the diversity in node thresholds clearly plays an important role that in-
teracts in complex ways with the structure of the network. For example, despite node 1’s
“central” position, it would not have succeeded in converting anyone at all to A were it not
for the extremely low threshold on node 3. This relates closely to a point made in work by
Watts and Dodds [409], who argue that for understanding the spread of behaviors in social
networks, we need to take into account not just the power of influential nodes, but also the
extent to which these influential nodes have access to easily influenceable people.
It is also instructive to look at how the spread of A comes to a stop in Figure 19.13, and
to ask whether the notion of clusters as obstacles to cascades can be extended to hold even
in the case when thresholds are heterogeneous. In fact, this is possible, by formulating the
notion of a cluster in this setting as follows. Given a set of node thresholds, let’s say that a
blocking cluster in the network is a set of nodes for which each node v has more than a 1�q v
fraction of its friends also in the set. (Notice how the notion of cluster density — like the
notion of thresholds — becomes heterogeneous as well: each node has a di↵erent requirement
for the fraction of friends it needs to have in the cluster.) By a fairly direct adaptation of the
analysis from Section 19.3, one can show that a set of initial adopters will cause a complete
582 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
3
2
1
5
4
6
7
8
9
10
.1
.1
.1
.3
.4
.4
.4
.5
.7
(a) One node is the initial adopter
3
2
1
5
4
6
7
8
9
10
.1
.1
.1
.3
.4
.4
.4
.5
.7
(b) The process ends after four steps
Figure 19.13: Starting with node 1 as the unique initial adopter, the new behavior A spreads to some but not all of the remaining nodes.
19.6. KNOWLEDGE, THRESHOLDS, AND COLLECTIVE ACTION 583
cascade — with a given set of node thresholds — if and only if the remaining network does
not contain a blocking cluster.
19.6 Knowledge, Thresholds, and Collective Action
We now switch our discussion to a related topic that integrates network e↵ects at both the
population level and the local network level. We consider situations where coordination
across a large segment of the population is important, and the underlying social network is
serving to transmit information about people’s willingness to participate.
Collective Action and Pluralistic Ignorance. A useful motivating example is the prob-
lem of organizing a protest, uprising, or revolt under a repressive regime [109, 110, 192].
Imagine that you are living in such a society, and are aware of a public demonstration against
the government that is planned for tomorrow. If an enormous number of people show up,
then the government will be seriously weakened, and everyone in society — including the
demonstrators — will benefit. But if only a few hundred show up, then the demonstrators
will simply all be arrested (or worse), and it would have been better had everyone stayed
home. In such circumstances, what should you do?
This is an example of a collective action problem, where an activity produces benefits
only if enough people participate. In this way, it is reminiscent of our analysis in Chapter 17
of population-level network e↵ects: as with joining a large-scale demonstration, you only
want to buy a fax machine if enough other people do. The starker setting of the present
example highlights a few points, however. In the case of a fax machine, you can watch the
experience of early adopters; you can read reviews and advertisements; you can canvass a
wide array of friends and colleagues to see what they plan to do. Due to the much stronger
negative payo↵s associated with opposing a repressive government, many of these options
are closed to you — you can talk about the idea with a small number of close friends whom
you trust, but beyond this your decision about whether to show up for the demonstration is
made di�cult by a lack of knowledge of other people’s willingness to participate, or of their
criteria for deciding whether to participate.
These considerations illustrate some of the reasons why repressive governments work so
hard to limit communication among their citizens. It is possible, for example, that a large
fraction of the population is strong enough in its opposition to be willing to take extreme
measures, but that most of these people believe they’re in a small minority — and hence
view opposition as too risky. In this way, a government could survive long after there is
enough strong opposition in principle to get rid of it.
This phenomenon is known as pluralistic ignorance [330], in which people have wildly
erroneous estimates about the prevalence of certain opinions in the population at large. It
584 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
w
u v
2 3
4
(a) An uprising will not occur
w x
u v
3 3
3 3
(b) An uprising will not occur
w x
u v
3 3
3 3
(c) An uprising can occur
Figure 19.14: Each node in the network has a threshold for participation, but only knows the threshold of itself and its neighbors.
is a principle that applies widely, not just in settings where a central authority is actively
working to restrict information. For example, a survey conducted in the U.S. in 1970 (and
replicated several times in the surrounding years with similar results) showed that while
only a minority of white Americans at that point personally favored racial segregation,
significantly more than 50% believed that it was favored by a majority of white Americans
in their region of the country [331].
A Model for the E↵ect of Knowledge on Collective Action. Let’s consider how the
structure of the underlying social network can a↵ect the way people make decisions about
collective action, following a model and a set of illustrative examples proposed by Michael
Chwe [109, 110]. Suppose that each person in a social network knows about a potential
upcoming protest against the government, and she has a personal threshold which encodes
her willingness to participate. A threshold of k means, “I will show up for the protest if I
am sure that at least k people in total (including myself) will show up.”
The links in the social network encode strong ties, where the two endpoints of each link
trust each other. Thus, we assume that each person in the network knows the thresholds
of all her neighbors in the network, but — due to the risky nature of communication about
dissent in this society — does not know the thresholds of anyone else. Now, given a network
with a set of thresholds, how should we reason about what is likely to happen?
Let’s consider the examples in Figure 19.14, which show some of the subtleties that arise
here. Scaling down our notion of “uprising” to a size commensurate with these 3-4 person
examples, suppose that each node represents one of the senior vice-presidents at a company,
each of whom must decide whether to actively confront the unpopular CEO at the next day’s
board meeting. It would be disastrous to do so without reasonable support from the others,
so each is willing to confront the CEO provided that at least a certain number of them do
19.6. KNOWLEDGE, THRESHOLDS, AND COLLECTIVE ACTION 585
so in total. We’ll also assume that each node knows what the social network looks like.
First, Figure 19.14(a) indicates some of the reasoning that nodes must do about the
decisions being made by other nodes. Here, node w would only join the protest if at least
four people do; since there are only three people in total, this means he will never join. Node
v knows that w’s threshold is four, so v knows that w won’t participate. Since v requires
three people in order to be willing to join, v won’t participate either. Finally, u only requires
two people in order to participate, but she knows the thresholds of both other nodes, and
hence can determine that neither will participate. So she doesn’t either. Hence, the protest
doesn’t happen.
Figure 19.14(b) introduces even more subtle considerations, in which nodes must reason
about what other nodes know in order to reason about what they will do. In particular,
consider the situation from u’s perspective (since it’s symmetric for all nodes). She knows
that v and w each have a threshold of three, and so each of u, v, and w would feel safe taking
part in a protest that contained all three of them. But she also knows that v and w don’t
know each other’s thresholds, and so they can’t engage in the same reasoning that she can.
Is it safe for u to join the protest? The answer is no, for the following reason. Since u
doesn’t know x’s threshold, there’s the possibility that it’s something very high, like 5. In
this case, node v, seeing neighbors with thresholds of 3 and 5, would not join the protest.
Neither would w. So in this case, if u joined the protest, she’d be the only one — a disaster
for her. Hence, u can’t take this chance, and so she doesn’t join the protest.
Since the situation is symmetric for all four nodes in Figure 19.14(b), we can conclude that
no node will join the protest, and so no protest happens. There is something striking about
this: each node in the network knows the fact that there are three nodes with thresholds of
3 — enough for a protest to form — but each holds back because they cannot be sure that
any other nodes know this fact.
Things would turn out very di↵erently if the link from v to x were shifted to instead
connect v and w, resulting in the network of Figure 19.14(c). Now, each of u, v, and w not
only knows the fact that there are three nodes with thresholds of 3, but this fact is common
knowledge [29, 154, 276]: among the set of nodes consisting of u, v, and w, each node knows
this fact, each node knows that each node knows it, each node knows that each node knows
that each node knows it, and so on indefinitely. We touched on common knowledge briefly
in the context of game theory in Chapter 6; as we see here, it also plays an important role
in interactions designed to achieve coordination.
So the di↵erences between the examples in Figures 19.14(b) and 19.14(c) are subtle, and
come down to the di↵erent networks’ consequences for the knowledge that nodes have about
what others know. This contrast also highlights another way of thinking about the power of
strong ties and tightly-knit communities for encouraging participation in high-risk activities,
a topic that we discussed in Section 19.4. Weak ties have informational advantages since
586 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
your strong ties are to people who knows things that heavily overlap with what you know.
But for collective action, such overlaps in knowledge can be precisely what is needed.
This model for common knowledge and coordination is developed further in [110]; under-
standing the precise interaction of knowledge with collective action remains an interesting
direction for further research.
Common Knowledge and Social Institutions. Building on these models, Chwe and
others have argued that a broad range of social institutions in fact serve the role of helping
people achieve common knowledge [111]. A widely-publicized speech, or an article in a high-
circulation newspaper, has the e↵ect not just of transmitting a message, but of making the
listeners or readers realize that many others have gotten the message as well.
This is a useful context for thinking about freedom of the press and freedom of assembly,
and their relationship to open societies. But institutions relatively far from the political
sphere can also have strong roles as generators of common knowledge. For example, Chwe
argues that Super Bowl commercials are often used to advertise products where there are
strong network e↵ects — things like cell-phone plans and other goods where it’s in your
interest to be one of a large population of adopters [111]. For example, the Apple Macintosh
was introduced in a Ridley-Scott-directed commercial during the 1984 Super Bowl. (Years
later, it was declared the “Greatest Television Commercial of All Time” by both TV Guide
and Advertising Age magazine.) As Chwe writes of the event, “The Macintosh was completely
incompatible with existing personal computers: Macintosh users could easily exchange data
only with other Macintosh users, and if few people bought the Macintosh, there would be
little available software. Thus a potential buyer would be more likely to buy if others bought
them also; the group of potential Macintosh buyers faced a coordination problem. By airing
the commercial during the Super Bowl, Apple did not simply inform each viewer about the
Macintosh; Apple also told each viewer that many other viewers were informed about the
Macintosh” [111].
Recently, David Patel has used principles of common knowledge to argue that di↵erences
between the organization of Sunni and Shiite religious institutions can help explain much
about the power dynamics that followed the 2003 U.S. invasion of Iraq [339]. In particular,
strong organizational structures enabled Friday sermons at Shiite mosques to be centrally
coordinated, while the Sunni religious establishment lacked comparable structures: “Shiite
Ayatollahs, controlling hierarchical networks of clerical deputies, can reliably and consis-
tently disseminate similar messages in di↵erent mosques, generating common knowledge and
coordination across dispersed Shiite congregations on national-level issues like federalism
and voting strategies. Through mosque networks, Shiites reliably know what Shiites in far
distant areas know” [339]. Patel thus argues that these mechanisms for facilitating shared
knowledge enabled Shiites to achieve coordination on goals at a national scale, in a way that
19.7. ADVANCED MATERIAL: THE CASCADE CAPACITY 587
other groups in post-invasion Iraq lacked the institutional power to do.
Through all of this, we’re seeing that social networks don’t simply allow for interaction
and the flow of information, but that these processes in turn allow individuals to base
decisions on what others know, and on how they expect others to behave as a result. The
potential of this framework for studying social processes and social institutions is still being
actively explored.
19.7 Advanced Material: The Cascade Capacity
If we go back to the basic model of this chapter, in which nodes choose between behaviors
A and B based on thresholds derived from a networked coordination game, an interesting
perspective is to understand how di↵erent network structures are more or less hospitable to
cascades. A first version of this perspective is the analysis in Section 19.3, where we showed
that clusters in the network structure form the natural obstacles to cascades. Here we take
a di↵erent approach; given a network, we ask: what is the largest threshold at which any
“small” set of initial adopters can cause a complete cascade? This maximum threshold is
thus an inherent property of the network, indicating the outer limit on its ability to support
cascades; we will refer to it as the cascade capacity of the network.
In order to make this idea work at a technical level, we clearly need to be careful about
what we mean by a “small” set. For example, clearly if we take the set of initial adopters
to be the full set of nodes, or (in most cases) something that is almost the full set of nodes,
then we can get cascades even at thresholds approaching or equal to 1.
It turns out that the cleanest way to formalize the question is in fact to consider infinite
networks in which each node has a finite number of neighbors. We can then define the
cascade capacity as the largest threshold at which a finite set of nodes can cause a complete
cascade. In this way, “small” will mean finite, in the context of a network where the full
node set is infinite.
A. Cascades on Infinite Networks
With this goal in mind, we now describe the model in general. The social network will be
modeled as a connected graph on an infinite set of nodes; although the node set is infinite,
each individual node is only connected to a finite number of other nodes.
The model of node behavior is the same one that we defined earlier in the chapter — the
fact that the node set is infinite doesn’t pose any problems, since each node only has a finite
set of neighbors, and it is only making decisions based on the behavior of these neighbors.
To be concrete, initially, a finite set S of nodes has behavior A (this is the small set of early
adopters), and all other nodes adopt B. Time then runs forward in steps t = 1, 2, 3, . . ..
In each step t, each node other than those in S uses the decision rule with threshold q to
588 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
u wvx
Figure 19.15: An infinite path with a set of early adopters of behavior A (shaded).
decide whether to adopt behavior A or B. (As before, we assume that the nodes in S are
committed to A, and never re-evaluate this decision.) Finally, we say that the set S causes
a complete cascade if, starting from S as the early adopters of A, every node in the network
eventually switches permanently to A. (Given the fact that the node set is infinite, we must
be careful to be clear on what this means: for every node v, there is some time t after which
v is always using behavior A.)
The Cascade Capacity. The key definition is now the following. We say that the cascade
capacity of the network is the largest value of the threshold q for which some finite set of
early adopters can cause a complete cascade. To illustrate this definition, let’s consider two
simple examples. First, in Figure 19.15, we have a network consisting of a path that extends
infinitely in both directions. Suppose that the two shaded nodes are early adopters of A, and
that all other nodes start out adopting B. What will happen? It’s not hard to check that if
q 1 2
, then nodes u and v will switch to A, after which nodes w and x will switch, and the
switches will simply propagate all the way down the path: for each node, there will come
some time at which it chooses to switch permanently to A. So the cascade capacity of the
infinite path is at least 1 2
, since we have just seen a finite set of initial adopters that causes
a complete cascade at threshold 1 2
. In fact, 1 2
is the exact value of the cascade capacity of
the infinite path: with q > 1 2
, no finite set of initial adopters can get any node to their right
to switch to A, and so A clearly cannot spread to all nodes.
Figure 19.16 shows a second simple example, a network consisting of an infinite grid in
which each node is connected to its eight nearest neighbors. Suppose that the nine shaded
nodes are early adopters of A, and that all other nodes start out adopting B. You can check
that if the threshold q is at most 3 8
, then behavior A gradually pushes its way out to the
neighbors of the shaded nodes: first to the nodes labeled c, h, i, and n; then to the nodes b,
d, f , g, j, k, m, and o; and then to other nodes from there, until every node in the grid is
eventually converted to A. (With a smaller threshold — when q 2 8
for example — behavior
A spreads even faster.) We can check that in fact 3 8
is the cascade capacity of the infinite
grid: given any finite set of initial adopters, they are contained in some rectangle of the grid,
and if q > 3 8
, no node outside this rectangle will ever choose to adopt A.
Note that the cascade capacity is an intrinsic property of the network itself. A network
with a large cascade capacity is one in which cascades happen more “easily” — in other words,
19.7. ADVANCED MATERIAL: THE CASCADE CAPACITY 589
cba
f
ed
g
h
j
i
k
m nl po
Figure 19.16: An infinite grid with a set of early adopters of behavior A (shaded).
they happen even for behaviors A that don’t o↵er much payo↵ advantage over the default
behavior B. As we discussed in Section 19.2, the fact that a small set of initial adopters
can eventually cause the whole population to switch illustrates how a better technology (A,
when q < 1 2
) can displace an existing, inferior one (B). Viewed in this sense, the example
of the grid in Figure 19.16 can be viewed as a kind of failure of social optimality. The fact
that the cascade capacity on the grid is 3 8
means that when q is strictly between 3 8
and 1 2
,
A is the better technology, but the structure of the network makes B so heavily entrenched
that no finite set of initial adopters of A can cause A to win.
We now consider the following fundamental question: how large can a network’s cascade
capacity be? The infinite path shows that there are networks in which the cascade capacity
can be as large as 1 2
: this means that a new behavior A can displace an existing behavior
B even when the two confer essentially equivalent benefits (with A having only the “tie-
breaking” advantage that when a node has an equal number of neighbors using A and B, it
chooses A). Does there exist any network with a higher cascade capacity? This would be
a bit surprising, since such a network would have the property that an inferior technology
590 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
can displace a superior one, even when the inferior technology starts at only a small set of
initial adopters.
In fact, we will show that no network has a cascade capacity larger than 1 2
. In other
words, regardless of the structure of the underlying network, if a new behavior requires 51%
of someone’s friends to adopt it before they do, then it can’t spread very far through the
population. Despite the fact that this is perhaps an intuitively natural fact, proving it is a
bit subtle, requiring a way to bound the extent of a behavior that is spreading at a threshold
beyond 1 2
.
B. How Large Can the Cascade Capacity Be?
We now formulate and prove this basic fact about the cascade capacity.
Claim: There is no network in which the cascade capacity exceeds 1 2
.
Although we motivated this claim as a natural one just above, it is less clear why it is true.
After all, it’s certainly imaginable a priori that there could be some cleverly constructed
network, set up in just the right way, so that even though each node needs 51% of its
neighbors to adopt before it does, the cascade rolls on steadily, eventually causing everyone
to switch. What we really need to show is the following: if q > 1 2
, then regardless of what
the underlying network looks like, a new behavior starting at a finite set of nodes will not
spread to every other node.
Analyzing the Interface. We’re going to approach this question by tracking the “inter-
face” where adopters of A are linked to adopters of B. At a very high level, we’re going
to show that as the process runs, this interface becomes narrower and narrower, eventually
shrinking to the point where the process must stop, having failed to reach all nodes.
More precisely, suppose the behavior A spreads from a finite initial set S with threshold
q > 1 2
. As time moves forward in steps t = 1, 2, 3, . . ., potentially larger and larger sets
become adopters of A. At any given point in time, each edge in the network can be described
as an A-A edge (connecting two adopters of A), a B-B edge (connecting two adopters of B),
or an A-B edge (connecting an adopter of A to an adopter of B). We define the interface to
be the set of A-B edges. Figure 19.17 shows a useful way to picture the interface: if the set
of adopters of A consists of the nodes inside the dark oval, then the edges in the interface
are the ones that cross the oval.
What we’re going to show is that, in each step, the size of the interface — i.e., the
number of edges it contains — must strictly decrease. This will be enough to show what we
need, for the following reason. The size of the interface clearly starts at some number I 0
:
since there is some finite set of initial adopters S, and since each of these has a finite set of
neighbors, the set of A-B edges is finite, and has some size I 0
. The size of the interface is
19.7. ADVANCED MATERIAL: THE CASCADE CAPACITY 591
u
v
w
x
(a) Before v and w adopt A
u
v
w
x
(b) After v and w adopt A
Figure 19.17: Let the nodes inside the dark oval be the adopters of A. One step of the process is shown, in which v and w adopt A: after they adopt, the size of the interface has strictly decreased. In general, the size of the interface strictly decreases with each step of the process when q > 1
2
.
always a non-negative whole number, so if it strictly decreases in each step, the spread of
A can run for at most I 0
steps before terminating. Since each step only results in a finite
number of nodes converting to A, the process will terminate with only a finite set of nodes
having adopted A. (So in fact we’ll get something stronger than we needed: not only does
A not spread everywhere, it only reaches a finite set starting from S.)
The Size of the Interface Decreases in Each Step. So the crux of this is to consider
one step of the process, and show that the size of the interface strictly decreases. What
happens in one step of the process? Figure 19.17 illustrates a way to think about this.
Certain nodes that are currently adopters of B discover, for the first time, that at least a q
fraction of their neighbors are now adopters of A, and so they too switch to A.
This causes the interface to change in the following way. When a node w switches from
B to A, its edges to nodes that remain with B change from being B-B edges to being A-B
edges — so this causes them to join the interface. (An example is the edge linking w and x
in Figure 19.17.) On the other hand, the edges from w to nodes that were already with A
change from being A-B edges to being A-A edges; in other words, they leave the interface.
(See for example the edge linking u and w.) Each edge that joins or leaves the interface in
this step can be accounted for in this way by exactly one node that switches from B to A.
So to analyze the change in the size of the interface, we can separately consider the
contribution from those edges accounted for by each individual node that switches. Thus,
592 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
consider a node w that switches; suppose that before the switch, it had a edges to nodes
that were already adopters of A, and b edges to nodes that will remain adopters of B at the
end of the step. So node w accounts for b edges joining the interface and a edges leaving
it. But since q > 1 2
, and node w decided to switch to A in this step, it must be that w had
more edges to adopters of A than to adopters of B — so a > b, and hence w accounts for
more edges leaving the interface than edges joining the interface. But this is true for each
node that switches in this step, and so the overall size of the interface goes down.
This is what we needed to show. Chaining back through the earlier arguments, since the
interface starts at some fixed size I 0
, the process can only go for at most I 0
steps before
running out of steam and stopping, not having reached all nodes.
Some Final Thoughts. We’ve shown that when q > 1 2
, no finite set of nodes can cause
a complete cascade, in any network. In terms of an underlying story about users choosing
between technologies A and B, the situation in which q > 1 2
corresponds intuitively to the
case in which the new technology A is in fact worse — the payo↵ from an A-A interaction is
lower than that of a B-B interaction, and so you’ll only switch to A in cases where more than
half your friends already have. So at least in the simple model we’ve been studying here,
a worse technology will not displace a better technology that’s already in widespread use.
(However, recall the connection with our earlier discussion of network e↵ects: in networks
where the cascade capacity is strictly less than 1 2
, it is possible for a better technology to be
unable to displace a worse one that is already in widespread use.)
It is also interesting to reflect a bit on the way in which we argued that A can’t spread
to all nodes when q > 1 2
; there’s a methodological parallel here to our discussion of matching
markets (though again the details are completely di↵erent). There too we had a process
— the bipartite auction procedure that updated prices — and we wanted to show that it
must come to a halt. Lacking any obvious measure of progress on the process, we invented a
non-obvious one — a kind of “potential energy” that steadily drained out of the process as
it ran, eventually forcing it to terminate. In retrospect, we used a very similar strategy here,
with the size of the interface serving as the potential energy function that steadily decreases
until the process had to stop.
C. Compatibility and its Role in Cascades
We’ve gotten a lot of mileage in this chapter from taking a game that is fundamentally
very simple — a coordination game with two possible strategies — and analyzing how it
is played across the edges of a potentially complex network. There are many directions in
which the game can be extended and generalized, and most of these lead quickly to current
research and open questions. To illustrate how even small extensions to the underlying game
can introduce new sources of subtlety, we discuss here an extension that takes into account
19.7. ADVANCED MATERIAL: THE CASCADE CAPACITY 593
the notion that a single individual can sometimes choose a combination of two available
behaviors. [225].
To illustrate what we mean by this, let’s go back to the extended example we considered
in Figure 19.5, and the discussion at the end of Section 19.2 of how behaviors A and B
ended up coexisting in the network. Coexistence is a common outcome, and it is interesting
to ask what things look like along the boundaries between A and B. For example, A and
B could be di↵erent languages coexisting along a national border, or A and B could be
social-networking sites that appeal respectively to students in college and to students in
high school. Our current model says that anyone positioned along the interface between
A and B in the network — for example, nodes 8–14 in Figure 19.5 — will receive positive
payo↵s from neighbors who adopt the same behavior, but payo↵s of 0 from their interactions
with neighbors who adopt di↵erent behaviors.
Experience suggests that when people are actually faced with such situations, they often
choose an option that corresponds to neither A nor B — rather, they become bilingual,
adopting both A and B. In some cases, bilinguality is meant literally: for example, someone
who lives near speakers of both French and German is reasonably likely to speak (some
amount of) both. But technological versions of bilinguality abound as well: people with
friends on two incompatible IM systems, or two di↵erent social-networking sites, will likely
have accounts on both; people whose work requires dealing with two di↵erent computer
operating systems will likely have a way to run both. The common feature of all these
examples is that an individual chooses to use some form of both available behaviors, trading
o↵ the greater ease of interaction with people of multiple types against the cost of having to
acquire and maintain both forms of behavior (i.e. the costs of having to learn an additional
language, maintain two di↵erent versions of a technology, and so forth). What e↵ect does
this bilingual option have on the spread of a behavior through a network?
Modeling the Bilingual Option. In fact, it is not hard to set up a model that captures
the possibility that a node will choose to be bilingual. On each edge, connecting two nodes v
and w, we still imagine a game being played, but now there are three available strategies: A,
B, and AB. The strategies A and B are the same as before, while the strategy AB represents
a decision to adopt both behaviors. The payo↵s follow naturally from the intuition discussed
above: the nodes can interact with each other using any behavior that is available to both
of them. If they interact using A, they each get a payo↵ of a, while if they interact using
B, they each get a payo↵ of b. In other words, two bilingual nodes can interact using the
better of the two behaviors; a bilingual node and a monolingual node can only interact using
the monolingual node’s behavior; and two monolingual nodes can only interact at all if they
have the same behavior. Written as a payo↵ matrix, the game is shown in Figure 19.18,
where we use the notation (a, b)+ to denote the larger of a and b.
594 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
v
w A B AB
A a, a 0, 0 a, a B 0, 0 b, b b, b
AB a, a b, b (a, b)+, (a, b)+
Figure 19.18: A Coordination Game with a bilingual option. Here the notation (a, b)+
denotes the larger of a and b.
r s u w yvxz
Figure 19.19: An infinite path, with nodes r and s as initial adopters of A.
It’s easy to see that AB is a dominant strategy in this game: why not be bilingual when
it gives you the best of both worlds? However, to model the trade-o↵ discussed earlier, we
need to also incorporate the notion that bilinguality comes with a cost — the meaning of
the cost varies with the context, but the cost in general corresponds to the additional e↵ort
and resource expenditure needed to maintain two di↵erent behaviors. Thus, we assume that
each node v will play a copy of this three-strategy Bilingual Coordination Game with each
of its neighbors; as in our models earlier in the chapter, v must use the same strategy in
each copy of the game it plays. Its payo↵ will be equal to the sum of its payo↵s in its game
with each neighbor, minus a single cost of c if v chooses to play the strategy AB. It is this
cost that creates incentives not to play AB, balancing the incentives that exist in the payo↵
matrix to play it.
The remainder of the model works as before. We assume that every node in an infinite
network starts with the default behavior B, and then (for non-strategic reasons) a finite set S
of initial adopters begins using A. We now run time forward in steps t = 1, 2, 3, . . .; in each of
these steps, each node outside S chooses the strategy that will provide it the highest payo↵,
given what its neighbors were doing in the previous step. We are interested in how nodes
will choose strategies as time progresses, and particularly which nodes eventually decide to
switch permanently from B to A or AB.
An Example. To get some practice with the model, let’s try it on the infinite path shown
in Figure 19.19. Let’s suppose that nodes r and s are the initial adopters of A, and that the
payo↵s are defined by the quantities a = 2, b = 3, and c = 1.
Here is how nodes behave as time progresses. In the first time step, the only interesting
decisions are the ones faced by nodes u and v, since all other nodes are either initial adopters
(who are hard-wired to play A) or nodes that have all neighbors using B. The decisions faced
19.7. ADVANCED MATERIAL: THE CASCADE CAPACITY 595
r s u w yvxz
Start B B B A A B B B
Step 1 B B AB A A AB B B
Step 2 B AB AB A A AB AB B
Step 3 AB AB A A A A AB AB
Step 4 AB A A A A A A AB
Figure 19.20: With payo↵s a = 5 and b = 3 for interaction using A and B respectively, and a cost c = 1 for being bilingual, the strategy A spreads outward from the initial adopters r and s through a two-phase structure. First, the strategy AB spreads, and then behind it, nodes switch permanently from AB to A.
by u and v are symmetric; for each of them, we can check that the strategy AB provides
the highest payo↵. (It yields a payo↵ of 2 + 3�1 = 4 from being able to interact with both neighbors, but having to pay a cost of 1 to be bilingual.) In the second time step, nodes w
and x have a fresh decision to make, since they now have neighbors using AB, but we can
check that B still yields the highest payo↵ for each of them. From here on, no node will
change its behavior in any future time steps. So with these payo↵s, the new behavior A does
not spread very far: the decision by the initial adopters to use A caused their neighbors to
become bilingual, but after that further progress stopped.
We can further experiment with this example by keeping the network the same, but
changing the payo↵s so that A becomes much more desirable: specifically, let’s set a = 5,
and keep b = 3 and c = 1. What happens in this case is more complex, and is depicted in
Figure 19.20. (For the discussion below, we will only talk about what happens to the right
of the initial adopters, since what’s going on to the left is symmetric.)
• In the first step, node u will switch to AB, since it receives a payo↵ of 5 + 3 � 1 = 7 from doing this. As a result, in the second step, node w also switches to AB.
• From the third step onward, the strategy AB continues to move to the right, one node at a time. However, something additional happens starting in the third step. Because
node w switched to AB in the second step, node u faces a new decision: it has one
neighbor using A and the other using AB, and so now u’s best choice is to switch from
596 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
AB to A. Essentially, there’s no point in being bilingual anymore if all your neighbors
now have the higher-payo↵ behavior available to them (A in this case).
• In the fourth step, node w also switches from AB to A, and more generally, the strategy A moves to the right, two steps behind the strategy AB. No other changes in strategy
happen, so each node switches first to AB (as the wave of billinguality passes through
it), and then permanently switches to A (the higher-payo↵ monolingual option) two
steps later.
Here is one way to view what is happening in this version of the example: as AB spreads
through the nodes, B becomes vestigial — there is no longer any point for a node to use it.
Thus, nodes abandon B completely over time, and so in the long run only A persists.
A Two-Dimensional Version of the Cascade Capacity. In the basic model earlier in
this chapter, with the underlying coordination game based just on strategies A and B, we
formulated the following question. We are given an infinite graph; for which payo↵ values a
and b is it possible for a finite set of nodes to cause a complete cascade of adoptions of A?
Phrased this way, the question appears to depend on two numbers (a and b), but we saw
earlier that in fact it depends only on the single number q = b/(a + b).
We can ask the analogous question for our model that includes the strategy AB: given
an infinite graph, for which payo↵ values a, b, and c is it possible for a finite set of nodes to
cause a complete cascade of adoptions of A? As with our earlier question, we can eliminate
one of the numbers from this question quite easily. The easiest way to do this is to note
that the answer to our question remains the same if we were to multiply each of a, b, and
c by the same fixed factor. (For example, it does not matter if we multiply each of a, b,
and c by 100 and measure the payo↵s in cents instead of dollars.) Therefore, we can assume
that b = 1 — fixing this as our basic “unit of currency” — and ask how the possibility of a
cascade depends on a and c. Choosing b as the number that we fix equal to 1 makes some
intuitive sense, since it is the payo↵ from using the default behavior B; in this way, we’re
essentially asking: how much better does the new behavior A have to be (the payo↵ a) and
how compatible should it be with B (the payo↵ c) in order for a cascade to have a possibility
of forming?
This question has recently been studied for graphs in general [225], and an interesting
qualitative conclusion arises from the model: A does better when it has a higher payo↵
(this is natural), but in general it has a particularly hard time cascading when the level
of compatibility is “intermediate” — when the value of c is neither too high nor too low.
Rather than describing the general analysis of this phenomenon, we show how it happens on
the infinite path, where the analysis is much simpler and where the main e↵ects are already
apparent. We then discuss some possible interpretations of this e↵ect.
19.7. ADVANCED MATERIAL: THE CASCADE CAPACITY 597
w
A ? B
payoff from choosing A: a
payoff from choosing B: 1
payoff from choosing AB: a + 1 - c
Figure 19.21: The payo↵s to a node on the infinite path with two neighbors using A and B.
When do Cascades Happen on an Infinite Path? The infinite path is an extremely
simple graph, and we saw earlier in this section that in the model with only the strategies
A and B, the condition for A to cascade is correspondingly very simple: a cascade of A’s
can occur precisely when the threshold q is at most 1/2 — or, equivalently, when a � b. In other words, a better technology will always spread on the path.
Once we add the strategy AB as an option, however, the situation becomes more subtle.
Since we are only concerned with whether some finite set of initial adopters can cause a
complete cascade of A’s, we can assume that this set of initial adopters forms a contiguous
interval of nodes on the path. (If not, we can take the leftmost and rightmost initial adopter,
and study the situation in which every node in between is also an initial adopter — this set
is still finite, and it will have just as good a chance of causing a complete cascade.) So
changes in nodes’ strategies will spread outward symmetrically to the left and right of the
initial adopters, and we simply need to account for the possible decisions that nodes make
in evaluating their strategies as this happens. Because of the symmetry, we will only think
about how strategy changes occur to the right of the initial adopters, since what is going on
to the left is the same.
There are two kinds of node-level decisions that are particularly useful for our analysis.
• First, we’ll have to think about nodes like w in Figure 19.21, with a left neighbor using A and a right neighbor using B. (For example, this happens in the first step of
the cascade with the node immediately to the right of the initial adopters.) In this
situation, node w receives a payo↵ of a from choosing A (because it can interact with
its left neighbor), a payo↵ of 1 from choosing B (because it can interact with its right
neighbor), and a payo↵ of a + 1 � c from choosing AB (because it can interact with both neighbors, but pays a cost of c to be bilingual).
Node w will choose the strategy that provides the highest payo↵, and that’s determined
by the relationship between a and c. In other words, we should be asking: for which
598 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
c
a
A vs. B AB vs. B
A vs. AB1
1
(a) Lines showing break-even points between strategies.
c
a
1
1
B
A
AB
(b) Regions defining the best choice of strategy.
Figure 19.22: Given a node with neighbors using A and B, the values of a and c determine which of the strategies A, B, or AB it will choose. (Here, by re-scaling, we can assume b = 1.) We can represent the choice of strategy as a function of a and c by dividing up the (a, c)-plane into regions corresponding to di↵erent choices.
values of a and c will node w choose A, for which will it choose B, and for which will it
choose AB? This can be answered easily if we plot the comparisons among the payo↵s
in the (a, c)-plane as shown in Figure 19.22(a), with the value of a on the x-axis and
the value of c on the y-axis. The break-even point between strategies AB and B, for
example, is given by the line defined by setting the two payo↵s equal: a + 1�c = 1, or equivalently a � c = 0. This is the diagonal line in the figure. Similarly, we draw lines for the break-even point between strategies A and B (a = 1) and between strategies A
and AB (a = a + 1 � c, or equivalently c = 1).
These three lines all meet at the point (1, 1), and so we see that they divide the (a, c)-
plane into six regions. As shown in Figure 19.22(b), A is the best strategy in two of
these regions, B is the best strategy in two of them, and AB is the best strategy in
two of them.
• If AB begins to spread, then we’ll also have to think about the situation pictured in Figure 19.23: a node whose left neighbor is using AB and whose right neighbor is using
B.
Now, if a < 1, then B will provide w with the highest payo↵ regardless of the value of
the cost c (as long as it is positive). So let’s consider the more interesting alternative,
when a � 1. This is very similar to the previous case, when w’s left-hand neighbor was using A; the one change is that the payo↵ to w for using B has now gone up to 2,
since now w can use B to interact with both neighbors rather than just one.
19.7. ADVANCED MATERIAL: THE CASCADE CAPACITY 599
w
AB ? B
payoff from choosing A: a
payoff from choosing B: 2
payoff from choosing AB: a + 1 - c (if A is better)
Figure 19.23: The payo↵s to a node on the infinite path with two neighbors using AB and B.
As a result of this, the lines in the (a, c)-plane defining the break-even points between
B and the other strategies shift to the right (they are now a = 2 and a + 1 � c = 2). This in turn shifts the three regions of the (a, c)-plane that define which strategy will
be chosen by w. We show this in Figure 19.24.
We are now in a position to determine the values of a and c for which a cascade of A’s
can occur. We start with a contiguous interval of initial adopters of A, and we consider the
node u immediately to the right of the initial adopters. (Again, everything here also applies
to the left of the initial adopters by symmetry.)
• If we are in the B region of Figure 19.22(b), then node u will favor B as its strategy, so it will stick with this and the new strategy A will not spread at all.
• If we are in the A region of Figure 19.22(b), then node u will favor A as its strategy, and it will switch to A. So in the next time step we will have exactly the same situation
shifted one node to the right, and as a result the new strategy A will spread all the
way down the path: a cascade will occur.
• Most interestingly, suppose we are in the AB region of Figure 19.22(b). Then, in the next time step, the situation will look di↵erent: the crucial decision will now be faced
by the next node w to the right of u, who will have its left neighbor (u) now using AB,
and its right neighbor still using B.
To understand what w will do, based on values of a and c, the we consult the regions in
Figure 19.24(b). But crucially, since we know that AB was the best choice in the first
step, we know that the values of a and c lie in the AB region from Figure 19.22(b) — so
when we consider Figure 19.24(b), we are concerned not with how its regions carve up
the full (a, c)-plane, but only how they carve up the AB region from Figure 19.22(b).
600 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
c
a
A vs. B AB vs. B
A vs. AB1
1 2
(a) Lines showing break-even points between strategies.
c
a
1
2
B
A
AB
1
(b) Regions defining the best choice of strategy.
Figure 19.24: Given a node with neighbors using AB and B, the values of a and c determine which of the strategies A, B, or AB it will choose, as shown by this division of the (a, c)-plane into regions.
In fact, they divide the AB region from Figure 19.22(b) by a diagonal line segment
from the point (1, 0) to the point (2, 1), as shown in Figure 19.25. To the left of this
line segment, B wins and the cascade stops. To the left of this line segment, AB wins
— so AB continues spreading to the right, and behind this wave of AB’s, nodes will
steadily drop B and use only A. This is the scenario that we saw in our example,
where B fails to persist because it becomes vestigial in a bilingual world.
Figure 19.25 in fact summarizes the four possible cascade outcomes, based on the values
of a and c (i.e. where they lie in the (a, c)-plane). Either (i) B is favored by all nodes outside
the initial adopter set, (ii) A spreads directly without help from AB, (iii) AB spreads for
one step beyond the initial adopter set, but then B is favored by all nodes after that, or (iv)
AB spreads indefinitely to the right, with nodes subsequently switching to A.
So a cascade of A’s can occur if the pair of values (a, c) lies in one of the two regions
described by (ii) and (iv). This means that the portion of the (a, c) plane where a cascade
can occur looks as depicted in Figure 19.26: it lies to the right of a vertical line with a
strange triangular “cut-out.” The vertical line makes a lot of sense: it corresponds to a � 1, or in other words, the requirement that interaction using A produces a higher payo↵ than
interaction using B. But what does the triangular cut-out mean? Formally, it says that when
the cost of bilinguality is neither too high nor too low, the new strategy A has to be “extra
good” — i.e. produce a payo↵ a significantly higher than 1 — in order to spread. Moreover,
although we won’t consider more complex graphs here, the region of the (a, c)-plane where a
cascade of A’s can occur in any graph turns out to have some kind of indentation analogous
to the triangular cut-out — though the particular boundary of the indentation depends on
19.7. ADVANCED MATERIAL: THE CASCADE CAPACITY 601
c
a
1
1 2
neither A nor
AB spreads
A spreads directly (no adoption of AB)
AB spreads
but then
stops
AB spreads indefinitely,
followed by A
(B becomes vestigial)
Figure 19.25: There are four possible outcomes for how A spreads or fails to spread on the infinite path, indicated by this division of the (a, c)-plane into four regions.
the structure of the graph [225].
This triangular cut-out region has a natural qualitative interpretation that provides po-
tential insight into how compatibility and bilinguality a↵ect the process of di↵usion in a
network. We discuss this interpretation now.
Interpretations of the Cascade Region. One way to appreciate what’s going on in the
triangular cut-out region is to consider the following question, phrased in terms of technology
adoption. Suppose that you’re the firm manufacturing the default technology B, and the
payo↵ from interacting via B is equal to 1. Now a new technology A with payo↵ a = 1.5
begins to appear. For which values of the bilinguality cost c should you expect B to survive?
Even without performing any concrete calculations, you could reason as follows. If it’s
extremely easy to maintain both technologies simultaneously, then adoption of AB will
become widespread — and once it is su�ciently widespread, people will begin dropping B
altogether, since A is better and it’s possible to interact with everyone using A. Essentially, A
will have won through “infiltration,” working its way into the population via coexistence with
B. On the other hand, if it’s extremely hard to maintain both technologies simultaneously,
then people on the boundary between the two user populations — those who have friends
602 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
c
a
1
1 2
The region
where a
cascade
of A's
can occur
Figure 19.26: The set of values for which a cascade of A’s can occur defines a region in the (a, c)-plane consisting of a vertical line with a triangular “cut-out.”
using both technologies — will have to simply choose one or the other. And in this case,
you could expect that they may well choose A, since it’s in fact better. In this case, A will
win through a kind of “direct conquest,” simply eliminating B as it goes.
But in between — when it’s neither extremely easy nor extremely hard to maintain both
technologies — something more favorable to B can happen. Specifically, a bilingual “bu↵er
zone” may form between people who adopt only A and those who adopt only B. On the
B-side of this bu↵er zone, no one will have an incentive to change what they’re doing, since
by using B they can interact with all their neighbors — the bilingual ones and the ones using
only B — rather than interacting with only a fraction of their neighbors by switching to
the marginally better technology A. In other words, the inferior technology B has survived
because it was neither too compatible nor too incompatible with A — rather, by partially
accommodating A, it prevented A from spreading too far.2
2On the infinite path, the bilingual bu↵er zones that form are very simple — just one node thick. But in general graphs, the bu↵er zones can have a more complex structure. In fact, it is possible to prove an analogue of the result from Section 19.3, where we showed that clusters are the only obstacle to cascades in the two-strategy model. The more general result is that with an additional bilingual option AB, a structure consisting of a cluster and a bilingual bu↵er zone accompanying it is the only obstacle to a cascade of A’s [225].
19.8. EXERCISES 603
One can tell this story about non-technological settings as well: for example, how dis-
course in a succession of geographically adjacent towns may switch from a traditional lan-
guage B to a more global language A that confers benefits beyond the immediate community
— or it may end up with bilingual inhabitants who use both. In related vein, one could even
consider how a more traditional set of cultural practices (B) may persist in the face of more
modern ones (A), depending on how easy it is for a person to observe both.
Of course, the model we are discussing is extremely simple, and the full story in any of
these scenarios will include many additional factors. For example, in studying competition
between technology firms, there has been a long line of work on the role that compatibility
and incompatibility can play [143, 235, 415], including case studies of technologies includ-
ing instant messaging [158] and electronic imaging [283]. But as with many of our earlier
analyses, the streamlined nature of the model helps provide insight into principles that have
reflections in more complex settings as well. In this particular case, the model also shows
how detailed network structure can play a role in a setting that has otherwise been analyzed
primarily at the population level, treating individuals as interacting in aggregate.
Finally, the discussion shows how the basic di↵usion model — based on a simple coordi-
nation game — is amenable to extensions that capture additional features of real situations
where di↵usion can take place. Even small extensions such as the one considered here can in-
troduce significant new sources of complexity, and the development of even richer extensions
is an open area of research.
19.8 Exercises
1. Consider the network depicted in Figure 19.27; suppose that each node starts with the
behavior B, and each node has a threshold of q = 1 2
for switching to behavior A.
(a) Now, let e and f form a two-node set S of initial adopters of behavior A. If other
nodes follow the threshold rule for choosing behaviors, which nodes will eventually
switch to A?
(b) Find a cluster of density greater than 1 � q = 1 2
in the part of the graph outside
S that blocks behavior A from spreading to all nodes, starting from S, at threshold q.
2. Consider the model from Chapter 19 for the spread of a new behavior through a social
network. Suppose we have the social network depicted in Figure 19.28; suppose that
each node starts with the behavior B, and each node has a threshold of q = 2 5
for
switching to behavior A.
(a) Now, let c and d form a two-node set S of initial adopters of behavior A. If other
nodes follow the threshold rule for choosing behaviors, which nodes will eventually
switch to A? Give a brief (1-2 sentence) explanation for your answer.
604 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
e
c
f
i
g
d
h
j
k
Figure 19.27: Starting from nodes e and f , the new behavior A fails to spread to the entire graph.
(b) Find a cluster of density greater than 1 � q = 3 5
in the part of the graph outside
S that blocks behavior A from spreading to all nodes, starting from S, at threshold q.
Give a brief (1-2 sentence) explanation for your answer.
(c) Suppose you were allowed to add a single edge to the given network, connecting
one of nodes c or d to any one node that it is not currently connected to. Could you
do this in such a way that now behavior A, starting from S and spreading with a
threshold of 2 5
, would reach all nodes? Give a brief explanation for your answer.
3. Consider the model from Chapter 19 for the di↵usion of a new behavior through a
social network. Recall that for this we have a network, a behavior B that everyone
starts with, and a threshold q for switching to a new behavior A — that is, any node
will switch to A if at least a q fraction of its neighbors have adopted A.
Consider the network depicted in Figure 19.29; suppose that each node starts with the
behavior B, and each node has a threshold of q = 2 5
for switching to behavior A.
Now, let e and f form a two-node set S of initial adopters of behavior A. If other nodes
follow the threshold rule for choosing behaviors, which nodes will eventually switch to
A?
(b) Find a cluster of density 1�q = 3 5
in in the part of the graph outside S that blocks
behavior A from spreading to all nodes, starting from S, at threshold q.
(c) Suppose you’re allowed to add one node to the set S of initial adopters, which
currently consists of e and f . Can you do this in such a way that the new 3-node set
causes a cascade at threshold q = 2 5
?
19.8. EXERCISES 605
c
d
e
f
g
h
i
j
k
Figure 19.28: Starting from nodes c and d, the new behavior A fails to spread to the entire graph.
Provide an explanation for your answer, either by giving the name of a third node that
can be added, together with an explanation for what will happen, or by explaining
why there is no choice for a third node that will work to cause a cascade.
4. Consider the model from Chapter 19 for the di↵usion of a new behavior through a
social network.
Suppose that initially everyone is using behavior B in the social network in Fig-
ure 19.30, and then a new behavior A is introduced. This behavior has a threshold of
q = 1/2: any node will switch to A if at least 1/2 of its neighbors are using it.
(a) Find a set of three nodes in the network with the property that if they act as the
three initial adopters of A, then it will spread to all nodes. (In other words, three
nodes who are capable of causing a cascade of adoptions of A.)
(b) Is the set of three nodes you found in (a) the only set of three initial adopters
capable of causing a cascade of A, or can you find a di↵erent set of three initial
adopters who could also cause a cascade of A?
(c) Find three clusters in the network, each of density greater than 1/2, with the
property that no node belongs to more than one of these clusters.
(d) How does your answer to (c) help explain why there is no set consisting of only
two nodes in the network that would be capable of causing a cascade of adoptions of
A? (I.e., only two nodes that could cause the entire network to adopt A.)
606 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
fe
k l
c d
i
nm
hg j
Figure 19.29: A social network in which a new behavior is spreading.
5. Continuing with the di↵usion model from Chapter 19, recall that the threshold q was
derived from a coordination game that each node plays with each of its neighbors.
Specifically, if nodes v and w are each trying to decide whether to choose behaviors A
and B, then:
• if v and w both adopt behavior A, they each get a payo↵ of a > 0;
• if they both adopt B, they each get a payo↵ of b > 0; and
• if they adopt opposite behaviors, they each get a payo↵ of 0.
The total payo↵ for any one node is determined by adding up the payo↵s it gets from
the coordination game with each neighbor.
Let’s now consider a slightly more general version of the model, in which the payo↵ for
choosing opposite behaviors is not 0, but some small positive number x. Specifically,
suppose we replace the third point above with:
• if they adopt opposite behaviors, they each get a payo↵ of x, where x is a positive number that is less than both a and b.
Here’s the question: in this variant of the model with these more general payo↵s, is
each node’s decision still based on a threshold rule? Specifically, is it possible to write
down a formula for a threshold q, in terms of the three quantities a, b, and x, so that
each node v will adopt behavior A if at least a q fraction of its neighbors are adopting
A, and it will adopt B otherwise?
19.8. EXERCISES 607
4
1
5
10
6 7
11
3
8
12
9
13 14
15
16
2
Figure 19.30: A social network on which a new behavior di↵uses.
In your answer, either provide a formula for a threshold q in terms of a, b, and x; or
else explain why in this more general model, a node’s decision can’t be expressed as a
threshold in this way.
6. A group of 20 students living on the third and fourth floors of a college dorm like to
play on-line games. When a new game appears on campus, each of these students
needs to decide whether to join, by registering, creating a player account, and taking
a few other steps necessary in order to start playing.
When a student evaluates whether to join a new on-line game, she bases her decision
on how many of her friends in this group are involved in the game as well. (Not all
pairs of people in this 20-person group are friends, and it is more important whether
your friends are playing than whether many people in the group overall are playing.)
To make the story concrete, let’s suppose that each game goes through the following
“life cycle” within this group of students:
(a) The game has some initial players in the group, who have discovered it and are
already involved in it.
(b) Each other student outside this set of initial players is willing to join the game if
at least half of her friends in the group are playing it.
608 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
(c) Rule (b) is applied repeatedly over time, as in our model from Chapter 19 for the
di↵usion of a new behavior through a social network.
Suppose that in this group of 20 students, 10 live on the third floor of the dorm and
10 live on the fourth floor. Suppose that each student in this group has two friends on
their own floor, and one friend on the other floor. Now, a new game appears, and five
students all living on the fourth floor each begin playing it.
The question is: if the other students use the rule above to evaluate whether to join
the game, will this new game eventually be adopted by all 20 students in the group?
There are three possible answers to this question: yes, no, or there is not information
in the set-up of the question to be able to tell. Say which answer you think is correct,
and explain.
7. Some friends of yours have gone to work at a large on-line game company, and they’re
hoping to draw on your understanding of networks to help them better understand the
user population in one of their games.
Each character in the game chooses a series of quests to go on, generally as part of a
group of characters who work together on them; there are many options for quests to
choose from, but once a character goes on a quest with a group, it can generally last
for a couple of weeks.
Your friends working at the game company have also mapped the social network of the
game, and they’ve invented what they find is a useful way of classifying each player’s
friends: a reinforced friend is one with whom the player has at least one other friend in
common, and an unreinforced friend is one with whom the player has no other friends
in common. For example, the figure below shows the friends of a player A: players B,
C, and D would count as reinforced friends, while player E would be an unreinforced
friend.
Now, your friends are particularly interested in what causes players to choose partic-
ular quests instead of others; and they are also interested in how players learn about
particular methods of cheating along the way — general tricks outside the rules of the
game that make it easier to accumulate points, usually regardless of which particular
quest they’re on. To do some market research on this, they’ve anonymously surveyed
players of the game, asking them two questions:
(a) How did you first learn about the current quest that you’re taking part in?
(b) How have you learned about ways of cheating in the game?
To their surprise, the answers to these questions were quite di↵erent. For (a), 80%
of respondents said that they first found out about the current quest they’re on from
19.8. EXERCISES 609
A
E
B
C
D
Figure 19.31: A small portion of the social network in an online game.
a reinforced friend, while for (b), 60% of respondents said that they found out about
ways of cheating from an unreinforced friend.
Your friends thought you might be able to shed some light on these findings. Why did
the answers to these two questions turn out di↵erently? Is the di↵erence specific to this
particular game, or could it be predicted from general principles of social networks?
In 1-2 paragraphs, describe how particular ideas from the book can shed light on why
the answers to these questions turned out the way they did.
610 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS
Chapter 20
The Small-World Phenomenon
20.1 Six Degrees of Separation
In the previous chapter, we considered how social networks can serve as conduits by which
ideas and innovations flow through groups of people. To develop this idea more fully, we
now relate it to another basic structural issue — the fact that these groups can be connected
by very short paths through the social network. When people try to use these short paths
to reach others who are socially distant, they are engaging in a kind of “focused” search
that is much more targeted than the broad spreading pattern exhibited by the di↵usion of
information or a new behavior. Understanding the relationship between targeted search and
wide-ranging di↵usion is important in thinking more generally about the way things flow
through social networks.
As we saw in Chapter 2, the fact that social networks are so rich in short paths is known
as the small-world phenomenon, or the “six degrees of separation,” and it has long been the
subject of both anecdotal and scientific fascination. To briefly recapitulate what we discussed
in that earlier chapter, the first significant empirical study of the small-world phenomenon
was undertaken by the social psychologist Stanley Milgram [297, 391], who asked randomly
chosen “starter” individuals to each try forwarding a letter to a designated “target” person
living in the town of Sharon, MA, a suburb of Boston. He provided the target’s name,
address, occupation, and some personal information, but stipulated that the participants
could not mail the letter directly to the target; rather, each participant could only advance
the letter by forwarding it to a single acquaintance that he or she knew on a first-name
basis, with the goal of reaching the target as rapidly as possible. Roughly a third of the
letters eventually arrived at the target, in a median of six steps, and this has since served as
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
611
612 CHAPTER 20. THE SMALL-WORLD PHENOMENON
basic experimental evidence for the existence of short paths in the global friendship network,
linking all (or almost all) of us together in society. This style of experiment, constructing
paths through social networks to distant target people, has been repeated by a number of
other groups in subsequent decades [131, 178, 257].
Milgram’s experiment really demonstrated two striking facts about large social networks:
first, that short paths are there in abundance; and second, that people, acting without any
sort of global “map” of the network, are e↵ective at collectively finding these short paths.
It is easy to imagine a social network where the first of these is true but the second isn’t
— a world where the short paths are there, but where a letter forwarded from thousands
of miles away might simply wander from one acquaintance to another, lost in a maze of
social connections [248]. A large social-networking site where everyone was known only by
9-digit pseudonyms would be like this: if you were told, “Forward this letter to user number
482285204, using only people you know on a first-name basis,” the task would clearly be
hopeless. The real global friendship network contains enough clues about how people fit
together in larger structures — both geographic and social — to allow the process of search
to focus in on distant targets. Indeed, when Killworth and Bernard performed follow-up
work on the Milgram experiment, studying the strategies that people employ for choosing
how to forward a message toward a target, they found a mixture of primarily geographic
and occupational features being used, with di↵erent features being favored depending on the
characteristics of the target in relation to the sender [243].
We begin by developing models for both of these principles — the existence of short paths
and also the fact that they can be found. We then look at how some of these models are
borne out to a surprising extent on large-scale social-network data. Finally, in Section 20.6,
we look at some of the fragility of the small-world phenomenon, and the caveats that must
be considered in thinking about it: particularly the fact that people are most successful
at finding paths when the target is high-status and socially accessible [255]. The picture
implied by these di�culties raises interesting additional points about the global structure of
social networks, and suggests questions for further research.
20.2 Structure and Randomness
Let’s start with models for the existence of short paths: Should we be surprised by the
fact that the paths between seemingly arbitrary pairs of people are so short? Figure 20.1(a)
illustrates a basic argument suggesting that short paths are at least compatible with intuition.
Suppose each of us knows more than 100 other people on a first-name basis (in fact, for most
people, the number is significantly larger). Then, taking into account the fact that each of
your friends has at least 100 friends other than you, you could in principle be two steps
away from over 100 · 100 = 10, 000 people. Taking into account the 100 friends of these
20.2. STRUCTURE AND RANDOMNESS 613
you
your friends
friends of your friends
(a) Pure exponential growth produces a small world
you
your friends
friends of your friends
(b) Triadic closure reduces the growth rate
Figure 20.1: Social networks expand to reach many people in only a few steps.
people brings us to more than 100 · 100 · 100 = 1, 000, 000 people who in principle could be three steps away. In other words, the numbers are growing by powers of 100 with each step,
bringing us to 100 million after four steps, and 10 billion after five steps.
There’s nothing mathematically wrong with this reasoning, but it’s not clear how much
it tells us about real social networks. The di�culty already manifests itself with the second
step, where we conclude that there may be more than 10, 000 people within two steps of you.
As we’ve seen, social networks abound in triangles — sets of three people who mutually
know each other — and in particular, many of your 100 friends will know each other. As a
result, when we think about the nodes you can reach by following edges from your friends,
many of these edges go from one friend to another, not to the rest of world, as illustrated
schematically in Figure 20.1(b). The number 10, 000 came from assuming that each of your
100 friends was linked to 100 new people; and without this, the number of friends you could
reach in two steps could be much smaller.
So the e↵ect of triadic closure in social networks works to limit the number of people
you can reach by following short paths, as shown by the contrast between Figures 20.1(a)
614 CHAPTER 20. THE SMALL-WORLD PHENOMENON
(a) Nodes arranged in a grid (b) A network built from local structure and random edges
Figure 20.2: The Watts-Strogatz model arises from a highly clustered network (such as the grid), with a small number of random links added in.
and 20.1(b). And, indeed, at an implicit level, this is a large part of what makes the small-
world phenomenon surprising to many people when they first hear it: the social network
appears from the local perspective of any one individual to be highly clustered, not the kind
of massively branching structure that would more obviously reach many nodes along very
short paths.
The Watts-Strogatz model. Can we make up a simple model that exhibits both of the
features we’ve been discussing: many closed triads, but also very short paths? In 1998,
Duncan Watts and Steve Strogatz argued [411] that such a model follows naturally from a
combination of two basic social-network ideas that we saw in Chapters 3 and 4: homophily
(the principle that we connect to others who are like ourselves) and weak ties (the links to
acquaintances that connect us to parts of the network that would otherwise be far away).
Homophily creates many triangles, while the weak ties still produce the kind of widely
branching structure that reaches many nodes in a few steps.
Watts and Strogatz made this proposal concrete in a very simple model that generates
random networks with the desired properties. Paraphrasing their original formulation slightly
(but keeping the main idea intact), let’s suppose that everyone lives on a two-dimensional
grid — we can imagine the grid as a model of geographic proximity, or potentially some
more abstract kind of social proximity, but in any case a notion of similarity that guides the
formation of links. Figure 20.2(a) shows the set of nodes arranged on a grid; we say that
20.2. STRUCTURE AND RANDOMNESS 615
Figure 20.3: The general conclusions of the Watts-Strogatz model still follow even if only a small fraction of the nodes on the grid each have a single random link.
two nodes are one grid step apart if they are directly adjacent to each other in either the
horizontal or vertical direction.
We now create a network by giving each node two kinds of links: those explainable purely
by homophily, and those that constitute weak ties. Homophily is captured by having each
node form a link to all other nodes that lie within a radius of up to r grid steps away, for
some constant value of r: these are the links you form to people because you are similar to
them. Then, for some other constant value k, each node also forms a link to k other nodes
selected uniformly at random from the grid — these correspond to weak ties, connecting
nodes who lie very far apart on the grid.
Figure 20.2(b) gives a schematic picture of the resulting network — a hybrid structure
consisting of a small amount of randomness (the weak ties) sprinkled onto an underlying
structured pattern (the homophilous links). Watts and Strogatz observe first that the net-
work has many triangles: any two neighboring nodes (or nearby nodes) will have many
common friends, where their neighborhoods of radius r overlap, and this produces many
triangles. But they also find that there are — with high probability — very short paths
connecting every pair of nodes in the network. Roughly, the argument is as follows. Suppose
616 CHAPTER 20. THE SMALL-WORLD PHENOMENON
we start tracing paths outward from a starting node v, using only the k random weak ties
out of each node. Since these edges link to nodes chosen uniformly at random, we are very
unlikely to ever see a node twice in the first few steps outward from v. As a result, these first
few steps look almost like the picture in Figure 20.1(a), when there was no triadic closure,
and so a huge number of nodes are reached in a small number of steps. A mathematically
precise version of this argument was carried out by Bollobás and Chung [67], who determined
the typical lengths of paths that it implies.
Once we understand how this type of hybrid network leads to short paths, we in fact find
that a surprisingly small amount of randomness is needed to achieve the same qualitative
e↵ect. Suppose, for example, that instead of allowing each node to have k random friends, we
only allow one out of every k nodes to have a single random friend — keeping the proximity-
based edges as before, as illustrated schematically in Figure 20.3. Loosely speaking, we can
think of this model with fewer random friends as corresponding to a technologically earlier
time, when most people only knew their near neighbors, and a few people knew someone far
away. Even this network will have short paths between all pairs of nodes. To see why, suppose
that we conceptually group k ⇥ k subsquares of the grid into “towns.” Now, consider the small-world phenomenon at the level of towns. Each town contains approximately k people
who each have a random friend, and so the town collectively has k links to other towns
selected uniformly at random. So this is just like the previous model, except that towns are
now playing the role of individual nodes — and so we can find short paths between any pair
of towns. But now to find a short path between any two people, we first find a short path
between the two towns they inhabit, and then use the proximity-based edges to turn this
into an actual path in the network on individual people.
This, then, is the crux of the Watts-Strogatz model: introducing a tiny amount of ran-
domness — in the form of long-range weak ties — is enough to make the world “small,” with
short paths between every pair of nodes.
20.3 Decentralized Search
Let’s now consider the second basic aspect of the Milgram small-world experiment — the fact
that people were actually able to collectively find short paths to the designated target. This
novel kind of “social search” task was a necessary consequence of the way Milgram formulated
the experiment for his participants. To really find the shortest path from a starting person
to the target, one would have to instruct the starter to forward a letter to all of his or her
friends, who in turn should have forwarded the letter to all of their friends, and so forth.
This “flooding” of the network would have reached the target as rapidly as possible — it
is essentially the breadth-first search procedure from Chapter 2 — but for obvious reasons,
such an experiment was not a feasible option. As a result, Milgram was forced to embark
20.3. DECENTRALIZED SEARCH 617
Figure 20.4: An image from Milgram’s original article in Psychology Today, showing a “com- posite” of the successful paths converging on the target person. Each intermediate step is positioned at the average distance of all chains that completed that number of steps. (Image from [297].)
on the much more interesting experiment of constructing paths by “tunneling” through the
network, with the letter advancing just one person at a time — a process that could well
have failed to reach the target, even if a short path existed.
So the success of the experiment raises fundamental questions about the power of collec-
tive search: even if we posit that the social network contains short paths, why should it have
been structured so as to make this type of decentralized search so e↵ective? Clearly the net-
work contained some type of “gradient” that helped participants guide messages toward the
target. As with the Watts-Strogatz model, which sought to provide a simple framework for
thinking about short paths in highly clustered networks, this type of search is also something
we can try to model: can we construct a random network in which decentralized routing
succeeds, and if so, what are the qualitative properties that are crucial for success?
A model for decentralized search. To begin with, it is not di�cult to model the kind
of decentralized search that was taking place in the Milgram experiment. Starting with the
grid-based model of Watts and Strogatz, we suppose that a starting node s is given a message
that it must forward to a target node t, passing it along edges of the network. Initially s
only knows the location of t on the grid, but, crucially, it does not know the random edges
out of any node other than itself. Each intermediate node along the path has this partial
information as well, and it must choose which of its neighbors to send the message to next.
These choices amount to a collective procedure for finding a path from s to t — just as the
participants in the Milgram experiment collectively constructed paths to the target person.
618 CHAPTER 20. THE SMALL-WORLD PHENOMENON
(a) A small clustering exponent (b) A large clustering exponent
Figure 20.5: With a small clustering exponent, the random edges tend to span long distances on the grid; as the clustering exponent increases, the random edges become shorter.
We will evaluate di↵erent search procedures according to their delivery time — the expected
number of steps required to reach the target, over a randomly generated set of long-range
contacts, and randomly chosen starting and target nodes.
Unfortunately, given this set-up, one can prove that decentralized search in the Watts-
Strogatz model will necessarily require a large number of steps to reach a target — much
larger than the true length of the shortest path [248]. As a mathematical model, the Watts-
Strogatz network is thus e↵ective at capturing the density of triangles and the existence of
short paths, but not the ability of people, working together in the network, to actually find
the paths. Essentially, the problem is that the weak ties that make the world small are “too
random” in this model: since they’re completely unrelated to the similarity among nodes
that produces the homophily-based links, they’re hard for people to use reliably.
One way to think about this is in terms of Figure 20.4, a hand-drawn image from Mil-
gram’s original article in Psychology Today. In order to reach a far-away target, one must
use long-range weak ties in a fairly structured, methodical way, constantly reducing the dis-
tance to the target. As Milgram observed in the discussion accompanying this picture, “The
geographic movement of the [letter] from Nebraska to Massachusetts is striking. There is a
progressive closing in on the target area as each new person is added to the chain” [297]. So
it is not enough to have a network model in which weak ties span only the very long ranges;
it is necessary to span all the intermediate ranges of scale as well. Is there a simple way to
adapt the model to take this into account?
20.4. MODELING THE PROCESS OF DECENTRALIZED SEARCH 619
20.4 Modeling the Process of Decentralized Search
Although the Watts-Strogatz model does not provide a structure where decentralized search
can be performed e↵ectively, a mild generalization of the model in fact exhibits both prop-
erties we want: the networks contain short paths, and these short paths can be found using
decentralized search [248].
Generalizing the network model. We adapt the model by introducing one extra quan-
tity that controls the “scales” spanned by the long-range weak ties. We have nodes on a grid
as before, and each node still has edges to each other node within r grid steps. But now,
each of its k random edges is generated in a way that decays with distance, controlled by a
clustering exponent q as follows. For two nodes v and w, let d(v, w) denote the number of
grid steps between them. (This is their distance if one had to walk along adjacent nodes on
the grid.) In generating a random edge out of v, we have this edge link to w with probability
proportional to d(v, w)�q.
So we in fact have a di↵erent model for each value of q. The original grid-based model
corresponds to q = 0, since then the links are chosen uniformly at random; and varying q
is like turning a knob that controls how uniform the random links are. In particular, when
q is very small, the long-range links are “too random,” and can’t be used e↵ectively for
decentralized search (as we saw specifically for the case q = 0 above); when q is large, the
long-range links are “not random enough,” since they simply don’t provide enough of the
long-distance jumps that are needed to create a small world. Pictorially, this variation in q
can be seen in the di↵erence between the two networks in Figure 20.5. Is there an optimal
operating point for the network, where the distribution of long-range links is su�ciently
balanced between these extremes to allow for rapid decentralized search?
In fact there is. The main result for this model is that, in the limit of large network
size, decentralized search is most e�cient when q = 2 (so that random links follow an
inverse-square distribution). Figure 20.6 shows the performance of a basic decentralized
search method across di↵erent values of q, for a network of several hundred million nodes.
In keeping with the nature of the result — which only holds in the limit as the network size
goes to infinity — decentralized search has about the same e�ciency on networks of this
size across all exponents q between 1.5 and 2.0. (And at this size, it’s best for a value of q
slightly below 2.) But the overall trend is already clear, and as the network size increases,
the best performance occurs at exponents q closer and closer to 2.
A Rough Calculation Motivating the Inverse-Square Network. It is natural to
wonder what’s special about the exponent q = 2 that makes it best for decentralized search.
In Section 20.7 at the end of this chapter, we describe a proof that decentralized search is
e�cient when q = 2, and sketch why search is more e�cient with q = 2 — in the limit of
620 CHAPTER 20. THE SMALL-WORLD PHENOMENON
7.0
6.0
5.0
0.0 1.0 2.0
ln T
exponent q
Figure 20.6: Simulation of decentralized search in the grid-based model with clustering exponent q. Each point is the average of 1000 runs on (a slight variant of) a grid with 400 million nodes. The delivery time is best in the vicinity of exponent q = 2, as expected; but even with this number of nodes, the delivery time is comparable over the range between 1.5 and 2 [248].
large network size — than with any other exponent. But even without the full details of the
proof, there’s a short calculation that suggests why the number 2 is important. We describe
this now.
In the real world where the Milgram experiment was conducted, we mentally organize
distances into di↵erent “scales of resolution”: something can be around the world, across
the country, across the state, across town, or down the block. A reasonable way to think
about these scales of resolution in a network model — from the perspective of a particular
node v — is to consider the groups of all nodes at increasingly large ranges of distance from
v: nodes at distance 2-4, 4-8, 8-16, and so forth. The connection of this organizational
scheme to decentralized search is suggested by Figure 20.4: e↵ective decentralized search
“funnels inward” through these di↵erent scales of resolution, as we see from the way the
letter depicted in this figure reduces its distance to the target by approximately a factor of
two with each step.
So now let’s look at how the inverse-square exponent q = 2 interacts with these scales of
resolution. We can work concretely with a single scale by taking a node v in the network,
and a fixed distance d, and considering the group of nodes lying at distances between d and
2d from v, as shown in Figure 20.7.
Now, what is the probability that v forms a link to some node inside this group? Since
area in the plane grows like the square of the radius, the total number of nodes in this group
is proportional to d2. On the other hand, the probability that v links to any one node in
the group varies depending on exactly how far out it is, but each individual probability
is proportional to d�2. These two terms — the number of nodes in the group, and the
20.4. MODELING THE PROCESS OF DECENTRALIZED SEARCH 621
v
number of nodes is
proportional to d 2
probability of linking to
each is proportional to d -2
2d d
Figure 20.7: The concentric scales of resolution around a particular node.
probability of linking to any one of them — approximately cancel out, and we conclude: the
probability that a random edge links into some node in this ring is approximately independent
of the value of d.
This, then, suggests a qualitative way of thinking about the network that arises when
q = 2: long-range weak ties are being formed in a way that’s spread roughly uniformly over
all di↵erent scales of resolution. This allows people fowarding the message to consistently
find ways of reducing their distance to the target, no matter how near or far they are from it.
In this way, it’s not unlike how the U.S. Postal Service uses the address on an envelope for
delivering a message: a typical postal address exactly specifies scales of resolution, including
the country, state, city, street, and finally the street number. But the point is that the postal
system is centrally designed and maintained at considerable cost to do precisely this job; the
corresponding patterns that guide messages through the inverse-square network are arising
spontaneously from a completely random pattern of links.
622 CHAPTER 20. THE SMALL-WORLD PHENOMENON
Figure 20.8: The population density of the LiveJournal network studied by Liben-Nowell et al. (Image from [277].)
20.5 Empirical Analysis and Generalized Models
The results we’ve seen thus far have been for stylized models, but they raise a number of
qualitative issues that one can try corroborating with data from real social networks. In
this section we discuss empirical studies that analyze geographic data to look for evidence
of the exponent q = 2, as well as more general versions of these models that incorporate
non-geographic notions of social distance.
Geographic Data on Friendship. In the past few years, the rich data available on social
networking sites has made it much easier to get large-scale data that provides insight into
how friendship links scale with distance. Liben-Nowell et al. [277] used the blogging site
LiveJournal for precisely this purpose, analyzing roughly 500,000 users who provided a U.S.
ZIP code for their home address, as well as links to their friends on the system. Note that
LiveJournal is serving here primarily as a very useful “model system,” containing data on
the geographic basis of friendship links on a scale that would be enormously di�cult to
obtain by more traditional survey methods. From a methodological point of view, it is an
interesting and fairly unresolved issue to understand how closely the structure of friendships
defined in on-line communities corresponds to the structure of friendships as we understand
them in o↵-line settings.
A number of things have to be done in order to align the LiveJournal data with the
basic grid model, and perhaps the most subtle involves the fact that the population density
of the users is extremely non-uniform (as it is for the U.S. as a whole). See Figure 20.8
for a visualization of the population density in the LiveJournal data. In particular, the
20.5. EMPIRICAL ANALYSIS AND GENERALIZED MODELS 623
v
w
rank 7
(a) w is the 7th closest node to v.
distance d
rank ~ d 2
(b) Rank-based friendship with uniform population den- sity.
Figure 20.9: When the population density is non-uniform, it can be useful to understand how far w is from v in terms of its rank rather than its physical distance. In (a), we say that w has rank 7 with respect to v because it is the 7th closest node to v, counting outward in order of distance. In (b), we see that for the original case in which the nodes have a uniform population density, a node w at distance d from v will have a rank that is proportional to d2, since all the nodes inside the circle of radius d will be closer to v than w is.
inverse-square distribution is useful for finding targets when nodes are uniformly spaced in
two dimensions; what’s a reasonable generalization to the case in which they can be spread
very non-uniformly?
Rank-Based Friendship. One approach that works well is to determine link probabilities
not by physical distance, but by rank. Let’s suppose that as a node v looks out at all other
nodes, it ranks them by proximity: the rank of a node w, denoted rank(w), is equal to the
number of other nodes that are closer to v than w is. For example, in Figure 20.9(a), node
w would have rank seven, since seven others nodes (including v itself) are closer to v than
w is. Now, suppose that for some exponent p, node v creates a random link as follows: it
chooses a node w as the other end with probability proportional to rank(w)�p. We will call
this rank-based friendship with exponent p.
Which choice of exponent p would generalize the inverse-square distribution for uniformly-
spaced nodes? As Figure 20.9(b) shows, if a node w in a uniformly-spaced grid is at distance
d from v, then it lies on the circumference of a disc of radius d, which contains about d2 closer
nodes — so its rank is approximately d2. Thus, linking to w with probability proportional
to d�2 is approximately the same as linking with probability rank(w)�1, so this suggests
that exponent p = 1 is the right generalization of the inverse-square distribution. In fact,
Liben-Nowell et al. were able to prove that for essentially any population density, if random
624 CHAPTER 20. THE SMALL-WORLD PHENOMENON
(a) Rank-based friendship on LiveJournal (b) Rank-based friendship: East and West coasts
Figure 20.10: The probability of a friendship as a function of geographic rank on the blogging site LiveJournal. (Image from [277].)
links are constructed using rank-based friendship with exponent 1, the resulting network
allows for e�cient decentralized search with high probability. In addition to generalizing the
inverse-square result for the grid, this result has a nice qualitative summary: to construct
a network that is e�ciently searchable, create a link to each node with probability that is
inversely proportional to the number of closer nodes.
Now one can go back to LiveJournal and see how well rank-based friendship fits the
distribution of actual social network links: we consider pairs of nodes where one assigns
the other a rank of r, and we ask what fraction f of these pairs are actually friends, as a
function of r. Does this fraction decrease approximately like r�1? Since we’re looking for a
power-law relationship between the rank r and the fraction of edges f , we can proceed as
in Chapter 18: rather than plotting f as a function of r, we can plot log f as a function of
log r, see if we find an approximately straight line, and then estimate the exponent p as the
slope of this line.
Figure 20.10(a) shows this result for the LiveJournal data; we see that much of the body
of the curve is approximately a straight line sandwiched between slopes of �1.15 and �1.2, and hence close to the optimal exponent of �1. It is also interesting to work separately with the more structurally homogeneous subsets of the data consisting of West-Coast users and
East-Coast users, and when one does this the exponent becomes very close to the optimal
value of �1. Figure 20.10(b) shows this result: The lower dotted line is what you should see if the points followed the distribution rank�1, and the upper dotted line is what you
should see if the points followed the distribution rank�1.05. The proximity of the rank-
based exponent on real networks to the optimal value of �1 has also been corroborated by subsequent research. In particular, as part of a recent large-scale study of several geographic
phenomena in the Facebook social network, Backstrom et al. [33] returned to the question
of rank-based friendship and again found an exponent very close to �1; in their case, the
20.5. EMPIRICAL ANALYSIS AND GENERALIZED MODELS 625
bulk of the distribution was closely approximated by rank�0.95.
The plots in Figure 20.10, and their follow-ups, are thus the conclusion of a sequence of
steps in which we start from an experiment (Milgram’s), build mathematical models based
on this experiment (combining local and long-range links), make a prediction based on the
models (the value of the exponent controlling the long-range links), and then validate this
prediction on real data (from LiveJournal and Facebook, after generalizing the model to
use rank-based friendship). This is very much how one would hope for such an interplay
of experiments, theories, and measurements to play out. But it is also a bit striking to see
the close alignment of theory and measurement in this particular case, since the predictions
come from a highly simplified model of the underlying social network, yet these predictions
are approximately borne out on data arising from real social networks.
Indeed, there remains a mystery at the heart of these findings. While the fact that
the distributions are so close does not necessarily imply the existence of any particular
organizing mechanism [70], it is still natural to ask why real social networks have arranged
themselves in a pattern of friendships across distance that is close to optimal for forwarding
messages to far-away targets. Furthermore, whatever the users of LiveJournal and Facebook
are doing, they are not explicitly trying to run versions of the Milgram experiment — if
there are dynamic forces or selective pressures driving the network toward this shape, they
must be more implicit, and it remains a fascinating open problem to determine whether such
forces exist and how they might operate. One intriguing approach to this question has been
suggested by Oskar Sandberg, who analyzes a model in which a network constantly re-wires
itself as people perform decentralized searches in it. He argues that over time, the network
essentially begins to “adapt” to the pattern of searches; eventually the searches become more
e�cient, and the arrangement of the long-range links begins to approach a structure that
can be approximated by rank-based friendship with the optimal exponent [361].
Social Foci and Social Distance. When we first discussed the Watts-Strogatz model in
Section 20.2, we noted that the grid of nodes was intended to serve as a stylized notion of
similarity among individuals. Clearly it is most easily identified with geographic proximity,
but subsequent models have explored other types of similarity and the ways in which they
can produce small-world e↵ects in networks [250, 410].
The notion of social foci from Chapter 4 provides a flexible and general way to produce
models of networks exhibiting both an abundance of short paths and e�cient decentralized
search. Recall that a social focus is any type of community, occupational pursuit, neighbor-
hood, shared interest, or activity that serves to organize social life around it [161]. Foci are
a way of summarizing the many possible reasons that two people can know each other or
become friends: because they live on the same block, work at the same company, frequent
the same cafe, or attend the same kinds of concerts. Now, two people may have many possi-
626 CHAPTER 20. THE SMALL-WORLD PHENOMENON
v
Figure 20.11: When nodes belong to multiple foci, we can define the social distance between two nodes to be the smallest focus that contains both of them. In the figure, the foci are represented by ovals; the node labeled v belongs to five foci of sizes 2, 3, 5, 7, and 9 (with the largest focus containing all the nodes shown).
ble foci in common, but all else being equal, it is likely that the shared foci with only a few
members are the strongest generators of new social ties. For example, two people may both
work for the same thousand-person company and live in the same million-person city, but it
is the fact that they both belong to the same twenty-person literacy tutoring organization
that makes it most probable they know each other. Thus, a natural way to define the social
distance between two people is to declare it to be the size of the smallest focus that includes
both of them.
In the previous sections, we’ve used models that build links in a social network from an
underlying notion of geographic distance. Let’s consider how this might work with this more
general notion of social distance. Suppose we have a collection of nodes, and a collection
of foci they belong to — each focus is simply a set containing some of the nodes. We let
dist(v, w) denote the social distance between nodes v and w as defined in terms of shared
foci: dist(v, w) is the size of the smallest focus that contains both v and w. Now, following
the style of earlier models, let’s construct a link between each pair of nodes v and w with
probability proportional to dist(v, w)�p. For example, in Figure 20.11, the node labeled v
construct links to three other nodes at social distances 2, 3, and 5. One can now show, subject
to some technical assumptions on the structure of the foci, that when links are generated
this way with exponent p = 1, the resulting network supports e�cient decentralized search
with high probability [250].
There are aspects of this result that are similar to what we’ve just seen for rank-based
20.5. EMPIRICAL ANALYSIS AND GENERALIZED MODELS 627
Figure 20.12: The pattern of e-mail communication among 436 employees of Hewlett Packard Research Lab is superimposed on the o�cial organizational hierarchy, show- ing how network links span di↵erent social foci [6]. (Image from http://www- personal.umich.edu/ ladamic/img/hplabsemailhierarchy.jpg)
friendship. First, as with rank-based friendship, there is a simple description of the underly-
ing principle: when nodes link to each other with probability inversely proportional to their
social distance, the resulting network is e�ciently searchable. And second, the exponent
p = 1 is again the natural generalization of the inverse-square law for the simple grid model.
To see why, suppose we take a grid of nodes and define a set of foci as follows: for each loca-
tion v on the grid, and each possible radius r around that location, there is a focus consisting
of all nodes who are within distance r of v. (Essentially, these are foci consisting of everyone
who live together in neighborhoods and locales of various sizes.) Then for two nodes who are
a distance d apart, their smallest shared focus has a number of nodes proportional to d2, so
this is their social distance. Thus, linking with probability proportional to d�2 is essentially
the same as linking with probability inversely proportional to their social distance.
Recent studies of who-talks-to-whom data has fit this model to social network structures.
In particular, Adamic and Adar analyzed a social network on the employees of Hewlett
Packard Research Lab that we discussed briefly in Chapter 1 connecting two people if they
exchanged e-mail at least six times over a three-month period [6]. (See Figure 20.12.) They
then defined a focus for each of the groups within the organizational structure (i.e. a group
of employees all reporting a common manager). They found that the probability of a link
628 CHAPTER 20. THE SMALL-WORLD PHENOMENON
between two employees at social distance d within the organization scaled proportionally
to d�3/4. In other words, the exponent on the probability for this network is close to, but
smaller than, the best exponent for making decentralized search within the network e�cient.
These increasingly general models thus provide us with a way to look at social-network
data and speak quantitatively about the ways in which the links span di↵erent levels of
distance. This is important not just for understanding the small-world properties of these
networks, but also more generally for the ways in which homophily and weak ties combine
to produce the kinds of structures we find in real networks.
Search as an Instance of Decentralized Problem-Solving. While the Milgram ex-
periment was designed to test the hypothesis that people are connected by short paths in
the global social network, our discussion here shows that it also served as an empirical study
of people’s ability to collectively solve a problem — in this case, searching for a path to a
far-away individual — using only very local information, and by communicating only with
their neighbors in the social network. In addition to the kinds of search methods discussed
here, based on aiming as closely to the target as possible in each step, researchers have also
studied the e↵ectiveness of path-finding strategies in which people send messages to friends
who have a particularly large number of edges (on the premise that they will be “better
connected” in general) [6, 7], as well as strategies that explicitly trade o↵ the proximity of a
person against their number of edges [370].
The notion that social networks can be e↵ective at this type of decentralized problem-
solving is an intriguing and general premise that applies more broadly than just to the
problem of path-finding that Milgram considered. There are many possible problems that
people interacting in a network could try solving, and it is natural to suppose that their
e↵ectiveness will depend both on the di�culty of the problem being solved and on the
network that connects them. There is a long history of experimental interest in collective
problem-solving [47], and indeed one way to view the bargaining experiments described in
Chapter 12 is as an investigation of the ability of a group of people to collectively find a
mutually compatible set of exchanges when their interaction is constrained by a network.
Recent experiments have explored this issue for a range of basic problems, across multiple
kinds of network structures [236, 237], and there is also a growing line of work in the design
of systems that can exploit the power of collective human problem-solving by very large
on-line populations [402, 403].
20.6. CORE-PERIPHERY STRUCTURES AND DIFFICULTIES IN DECENTRALIZED SEARCH629
core
periphery
Figure 20.13: The core-periphery structure of social networks.
20.6 Core-Periphery Structures and Di�culties in De- centralized Search
In the four decades since the Milgram experiment, the research community has come to
appreciate both the robustness and the delicacy of the “six degrees” principle. As we noted in
Chapter 2, many studies of large-scale social network data have confirmed the pervasiveness
of very short paths in almost every setting. On the other hand, the ability of people to find
these paths from within the network is a subtle phenomenon: it is striking that it should
happen at all, and the conditions that facilitate it are not fully understood.
As Judith Kleinfeld has noted in her recent critique of the Milgram experiment [255],
the success rate at finding targets in recreations of the experiment has often been much
lower than it was in the original work. Much of the di�culty can be explained by lack of
participation: many people, asked to forward a letter as part of the experiment, will simply
throw it away. This is consistent with lack of participation in any type of survey or activity
carried out by mail; assuming this process is more or less random, it has a predictable e↵ect
on the results, and one can correct for it [131, 416].
But there are also more fundamental di�culties at work, pointing to questions about
large social networks that may help inform a richer understanding of network structure. In
630 CHAPTER 20. THE SMALL-WORLD PHENOMENON
particular, Milgram-style search in a network is most successful when the target person is
a✏uent and socially high-status. For example, in the largest small-world experiment to date
[131], 18 di↵erent targets were used, drawn from a wide range of backgrounds. Completion
rates to all targets were small, due to lack of participation in the e-mail based forwarding of
messages, but they were highest for targets who were college professors and journalists, and
particularly small for low-status targets.
Core-Periphery Structures. This wide variation in the success rates of search to di↵er-
ent targets does not simply arise from variations in individual attributes of the respective
people — it is based on the fact that social networks are structured to make high-status
individuals much easier to find than low-status ones. Homophily suggests that high-status
people will mainly know other high-status people, and low-status people will mainly know
other low-status people, but this does not imply that the two groups occupy symmetric or
interchangeable positions in the social network. Rather, large social networks tend to be
organized in what is called a core-periphery structure [72], in which the high-status people
are linked in a densely-connected core, while the low-status people are atomized around the
periphery of the network. Figure 20.13 gives a schematic picture of such a structure. High-
status people have the resources to travel widely; to meet each other through shared foci
around clubs, interests, and educational and occupational pursuits; and more generally to
establish links in the network that span geographic and social boundaries. Low-status people
tend to form links that are much more clustered and local. As a result, the shortest paths
connecting two low-status people who are geographically or socially far apart will tend to go
into the core and then come back out again.
All this has clear implications for people’s ability to find paths to targets in the network.
In particular, it indicates some of the deep structural reasons why it is harder for Milgram-
style decentralized search to find low-status targets than high-status targets. As you move
toward a high-status target, the link structure tends to become richer, based on connections
with an increasing array of underlying social reasons. In trying to find a low-status target,
on the other hand, the link structure becomes structurally more impoverished as you move
toward the periphery.
These considerations suggest an opportunity for richer models that take status e↵ects
more directly into account. The models we have seen capture the process by which people
can find each other when they are all embedded in an underlying social structure, and
motivated to continue a path toward a specific destination. But as the social structure
begins to fray around the periphery, an understanding of how we find our way through it
has the potential to shed light not just on the networks themselves, but on the way that
network structure is intertwined with status and the varied positions that di↵erent groups
occupy in society as a whole.
20.7. ADVANCED MATERIAL: ANALYSIS OF DECENTRALIZED SEARCH 631
a b
c
d
e
f
g
h i
j
k
l
m
n
o
p
(a) A set of nodes arranged in a ring.
a b
c
d
e
f
g
h i
j
k
l
m
n
o
p
(b) A ring augmented with random long- range links.
Figure 20.14: The analysis of decentralized search is a bit cleaner in one dimension than in two, although it is conceptually easy to adapt the arguments to two dimensions. As a result, we focus most of the discussion on a one-dimensional ring augmented with random long-range links.
20.7 Advanced Material: Analysis of Decentralized Search
In Section 20.4, we gave some basic intuition for why an inverse-square distribution of links
with distance makes e↵ective decentralized search possible. Even given this way of thinking
about it, however, it still requires further work to really see why search succeeds with this
distribution. In this section, we describe the complete analysis of the process [249].
To make the calculations a bit simpler, we vary the model in one respect: we place the
nodes in one dimension rather than two. In fact, the argument is essentially the same no
matter how many dimensions the nodes are in, but one dimension makes things the cleanest
(even if not the best match for the actual geographic structure of a real population). It turns
out, as we will argue more generally later in this section, that the best exponent for search is
equal to the dimension, so in our one-dimensional analysis we will be using an exponent of
q = 1 rather than q = 2. At the end, we will discuss the minor ways in which the argument
needs to be adapted in two or higher dimensions.
We should also mention, recalling the discussion earlier in the chapter, that there is a
second fundamental part of this analysis as well — showing that this choice of q is in fact the
best for decentralized search in the limit of increasing network size. At the end, we sketch
why this is true, but the full details are beyond what we will cover here.
632 CHAPTER 20. THE SMALL-WORLD PHENOMENON
a b
c
d
e
f
g
h i
j
k
l
m
n
o
p
Figure 20.15: In myopic search, the current message-holder chooses the contact that lies closest to the target (as measured on the ring), and it forwards the message to this contact.
A. The Optimal Exponent in One Dimension
Here, then, is the model we will be looking at. A set of n nodes are arranged on a one-
dimensional ring as shown in Figure 20.14(a), with each node connected by directed edges
to the two others immediately adjacent to it. Each node v also has a single directed edge
to some other node on the ring; the probability that v links to any particular node w is
proportional to d(v, w)�1, where d(v, w) is their distance apart on the ring. We will call the
nodes to which v has an edge its contacts: the two nodes adjacent to it on the ring are its
local contacts, and the other one is its long-range contact. The overall structure is thus a ring
that is augmented with random edges, as shown in Figure 20.14(b). Again, this is essentially
just a one-dimensional version of the grid with random edges that we saw in Figure 20.5.1
Myopic Search. Let’s choose a random start node s and a random target node t on this
augmented ring network. The goal, as in the Milgram experiment, is to forward a message
from the start to the target, with each intermediate node on the way only knowing the
locations of its own neighbors, and the location of t, but nothing else about the full network.
The forwarding strategy that we analyze, which works well on the ring when q = 1, is a
1We could also analyze a model in which nodes have more outgoing edges, but this only makes the search problem easier; our result here will show that even when each node has only two local contacts and a single long-range contact, search can still be very e�cient.
20.7. ADVANCED MATERIAL: ANALYSIS OF DECENTRALIZED SEARCH 633
simple technique that we call myopic search: when a node v is holding the message, it passes
it to the contact that lies as close to t on the ring as possible. Myopic search can clearly
be performed even by nodes that know nothing about the network other than the locations
of their friends and the location of t, and it is a reasonable approximation to the strategies
used by most people in Milgram-style experiments [243].
For example, Figure 20.15 shows the myopic path that would be constructed if we chose
a as the start node and i as the target node in the network from Figure 20.14(b).
1. Node a first sends the message to node d, since among a’s contacts p, b, and d, node
d lies closest to i on the ring.
2. Then d passes the message to its local contact e, and e likewise passes the message to
its local contact f , since the long-range contacts of both d and e lead away from i on
the ring, not closer to it.
3. Node f has a long-range contact h that proves useful, so it passes it to h. Node h
actually has the target as a local contact, so it hands it directly to i, completing the
path in five steps.
Notice that this myopic path is not the shortest path from a to i. If a had known that its
friend b in fact had h as a contact, it could have handed the message to b, thereby taking
the first step in the three-step a-b-h-i path. It is precisely this lack of knowledge about the
full network structure that prevents myopic search from finding the true shortest path in
general.
Despite this, however, we will see next that in expectation, myopic search finds paths
that are surprisingly short.
Analyzing Myopic Search: The Basic Plan. We now have a completely well-defined
probabilistic question to analyze, as follows. We generate a random network by adding
long-range edges to a ring as above. We then choose a random start node s and random
target node t in this network. The number of steps required by myopic search is now a
random variable X, and we are interested in showing that E [X], the expected value of X,
is relatively small.
Our plan for putting a bound on the expected value of X follows the idea contained in
Milgram’s picture from Figure 20.4: we track how long it takes for the message to reduce its
distance by factors of two as it closes in on the target. Specifically, as the message moves
from s to t, we’ll say that it’s in phase j of the search if its distance from the target is
between 2j and 2j+1. See Figure 20.16 for an illustration of this. Notice that the number of
di↵erent phases is at most log 2
n — that is, the number of doublings needed to go from 1 to
n. (In what follows, we will drop the base of the logarithm and simply write log n to denote
log 2
n.)
634 CHAPTER 20. THE SMALL-WORLD PHENOMENON
t
2
2
2
j-1
j
j+1
s
phase j
phase j-1
Figure 20.16: We analyze the progress of myopic search in phases. Phase j consists of the portion of the search in which the message’s distance from the target is between 2j and 2j+1.
We can write X, the number of steps taken by the full search, as
X = X 1
+ X 2
+ · · · + X log n
;
that is, the total time taken by the search is simply the sum of the times taken in each phase.
Linearity of expectation says that the expectation of a sum of random variables is equal to
the sum of their individual expectations, and so we have
E [X] = E [X 1
+ X 2
+ · · · + X log n
] = E [X 1
] + E [X 2
] + · · · + E [X log n
] .
We will now show — and this is the crux of the argument — that the expected value of each
X j
is at most proportional to log n. In this way, E [X] will be a sum of log n terms, each at
most proportional to log n, and so we will have shown that E [X] is at most proportional to
(log n)2.
This will achieve our overall goal of showing that myopic search is very e�cient with the
given distribution of links: the full network has n nodes, but myopic search constructs a
path that is exponentially smaller: proportional to the square of log n.
20.7. ADVANCED MATERIAL: ANALYSIS OF DECENTRALIZED SEARCH 635
Figure 20.17: Determining the normalizing constant for the probability of links involves evaluating the sum of the first n/2 reciprocals. An upper bound on the value of this sum can be determined from the area under the curve y = 1/x.
Intermediate Step: The Normalizing Constant In implementing this high-level strat-
egy, the first thing we need to work out is in fact something very basic: we’ve been saying
all along that v forms its long-range link to w with probability proportional to d(v, w)�1, but
what is the constant of proportionality? As in any case when we know a set of probabilities
up to a missing constant of proportionality 1/Z, the value of Z is here simply the sum of
d(v, u)�1 over all nodes u 6= v on the ring. Dividing everything down by this normalizing constant Z, the probability of v linking to w is then equal to 1
Z
d(v, w)�1.
To work out the value of Z, we note that there are two nodes at distance 1 from v, two
at distance 2, and more generally two at each distance d up to n/2. Assuming n is even,
there is also a single node at distance n/2 from v — the node diametrically opposite it on
the ring. Therefore, we have
Z 2 ✓
1 + 1
2 +
1
3 +
1
4 + · · · +
1
n/2
◆ . (20.1)
The quantity inside parentheses on the right is a common expression in probabilistic calcu-
lations: the sum of the first k reciprocals, for some k, in this case n/2. To put an upper
bound on its size, we can compare it to the area under the curve y = 1/x, as shown in
Figure 20.17. As that figure indicates, a sequence of rectangles of unit widths and heights
1/2, 1/3, 1/4, . . . , 1/k fits under the curve y = 1/x as x ranges from 1 to k. Combined with
a single rectangle of height and width 1, we see that
1 + 1
2 +
1
3 +
1
4 + · · · +
1
k 1 +
Z k
1
1
x dx = 1 + ln k.
636 CHAPTER 20. THE SMALL-WORLD PHENOMENON
t
2
2
2
j-1
j
j+1
s
v
w distance d
distance at most d/2?
Figure 20.18: At any given point in time, the search is in some phase j, with the message residing at a node v at distance d from the target. The phase will come to an end if v’s long- range contact lies at distance d/2 from the target t, and so arguing that the probability of this event is large provides a way to show that the phase will not last too long.
Plugging in k = n/2 to the expression on the right-hand side of inequality (20.1) above, we
get
Z 2(1 + ln(n/2)) = 2 + 2 ln(n/2).
For simplicity, we’ll use a slightly weaker bound on Z, which follows simply from the obser-
vation that ln x log 2
x:
Z 2 + 2 log 2
(n/2) = 2 + 2(log 2
n) � 2(log 2
2) = 2 log 2
n.
Thus, we now have an expression for the actual probability that v links to w (including its
constant of proportionality): the probability v links to w is
1
Z d(v, w)�1 �
1
2 log n d(v, w)�1.
20.7. ADVANCED MATERIAL: ANALYSIS OF DECENTRALIZED SEARCH 637
t
s
v
w distance d/2distance d/2
distance d
there are d+1 nodes within distance
d/2 of t, and each has prob. at least
proportional to 1/(d log n)
Figure 20.19: Showing that, with reasonable probability, v’s long-range contact lies within half the distance to the target.
Analyzing the Time Spent in One Phase of Myopic Search. Finally, we come to
the last and central step of the analysis: showing that the time spent by the search in any
one phase is not very large. Let’s choose a particular phase j of the search, when the message
is at a node v whose distance to the target t is some number d between 2j and 2j+1. (See
Figure 20.18 for an illustration of all this notation in context.) The phase will come to an
end once the distance to the target decreases below 2j, and we want to show that this will
happen relatively quickly.
One way for the phase to come to an end immediately would be for v’s long-range contact
w to be at distance d 2
from t. In this case, v would necessarily be the last node to belong
to phase j. So let’s show that this immediate halving of the distance in fact happens with
reasonably large probability.
The argument is pictured in Figure 20.19. Let I be the set of nodes at distance d 2
from
638 CHAPTER 20. THE SMALL-WORLD PHENOMENON
t; this is where we hope v’s long-range contact will lie. There are d + 1 nodes in I: this
includes node t itself, and d/2 nodes consecutively on each side of it. Each node w in I has
distance at most 3d/2 from v: the farthest one is on the “far side” of t from v, at distance
d + d/2. Therefore, each node w in I has probability at least
1
2 log n d(v, w)�1 �
1
2 log n ·
1
3d/2 =
1
3d log n
of being the long-range contact of v. Since there are more than d nodes in I, the probability
that one of them is the long-range contact of v is at least
d · 1
3d log n =
1
3 log n .
If one of these nodes is the long-range contact of v, then phase j ends immediately in this
step. Therefore, in each step that it proceeds, phase j has a probability of at least 1/(3 log n)
of coming to an end, independently of what has happened so far. To run for at least i steps,
phase j has to fail to come to an end i�1 times in a row, and so the probability that phase j runs for at least i steps is at most
✓ 1 �
1
3 log n
◆ i�1
.
Now we conclude by just using the formula for the expected value of a random variable:
E [X j
] = 1 · Pr [X j
= 1] + 2 · Pr [X j
= 2] + 3 · Pr [X j
= 3] + · · · (20.2)
There is a useful alternate way to write this: notice that in the expression
Pr [X j
� 1] + Pr [X j
� 2] + Pr [X j
� 3] + · · · (20.3)
the quantity Pr [X j
= 1] is accounted for once (in the first term only), the quantity Pr [X j
= 2]
is accounted for twice (in the first two terms only), and so forth. Therefore the expressions
in (20.2) and (20.3) are the same thing, and so we have
E [X j
] = Pr [X j
� 1] + Pr [X j
� 2] + Pr [X j
� 3] + · · · (20.4)
Now, we’ve just argued above that
Pr [X j
� i] ✓
1 � 1
3 log n
◆ i�1
,
and so
E [X j
] 1 + ✓
1 � 1
3 log n
◆ +
✓ 1 �
1
3 log n
◆ 2
+
✓ 1 �
1
3 log n
◆ 3
+ · · ·
20.7. ADVANCED MATERIAL: ANALYSIS OF DECENTRALIZED SEARCH 639
t
v
w
distance d
radius d/2
Figure 20.20: The analysis for the one-dimensional ring can be carried over almost directly to the two-dimensional grid. In two dimensions, with the message at a current distance d from the target t, we again look at the set of nodes within distance d/2 of t, and argue that the probability of entering this set in a single step is reasonably large.
The right-hand side is a geometric sum with multiplier
✓ 1 �
1
3 log n
◆ , and so it converges
to
1
1 � ⇣ 1 � 1
3 log n
⌘ = 3 log n.
Thus we have
E [X j
] 3 log n.
And now we’re done. E [X] is a sum of the log n terms E [X 1
] + E [X 2
] + · · · + E [X log n
] ,
and we’ve just argued that each of them is at most 3 log n. Therefore, E [X] 3(log n)2, a quantity proportional to (log n)2 as we wanted to show.
640 CHAPTER 20. THE SMALL-WORLD PHENOMENON
B. Higher Dimensions and Other Exponents
Using the analysis we’ve just completed, we now discuss two further issues. First, we sketch
how it can be used to analyze networks built by adding long-range contacts to nodes arranged
in two dimensions. Then we show how, in the limit of increasing network size, search is more
e�cient when q is equal to the underlying dimension than when it is equal to any other
value.
The Analysis in Two Dimensions. It’s not hard to adapt our analysis for the one-
dimensional ring directly to the case of the two-dimensional grid. Essentially, we only used
the fact that we were in one dimension in two distinct places in the analysis. First, we used
it when we determined the normalizing constant Z. Second, and in the end most crucially,
we used it to argue that there were at least d nodes within distance d/2 of the target t.
This factor of d canceled the d�1 in the link probability, allowing us to conclude that the
probability of halving the distance to the target in any given step was at least proportional
to 1/(log n), regardless of the value of d.
At a qualitative level, this last point is the heart of the analysis: with link probability
d�1 on the ring, the probability of linking to any one node exactly o↵sets the number of
nodes close to t, and so myopic search makes progress at every possible distance away from
the target.
When we go to two dimensions, the number of nodes within distance d/2 of the target
will be proportional to d2. This suggests that to get the same nice cancellation property,
we should have v link to each node w with probability proportional to d(v, w)�2, and this
exponent �2 is what we will use. With the above ideas in mind, and with this change in the exponent to �2, the analysis
for two dimensions is almost exactly the same as what we just saw for the one-dimensional
ring. First, while we won’t go through the calculations here, the normalizing constant Z is
still proportional to log n when the probability of v linking to w is proportional to d(v, w)�2.
We then consider log n di↵erent phases as before; and as depicted in Figure 20.20, we consider
the probability that at any given moment, the current message-holder v has a long-range
contact w that halves the distance to the target, ending the phase immediately. Now we
use the calculation foreshadowed in the previous paragraph: the number of nodes within
distance d/2 of the target is proportional to d2, and the probability that v links to each is
proportional to 1/(d2 log n). Therefore, the probability that the message halves its distance
to the target in this step is at least d2/(d2 log n) = 1/(log n), and the rest of the analysis
then finishes as before.
A similarly direct adaptation of the analysis shows that decentralized search is e�cient
for networks built by adding long-range contacts to grids in D > 2 dimensions, when the
exponent q is equal to D.
20.7. ADVANCED MATERIAL: ANALYSIS OF DECENTRALIZED SEARCH 641
t
s
v
distancedistance
It takes a long-time for the search to find
a long-range link into K, and crossing K
via local contacts is slow too.
K
nn
Figure 20.21: To show that decentralized search strategies require large amounts of time with exponent q = 0, we argue that it is di�cult for the search to cross the set of
p n nodes
closest to the target. Similar arguments hold for other exponents q < 1.
Why Search is Less E�cient with Other Exponents. Finally, let’s sketch why de-
centralized search is less e�cient when the exponent is anything else. For concreteness, we’ll
focus on why search doesn’t work well when q = 0 — the original Watts-Strogatz model
when long-range links are chosen uniformly at random. Also, we’ll talk again about the
one-dimensional ring rather than the two-dimensional grid, since things are a bit cleaner in
one dimension, although again the analysis in two dimensions is essentially the same.
The key idea, as with the “good” exponent q = 1, is to consider the set of all nodes
within some distance of the target t. But whereas in the case of q = 1 we wanted to argue
that it is easy to enter smaller and smaller sets centered around t, here we want to identify
a set of nodes centered at t that is somehow “impenetrable” — a set that is very hard for
the search to enter.
In fact, it is not di�cult to do this. The basic idea is depicted in Figure 20.21; we sketch
642 CHAPTER 20. THE SMALL-WORLD PHENOMENON
how the argument works, but without going into all the details. (The details can be found
in [249].) Let K be the set of all nodes within distance less than p
n of the target t. Now,
with high probability, the starting point of the search lies outside K. Because long-range
contacts are created uniformly at random (since q = 0), the probability that any one node
has a long-range contact inside K is equal to the size of K divided by n: so it is less than
2 p
n/n = 2/ p
n. Therefore, any decentralized search strategy will need at least p
n/2 steps
in expectation to find a node with a long-range contact in K. On the other hand, as long
as it doesn’t find a long-range link leading into K, it can’t reach the target in less than p
n steps, since it would take this long to “walk” step-by-step through K using only the
connections among local contacts. From this, one can show that the expected time for any
decentralized search strategy to reach t must be at least proportional to p
n.
There are similar arguments for every other exponent q 6= 1. When q is strictly between 0 and 1, a version of the argument above works, with a set K centered at t whose width depends
on the value of q. And when q > 1, decentralized search is ine�cient for a di↵erent reason:
since even the long-range links are relatively short, it takes a long time for decentralized
search to find links that span su�ciently long distances. This makes it hard to quickly
traverse the distance from the starting node to the target.
Overall, one can show that for every exponent q 6= 1, there is a constant c > 0 (depending on q), so that it takes at least proportional to nc steps in expectation for any decentralized
search strategy to reach the target in a network generated with exponent q. So in the limit,
as n becomes large, decentralized search with exponent q = 1 requires time that grows like
a polynomial in log n, while decentralized search at any other exponent requires a time that
grows like a polynomial in n — exponentially worse.2 The exponent q = 1 on the ring —
or q = 2 in the plane — is optimally balanced between producing networks that are “too
random” for search, and those that are not random enough.
20.8 Exercises
1. In the basic “six degrees of separation” question, one asks whether most pairs of people
in the world are connected by a path of at most six edges in the social network, where
an edge joins any two people who know each other on a first-name basis.
Now let’s consider a variation on this question. Suppose that we consider the full
population of the world, and suppose that from each person in the world we create a
directed edge only to their ten closest friends (but not to anyone else they know on a
first-name basis). In the resulting “closest-friend” version of the social network, is it
possible that for each pair of people in the world, there is a path of at most six edges
2Of course, it can take very large values of n for this distinction to become truly pronounced; recall Figure 20.6, which showed the results of simulations on networks with 400 million nodes.
20.8. EXERCISES 643
connecting this pair of people? Explain.
2. In the basic “six degrees of separation” question, one asks whether most pairs of people
in the world are connected by a path of at most six edges in the social network, where
an edge joins any two people who know each other on a first-name basis.
Now let’s consider a variation on this question. For each person in the world, we ask
them to rank the 30 people they know best, in descending order of how well they know
them. (Let’s suppose for purposes of this question that each person is able to think of
30 people to list.) We then construct two di↵erent social networks:
(a) The “close-friend” network: from each person we create a directed edge only to
their ten closest friends on the list.
(b) The “distant-friend” network: from each person we create a directed edge only to
the ten people listed in positions 21 through 30 on their list.
Let’s think about how the small-world phenomenon might di↵er in these two networks.
In particular, let C be the average number of people that a person can reach in six
steps in the close-friend network, and let D be the average number of people that a
person can reach in six steps in the distant-friend network (taking the average over all
people in the world).
When researchers have done empirical studies to compare these two types of networks
(the exact details often di↵er from one study to another), they tend to find that one
of C or D is consistently larger than the other. Which of the two quantities, C or D,
do you expect to be larger? Give a brief explanation for your answer.
3. Suppose you’re working with a group of researchers studying social communication
networks, with a particular focus on the distances between people in such networks,
and the broader implications for the small-world phenomenon.
The research group is currently negotiating an agreement with a large mobile phone
carrier to get a snapshot of their “who-calls-whom” graph. Specifically, under a strict
confidentiality agreement, the carrier is o↵ering to provide a graph in which there is
a node representing each of the carrier’s customers, and each edge represents a pair
of people who called each other over a fixed one-year period. (The edges will be
annotated with the number of calls and the time at which each one happened. No
personal identification will be provided with the nodes.)
Recently, the carrier has proposed that instead of providing all the data, they’ll only
provide edges corresponding to pairs of people who called each other at least once a
week on average over the course of the year. (That is, all nodes will be present, but
there will only be edges for pairs of people who talked at least 52 times.) The carrier
644 CHAPTER 20. THE SMALL-WORLD PHENOMENON
understands that this is not the full network, but they would prefer to release less data
and they argue that this will be a good approximation to the full network.
Your research group objects, but the carrier is not inclined to change its position unless
your group can identify specific research findings that are likely to be misleading if
they are drawn from this reduced dataset. The leader of your research group asks
you to prepare a brief response to the carrier, identifying some concrete ways in which
misleading conclusions might be reached from the reduced dataset.
What would you say in your response?
Chapter 21
Epidemics
The study of epidemic disease has always been a topic where biological issues mix with
social ones. When we talk about epidemic disease, we will be thinking of contagious diseases
caused by biological pathogens — things like influenza, measles, and sexually transmitted
diseases, which spread from person to person. Epidemics can pass explosively through a
population, or they can persist over long time periods at low levels; they can experience
sudden flare-ups or even wave-like cyclic patterns of increasing and decreasing prevalence. In
extreme cases, a single disease outbreak can have a significant e↵ect on a whole civilization,
as with the epidemics started by the arrival of Europeans in the Americas [130], or the
outbreak of bubonic plague that killed 20% of the population of Europe over a seven-year
period in the 1300s [293].
21.1 Diseases and the Networks that Transmit Them
The patterns by which epidemics spread through groups of people is determined not just by
the properties of the pathogen carrying it — including its contagiousness, the length of its
infectious period, and its severity — but also by network structures within the population
it is a↵ecting. The social network within a population — recording who knows whom —
determines a lot about how the disease is likely to spread from one person to another. But
more generally, the opportunities for a disease to spread are given by a contact network:
there is a node for each person, and an edge if two people come into contact with each other
in a way that makes it possible for the disease to spread from one to the other.
This suggests that accurately modeling the underlying network is crucial to understanding
the spread of an epidemic. This has led to research studying how travel patterns within
a city [149, 295] or via the worldwide airline network [119] could a↵ect the spread of a
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
645
646 CHAPTER 21. EPIDEMICS
fast-moving disease. Contact networks are also important in understanding how diseases
spread through animal populations — with researchers tracing out the interactions within
livestock populations during epidemics such as the 2001 foot-and-mouth outbreak in the
United Kingdom [211] — as well as plant populations, where the a↵ected individuals occupy
fixed locations and diseases tend to have a much clearer spatial footprint [139]. And similar
models have been employed for studying the spread of computer viruses, with malicious
software spreading between computers across an underlying communication network [241].
The pathogen and the network are closely intertwined: even within the same population,
the contact networks for two di↵erent diseases can have very di↵erent structures, depending
on the diseases’ respective modes of transmission. For a highly contagious disease, involving
airborne transmission based on coughs and sneezes, the contact network will include a huge
number of links, including any pair of people who sat together on a bus or an airplane. For
a disease requiring close contact, or a sexually transmitted disease, the contact network will
be much sparser, with many fewer pairs of people connected by links. Similar distinctions
arise in studying computer viruses, where a piece of software infecting computers across the
Internet will have a much broader contact network than one that spreads by short-range
wireless communication between nearby mobile devices [251].
Connections to the Di↵usion of Ideas and Behaviors. There are clear connections
between epidemic disease and the di↵usion of ideas through social networks. Both diseases
and ideas can spread from person to person, across similar kinds of networks that connect
people, and in this respect, they exhibit very similar structural mechanisms — to the extent
that the spread of ideas is often referred to as “social contagion” [85]. Having considered the
di↵usion of ideas, innovations, and new behaviors in Chapter 19, why then are we revisiting
this topic afresh in the context of diseases?
In the context of our discussions here about networks, the biggest di↵erence between
biological and social contagion lies in the process by which one person “infects” another.
With social contagion, people are making decisions to adopt a new idea or innovation, and
our models in Chapter 19 were focused on relating the underlying decision-making processes
to the larger e↵ects at the network level. With diseases, on the other hand, not only is there
a lack of decision-making in the transmission of the disease from one person to another,
but the process is su�ciently complex and unobservable at the person-to-person level that
it is most useful to model it as random. That is, we will generally assume that when two
people are directly linked in the contact network, and one of them has the disease, there is
a given probability that he or she will pass it to the other. This use of randomness allows
us to abstract away questions about the mechanics of how one person catches a disease from
another for which we have no useful simple models.
This, then, will be the concrete di↵erence in our discussion of biological as opposed to
21.2. BRANCHING PROCESSES 647
social contagion — not so much the new context as the new classes of models, based on
random processes in networks, that will be employed. In the next three sections, we discuss
some of the most basic probabilistic models for epidemics in networks; we then consider
how these models provide insight into some basic qualitative issues in the spread of disease,
including synchronization, timing, and concurrency in transmission. Finally, we discuss how
some of the models developed here are related to similar issues in genetic inheritance, where
a kind of randomized propagation takes place through genealogical networks.
Before moving on to this, it is worth noting that randomized models can also sometimes
be useful in studying social contagion, particularly in cases where the underlying decision
processes of the individuals are hard to model and hence more usefully abstracted as random.
Often the two approaches — decision-based and probabilistic — produce related results, and
they can sometimes be used in conjunction [62, 408]. Understanding the relationship between
these methodologies at a deeper level is an interesting direction for further research.
21.2 Branching Processes
We begin with perhaps the simplest model of contagion, which we refer to as a branching
process. It works as follows.
• (First wave.) Suppose that a person carrying a new disease enters a population, and transmits it to each person he meets independently with a probability of p. Further,
suppose that he meets k people while he is contagious; let’s call these k people the
first wave of the epidemic. Based on the random transmission of the disease from the
initial person, some of the people in the first wave may get infected with the disease,
while others may not.
• (Second wave.) Now, each person in the first wave goes out into the population and meets k di↵erent people, resulting in a second wave of k · k = k2 people. Each infected person in the first wave passes the disease independently to each of the k second-wave
people they meet, again independently with probability p.
• (Subsequent waves.) Further waves are formed in the same way, by having each person in the current wave meet k new people, passing the disease to each independently with
probability p.
Thus the contact network for this epidemic can be drawn as in Figure 21.1(a) (with k = 3
and only the first three waves shown). We refer to such a network as a tree: it has a single
node at the top called the root; every node is connected to a set of nodes in the level below
it; and every node but the root is also connected to a single node in the level above it. The
tree that forms the contact network for the branching process is in fact infinite, since we
continue defining waves indefinitely.
648 CHAPTER 21. EPIDEMICS
(a) The contact network for a branching process
(b) With high contagion probability, the infection spreads widely
(c) With low contagion probability, the infection is likely to die out quickly
Figure 21.1: The branching process model is a simple framework for reasoning about the spread of an epidemic as one varies both the amount of contact among individuals and the level of contagion.
21.2. BRANCHING PROCESSES 649
Now, what is the behavior of an epidemic in this model? We can picture the spread of
the epidemic by highlighting the edges of the contact network on which the disease passes
successfully from one person to another — recall that each of these infections happens
independently with probability p. Thus, Figure 21.1(b) shows an aggressive epidemic that
infects two people in the first wave, three in the second wave, five in the third wave, and
presumably more in future waves (not shown in the picture). Figure 21.1(c), on the other
hand, shows a much milder epidemic (for a less contagious disease, with a smaller value of
p): of the two people infected in the first wave, one doesn’t infect anyone else, and the other
infects only one further person who in turn doesn’t pass it on. This disease has completely
vanished from the population after the second wave, having infected only four people in
total.
The Basic Reproductive Number and a Dichotomy for Branching Processes. Our
last observation about Figure 21.1(c) reflects a fundamental property of branching processes:
if the disease in a branching process ever reaches a wave where it fails to infect anyone, then
it has died out: since people in future waves can only catch the disease from others higher
up in the tree, no one in any future wave will be infected either.
So there are really only two possibilities for a disease in the branching process model: it
reaches a wave where it infects no one, thus dying out after a finite number of steps; or it
continues to infect people in every wave, proceeding infinitely through the contact network.
And it turns out that there is a simple condition to tell these two possibilities apart, based
on a quantity called the basic reproductive number of the disease.
The basic reproductive number, denoted R 0
, is the expected number of new cases of the
disease caused by a single individual. Since in our model everyone meets k new people and
infects each with probability p, the basic reproductive number here is given by R 0
= pk.
The outcome of the disease in a branching process model is determined by whether the basic
reproductive number is smaller or larger than 1.
Claim: If R 0
< 1, then with probability 1, the disease dies out after a finite
number of waves. If R 0
> 1, then with probability greater than 0 the disease
persists by infecting at least one person in each wave.
We give a proof of this claim in Section 21.8. Even without the details of the proof, however,
we can see that the basic condition expressed in the claim — comparing R 0
to 1 — has a
natural intuitive basis. When R 0
< 1, the disease isn’t able to replenish itself: each infected
person produces less than one new case in expectation, and so — even if it grows briefly
due to the outcome of random fluctuations — the size of the outbreak is constantly trending
downward. When R 0
> 1, on the other hand, the size of outbreak is constantly trending
upward. Notice, however, that even when R 0
> 1, the conclusion is simply that the disease
persists with positive probability, not with absolute certainty: whenever p < 1, then there
650 CHAPTER 21. EPIDEMICS
is always some chance that none of the first few infected people will succeed in infecting
anyone else, causing the disease to die out. In other words, even an ultra-contagious disease
can simply get “unlucky” and vanish from the population before it has a chance to really
get going.
The dichotomy expressed by this condition has an interesting “knife-edge” quality to it
when R 0
is close to 1. In particular, suppose we have a branching process where R 0
is very
slightly below 1, and we increase the contagion probability p by a little bit; the result could
push R 0
above 1, suddenly resulting in a positive probability of an enormous outbreak.
The same e↵ect can happen in the reverse direction as well, where slightly reducing the
contagiousness of a disease to push R 0
below 1 can eliminate the risk of a large epidemic.
And since R 0
is the product of p and k, small changes in the number of people k that each
person comes into contact with can also have a large e↵ect when R 0
is near 1.
All this suggests that around the critical value R 0
= 1, it can be worth investing large
amounts of e↵ort even to produce small shifts in the basic reproductive number. Since R 0
is the product of the two terms p and k, it is in fact easy to interpret two basic kinds of
public-health measures in terms of reductions to R 0
: quarantining people, which reduces the
quantity k, and encouraging behavioral measures such as better sanitary practices to reduce
the spread of germs, which reduces the quantity p.
The branching process model is clearly a very simplified model of disease-spreading; the
structure of the contact network, with no triangles at all, is reminiscent of our first pass at
a model for the small-world phenomenon in Chapter 20. Thus, in the next few sections, we
will look at models that can handle more complex contact networks. For these models, a
dichotomy as simple as the one in the Claim above does not hold. However, the notion of the
basic reproductive number is still a useful heuristic guide to the behavior of more complex
models; even when epidemiological modelers do not have a precise condition governing when
an epidemic will persist and when it will die out, they find the reproductive number R 0
to
be a useful approximate indication of the spreading power of the disease.
21.3 The SIR Epidemic Model
We now develop an epidemic model that can be applied to any network structure. To
do this, we preserve the basic ingredients of the branching process model at the level of
individual nodes, but make the contact structure much more general. An individual node
in the branching process model goes through three potential stages during the course of the
epidemic:
• Susceptible: Before the node has caught the disease, it is susceptible to infection from its neighbors.
21.3. THE SIR EPIDEMIC MODEL 651
• Infectious: Once the node has caught the disease, it is infectious and has some proba- bility of infecting each of its susceptible neighbors.
• Removed: After a particular node has experienced the full infectious period, this node is removed from consideration, since it no longer poses a threat of future infection.
Using this three-stage “life cycle” for the disease at each node, we now define a model for
epidemics on networks. We are given a directed graph representing the contact network; so
an edge pointing from v to w in the graph means that if v becomes infected at some point,
the disease has the potential to spread directly to w. To represent a symmetric contact
between people, where either has the potential to directly infect the other, we can put in
directed edges pointing each way: both from v to w and also from w to v. Since contacts
between people are often symmetric, it is fine to use networks where most edges appear in
each direction, but it is sometimes convenient to be able to express asymmetric contacts as
well.
Now, each node has the potential to go through the Susceptible-Infectious-Removed
cycle, where we abbreviate these three states as S, I, and R. The progress of the epidemic
is controlled by the contact network structure and by two additional quantities: p (the
probability of contagion) and t I
(the length of the infection).
• Initially, some nodes are in the I state and all others are in the S state.
• Each node v that enters the I state remains infectious for a fixed number of steps t I
.
• During each of these t I
steps, v has a probability p of passing the disease to each of its
susceptible neighbors.
• After t I
steps, node v is no longer infectious or susceptible to further bouts of the
disease; we describe it as removed (R), since it is now an inert node in the contact
network that can no longer either catch or transmit the disease.
This describes the full model; we refer to it as the SIR model, after the three disease states
that nodes experience. Figure 21.2 shows an example of the SIR model unfolding on a
particular contact network through successive steps; in each step, shaded nodes with dark
borders are in the I state and shaded nodes with thin borders are in the R state.
The SIR model is clearly most appropriate for a disease that each individual only catches
once in their lifetime; after being infected, a node is removed either because it has acquired
lifetime immunity or because the disease has killed it. In the next section, we will consider
a related model for diseases that can be caught multiple times by the same person. Notice
also that the branching process model from Section 21.2 is a special case of the SIR model:
it simply corresponds to the SIR model where t I
= 1 and the contact network is an infinite
tree, with each node connected to a fixed number of neighbors in the level below.
652 CHAPTER 21. EPIDEMICS
y
x z
t
r v
u
w
s
(a)
y
x z
t
r v
u
w
s
(b)
y
x z
t
r v
u
w
s
(c)
y
x z
t
r v
u
w
s
(d)
Figure 21.2: The course of an SIR epidemic in which each node remains infectious for a number of steps equal to t
I
= 1. Starting with nodes y and z initially infected, the epidemic spreads to some but not all of the remaining nodes. In each step, shaded nodes with dark borders are in the Infectious (I) state and shaded nodes with thin borders are in the Removed (R) state.
Extensions to the SIR model. Although the contact network in the general SIR model
can be arbitrarily complex, the disease dynamics are still being modeled in a simple way.
Contagion probabilities are set to a uniform value p, and contagiousness has a kind of “on-o↵”
property: a node is equally contagious for each of the t I
steps while it has the disease.
However, it is not di�cult to extend the model to handle more complex assumptions.
First, we can easily capture the idea that contagion is more likely between certain pairs of
nodes by assigning a separate probability p v,w
to each pair of nodes v and w for which v
links to w in the directed contact network. Here, higher values of p v,w
correspond to closer
contact and more likely contagion, while lower values indicate less intensive contact. We
can also choose to model the infectious period as random in length, by assuming that an
infected node has a probability q of recovering in each step while it is infected, while leaving
21.3. THE SIR EPIDEMIC MODEL 653
Figure 21.3: In this network, the epidemic is forced to pass through a narrow “channel” of nodes. In such a structure, even a highly contagious disease will tend to die out relatively quickly.
the other details of the model as they are.
More elaborate extensions to the model involve separating the I state into a sequence
of several states (e.g. early, middle, and late periods of the infection), and allowing the
contagion probabilities to vary across these states [238]. This could be used, for example,
to model a disease with a highly contagious incubation period, followed by a less contagious
period while symptoms are being expressed. Researchers have also considered variations on
the SIR model in which the disease-causing pathogen is mutating (and thus changing its
disease characteristics) over the course of the outbreak [183].
The Role of the Basic Reproductive Number. We now discuss some observations
about the SIR model, focusing on the most basic version of the model in an arbitrary
network. First, let’s recall the claim made at the end of Section 21.2, that in networks
that do not have a tree structure, the simple dichotomy in epidemic behavior determined
by the basic reproductive number R 0
does not necessarily hold. In fact, it is not hard to
construct an example showing how this dichotomy breaks down. To do this, let’s start with
the network depicted in Figure 21.3, and suppose that these layers of two nodes at a time
continue indefinitely to the right. Let’s consider an SIR epidemic in which t I
= 1, the
infection probability p is 2/3, and the two nodes at the far left are the ones that are initially
infected.
When we don’t have a tree network, we need to decide how to define an analogue of
the basic reproductive number. In a network as highly structure as the one in Figure 21.3,
we can work directly from the definition of R 0
as the expected number of new cases of the
disease caused by a single individual. (For less structured networks, one can consider R 0
to be the expected number of new cases caused by a randomly chosen individual from the
population.) In Figure 21.3, each infected node has edges to two nodes in the next layer;
since it infects each with probability 2/3, the expected number of new cases caused by this
654 CHAPTER 21. EPIDEMICS
y
x z
t
r v
u
w
s
Figure 21.4: An equivalent way to view an SIR epidemic is in terms of percolation, where we decide in advance which edges will transmit infection (should the opportunity arise) and which will not.
node is 4/3.
So in our example, R 0
> 1. Despite this, however, it is easy to see that the disease
will die out almost surely after reaching only a finite number of steps. In each layer, there
are four edges leading to the next layer, and each will independently fail to transmit the
disease with probability 1/3. Therefore, with probability (1/3)4 = 1/81, all four edges will
fail to transmit the disease — and at this point, these four edges become a “roadblock”
guaranteeing the disease can never reach the portion of the network beyond them. Thus, as
the disease moves along layer-by-layer, there is a probability of at least 1/81 that each layer
will be its last. Therefore, with probability 1, it must come to an end after a finite number
of layers.
This is a very simple example, but it already indicates how di↵erent network structures
can be more or less conducive to the spread of a disease — even taking contagiousness and
other disease properties as given. Whereas the contact network of the simple branching
process from Section 21.2 was a tree that expanded rapidly in all directions, the network in
Figure 21.3 forces the disease to pass through a narrow “channel” in which a small break-
down in contagion can wipe it out. Understanding how specific types of network structure
interact with disease dynamics remains a challenging research question, and one that a↵ects
predictions about the course of real epidemics.
21.3. THE SIR EPIDEMIC MODEL 655
SIR Epidemics and Percolation. Thus far we have been thinking about SIR epidemics
as dynamic processes, in which the state of the network evolves step-by-step over time. This
captures the temporal dynamics of the disease itself as it spreads through a population.
Interestingly, however, there is an equivalent and completely static view of these epidemics
that is often very useful from a modeling point of view [44, 173].
We now describe how to arrive at this static view of the process, focusing on the basic
SIR model in which t I
= 1. Consider a point in an SIR epidemic when a node v has just
become infectious, and it has a susceptible neighbor w. Node v has one chance to infect
w (since t I
= 1), and it succeeds with probability p. We can view the outcome of this
random event as being determined by flipping a coin that has a probability p of coming up
“heads”, and observing the outcome. From the point of view of the process, it clearly does
not matter whether the coin was flipped at the moment that v first became infectious, or
whether it was flipped at the very beginning of the whole process and is only being revealed
now. Continuing this reasoning, we can in fact assume that for each edge in the contact
network — from a node v to a node w — a coin with “heads” probability p is flipped at the
very beginning of the process (independently of the coins for all other pairs of neighbors),
and the result is stored so that it can be later checked in the event that v becomes infectious
while w is susceptible.
With all the coins flipped in advance, the SIR process can be viewed as follows. The
edges in the contact network for which the coin flip is successful are declared to be open; the
remaining edges are declared to be blocked. The situation is now as pictured in Figure 21.4,
which shows a sample result of coin flips consistent with the pattern of infections in the
example from Figure 21.2. And we can now see how to make use of the open and blocked
edges to represent the course of the epidemic: A node v will become infected during the
epidemic if and only if there is a path to v from one of the initially infected nodes that
consists entirely of open edges. Thus, while Figure 21.4 looks superficially di↵erent from the
sequence of stages in Figure 21.2, it is in fact a beautifully compact way to summarize the
course of the epidemic: the nodes that are eventually infected are precisely those that can
be reached from the initially infected nodes along a sequence of open edges in the network.
This static view of the model is often referred to as percolation, due to the following
physical analogy. If we think of the contact network as a system of pipes, and the pathogen
as a fluid moving through these pipes, then the edges in the contact network on which
contagion succeeds are the “open” pipes and the edges on which it fails are the “blocked”
pipes. We now want to know which nodes the fluid will reach, given that it can only pass
through open pipes. In fact, this is not simply an illustrative metaphor; percolation is a
topic that has been extensively studied by physicists and mathematicians as a model for the
flow of fluids through certain types of porous media [69, 173]. It is both an interesting topic
in its own right, and useful for its role as an equivalent view of the progress of an epidemic.
656 CHAPTER 21. EPIDEMICS
v
u
w
(a)
v
u
w
(b)
v
u
w
(c)
v
u
w
(d)
v
u
w
(e)
Figure 21.5: In an SIS epidemic, nodes can be infected, recover, and then be infected again. In each step, the nodes in the Infectious state are shaded.
21.4 The SIS Epidemic Model
In the previous sections we have been considering models for epidemics in which each in-
dividual contracts the disease at most once. However, a simple variation on these models
allows us to reason about epidemics where nodes can be reinfected multiple times.
To represent such epidemics, we have nodes that simply alternate between two possible
states: Susceptible (S) and Infectious (I). There is no Removed state here; rather, after a
node is done with the Infectious state, it cycles back to the Susceptible state and is ready to
catch the disease again. Because of this alternation between the S and I states, we refer to
the model as the SIS model.
Aside from the lack of an R state, the mechanics of the model follow the SIR process
very closely.
• Initially, some nodes are in the I state and all others are in the S state.
• Each node v that enters the I state remains infectious for a fixed number of steps t I
.
• During each of these t I
steps, v has a probability p of passing the disease to each of its
susceptible neighbors.
• After t I
steps, node v is no longer infectious, and it returns to the S state.
Figure 21.5 shows an example of the SIS model unfolding on a three-node contact network
with t I
= 1. Notice how node v starts out infected, recovers, and later becomes infected
again — we can imagine this as the contact network within a three-person apartment, or a
three-person family, where people pass a disease on to others they’re living with, and then
get it back from them later.
As with the SIR model, the SIS model can be extended to handle more general kinds of
assumptions: di↵erent contagion probabilities between di↵erent pairs of people; probabilistic
21.4. THE SIS EPIDEMIC MODEL 657
recovery from the disease, in which each infected node transitions back to the susceptible
state with probability q each step; and multiple stages of infection, with varying disease
properties across them.
Life Cycles of SIR and SIS Epidemics. The examples in this section and preceding
one suggest that the overall “trajectories” of SIR and SIS epidemics on (finite-size) graphs
are qualitatively quite di↵erent. An SIR epidemic on a finite graph is burning through a
bounded supply of nodes — since nodes can never be reinfected — and therefore it must
come to an end after a relatively small number of steps. An SIS epidemic, on the other
hand, can run for an extremely long time as it cycles through the nodes potentially multiple
times. But as Figure 21.5(e) illustrates, if there ever comes a point in an SIS epidemic when
all nodes are simultaneously free of the disease, then the epidemic has died forever: there
are no longer any infected individuals to pass the disease to others. And on a finite graph,
there will eventually (with probability 1) come a point in time when all contagion attempts
simultaneously fail for t I
steps in a row, and at this point it will be over. Thus a key question
with an SIS epidemic on a given contact network is to understand how long the outbreak
will last, and how many individuals will be a↵ected at di↵erent points in time.
For contact networks where the structure is mathematically tractable, researchers have
in fact proved “knife-edge” results for the SIS model similar to our dichotomy for branching
processes. These results, on particular classes of contact networks, show that at a particular
critical value of the contagion probability p, an SIS epidemic on the network will undergo a
rapid shift from one that dies out quickly to one that persists for a very long time [52, 278].
This type of analysis tends to be mathematically quite complex, with this critical value of
the contagion probability p depending in subtle ways on the structure of the network.
A Connection Between SIR and SIS Epidemics. Despite the di↵erences between the
SIR and SIS models, in fact it is possible to represent some of the basic variants of the SIS
model as special cases of the SIR model. This surprising relationship is further evidence of
the flexibility of the basic epidemic models, in which formalisms defined in di↵erent ways
turn out to have very close connections to each other.
We describe the relationship for the SIS model with t I
= 1, when each node is infectious
for a single step before recovering. The key insight is that if we think about a node v as in
fact being a “di↵erent individual” at each time step, then we can represent things so that
nodes are never reinfected. Specifically, given an instance of the SIS model with t I
= 1, we
create a separate copy of each node for each time step t = 0, 1, 2, 3 and onward. We will call
this the time-expanded contact network. Now, for each edge in the original contact network,
linking a node v to a node w, we create edges in the time-expanded contact network from
the copy of v at time t to the copy of w at time t + 1; this simply encodes the idea that w
658 CHAPTER 21. EPIDEMICS
v
u
w
v
u
w
v
u
w
v
u
w
v
u
w
step 0 step 1 step 2 step 3 step 4
(a) To represent the SIS epidemic using the SIR model, we use a “‘time-expanded” contact network
v
u
w
v
u
w
v
u
w
v
u
w
v
u
w
step 0 step 1 step 2 step 3 step 4
(b) The SIS epidemic can then be represented as an SIR epidemic on this time-expanded network.
Figure 21.6: An SIS epidemic can be represented in the SIR model by creating a separate copy of the contact network for each time step: a node at time t can infect its contact neighbors at time t + 1.
can potentially catch the disease at time t + 1 if v is infected at time t. Figure 21.6(a) shows
this construction applied to the contact network from Figure 21.5.
The point is that the same SIS disease dynamics that previously circulated around in the
original contact network can now flow forward in time through the time-expanded contact
network, with copies of nodes that are in the I state at time t producing new infections in
copies of nodes at time t + 1. But on this time-expanded graph we have an SIR process,
since any copy of a node can be treated as removed (R) once its one time step of infection
is over; and with this view of the process, we have the same distribution of outcomes as the
original SIS process. Figure 21.6(b) shows the course of the SIR epidemic that corresponds
to the SIS epidemic in Figure 21.5.
21.5. SYNCHRONIZATION 659
21.5 Synchronization
The models we’ve developed give us a framework for thinking about various broader issues
in the spread of disease. We already encountered one of these issues in the dichotomy for
branching processes, which provided a formal basis for the sensitivity of outbreaks to small
variations in contagiousness, and for the crucial role of the basic reproductive number. We
now look at a related issue in the global dynamics of a disease — the tendency of epidemics for
certain diseases to synchronize across a population, sometimes producing strong oscillations
in the number of a↵ected individuals over time. Such e↵ects are well-known for diseases
including measles [196, 213] and syphilis [195].
When looking at public-health data, it is natural to look at periodic oscillations in the
number of cases of a disease and to try positing external causes for the e↵ect. For example,
cycles in the prevalence of syphilis across the U.S. over the past 50 years have traditionally
been attributed to large-scale societal changes, including changes in sexual mores and other
forces [195]. While such factors clearly play a role, recent research has shown that oscillations
and synchronization over time can in fact result largely from the contagion dynamics of the
disease itself, and that similar patterns can be created in direct simulations of the disease
using the types of models we have been considering here [195, 267].
We now describe how such e↵ects can be produced using simple epidemic models. The
crucial ingredients appear to be a combination of temporary immunity and long-range links
in the contact network. Roughly, long-range links produce coordination in the timing of
flare-ups across dispersed parts of the network; when these subside, the temporary immunity
produces a network-wide deficit in the number and connectivity of susceptible individuals,
yielding a large “trough” in the size of the outbreak that directly follows the “peak” from the
earlier flare-ups. We now describe how to make this intuitive picture concrete using simple
models.
The SIRS Epidemic Model. The first step in producing a model with oscillations is to
allow the disease to confer temporary but not permanent immunity on infected individuals
— a feature of many real diseases. To do this, we combine elements of the SIR and SIS
models in a simple way, so that after an infected node recovers, it passes briefly through
the R state on its way back to the S state. We call the resulting model the SIRS model
[267], since nodes pass through the sequence S-I-R-S as the epidemic proceeds. In detail,
the model works as follows.
• Initially, some nodes are in the I state and all others are in the S state.
• Each node v that enters the I state remains infectious for a fixed number of steps t I
.
• During each of these t I
steps, v has a probability p of passing the disease to each of its
susceptible neighbors.
660 CHAPTER 21. EPIDEMICS
• (The new feature of the model.) After t I
steps, node v is no longer infectious. It then
enters the R state for a fixed number of steps t R
. During this time, it cannot be infected
with the disease, nor does it transmit the disease to other nodes. After t R
steps in the
R state, node v returns to the S state.
For an SIRS epidemic, the course of the disease through a population is clearly a↵ected not
just by the quantities p and t I
, but also by the length t R
of the temporary immunity that is
conferred.
Small-World Contact Networks. Temporary immunity can produce oscillations in very
localized parts of the network, with patches of immunity following large numbers of infections
in a concentrated area. But for this to produce large fluctuations that can be seen at the
level of the full network, the flare-ups of the disease have to be coordinated so that they
happen at roughly the same time in many di↵erent places. A natural mechanism to produce
this kind of coordination is to have a network that is rich in long-range connections, linking
otherwise far-apart sections of the network.
This kind of structure is familiar from our discussion of small-world properties in Chap-
ter 20. There, we considered network models where many of the links were “local” and
clustered — connecting nodes with very similar social and geographic characteristics, ac-
cording to the principle of homophily — while some were long-range links, corresponding
to weak ties that link very di↵erent parts of the network. In Chapter 20 we focused on
the e↵ect this kind of structure has on the distances between nodes. But there is a closely
related consequence: long-range links make it possible for things that happen in one part of
the network to quickly a↵ect what is happening elsewhere.
Watts and Strogatz observed the relevance of small-world properties to synchronization
in their original paper on the topic [411], and Kuperman and Abramson showed how it could
naturally lead to synchronization and oscillation in epidemics [267]. For their analysis they
constructed random networks with small-world properties, in a manner very similar to the
grid-plus-random-edges construction discussed in Chapter 20; instead of the version from
that chapter, they more closely followed the original construction of Watts and Strogatz, in
which a ring network is rewired to produce random shortcuts [411]. Specifically, they started
with a graph in which the nodes are arranged in a ring, and each node is connected to its
neighbors for some number of steps in each direction. These are all homophilous links, in
that they connect nodes that are very close together on the ring. Then, independently with
some probability c, they turned each edge into a weak tie by rewiring one end of it to a node
chosen uniformly at random. Thus, the probability c controls the fraction of links in the
network that serve as long-range weak ties.
When the SIRS model is run on this kind of network, one finds very di↵erent behavior
depending on the value of c, as indicated in Figure 21.7. When c is very small, disease
21.5. SYNCHRONIZATION 661
Figure 21.7: These plots depict the number of infected people over time (the quantity n inf
(t) on the y-axis) by SIRS epidemics in networks with di↵erent proportions of long-range links. With c representing the fraction of long-range links, we see an abscence of oscillations for small c (c = 0.01), wide oscillations for large c (c = 0.9), and a transitional region (c = 0.2) where oscillations intermittently appear and then disappear. (Results and image from [267].)
transmission through the network occurs mainly via the short-range local edges, and so
flare-ups of the disease in one part of the network never become coordinated with flare-ups
in other parts. As c increases, these flare-ups start to synchronize, and since each burst
produces a large number of nodes with temporary immunity, there is a subsequent trough
as the disease has di�culty making its way through the sparser set of available targets. For
very large values of c (such as c = 0.9 in Figure 21.7), there are clear waves in the number
of a↵ected individuals; for intermediate values of c (such as c = 0.2) one observes interesting
e↵ects in which the system achieves network-wide synchronization for a period, and then
seems to fall back “out of sync” for reasons that are hard to quantify.
These results show how fairly complex epidemic dynamics can arise from simple models of
contagion and contact structure. There are, however, a number of interesting open questions;
662 CHAPTER 21. EPIDEMICS
the results discussed here have been primarily found through simulation, and analyzing the
onset of synchronization mathematically in this model remains largely unexplored.
Sychronization in Epidemic Data. It is possible to study these e↵ects empirically —
and evaluate proposed models — using extensive records of disease prevalence that reach
back many years. Grassly, Fraser, and Garnett [195] performed an instructive compari-
son of syphilis and gonorrhea that illustrates a number of synchronization principles. The
prevalence of syphilis exhibits prominent oscillations on an 8-11-year cycle, while gonor-
rhea exhibits very little in the way of periodic behavior. Yet the two diseases a↵ect similar
populations, and are presumably subject to very similar societal forces.
These di↵erences are consistent, however, with the fact that syphilis confers limited
temporary immunity after infection, while gonorrhea does not. Moreover, the timing of the
syphilis cycles fit well with the timing of the immune properties associated with it. And from
the cyclic patterns, one finds that the extent of synchronization between di↵erent regions
of the United States increases over time, suggesting that the contact network on which it
spread became increasingly connected with cross-country links over the second half of the
20th century [195].
There are many further directions in which research on epidemic synchronization is pro-
ceeding, including attempts to model more complex temporal phenomena. For example,
data for some diseases such as measles shows that epidemics in di↵erent cities can synchro-
nize so as to be out of phase, with the flare-ups in one city consistently coinciding with
troughs in the other [196]. One needs more than simply long-range contacts to explain such
properties [213]. There is also the question of how immunization, prevention programs, and
other medical interventions can take advantage of these timing properties — another way in
which insights from even simple models can help to inform decision-making in this area.
21.6 Transient Contacts and the Dangers of Concur- rency
Thus far, our epidemic models have taken the underlying contact network to be a relatively
static object, in which all the links in the contact network are present throughout the course
of the epidemic. This is a reasonable simplifying assumption for diseases that are relatively
contagious and spread quickly, at a rate faster than the typical creation or dissolution of a
contact.
But as we move down the spectrum toward diseases that spread through a population over
longer time scales, it is useful to revisit these assumptions. For a disease like HIV/AIDS, the
epidemic progresses over many years, and its course is heavily dependent on the properties
of the sexual contact network. Most people have zero, one, or very few contacts at any single
21.6. TRANSIENT CONTACTS AND THE DANGERS OF CONCURRENCY 663
v
u x
w
y
[1,5]
[7,11]
[2,6]
[12,16]
(a) In a contact network, we can annotate the edges with time windows during which they existed.
v
u x
w
y
[1,5]
[12,16]
[2,6]
[7,11]
(b) The same network as in (a), except that the timing of the w-v and w-y partnerships have been reversed.
Figure 21.8: Di↵erent timings for the edges in a contact network can a↵ect the potential for a disease to spread among individuals. For example, in (a) the disease can potentially pass all the way from u to y, while in (b) it cannot.
point in time (a few people have many, which is important as well); and the identities of
these contacts can shift significantly while the disease progresses, as new sexual partnerships
are formed and others break up.
So for modeling the contact network in such diseases, it is important to take into account
the fact that contacts are transient — they do not necessarily last through the whole course
of the epidemic, but only for particular windows of time. Thus, we will consider contact
networks in which each edge is annotated with the period of time during which it existed
— that is, the time range over which it was possible for one endpoint of the edge to have
passed the disease directly to the other.
Figure 21.8(a) shows an example of this, with the numbers inside square brackets indi-
cating the time ranges when each edge exists. Thus the u-v and w-x partnerships happen
first, and they overlap in time; after this, w has a partnership with v and then later with y.
Note also that for this section — in keeping with the motivation from HIV/AIDS and similar
diseases — we assume the edges to be undirected rather than directed, to indicate that in-
fection can pass in either direction between a pair of people in a partnership. (As in previous
sections, we could also accomplish this by having directed edges pointing in both directions
between each pair of connected people, but since everything here will be symmetric, it is
more convenient to use undirected edges.)
The Consequences of Transient Contacts. A little experimentation with the example
in Figure 21.8(a) indicates how the timing of di↵erent edges can a↵ect the spread of a disease.
664 CHAPTER 21. EPIDEMICS
vu w
[1,5] [6,10]
(a) v’s two partnerships happen serially
vu w
[1,5] [2,6]
(b) v’s two partnership’s happen concurrently
Figure 21.9: A disease tends to be able to spread more widely with concurrent partnerships (b) than with serial partnerships (a).
For example, if u has the disease at time 1, it is possible for it to spread all the way to y,
through v and w as intermediaries. (Of course, if contagion is probabilistic as before, it will
not necessarily succeed in spreading; but it has the potential to do so.) On the other hand,
u cannot spread the disease to x: node u could pass the disease to v, who could pass it to
w; but by the time it reaches w, the partnership of w and x is long over.
Moreover, changing the timing of partnerships can change the possible transmission path-
ways, even as the set of underlying contacts remains the same. For instance, the example in
Figure 21.8(b) di↵ers from the one in Figure 21.8(a) only in that the temporal order of the
w-v and w-y partnerships has been reversed. But notice that while u was able to pass the
disease all the way to y in Figure 21.8(a), it cannot do so in Figure 21.8(b): in the latter
case, the w-y partnership is over by the time the disease could possibly get from u to w.
Such considerations are crucial as health workers and epidemiologists map out the contact
networks associated with a disease such as HIV/AIDS. For example, we can see from the
di↵erence between Figures 21.8(a) and 21.8(b) that in order for y to know whether he or she
is at risk from a disease carried by u, it is not enough even to map out the full set of sexual
partnerships; it is crucial to know information about the order of events as well. Or if we
go back to the striking Figure 2.7 from Chapter 2, mapping out the relationships within a
high school, we can appreciate that the image itself is not enough to fully chart the potential
spread of diseases through this population — we would also need to know the timing of these
relationships.
Networks in which the edges only exist for specific periods of time have been the subject of
modeling e↵orts in many areas, including sociology [182, 305, 258], epidemiology [307, 406],
mathematics [106], and computer science [53, 239]. It is an issue that is relevant not just to
the spread of disease, but also to a wide range of settings that are modeled by networks. For
example, the di↵usion of information, ideas, and behaviors through social networks clearly
also depends on how the timing of di↵erent communications between people either enables
or blocks the flow of information to di↵erent parts of the population.
Concurrency. Di↵erences in the timing of contacts do not just a↵ect who has the potential
to spread a disease to whom; the pattern of timing can influence the severity of the overall
21.6. TRANSIENT CONTACTS AND THE DANGERS OF CONCURRENCY 665
v
u x
w
y
[1,5]
[12,16]
[2,6]
[7,11]
(a) No node is involved in any concurrent partner- ships
v
u x
w
y
[1,5]
[3,7]
[2,6]
[1,5]
(b) All partnerships overlap in time
Figure 21.10: In larger networks, the e↵ects of concurrency on disease spreading can become particularly pronounced.
epidemic itself. A timing pattern of particular interest — and concern — to HIV researchers
is concurrency [307, 406].
A person is involved in concurrent partnerships if he or she has two or more active
partnerships that overlap in time. For example, in each of Figures 21.9(a) and 21.9(b), node
v has partnerships with each of u and w. But in the first of these figures, the partnerships
happen serially — first one, then the other — while in the second, they happen concurrently,
overlapping in time. The concurrent pattern causes the disease to circulate more vigorously
through this three-person network. u and w may not be aware of each other’s existence,
but the concurrent partnerships make it possible for either of u or w to spread the disease
to the other; the serial partnerships only allow spreading from u to w, but not the other
way. In larger examples one can find more extreme e↵ects; for example, Figure 21.10(b)
di↵ers from Figure 21.10(a) only in that the time windows of the partnerships have been
“pushed together” so that they all overlap. But the e↵ect is considerable: where the pattern
in Figure 21.10(a) allowed di↵erent parts of the network to be “walled o↵” from each other
by the timing e↵ects, the concurrent partnerships make it possible for any node with the
disease to potentially spread it to any other.
In simulations with various notions of concurrency, Morris and Kretzschmar found that
small changes in the amount of concurrency — keeping other variables like the average
number and duration of partnerships fixed — could produce large changes in the size of
the epidemic [307]. Qualitatively, this aligns well with the intuition from earlier sections,
that changing the average number of new cases of a disease caused by an infected individual
even slightly can sometimes have significant consequences. For some of the simplest models,
such as the branching process, it is possible to make this intuition precise; for more complex
666 CHAPTER 21. EPIDEMICS
models such as the present one involving concurrency in arbitrary networks, it remains the
topic of ongoing research.
Concurrency is just one particular kind of pattern to be found in the timing of rela-
tionships in a contact network. Further research in this area could possibly uncover more
subtle patterns as well; the interaction of timing and network structure has the potential to
provide further insights into the way diseases spread through the changing contacts within
a population.
21.7 Genealogy, Genetic Inheritance, and Mitochon- drial Eve
Our discussion of epidemics has provided us with a way of thinking about processes that
spread randomly over time through a network. As mentioned earlier, this is a useful frame-
work for modeling many kinds of things that spread, not just diseases. The spread of
information can be modeled this way, as an alternative to the approaches based on explicit
decision rules discussed in Chapter 19. In such settings, adapting the ideas from this chapter
can be relatively straightforward but still very informative.
In this section, instead, we apply the perspective of random spreading to a situation
where the connection is at first a bit more subtle; it takes a little work to precisely identify
the network and the process that is spreading through it. The setting is that of genetic
inheritance. What we will find is that when we view inheritance of traits as a random
process that takes place on a network linking organisms in successive generations — in other
words, with edges connecting parents to their o↵spring — then we can obtain insight into
some fundamental hereditary processes. We start with a story that illustrates some of the
basic genetic issues we’ll consider.
Mitochondrial Eve. In 1987, Rebecca Cann, Mark Stoneking, and Allan Wilson pub-
lished a paper in the journal Nature [94] where they provided evidence for a rather striking
proposition. Consider following your maternal ancestry backward in time through human
history, producing a trail that goes from you to your mother, to her mother (i.e. your mater-
nal grandmother), to her mother, and so on indefinitely. Each of us in principle can produce
such a maternal ancestry trail, which we’ll call a maternal lineage. Now, the claim of Cann,
Stoneking, and Wilson was that all these lineages in fact meet at a single woman who lived
between 100,000 and 200,000 years ago, probably in Africa. She is at the root of all our
maternal ancestries.
Let’s first ask how they reached this conclusion, and then consider what it signifies.
One way to infer facts about maternal ancestries is to study the DNA found not in our
cells’ nuclei, but in the much smaller, separate genome that each of us has in our cells’
21.7. GENEALOGY, GENETIC INHERITANCE, AND MITOCHONDRIAL EVE 667
mitochondria. Unlike nuclear DNA, which contains parts of both our parents’ genomes,
this mitochondrial DNA is (to a first approximation) passed to children entirely from their
mothers. So roughly speaking, aside from random mutations, you have your mother’s DNA,
she has her mother’s DNA, and so on through your maternal ancestry. With this in mind,
Cann, Stoneking, and Wilson analyzed the mitochondrial DNA of people drawn from a wide
sample of geographic and ethnic backgrounds; using standard techniques to estimate the rate
at which genetic sequences will diverge through random mutations over many generations,
they concluded that all the mitochondrial DNA in this population likely had a common
origin roughly 100,000-200,000 years ago. By “common origin” here, we mean a single
mitochondrial genome belonging to a single human being; because she is the source of the
mitochondrial DNA of everyone on earth, researchers standardly refer to this woman as
Mitochondrial Eve.
This finding caught the public imagination when it was first announced; it received a
fair amount of media attention at the time, and its implications have been nicely explored
in general books about human history [333]. The analysis involved in the original finding
has since been refined by a number of other research groups; caveats have been introduced
due to the fact that the inheritance of mitochondrial DNA may be more complicated than
originally thought; but the basic conclusion has been mainly accepted at a general level.
As to what this finding signifies: on first hearing, it takes a bit of thought to sort out
what it implies and what it doesn’t. It is indeed striking to be able to posit the existence
of a single person from the not-so-distant evolutionary past who is an ancestor of everyone.
Mitochondrial Eve (in contrast to her namesake Eve from the Bible) was not asserted to be
the only living woman in her time; there were presumably many other women living at the
same time as her, but from the point of view of present-day mitochondrial DNA, all these
women are genetically irrelevant: somewhere along the line from then to now, each of their
lines of mitchondrial DNA died out.
On the other hand, one also needs to be careful before attributing too much to the
relatively recent existence of Mitochondrial Eve. In particular, while her contemporaries were
genetically irrelevant to our mitochondrial DNA, they are not irrelevant to the remainder of
our genomes; each of us has genetic contributions from a large number of ancestors. (Though
even here there is more going on than meets the eye, as we’ll discuss shortly.) Moreover, the
overlapping patterns of our respective ancestries are complex and still not well understood;
what we learn from Mitochondrial Eve is that all our ancestries are pinned together along
their maternal lines, a couple of hundred thousand years into the past.
Ultimately, the identification of Mitochondrial Eve was in a sense a showcase of ideas
that had been emerging in the genetics community over the previous decade [245, 325].
These ideas were based on models that could predict the existence of common ancestors
and make estimates about their recency. And they showed that at a mathematical level,
668 CHAPTER 21. EPIDEMICS
current generation
new generation
each offspring comes from
a single parent chosen
uniformly at random
Figure 21.11: In the basic Wright-Fisher model of single-parent ancestry, time moves step- by-step in generations; there are a fixed number of individuals in each generation; and each o↵spring in a new generation comes from a single parent in the current generation.
independent of the di�culty of establishing evidence from genetic data, the existence of
someone like Mitochondrial Eve was not only natural, but in fact — as we will see next —
essentially inevitable. At their core, these models were built from a probabilistic formal-
ism involving networks; indeed, even in a qualitative sense, one can appreciate something
epidemic-like about the way in which copies of di↵erent people’s mitochondrial DNA spread
through subsequent generations, inhabiting future o↵spring, until one eventually crowds out
all the others. We now describe the basic versions of these models, and how they connect to
questions about ancestry.
A Model of Single-Parent Ancestry We use a fundamental model of ancestry known
in population genetics as the Wright-Fisher model [325]. To remain tractable, the model
involves a number of simplifying assumptions. Consider a population that is constrained
by resources to maintain a fixed size N in each generation. Time moves step-by-step from
one generation to the next; each new generation is formed by having the current set of N
individuals produce N o↵spring in total. Each o↵spring in this new generation is produced
from a single parent, and this parent is selected independently and uniformly at random
from among those in the current generation. Figure 21.11 depicts this process; as shown
there, we can draw the relationship of one generation to the next as a graph, with a node for
each individual, and an edge connecting each o↵spring to their parent chosen uniformly at
random from the previous generation. Notice that because of this rule for selecting parents,
certain individuals in the upper generation can have multiple children (such as the first and
last in Figure 21.11), while others may have none.
21.7. GENEALOGY, GENETIC INHERITANCE, AND MITOCHONDRIAL EVE 669
s t u v w x y z
Figure 21.12: We can run the model forward in time through a sequence of generations, ending with a set of present-day individuals. Each present-day individual can then follow its single-parent lineage by following edges leading upward through the network.
The structure of this model reflects a few underlying assumptions. To begin with, we’re
assuming a neutral model in which no individual has a selective advantage in reproduction;
everyone has the same chance of producing o↵spring. Furthermore, we’re modeling a situa-
tion in which each individual is produced from a single parent, as opposed to two parents in
a sexually reproducing population. This is consistent with several possible interpretations.
• First, and most directly, it can be used to model species that engage in asexual repro- duction, with each organism arising from a single parent.
• Second, it can be used to model single-parent inheritance even in sexually reproducing populations, including the inheritance of mitochondrial DNA among women as in our
discussion above. In this interpretation, each node represents a human woman, with
women linked to their mothers in the previous generation. Moreover, as we will discuss
later, there is in fact a much more general way to use this model to think about
inheritance in sexually reproducing populations.
670 CHAPTER 21. EPIDEMICS
s t u v w x y z
Figure 21.13: A re-drawing of the single-parent network fom Figure 21.12. As we move back in time, lineages of di↵erent present-day individuals coalesce until they have all converged at the most recent common ancestor.
• Third, it can be used to model purely “social” forms of inheritance, such as master- apprentice relationships. For example, if you receive a Ph.D. in an academic field, you
generally have a single primary advisor. If you model students as being “descended”
from advisors, than we can trace ancestries through sequences of advisors back into
the past — just as we traced maternal lineages.
Now, if we run this model forward in time through multiple generations, we get a network
such as the one pictured in Figure 21.12. Each individual is connected to one parent in the
previous generation; time runs from top to bottom, with N present-day individuals in the
lowest layer (named s through z in the figure). Notice that from any one of these individuals
at the bottom, we can trace its single-parent lineage backward in time by following edges
upward, always taking the single edge leading up out of each node we encounter.
If we imagine the individuals in the bottom row of Figure 21.12 to be present-day women,
then Mitochondrial Eve would be the lowest node in the figure where all the maternal lineages
first fully converge. It’s a bit tricky, visually, to find this node in Figure 21.12, but we can re-
21.7. GENEALOGY, GENETIC INHERITANCE, AND MITOCHONDRIAL EVE 671
draw the same ancestries with the paths unscrambled in Figure 21.13, and then the location
of Eve becomes easy to see: she’s the third node in the second row from the top (with the
lineages leading back to her consisting of the darkened edges).
These examples indicate how the existence of common ancestors — and even the number
of generations needed to reach them — can be predicted from the Wright-Fisher model. To
do this, we employ a useful trick for reasoning about the model: we think of the ancestries
as being built backward in time, rather than forward. In other words, an equivalent view
of the model is to take a set of present-day individuals, and construct earlier generations
one at a time by having each current individual choose its parent in the previous generation
independently and uniformly at random.
We can see how this works by moving upward through the levels of Figure 21.13. When-
ever two individuals happen to choose the same parent, then their lineages coalesce into a
common lineage from that point onward. Thus, we start with N distinct lineages in the
present, but as we build generations going backward in time, the number of distinct lineages
decreases whenever individuals on distinct lineages choose the same parent. This coalescence
will happen rapidly at first, when there are many lineages and the probability of a collision
between lineages is high; as time goes on, the number of distinct lineages of present-day indi-
viduals shrinks more and more slowly. But whenever there is more than one distinct lineage,
there is a finite expected time until two of them collide, and so the process must eventually
reach a single lineage. The node at which this first happens is called the most recent common
ancestor — the analogue of Mitochondrial Eve in this model. The model is simple enough
that one can estimate the expected time until the collisions among lineages, and hence the
expected number of generations to the most recent common ancestor [245, 325].
Genetic Interpretations. Although the maternal inheritance of mitochondrial DNA makes
for a very simple single-parent process, the Wright-Fisher model is relevant to sexually re-
producing populations for a much more fundamental reason. While the chromosomes of
your parents recombined to produce your genome, making your chromosomes a patchwork
of theirs, any single point in your genome — a single nucleotide on one of your chromosomes
— was inherited from just one of your mother or father. They, in turn, inherited it from just
one of their mother or father, and so on. As a result, if we want to trace the ancestry of a
single point in your genome, we are following a single-parent lineage, even though o↵spring
are produced by sexual reproduction. The most recent common ancestor for this particular
point, looking across a population of N individuals, will thus follow from the same analysis
we’ve seen above, as it did for mitochrondrial DNA.
Because of recombination, the lineages for one point in the genome may di↵er from the
lineages for even a nearby point, and hence the most recent common ancestors may di↵er as
well. One can develop probabilistic models for how these lineages relate to each other, but
672 CHAPTER 21. EPIDEMICS
the analysis becomes much more complex [418].
There are many other issues that arise when extending these simplified models to more
complex genetic applications. For example, geographic barriers in a population can isolate
individuals from each other, and this can have an e↵ect on the patterns of interaction among
lineages [354]. More generally, spatial constraints on the interactions among individuals can
a↵ect these patterns, providing another setting in which network properties can potentially
inform broader conclusions about genetic outcomes.
21.8 Advanced Material: Analysis of Branching and Coalescent Processes
In this section, we analyze two of the basic processes discussed in this chapter: the branching
process for the spread of an epidemic with simplified contact network structure, and the
coalescent process for the merging of lineages back to a common ancestor. Both of these are
based on probabilistic reasoning involving branching tree structures: the first as the epidemic
spreads forward through individuals, and the second as the lineages travel backward in time.
A. Analysis of Branching Processes
Recall the branching process model that we considered in Section 21.2: each infected indi-
vidual meets k others and infects each with probability p. Thus, the expected number of
new cases of the disease caused by each infected individual is R 0
= pk, the basic reproductive
number. We want to show that the persistence of the disease depends critically on whether
R 0
is smaller or larger than 1, a notion that we will formulate as follows.
Recall that the population in this model is organized into a tree (as shown in Fig-
ure 21.1(a)) in which every node is connected to k nodes just below it. Let q n
denote
the probability that the epidemic survives for at least n waves — in other words, that some
individual in the nth level of the tree becomes infected. Let q⇤ be the limit of q n
as n goes
to infinity; we can think of this as the probability that the disease persists indefinitely. We
will prove the following claim.
Claim: (a) If R 0
< 1 then q⇤ = 0. (b) If R 0
> 1 then q⇤ > 0.
This establishes the “knife-edge” quality of R 0
that we discussed in Section 21.2.
The Expected Number of Infected Individuals. We start by considering an approach
to this problem that gets us partway to a proof of the claim: considering the expected number
of infected individuals at each level of the tree.
First, let’s consider the total number of individuals at each level. The number of indi-
viduals at any given level exceeds the number at the previous level by a factor of k, and
21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES673
Individual j is infected if each
contact from the root to j
successfully transmits the disease
j
Figure 21.14: To determine the probability that a particular node is infected, we multiply the (independent) probabilities of infection on each edge leading from the root to the node.
therefore the number who are at level n is kn. (This is also true at level n = 0: the top level
consists of just the root, and k0 = 1.)
Now, let X n
be a random variable equal to the number of infected individuals at level
n. One way to think about the expected value E [X n
] is to write X n
as a sum of simpler
random variables as follows. For each individual j at level n, let Y nj
be a random variable
equal to 1 if j is infected, and equal to 0 otherwise. Then
X n
= Y n1
+ Y n2
+ · · · + Y nm
,
where m = kn, since the right-hand side simply counts up, one-by-one, the number of infected
individuals at level n. Linearity of expectation says that the expectation of the sum of a set
of random variables is equal to the sum of their expectations, and so
E [X n
] = E [Y n1
+ Y n2
+ · · · + Y nm
] = E [Y n1
] + E [Y n2
] + · · · + E [Y nm
] . (21.1)
The reason to write things this way is that each expectation on the right-hand side is ex-
tremely easy to work out: E [Y nj
] = 1 · Pr [Y nj
= 1] + 0 · Pr [Y nj
= 0] = Pr [Y nj
= 1], and so
the expectation of each Y nj
is just the probability that individual j gets infected.
674 CHAPTER 21. EPIDEMICS
k individuals
n
n
probability p of each being infected
Figure 21.15: The expected number of individuals infected at level n is the product of the number of individuals at that level (kn) and the probability that each is infected (pn).
Individual j at depth n gets infected precisely when each of the n contacts leading
from the root to j successfully transmit the disease, as shown in Figure 21.14. Since each
contact transmits the disease independently with probability p, individual j is infected with
probability pn. Therefore E [Y nj
] = pn. We have already concluded that there are kn
individuals at level n of the tree, and hence kn terms on the right-hand side of Equation
(21.1). Therefore, as summed up in Figure 21.15, we conclude that
E [X n
] = pnkn = (pk)n = Rn 0
. (21.2)
From Expected Values to Probabilities of Persistence. Equation (21.2) suggests
the importance of the basic reproductive number R 0
in reasoning about the spread of an
epidemic in the branching process model. Now let’s consider what this tells us about q⇤, the
probability that the epidemic persists indefinitely.
First, the fact that E [X n
] = Rn 0
immediately establishes part (a) of the Claim, that when
R 0
< 1 we have q⇤ = 0. To see why, we go back to the definition of E [X n
] and apply a fact
that we also found useful in Section 20.7. To recap the discussion there, the definition for
the expected value is
E [X n
] = 1 · Pr [X n
= 1] + 2 · Pr [X n
= 2] + 3 · Pr [X n
= 3] + · · · (21.3)
21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES675
and an alternate but equivalent way to write the right-hand side is as
Pr [X n
� 1] + Pr [X n
� 2] + Pr [X n
� 3] + · · · (21.4)
since we observe that each term Pr [X n
= i] contributes exactly i copies of itself to the sum
in (21.4). Therefore we have
E [X n
] = Pr [X n
� 1] + Pr [X n
� 2] + Pr [X n
� 3] + · · · (21.5)
From (21.5) we observe that E [X n
] must be at least as large as the first term on the
right-hand side, and so E [X n
] � Pr [X n
� 1]. Notice also that Pr [X n
� 1] is precisely the definition of q
n
, and so E [X n
] � q n
. But E [X n
] = Rn 0
which is converging to 0 as n grows,
and hence q n
must also be converging to 0. This shows that q⇤ = 0 when R 0
< 1.
Now, when R 0
> 1, the expected values E [X n
] = Rn 0
go to infinity as n grows. However,
this fact by itself is not enough to show that q⇤ > 0. It is entirely possible to have a sequence
of random variables for which E [X n
] goes to infinity but Pr [X n
> 0] converges to 0 as n
grows. (As a simple example, suppose that X n
were a random variable taking the value 4n
with probability 2�n, and taking the value 0 otherwise. Then E [X n
] = (4/2)n = 2n, which
goes to infinity, while Pr [X n
> 0] = 2�n, which goes to 0.)
This won’t happen in our case, but these considerations do say that to establish q⇤ > 0
when R 0
> 1, we’ll need to use something more specific about the process than simply the
expected number of infected individuals. We do this now, developing a formula for q n
that
in the end will allow us to determine the value of q⇤ exactly.
A Formula for q n
. The quantity q n
depends on three more fundamental quantities: the
number of contacts per individual k, the contagion probability p, and the level of the tree n.
In fact, it’s di�cult to write down a direct formula for q n
in terms of these quantities, but
it’s not hard to express q n
in terms of q n�1. This is what we’ll do first.
Consider the root node, and let’s first ask what it would take for the following event to
hold:
(⇤) The disease spreads through the root node’s first contact j and then continues to persist down to n levels in the part of the tree reachable through j.
This is illustrated in Figure 21.16. First, for the event (⇤) to hold, it would require that j catches the disease directly from the root, which happens with probability p. At this point,
j becomes completely analogous to the root node of its own branching process, consisting
of all nodes reachable from it downward in the tree. So for event (⇤) to hold, after j is infected, it is then necessary that the disease persists for n � 1 levels in the version of the branching process in which we view node j as the root. This happens with probability q
n�1,
676 CHAPTER 21. EPIDEMICS
j
n-1 levels
prob. q
prob p
n-1
Figure 21.16: In order for there to be an infection at level n, the root must infect one of its immediate descendants, and then this descendant must, recursively, produce an infection at level n � 1.
by the definition of q n�1. Therefore, the probability of the event (⇤) is pqn�1. Or, taking the
complementary view, event (⇤) fails to hold with probability
1 � pq n�1.
Now, there is a copy of event (⇤) for each of the direct contacts of the root node, and each fails to hold with probability 1�pq
n�1. Since they’re independent, the probability that
they all fail to hold is
(1 � pq n�1)
k.
At this point, we’re almost done. The disease fails to persist down to level n of the tree,
starting at the root, if it fails to reach level n through any of the root’s direct contacts.
In other words, the disease fails to persist to level n precisely when all the copies of event
(⇤), for each direct contact of the root, fail to hold. Again, we just determined that this probability is (1 � pq
n�1) k. But this probability is also 1 � q
n
, since by the definition of q n
,
the quantity 1 � q n
is exactly the probability that the disease fails to persist to n levels.
Therefore,
1 � q n
= (1 � pq n�1)
k
and solving for q n
we get
q n
= 1 � (1 � pq n�1)
k. (21.6)
21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES677
y = f(x)
y = x1
1
0
Figure 21.17: To determine the limiting probability of an infection at depth n, as n goes to infinity, we need to repeatedly apply the function f (x) = 1 � (1 � px)k, which is the basis for the recurrence q
n
= f (q n�1).
Since we are assuming that the root is infected, and we can treat the root as level 0 of the
tree, we have q 0
= 1; this simply says that the root is infected with probability 1. Starting
from q 0
= 1, we can then build up the values q 1
, q 2
, q 3
, . . . in order, determining each from
the previous one in the list using Equation (21.6). Simply being able to determine the values
of each q n
this way, however, doesn’t immediately tell us where they’re headed as n goes to
infinity. For that we need a technique that looks at the limiting value of this sequence.
Following the values q n
to a limit. If we define the function f (x) = 1�(1�px)k, then we can write Equation (21.6) as follows: q
n
= f (q n�1). This suggests a very clean, purely
algebraic way of formulating our question about q⇤. We have a function f (x) = 1�(1�px)k, and we simply want to study the sequence of values 1, f (1), f (f (1)), f (f (f (1))), . . ., obtained
by applying f repeatedly.
To get started thinking about this, let’s plot the function f on a pair of x-y axes, as in
Figure 21.17. Here are some basic facts about f that help in producing this plot.
• First, f (0) = 0 and f (1) = 1 � (1 � p)k < 1. This means that the plot of f passes through the origin, but lies below the line y = x once x = 1, as shown in Figure 21.17.
• Second, the derivative of f is f 0(x) = pk(1 � px)k. Notice that as x ranges between 0 and 1, the quantity f 0(x) is positive but monotonically decreasing. This means that f
has the increasing but concave shape depicted in Figure 21.17.
678 CHAPTER 21. EPIDEMICS
y = f(x)
y = x1
1
0
Figure 21.18: When we repeatedly apply the function f (x), starting at x = 1, we can follow its trajectory by tracing out the sequence of steps between the curves y = f (x) and y = x.
• Finally, the slope of f at x = 0 is equal to f 0(0) = pk = R 0
. So in the case when
R 0
> 1, which is what we’re focusing on now, the function f starts out above the line
y = x for small positive values of x.
When R 0
> 1, we can take these points together — that y = f (x) starts out above y = x
for small positive values of x but ends up below it by the time we get to x = 1 — and
conclude that y = f (x) must cross y = x somewhere in the interval between 0 and 1, at a
point x⇤ > 0.
Now, using this plot, let’s take a geometric view of the sequence of values
1, f (1), f (f (1)), f (f (f (1))), . . .
that we’re analyzing. In particular, let’s track this sequence on the line y = x. If we’re
currently at a particular point (x, x) on the line y = x, and we want to get to the point
(f (x), f (x)), we can do that as follows. We first move vertically to the curve y = f (x); this
puts us at the point (x, f (x)). We then move horizontally back to the line y = x; this puts
us at the point (f (x), f (x)) as desired. This two-step vertical-horizontal motion is depicted
as the first two parts of the dashed line in Figure 21.18. Continuing this process, we pass
through all the points in the sequence x, f (x), f (f (x)), . . . along the line y = x.
If we start this from x = 1, as indicated in Figure 21.18, the process converges to the
point (x⇤, x⇤) where the line y = x meets the curve y = f (x). Now we can go back to
the interpretation of all this in terms of the branching process. The sequence of values
21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES679
y = f(x)
y = x1
1
0
Figure 21.19: When y = f (x) only intersects y = x at zero, the repeated application of f (x) starting at x = 1 converges to 0.
1, f (1), f (f (1)), . . . is precisely the sequence q 0
, q 1
, q 2
, . . ., as we argued above, and so we
have concluded that it converges to x⇤ > 0: the unique point at which f (x) = x in the
interval strictly between 0 and 1.
This concludes the argument that when R 0
> 1, the probability that the epidemic persists
for n levels converges to a positive value as n goes to infinity.
It is also worth noticing that this style of analysis shows that q⇤ = 0 when R 0
< 1.
Indeed, when R 0
< 1, the curve y = f (x) looks much like it does in Figure 21.17, except that
its derivative at 0 is R 0
< 1, and so it lies out below the line y = x for the whole interval
between 0 and 1. This means that when we follow the sequence of values 1, f (1), f (f (1)), . . .
as the dashed lines do in Figure 21.18, it descends all the way to x = 0 without stopping at
any intermediate point. (See Figure 21.19.) This shows that the resulting limit, which is q⇤,
is equal to 0 in this case.
B. Analysis of Coalescent Processes
We now analyze a di↵erent process arising from earlier in the chapter — the merging of
ancestral lineages discussed in Section 21.7. In particular, we will derive an estimate of the
expected number of generations one has to go back in order to find the most recent common
ancestor for a set of individuals in the model from that section [245, 325]. Like the analysis
of branching processes, this will require probabilistic calculations on trees. In this case,
however, it is tricky to get an exact answer, and we will make use of two approximations as
680 CHAPTER 21. EPIDEMICS
Figure 21.20: We can view the search for coalescence as a backward walk through a sequence of earlier generations, following lineages as they collide with each other.
we estimate the required number of generations. (In fact, these approximations still allow
for a very accurate estimate.)
In addition to the approximations, which we’ll specify in context later, we start by varying
the statement of the problem slightly, following the original work on the topic. Specifically,
we will focus on a small sample of k individuals in a large population of size N ; rather than
analyzing the time until all lineages in the full population merge into a common ancestor,
we will consider the time until the lineages of these k merge into a common ancestor. This
is reasonable from the point of view of applications, since generally one is only ever studying
a fixed-size sample of a large population; also, the calculations involved provide insight into
the question for the full population as well.
To recall the model from Section 21.7, adapted to the plan of looking at fixed-size samples
21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES681
of k individuals, we can pose the question as follows. There are N individuals in each
generation. For each of the k individuals in the initial sample, we choose a parent for each
uniformly at random from the previous generation. We continue working backward in time
this way, extending each of the k lineages through earlier generations. Whenever we get
to a generation where two individuals happen to choose the same parent, their lineages
merge (since their ancestors will now be the same), and so the process continues with fewer
distinct lineages to track. Finally, we stop when we first reach a point where the number of
lineages has been reduced to one, a moment that we call coalescence. We want to estimate
the expected time until coalescence occurs. Figure 21.20 illustrates this on an example with
k = 6 initial present-day individuals (in the bottom row) and a population size of N = 27
(the number of nodes in each row).
The Probability that Lineages Collide in One Step. The key to this analysis is to
consider a single step, in which we have a set of j distinct lineages that we’re tracking, and
we want to estimate the probability that at least two of them will choose a common parent
in the previous generation. (We will call this a collision between two lineages.)
The easiest case to think about is j = 2. Suppose we view the random choice as being
made sequentially by the two lineages under consideration. The first lineage chooses a parent
uniformly at random, and then there is only a collision if the second lineage picks the same
parent uniformly at random from the N available choices. The probability that this occurs
is therefore exactly 1/N .
Things get more complicated when j is larger than 2. First, let’s compute the probability
that no two lineages collide by imagining that the lineages choose their parents one at a time.
For no two lineages to collide, it must be the case that after the first lineage chooses a parent,
the second chooses a parent distinct from this, the third chooses a parent distinct from these
two, and so forth, up to the jth lineage, who must choose a parent distinct from the first
j � 1. The probability that this happens is ✓
1 � 1
N
◆ ✓ 1 �
2
N
◆ ✓ 1 �
3
N
◆ · · ·
✓ 1 �
j � 1 N
◆ .
Expanding out this product, we see that it is equal to
1 � ✓
1 + 2 + 3 + · · · + j � 1 N
◆ + (terms with N 2 or higher in the denominator).
In particular, it is at most
1 � ✓
1 + 2 + 3 + · · · + j � 1 N
◆ +
g(j)
N 2
for a function g(·) that depends only on j. So far this calculation has been exact, but here we come to the first of two approximations, following [245]: rather than deal with the
682 CHAPTER 21. EPIDEMICS
complexity of this last term, we observe that when the population size N is much larger than
j, expressions of the form g(j)/N 2 are negligible compared to (1 + 2 + · · · + j � 1)/N . We therefore ignore them, and approximate the probability that no two lineages collide by
1 � ✓
1 + 2 + 3 + · · · + j � 1 N
◆ = 1 �
j(j � 1) 2N
. (21.7)
Now, when two lineages do in fact collide, there are a number of possibilities: it could
be that there is simply a two-way collision between two of the lineages while all the others
remain distinct, or it could be that more than two lineages collide in a single generation.
We now describe how the latter scenario can come to pass, and then argue that it is very
unlikely.
• First, it could be that three lineages all choose the same parent in a single generation. For any particular set of three lineages, the probability this happens is exactly 1/N 2:
imagining the choice being made sequentially, the first lineage can pick any parent,
and then the second and third must independently pick this same parent from the N
available choices. Now, since there are fewer than j3 sets of three lineages, the prob-
ability that any three-way collision happens in a given generation is less than j3/N 2.
When N is much larger than j, this quantity is negligible compared to expressions as
in (21.7) that only have N in the denominator.
• Alternately, it could be that two di↵erent pairs of lineages each have a separate, two- way collision in the same generation: suppose that lineage A collides with lineage B,
and lineage C collides with lineage D. The collision of A and B has probability 1/N ,
and the collision of C and D is an event independent from this, also with probability
1/N . Therefore, for this particular choice of four lineages, both collisions happen with
probability 1/N 2. Since there are less than j4 ways of choosing A, B, C, and D,
the probability that there are simultaneous two-way collisions involving any choice of
four lineages is less than j4/N 2. Again, with N much larger than j, this quantity is
negligible compared to expressions that only have N in the denominator.
These arguments lead to our second approximation: we will assume there is never a genera-
tion prior to the most recent common ancestor in which we have more than a single two-way
collision among lineages.
This means that when we have a generation in which the j current lineages fail to re-
main completely distinct, it happens because exactly two of them choose a common parent,
reducing the number of lineages from j to j � 1.
The Expected Time Until Coalescence. Our two approximations have led to a very
clean view of the process as it works backward in time. In this approximate view, we start
21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES683
From 6 to 5: waiting for an event
of prob. 15/N
From 5 to 4: waiting for an event
of prob. 10/N
From 4 to 3: waiting for an event
of prob. 6/N
From 3 to 2: waiting for an event
of prob. 3/N
From 2 to 1: waiting for an event
of prob. 1/N
Figure 21.21: Assuming that no three lineages ever collide simultaneously, the time to co- alescence can be computed as the time for a sequence of distinct collision events to occur.
with k distinct lineages and wait until two of them collide. This happens with probability k(k�1)
2N
in each generation. Once a collision happens, we have k � 1 distinct lineages, and we wait for two of them to collide with probability (k�1)(k�2)
2N
per generation. Things continue
this way until we are down to two distinct lineages, at which point we wait for them to collide
with probability 2 2N
= 1 N
per generation. The overall process is shown, for our example with
k = 6, in Figure 21.21.
Guided by this view of the process, we can analyze it as follows. Let W be a random
variable equal to the number of generations back until coalescence. We can write
W = W k
+ W k�1 + Wk�2 + · · · + W2,
where W j
is a random variable equal to the number of generations during which there are
684 CHAPTER 21. EPIDEMICS
exactly j distinct lineages. By linearity of expectation, we have
E [W ] = E [W k
] + E [W k�1] + · · · + E [W2] .
So it remains to figure out the terms on the right-hand side.
Each random variable of the form W j
can be viewed in the following way: when we
have j distinct lineages, we move backward through successive generations, waiting until a
particular event (a collision) first happens. We now make use of our approximations: W j
is
very close to a simpler random variable in which we have j lineages, we count the number of
steps until this number of lineages is reduced to j �1, and in each generation this reduction of lineages occurs with probability exactly equal to p = j(j�1)
2N
. We let X j
denote this closely
related, simpler random variable; we write
X = X k
+ X k�1 + Xk�2 + · · · + X2,
and we will be interested in determining the expectation
E [X] = E [X k
] + E [X k�1] + · · · + E [X2]
rather than the expectation E [W ].
How should we think about the expectation of one of these simpler random variables
X j
? It is precisely as though we have a coin that comes up “heads” with a fixed probability
p = j(j�1) 2N
per flip, and we want to know the expected number of flips until we see the first
heads. To compute this expectation, we recall Equation (21.5) from earlier in this section,
applied to the current random variable X j
:
E [X j
] = Pr [X j
� 1] + Pr [X j
� 2] + Pr [X j
� 3] + · · ·
The probability that X j
is at least some value i is just the probability that the coin comes
up “tails” on its first i flips, which is (1 � p)i. Therefore,
E [X j
] = 1 + (1 � p) + (1 � p)2 + (1 � p)3 + · · · = 1
1 � (1 � p) =
1
p .
This is a very intuitive relationship: the expected time to see the first heads on a coin with
a heads probability of p is simply 1 p
.
The random variable X j
describes precisely this process with p = j(j�1) 2N
. Therefore,
E [X j
] = 2N
j(j � 1) ,
and so
E [X] = 2N
2 · 1 +
2N
3 · 2 +
2N
4 · 3 + · · · +
2N
j(j � 1) + · · · +
2N
k(k � 1) (21.8)
= 2N
✓ 1
2 · 1 +
1
3 · 2 +
1
4 · 3 + · · · +
1
j(j � 1) + · · · +
1
k(k � 1)
◆ . (21.9)
21.9. EXERCISES 685
This last sum can be evaluated by noticing that
1
j(j � 1) =
1
j � 1 �
1
j
and applying this identity to each term in (21.9) we get
E [X] = 2N
✓ 1
1 �
1
2
� +
1
2 �
1
3
� + · · · +
1
j � 1 �
1
j
� +
1
j �
1
j + 1
� + · · · +
1
k � 1 �
1
k
�◆ .
In this new way of writing the sum, almost all the terms inside the parentheses cancel each
other out, and the only two that survive are 1 and �1 k
. Hence we conclude that
E [X] = 2N
✓ 1 �
1
k
◆ .
This gives us the result we were looking for — the approximate number of generations
until coalescence — and so we can conclude with just a few final observations. First, once
k becomes moderately large, the expected time to coalescence depends only very weakly on
k; it is roughly 2N as k grows. Second, the breakdown of X into X k
+ X k�1 + · · · + X2 lets
us appreciate where most of the time is being spent in the merging to a common ancestor.
As we begin moving backward in time, collisions happen relatively quickly at first; but as
we continue moving backward, we find that essentially half the expected time is being spent
once the lineages have merged down to just two, and these two are searching for a final
collision point at the most recent common ancestor. Third, the approximations provide us
with a very simple way to build trees according to this process: we simply follow the recipe
in Figure 21.21, drawing parallel lines backward in time until the outcome of a coin-flip tells
us to pick two of the lines uniformly at random and merge them into one.
Finally, we should note that although we have introduced some approximations into the
original formulation of the problem, subsequent work has shown that the final estimates are
very close to the exact results one gets through much more intricate analysis [91, 174].
21.9 Exercises
1. Suppose you are studying the spread of a rare disease among the set of people pictured
in Figure 21.22. The contacts among these people are as depicted in the network in the
figure, with a time interval on each edge showing when the period of contact occurred.
We assume that the period of observation runs from time 0 to time 20.
(a) Suppose that s is the only individual who had the disease at time 0. Which nodes
could potentially have acquired the disease by the end of the observation period,
at time 20?
686 CHAPTER 21. EPIDEMICS
v
u x
y
z
[1,3]
[5,9]
[14,18]
[12,16]
w
s
[7,12]
[4,8]
[10,16]
Figure 21.22: Contacts among a set of people, with time intervals showing when the contacts occurred.
(b) Suppose that you find, in fact, that all nodes have the disease at time 20. You’re
fairly certain that the disease couldn’t have been introduced into this group from
other sources, and so you suspect instead that a value you’re using as the start
or end of one of the time intervals is incorrect. Can you find a single number,
designating the start or end of one of the time intervals, that you could change
so that in the resulting network, it’s possible for the disease to have flowed from
s to every other node?
a b c d e
Figure 21.23: A contact graph on five people.
2. Imagine that you know a contact graph on a set of people, but you don’t know exactly
the times during which contacts happened. Suppose you have a hypothesis that a
particular disease passed between certain pairs of people, but not between certain other
pairs. (Let’s call the first set of pairs positive, and the second set of pairs negative.)
It’s natural to ask whether it’s possible to find a set of time intervals for the edges that
support this hypothesis in a strong sense: they make it possible for the disease to flow
between the positive pairs, but not between the negative pairs.
Let’s try this genre of question out on the simple contact graph among five people
shown in Figure 21.23.
21.9. EXERCISES 687
(a) Can you find time intervals for the edges that make it possible for the disease to
flow from every node to every other node, with the one exception that it is not
possible for it to flow from node a to node e? If you think it is possible, describe
such a set of time intervals; if you think it is not possible, explain why no such
set of time intervals exists.
(b) Can you find time intervals for the edges that make it possible for the disease to
flow from a to d and from b to e, but not from a to c? Again, if you think it
is possible, describe such a set of time intervals; if you think it is not possible,
explain why no such set of time intervals exists.
3. Imagine that you’re advising a group of agricultural o�cials who are investigating
measures to control the outbreak of an epidemic in its early stages within a livestock
population. On short notice, they are able to try controlling the extent to which the
animals come in contact with each other, and they are also able to introduce higher
levels of sanitization to reduce the probability that one animal passes the disease to
another.
Both of these measures cost money, and the estimates of the costs are as follows. If
the o�cials spend x dollars controlling the extent to which animals come into contact
with each other, then they expect each animal to come into contact with
40 � x
200, 000
others. If the o�cials spend y dollars introducing sanitization measures to reduce the
probability of transmission, then they expect the probability an infected animal passes
it to another animal contact to be
.04 � y
100, 000, 000 .
The o�cials have two million dollars budgeted for this activity. Their current plan is
to spend one million on each of the two kinds of measures. Using what you know about
epidemics, would you advise them that this is a good use of the available money? If
so, why? If not, can you suggest a better way to allocate the money?
688 CHAPTER 21. EPIDEMICS
Part VII
Institutions and Aggregate Behavior
689
Chapter 22
Markets and Information
In this final part of the book, we build on the principles developed thus far to consider the
design of institutions, and how di↵erent institutions can produce di↵erent forms of aggregate
behavior. By an institution here, we mean something very general — any set of rules,
conventions, or mechanisms that synthesizes individual behavior across a population into an
overall outcome. In the next three chapters, we will focus on three fundamental classes of
institutions: markets, voting, and property rights.
We begin by discussing markets, and specifically their role in aggregating and conveying
information across a population. Each individual participant in the market arrives with
certain beliefs and expectations — about the value of assets or products, and about the
likelihood of events that may a↵ect these values. The markets we study will be structured so
as to combine this set of beliefs into an overall outcome — generally in the form of market
prices — that represents a kind of synthesis of the underlying information.
This is part of a broad issue we have seen several times so far: the fact that individuals’
expectations a↵ect their behavior. For example, we saw this in Chapter 8 on Braess’s
Paradox, where the optimal route depends on which routes others are expected to choose;
in Chapter 16 on information cascades, where people draw inferences about the unknown
desirability of alternatives (restaurants or fashions) from the behavior of others; and in
Chapter 17 on network e↵ects, where the unknown value of a product (a fax machine or a
social-networking site) depends on how many others are also expected to use the product. In
each of these cases, individuals have to decide what to do without knowing exactly what will
happen. Will the route be crowded or not? Is the restaurant good or bad? Will others also
join the social networking site? In all of these situations, individuals’ expectations about
payo↵s matter for how they will choose.
Along with the similarities among these settings, there is also an important di↵erence
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
691
692 CHAPTER 22. MARKETS AND INFORMATION
that will be fundamental to our discussion here: whether the unknown desirability of the
di↵erent alternatives is exogenous or endogenous. Exogenous desirability means that a given
alternative is inherently a good idea or a bad idea, regardless of how the individuals make
their decisions. Thus, for example, in our model of information cascades, people decided
whether to accept or reject an option that was in fact fundamentally either good or bad,
and the desirability of the option wasn’t a↵ected by whether people accepted it or not.
Endogenous desirability is di↵erent, and somewhat more subtle: it means that the desirability
of an alternative depends on the actual decisions people make about it. In our model of
network tra�c and Braess’s Paradox, no particular route is a priori crowded or not; a route
becomes crowded if many people choose it. Similarly, we can’t tell whether a product with
network e↵ects — like a fax machine — is worth purchasing or not until we know whether
many people in fact purchase it.
We will consider both types of cases in this chapter. First we will look at what happens in
asset markets where the desirability of the assets is exogenous, but unknown. We will begin
this analysis by focusing on betting markets as a simple, stylized domain with exogenous but
uncertain outcomes. We describe how individuals behave and how prices are set in betting
markets, and then discuss how the ideas we develop about betting markets provide insight
into more complex settings like stock markets. After this, we will consider what happens in
markets where the desirability of the items is endogenous. The issue we focus on in this case
is the role of asymmetric information.
The next two chapters in this part of the book will discuss voting and the role of property
rights. Markets and voting mechanisms are alternative institutions that aggregate individual
behavior into outcomes for the group. One important di↵erence between them is that voting
mechanisms are typically used to produce a single group decision while in markets each
individual may choose a di↵erent outcome. The final chapter discusses the role of property
rights in influencing what outcomes are possible.
22.1 Markets with Exogenous Events
In this section we begin by examining how markets aggregate opinions about events in
settings where the underlying events are exogenous — the probabilities of the events are
not a↵ected by the outcomes in the market. Prediction markets are one basic example of
this setting. These are markets for (generally very simple) assets which have been created
to aggregate individuals’ predictions about a future event into a single group, or market,
opinion. In a prediction market, individuals bet on the outcome of some event by trading
claims to monetary amounts that are conditional on the outcome of the event.
One of the most well-known uses of prediction markets has been for the forecasting of
22.1. MARKETS WITH EXOGENOUS EVENTS 693
election results. For example, the Iowa Electronic Markets1 ran a market (one of many with
this structure) in which individuals could buy or sell a contract that would pay $1 in the
event that a Democrat won the 2008 U.S. Presidential election, and would pay nothing if this
event did not occur. An individual who bought this contract was betting that a Democrat
would win the election. The corresponding contract that paid $1 if a Republican won was
also available. In Figure 1.13 from Chapter 1, we saw a plot of the prices for these two
contracts over time, and we saw how the movement of the prices followed the — exogenous
— course of events a↵ecting the perceived likelihood of the election outcome.
In a prediction market, or any other market, there are two sides to any trade: what
someone buys, someone else sells. So a trade in a prediction market means that two people
disagree about which side of the bet they want to take. But note that at the price where the
trade actually occurs, both the buyer and the seller find the trade desirable. In a sense that
we will make precise later, the price separates their beliefs: their beliefs are on opposite sides
of the price, and we can view the price as an average of their beliefs. This is the motivation
for the usual interpretation that the price in a prediction market is an average prediction
about the probability of the event occurring. So, if the price of the Democrat-wins contract
is 60 cents, the usual interpretation is that “the market” believes that the probability of a
Democrat winning the election is 0.6. Of course, the market itself does not have beliefs —
it’s simply an institution, a place where trade is conducted under a particular set of rules.
So when we say that the market believes something about a future event, this phrase really
means that the market price represents an average belief.
Betting markets for sporting events such as horse races are also markets that aggregate
diverse opinions into a price. As is the case with prediction markets, the outcome of the
sporting event is independent of the betting behavior of the participants. Of course, some
bettors may have more accurate beliefs than other bettors. But, assuming that there is no
cheating, what happens in the betting market does not a↵ect the outcome of the sporting
event.
Markets for stocks are similar to prediction markets or betting at horse races, and we
will use the understanding we develop for these betting markets to help us understand how
the stock market works. In both betting markets and the stock market, individuals make
decisions under uncertainty about the value of a contract, bet, or stock, and the market
aggregates their diverse opinions about the value of the asset. But there is also an important
di↵erence between the stock market and gambling. The price set in a gambling market, and
who holds what bets, are both interesting, but they do not a↵ect the allocation of real capital.
On the other hand the stock market allocates the available shares of stock in a company.
The market price for these shares determines the cost of equity capital for the company; it
is the expected rate of return that investors demand in order to hold the existing shares of
1www.biz.uiowa.edu/iem/
694 CHAPTER 22. MARKETS AND INFORMATION
stock or to buy new shares of stock. The financial capital that the company receives for its
shares of stock a↵ects its real investment decisions and thus the future value of the stock.
So there is an indirect link between the aggregate opinion in the market about the company,
its stock price, and the actual value of the company. But this link is really quite indirect,
and for a first pass at understanding stock market prices it is reasonable to ignore it. (In
fact, much of the academic literature on asset pricing also ignores this e↵ect.)
As we will see, we can interpret the price of the asset being traded, whether it’s a stock,
a contract that pays out if a Democrat wins, or a betting ticket at a race-track, as a market
prediction about some event. In the next section, we will examine how these markets work
and we will build an understanding of the circumstances under which they do a good or bad
job of producing a useful aggregate prediction.
22.2 Horse Races, Betting, and Beliefs
It is easiest to understand what goes on in these markets if we begin with the simple example
of betting on a two-horse race [64]. Suppose that two horses, whom we’ll call A and B, will
run a race which one of them will win (we will ignore the possibility of a tie). How should a
bettor who has w dollars available to bet allocate his wealth between bets on the two horses?
We will make the assumption that the bettor plans to bet all of this money w on the two
horses in some fashion: We will let r be a number between 0 and 1 representing the fraction
of his wealth that he bets on horse A; the remaining 1 � r fraction of his wealth will be bet on horse B. The bettor could bet all of his money on horse A (r = 1), all of it on horse B
(r = 0), or he could split it up and bet some on each horse (by choosing r strictly between 0
and 1). The only thing the bettor cannot do is save some of the money and not bet it. We
will see later that in our model there is a betting strategy that returns his wealth for sure,
so this lack of a direct way to not bet is not really a constraint.
It seems reasonable to expect a bettor’s choice of bet will depend on what the bettor
believes about the likelihood of each horse winning the race. Let’s suppose that the bettor
believes that horse A will win with probability a, and that horse B will win with probability
b = 1 � a. It seems sensible to suppose that the fraction of wealth r bet on horse A won’t decrease if the probability of A winning increases, and r should be equal to one if a = 1
because then betting on horse A is a sure thing. But if neither horse is a sure thing, what
should the bet look like? The answer to this question depends on more than the probability
of A or B winning the race; in particular, it may depend on two other factors.
First, the bettor’s choice of bet may depend on the odds. If the odds on horse A are
for example three-to-one, then a one-dollar bet on horse A will pay three dollars if horse A
wins, and will pay nothing if horse A loses. More generally, if the odds on horse A are o A
,
and the odds on horse B are o B
, then a bet of x dollars on horse A will pay o A
x dollars if A
22.2. HORSE RACES, BETTING, AND BELIEFS 695
wins, and a bet of y dollars on horse B will pay o B
y dollars if horse B wins. A bettor might
find high odds attractive and bet a lot on a horse with high odds in the hope of winning a
large sum of money. But if he does this then he will have little left to bet on a horse with
low odds, and if that horse wins the race he will be left with very little money. How a bettor
evaluates the di↵erent levels of risk in these options is the topic we turn to next.
Modeling Risk and Evaluating the Utility of Wealth. A bettor’s reaction to risk is
the second factor that influences his choice of bets. It seems reasonable to suppose that a
bettor who is very risk-averse will bet so as to have some money left no matter which horse
wins, by betting some money on each horse. Someone who does not care as much about
risk may place a bet more skewed toward one of the horses, and a person who does not
care about risk at all might even bet everything on one horse. This issue of characterizing
risk will become even more important when we move from the simple example of betting
on horse races to investing in financial markets. Individuals invest significant amounts of
their wealth in a wide variety of assets, all of which are subject to some risk, and it is very
natural to assume that most people will not want to choose investment strategies where
there is a plausible scenario in which their savings are reduced to zero. We can formulate the
same issues in our current example, considering the bets on horse A or B as the alternatives
that contain risk, while also keeping in mind that the whole example is a simply-formulated
metaphor for markets with exogenous events in general.
How do we model the bettor’s attitude toward risk? We saw a simple version of this
question in Chapter 6 in which we asked how a player in a game evaluates the payo↵ of
a strategy with random payo↵s. Our answer was that the player evaluates each strategy
according to the expected value of its payo↵, and we will use the same idea here. We assume
that the bettor evaluates a bet according to the expected value of the payo↵ on the bet. But
here we need to be a bit careful. What really is the payo↵ on the bet? Is it the amount of
money won or lost, or is it how the bettor feels about the amount of money?
Presumably the bettor prefers outcomes in which he obtains larger amounts of money,
but how does his actual evaluation of the outcome depend on the amount of wealth he
acquires? To make this precise, we need to define a numerical way of specifying the bettor’s
evaluation of the outcome as a function of his wealth, and then use this numerical measure
as the bettor’s payo↵. We will do this with a utility function U (·): when a bettor has a wealth w, his evaluation of the outcome — i.e., his payo↵ — is equal to the quantity U (w).
The simplest example of a utility function is the linear function U (w) = w, in which a
bettor’s utility for wealth is exactly its value. We could also consider more general linear
utility functions, of the form U (w) = aw+b for some positive number a. With such functions,
the bettor’s utility increase from gaining a dollar is precisely equal to his utility decrease from
losing a dollar. At first glance, it might seem strange to use any other utility function, but
696 CHAPTER 22. MARKETS AND INFORMATION
in fact linear utility functions predict behavior that doesn’t align well with either empirical
evidence or common sense.
An easy way to see this is to ask whether a bettor would accept a fair gamble. We
consider a particular instance of a fair gamble to illustrate the issue; other examples show
similar e↵ects. Suppose that a bettor’s current total wealth is w, and he is o↵ered a gamble
in which he gains w dollars with probability 1 2
, and loses w dollars with probability 1 2
. In
other words, with probability 1 2
, his wealth after the bet will be 2w, and with probability 1 2
,
his wealth after the bet will be 0. We call this “fair” in the sense that the expected value of
the bettor’s wealth after the bet is 1 2
· (2w) + 1 2
· 0 = w. Now, a bettor with the linear utility function U (w) = w would be indi↵erent between accepting and rejecting this gamble, since
his expected utility from accepting the gamble would be
1
2 U (2w) +
1
2 U (0) =
1
2 · (2w) +
1
2 · 0 = w,
which is the same as his expected utility from passing up the opportunity to gamble. The
same would hold for any linear utility function. But if we imagine our scenario serving as
a model for the behavior of an individual investing his wealth in a financial market, then
our calculation corresponds to the following premise: an investor whose total net worth is
$1 million would be indi↵erent between accepting and rejecting an investment strategy in
which he gains or loses a million dollars with equal probability. This isn’t a good model for
investor behavior; we’d much prefer a model in which the investor views the downside of
such a strategy — having your net worth reduced to zero — as far outweighing the upside
of potentially doubling your net worth. In other words, we want to model the idea that the
investor, at least to some extent, views this strategy as highly risky despite its expected net
change of zero to his wealth.
We can capture this kind of behavior by assuming that the bettor’s utility grows at a
decreasing rate as a function of his wealth w. Examples of utility functions that have this
property are U (w) = w1/2 and U (w) = ln(w), the natural logarithm of wealth. In these
cases utility still increases as a function of wealth; it is simply that the rate of increase slows
down as wealth increases: the appeal of gaining each extra dollar goes down as you become
wealthier. With a bit of analysis, we can check that a bettor with this type of utility would
reject a fair gamble. For example, if U (w) = w1/2, then he would evaluate the expected
utility from accepting the bet discussed above as
1
2 U (2w) +
1
2 U (0) =
1
2 · (2w)1/2 +
1
2 · 0 = 2�1/2 · w1/2.
This is less than his current utility of w1/2, which he could maintain simply by refusing to
bet. Thus, with this kind of utility function, which grows at a decreasing rate in the wealth
w, the bettor would reject the gamble.
22.2. HORSE RACES, BETTING, AND BELIEFS 697
w
payoff
Figure 22.1: When we assume that an individual’s utility is logarithmic in his wealth, this means that utility grows at a decreasing rate as wealth increases.
We will be using these types of utility functions in our analysis here. It’s important
to remember, however, that regardless of how we model utility as a function of wealth, we
assume that the bettor evaluates bets according to the expected value of the utility; the
di↵erence is simply in the shape of the bettor’s utility function.
Logarithmic Utility. To build a simple model of how a bettor might behave, we’ll suppose
in particular that the bettor’s utility function is the natural logarithm of wealth, ln(w), where
w > 0 is the bettor’s wealth. This utility function is plotted in Figure 22.1; as noted above,
it grows with the wealth, but the rate of growth slows down with increasing wealth. This
logarithmic form for utility has a simple intuitive property: the bettor receives the same
utility benefit from doubling his wealth, regardless of how much he currently has. In other
words, the value of each additional dollar declines as wealth increases, but the value of
doubling one’s wealth is always the same. To see why this is true, we use the following basic
fact about logarithms:
ln(x) � ln(y) = ln(x/y) (22.1)
698 CHAPTER 22. MARKETS AND INFORMATION
0 0.2 0.4 0.6 0.8 1
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Figure 22.2: A plot of a ln(r) + b ln(1�r) as a function of r, when a = 0.75. The maximum is achieved when r = a.
for any x and y. Given this fact, the increase in utility from doubling your money, when
your current wealth is w, is equal to
ln(2w) � ln(w) = ln(2w/w) = ln(2),
where the first equality follows by plugging x = 2w and y = w into Equation (22.1). A similar
argument would hold for any multiplicative increase or decrease in the bettor’s wealth: the
change in utility doesn’t depend on the current wealth.
The logarithmic utility function will make our analysis particularly clean, although the
general ideas of the analysis apply equally well to any utility function that grows at a
decreasing rate in wealth.
The Optimal Strategy: Betting Your Beliefs. Let’s now figure out the optimal strat-
egy for our bettor, given the logarithmic utility of wealth, the odds being o↵ered, and the
bettor’s beliefs about the respective probabilities that horses A and B will win.
22.2. HORSE RACES, BETTING, AND BELIEFS 699
Recall that the odds on horse A are o A
and the odds on horse B are o B
. Suppose that
the bettor bets a fraction r of his wealth on horse A. Then since the amount bet on A is rw,
the bettor’s wealth will be rwo A
if horse A wins. The amount bet on B is (1 � r)w, so the bettor’s wealth will be (1�r)wo
B
if horse B wins. The bettor believes there is a probability
a that horse A will win, and a probability b = 1 � a that horse B will win. So with the given betting strategy r and with these probabilities, the bettor ends up with a utility of
ln(rwo A
) with probability a (in the event horse A wins), and a utility of ln((1�r)wo B
) with
probability 1 � a (in the event horse B wins). Adding these up, the expected utility after the bet is
a ln(rwo A
) + (1 � a) ln((1 � r)wo B
). (22.2)
The bettor wants to choose r to maximize the value in this expression.
As a step toward maximizing this, we can use another basic fact about logarithms, closely
related to Equation (22.1):
ln(x) + ln(y) = ln(xy) (22.3)
for any x and y. As a result, we can unpack the products of variables inside the logarithms
in Formula (22.2), arriving at an equivalent way to write the expected utility that the bettor
wants to maximize:
a ln(r) + (1 � a) ln(1 � r) + a ln(wo A
) + (1 � a) ln(wo B
). (22.4)
Something interesting is already happening here. The third and fourth terms of this expres-
sion do not contain the value r, and this value r is the only thing the bettor has control
over. So the bettor’s maximization problem is really just to maximize the first two terms:
he needs to choose r to maximize
a ln(r) + (1 � a) ln(1 � r). (22.5)
This leads to a counter-intuitive conclusion, but one that follows directly from our as-
sumption of a logarithmic utility function: the formula (22.5) does not contain the odds
o A
and o B
, and so when the bettor determines the optimal choice of r by maximizing this
formula, it will not depend on the value of the odds. This makes sense once we think further
about what logarithmic utility really means. We can interpret the odds of o A
on horse A
as follows: In the event that horse A wins, you will first be paid rw dollars, and then your
wealth will be further increased by a factor of o A
. But we just argued that with logarithmic
utilities, the benefit from a multiplicative increase in your wealth is a fixed amount indepen-
dent of how much you have. So while this final multiplicative boost of o A
is a nice bonus, the
value you assign to it is independent of how much money you have at the time, and hence
it does not a↵ect your choice of r.
So let’s go back to the bettor’s problem of maximizing Formula (22.5). Its typical shape
as a function of r is shown in Figure 22.2: it drops steeply near r = 0 and r = 1, and it
700 CHAPTER 22. MARKETS AND INFORMATION
assumes a unique maximum in between. With some very simple calculus, one can show that
it is maximized at r = a. In what follows, we’ll use this result in a “black-box” fashion,
without worrying how it is obtained, but the argument is very short. The derivative of the
expression in Formula (22.5) with respect to r is
a
r �
1 � a 1 � r
(22.6)
If we set this to zero, we get an equation that is solved simply by setting r = a, and this is
the maximum point.
This result has a nice interpretation: the bettor bets his beliefs. The fraction of wealth
bet on horse A is the bettor’s belief about the probability that horse A wins. Note also
that the optimal bet has the sensible property that the amount bet on A increases with
the probability of A winning, and it approaches the bettor’s full wealth as this probability
approaches 1.
We now use this basic result to study what happens in markets with many participants.
Throughout this analysis we will continue to assume that all bettors have logarithmic utility.
If we were to use a di↵erent utility function, the overall results to follow would still have the
same qualitative behavior, but the analysis would become much more elaborate and some of
the specific facts we use would no longer hold. In particular, with a di↵erent utility function,
the bettors’ decisions would no longer necessarily be independent of the odds, and this would
lead to more complex reasoning about bettor behavior.
22.3 Aggregate Beliefs and the “Wisdom of Crowds”
When there is only one bettor, we can learn about the bettor’s beliefs by observing his
optimal strategy, but with only one bettor the race-track cannot be said to be aggregating
multiple opinions. To understand how aggregation works, we now consider systems where
there are multiple bettors.
Let’s suppose that there are N bettors named 1, 2, 3, . . . , N , and that each bettor n
believes there is a probability of a n
that horse A will win, and thus a probability of b n
= 1�a n
that B will win. We will allow the bettors to disagree about the probability of winning, but
we don’t require that they actually do disagree.2 As we will see, if they agree, then although
they wouldn’t be willing to take opposite sides of a bet with respect to each other, we can
determine the market odds and the aggregate opinion — it will simply be the commonly
held opinion.
2We take bettors’ disagreement about the probabilities as exogenously given. It is also interesting to ask what happens if any disagreements are generated by di↵ering information. But this case is much more complex as here bettors also need to make inferences about the information of others from whatever market statistics they can observe [345].
22.3. AGGREGATE BELIEFS AND THE “WISDOM OF CROWDS” 701
There is no reason to assume that the bettors all have the same wealth, and once they
begin betting there will be winners and losers, so eventually their wealths will have to di↵er.
So we might as well allow for di↵erent wealths at the beginning. Suppose that bettor n has
wealth w n
and thus the total wealth is the sum of all w n
, which we write
w = w 1
+ w 2
+ · · · + w N
.
We will assume that all of the bettors evaluate wealth using the same utility function, and
we will continue to use the natural logarithm of wealth, ln(w), for this utility.
In Section 22.2, we saw that the optimal betting strategy for bettor n with belief a n
is
r n
= a n
: that is, bettor n will bet a n
w n
on horse A. Correspondingly, bettor n will bet
(1 � a n
)w n
on horse B. So the amount that all the bettors together bet on horse A is the
sum
a 1
w 1
+ a 2
w 2
+ · · · + a N
w N
,
and the total amount bet on horse B is the sum
b 1
w 1
+ b 2
w 2
+ · · · + b N
w N
.
Since each bettor bets all of his wealth, the total amount bet is the aggregate wealth w.
The Odds Determined by the Race-Track. Now we would like to determine the odds
that the race-track should o↵er on horses A and B if it wants to break even — that is, if it
wants to pay out to the bettors exactly the total amount bet, no matter which horse wins.
We assume that the race-track collects the bets from the bettors, and so it has w in total
bets. It then uses this money to pay o↵ the winning bets. We will assume that no matter
which horse wins the race, the race-track pays out everything that it collects. That is, it has
no cost and makes no profit.
If horse A wins, the amount that is owed to bettor n is a n
w n
o A
. The total amount owed
to the bettors is the sum of their winnings, which is equal to
a 1
w 1
o A
+ · · · + a N
w N
o A
.
In order to have the amount that is paid out to bettors in the event that horse A wins equal
the amount of money w that the track collected, the odds on horse A must solve
a 1
w 1
o A
+ · · · + a N
w N
o A
= w. (22.7)
That is, the equilibrium odds on horse A are determined so that the track just breaks even
if A wins. Solving for the inverse of the equilibrium odds on horse A (which will produce a
somewhat nicer expression than the formula for the actual odds), we get
a 1
w 1
w + · · · +
a N
w N
w = o�1
A
. (22.8)
702 CHAPTER 22. MARKETS AND INFORMATION
If we write f n
= w n
/w for the share of the total wealth held by bettor n, then this can be
written as
a 1
f 1
+ · · · + a N
f N
= o�1 A
. (22.9)
An analogous calculation shows that the equilibrium inverse odds on horse B are
b 1
f 1
+ · · · + b N
f N
= o�1 B
. (22.10)
These inverse odds have a nice interpretation. If the odds on horse A are 4 (i.e. four-to-
one, with a one-dollar bet paying four-dollars), then in order to have a one-dollar payout in
the event that horse A wins the bettor would need to bet one-fourth of a dollar on horse A.
This amount is the inverse of the odds for A: that is, a bet of o�1 A
dollars on horse A will
result in a payment of $1 in the event that horse A wins the race. Thus the inverse odds
on A are the price of a dollar to be paid in the event that A wins, and similarly the inverse
odds on B are the price of a dollar to be paid in the event that B wins.
State Prices. Let’s denote these “prices of a dollar” by ⇢ A
= o�1 A
for the event that horse
A wins and ⇢ B
= o�1 B
for the event that horse B wins. These prices are usually called state
prices, as they are the price of a dollar in the event that a certain future state of the world
is reached [24].
There is one more feature of equilibrium odds that is important. Let’s ask how much a
bettor would have to pay now to get one dollar for sure after the race. To do this the bettor
needs to bet enough on horse A so as to receive one dollar if A wins, and enough on horse
B so as to receive one dollar if horse B wins. These amounts are, as we saw before, o�1 A
and
o�1 B
. So the amount needed to guarantee receiving one dollar in either case — i.e., regardless
of the outcome — is o�1 A
+ o�1 B
. Let’s use the equilibrium values of odds to figure out how
much money this takes.
o�1 A
+ o�1 B
= (a 1
f 1
+ · · · + a N
f N
) + (b 1
f 1
+ · · · + b N
f N
)
= (a 1
+ b 1
)f 1
+ · · · + (a N
+ b N
)f N
= (1 · f 1
) + · · · + (1 · f N
)
= 1.
This calculation shows there is a betting strategy that will turn one dollar before the race
into one dollar for sure after the race. This is the sense in which our assumption that the
bettors actually bet all of their wealth is not a constraint. Any portion of their wealth that
they don’t want to risk they can bet according to the inverse odds. This calculation also
gives us the very useful property that inverse odds, or state prices, sum to one.
Now having done all of these calculations we are in a position to interpret the state prices.
First, note that if every bettor believes that the probability that horse A will win is a, then
22.3. AGGREGATE BELIEFS AND THE “WISDOM OF CROWDS” 703
⇢ A
= a. That is, if the bettors agree about the probabilities then the market accurately
reflects these beliefs, with the state price equal to the common belief. Second, since the
wealth shares sum to one, the state prices are weighted averages of the bettors’ beliefs. The
weight on bettor n’s beliefs is bettor n’s share, f n
, of the aggregate wealth. In particular, if
a bettor has no wealth then the state price is not influenced by his beliefs as he cannot bet.
Alternatively, if a bettor controls all of the wealth then the state price is his probability.
More generally, how much influence a bettor’s beliefs have on the state price depends on
how much of the aggregate wealth is controlled by that bettor.
So it really does make sense to think of the state prices as the market’s averaging of in-
dividual beliefs — or, in the more typical phrasing, they can be interpreted as the market’s
beliefs. For our horse-race market (with the logarithmic utility function) this market proba-
bility is the weighted average of the investors’ beliefs, with each investor’s weight determined
by his share of the wealth.3
The Relationship to the “Wisdom of Crowds.” What does this analysis say about
the intuition popularized by recent books such as James Surowiecki’s The Wisdom of Crowds
[383]? The basic argument there, drawing on a long history of intuition about markets, is
that the aggregate behavior of many people, each with limited information, can produce very
accurate beliefs.
Our results on state prices illustrate some of the technical basis for this intuition. In
particular, we found that the crowd at the racetrack determines the odds, or the state
prices, and these odds are an average of the opinions in the crowd. If the opinions in the
crowd about the probability of horse A winning are independently drawn from a distribution
whose mean is equal to the true probability of horse A winning, and if wealth shares are
equal, then the state prices actually do converge to the true probabilities as the size of the
crowd grows. This occurs because the state prices are actually the average belief in the
crowd, and this average converges to the truth with the size of the crowd.4
But these claims have two important qualifications embedded in them, both of which are
important for understanding the limitations of the wisdom of crowds. First, it is important
that the opinions are independent. We explored the subtleties of non-independent opinions in
Chapter 16, noting the di�culties they pose in reasoning about the behavior of crowds, and
the fact that they can lead to poor aggregate predictions even when many people participate.
Second, it is important that all beliefs are equally weighted. If some bettors have more
wealth than others, then state prices place more weight on their beliefs than on the beliefs of
3The logarithmic utility function is important for the exact form of this relationship. With other utility functions, state prices also depend on individuals’ attitudes toward risk.
4In this discussion we treat bettors’ beliefs as fixed and exogenously given. If instead, bettors’ beliefs di↵er because they have di↵ering information, then bettors should learn from prices. In the case in which beliefs are independent draws from a distribution with mean equal to the true probability, the market price reveals the average belief, and all bettors should use it to update their individual beliefs via the market price.
704 CHAPTER 22. MARKETS AND INFORMATION
those who have little wealth. Whether this reduces or improves the accuracy of state prices
depends on whether the beliefs of these wealthy bettors are more or less accurate than those
of the poorer bettors. One might expect that over time those bettors with more accurate
beliefs would become rich as they tend to make better bets than do those with less accurate
beliefs. If this occurs then more accurate beliefs will have more weight placed on them and
the market price will become a better and better predictor. We investigate this idea more
fully in Section 22.10.
It is also interesting to think about what happens over time to state prices as the bettors
watch horse races and learn about the likelihood of each horse winning. Suppose, for example,
that horses A and B run a race against each other every week, and that the outcomes of
the races are independent. If the true probability of horse A winning is a, then the fraction
of times that A wins the race will converge to a. A bettor who watches the races, and who
initially forecasts a winning probability for A that is not a, should modify his beliefs in light
of his experience. In our examination of Bayesian learning in Chapter 16 we argued that
over time an observer who watches independent events and employs Bayes’ Rule learns the
true probability. (This result is reviewed and expanded on in Section 22.10.) So over time,
each bettor’s belief about the probability of A winning will converge to a, and similarly each
bettor’s belief about the probability of B winning will converge to b. The state prices are
weighted averages of these beliefs, so they too converge to a and b.
22.4 Prediction Markets and Stock Markets
Thus far we have been telling a story about horse races, but there is a direct analogy to
any market where participants purchase assets whose future value depends on the outcome
of uncertain events. Two specific examples are prediction markets and — by far the most
consequential application of these ideas — stock markets. In both cases, we will see that
state prices play a key role in how we reason about what takes place in the market.
Prediction Markets. In a prediction market, individuals trade claims to a one-dollar
return conditional on the occurrence of some event. For example, as we discussed at the
beginning of the chapter, participants might trade claims to a one-dollar return in the event
that a Democrat wins the next U.S. Presidential election. The institutional structure of
prediction markets di↵ers from the structure of a betting market at a race-track. In a
prediction market individuals trade with each other through the market, while at a race
track individuals place bets directly with the track. Nonetheless, prices play the same role
in both markets. The inverse odds on a horse are the cost of a one-dollar return in the event
that the horse wins the race. Similarly, the price of a contract in a prediction market is
the price of a one-dollar return in the event specified in the contract (such as a particular
22.4. PREDICTION MARKETS AND STOCK MARKETS 705
outcome of an election). In both cases the prices reflect an averaging of the beliefs of the
participants in the market.
Here, we will ignore the various institutional structures of prediction markets and instead
see how much we can discover about them by applying our analysis of horse races via state
prices. Consider, for example, the prediction market for the 2008 U.S. Presidential election
with two possible outcomes: a Democrat wins or a Republican wins. (The same analysis
can handle prediction markets with many plausible outcomes, such as the earlier prediction
market for the identity of the Democratic and Republican nominees for President in 2008.)
Let f n
be the share of the total wealth bet in the prediction market that is bet by trader
n. Let a n
and b n
be trader n’s probabilities of a Democrat and a Republican winning, respec-
tively. Then, just as was the case for horse races, the market price ⇢D for the contract on a
Democratic winner will be the wealth-share weighted average of the investors’ probabilities
of a Democrat winning the election. That is,
⇢D = a 1
f 1
+ · · · + a N
f N
. (22.11)
Similarly, the price of the contract on a Republican winner will be the wealth-share weighted
average of the investors’ probabilities of a Republican winning the election. In this discussion
we are looking at a snapshot of the prediction market at one point in time, which we can
think of as a point just before the event occurs. We can also examine the dynamic behavior
of the market over periods of time during which beliefs and wealth shares are both likely to
change; such dynamic questions are the focus of Section 22.10 at the end of this chapter.
Are the prices of these contracts good predictors of the outcome of the election? They
are weighted averages of the beliefs of the investors in the markets, but as we saw with
horse races this doesn’t necessarily make them either good or bad predictors. It depends
on the dispersion of beliefs in the population of investors and on how wealth shares are
distributed across those investors. One way to address this question empirically is to look
at the predictions made by real prediction markets and ask how well they have done at
predicting the outcome of actual events. An interesting paper by Berg, Nelson, and Rietz
[51] shows that for the period 1988–2004, the Iowa Electronic Markets did a significantly
better job of predicting the outcome of U.S. Presidential elections than was done by an
average of the major national polls.
Stock Markets. Stock markets also provide individuals with the opportunity to bet on
future states of the world, but these are more complicated bets, since stocks don’t just o↵er
a one-dollar return in the event that a particular state occurs. Instead a share of stock in
a company o↵ers a monetary amount that will vary depending on which of possibly many
states occurs. These states may be things like “the company’s current investment in research
and development succeeds”, “a strong new competitor enters the market”, “the demand for
706 CHAPTER 22. MARKETS AND INFORMATION
the goods produced by the company grows more rapidly than expected”, or “the workers
go on strike”. Each of these states would have an impact on the future value of the stock
in the company, and so it’s reasonable to think of the stock as providing a di↵erent amount
conditional on each state. The di↵erence between a bet on a two-horse race and a share of
stock, then, is that there are many states, as there would be in a many-horse race, and the
amount of money that the owner of a share of stock has a claim to is not given by explicit
odds but instead is determined by the value of the stock in each of these states.5
If we knew the price of a dollar in each of these states (the state prices), and the value
of the stock in each state, we would know how much investors should be willing to pay for
the stock conditional on each state: it would be the value of the stock in that state times
the price of a dollar in that state. The price of the stock today would be the sum of these
terms across all the states. As long as the collection of stocks traded are rich enough, it is
possible to determine the state prices from the prices of stocks, and conversely to determine
stock prices from state prices. We now give a sense for how this works by means of a simple
example.
State Prices in the Stock Market. The general framework for determining state prices
from stock prices and vice versa, and for specifying when a set of stocks is “rich enough”
to be able to perform this determination, is complex. Here is a streamlined example that
conveys the central idea.
Suppose there are two companies, named 1 and 2, whose stock is being traded. There
are also two possible states, which we’ll call s 1
and s 2
. To be concrete, we can imagine that
state s 1
is “Company 1 does well,” and state s 2
is “Company two does well”. Suppose that
stock in Company 1 is worth one dollar in state s 1
and worth nothing in state s 2
; and stock
in Company 2 is worth one dollar in state s 2
and nothing in state s 1
. Then the stocks are
equivalent to the contracts traded in a prediction market and, just as we saw in a prediction
market, their prices are the market probabilities of the states.
Now let’s suppose, more realistically, that each stock is worth something in each state.
Suppose that stock in Company 1 is worth two dollars in state s 1
and one dollar in state
s 2
; and stock in Company 2 is worth one dollar in state s 1
and two dollars in state s 2
. If
we know the state prices we can determine the price of each stock. Let’s call these prices v 1
for stock in Company 1 and v 2
for stock in Company 2, and let’s write ⇢ 1
and ⇢ 2
to denote
the state prices for states s 1
and s 2
. The price of a share of stock in Company 1 is the
value now of the future worth of the company, which is 2⇢ 1
+ 1⇢ 2
. Intuitively, this is because
we can think of a share of stock in Company 1 as simply a “package deal” that o↵ers two
dollars in state 1 and one dollar in state 2; the price of this package is just the price of its
5Here, just as in the case of horse races we are taking the value of the stock in each event as being given exogenously. This is an important simplification, since in reality the value of the stock in each event is endogenous and is determined in the market equilibrium that arises in that state.
22.4. PREDICTION MARKETS AND STOCK MARKETS 707
constituent ingredients, which if sold separately would require the purchase of two one-dollar
contracts for state 1, at a price of 2⇢ 1
, plus the purchase of a single one-dollar contract for
state 2, at a price of ⇢ 2
. (Of course, these contracts based on the states themselves are not
sold separately; the point is that they only come implicitly “bundled” into the price of the
stock.) Similarly, the price of a share of Company 2 is 1⇢ 1
+ 2⇢ 2
.
Conversely, if we know the price of each stock we can determine the state prices by solving
the system of equations
v 1
=2⇢ 1
+ 1⇢ 2
v 2
=1⇢ 1
+ 2⇢ 2
for the state prices ⇢ 1
and ⇢ 2
. The solutions are
⇢ 1
= 2v
1
� v 2
3
⇢ 2
= 2v
2
� v 1
3 .
With these examples in mind we can now get some idea of what is meant by “a rich
enough set of stocks”. Essentially, we need a set of stocks such that when we write down
their prices as functions of the underlying state prices, we get a system of equations like the
one above that can be solved by a unique set of state prices. If we can do this, then stock
prices determine state prices and investors will be able to use trades in stocks to move money
across states in any way that they want. Indeed, in this case, we can essentially imagine that
there is a big prediction market with a contract for each state, find the equilibrium state
prices, and then determine stock prices from these state prices.
The conclusion of our analysis here is that stock markets, prediction markets, and betting
markets are all essentially the same. They each give individuals the opportunity to place
bets, and they all produce prices which can be interpreted as aggregate predictions about
the likelihood of future states. This point of view also provides us with some intuition
about what causes prices to change: They change if the distribution of wealth changes (and
individuals’ beliefs di↵er) or if individuals’ beliefs about the probability of states change.
If individuals suddenly believe, for whatever reason, that states with high payo↵s are less
likely, then prices will fall. Conversely, if individuals become more optimistic then prices
will rise. Of course, this leaves us with the question of why individuals’ beliefs change. One
possibility is that they are learning about the likelihood of states using Bayes’ Rule.6 If their
observations cause them to predict a less optimistic probability, then prices will fall. If they
are in an environment in which information cascades can occur, then as we saw in Chapter
16, even small events may be able to cause large changes in predictions.
6This topic is explored in more detail in Section 22.10 at the end of this chapter.
708 CHAPTER 22. MARKETS AND INFORMATION
22.5 Markets with Endogenous Events
As we noted at the beginning of this chapter, sometimes the events that individuals have
beliefs about are endogenous — that is, whether they come true depends on the aggregate
behavior of the individuals themselves. To take an example from our discussion of network
e↵ects, if no one expects anyone else to join a particular social-networking site, then no one
expects a positive payo↵ from joining the site; consequently, no one joins and the payo↵ to
joining is indeed low. Alternately, if many people expect a large membership at the social-
networking site, then they expect a large payo↵; as a result, many people join and the payo↵
is indeed high.
Here’s a di↵erent example, from a setting in which there is a market with buyers and
sellers. Suppose that people expect used cars o↵ered for sale to be of uniformly low quality.
Then no one will be willing to pay a high price for a used car. As a result, no one with a
good used car will o↵er it for sale (since it would get a price below what it’s worth), and
so in fact the market will contain only low-quality used cars for sale. On the other hand, if
people expect used cars to have some reasonable average quality, then they may be willing
to pay a price high enough to induce sellers with both good and bad used cars to put them
on the market.
An important common theme in these two stories is the notion of self-fulfilling expec-
tations — and in particular, the presence of multiple di↵erent self-fulfilling expectations
equilibria. This concept was central to our discussion of network e↵ects in Chapter 17, and
we see it again in the example of the used-car market. With one set of expectations, the
world turns out in a certain way that makes the expectations come true; but with a di↵erent
set of expectations, the world would have turned out in a di↵erent way that would have
made those expectations come true.
Asymmetric Information. There is, however, an important di↵erence between these two
stories. In the case of the social-networking site, it seems reasonable to suppose that most
people have similar information about the payo↵s to joining the site. This information may
be more or less accurate, but there is no a priori reason to suppose that some large fraction
of the population is intrinsically more well-informed than some other large fraction. In the
case of the used-car market, however, each seller of a used car knows something about his or
her own car — its quality — that potential buyers do not know. This is an inherent feature
of how the market works: there is asymmetric information.
In Chapter 17 we studied self-fulfilling expectations equilibria in settings without asym-
metric information. In the rest of this chapter, we add asymmetric information to the picture;
this turns out to be a basic ingredient in the way that beliefs about endogenous events can
manifest themselves in markets. There is a fundamental reason for this: in many settings
where buyers and sellers interact, one side of the market has better information about the
22.6. THE MARKET FOR LEMONS 709
goods or services being traded than the other side does. In the market for used cars, sellers
know more than buyers do about the sellers’ cars. On electronic markets for goods, such as
eBay, sellers often know more than buyers do about the goods they are o↵ering for sale. In
the market for health insurance, on the other hand, buyers of insurance often know more
than sellers do about the value of the good (health insurance) being purchased, since a buyer
may well know more about his or her inherent health risks than the company o↵ering the
insurance does. In the stock market, either side of a transaction could have information
about the future value of the stock that is not known to the other side of the transaction (a
feature that we ignored in our earlier discussion of the stock market). In all of these cases,
uninformed traders need to form expectations about the value of the good being traded, and
these expectations should take into account the behavior of better-informed traders.
22.6 The Market for Lemons
At the beginning of the chapter, we started with a simple scenario involving horse-racing
and then showed how the resulting principles extended to much larger and more complex
systems such as the stock market. For considering the role of asymmetric information, we’ll
follow a similar strategy, first developing the case of the used-car market as a simple, stylized
example, and then showing how the same principles apply to a range of more complex and
fundamental markets.
In focusing first on used cars, we’re following the rhetorical lead of the economist George
Akerlof, who published a foundational paper on asymmetric information [9] for which he
shared the 2001 Nobel Prize in Economics. His leading example in the paper was the market
for used cars — or, as he called it, the “market for lemons.” (A used car that is particularly
bad is called a lemon.) The idea behind this phrase is old, probably as old as trading itself,
but Akerlof was the first to clearly articulate the underlying principle and its implications
for how markets work — or, in some cases, how they fail to work. Once we develop the basic
ideas for the example of the used car market, we will then discuss how to apply these ideas
to other markets.
Let’s suppose that there are two types of used cars: good cars, and bad cars. Each seller
knows the quality of his or her own car. Buyers do not know the quality of any seller’s
car, but they are aware of the fact that sellers know the quality of their own cars. Market
participants — the buyers and sellers — value used cars di↵erently. To keep the analysis
simple we will pick the following specific values for used cars.
• Sellers value good cars at 10 and bad cars at 4. (We can imagine these as multiples of a thousand dollars, for example.) These values can be interpreted as sellers’ reservation
prices for their cars. That is, a seller with a good car would be willing to sell it for a
price of at least 10, but at any lower price would prefer to hold onto it. Similarly, a
710 CHAPTER 22. MARKETS AND INFORMATION
a seller with a bad car would be willing to sell it for a price of at least 4, but at any
lower price would prefer to hold onto it.
• Buyers value good cars at 12 and bad cars at 6. These values can be interpreted as buyers’ reservation prices for cars. Thus, a buyer would be willing to buy a car that
is known to be a good car if and only if the price is no more than 12, and a bad car if
and only if the price is no more than 6.
Note that we have assumed that for any type of car, all buyers place the same value on it
and this value is more than the common value that all sellers place on it. This simplification
is not necessary for our analysis, but it will make our example of market failure more striking.
Let’s suppose that a fraction g of used cars are good cars, and hence a fraction 1 � g are bad cars. We’ll also assume that everyone knows g. Finally, let’s suppose that there are
more buyers than used cars (or more buyers than sellers, since each seller has one used car).
The Market with Symmetric Information. As an initial baseline, let’s consider the
simple case in which the type of each car is known to everyone. In this case, since there are
more buyers than sellers, every car could be sold to some buyer.
How would the market work under this assumption? We would expect to observe di↵erent
prices for good cars and bad cars. The price of good cars will clearly be between 10 and 12
because only at prices in this range can all good cars be sold. Similarly, the price of bad
cars will be between 4 and 6. Since there are more buyers than sellers, some buyers will not
be able to purchase a car, and so we would expect prices of each type of car to be bid up
to the upper limit of the range of prices. That is, the price of good cars will be 12 and the
price of bad cars will be 6.
The Market with Asymmetric Information. But what happens if buyers cannot tell
in advance of a purchase what type of car they are buying? Since cars are indistinguishable
to buyers, there can only be one price for a used car — all cars that trade will have to trade
at that price. Furthermore, because a buyer can’t tell the quality of the car she is buying,
the quality of the car she gets is random, based on the mixture of qualities on the market.
Given this random aspect to the outcome, we need to consider how buyers evaluate risk,
just as we did when considering bettors and horse races. To keep the analysis simple in the
current case of the used-car market, we will assume that buyers do not care about risk: they
simply evaluate the expected value of the used cars they are considering. We could also
introduce utility functions that capture the notion that buyers are concerned with risk, as
we did in the earlier parts of this chapter, but in the present case it would complicate the
model without significantly changing our qualitative conclusions.
Let’s consider what the market looks like in the case that buyers can’t distinguish among
di↵erent types of cars. First, the fraction of good cars in the population of used cars for sale
22.6. THE MARKET FOR LEMONS 711
is some number h. This fraction h may be the same as g — the fraction of good cars in the
overall used-car population — but it might not, since not all sellers of good used cars will
necessarily put their cars up for sale. Given this fraction h, the value that any buyer places
on a used car is
12h + 6(1 � h) = 6 + 6h. (22.12)
Thus, in order for buyers to know how much they should be willing to pay for a used car,
they need to have a prediction for the value of h.
This puts us into the domain of self-fulfilling expectations equilibria, similarly to what we
saw in discussing network e↵ects in Chapter 17 (but here with the added issue of information
asymmetry). We will look for a shared expectation h by the buyers that is self-fulfilling, in
that if each buyer expects a fraction h of the cars on the market to be good, then indeed an
h fraction of the cars on the market will be good.
Characterizing the Self-Fulfilling Expectations Equilibria. One candidate for an
equilibrium of this form is h = g. This would be a correct prediction by the buyers if all
sellers indeed choose to o↵er their car for sale. If this occurs, then we can plug h = g into
Equation (22.12) and see that buyers would be willing to pay 6 + 6g for a car. Let’s call this
price p⇤. For the prediction h = g to be correct, it must be the case that at a price of p⇤
both sellers who own good cars and sellers who own bad cars o↵er them for sale. A seller
with a good car would o↵er it for sale at p⇤ provided that
p⇤ = 6 + 6g � 10,
or, equivalently, if g � 2/3. It is easy to see that if a seller with a good car would sell it at p⇤ then a seller with a bad car would also be happy to make a sale. So, if g � 2/3, there is a self-fulfilling expectations equilibrium in which all cars are o↵ered for sale.
Now let’s consider what happens if g < 2/3. Could there be a self-fulfilling expectations
equilibrium in which h = g — that is, in which all cars are o↵ered for sale? We can analyze
this as follows. When g < 2/3, the price that buyers would be willing to pay if they believe
that all cars will be o↵ered for sale, using Equation (22.12), is
p⇤ = 6 + 6g < 10.
However, owners of good cars would not be willing to sell for a price below 10, and so when
g < 2/3 they will keep them o↵ the market — meaning that h, the fraction of good cars
on the market, would not be equal to g. So in this case, there cannot be a self-fulfilling
expectations equilibrium in which h = g.
However, for any value of g, there is always a self-fulfilling expectations equilibrium in
which h = 0 — that is, in which only bad cars are sold. To see why, note that if buyers
expect there to be only bad cars on the market, then they are willing to pay 6 for a car.
712 CHAPTER 22. MARKETS AND INFORMATION
At this price, sellers of bad cars would be willing to sell, but sellers of good cars would not,
and so the market would consist only of bad cars. Thus, this is a self-fulfilling expectations
equilibrium with h = 0.
So to summarize, the value g = 2/3 is a critical point in this example. If g � 2/3, there are two possible self-fulfilling expectations equilibria: one in which all cars are sold, and one
in which only bad cars are sold. If g < 2/3, on the other hand, then the only equilibrium
is the one in which only bad cars are sold. In this latter case, the abundance of bad cars,
combined with buyers’ inability to distinguish good cars from bad cars, has driven the good
cars out of the market.
Complete Market Failure. Our example with good and bad cars illustrates the basic
idea of how equilibria with asymmetric information work, but it doesn’t fully capture the
possible extent of market failure, or how bad the e↵ect can get. To explore this, let’s
consider an example in which there are now three types of used cars: good cars, bad cars,
and lemons. Good cars and bad cars still play the same basic role as in our previous example,
whereas lemons are completely worthless to both buyers and sellers: a market in which only
lemons are o↵ered for sale is in fact not a functioning market at all, since it consists only of
opportunities to trade items of value 0.
For this example with three types of used cars, let’s suppose:
• One-third of the cars are good, one-third are bad and one-third are lemons.
• Sellers value good cars at 10, bad cars at 4, and lemons at 0.
• Buyers value good cars at 12, bad cars at 6, and lemons at 0.
• There are more buyers than there are used cars.
So if there were complete information, we would expect all good and bad cars to be sold
at prices of 12 and 6 respectively, since there are more buyers than sellers, and the buyers
value each type of car at least as much as the sellers do, Whether the lemons are sold is a
question of buyer and seller indi↵erence, since they are worth 0 to everyone.
But with asymmetric information, we need to consider what the possible self-fulfilling
expectations equilibria are. There are three candidates for an equilibrium: (a) all cars are
o↵ered for sale; (b) only bad cars and lemons are o↵ered for sale; or (c) only lemons are
o↵ered for sale. Again, note that option (c) represents the complete failure of the market,
since all items on the market would have value 0. Let’s consider each of these in turn and
see which are actually possible.
(a) First, suppose buyers expect all cars to be on the market. Then the expected value of
a car to a buyer would be 12 + 6 + 0
3 = 6.
22.6. THE MARKET FOR LEMONS 713
This is less than the value that sellers of good cars places on their cars, and so they
would not put them on the market, meaning that this expectation would not be borne
out by what happens. Hence this is not an equilibrium.
(b) Alternately, suppose buyers expect bad cars and lemons to be on the market. Then
the expected value of a car to a buyer would be
6 + 0
2 = 3.
But this is less than the value that sellers of bad cars places on bad cars, and so they
would not put them on the market, meaning again that this expectation would not be
borne out by what happens. So this too is not an equilibrium.
(c) Finally, as in our previous example with two types of cars, it is clearly an equilibrium
if buyers expect only lemons to be sold. In this case, their expected value for a car is
0, and if this is what they are willing to pay, then the market will consist completely
of lemons.
Notice how the market has been subverted by a kind of chain reaction: good cars can’t
survive on the market because of the frequency of bad cars and lemons; and even without
the good cars, the bad cars can’t survive on the market because of the frequency of lemons.
It is not hard to produce this e↵ect with even larger numbers of types of cars: things can
chain together so that, in Akerlof’s words, “it is ... possible to have the bad driving out the
not-so-bad driving out the medium driving out the not-so-good driving out the good in such
a sequence of events that no market exists at all” [9].
Summary: Ingredients of the Market for Lemons. In the next section, we’ll take
the lessons from our used-car examples and apply them to markets that are much larger and
more fundamental. To do this, it’s useful to review the key features of the current examples
that led to market failure:
(i) The items that can be o↵ered for sale have varying qualities.
(ii) For any given level of quality, the buyers value the items of that quality at least as
much as the sellers do – so with complete information, the market would succeed in
allocating items from sellers to buyers, potentially with di↵erent prices for di↵erent
levels of quality.
(iii) There is asymmetric information about the quality of the items — only one side in
a transaction can reliably determine the quality what is being sold. (In the used-car
example, the seller in a potential buyer-seller transaction could tell the quality of what
he was selling. In the next section, we’ll also talk about other basic markets in which
it is the buyer who has this power relative to the seller.)
714 CHAPTER 22. MARKETS AND INFORMATION
(iv) Because of (iii), the items all must be sold for the same uniform price, and sellers will
only put their items up for sale if they value them at or below this uniform price.
The market does not necessarily fail when these ingredients are present. It depends on
whether there is an equilibrium where the buyers expect a mixture of qualities, and hence a
price they are willing to pay, that induces the sellers to put their items up for sale. Market
failure becomes more likely when the fraction of low-quality items is higher, and also when
the di↵erence between buyer and seller values is smaller.
In our discussion we have implicitly compared the market outcome with an outcome
that can be achieved only with full information about each seller’s car. But only the seller
knows the value of his car, so any allocation procedure — not just a market — has to deal
with this problem. Any procedure has to at least implicitly reward sellers for revealing their
information, and this reward drives a wedge between the compensation needed to convince
sellers to participate and the amount that buyers are willing to pay. Determining exactly
which assignments are possible is complicated, but the optimal allocation that would be
possible with full information cannot always be achieved.
22.7 Asymmetric Information in Other Markets
The ideas behind the market for lemons turn out to be fundamental to some of society’s
most important markets. Once you start thinking about interactions in which one party
to the transaction knows something that the other party cares about, you realize that the
phenomenon we’re discussing is far from exceptional; in fact, it occurs all the time.
The Labor Market. One example where these ideas apply very naturally is to the labor
market, in which people seeking jobs play the role of the sellers, and companies seeking
employees play the role of the buyers. That is, we think of the process of employment as a
market where people o↵er their skills for sale to employers, who pay them wages in return.
Let’s consider the basic assumptions of the market for lemons — numbered (i)–(iv) at the
end of the previous section — in the context of the labor market.
(i) There are di↵erent qualities of workers — some are very productive while others are
less productive, and this a↵ects the value they will produce for the company that hires
them.
(ii) It is natural to consider a setting where there are di↵erent kinds of jobs at di↵erent levels
of wages, and where companies would be willing to hire any given prospective employee
if they could accurately determine which jobs and wage levels were appropriate for
them.
22.7. ASYMMETRIC INFORMATION IN OTHER MARKETS 715
(iii) There is asymmetric information: a person generally has a better sense for how pro-
ductive they are than a prospective employer does.
(iv) If we take a strong but plausible version of (iii), where employers can’t reliably de-
termine the quality of the people they are hiring, then employers can’t hire only pro-
ductive workers and wages can’t directly depend on the quality of the person being
hired. Rather, a uniform wage will be o↵ered, and only applicants who believe this
wage acceptably values their skills will take the job.
In this analysis we are assuming that although workers have di↵ering productivities,
each individual’s productivity is a fixed, given amount, and not a↵ected by anything that
the worker chooses to do. It is plausible that workers can a↵ect their productivity by varying
the amount of e↵ort that they put into their job, but we will ignore this issue for the sake
of the present formulation. Thus, the key issue is point (iv) above, which — as in the case
of the used-car market — can be viewed as a problem of adverse selection. The firm cannot
select for a population consisting only of high productivity workers; instead, if it hires any
workers at all, the only thing it can be sure of is getting those with low productivity.
It’s useful to work through the consequences of information asymmetry in the labor
market through a simple example whose structure closely parallels our used-car example.
Suppose a firm hires workers from a large pool of potential employees. Suppose further that
workers come in two types, productive and unproductive, and that half the workers in the
population are of each type. Each productive worker hired by the firm will generate $80,000
of revenue per year for the firm, while each unproductive worker will generate $40,000 of
revenue per year for the firm.
Each worker knows his own type. Also, each worker could choose not to work for the
firm, and instead generate an alternate income by being self-employed. Workers who are
more productive will get more value from being self-employed: suppose that each productive
worker could produce an income of $55,000 per year through self-employment, while each
unproductive worker could produce an income of $25,000 per year through self-employment.
So if the firm could accurately determine the type of each job applicant, the situation would
be straightforward: the firm could o↵er a salary between $55,000 and $80,000 to each pro-
ductive applicant, and a salary between $25,000 and $40,000 to each unproductive applicant,
all job o↵ers would be accepted, and both workers and the firm would benefit from each job
that is taken at the firm.
Unfortunately for the firm, it cannot reliably determine the type of each worker. So the
firm has to o↵er a uniform wage of w, and simply hire workers who are willing to work at
wage w. The firm is willing to o↵er a given wage w if and only if the average revenue it
receives from the workers it hires at this wage is at least w.
716 CHAPTER 22. MARKETS AND INFORMATION
Equilibria in the Labor Market. In our example, what wages can be o↵ered, and which
workers will be willing to work for the firm? The reasoning is very similar to the case of used
cars. We start by looking for a self-fulfilling expectations equilibrium. If the firm expects all
workers to be on the job market, then — since there are equal numbers of the two types of
workers — its expected revenue per employee will be
80, 000 + 40, 000
2 = 60, 000,
and hence it can o↵er a uniform wage of $60,000 per year. At this wage, both types of
workers will be willing to accept the firm’s o↵ers, and so the expectation will be confirmed
by what happens — we have an equilibrium in which all workers are hired.
By analogy with the used-car example, there is also another — less socially desirable —
equilibrium. If the firm expects only unproductive workers to be on the job market, then
it expects to make only $40,000 per year per employee, and so this is the maximum wage
it will o↵er. At this wage, only unproductive workers will be willing to accept jobs, and
so again the firm’s expectations are confirmed. So there are two possible equilibria here —
a high one and a low one, with di↵erent mixtures of workers in the applicant pool in the
two equilibria. Essentially, the firm’s a priori level of confidence in the quality of its job
applicants is self-fulfilling.
Things change if we shift the relative fractions of productive and unproductive workers
in the population. Suppose that only 1/4 of the workers are productive and 3/4 are unpro-
ductive. There is still an equilibrium in which only unproductive workers are hired, but is
there also one in which all workers are hired? If the firm were to expect all workers to be on
the market, then its expected revenue per employee would be
1
4 · 80, 000 +
3
4 · 40, 000 = 50, 000,
and so this is the highest wage it could o↵er. But at this wage, the productive workers
wouldn’t be willing to accept the firm’s o↵ers, and so in fact not all workers would be on the
market. In other words, there is no equilibrium in which the productive workers apply for
jobs at the firm — just as with good used cars, they have been driven out of the market by
the high frequency of unproductive workers.
The Market for Insurance. There are many markets that we can analyze in a similar
fashion. For example, asymmetric information plays an important role in the market for
health insurance. Health insurance companies generally know much less about the health of
those they insure than the insured know about their own health. Health insurance companies
are very good at predicting the average cost of insuring a pool of people, but it is di�cult
for them to predict the cost of insuring any particular individual. They group people into
risk categories based on their medical history, but within any group each individual knows
22.7. ASYMMETRIC INFORMATION IN OTHER MARKETS 717
more about his or her own history, and about how he or she will behave in the future, than
the insurance company knows.
So we have all the ingredients of the market for lemons: individuals in a given risk
category can be more or less costly to insure, but the insurance company cannot reliably
make these fine-grained distinctions. We should also notice an interesting twist in the case
of health insurance: it is the buyers of health insurance, rather than the sellers, who have the
additional information. But the consequence is the same. For any risk category, the insurance
company has to essentially charge a uniform price for the insurance that is su�cient to cover
the average cost of providing health care for the group. This means that the healthiest
individuals in the group are being charged a price that is greater than the expected cost
of providing care for them, and so they may be unwilling to buy insurance. Then, because
these relatively healthy people do not participate, the average quality of the remaining pool
goes down; the insurance company would need to set a higher price for this less healthy pool.
Now the healthiest people in this remaining pool may be unwilling to pay this higher price,
they too may chose not to buy insurance, and the average quality in the pool goes down
further. As in the case of the used-car market, the market for health insurance can unravel
to the point that no one buys insurance. Of course, whether this actually happens depends
on the actual numbers: how much it costs to provide the insurance, and how much people
value the insurance compared to their alternatives. But just as in our earlier examples, we
see how socially undesirable outcomes can occur in the market when there are imbalances
in information.
The information asymmetry we have focused on in the market for health insurance leads,
just as in the cases of used cars or employment, to a type of adverse selection. Insurance
companies cannot select for a population consisting only of healthy individuals; rather, if
anyone buys insurance at all, the only thing one can be sure of is that it will be bought by
those who are less healthy. There is another type of information asymmetry that occurs in
the market for health insurance that we have so far ignored in our discussion. As in the
previous examples, we have treated the health status of each individual, and thus his or her
cost to insure, as fixed and given. But individuals can take actions which a↵ect their health.
If these actions are not observable to the insurance company then we have a new source of
information asymmetry, since each individual knows more about his future behavior than
the insurance company does. Once an individual purchases health insurance, his incentive to
undertake costly actions to maintain his health is reduced, since he no longer bears the full
cost of poor health. This introduces an e↵ect known as moral hazard: when you’re shielded
from the full cost of your potential bad behavior, you have less incentive to avoid engaging
in it.
718 CHAPTER 22. MARKETS AND INFORMATION
Information Asymmetry in Trading and the Stock Market. It is useful to reflect
further on these examples in light of one of the basic lessons of this chapter: that in any
trade, each trader should ask why whoever is on the other side of the trade wants to make
the trade. As we noted at the beginning of this chapter, if one trader is buying then the other
trader is selling, and vice-versa. So the actions of the two traders are exactly the opposite of
each other. Understanding the motivation behind the other trader’s action may be crucial
to understanding whether the trade is actually a good idea. For example, in the used-car
market, a buyer should ask why any seller wants to sell. The same question can be asked
by sellers when they are at a potential information disadvantage: for example, companies
selling health insurance cannot be sure exactly why any one individual is seeking to buy
their insurance.
All these issues play an important role in another market we discussed earlier in the
chapter — the market for financial assets such as stocks or bonds. Here too, for every buyer
there is a seller, and each should be curious about the other’s motivation. A seller of a stock
could be selling because of a desire to adjust their portfolio or a need for cash. A seller of a
stock could be selling because their opinion di↵ers from the opinion reflected in the market
price (the market belief), even though they do not have private information. Alternatively,
a seller could be selling because they have some private information that suggests that the
price of the stock will fall in the future. Similarly, a buyer could want to buy because they
have extra cash to invest, because they simply happen to have a di↵erent belief about the
market, or because they are taking advantage of information that suggests the price of the
stock will rise in the future.
When one side in such stock trades has better information, the other party would value
the stock di↵erently if they too had the information. Determining what the other party
to the transaction knows is often impossible, but understanding that sometimes the other
party knows something is not impossible. Once each party to the transaction takes this into
account it is possible that no trade occurs, just as we saw with the example of used cars
[299].
22.8 Signaling Quality
Given how powerfully information asymmetry can a↵ect the operation of a market, it is
natural to consider methods for alleviating it. One fundamental approach, useful in a number
of settings, is to create a kind of certification mechanism: a way for a seller to provide a
signal about the quality of the good that he or she is o↵ering for sale.
To return to the case of used cars, for example, we can identify a variety of such possible
signals. One that dealers sometimes o↵er is a guarantee that a given car is a “certified used
car.” Dealers certify that these cars have been checked for a number of possible defects and
22.8. SIGNALING QUALITY 719
that any problems have been repaired. Another signaling mechanism is to o↵er a warranty
promising that if the car needs to be repaired during some period after the sale, then the
seller will pay for, or provide for free, the needed repair. Both of these quality assurances
are directly valuable to buyers, but their value is more than you might imagine. It is less
expensive for sellers who have good cars to provide these guarantees or warranties than
it is for sellers who have bad cars. Either fewer repairs are needed before the car is sold
or fewer repairs are expected to be needed after the car is sold. If it is too expensive for
sellers of bad cars to provide these signals, then only good cars will have the signals — or
at least, applying a milder form of this reasoning, the population of cars with these signals
will contain a higher proportion of good cars compared to the population as a whole. Thus,
buyers can make inferences about the quality of the car from the existence of the signal.
These inferences raise the expected value of the car to the buyer even more than the direct
value of the completed or promised future repairs.
Thus, the overall system of warranties might be crucial for breaking down information
asymmetries that could otherwise cause the market to fail.
Signaling in the Labor Market. This idea of signaling applies to many settings other
than just the used-car market. Perhaps its most important application is to the labor market,
in which education can serve as signal; Michael Spence developed this idea and shared the
2001 Nobel Prize in Economics (with George Akerlof and Joseph Stiglitz) for his work on
this topic [379].
Spence’s idea is easy to understand in the context of our earlier labor market example,
where firms cannot initially distinguish productive workers from unproductive ones. Suppose
that it is easier for productive workers to obtain education than it is for unproductive workers.
(Perhaps productive workers also perform better in school, and they can obtain a degree with
less e↵ort.) In this case, education provides a credible signal of productivity, and employers
would be willing to pay higher wages to workers with more education than to workers with
less education.
Notice that this signaling mechanism works even if education has no direct e↵ect on a
worker’s productivity. Of course, education is also intrinsically valuable, but when we take
information asymmetry into account, we see that education has a kind of two-fold power
in the market. It trains workers for future employment; but beyond this, it also reduces
information asymmetry about worker quality in a way that can potentially be necessary for
the labor market to function e↵ectively at all.
720 CHAPTER 22. MARKETS AND INFORMATION
22.9 Quality Uncertainty On-Line: Reputation Sys- tems and Other Mechanisms
Once we adopt the perspective that the availability of information is crucial in many markets,
we can begin to appreciate that many of the standard mechanisms used in Web sites for
on-line commerce are in fact motivated by considerations of asymmetric information and
signaling. In this section we will discuss two of these mechanisms: reputation systems, and
the role of ad-quality measures in sponsored-search advertising.
Reputation Systems. One of the clearest examples of these ideas at work in an on-line
setting is the development of reputation systems for sites like eBay [171]. Since eBay is
designed to facilitate trade between arbitrary people who have never met and may never
meet again, a buyer faces a risk that he is dealing with a bad seller (like a seller of a lemon)
who will provide an item of lower quality than advertised, or possibly fail to provide a
promised item at all. Thus we have a situation that corresponds closely to the market for
lemons: if buyers believe that the chance of receiving bad products (or of being cheated
outright) is too high, then the price they will be willing to pay for an arbitrary item on eBay
will be so low that no seller of reasonable items will want to participate. In this case, eBay’s
market could fail completely.
Reputation systems are a kind of feature provided by Web sites like eBay to o↵er a
certification mechanism for alleviating this problem. After each purchase, the buyer can
provide an evaluation of the seller, reporting whether the transaction and the item they
received met the expectations that were conveyed. The evaluations received by a seller are
synthesized by an algorithm at the heart of the system to provide an overall reputation score
for the seller. A seller’s reputation score evolves over time: favorable evaluations cause the
score to go up while unfavorable ones cause it to go down. Thus, a good reputation score
serves as a signal — in principle, it is costly to obtain, since it requires engaging in a sequence
of transactions that cause the respective buyers to be satisfied. If it’s cheaper for good sellers
to acquire a good reputation than it is for bad sellers to acquire the same reputation, then
reputation can serve as a signal of seller quality, just as a seller certifying his used car or a
worker paying for education serves as a signal. In this way, if a site like eBay can convince
buyers to have confidence in the reliability of the reputation system, then the resulting scores
can reduce some of the strong information asymmetries inherent in the site.
There are many challenges in creating a reputation system that functions e↵ectively, and
a number of these challenges arise from the on-line nature of the application itself [171].
In particular, participants on a site like eBay can generally create multiple identities by
registering for multiple user accounts on the system, and this leads to several approaches
for subverting the goals of the reputation system. First, a seller interested in misbehaving
22.9. QUALITY UNCERTAINTY ON-LINE: REPUTATION SYSTEMS AND OTHER MECHANISMS721
can build up the reputation of a particular identity so that buyers will trust it, then behave
badly until its reputation gets seriously damaged, discard the identity in favor of a freshly
created one, and start the process again. In other words, the reputational consequences
of bad behavior can be mitigated on-line if there is an easy way to “start over” by simply
registering a new identity on the site. This ability to start over adds a severe moral-hazard
feature to the on-line transaction problem, just like the ability of an individual to a↵ect his
health status adds a moral-hazard component to health insurance. This makes the problem
of creating a reliable reputation system more di�cult than it would be if there were only
an adverse-selection problem. In addition, the design of a reputation system is further
complicated by the potential for other kinds of misleading seller behavior. In particular, a
seller can operate several identities simultaneously, and have the di↵erent identities engage in
transactions with one another purely for the purpose of having them lavish positive feedback
on each other. The seller can thereby obtain identities with high reputation scores despite
no genuine history of good behavior.
In spirit, these types of strategies are reminiscent of what we saw in our discussion of
link analysis for Web search — they are extreme versions of the general principle that when
people’s behavior is being evaluated by an algorithm, we should expect that many people
will react and adapt to the criteria of the algorithm in ways that benefit them. Designing
reputation systems that are robust in the presence of these kinds of di�culties is an ongoing
research question.
Ad Quality in Keyword-Based Advertising. The ideas behind the market for lemons
also show up clearly in the systems that search engines use for keyword-based advertising,
and in fact this makes for an interesting case study in how these ideas have influenced a
large on-line market. Specifically, we talked in Chapter 15 about the problem of ad quality:
how the ranking of an ad on a page should not be based purely on the bid o↵ered by the
advertiser, but also on an estimate of the true clickthrough rates that this ad will have in a
given position, compared to other ads. Otherwise, an unappealing ad based on a high bid
could end up clogging the top slot on the page, generating very little revenue for the search
engine because almost no one clicks on it.
But when you look at how the search industry actually runs the market for advertising,
you quickly appreciate that the notion of “ad quality” is not just a proxy for the estimated
rate of clicks the ad will get — it is a more subtle concept that is based on a broader estimate
of overall user satisfaction with the ad. A common scenario here is as follows. There can be
an advertiser that bids very highly for an ad on a certain query, and this ad has enticing text
that causes it to get clicked on at a rapid rate by users from the search results page. A high
bid per click multiplied by a high rate of clicks generates significant revenue for the search
engine. However, the actual page the ad links to (the landing page that users reach when
722 CHAPTER 22. MARKETS AND INFORMATION
they click on the ad) is of low quality — not fraudulent, just not actually very relevant to
the query with which the ad is associated. Consider for example an advertiser that places a
high bid on an ad for the query “Caribbean vacations,” and includes ad text on the Google
results page saying “Dream vacations here” — but when you click on the ad, you get to a
page that’s trying to rent vacation properties in some completely di↵erent part of the world.
It would be natural for most users to be disappointed when they click on this ad.
In such scenarios, the current strategy of search engines is to significantly lower the
placement of such an ad on the page, or not to display it, even though this apparently causes
them to lose the high rate of price-per-click revenue that the ad would generate in a high
position. Their reason for this is the following: If users learn from experience that clicking on
ads often takes them to low-quality landing pages, then they won’t click on ads as much in
general, and this overall e↵ect on user behavior will have a huge negative e↵ect on revenue in
the long run. Essentially, the short-term gain in revenue from high-clickthrough low-quality
ads is being traded o↵ against the long-term losses due to user perceptions of quality.
The problem of asymmetric information is a fundamental issue behind this trade-o↵,
and in fact the market for search advertising exhibits the basic ingredients of the market
for lemons. Although clicking on a single search ad is a much less significant action than
purchasing a car or hiring a new employee — as in our earlier examples of the used-car
market and the labor market — it is still an activity that a user (the buyer) will undertake
only if she believes that what she will find at the other end of the resulting link (the item
being o↵ered by the seller) is worth her e↵ort. And just as a buyer can’t tell the true quality
of a used car until she purchases it, a user can’t tell how well the ad text reflects the true
quality of the landing page until she clicks on the ad. In this respect, advertisers have more
information about the quality of their landing pages than the users of search engines do, and
users implicitly form a mental estimate of how much they expect the ad text to reflect the
quality of the landing page.
Notice, therefore, that while the analogy to the market for lemons is quite natural, it is
also a bit subtle. In particular, it is not about the relationship between the advertisers and
the search engine (though one can look for information asymmetry there too), but between
the users and the advertisers, with user e↵ort in clicking on ads as the quantity being valued.
In aggregate, of course, all these user decisions to click are crucial, since they add up to a
large portion of the search industry’s revenue.
Now, we have seen in our earlier analyses that there can be multiple self-fulfilling ex-
pectations equilibria in these types of markets: some where buyers predict high average
quality and there are in fact high-quality items for sale; and some where buyers predict low
average quality and only low-quality items are on the market. Such equilibria are based on
the assumption that consumers make correct predictions, which makes sense in the case of
search advertising if users have time to learn the distribution of ad quality. Since the search
22.10. ADVANCED MATERIAL: WEALTH DYNAMICS IN MARKETS 723
engines have control over the ads they display, they are trying to maintain a mixture of
ads of reasonable quality, thereby selecting an equilibrium for the overall market in which
users expect high-quality ads, and advertisers with high-quality content are correspondingly
willing to advertise via search engines.
22.10 Advanced Material: Wealth Dynamics in Mar- kets
When we considered markets for assets such as stocks, shares in a prediction market, or bets
in a horse race, we observed that market prices serve to aggregate the beliefs of the mar-
ket participants — essentially, the market produces a weighted average of the participants’
beliefs, with the weights determined by the participants’ relative shares of wealth. Now,
if we were to watch the market as it runs over time, certain participants would do better
than others, their wealth shares would increase, and as a result their overall e↵ect on the
aggregate market price would increase. If we expect that people with more accurate beliefs
will do better in the market, then this re-weighting as wealth shifts toward them should in
fact produce more accurate market prices.
This intuition about market evolution over time was developed in the writings of a number
of economists in the mid-20th-century [11, 157, 172]. The basic argument is that markets
impose a kind of “natural selection” favoring traders whose decisions are closest to optimal.
Early writers used this idea to argue that one tends to find rational investors in markets
(because the others will have been driven out of the market) and that stock markets tend to
be e�cient (because prices are determined by the traders who have survived over the long
run).
It has only been relatively recently, however, that this general idea has been explored
more precisely, and its scope and limitations have begun to be understood. In this section,
we describe a basic mathematical analysis that formalizes the intuition at the heart of these
ideas [64]. The analysis will work by developing a close analogy between wealth dynamics
in a market and the use of Bayes’ Rule by an individual who learns over time. Recall from
Chapter 16 that Bayes’ Rule provides a systematic way to make use of new observations in
decision-making. We will see that as wealth moves between participants in a market, their
contributions to the aggregate market price change over time exactly like the probabilities
on di↵erent hypotheses would change according to Bayes’ Rule.
So in a precise sense, although the market is simply an institution that facilitates trade,
it can also be viewed as acting like an artificially intelligent Bayesian agent that aggregates
information. Moreover, if there is a set of traders who have correct beliefs, then over time
their fraction of the wealth converges to one, and the market price converges to reflect their
(correct) beliefs. This provides a concrete expression of the general idea that markets for
724 CHAPTER 22. MARKETS AND INFORMATION
assets tend to work well at synthesizing the information held by groups of people.
A. Bayesian Learning in a Market
We begin the analysis by considering how a Bayesian learner — that is, someone applying
Bayes’ Rule — would update his beliefs over time in a market. Once we’ve done this, we’ll
draw an analogy to how the wealth of participants changes over time.
We discussed Bayes’s Rule in Chapter 16, but here we’ll cast it in the notation of this
chapter, and also extend some of the conclusions. Since horse races have served as a useful
example to suggest the phenomena at work in more complex settings like the stock market,
we’ll continue to use horse races in the discussion here. Thus, suppose that two horses A
and B will run a race every week; suppose that the outcomes of these races are independent;
and suppose that A wins each one with probability a (and hence B wins with probability
b = 1 � a). Now, our Bayesian learner does not know the values of a and b; rather, he wants to
learn them over time by watching the outcomes of races. He begins with a set of N possible
hypotheses for the probabilities, which we will denote by
(a 1
, b 1
), (a 2
, b 2
), . . . , (a N
, b N
).
For now, let’s assume in fact that one of these hypotheses is correct (although the learner
does not know which it is); re-labeling them if necessary, we’ll suppose that (a 1
, b 1
) = (a, b).
The learner begins with a prior probability on each hypothesis; let f n
be the prior prob-
ability on hypothesis (a n
, b n
). We will assume that each prior probability f n
is greater than
zero, indicating that the learner considers it to be a possible description of the true proba-
bility. Since these prior probabilities form an initial weighted average over the hypotheses,
the learner’s initial predicted probability of A winning is
a 1
f 1
+ a 2
f 2
+ · · · + a N
f N
.
Now, suppose that T races are run, and we observe a sequence S of outcomes of these races,
in which horse A wins a total of k times and horse B wins a total of ` times. Then using
Bayes’ Rule as in Chapter 16, we can compute the posterior probability of the hypothesis
(a n
, b n
), conditional on the sequence S, as follows.
Pr [(a n
, b n
) | S] = f
n
· Pr [S | (a n
, b n
)]
Pr [S]
= f
n
· Pr [S | (a n
, b n
)]
f 1
· Pr [S | (a 1
, b 1
)] + f 2
· Pr [S | (a 2
, b 2
)] + · · · + f N
· Pr [S | (a N
, b N
)] .
Now, the probability of S given a hypothesis (a n
, b n
) is simply the probability it assigns to
the sequence of wins in S: since horse A wins a total of k times and horse B wins a total of
22.10. ADVANCED MATERIAL: WEALTH DYNAMICS IN MARKETS 725
` times, this probability is just ak n
b` n
. Therefore, we have
Pr [(a n
, b n
) | S] = f
n
ak n
b` n
f 1
ak 1
b` 1
+ f 2
ak 2
b` 2
+ · · · + f N
ak N
b` N
. (22.13)
Moreover, after this sequence S of observed outcomes, the learner’s predicted probability on
horse A is
a 1
Pr [(a 1
, b 1
) | S] + a 2
Pr [(a 2
, b 2
) | S] + · · · + a N
Pr [(a N
, b N
) | S] . (22.14)
This is the sense in which the learner is Bayesian: as he observes outcomes, he updates his
predicted probability according to Bayes’ Rule.
Convergence to the Correct Hypothesis. Now, let’s consider how the posterior proba-
bilities of the di↵erent hypotheses fare as horse races are run over a long period of time. The
easiest way to do this is to consider the ratios of these probabilities. After a sequence S of
observed outcomes, with k wins by A and ` wins by B, the ratio of the posterior probability
on hypothesis (a m
, b m
) to the posterior probability on hypothesis (a n
, b n
) can be computed
simply by taking the ratios of the respective expressions given by Equation (22.13), and
noticing that the two expressions have the same denominator:
Pr [(a m
, b m
) | S] Pr [(a
n
, b n
) | S] =
f m
ak m
b` m
f n
ak n
b` n
(22.15)
We will be particularly interested in ratios that compare the correct hypothesis (a 1
, b 1
)
to an alternate hypothesis (a n
, b n
):
Pr [(a 1
, b 1
) | S] Pr [(a
n
, b n
) | S] =
f 1
ak 1
b` 1
f n
ak n
b` n
(22.16)
We will call this ratio R n
[S]. Taking logarithms yields something known as the log odds ratio
of the two hypotheses, given the sequence of observed outcomes S:
ln(R n
[S]) = ln
✓ f
1
f n
◆ + k ln
✓ a
1
a n
◆ + ` ln
✓ b 1
b n
◆ .
Now, let’s divide both sides by the total number of observations T , obtaining
1
T ln(R
n
[S]) = 1
T ln
✓ f
1
f n
◆ +
k
T ln
✓ a
1
a n
◆ +
`
T ln
✓ b 1
b n
◆ . (22.17)
We’re interested in what happens as T goes to infinity, and in this case the right-hand
side of this equation can be simplified as follows. The first term is just a fixed constant
ln(f 1
/f n
) divided by T , so it converges to 0 as T grows. To analyze the second and third
terms, we observe, using the Law of Large Numbers, that k/T converges almost surely to the
726 CHAPTER 22. MARKETS AND INFORMATION
true probability of horse A winning, which is a, and `/T converges almost surely to the true
probability of horse B winning, which is b. So the entire right-hand side of Equation (22.17)
converges almost surely to
a ln
✓ a
1
a n
◆ + b ln
✓ b 1
b n
◆ = a ln(a
1
) + b ln(b 1
) � [a ln(a n
) + b ln(b n
)] (22.18)
We’d like to know whether this limit is positive, negative, or zero, since that will let us
reason about what’s happening to the left-hand side of Equation (22.17). Here is how we
can think about this limit. The first two terms have the form a ln(x) + (1 � a) ln(1 � x), with x = a
1
, and the third and fourth terms have this form as well, with x = a n
. But we
know from Section 22.2, and specifically the discussion around Equation (22.5), that the
expression a ln(x) + (1 � a) ln(1 � x) is maximized when x = a, and it is strictly smaller than this maximum for all other values of x. Since a
1
= a, the sum of the first two terms
therefore achieves this maximum, and since a n
6= a, the third and fourth terms that are being subtracted o↵ don’t achieve the maximum. Therefore, the expression in (22.18) is strictly
positive (since the first two terms outweigh the latter two), and so returning to Equation
(22.17), we conclude that 1
T ln(R
n
[S]) > 0
almost surely as T goes to infinity.
It follows that as T goes to infinity, ln(R n
[S]), and hence R n
[S] itself, must be diverging
to positive infinity. Moreover, this takes place for every n > 1 — that is, for every incorrect
hypothesis. How can this happen? Each R n
[S] is the ratio of two probabilities, so in order
for one of the probabilities (on (a 1
, b 1
)) to become larger than all the others by an arbitrary
factor, it must be that the probability on hypothesis (a 1
, b 1
) is converging to one while the
probability on each of the others is converging to zero.
The conclusion from this analysis is that the Bayesian learner will, in the limit, assign
a posterior probability of one to the correct hypothesis. Moreover, this means that his
predicted probability on horse A, as computed by Equation (22.14), is converging to a 1
= a.
Convergence without a Correct Hypothesis. If we think about it, the analysis above
in fact shows something stronger than we’ve claimed. In order for the learner to converge
to a posterior probability of one on the hypothesis (a 1
, b 1
), it is not necessary that (a 1
, b 1
)
actually be correct. We simply need that the expression in Equation(22.18) is positive for
all competing hypotheses n > 1.
The interpretation of this stronger claim is usually expressed in terms of a notion of
“distance” between hypotheses, as follows. For a given hypothesis (a n
, b n
), we define the
relative entropy D (a,b)
(a n
, b n
) between (a n
, b n
) and the true hypothesis (a, b) to be
D (a,b)
(a n
, b n
) = a ln(a) + b ln(b) � [a ln(a n
) + b ln(b n
)]. (22.19)
22.10. ADVANCED MATERIAL: WEALTH DYNAMICS IN MARKETS 727
By our earlier observation about the maximization of a ln(x) + (1�a) ln(1�x), we see that the contribution of the first two terms must always outweigh the negative e↵ect of the third
and fourth terms, and so D (a,b)
(a n
, b n
) is always a non-negative number, and it is zero when
(a n
, b n
) = (a, b). We can therefore interpret the relative entropy as a non-linear measure
of how far a given hypothesis is from the truth: smaller relative entropies indicate better
agreement with the true hypothesis.
Going back to Equations (22.17) and (22.18), we see that even when (a 1
, b 1
) is not the
correct hypothesis, the quantity ln(R n
[S])/T is converging almost surely to
D (a,b)
(a n
, b n
) � D (a,b)
(a 1
, b 1
).
Therefore, suppose that a 1
6= a, but that the hypothesis (a 1
, b 1
) is closer than any other
(a n
, b n
) to the true hypothesis in relative entropy: that is,
D (a,b)
(a 1
, b 1
) < D (a,b)
(a n
, b n
)
for all n > 1. Then just as before, we have
1
T ln(R
n
[S]) > 0
almost surely as T goes to infinity. And from this, we draw the same conclusion as before:
that the posterior probability the learner places on (a 1
, b 1
) converges to one.
In other words, when no hypothesis is correct but some hypothesis is uniquely closest to
the truth in relative entropy, a Bayesian learner will assign a posterior probability of one to
this hypothesis in the limit.
B. Wealth Dynamics
We have now seen how a Bayesian learner aggregates information about events taking place in
a market: the learner maintains a weighted average of the probabilities assigned by di↵erent
hypotheses, updating the weights using Bayes’ Rule. Earlier in the chapter, we saw that
the odds computed by the market are also a weighted average, in that case an average of
bettors’ beliefs weighted by their wealth shares. As time runs forward, the weights in this
weighted average are updated simply because the bettors are gaining and losing money. We
now show that this updating works exactly the way Bayes’ Rule does, which is why the
aggregate behavior of the market itself can be viewed as that of a Bayesian learner.
Evolution of Wealth Shares. Let’s use the framework of betting markets from Sec-
tions 22.2 and 22.3 (again as a stand-in for more complex settings like stock markets). There
are N bettors; each bettor n has a fixed belief that horse A will win with probability a n
(and
hence that horse B will win with probability b n
= 1 � a n
). Bettor n has an initial wealth of
728 CHAPTER 22. MARKETS AND INFORMATION
w n
; if the total wealth of all bettors is w, then this corresponds to a share f n
= w n
/w of the
total wealth.
Now, horses A and B race each other in each of time steps t = 1, 2, 3, . . .. At the start
of each time step t, before the tth race, the market determines odds o hti A
and o hti B
on horses
A and B; note that the odds may be di↵erent in each step, and as we saw in Section 22.3,
they may depend on who the bettors are and how much they are betting. Also at the start
of each step t, each bettor n has a current wealth w hti n
; he bets this wealth optimally given
his beliefs (a n
, b n
). As we saw in Section 22.2, this corresponds to putting a bet of a n
w hti n
on
horse A and a bet of b n
w hti n
on horse B. Consequently, bettor n’s new wealth w ht+1i n
after
this race is equal to a n
w hti n
o hti A
if A wins, and it is equal to b n
w hti n
o hti B
if B wins.
Let’s consider two bettors m and n, with initial wealth shares f m
and f n
, and suppose
that by step t, due to the results of their bets on the first t�1 races, their wealth shares are f hti m
and f hti n
respectively. Let’s consider the two possible outcomes of race t.
• If horse A wins race t, the wealth of bettor m is multiplied by a m
o hti A
and the wealth of
bettor n is multiplied by a n
o hti A
. Therefore, in this case, the ratio of the wealth shares
of m and n changes from f hti m
/f hti m
to a m
f hti m
/a n
f hti m
. (Notice that the odds cancel out
of this ratio, since they apply equally to both bettors.) In other words, the ratio is
multiplied by a m
/a n
.
• If horse B wins race t, the wealth of bettor m is multiplied by b m
o hti B
and the wealth of
bettor n is multiplied by b n
o hti B
. Therefore, in this case, the ratio of the wealth shares of
m and n changes from f hti m
/f hti m
to b m
f hti m
/b n
f hti m
. In other words, the ratio is multiplied
by b m
/b n
.
So we see that whenever horse A wins a race, the ratio of wealth shares of bettors m and n
changes by a factor of a m
/a n
, while whenever horse B wins a race, the ratio of their wealth
shares changes by a factor of b m
/b n
.
Suppose we apply these changes, starting from wealth shares f m
and f n
, over a sequence
of races S in which A wins k times and B wins ` times. Then we end up with a ratio of
wealth shares that’s equal to f
m
ak m
b` m
f n
ak n
b` n
. (22.20)
The point is that this is exactly the same as Equation (22.15), describing the ratio of posterior
probabilities that a Bayesian learner puts on the hypotheses (a m
, b m
) and (a n
, b n
), starting
from prior probabilities of f m
and f n
. So the analogy is perfect: the wealth shares of the
bettors evolve exactly like the posterior probabilities on hypotheses under Bayes’ Rule. That
is, the market treats each bettor as a hypothesis about the two horses, and in response to
the outcome of the race it adjusts that bettor’s wealth share in exactly the same way that a
Bayesian learner would adjust the probability on the hypothesis.
22.10. ADVANCED MATERIAL: WEALTH DYNAMICS IN MARKETS 729
We can draw two main conclusions from this.
• First, the inverse odds maintained by the market are computed from the wealth-share- weighted average of the bettors’ beliefs, using Equation (22.9) from Section 22.3. This
equation is parallel to the Bayesian learner’s Equation (22.14) by which he determines
the predicted probability on horse A. Hence, the market’s inverse odds follow the
results of Bayesian learning as well.
• Since the ratios of wealth shares evolve just as the posterior probabilities of hypotheses, we can conclude that if there is a unique bettor whose beliefs are closest in relative
entropy to the correct probabilities (a, b), then in the limit the wealth share of this
bettor will converge to 1. So the market is selecting for bettors with more accurate
beliefs, where “accuracy” here refers to the bettor’s distance from the truth in relative
entropy. Combined with our previous observation about the inverse odds, we see that
in the limit, the assets (i.e. the possible bets) are priced according to the most accurate
information held by any of the market participants.
It is also important to note that in the special case when one of the bettors has correct
beliefs, this bettor will acquire a wealth share of 1 in the limit, and the market will come to
reflect the bettor’s (correct) beliefs.
Extensions and Interpretations. We have kept the model very simple so as to make
the calculations clear. But it is possible to extend the model to incorporate a number of
further considerations.
First, we are assuming that the bettors have fixed beliefs and do not learn from observing
the outcomes of races. This makes it easy to isolate the e↵ect of wealth dynamics in the
market, distinguishing it from the learning dynamics of individual participants. But while it
is a bit messy, it is not particularly di�cult to combine this analysis of wealth dynamics with
Bayesian learning by the bettors. Second, our analysis assumes that each bettor reinvests
his entire wealth in the market in each time step. However, this too can be extended to a
model in which bettors must decide both how much to reinvest in the market, as well as how
to allocate this investment across the di↵erent options [64].
Our overall conclusion, that the market selects for the trader with the most accurate
beliefs, and asymptotically prices assets according to these beliefs, applies equally well in
other settings such as prediction markets. Notice that the argument here about the perfor-
mance of the market is not based on the benefits of averaging, as in our previous discussion
of the “wisdom of crowds.” Rather, in the analysis here, the crowd is exactly as smart as its
smartest participant in the limit, since in the limit it is only this participant whose beliefs af-
fect the market’s predictions. As noted earlier, this idea draws on a long history of economic
arguments for market e�ciency based on natural selection [11, 157, 172], in which smarter
730 CHAPTER 22. MARKETS AND INFORMATION
traders come to hold an increasingly large fraction of the wealth in the market, and thereby
exert an increasingly large influence on the market. The model here puts this intuition on
more precise footing [64], and subsequent research has expanded on it in important ways
[65, 362].
While these expanded models are too complex to describe in detail here, they relate in
interesting ways to some of the issues from earlier in the chapter. First, and rather surpris-
ingly, the more complex models show that the assumption of logarithmic utility, on which
the model here is based, is in fact not important for the general conclusion about market
selection. A more general and abstract analysis shows that we only need the assumption
that traders are risk-averse — that is, their utility gain from increased wealth decreases as
their wealth grows. The recent line of research also shows that these results apply to more
complex markets, provided that there is a rich enough set of assets being traded. Intuitively,
if there aren’t enough traded assets, then there may not be enough ways for traders with
better beliefs to take advantage of traders with worse beliefs, and thus the traders with worse
beliefs may not get driven out of the market. The richness condition that is needed for stock
markets is exactly the condition discussed in Section 22.4. The conclusion of this analysis is
that if there is a rich enough set of assets traded in the stock market, then in the long run
the market prices assets as correctly as possible given the collection of traders’ beliefs that
are made available to the market.
22.11 Exercises
1. Consider a betting market with two horses A and B and two bettors 1 and 2, as in
Section 22.3. Let’s suppose that each bettor has wealth w. Bettor 1 believes there is a
probability of 1/2 that horse A will win, and a probability of 1/2 that horse B will win.
Bettor 2 believes there is a probability of 1/4 that horse A will win, and a probability
of 3/4 that horse B will win. Both bettors have logarithmic utility for wealth, and
they each choose bets to maximize expected utility of wealth given their beliefs.
(a) How much money should bettors 1 and 2 each bet on horses A and B respectively?
(b) Find the equilibrium inverse odds on horse A and on horse B.
(c) How much money will bettor 1 have after the race if horse A wins? How about if
horse B wins?
2. Consider a betting market with two horses A and B and two bettors 1 and 2 as in
Section 22.3. Let’s suppose that each bettor has wealth w. Bettor 1’s beliefs are (a 1
, b 1
)
where the first number in the pair is bettor 1’s probability of horse A winning the race.
Both bettors have logarithmic utility for wealth. Bettor 1 chooses his bets to maximize
22.11. EXERCISES 731
his expected utility of wealth using his beliefs, as in the chapter. Bettor 2, however,
behaves di↵erently; he believes that the inverse odds are the correct probabilities and
he maximizes his expected utility using these inverse odds.
(a) Bettor 1’s optimal bet on horse A is some function of his wealth and his beliefs.
Let’s call this function f 1
(w, a 1
). Determine this function.
(b) Suppose that bettor 2 knows the equilibrium inverse odds on horse A, which we
will call ⇢ A
. Bettor 2’s optimal bet on horse A is some function of his wealth
and the equilibrium inverse odds on horse A. Let’s call this function f 2
(w, ⇢ A
).
Determine this function.
(c) If we take Equation (22.8) from Section 22.3, applied to the betting rules in
question, then we see that the equilibrium inverse odds on horse A must solve the
equation f
1
(w, a 1
)
2w +
f 2
(w, ⇢ A
)
2w = ⇢
A
.
Using this observation, find the equilibrium inverse odds on horse A.
(d) Now let’s generalize this idea to many bettors. Suppose that most bettors are
like better 2; they trust that the inverse odds are in some sense correct and they
use them in deciding how to bet. Only a few bettors are like bettor 1; they have
beliefs and they bet according to their beliefs. Would you expect the ”Wisdom of
Crowds” idea to be more or less likely to be true in this market than in a market
in which each bettor has beliefs and bets according to their own beliefs? Does
your answer depend on which bettors bet using inverse odds as their beliefs and
which ones use their own beliefs? (Think about which bettors are more likely to
have correct beliefs.)
3. Consider the model of the market for lemons. Suppose that there are three types of
used cars: good ones, medium ones and lemons, and that sellers know which type of
car they have. Buyers do not know which type of car a seller has. The fraction of used
cars of each type is 1 3
and buyers know this. Let’s suppose that a seller who has a
good car values it at $8,000, a seller with a medium car values it at $5,000 and a seller
with a lemon values the lemon at $1,000. A seller is willing to sell his car for any price
greater than or equal to his value for the car; the seller is not willing to sell the car
at a price below the value of the car. Buyers values for good cars, medium cars and
lemons are, $9,000, $8,000 and $4,000, respectively. As in Chapter 22 we will assume
that buyers are risk-neutral; that is, they are willing to pay their expected value of a
car.
(a) Is there an equilibrium in the used-car market in which all types of cars are sold?
Explain briefly.
732 CHAPTER 22. MARKETS AND INFORMATION
(b) Is there an equilibrium in the used-car market in which only medium quality cars
and lemons are sold? Explain briefly.
(c) Is there an equilibrium in the used-car market in which only lemons are sold?
Explain briefly.
4. Consider the model of the market for lemons from Chapter 22. Suppose that there are
two types of used cars — good ones and lemons — and that sellers know which type of
car they have. Buyers do not know which type of car a seller has. The fraction of used
cars of each type is 1 2
and buyers know this. Let’s suppose that a seller who has a good
car values it at $10,000 and a seller with a lemon values the lemon at $5,000. A seller
is willing to sell his car for any price greater than or equal to his value for the car; the
seller is not willing to sell the car at a price below the value of the car. Buyers’ values
for good cars and lemons are $14,000 and $8,000, respectively. As in Chapter 22 we
will assume that buyers are risk-neutral; that is, they are willing to pay their expected
value of a car.
(a) Is there an equilibrium in the used-car market in which all types of cars are sold?
Briefly explain.
(b) Is there an equilibrium in the used-car market in which only lemons are sold?
Briefly explain.
5. Consider the model of the market for lemons. Suppose that there are three types of
used cars: good ones, medium ones, and lemons, and that sellers know which type of
car they have. Buyers do not know which type of car a seller has. The fraction of used
cars of each type is 1 3
and buyers know this. Let’s suppose that a seller who has a
good car values it at $4,000, a seller with a medium car values it at $3,000 and a seller
with a lemon values it at $0. A seller is willing to sell his car for any price greater
than or equal to his value for the car; the seller is not willing to sell the car at a price
below the value of the car. Buyers’ values for good cars, medium cars, and lemons are
$10,000, $4,000 and $1,000, respectively. As in Chapter 22 we will assume that buyers
are willing to pay their expected value of a car. We will also assume that there are at
least as many buyers as used cars.
(a) Is there an equilibrium in the used car market in which all types of used cars are
sold? If so, find some equilibrium price for used cars such that all used cars are sold,
together with a brief explanation of why all cars are sold. If not, explain why not.
(b) Now suppose that someone develops a way for sellers of good used cars to certify
that their cars are good cars. All sellers of good used cars do this and they are no
longer part of the general market for uncertified used cars, which now consists only of
22.11. EXERCISES 733
medium used cars and lemons in equal numbers. Is there an equilibrium in the market
for these remaining, uncertified used cars in which both medium used cars and lemons
are sold? If so, find some equilibrium price for used cars such that medium used cars
and lemons are sold, together with a brief explanation. If not, explain why not.
6. Consider the model of the market for lemons. Suppose that there are two types of used
cars, good ones and lemons, and that sellers know which type of car they have. Buyers
do not know which type of car a seller has. The fraction of used cars that are good
cars is g and buyers know this fraction. Let’s suppose that a seller who has a good car
values it at $10,000 and that a seller with a lemon values the lemon at $4,000. A seller
is willing to sell his car for any price greater than or equal to his value for the car; the
seller is not willing to sell the car at a price below the value of the car. Buyers values
for good cars and lemons are, $12,000 and $5,000, respectively. As in Chapter 22 we
will assume that buyers are risk neutral; that is, they are willing to pay their expected
value of a car.
(a) Suppose that you observe that used cars sell for a price of $10,000. What can you
say about the fraction of used cars that are lemons?
(b) Suppose, instead that the fraction of used cars that are lemons is g = 0.5. What
is the maximum selling price for used cars?
7. In this question we are going to examine how a tax on the purchase of used cars might
a↵ect the price and quantity of used cars traded. Suppose that there are two types
of used cars: good ones and bad ones. Sellers of used cars know the type of car that
they own. Buyers do not know which type of car any particular seller has. Buyers do
know that there are good and bad used cars, and they know that of the 100 people
who own used cars and are interested in selling their car, 50 have good cars and 50
have bad cars. Let’s suppose that there are 200 possible buyers of used cars. [As in
Chapter 22 we want to assume that there are more buyers than sellers to make the
analysis straightforward.] A seller who has a good used car values it at $8,000 and a
seller who has a bad used car values it at $3,000. A seller is willing to sell his car for
any price greater than or equal to his value for the car; no seller is willing to sell his
car for a price less than his value for the car. Buyers values for good and bad used
cars are $10,000 and $6,000, respectively. As in Chapter 22 we will assume that buyers
each want at most one used car and they are willing to pay their expected value for a
used car.
(a) Find all of the equilibria in the market for used cars. For each equilibrium provide
the price of used cars and the number of used cars traded.
734 CHAPTER 22. MARKETS AND INFORMATION
(b) Now suppose that the government places a tax of $100 on the purchase of used
cars. That is, anyone who buys a used car must pay a tax of $100 on the purchase of
the car. This e↵ectively lowers the values that buyers place on any type of used car by
$100. Find all of the equilibria in the market for used cars.
(c) Now let’s change the setup of the problem a bit so that there are three types of
used cars: good ones, bad ones and lemons. There are 50 sellers with good cars, 50
with bad cars and 50 with lemons. Buyers and sellers values for good and bad used
cars are the same as before. Everyone (both buyers and sellers) values a lemon at 0.
There are still 200 buyers. (i) There is no tax on the purchase of used cars. Find all
of the equilibria in the market for used cars. (ii) Now the government imposes a tax
of $100 on the purchase of used cars. Find all of the equilibria in the market for used
cars.
8. A group of researchers have been investigating the quality and seaworthiness of five-
year-old boats in the U.S. They are using a classification of the boats into five possible
categories: excellent, good, medium, poor, and dangerous. They have concluded that
there are no excellent five-year-old boats, and that most of these boats are of medium
and/or lower quality. To conduct their study these researchers pretended to be poten-
tial buyers of five-year-old boats. They examined a very large number of five-year-old
boats o↵ered for sale both by private sellers (individuals) and by boat dealers. Based on
the results of their study this group of researchers has concluded that there should be
an investigation by the U. S. Coast Guard into the quality of these older boats. What
concerns do you have about the methodology the researchers used in their study? Can
you suggest an alternative approach that they might have used in order to draw a more
careful conclusion about the actual quality distribution of five-year-old boats?
Chapter 23
Voting
In the previous chapter, we saw a first example of an institution that can synthesize
information held by many people, through the ways in which markets serve to aggregate the
individual beliefs of investors. We now turn to a second fundamental institution: voting.
23.1 Voting for Group Decision-Making
Like markets, voting systems also serve to aggregate information across a group, and as a
result, it’s hard to draw a perfectly clear dividing line between these two kind of institutions.
But there are definite distinctions between the respective settings in which they are typically
applied. A first important distinction is that voting is generally used in situations where a
group of people is expressly trying to reach a single decision that in a sense will speak for
the group. When a population votes on a set of candidates or ballot initiatives, a legislative
body votes on whether to pass a bill, a jury votes on a verdict in a trial, a prize committee
votes on the recipient of an award, or a group of critics votes on the top movies of the past
century, the resulting decision is a single outcome that stands for the group, and has some
kind of binding e↵ect going forward. In contrast, markets synthesize the opinions of a group
more indirectly, as investors’ beliefs are conveyed implicitly through their transactions in
the market — choosing how much to invest or to bet, choosing whether to buy or not to
buy, and so forth. The overt goal of the market is to enable these transactions, rather than
any broader synthesis or group decision that might in fact arise from the transactions in
aggregate.
There are other important distinctions as well. A simple but important one is that the
choices in a market are often numerical in nature (how much money to transact in various
ways), and the synthesis that takes place generally involves arithmetic on these quantities —
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
735
736 CHAPTER 23. VOTING
weighted averages and other measures. Many of the key applications of voting, on the other
hand, take place in situations where there’s no natural way to “average” the preferences of
individuals — since the preferences are over di↵erent people, di↵erent policy decisions, or
di↵erent options under a largely subjective criterion. Indeed, as we will see in this chapter,
much of the richness of the theory of voting comes from precisely this attempt to combine
preferences in the absence of simple metaphors like averaging.
The notion of voting encompasses a broad class of methods for reaching a group decision.
For example, the methods to reach a jury verdict, an outcome in a U.S. Presidential Election,
or a winner of college football’s Heisman Trophy are all distinct, and these distinctions have
e↵ects both on the process and the result. Moreover, voting can be used in settings where
a single “winner” must be chosen, as well as in situations where the goal is to produce a
ranked list. Examples of the latter include the ranking of college sports teams by aggregating
multiple polls, or published rankings of the greatest movies, songs, or albums of all time by
combining the opinions of many critics.
Voting is often used in situations where the voters disagree because of genuine divergence
in their subjective evaluations. For example, film critics who disagree on whether to rank
Citizen Kane or The Godfather as the greatest movie of all time are generally not disagreeing
because they lack relevant information about the two movies — we can expect that they
are closely familiar with both — but because of di↵ering aesthetic evaluations of them. In
other cases, however, voting is used to achieve group decisions where the di�culty is a lack
of information — where the members of the group would likely be unanimous if they had all
the information relevant to the decision. For example, jury verdicts in criminal trials often
hinge on genuine uncertainty as to whether the defendant committed the crime; in such
cases one expects that jurors all have approximately the same goal in mind (determining
the correct verdict), and the di↵erences are in their access to and processing of the available
information. We will consider both of these settings in this chapter.
Ideas from the theory of voting have been adopted in a number of recent on-line appli-
cations [140]. Di↵erent Web search engines produce di↵erent rankings of results; a line of
work on meta-search has developed tools for combining these rankings into a single aggregate
ranking. Recommendation systems for books, music, and other items — such as Amazon’s
product-recommendation system — have employed related ideas for aggregating preferences.
In this case, a recommendation system determines a set of users whose past history indicates
tastes similar to yours, and then uses voting methods to combine the preferences of these
other users to produce a ranked list of recommendations (or a single best recommendation)
for you. Note that in this case, the goal is not a single aggregate ranking for the whole pop-
ulation, but instead an aggregate ranking for each user, based on the preferences of similar
users.
Across all of these di↵erent contexts in which voting arises, one sees a recurring set of
23.2. INDIVIDUAL PREFERENCES 737
questions. How should we produce a single ranking from the conflicting opinions provided
by multiple voters? Is some version of majority voting a good mechanism? Is there a better
one? And ultimately, what does it even mean for a voting system to be good? These are
some of the questions we address in this chapter.
23.2 Individual Preferences
The goal of a voting system, for our purposes, can be described as follows. A group of
people is evaluating a finite set of possible alternatives; these alternatives could correspond
to political candidates, possible verdicts in a trial, amounts of money to spend on national
defense, nominees for an award, or any other set of options in a decision. The people involved
wish to produce a single group ranking that orders the alternatives from best to worst, and
that in some sense reflects the collective opinion of the group. Of course, the challenge will
be to define what it means to “reflect” the multiple opinions held by members of the group.
To begin with, let’s consider how to model the opinions of any one member of the group.
We suppose that for each individual, he or she is able to determine a preference between any
two alternatives when presented with these two as a pair. If individual i prefers alternative
X to alternative Y , then we write X � i
Y . (Sometimes, for ease of discussion, we will say
that X “defeats” Y according to i’s preferences.) Thus, for example, if a set of film critics
are each given a large list of movies and asked to express preferences, we could write Citizen
Kane � i
The Godfather to express the fact that critic i prefers the former movie to the
latter. We will sometimes refer to an individual’s preferences over all pairs of alternatives,
represented by � i
, as this individual’s preference relation over the alternatives.
Completeness and Transitivity. We require that individual preferences satisfy two prop-
erties. The first is that each person’s preferences are complete: for each pair of distinct
alternatives X and Y , either she prefers X to Y , or she prefers Y to X, but not both. It
is possible to extend the theory here to consider the possibility that an individuals has ties
in her preferences (i.e. for some pairs of alternatives, she likes them equally), and also the
possibility that for some pairs of alternatives, an individual has no preference (perhaps be-
cause individual i has no knowledge of one of X or Y ). Both of these extensions introduce
interesting complications, but for this chapter we focus on the case in which each individual
has a preference between each pair of alternatives.
The second requirement is that each individual’s preferences be transitive: if an individual
i prefers X to Y and Y to Z, then i should also prefer X to Z. This seems like a very sensible
restriction to impose on preferences, since otherwise we could have situations in which an
individual had no apparent favorite alternative. In other words, suppose we were evaluating
preferences over flavors of ice cream, and we had an individual i for whom Chocolate � i
738 CHAPTER 23. VOTING
X
W
Suppose X defeats the most
other alternatives, but W
defeats X.
(a)
X
W
Then by transitivity, W defeats everything X defeats, plus X itself. So W defeats more
than X does.
(b)
Figure 23.1: With complete and transitive preferences, the alternative X that defeats the most others in fact defeats all of them. If not, some other alternative W would defeat X (as in (a)), but then by transitivity W would defeat more alternatives than X does (as in (b)).
Vanilla, and Vanilla � i
Strawberry — and, in a violation of transitivity, also Strawberry
� i
Chocolate. A simple informal argument for why such preferences seem pathological is
the following: if individual i were to walk up to the counter at an ice cream shop and see
all three flavors on display, which would she choose? This would in some sense have to
be her favorite, despite the fact that each of the three flavors is defeated by some other
flavor in her preferences. There has been a long line of work exploring the philosophical
and psychological basis for transitive preferences, as well as identifying natural situations in
which non-transitive preferences can in fact arise [12, 41, 163]. For our purposes here, we
will assume that each individual’s preferences are transitive.
Individual Rankings. Thus far we have been expressing an individual’s opinions about
a set of alternatives in terms of his or her preferences over pairs. An alternate model for
opinions would be to imagine that each individual produces a completely ranked list of all
the alternatives, ranking them from best to worst.
Notice that from such a ranked list, we could define a preference relation � i
very simply:
we’d say that X � i
Y if alternative X comes before alternative Y in i’s ranked list. In this
case, we’ll say that the preference relation arises from the ranked list. It’s not hard to see
that if a preference relation arises from a ranked list of the alternatives, then it must be
complete and transitive: completeness holds since for each pair of alternatives, one precedes
23.2. INDIVIDUAL PREFERENCES 739
the other in the list; and transitivity holds since if X precedes Y and Y precedes Z in the
list, then X must also precede Z.
What is somewhat less obvious is that this fact holds in the opposite direction as well:
If a preference relation is complete and transitive, then it arises from some ranked
list of the alternatives.
The way to see why this is true is to consider the following method for constructing a ranked
list from a complete and transitive preference relation. First, we identify the alternative X
that defeats the most other alternatives in pairwise comparisons — that is, the X so that
X � i
Y for the most other choices of Y . We claim that this X in fact defeats all the other
alternatives: X � i
Y for all other Y .
We’ll see why this is true in a moment; but first, let’s see why this fact lets us construct
the ranked list we want. To begin with, having established that X defeats all other alter-
natives, we can safely put it at the front of the ranked list. We now remove X from the
set of alternatives, and repeat exactly the same process on the remaining alternatives. The
preferences defined by � i
are still complete and transitive on the remaining alternatives,
so we can apply our claim again on this smaller set: for the alternative Y that defeats the
most others in this set, it in fact defeats every remaining alternative. Hence Y defeats every
alternative in the original set except for X, so we can put Y second in the list, remove it
too from the set of alternatives, and continue in this way until we exhaust the finite set
of alternatives. The way we’ve constructed the list, each alternative is preferred to all the
alternatives that come after it, and so � i
arises from this ranked list.
All of this depends on showing that for any complete and transitive preferences over a
set of alternatives (including the original preferences, and the ones we get as we remove
alternatives), the alternative X that defeats the most others in fact defeats all of them.
Here is an argument showing why this is true (also illustrated in Figure 23.1). We suppose,
for the sake of a contradiction, that it were not true; then there would be some alternative
W that defeats X. But then, for every Y that is defeated by X, we’d have W � i
X and
X � i
Y , and so by transitivity W � i
Y . The conclusion is that W would defeat everything
X defeats, and also defeat X — so W would defeat more alternatives than X does. This is
a contradiction, since we chose X as the alternative that defeats the most others. Therefore,
our assumption that some W defeats X cannot be correct, and so X in fact defeats all other
alternatives. This argument justifies our construction of the ranked list.
In view of all this, when we have a complete and transitive preference relation, we can
equally well view it as a ranked list. Both of these views will be useful in the discussion to
follow.
740 CHAPTER 23. VOTING
23.3 Voting Systems: Majority Rule
In the previous section we developed a way to talk about the individual preference relations
that we’re seeking to combine. We can now describe a voting system (also called an aggrega-
tion procedure) as follows: it is any method that takes a collection of complete and transitive
individual preference relations — or equivalently, a collection of individual rankings — and
produces a group ranking.
This is a very general definition, and at this level of generality it may be hard to see
what makes for a “reasonable” voting system. So we begin, in this section and the next,
by discussing two of the most common classes of voting systems. By considering these,
we’ll start to identify some of the principles — and pathologies — at work in voting more
generally.
Majority Rule and the Condorcet Paradox. When there are only two alternatives,
the most widely used voting system — and arguably the most natural — is majority rule.
Under majority rule, we take the alternative that is preferred by a majority of the voters
and rank it first, placing the other alternative second. For this discussion we will assume
that the number of voters is odd, so that we won’t have to worry about the possibility of
majority rule producing ties.
Since majority rule is so natural in the case of two alternatives, it is natural to try
designing a voting system based on majority rule when there are more than two alternatives.
This, however, turns out to be remarkably tricky. Probably the most direct approach is to
first create group preferences, by applying majority rule to each pair of alternatives, and
then trying to turn these group preferences into a group ranking. That is, we create a group
preference relation � out of all the individual preferences � i
as follows. For each pair of
alternatives X and Y , we count the number of individuals for whom X � i
Y and the number
of individuals for whom Y � i
X. If the first number is larger than the second, then we say
that the group preference � satisfies X � Y , since a majority of the voters prefer X to Y when these two alternatives are considered in isolation. Similarly, we say Y � X in the group preference if Y �
i
X for a majority of the individuals i. Since the number of voters
is odd, we can’t have equal numbers favoring X and favoring Y . So for every distinct pair
of alternatives we will have exactly one of X � Y or Y � X. That is, the group preference relation is complete.
There’s no problem getting this far; the surprising di�culty is that the group preferences
may not be transitive, even when each individual’s preferences are transitive. To see how this
can happen, suppose that we have three individuals named 1, 2, and 3, and three alternatives
named X, Y , and Z. Suppose further that individual 1’s ranking is
X � 1
Y � 1
Z, (23.1)
23.3. VOTING SYSTEMS: MAJORITY RULE 741
College National Ranking Average Class Size Scholarship Money O↵ered X 4 40 $3000 Y 8 18 $1000 Z 12 24 $8000
Figure 23.2: When a single individual is making decisions based on multiple criteria, the Condorcet Paradox can lead to non-transitive preferences. Here, if a college applicants wants a school with a high ranking, small average class size, and a large scholarship o↵er, it is possible for each option to be defeated by one of the others on a majority of the criteria.
individual 2’s ranking is
Y � 2
Z � 2
X, (23.2)
and individual 3’s ranking is
Z � 3
X � 3
Y. (23.3)
Then using majority-rule to define group preferences, we’d have X � Y (since X is preferred to Y by both 1 and 3), Y � Z (since Y is preferred to Z by both 1 and 2), and Z � X (since Z is preferred to X by both 2 and 3). This violates transitivity, which would require
X � Z once we have X � Y and Y � Z. The possibility of non-transitive group preferences arising from transitive individual pref-
erences is called the Condorcet Paradox, after the Marquis de Condorcet, a French political
philosopher who discussed it in the 1700s. And there’s something genuinely counter-intuitive
about it. If we recall our earlier discussion of non-transitive preferences as being somehow
“incoherent,” the Condorcet Paradox describes a simple scenario in which a set of people,
each with perfectly plausible preferences, manages to behave incoherently when forced to
express their collective preferences through majority rule. For example, let’s return to our
example of an individual who prefers Chocolate to Vanilla to Strawberry to Chocolate. Even
if we were to assume that no one individually behaves this way, the Condorcet Paradox shows
how this can arise very naturally as the group preferences of a set of ice-cream eating friends,
when they plan to share a pint of ice cream and decide on which flavor to buy using majority
rule.
The Condorcet Paradox has in fact also been used to show how a single person can natu-
rally be led to form non-transitive individual preferences [41, 163]. Consider, for example, a
student deciding which college to attend. She prefers to go to a college that is highly ranked,
that has a small average class size, and that o↵ers her a significant amount in scholarship
money. Suppose she has been admitted to the following three colleges, with characteristics
as described in Figure 23.2.
In comparing colleges, the student was planning to decide between pairs of colleges by
favoring the one that did better on a majority of these three criteria. Unfortunately, this
742 CHAPTER 23. VOTING
leads to the preferences X � i
Y (since X is better than Y on ranking and scholarship
money), Y � i
Z (since Y is better than Z on ranking and average class size), and Z � i
X
(since Z is better than X on average class size and scholarship money). It’s not hard to see
the analogy: each criterion is like a voter, and the student’s “individual preference relation”
is really the group preference relation synthesized from these three criteria. But it does show
some of the complications that arise even when one individual engages in decision-making
in the presence of multiple criteria.
Voting Systems based on Majority Rule. The Condorcet Paradox portends trouble
for the design of voting systems in general, but given that we need some way to produce an
actual group ranking (including an actual top-ranked alternative), it’s still worth exploring
what can be done using majority rule. We’ll focus on methods for selecting a top-ranked
alternative, which we’ll think of as the “group favorite”; to produce a full ranked list, one
could first select a group favorite, remove it from the available alternatives, and then apply
the procedure repeatedly to what’s left.
One natural approach for finding a group favorite is as follows. We arrange all the
alternatives in some order, and then eliminate them one-by-one in this order using majority
rule. Thus, we compare the first two alternatives by majority vote, compare the winner of
this vote to the third alternative, then compare the winner of that to the fourth alternative,
and so on. The winner of the final comparison is deemed to be the group favorite. We can
represent this in a pictorial way as in Figure 23.3(a), showing the sequence of eliminations
in a four-alternative example, with alternatives A and B compared first, then the winner
compared to C, and the winner of that compared to D. We can think of this as an agenda
for a meeting in which pairs of alternatives are proposed to the group, majority votes are
taken, and a group favorite emerges from this.
This is an example of a more general strategy for using majority rule over pairs of
alternatives to find a group favorite: we can arrange them in any kind of “elimination
tournament,” in which alternatives are paired o↵ against each other in some fashion, with
the winner advancing to a subsequent round while the loser is eliminated. The alternative
that eventually emerges as the overall winner of this tournament is declared to be the group
favorite. The system we were just discussing, shown in Figure 23.3(a), is one such way to
structure an elimination tournament; Figure 23.3(b) depicts another.
Pathologies in Voting Systems based on Majority Rule. These systems do produce
a group favorite (and, by repeatedly invoking the system on the remaining alternatives, also
produce a group ranking). The Condorcet Paradox, however, can be used to uncover an
important pathology that such systems exhibit: their outcomes are susceptible to a kind of
strategic agenda-setting. Let’s go back to our original example of the Condorcet Paradox,
23.3. VOTING SYSTEMS: MAJORITY RULE 743
A B
D
C
Winner of A vs. B
Outcome of A-B winner
vs. C
Overall group favorite
(a) Introducing new alternatives one at a time.
A B DC
Winner of A vs. B
Overall group favorite
Winner of C vs. D
(b) Pairing o↵ alternatives in a di↵erent order.
Figure 23.3: One can use majority rule for pairs to build voting systems on three or more alternatives. The alternatives are considered according to a particular “agenda” (in the form of an elimination tournament), and they are eliminated by pairwise majority vote according to this agenda. This produces an eventual winner that serves as the overall group favorite.
744 CHAPTER 23. VOTING
X Y
Z
X wins
here
Z wins
overall
(a) An agenda in which Z wins.
Z Y
X
Y wins
here
X wins
overall
(b) An agenda in which X wins.
Figure 23.4: With individual rankings as in the Condorcet Paradox, the winner of the elimination tournament depends entirely on how the agenda is set.
where three voters had the individual rankings over alternatives X, Y , and Z given by lists
(23.1)–(23.3). They decide to choose a group favorite using a version of the system shown
in Figure 23.3(a), scaled down to three alternatives: they will first perform majority vote
between two of the alternatives, and then perform majority vote between the winner of this
first vote and the remaining alternative.
The question then becomes how to set the agenda for this process. That is, which two
of the alternatives X, Y , and Z will be voted on first, and which one will be held until the
final vote? Because of the structure of the individual preferences, the choice of agenda in
this case has a decisive e↵ect on the outcome. If, as in Figure 23.4(a), alternatives X and Y
are paired o↵ first, then X will win this first vote but then be defeated by Z as the group
favorite. On the other hand, if alternatives Y and Z are paired o↵ first (as in Figure 23.4(b)),
then Y will win this first vote but then be defeated by X as the group favorite. (We could
do a similar thing to have Y be the group favorite.)
So with individual preferences as in the Condorcet paradox, the overall winner is deter-
mined entirely by how the votes between pairs are sequenced. To put it a di↵erent way, if
the voter who likes Z best gets to set the agenda, then she can sequence the votes so Z wins;
but if the voter who likes X or Y best gets to set the agenda, then he can sequence the
votes so his respective favorite wins. The group favorite is thus determined by the individual
who controls the agenda. Nor can this be remedied by using a system in which voters can
re-introduce alternatives once they’ve been eliminated: with preferences as in the Condorcet
Paradox, there’s always an alternative that can be re-introduced to defeat the current can-
didate for the group favorite, and so a process in which alternatives can be re-introduced for
23.4. VOTING SYSTEMS: POSITIONAL VOTING 745
consideration would never come to an end.
Earlier in this section — using an example of a student choosing colleges based on mul-
tiple criteria, in Figure 23.2 — we observed that the Condorcet Paradox can also capture
pathologies in the decisions made by a single individual, rather than a group. The problem of
agenda-setting has an analogue in the context of such individual decisions as well. Suppose,
for example, that the student in our earlier college-choice example makes the natural deci-
sion to eliminate choices one at a time as acceptance o↵ers come in. Then if the acceptances
arrive in the order X, Y , Z, she will eliminate Y in favor of X when Y arrives (X has a
higher ranking and a higher scholarship o↵er) and then eliminate X in favor of Z when the
acceptance from Z arrives (since Z has a smaller average class size and a higher scholarship
o↵er). Each of these makes sense as a pairwise decision, and it leads to Z as her overall
decision — but it has the property that Y , which she eliminated first, is in fact a choice she
preferred to Z. This is precisely the problem of having a final decision that depends on the
agenda by which alternatives are considered.
23.4 Voting Systems: Positional Voting
A di↵erent class of voting systems tries to produce a group ranking directly from the in-
dividual rankings, rather than building up the group ranking from pairwise comparisons
of alternatives. In this type of system, each alternative receives a certain weight based on
its positions in all the individual rankings, and the alternatives are then ordered according
to their total weight. A simple example of such a system is the Borda Count, named for
Jean-Charles de Borda, who proposed it in 1770. The Borda Count is often used to choose
the winners of sports awards, such as the Heisman trophy in college football; a variant of
it is used to select the Most Valuable Player in professional baseball; and it is used by the
Associated Press and United Press International to rank sports teams.
In the Borda Count, if there are k alternatives in total, then individual i’s ranking confers
a weight of k � 1 on her first-ranked alternative, a weight of k � 2 on her second-ranked alternative, and so on down to a weight of 1 on her second-to-last alternative, and a weight of
0 on her last alternative. In other words, each alternative receives a weight from individual
i equal to the number of other alternatives ranked lower by i. The total weight of each
alternative is simply the sum of the weights it receives from each of the individuals. The
alternatives are then ordered according to their total weights. (We will suppose that if
two alternatives receive the same total weight, then some tie-breaking system arranged in
advance is used to decide which of these two alternatives to place in front of the other.)
For example, suppose there are four alternatives, named A, B, C, and D, and there are
two voters with the individual rankings
A � 1
B � 1
C � 1
D
746 CHAPTER 23. VOTING
and
B � 2
C � 2
A � 2
D.
Then the weight assigned by the Borda Count to alternative A is 3 + 1 = 4, the weight
assigned to B is 5, the weight assigned to C is 3, and the weight assigned to D is 0.
Therefore, sorting the weights in descending order, the group ranking is
B � A � C � D.
It is easy to create variants of the Borda Count that retain its basic flavor: we can assign
any number of “points” to each position in each list, and then rank the alternatives by the
total number of points they receive based on their positions in all lists. The Borda Count
assigns k � 1 points for first, k � 2 points for second, and so forth, but one could imagine versions that assign points di↵erently: for example, to make only the top three positions in
each individual ranking matter, one could assign 3 points for first, 2 for second, 1 for third,
and 0 points for all other positions, with the group ranking still determined by the total
number of points the alternatives receive. We refer to any system of this type as a positional
voting system, since the alternatives receive numerical weights based on their positions on
the individual rankings.
A key appealing feature of the Borda Count is that — ignoring ties — it always produces
a complete, transitive ranking for a set of alternatives. This is simply by its definition, since
it creates a single numerical criterion along which to sort the alternatives (including, as
noted previously, a rule for tie-breaking). But the Borda Count also has some fundamental
pathologies, as we now discuss.
Pathologies in Positional Voting Systems. Most of the problems with the Borda
Count, and with positional voting systems more generally, arise from the fact that competi-
tion for top spots in the group ranking can depend critically on the rankings of alternatives
that are further down in the list.
Here’s a hypothetical scenario illustrating how this can happen. Suppose that a magazine
writes a column in which it asks five film critics to discuss their choice for the greatest movie
of all time; the two movies discussed in the column are Citizen Kane and The Godfather,
and the column ends with a majority-vote decision on the winner. Critics 1, 2, and 3 favor
Citizen Kane, while critics 4 and 5 favor The Godfather.
At the last minute, however, the editors decide the column needs a more “modern” feel,
so they introduce Pulp Fiction as a third option that needs to be discussed and evaluated.
Since there are now three options, the magazine decides to have each critic produce a ranking,
and then use the Borda Count for the overall decision that will serve as the punchline of the
column. The first three critics (who all prefer older movies) each report the ranking
Citizen Kane � i
The Godfather � i
Pulp Fiction.
23.4. VOTING SYSTEMS: POSITIONAL VOTING 747
Critics 4 and 5 (who only like movies made in the last 40 years) each report the ranking
The Godfather � i
Pulp Fiction � i
Citizen Kane.
Applying the Borda Count, we see that Citizen Kane receives a weight of 2 from each of
the first three critics, and a weight of 0 from the last two, for a total of 6. The Godfather
receives a weight of 1 from each of the first three critics, and a weight of 2 from the last two,
for a total of 7. Pulp Fiction receives a weight of 0 from each of the first three critics, and a
weight of 1 from the last two, for a total of 2. As a result, the Borda Count produces The
Godfather as the overall group favorite.
Notice what’s happened here. The outcome of the head-to-head comparison between
Citizen Kane and The Godfather remains the same as before — Citizen Kane is favored by
a vote of three to two. But because a third alternative was introduced, the identity of the
group favorite has changed. Moreover, this is not because the group was particularly fond of
this new, third alternative — the third alternative loses in a head-to-head vote against each
of the two existing alternatives. To put the di�culty in another way: Citizen Kane fails to
rank first in the Borda Count even though it defeats each of the other two alternatives in a
head-to-head comparison under majority rule. So what we find is that the outcome in the
Borda Count can depend on the presence of alternatives that intuitively seem “irrelevant”
— weak alternatives that essentially act as “spoilers” in shifting the outcome from one
higher-ranked alternative to another.
The possibility of such a result suggests further di�culties with the Borda Count —
specifically, the problem of strategic misreporting of preferences. To see how this happens,
let’s consider a slightly di↵erent scenario. Suppose in our previous story that critics 4 and 5
actually had the true ranking
The Godfather � i
Citizen Kane � i
Pulp Fiction.
In other words, in this version of the story, all five critics agree that Pulp Fiction should
be ranked last among these three movies. If we were to run the Borda Count on this set of
five individual rankings, the group ranking would place Citizen Kane first (it would receive
a total weight of 3 · 2 + 2 · 1 = 8 to The Godfather’s 3 · 1 + 2 · 2 = 7). However, suppose that critics 4 and 5 understand the pathologies that are possible with the Borda Count, and they
decide in advance to misreport their rankings as
The Godfather � i
Pulp Fiction � i
Citizen Kane.
Then we have the individual rankings from the previous scenario, and The Godfather ends
up being ranked first.
The underlying point is that voters in the Borda Count can sometimes benefit by lying
about their true preferences, particularly so as to downgrade the overall group ranking of an
alternative that many other voters will put at the top of their individual rankings.
748 CHAPTER 23. VOTING
Examples from U.S. Presidential Elections. Versions of these pathologies are familiar
from U.S. Presidential elections as well. The full Presidential Election process in the United
States has a complex specification, but if we think about how states choose their electors in
the general election — i.e. how they choose which candidate will receive the state’s electoral
votes — it is generally done using plurality voting: the candidate who is top-ranked by
the most voters wins. (The U.S. Constitution doesn’t require this, and some states have
considered other methods and used others in the past, but this is the typical system.)
If we think about it, plurality voting is in fact a positional voting system, since an
equivalent way to run it is as follows. We ask each voter to report an individual ranking of
all the candidates. Each individual ranking then confers a weight of 1 to the candidate at
the top of the ranking, and a weight of 0 to all the other candidates. The candidate with
the greatest total weight from these rankings is declared the winner. Note that this is just
a di↵erent way of saying that the candidate who is top-ranked by the most voters wins, but
it makes it clear that this system fits the structure of a positional method.
Plurality voting exhibits di�culties analogous to what we observed for the Borda Count.
With only two candidates, plurality voting is the same as majority rule; but with more than
two candidates, one sees recurring “third-party” e↵ects, where an alternative that is the
favorite of very few people can potentially shift the outcome from one of the two leading
contenders to the other. In turn, this causes some voters to make their choices strategically,
misreporting their top-ranked choice so as to favor a candidate with a better chance of
winning. Such issues have been present in recent U.S. Presidential elections, and their
e↵ects in important earlier elections, such as the election of Abraham Lincoln in 1860, have
also been studied [384].
23.5 Arrow’s Impossibility Theorem
We have now looked at a number of di↵erent voting systems, and we’ve seen that when there
are more than two alternatives under consideration, they all exhibit pathological behavior.
If we were to consider further voting systems used in practice, we’d find they too su↵ered
from inherent problems in the way they produce a group ranking. At some point, however,
it makes sense to step back from specific voting systems and ask a more general question: is
there any voting system that produces a group ranking for three or more alternatives, and
avoids all of the pathologies we’ve seen thus far?
Making this question concrete requires that we precisely specify all the relevant defini-
tions. We’ve already discussed the precise definition of a voting system: for a fixed number
of voters k, it is any function that takes a set of k individual rankings and produces a group
ranking. The other thing we need to do is to specify what it means for a voting system to
be free of pathologies. We will do this by specifying two properties that we would like a
23.5. ARROW’S IMPOSSIBILITY THEOREM 749
reasonable voting system to satisfy. They are the following.
• First, if there is any pair of alternatives X and Y for which X � i
Y in the rankings
of every individual i, then the group ranking should also have X � Y . This is a very natural condition, known as the Pareto Principle, or Unanimity; it simply requires
that if everyone prefers X to Y , then the group ranking should reflect this. One can
think of Unanimity as ensuring that the group ranking be responsive to the individual
rankings in at least a minimal way.
• Second, we require that for each pair of alternatives, the ordering of X and Y in the group ranking should depend only on how each individual ranks X and Y relative to
each other. In other words, suppose we have a set of individual rankings that produces
a group ranking in which X � Y . If we then take some third alternative Z and shift its position in some of the individual rankings, while leaving the relative ordering of
X and Y unchanged, then the voting system should still produce X � Y for this new set of individual rankings.
This condition is called Independence of Irrelevant Alternatives (IIA), since it requires
that the group ranking of X and Y should depend only on voter preferences between X
and Y , not on how they evaluate other alternatives. IIA is more subtle than Unanimity,
but the failure of IIA is in fact responsible for most of the pathological behavior we
saw in our earlier discussions of specific voting systems. It was clearly at work in
the strategic misreporting of preferences for the Borda Count, since there the shift in
ranking of a third alternative Z was su�cient to change the outcome between two other
alternatives X and Y . It also plays a role in the problem of strategic agenda-setting
for elimination systems based on majority rule: the key idea there was to choose an
agenda that eliminated one alternative X early, before it could be paired against an
alternative Y that it would in fact defeat.
Voting Systems that Satisfy Unanimity and IIA. Since Unanimity and IIA are both
reasonable properties, it’s natural to ask what voting systems satisfy them. When there are
only two alternatives, majority rule clearly satisfies both: it favors X to Y when all voters
do, and — since there are only two alternatives — the group ranking of X and Y clearly
does not depend on any other alternatives.
When there are three or more alternatives, it’s trickier to find a voting system that satis-
fies these two properties: neither the positional systems nor the systems based on majority
rule that we’ve considered will work. There is, however, a voting system that satisfies the
two properties: dictatorship. That is, we pick one of the individuals i, and we simply declare
the group ranking to be equal to the ranking provided by individual i. Notice that there are
really k di↵erent possible voting systems based on dictatorship — one in which each of the
750 CHAPTER 23. VOTING
k possible voters is chosen as the dictator.
We can easily check that each of these k dictatorship systems satisfies Unanimity and
IIA. First, if everyone prefers X to Y , then the dictator does, and hence the group ranking
does. Second, the group ranking of X and Y depends only how the dictator ranks X and
Y , and does not depend on how any other alternative Z is ranked.
Arrow’s Theorem. In the 1950s, Kenneth Arrow proved the following remarkable result
[22, 23], which clarifies why it’s so hard to find voting systems that are free of pathological
behavior.
Arrow’s Theorem: If there are at least three alternatives, then any voting system
that satisfies both Unanimity and IIA must correspond to dictatorship by one
individual.
In other words, dictatorship is the only voting system that satisfies both Unanimity and IIA.
Since dictatorship is generally viewed as an undesirable property too, Arrow’s Theorem
is often phrased as an impossibility result. That is, suppose we say a voting system satisfies
non-dictatorship if there is no individual i for which the group ranking always coincides with
i’s ranking. Then we can phrase Arrow’s Theorem as follows.
Arrow’s Theorem (equivalent version): If there are at least three alternatives, then
there is no voting system that satisfies Unanimity, IIA, and non-dictatorship.
Ultimately, what Arrow’s Theorem shows us is not that voting is necessarily “impossible,”
but that it is subject to unavoidable trade-o↵s — that any system we choose will exhibit
certain forms of undesirable behavior. It therefore helps to focus discussions of voting on
how to manage these trade-o↵s, and to evaluate di↵erent voting systems in light of them.
23.6 Single-Peaked Preferences and the Median Voter Theorem
Condorcet’s Paradox and Arrow’s Theorem are facts of nature; we cannot make them go
away. However, a common approach when faced with an impossibility result is to consider
reasonable special cases of the problem where the underlying di�culties do not arise. There
has been a long line of research in voting that follows this direction.
The starting point for this line of research is the observation that there’s something a bit
unusual about the individual rankings used in the set-up of the Condorcet Paradox. Recall
that with three alternatives X, Y , and Z, and three voters 1, 2, and 3, we had
X � 1
Y � 1
Z
23.6. SINGLE-PEAKED PREFERENCES AND THE MEDIAN VOTER THEOREM 751
Y � 2
Z � 2
X
Z � 3
X � 3
Y
Suppose that X, Y , and Z correspond to amounts of money to spend on education or
national defense, with X corresponding to a small amount, Y to a medium amount, and Z
to a large amount. Then the preferences of voter 1 make sense: she is happiest with the
smallest amount, and second-happiest with a medium amount. The preferences of voter 2
also make sense: he is happiest with a medium amount, but if not medium, then he prefers
a large amount. The preferences of voter 3, on the other hand, are harder to justify in a
simple way: he prefers a large amount, but his second choice is a small amount, with medium
coming last. In other words, the first two voters have preferences that can be explained by
proximity to a fixed number: each of them has an “ideal” amount that they’d like, and they
evaluate the alternatives by how close they come to this ideal. The third voter’s preferences
can’t be explained this way: there’s no “ideal” quantity such that both large and small are
close to it, but medium isn’t. This is not to say that a person couldn’t hold these preferences
(e.g. “if we’re not willing to invest enough in education to do it right, we shouldn’t spend
anything at all”), but they’re more unusual.
Similar reasoning would apply if X, Y , and Z were political candidates arranged in order
on a political spectrum, with X the liberal candidate, Y the moderate candidate, and Z
the conservative candidate. In this case, voter 1 prefers more liberal candidates; voter 2
prefers moderates and leans conservative when forced to choose between extremes; but voter
3 favors the conservative candidate, followed next by the liberal candidate, with the moderate
candidate last. Again, the preferences of voters 1 and 2 can be explained by assuming that
each evaluates candidates by their proximity to a personal “ideal” point on the political
spectrum, but voter 3’s preferences are less natural in that they can’t be explained this way.
We now describe a way to formalize the “unusualness” in voter 3’s ranking, and we then
show that for rankings that do not contain this structure, the Condorcet Paradox cannot
arise.
Single-peaked preferences. For alternatives corresponding to numerical quantities or
linear orderings like a political spectrum, it is reasonable to assume that individual prefer-
ences tend to look like the preferences of voters 1 and 2 in our example: each has a particular
favorite point in the range of alternatives, and they evaluate alternatives by their proximity
to this point. In fact, for our discussion here, it is enough to assume something weaker:
simply that each voter’s preferences “fall away” consistently on both sides of their favorite
alternative.
To make this precise, let’s assume that the k alternatives are named X 1
, X 2
, . . . , X k
,
and that voters all perceive them as being arranged in this order. (Again, we’ll think of
the examples of numerical quantities or candidates on a political spectrum.) We say that a
752 CHAPTER 23. VOTING
rank
1
2
3
4
5
X 1
X 2
X 3
X 4
X 5
alternatives
(a) Voter 1’s ranking.
rank
1
2
3
4
5
X 1
X 2
X 3
X 4
X 5
alternatives
(b) Voter 2’s ranking.
rank
1
2
3
4
5
X 1
X 2
X 3
X 4
X 5
alternatives
(c) Voter 3’s ranking.
Figure 23.5: With single-peaked preferences, each voter’s ranking of alternatives decreases on both sides of a “peak” corresponding to her favorite choice.
voter has single-peaked preferences if there is no alternative X s
for which both neighboring
alternatives X s�1 and Xs+1 are ranked above Xs. In other words, a voter never prefers
two options that lie on opposite sides of a middle option. (Since we are assuming voters
have complete and transitive preferences, we will also refer to single-peaked preferences as
single-peaked rankings.)
Such preferences are called single-peaked because the condition we impose is equivalent
to the following one: each voter i has a top-ranked option X t
, and her preferences fall o↵ on
both sides of X t
:
X t
� i
X t+1
� i
X t+2
� i
· · ·
and
X t
� i
X t�1 �i Xt�2 �i · · ·
Pictorially, this can be represented as in Figure 23.5. The example shown there has three
23.6. SINGLE-PEAKED PREFERENCES AND THE MEDIAN VOTER THEOREM 753
voters with preferences
X 1
� 1
X 2
� 1
X 3
� 1
X 4
� 1
X 5
X 2
� 2
X 3
� 2
X 4
� 2
X 1
� 2
X 5
X 3
� 3
X 2
� 3
X 1
� 3
X 4
� 3
X 5
and each of the three plots shows one of these sets of individual preferences: In the plots,
there is an oval for each alternative, and its height corresponds to its position in the list. As
drawn, the single peak in an individual’s ranking emerges visually as a peak in the plot.
Majority Rule with Single-Peaked Preferences. Single-peaked preferences are natu-
ral as a model for many kinds of rankings, but their significance in the theory of voting lies
in the following observation, made by Duncan Black in 1948 [61].
Recall our first, most basic attempt at synthesizing a group ranking from a set of individ-
ual rankings, back in Section 23.3: we would compare each pair of alternatives X and Y to
each other, using majority rule to produce a group preference of the form X � Y or Y � X (depending on which alternative is preferred by more voters). As before, we’ll suppose that
the number of voters is odd, so that we don’t have to worry about the possibility of ties. Our
hope was that the resulting group preference relation � would be complete and transitive, so that we could produce a group ranking from it. Unfortunately, the Condorcet Paradox
showed that this hope was in vain: transitive individual preferences can give rise to group
preferences that are non-transitive.
But here’s the point of the framework we’ve developed in this section: with single-peaked
preferences, our original plan works perfectly. This is the content of the following result.
Claim: If all individual rankings are single-peaked, then majority rule applied to
all pairs of alternatives produces a group preference relation � that is complete and transitive.
It is not initially clear why this striking fact should be true, but in fact it follows for an
intuitively natural reason, as we now describe.
The Median Individual Favorite. As in other attempts to construct group rankings,
we start by figuring out how to identify a group favorite — an alternative that can be placed
at the top of the ranking — and then proceed to fill in further slots of the ranking. Finding
a group favorite is the crux of the problem, since that requires us to identify an alternative
that defeats every other alternative in a pairwise majority vote.
Let’s consider the top-ranked alternative for each voter, and sort this set of individual
favorites from left to right, along our linear order. Notice that if several voters have the same
alternative as their respective individual favorite, then this alternative will appear multiple
754 CHAPTER 23. VOTING
rank
X m
X t
alternatives
All voters with peaks left
of X m
prefer X m
to X t
Figure 23.6: The proof that the median individual favorite X m
defeats every other alternative X
t
in a pairwise majority vote: if X t
is to the right of X m
, then X m
is preferred by all voters whose peak is on X
m
or to its left. (The symmetric argument applies when X t
is to the left of X
m
.)
times in the sorted list: it is fine for the list to have repetitions. Now consider the individual
favorite that forms the median of this list — that is, the individual favorite that lies exactly
at the halfway point in the sorted order. For example, in the preferences from Figure 23.5,
the sorted list of individual favorites would be X 1
, X 2
, X 3
, and so the median is X 2
. With
more voters, if the individual favorites were (for example) X 1
, X 1
, X 2
, X 2
, X 3
, X 4
, X 5
, then
the median would also be X 2
, since we are considering the median of the list with all the
repetitions included.
The median individual favorite is a natural idea to consider as a potential group favorite,
since it naturally “compromises” between more extreme individual favorites on either side.
And in fact it works very well for this purpose:
The Median Voter Theorem: With single-peaked rankings, the median individual
23.6. SINGLE-PEAKED PREFERENCES AND THE MEDIAN VOTER THEOREM 755
favorite defeats every other alternative in a pairwise majority vote.
To see why this is true, let X m
be the median individual favorite, and let X t
be any other
alternative. Let’s suppose that X t
lies to the right of X m
— that is, t > m. (The case in
which it lies to the left has a completely symmetric argument.) Let’s also order the voters
in the sorted order of their individual favorites.
The argument is now depicted schematically in Figure 23.6. The number of voters k is
odd, and we know that — since it is the median — X m
is in position (k + 1)/2 of the sorted
list of individual favorites. This means that for everyone in the first (k + 1)/2 positions, X m
is either their favorite, or their favorite lies to the left of X m
. For each voter in this latter
group, X m
and X t
are both on the right-hand “down-slope” of this voter’s preferences, but
X m
is closer to the peak than X t
is, so X m
is preferred to X t
. It follows that everyone in the
first (k + 1)/2 positions prefers X m
to X t
. But this is a strict majority of the voters, and so
X m
defeats X t
in a pairwise majority vote.
To put it succinctly: the median individual favorite X m
can always count on gathering a
majority of support against any other alternative X t
, because for more than half the voters,
X m
lies between X t
and each of their respective favorites.
From this fact about the median individual favorite, it is easy to see why majority rule
among all pairs produces a complete and transitive group ranking: we simply build up the
group ranking by identifying group favorites one at a time. That is, we start by finding the
median individual favorite and placing it at the top of the group ranking. This is safe to do
since the Median Voter Theorem guarantees that it defeats all other alternatives that will
come later in the list. Now we remove this alternative from each individual ranking. Notice
that when we do this, the rankings all remain single-peaked: essentially, we have simply
“decapitated” the peak from each ranking, and the second item in each voter’s ranking
becomes their new peak. We now have a version of the same problem we faced before, with
single-peaked rankings on a set of alternatives that is one smaller. So we find the median
individual favorite on the remaining alternatives, place it second in the group ranking, and
continue in this way until we exhaust the finite set of alternatives.
For example, applying this to the three voters in Figure 23.5, we would identify X 2
as
the median individual favorite, and we’d place it first in the group ranking. Once we remove
this alternative, we have three single-peaked rankings on the alternatives X 1
, X 3
, X 4
, and
X 5
. The individual favorites in this reduced set are X 1
, X 3
, and X 3
, so X 3
is the new median
individual favorite, and we place it second in the group ranking. Proceeding in this way, we
end up with the group ranking
X 2
� X 3
� X 1
� X 4
� X 5
.
Since voter 2 was the original “median voter” in the sense of having the original median
individual favorite, the start of the group ranking necessarily agrees with the start of voter
756 CHAPTER 23. VOTING
2’s individual ranking: they both place X 2
first. However, the full group ranking does not
coincide with voter 2’s full individual ranking: for example, voters 1 and 3 both prefer X 1
to X 4
, even through voter 2 doesn’t, and the group ranking reflects this.
23.7 Voting as a Form of Information Aggregation
Thus far, we have focused primarily on situations in which voting is used to aggregate
fundamentally and genuinely di↵erent opinions within a group of people. But there are
contexts in which voting is used by a group of people who have a shared goal — where it is
reasonable to suppose that there is a true best ranking of the alternatives, and the purpose
of the voting system is to discover it. This is unlikely to be appropriate for the ranking of
political candidates or works of art, but it can be a good model of jury deliberations in cases
where the decision hinges on genuine uncertainty about the facts. It can also be a good
model for decisions made by a board of advisers to a company, evaluating business plans
that each yield an uncertain future payo↵.
In settings like these, where we imagine that there is a true best ranking, it’s reasonable
to suppose that the individual rankings di↵er only because they are based on di↵erent infor-
mation, or based on di↵erent evaluations of the available information. If everyone in such a
case had the same information and evaluated it in the same way, they would have the same
ranking.
We will see that these considerations can lead to some potentially complex e↵ects in the
way individuals reason about their votes. As a starting baseline, we begin with a simple
model in which individuals vote simultaneously, and based purely on their own individual
rankings. We then discuss what happens in situations where one or both of these assumptions
do not hold — when voting is done sequentially, or when knowledge of other rankings would
cause an individual to change her own ranking.
Simultaneous, Sincere Voting: The Condorcet Jury Theorem. We begin with a
simple setting in which there are two alternatives X and Y ; one of these two is genuinely
the best choice, and each voter will cast a vote for what she believes to be this best choice.
To model the idea that voters possess di↵erent but uncertain information, we use a general
framework that worked well in Chapter 16 on information cascades. We suppose first that
there is a prior probability of X being the best choice, and that this is known to all voters.
For simplicity, we’ll take this prior probability to be 1/2 in our analysis here; this means
that initially, X and Y are equally likely to be the best choice. Then, each voter receives an
independent, private signal about which of X or Y is better. For some value q > 1/2, signals
favoring the best choice occur at a rate of q; writing this in terms of conditional probability
23.7. VOTING AS A FORM OF INFORMATION AGGREGATION 757
as in Chapter 16, we have
Pr [X-signal is observed | X is best] = q
and
Pr [Y-signal is observed | Y is best] = q.
(We can imagine each voter’s signal as behaving like the flip of a biased coin: for each voter,
it lands on the side indicating the better alternative with probability q.)
Unlike the case in Chapter 16, all the votes in our current analysis are being made
simultaneously: no voter is able to see the decisions made by any other voter before reaching
her own decision. Also, we assume that everyone is voting sincerely: each voter will choose
the alternative she believes to be better, based on the information she has available (in the
form of her private signal). To model sincere voting by an individual, we can use conditional
probabilities just as in Chapter 16. When a voter observes a signal favoring X, she first
evaluates the conditional probability
Pr [X is best | X-signal is observed] .
She then decides to vote in favor of X if this probability is greater than 1/2, and in favor
of Y if this probability is less than 1/2. The analogous reasoning applies if she observes a
signal favoring Y ; we focus just on the case of an X-signal since the analysis is symmetric
in the two cases.
We can evaluate the conditional probability underlying the voter’s decision using Bayes’
Rule, by strict analogy with the calculations we did in Section 16.3. We have
Pr [X is best | X-signal is observed] = Pr [X is best] · Pr [X-signal is observed | X is best]
Pr [X-signal is observed] .
By our assumption about the prior probability, we know that Pr [X is best] = 1/2. By the
definition of the signals, we know that Pr [X-signal is observed | X is best] = q. Finally, there are two ways for an X-signal to be observed: if X is best, or if Y is best. Therefore,
Pr [X-signal is observed] = Pr [X is best] · Pr [X-signal is observed | X is best] + Pr [Y is best] · Pr [X-signal is observed | Y is best]
= 1
2 · q +
1
2 (1 � q) =
1
2 .
Putting all this together, we get
Pr [X is best | X-signal is observed] = (1/2)q
1/2 = q.
The conclusion — which is completely natural — is that the voter will favor the alternative
that is reinforced by the signal she receives. In fact, the calculation using Bayes’ Rule gives
758 CHAPTER 23. VOTING
us more than just this conclusion; it also shows how much probability she should assign to
this favored alternative based on the signal.
The Marquis de Condorcet wrote about this type of scenario in 1785. In his version,
he took as a given the assumption that each voter chooses the best alternative with some
probability q > 1/2, rather than deriving it from the assumption of a private signal — but
the model based on either of these starting assumptions is e↵ectively the same. Condorcet’s
interest was in showing that majority rule is e↵ective when there are many voters who favor
the better of two choices at a rate slightly better than half. His probabilistic formulation
of individuals’ decisions was a novel step — probability was still a relatively new idea in
his time — and his main observation, now known as the Condorcet Jury Theorem, is the
following. Suppose that X is the best alternative (the case for Y being symmetric). Then
as the number of voters increases, the fraction of voters choosing X will converge almost
surely to the probability of receiving an X-signal, which is q > 1/2. In particular, this
means that the probability of the majority reaching a correct decision converges to 1 as the
number of voters grows. In this sense, Condorcet’s Jury Theorem is one of the oldest explicit
formulations of the “wisdom of crowds” idea: aggregating the estimates of many people can
lead to a decision of higher quality than that of any individual expert.
23.8 Insincere Voting for Information Aggregation
One of the assumptions behind the Condorcet Jury Theorem in the previous section is that
all individuals are voting sincerely: each is choosing the alternative he or she believes to be
best, given the information available. On the surface, this seems like a mild assumption. If
the voters could share all their signals, they would reach a unanimous evaluation of the best
alternative; but since they aren’t able to communicate with each other and have access only
to their own private signals, why should a voter do anything anything other than follow her
best guess based on her own signal?
In fact, however, there are very natural situations in which an individual should actually
choose to vote insincerely — favoring the alternative she believes to be worse — even though
her goal is to maximize the probability that the group as a whole selects the best alternative.
This is clearly a counter-intuitive claim, and its underlying basis has only been elucidated
relatively recently [30, 159, 160]. To explain how this phenomenon can arise, we begin with a
hypothetical experiment, modeled on a scenario described by Austen-Smith and Banks [30].
An Experiment that Encourages Insincere Voting. Here’s how the experiment works.
An experimenter announces that an urn with 10 marbles will be placed at the front of a room;
there is a 50% chance the urn will contain ten white marbles, and a 50% chance that it will
contain nine green marbles and one white marble. (We describe the first kind of urn as
23.8. INSINCERE VOTING FOR INFORMATION AGGREGATION 759
“pure” and the second kind as “mixed.”)
The experimenter asks a group of three people to collectively guess which kind of urn
it is. Their group decision will be made according to the following protocol. First, each of
the three people is allowed to draw one marble from the urn, look at it (without showing
it to the other two), and then replace it in the urn. Then, the three people are asked to
cast simultaneous votes, without communicating, each guessing which type of urn they think
it is. If a majority of the votes are for the correct type of urn, then all three people win
a monetary prize; if a majority of the votes are for the wrong type of urn, then all three
people get nothing. (Note that each person gets nothing if the majority is wrong, even if
they personally voted for the correct alternative.)
One can see that the experiment is designed to create a set of independent private signals
for the voters: the color of the marble drawn by each voter is her private signal, and she
cannot communicate this signal to any of the other voters. Rather, the group decision must
be reached by majority vote, with each voter having access to these di↵erent and potentially
conflicting probabilistic signals.
We now ask how an individual should reason about the conditional probabilities of the
di↵erent urn types based on the marble she draws; after this, we’ll consider how she should
actually vote.
Conditional Probabilities and Decisions about Voting. First, suppose that you (as
one of the three people in the experiment) draw a white marble. While we won’t go through
the precise calculations, it’s not hard to work out, using Bayes’ Rule just as we did in the
previous section, that the urn in this case is significantly more likely to be pure than mixed.
(Intuitively, if you see a white marble, it’s much more likely to have been from the all-white
pure urn than to be the single white marble in the mixed urn.) On the other hand, if you
draw a green marble, then in fact you know for certain that the urn is mixed — since green
marbles are only found in mixed urns.
Therefore, if you were to vote sincerely, you would vote “pure” on drawing a white marble
and “mixed” if you draw a green marble. But suppose you knew that the other two people in
the group were going to vote sincerely, and you wanted to choose your vote to maximize the
chance that the majority among the three of you produced the right answer. Then a useful
question to ask yourself is, “In what situations does my vote actually a↵ect the outcome?” If
you think about it, your vote only a↵ects the outcome when the other two (sincere) votes are
split — when there is one vote for pure and one vote for mixed. In this case, however, one
of your two partners in the experiments actually drew a green marble, and so the urn must
be mixed. Here’s the conclusion from this reasoning: whenever your vote actually matters
to the outcome, the urn is mixed!
So if you know your two partners will be voting sincerely, you can best help the group
760 CHAPTER 23. VOTING
by always voting “mixed,” so as to give a single draw of a green marble the chance to
sway the majority outcome. In other words, you’re manipulating the group choice by voting
strategically. You’re not doing this to take advantage of the other voters; indeed, you’re
doing it to make it more likely the group will make the best choice. But nonetheless, it is
not optimal for you to vote sincerely in this case, if your two partners are voting sincerely.
Interpretations of the Voting Experiment. Once we appreciate what’s going on here,
it becomes natural to think of voting with a shared objective in terms of game theory.
The voters correspond to players, their possible strategies are the possible ways of choosing
votes based on private information, and they receive payo↵s based on the votes selected by
everyone. The experiment we’ve just considered constructs a scenario in which sincere voting
is not an equilibrium. Notice that while our analysis has therefore ruled out the most natural
candidate for an equilibrium, it hasn’t actually determined what an equilibrium looks like
for this game. In fact, there are multiple equilibria, some of which are a bit complicated to
compute, and we won’t try to work them out here.
There are a few further points worth reflecting on from this discussion. First, the ex-
periment presents the phenomenon of insincere voting in a very clean and stylized form,
which has the advantage of clearly exposing what’s going on. But versions of this scenario
arise in real-world situations as well, when a highly symmetric decision process like majority
vote clashes with a pair of alternatives that has an asymmetric structure — like the pure
and mixed urns here. Suppose, for example, that a corporate advisory board has to decide
between a risky and a safe course of action for the company, and they decide to use majority
vote. Suppose further that board members have their own private evidence in favor of one
option or the other, and in fact if anyone were to have genuine evidence in favor of the risky
option, then it would be clearly the better choice. If you’re a board member in this case,
and you know that the other board members will be voting sincerely, then your vote will
only matter in the case when half of the rest of the board has evidence in favor of the risky
option — in which case, the risky option is the better idea. So the group would be better
served if you voted insincerely in favor of the risky option, to improve its chances of being
chosen when it should be. Of course, viewing the process of voting as a game, you should
appreciate that the situation is in fact more complicated: rather than assuming that the
other board members will vote sincerely, you may want to assume that they are also going
through this reasoning. Determining how to behave, given this, is a complex problem.
Finally, it’s worth highlighting a key methodological point in this analysis — the underly-
ing principle in which you evaluate the consequences of your actions only in the cases where
they actually a↵ect the outcome. This was the clarifying insight that exposed why insincere
voting was the right decision. Researchers have observed that the use of this principle for
voting forms a parallel with reasoning in other game-theoretic contexts as well, including
23.9. JURY DECISIONS AND THE UNANIMITY RULE 761
the “winner’s curse” for auctions that we saw in Chapter 9 [159]. There, when many people
bid on an item that has a common value (such as oil-drilling rights for a tract of land), the
value of your bid only matters if you win, in which case your estimate of the true value of
the item is more likely to be an over-estimate than an under-estimate. Hence you should
take this into account when bidding, and bid lower than your estimate of the true value.
This type of insincerity in bidding is analogous to the insincerity in voting that we’ve been
discussing here; in both cases, they arise because you’re evaluating your decision contingent
on its actually a↵ecting the outcome, which provides additional implicit information that
needs to be taken into account.
23.9 Jury Decisions and the Unanimity Rule
Jury decisions in criminal trials were an important initial example to motivate this discussion:
they form a natural class of instances where a group of voters (the jurors) agree in principle
that there is a “best” decision for the group — the defendant should be convicted if guilty and
acquitted if innocent — and they want to aggregate their individual opinions to try arriving
at this best decision. Given what we’ve just seen, it is natural to ask: can insincere voting
arise in this case, and if so, what are its consequences? As Feddersen and Pesendorfer have
argued, insincere voting in fact can arise naturally as a strategy for jurors who want their
vote to contribute to the best overall group decision [160]. We describe the basic structure
of their analysis here.
Verdicts, Unanimity, and Private Signals. If we compare jury decisions in criminal
trials with the set-up for the Condorcet Jury Theorem from Section 23.7, we notice two basic
di↵erences, both of which arise from institutional features of the criminal-justice system
designed to help avoid convicting innocent defendants.
The first di↵erence is that it generally requires a unanimous vote in order to convict a
defendant. So if we have k jurors, and the two options acquittal and conviction, each juror
votes for one of these options, and conviction is chosen by the group only if each juror votes
for it. The second di↵erence is in the criterion that jurors are asked to use for evaluating the
two alternatives. In the model from Section 23.7, if each voter could observe all the available
information, she would choose alternative X if
Pr [X is best | all available information] > 1
2 .
In a criminal trial, however, the instructions to a jury are not, “The defendant should be
convicted if he is more likely to be guilt than innocent,” but instead “The defendant should
be convicted if he is guilty beyond a reasonable doubt.” This means that jurors should not
762 CHAPTER 23. VOTING
be asking whether
Pr [defendant is guilty | all available information] > 1
2 ,
but whether
Pr [defendant is guilty | all available information] > z for some larger number z.
We now consider how to model the information available to each juror. Following the
framework used for the Condorcet Jury Theorem in Section 23.7, we assume that each
juror receives an independent private signal suggesting guilt (a G-signal) or innocence (an
I-signal). The defendant, in reality, is of course either guilty or innocent, and we assume
that signals favoring the truth are more abundant than signals favoring the wrong answer:
for some number q > 1 2
, we have
Pr [G-signal | defendant is guilty] = q
and
Pr [I-signal | defendant is innocent] = q. A juror who observes a G-signal is interested in the conditional probability of guilt given
the signal, namely Pr [defendant is guilty | G-signal]. Let’s assume a prior probability of 1/2 that the defendant is guilty — i.e., in the absence of any signals. Then the argument using
Bayes’ Rule from Section 23.7 (with conviction and acquittal playing the roles of the two
alternatives X and Y from that section) applies directly here, showing that
Pr [defendant is guilty | G-signal] = q,
and similarly that
Pr [defendant is innocent | I-signal] = q. The conclusions of the analysis to follow would remain essentially the same, with slightly
di↵erent calculations, if we were to assume any prior probability between 0 and 1.
Before proceeding with the analysis, it’s fair to ask whether the modeling assumption
that jurors receive independent, private signals about guilt or innocence is reasonable — after
all, they sit through the trial together, and they all see the same evidence being presented.
Clearly, the assumption of private signals is a simplified approximation, but it is also clear
that jurors in real trials can and do form widely divergent views of the facts in a case.
This is natural: despite seeing the same evidence, jurors form di↵erent interpretations and
inferences based on their own personal intuitions and decision-making styles — things that
cannot necessarily be transmitted as facts from one person to another [160]. So in this case
we can think of the private signals as representing private interpretations of the information
presented, rather than some personal source of additional information A rational juror is
thus guided by her own signal, but she would also be influenced by knowledge of the signals
of others — i.e. by knowledge that others had interpreted things the same or di↵erently.
23.9. JURY DECISIONS AND THE UNANIMITY RULE 763
Modeling a Juror’s Decision. As noted above, the unanimity rule is designed to make
it hard for an innocent defendant to be convicted, since such a result would require every
single juror to “erroneously” favor conviction. On the surface, this informal principle makes
sense — but as we saw in Section 23.8, reasoning about such principles can become subtle
when we assume that individuals are choosing their votes with the overall group decision in
mind.
In particular, things become complicated for the following reason. Suppose that you’re
one of the k jurors, and you received an I-signal. At first, it seems clear that you should
vote to acquit: after all, your I-signal on its own gives you a conditional probability of q > 1 2
that the defendant is innocent. But then you remember two things. First, the criterion for
conviction by the group is
Pr [defendant is guilty | available information] > z,
which means that in principle the unobserved signals of everyone else — if only you knew
what they were — could be enough to push the conditional probability of guilt above z,
despite your I-signal. Second, you ask yourself the key question from Section 23.8: “In what
situations does my vote actually a↵ect the outcome?” Given the unanimity rule, your vote
only a↵ects the outcome when every juror but you is voting to convict. If you believe that
everyone else’s vote will reflect the signal they received, then you can work out exactly what
the full set of signals is in the event that your vote a↵ects the outcome: it consists of k � 1 G-signals and your one I-signal.
What is the probability the defendant is guilty in this case? We can use Bayes’ Rule to
say that
Pr [defendant is guilty | you have the only I-signal]
= Pr [defendant is guilty] · Pr [you have the only I-signal | defendant is guilty]
Pr [you have the only I-signal] .
Our assumption is that Pr [defendant is guilty] = 1/2, and since the G-signals are indepen-
dent, we have Pr [you have the only I-signal | defendant is guilty] = qk�1(1 � q). (For this latter calculation, there is a probability of qk�1 that each of the k � 1 other jurors gets a G-signal, times a probability of 1 � q that you get an I-signal.) Finally, as usual in Bayes’ Rule calculations, we determine the two di↵erent ways in which all jurors but you receive
G-signals: if the defendant is guilty, or if he is innocent:
Pr [you have the only I-signal]
= Pr [defendant is guilty] · Pr [you have the only I-signal | defendant is guilty] + Pr [defendant is innocent] · Pr [you have the only I-signal | defendant is innocent]
= 1
2 · qk�1(1 � q) +
1
2 (1 � q)k�1q.
764 CHAPTER 23. VOTING
(The second term in the last expression arises from an analogous calculation to what we
used for the first term: if the defendant is innocent, there is a probability of (1 � q)k�1 that each of the k �1 jurors other than you gets a G-signal, times a probability of q that you get an I-signal.) Putting these quantities together, we have
Pr [defendant is guilty | you have the only I-signal] = 1
2
qk�1(1 � q) 1
2
qk�1(1 � q) + 1 2
(1 � q)k�1q
= qk�2
qk�2 + (1 � q)k�2 ,
where the second quality follows just by canceling q(1 � q)/2 from both the numerator and denominator.
Now, since q > 1/2, the term (1 � q)k�2 represents an arbitrarily small portion of the total denominator as the jury size k goes to infinity — and so in particular
Pr [defendant is guilty | you have the only I-signal]
converges to 1 as k goes to infinity. Hence if the jury size k is large enough, it follows that
Pr [defendant is guilty | you have the only I-signal] > z. We conclude from this that if you believe everyone else is voting their signals, and if there
are enough other jurors, then in the only case where your vote to acquit a↵ects the outcome,
the defendant is in fact guilty beyond a reasonable doubt. So if you were to vote with the
actual instructions to the jury in mind, you should ignore your signal and vote to convict.
Of course, you should do this with even more confidence in the event that you receive a
G-signal, and so we can summarize the conclusion even more starkly: if you believe everyone
else is voting their signals, and the jury size is large enough, you should always ignore your
signal and vote to convict.
Intuitively, what’s going on is that you only a↵ect the outcome of a unanimous vote
when everyone else holds the opposite opinion; on the assumption that everyone else is
as well-informed as you are, and voting their true opinion, the conclusion is that they’re
probably (collectively) right, and you’re wrong. As with our earlier example in Section 23.8,
this serves as an interesting reminder that when you design a procedure or protocol for a
group of people to follow, you should expect that they’ll adapt their behavior in light of the
rules you define. Here, the voting system based on unanimity was designed to help prevent
erroneous convictions, but in fact it creates an incentive for people to disregard signals that
the defendant is innocent.
Equilibria for Voting under Unanimity and Other Systems. As in Section 23.8,
we’ve shown that (for large enough juries) voting your signal is not an equilibrium — if
everyone else is doing it, then you should always vote to convict. In their analysis of this
23.9. JURY DECISIONS AND THE UNANIMITY RULE 765
problem, Feddersen and Pesendorfer went further and worked out what the equilibria for
jury voting in this model actually look like.
First, there’s an equilibrium that’s easy to find but a bit pathological: if everyone decides
to ignore their signals and vote to acquit, this is an equilibrium. To see why, notice that no
juror can a↵ect the outcome by changing her behavior, and hence there is no incentive for
any juror to change what she is doing.
More interestingly, there is a unique equilibrium with the properties that (i) all jurors
use the same strategy, and (ii) each juror’s behavior actually depends on the signal she
receives. This is a mixed-strategy equilibrium, in which each juror always votes to convict
on a G-signal, and votes to convict with some probability between 0 and 1 on an I-signal.
The idea is that each juror with an I-signal may randomly choose to disregard it, e↵ectively
correcting for the possibility that she is wrong. One can show that when jurors follow this
equilibrium, the probability that their group decision convicts an innocent defendant is a
positive number that does not converge to zero as the size of the jury goes to infinity. This
forms a sharp contrast to the Condorcet Jury Theorem, where the probability of a correct
decision is converging to 1 as the number of voters grows. The problem here is that the
unanimity rule encourages voters to “over-correct” so strongly for the chance that they
might be wrong, it leads to a noticeable probability that the group as a whole reaches the
wrong decision.
Moreover, the unanimity rule is particularly bad in this regard. Specifically, with further
analysis, we can study voting systems in which convicting a defendant requires only that an
f fraction of the jurors vote for conviction, for di↵erent values of f with 0 < f < 1. For a
given choice of f , we’ll call such a system the f -majority rule. There is still an equilibrium
here in which jurors employ randomization, sometimes disregarding their signals to correct
for the possibility that they are wrong. But with the f -majority rule, a juror’s vote a↵ects
the outcome when the remaining jurors are divided between convicting and acquitting in a
ratio of f to (1 � f ) — a much less extreme split than under the unanimity rule, where a juror’s vote a↵ects the outcome only in the event that she is singular in her opposition to
convicting. As a result of this, the randomized correction used by jurors is correspondingly
less extreme, and one can show that as the jury size goes to infinity, the probability of the
group decision being wrong goes to 0 [160].
This result o↵ers a further reason to question the appropriateness of the unanimity rule
— the result suggests that a decision rule for juries requiring conviction by a wide majority,
rather than a unanimous vote, might actually induce behavior in which there is a lower
probability of erroneous convictions. It is again an indication of the subtle issues that arise
when one evaluates the trade-o↵s between di↵erent social institutions in light of the behaviors
they induce in the people who take part in them.
766 CHAPTER 23. VOTING
23.10 Sequential Voting and the Relation to Informa- tion Cascades
Let’s return to the original formulation of the Condorcet Jury Theorem, with individuals
who vote simultaneously and sincerely over two alternatives X and Y . In the last two
sections, we’ve examined what happens when we remove the assumption of sincerity. It’s
also interesting to instead remove the assumption of simultaneity and see what happens.
We’ll keep sincerity in this discussion, since this simplifies the analysis by only changing one
aspect of the model at a time. So each voter will cast a vote for the alternative she believes
to be the best choice.
When we assume that voters act sincerely but sequentially, we have a model that closely
aligns with the formulation of information cascades from Chapter 16. In our model for
information cascades, we assumed that voters make choices sequentially: they are able to
observe the choices (but not the private signals) of earlier voters; and they can choose to
disregard their own signals if it increases the chance that they personally choose the better
alternative. Note that in this model of cascades, voters are still behaving sincerely in the
sense that they are trying to choose the alternative that is more likely to be correct, based
on everything they are able to observe.
Aside from this distinction between simultaneous and sequential voting, the set-up for
the Condorcet Jury Theorem from Section 23.7 is otherwise quite similar to the model for
information cascades from Chapter 16. In both models, there is a given prior probability for
X to be correct, and there are private signals favoring the correct alternative with probability
greater than 1/2. Therefore, we can invoke our analysis from Section 16.5 to argue that if
voters act sequentially, two initial votes in favor of X will cause a cascade in which all
subsequent votes are for X as well — regardless of whether X is the correct decision. More
generally, once the number of votes for one alternative first exceeds the number of votes for
the other alternative by at least two, a cascade will form in which all subsequent voters will
strategically choose to disregard their own signals.
The fact that cascades begin when one alternative leads the other by exactly two votes
depends on the specific structure of our simplified model from Chapter 16. The broader
principle, however, is quite general. In sequential voting of the type we’re describing, a
cascade will eventually develop. And cascades can be wrong: even if Y is the best alternative,
a cascade for X can develop. Moreover, increasing the number of voters does essentially
nothing to stop this cascade. So the principle behind the Condorcet Jury Theorem does not
apply in this setting: There is no reason to expect that a large crowd of sequential voters
will get the answer right.
23.11. ADVANCED MATERIAL: A PROOF OF ARROW’S IMPOSSIBILITY THEOREM767
Profile 1:
Individual Ranking Ranking restricted to X and Y 1 W � X � Y � Z X � Y 2 W � Z � Y � X Y � X 3 X � W � Z � Y X � Y
Profile 2:
Individual Ranking Ranking restricted to X and Y 1 X � Y � W � Z X � Y 2 Z � Y � X � W Y � X 3 W � X � Y � Z X � Y
Figure 23.7: The two profiles above involve quite di↵erent rankings, but for each individual, her ranking restricted to X and Y in the first profile is the same as her ranking restricted to X and Y in the second profile. If the voting system satisfies IIA, then it must produce the same ordering of X and Y in the group ranking for both profiles.
23.11 Advanced Material: A Proof of Arrow’s Impos- sibility Theorem
In this section we give a proof of Arrow’s Theorem [22, 23], which was stated in Section 23.5.
The proof we present is not Arrow’s original one; instead, we follow a shorter proof found
more recently by John Geanakoplos [179].
Let’s begin by stating the theorem in a language that will help in discussing the ideas
in the proof. We start with a finite set of alternatives. We have a set of k individuals,
whom we can assume to be numbered 1, 2, 3, . . . , k; each individual has a ranking of the
possible alternatives. We’ll call the collection of all k rankings a profile. In this terminology,
a voting system is simply a function that takes a profile and produces a group ranking: a
single ranking of the alternatives.1 The voting system satisfies Unanimity if it puts X � Y in the group ranking whenever X �
i
Y according to the ranking of each individual i. The
voting system satisfies the Independence of Irrelevant Alternatives (abbreviated IIA) if the
ordering of alternatives X and Y in the group ranking depends only on the ordering of X
and Y in each individual ranking, and not on their position relative to any other alternatives.
Here’s a slightly di↵erent but equivalent way to describe IIA, which will be useful in our
discussion. Consider a profile of rankings, and any two alternatives X and Y . We say that
an individual’s ranking restricted to X and Y consists of a copy of his or her ranking in
which we erase all the alternatives other than X and Y . A profile restricted to X and Y is
1As in earlier sections of this chapter, we will consider the case in which individual rankings have no ties, and the voting system is required to produce a group ranking that has no ties either.
768 CHAPTER 23. VOTING
the profile consisting of all individual rankings restricted to X and Y . Then, as illustrated
in Figure 23.7, if a voting system satisfies IIA, it must produce the same ordering of X and
Y for any two profiles that are the same when restricted to X and Y . (In other words, the
profile restricted to X and Y is the only “data” the voting system can look at in ordering
X and Y in the group ranking.)
Recall from Section 23.5 that a voting system can satisfy both Unanimity and IIA via
dictatorship: it selects some individual j in advance, and for any profile of individual rankings,
it simply declares the group ranking to be j’s ranking. There are k di↵erent dictatorship
procedures, depending on which of the k individuals is chosen in advance to be the dictator.
Arrow’s Theorem is that the k dictatorship procedures are the only voting systems that
satisfy Unanimity and IIA. This is the statement we prove here.
The challenge in proving Arrow’s Theorem is that the Unanimity and IIA conditions are
both quite simple, and hence give us relatively little to work with. Despite this, we need
to take an arbitrary voting system satisfying these two properties, and show that it in fact
coincides with dictatorship by a single individual.
Our proof will consist of three main steps. First we show the following interesting fact;
its utility in the proof is not immediately apparent, but it plays a crucial role. Let’s call
X a polarizing alternative if it is ranked either first or last by every individual. Profiles P
and P 0 in Figure23.8 are examples of profiles in which X is a polarizing alternative. We’ll
show that if a voting system satisfies Unanimity and IIA, then it must place any polarizing
alternative in either first or last place in the group ranking. In other words, such a voting
system can’t find a way to “average” and place a polarizing alternative somewhere in the
middle of the group ranking. Note that many profiles don’t contain a polarizing alternative;
this fact only applies to those that do. In the second step of the proof, we then use this fact
to identify a natural candidate for the role of dictator, and in the third step, we prove that
this individual is in fact a dictator.
First Step: Polarizing Alternatives. For the remainder of the proof, let F be a voting
system satisfying Unanimity and IIA. We will use P to denote a profile of individual rankings,
and use F (P ) to denote the group ranking that F produces, as a function of this profile.
We will work toward identifying an individual j with the property that F simply consists of
dictatorship by j.
First, let P be a profile in which X is a polarizing alternative, and suppose by way of
contradiction that F does not place X in either first or last place in the group ranking F (P ).
This means that there are other alternatives Y and Z so that Y � X � Z in the group ranking F (P ).
Now, for any individual ranking that puts Y ahead of Z, let’s change it by sliding Z to the
position just ahead of Y . This produces a new profile P 0, as sketched in Figure 23.8. Since
23.11. ADVANCED MATERIAL: A PROOF OF ARROW’S IMPOSSIBILITY THEOREM769
Profile P :
Individual Ranking 1 X � · · · � Y � · · · � Z � · · · 2 X � · · · � Z � · · · � Y � · · · 3 · · · � Y � · · · � Z � · · · � X
Profile P 0:
Individual Ranking 1 X � · · · � Z � Y � · · · 2 X � · · · � Z � · · · � Y � · · · 3 · · · � Z � Y � · · · � X
Figure 23.8: A polarizing alternative is one that appears at the beginning or end of every individual ranking. A voting system that satisfies IIA must put such an alternative at the beginning or end of the group ranking as well. The figure shows the key step in the proof of this fact, based on rearranging individual rankings while keeping the polarizing alternative in its original position.
X is a polarizing alternative, the relative order of X and Z does not change in any individual
ranking when we do this, nor does the relative order of X and Y . Therefore, by IIA, we
still have Y � X � Z in the group ranking F (P 0). But in P 0, alternative Z is ahead of alternative Y in every individual ranking, and so by Unanimity we have Z � Y in the group ranking F (P 0). Putting these together, the group ranking F (P 0) has Y � X � Z � Y , which contradicts the fact that the voting system F always produces a transitive group ranking.
This contradiction shows that our original assumption of alternatives Y and Z with
Y � X � Z in F (P ) cannot be correct, and so X must appear in either the first or last position in the group ranking F (P ).
Second Step: Identifying a Potential Dictator. In the next step, we create a sequence
of profiles with the property that each di↵ers from the next by very little, and we watch how
the group ranking (according to F ) changes as we move through this sequence. As we track
these changes, a natural candidate for the dictator will emerge.
Here is how the sequence of profiles is constructed. We pick one of the alternatives, X,
and we start with any profile P 0
that has X at the end of each individual ranking. Now,
one individual ranking at a time, we move X from last place to first place while leaving all
other parts of the rankings the same, as shown in Figure 23.9. This produces a sequence of
rankings P 0
, P 1
, P 2
, . . . , P k
, where P i
(i) has X at the front of the individual rankings of 1, 2, . . . , i;
770 CHAPTER 23. VOTING
Profile P 0
:
Individual Ranking 1 · · · � Y � · · · � Z � · · · � X 2 · · · � Z � · · · � Y � · · · � X 3 · · · � Y � · · · � Z � · · · � X
Profile P 1
:
Individual Ranking 1 X � · · · � Y � · · · � Z � · · · 2 · · · � Z � · · · � Y � · · · � X 3 · · · � Y � · · · � Z � · · · � X
Profile P 2
:
Individual Ranking 1 X � · · · � Y � · · · � Z � · · · 2 X � · · · � Z � · · · � Y � · · · 3 · · · � Y � · · · � Z � · · · � X
Profile P 3
:
Individual Ranking 1 X � · · · � Y � · · · � Z � · · · 2 X � · · · � Z � · · · � Y � · · · 3 X � · · · � Y � · · · � Z � · · ·
Figure 23.9: To find a potential dictator, one can study how a voting system behaves when we start with an alternative at the end of each individual ranking, and then gradually (one person at a time) move it to the front of people’s rankings.
(ii) has X at the end of the individual rankings of i + 1, i + 2, . . . , k; and
(iii) has the same order as P 0
on all other alternatives.
So in other words, P i�1 and Pi di↵er only in that individual i ranks X last in Pi�1, and he
ranks it first in P i
.
Now, by Unanimity, X must be last in the group ranking F (P 0
), and it must be first in
the group ranking F (P k
). So somewhere along this sequence there is a first profile in which
X is not in last place in the group ranking; suppose this first profile is P j
. Since X is a
polarizing alternative in P j
, and it is not in last place in F (P j
), it must be in first place.
So individual j has a huge amount of power over the outcome for alternative X, at least
in this sequence of rankings: by switching her own ranking of X from last to first, she causes
23.11. ADVANCED MATERIAL: A PROOF OF ARROW’S IMPOSSIBILITY THEOREM771
X to move from last to first in the group ranking. In the final step of the proof, we will show
that j is in fact a dictator.
Third Step: Establishing that j is a Dictator. The key argument in showing that j is
a dictator is to show that for any profile Q, and any alternatives Y and Z that are di↵erent
from X, the ordering of Y and Z in the group ranking F (Q) is the same as the ordering of Y
and Z in j’s individual ranking in Q. After that we’ll show that the same also holds for pairs
of alternatives in which one of the alternatives is X. In this way, we’ll have established that
the ordering of each pair is determined entirely by j’s ordering, and hence j is a dictator.
So let Q be any profile, and let Y and Z be alternatives not equal to X such that j ranks
Y ahead of Z. We will show that F (Q) puts Y ahead of Z as well.
We create an additional profile Q0 that is a variant of Q; this new profile will help us
understand how j controls the ordering of Y and Z. First, we take Q, move X to the front
of the individual rankings of 1, 2, . . . , j, and move X to the end of the individual rankings
of j + 1, j + 2, . . . , k. Then, we move Y to the front of j’s individual ranking (just ahead of
X). We call the resulting profile Q0.
Now, we make the following observations.
• We know that X comes first in the group ranking F (P j
). Since Q0 and P j
are the same
when restricted to X and Z, it follows from Independence of Irrelevant Alternatives
that X � Z in F (Q0).
• We know that X comes last in the group ranking F (P j�1). Since Q
0 and P j�1 are the
same when restricted to X and Y , it follows from IIA that Y � X in F (Q0).
• By transitivity, we conclude that Y � Z in F (Q0).
• Q and Q0 are the same when restricted to Y and Z, since we produced Q0 from Q without ever swapping the order of Y and Z in any individual ranking. By IIA, it
follows that Y � Z in F (Q).
• Since Q was any profile, and Y and Z were any alternatives (other than X) subject only to the condition that j ranks Y ahead of Z, it follows that the ordering of Y and
Z in the group ranking is always the same as j’s.
Thus we’ve shown that j is a dictator over all pairs that do not involve X. We’re almost
done; we just have to show that j is also a dictator over all pairs involving X as well.
To show this, first observe that we can run the argument thus far with respect to any
other alternative W di↵erent from X, and thereby establish that there is also an individual
` who is a dictator over all pairs not involving W . Suppose that ` is not equal to j. Now,
for X and some third alternative Y di↵erent from X and W , we know that the profiles P j�1
772 CHAPTER 23. VOTING
and P j
di↵er only in j’s individual ranking, yet the ordering of X and Y is di↵erent between
the group rankings F (P j�1) and F (Pj). In one of these two group rankings, the ordering
of X and Y must therefore di↵er from the ordering of X and Y in `’s individual ranking,
contradicting the fact that ` is a dictator for the pair X and Y . Hence our assumption that
` is di↵erent from j must be false, and thus j is in fact a dictator over all pairs.
23.12 Exercises
1. In this chapter, we discussed how voting systems based on majority rule are suscep-
tible to strategic agenda-setting. Let’s explore how one might do this on some basic
examples.
(a) Suppose there are four alternatives, named A, B, C, and D. There are three
voters who have the following individual rankings:
B � 1
C � 1
D � 1
A
C � 2
D � 2
A � 2
B
D � 3
A � 3
B � 3
C
You’re in charge of designing an agenda for considering the alternatives in pairs
and eliminating them using majority vote, via an elimination tournament in the
style of the examples shown in Figure 23.3.
You would like alternative A to win. Can you design an agenda (i.e. an elimination
tournament) in which A wins? If so, describe how you would structure it; if not,
explain why it is not possible.
(b) Now, consider the same question, but for a slightly di↵erent set of individual
rankings in which the last two positions in voter 3’s ranking have been swapped.
That is, we have:
B � 1
C � 1
D � 1
A
C � 2
D � 2
A � 2
B
D � 3
A � 3
C � 3
B
We now ask the same question: Can you design an agenda in which A wins? If
so, describe how you would structure it; if not, explain why it is not possible.
2. The Borda Count is susceptible to strategic misreporting of preferences. Here are some
examples to practice how this works.
23.12. EXERCISES 773
(a) Suppose you are one of three people voting on a set of four alternatives named
A, B, C, and D. The Borda Count will be used as the voting system. The other
two voters have the rankings
D � 1
C � 1
A � 1
B
D � 2
B � 2
A � 2
C
You are voter 3 and would like alternative A to appear first in the group ranking,
as determined by the Borda Count. Can you construct an individual ranking for
yourself so that this will be the result? If so, explain how you would choose your
individual ranking; if not, explain why it is not possible.
(b) Let’s consider the same question, but with di↵erent rankings for the other two
voters, as follows:
D � 1
A � 1
C � 1
B
B � 2
D � 2
A � 2
C
Again, as voter 3, you would like alternative A to appear first in the group ranking
determined by the Borda Count. Can you construct an individual ranking for
yourself so that this will be the result? If so, explain how you would choose your
individual ranking; if not, explain why it is not possible.
3. In Section 23.6, we considered a setting in which alternatives are arranged on a line.
Each voter has an “ideal” point on the line, and she ranks alternatives by the distance of
these alternatives to her ideal point. An interesting property of this setting is that the
Condorcet Paradox cannot arise; more strongly, majority vote over pairs of alternatives
always produces group preferences that are complete and transitive.
Suppose we try to generalize this by allowing alternatives and voters to be positioned
in two dimensions rather than one. That is, suppose that each alternative corresponds
to a point in two-dimensional space. (For example, perhaps the alternatives are dif-
ferent versions of a piece of legislation, and they di↵er in two distinct characteristics,
corresponding to the two dimensions.) As before, each voter has an “ideal” point in the
two-dimensional plane where the alternatives reside, and she evaluates the alternatives
by their respective distances (in the plane) to this ideal point.
Unfortunately, the desirable properties that applied to one-dimensional preferences no
longer hold here. Show how to construct a set of three alternatives in two dimensions,
and a set of three voters, each with an ideal point, so that the resulting set of individual
preferences produces the preferences that we saw in the Condorcet Paradox.
774 CHAPTER 23. VOTING
Chapter 24
Property Rights
The final broad class of social institutions we consider is concerned with the allocation
of resources in a society via property rights. Property rights give the holder of the right the
ability to use a resource, the ability to exclude others from using it, and usually the right to
sell or transfer the resource to another person. Property can take many forms, ranging from
physical property such as a plot of land or a can of Diet Coke, to intellectual property such
as a song or a manufacturing process. In this chapter we will examine how the existence
and form of property rights, or the lack of property rights, a↵ects social outcomes for each
of these types of property. The central message of this chapter is that the property rights a
society chooses to establish will a↵ect the allocations that occur, and some property rights
are more likely than others to result in socially optimal allocations.
24.1 Externalities and the Coase Theorem
In Chapter 17 we argued that the allocation of goods that arises in a market equilibrium
(for an economy without network e↵ects) is socially optimal. In a market equilibrium, the
goods that are produced are assigned to the consumers who value them the most, and any
unit of a good that is produced costs society less to produce than it is worth to the consumer
who receives the good. This results in maximum total social surplus. The intuition for this
fact comes from the observation that at a market equilibrium allocation, each person who
consumes a unit of a good pays the cost to society of producing a unit of the good, and
anyone who is not consuming the good is unwilling to pay the cost of producing a unit of
the good. In this discussion, and in Chapter 17, we have assumed (implicitly) that: the cost
of producing the good correctly reflects the true cost to society of producing the good; an
individual’s willingness to pay for a unit of the good correctly reflects the value to society
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010. Draft version: June 10, 2010.
775
776 CHAPTER 24. PROPERTY RIGHTS
of allowing that individual to consume the good; a producer of a good owns it (has the
property right to it) and can sell it at the market price; and, in order to consume the good
an individual must buy the good at the market price.
These are important qualifications to the social optimality of market equilibria. To see
why getting the values right is so important, let’s tell the story of production and trade in
slightly di↵erent terms. When an individual consumes a can of Diet Coke the individual
creates a personal benefit (otherwise she would not voluntarily consume it) and she creates a
harm to the rest of society as there is now one less can of Diet Coke that could be consumed
by another member of the society. However, if the price that the consumer pays for the
can of Diet Coke is equal to the cost to society of producing another can of Diet Coke,
then the consumer who buys and consumes the can of Diet Coke compensates the rest of
society correctly for the harm she imposes. Well-defined property rights play an important
role beneath the surface of this story. One important role that they play is that every good
that is produced or consumed is covered by a clear property right. If property rights to
the can of Diet Coke are clear, and if no one else is a↵ected by the actions of the Diet
Coke producer and the Diet Coke consumer, then property rights cover this transaction
completely. If instead the actions of either the producer or the consumer of our can of Diet
Coke a↵ect others in a way that is not covered by some property right, then the resulting
equilibrium need not be socially optimal. When the welfare of some individuals or firms is
a↵ected by the actions of other individuals or firms without a property right that requires
mutually agreeable compensation, then we say that an externality occurs. Externalities can
be negative, as we saw in Chapter 8 where we discussed tra�c congestion, or positive as we
focused on in Chapter 17 where we discussed goods with network e↵ects. In this chapter we
will discuss externalities at a more general level.
Externalities and Non-Optimal Allocations. Let’s explore several examples to see how
externalities might arise and why they can create non-optimal allocations. First, suppose
someone decides to smoke a cigar in a restaurant in which there is one other diner. The
smoker purchased the cigar at a price which presumably covered the cost of producing the
cigar, so at least between the smoker and the producer there is no externality created by the
sale of the cigar. But in the act of consuming the cigar in the restaurant the smoker imposes
a harm on the other diner without compensating this person for the harm. Whether the
resulting allocation is socially optimal or not depends on the amounts of harm and benefit
created.
Suppose that the amount of harm su↵ered by the other diner is $10; that is, if the other
diner received a compensation of $10 for the harm created by the smoke, then this person
would be just as well o↵ as he would be if the other diner didn’t smoke the cigar in the
restaurant. On the other hand, suppose the benefit that the smoker receives from smoking
24.1. EXTERNALITIES AND THE COASE THEOREM 777
the cigar, above the price he paid for it, is only $5. Then smoking the cigar reduces total
social surplus by $5: the di↵erence between the harm of $10 created by the smoke and the
benefit of $5 received from creating the smoke. In this case, social optimality requires a
smoke-free environment in the restaurant. One mechanism that would achieve this goal is a
law that prohibits smoking in restaurants.
An alternative mechanism that would achieve the same goal would be to establish a
property right to smoke-free air in restaurants and to make this property right tradeable. In
this case, the other diner can choose whether or not to allow the smoker to smoke his cigar
by agreeing to abandon the property right in exchange for suitable compensation — that
is by selling the property right. Since we’re assuming the smoker only values the ability to
smoke at $5, there would be no smoking in the restaurant in this case, because the smoker
would not be willing to pay enough to compensate the other diner for the ($10) harm caused
by the smoke. Of course, if the benefit the smoker received from smoking were $15, rather
than $5, then there would be a trade. The smoker would pay the other diner an amount
between $10 and $15 for the clean air right, the smoker would smoke his cigar, both parties
would be happy with this outcome, and we would have a socially optimal allocation.1
In the smoking example, establishing a property right to smoke-free air results in a socially
optimal allocation no matter what the individuals’ values are for smoking and smoke-free
air. Alternatively, a property right that allows the smoker to smoke will also work, because
after negotiation between the smoker and the other diner, smoking will occur exactly when
it is socially optimal for it to occur. The possible lack of social optimality arises when
there is no clear property right or no property right at all. In this case, the individuals
may simply disagree about whether smoking is allowed or not and negotiation to a socially
optimal allocation seems unlikely to occur.
Finally, a law that prohibits smoking in restaurants will result in a socially optimal
allocation if the values are such that the optimal allocation requires no smoking, but it will
fail to provide an optimal allocation in the case in which smoking is optimal. In practice,
smoking in restaurants is banned in parts of the U.S., and it is useful to relate this to
the issue of optimal allocations. There are several possible motivations for a smoking ban,
phrased in terms of the discussion above. Maybe the underlying values are such that the
optimal allocation always or almost always requires smoke-free air; maybe policy makers
believe that individuals consistently undervalue smoke-free air and so would make mistakes
if they were allowed to trade; or maybe the costs of enforcing and trading a property right to
smoke-free air in restaurants would be so expensive that it is better to just outlaw smoking
in restaurants.
Let’s explore the last of these motivations — the cost of establishing the property right
1In this discussion, and in the rest of this chapter, we will assume that individual’s values are independent of their wealth.
778 CHAPTER 24. PROPERTY RIGHTS
— in a bit more detail. In our example there was only one other diner. What if instead there
were many other diners and employees in the restaurant? Then no matter who owns the
property right (the smoker or the others in the restaurant) a complex negotiation would be
required, and if diners come and go, the negotiation would have to be conducted repeatedly.
This could easily be so costly as to simply be infeasible, and instead establishing a law
banning smoking could be the next best alternative. It is quite likely to be the best we can
do if social optimality would typically result in a smoke-free environment anyhow.
Our example of smoking in a restaurant is a simplified story capturing some of the issues
that arise in a broad and important domain where property rights play a role: the problem
of environmental impact from activities like industrial production. Similar issues can arise
here when property rights are not clearly defined and enforced. For example, consider a
power-generating plant that pollutes the air and water. The power plant pays for many
of the goods that it uses in the process of generating power; goods such as labor, capital
equipment, and fuel are purchased on the market at prices that compensate the sellers of
these goods for the harm they su↵er in giving them up. But the power plant also implicitly
uses up clean air and clean water in the process of production. If the power plant had to
pay a price that reflected the harm it causes to others — both individuals and other firms
— by converting clean air and water into dirty air and water, then the allocation of power,
air, and water would be socially optimal. Just as in the case of smoking in a restaurant,
establishing a property right to either clean air and water or a property right giving the
power plant the right to pollute the air and water, would in principle work to produce a
socially optimal allocation of both power and the degree of pollution of air and water. It is
worth pointing out that social optimality is unlikely to mean there would be no pollution.
Instead, it simply requires that the amount of pollution, like the amount of all other goods,
is determined so that there is no reallocation that improves welfare. But also just as in the
case of the restaurant, the transaction costs involved in negotiation between the power plant
and all those a↵ected by its activities may be prohibitive.
Mechanisms for Determining Socially Optimal Allocations. One di�culty with
using property rights and mutually agreeable compensation to determine the socially optimal
allocation in our power plant example is the following: how do we discover the true amount
of harm created by the pollution? If we simply ask people how much harm they su↵er from
pollution, and attempt to use this to decide whether or not to allow the pollution, then
each person who is harmed has an incentive to overstate the amount of harm. Similarly, the
firm creating the pollution has an incentive to overstate the cost of reducing its pollution.
However, there is a procedure that can be used to address this incentive problem and we
have already analyzed a special case of it.
In Chapter 15, we demonstrated that the VCG procedure results in an e�cient matching
24.1. EXTERNALITIES AND THE COASE THEOREM 779
of sellers to buyers in a matching market (in the specific context of advertising slots and
advertisers), even when the buyers’ valuations for items is not known. This occurs because
VCG pricing makes truth-telling a dominant strategy for buyers. A similar mechanism can
be used to induce truth-telling for polluters and for those who su↵er the harm created by
pollution. The pollution setting is a bit more complex because the valuations of both the
buyers (the polluters) and the sellers (the a↵ected individuals) are unknown. Here we might
imagine the government running the mechanism, collecting revenue from the polluter, and
providing compensation to individuals harmed by the pollution. The goal of this mechanism
is to determine the socially optimal amount of pollution; it is not to use the money collected
from the polluter to fully compensate those who are harmed by the pollution. In fact,
individuals may be better or worse o↵ once the mechanism is run and the payments occur.
In addition, the amount of revenue collected may not equal the amount of compensation, so
the government may run either a surplus or a deficit.2
Actually running a VCG mechanism to determine the optimal amount of pollution would
be di�cult and costly. The first problem is to determine who is potentially harmed by the
pollution and thus who should be included in the mechanism. Next, the mechanism would
have to be re-run every time the group of a↵ected individuals changes and every time the
polluter wants to change the amount of pollution. This would have to be done for every
polluter. The cost of running these mechanisms over and over would be large. Instead,
some governments use a more market-based approach in which firms can buy the right to
pollute at market prices. These are called cap and trade systems. The U.S. uses a cap and
trade system for sulphur dioxide emissions [394]. In a cap and trade system the government
provides a number of pollution emissions permits, allows the firms to trade these permits,
and requires that any firm emitting pollution own a number of permits equal to the pollution
it emits. If the initial number of permits is set correctly, then this too achieves a socially
optimal allocation of pollution.
The central idea behind the use of property rights or tradeable pollution permits as a
device to solve problems created by externalities is Coase’s Theorem [113], which roughly
says that if tradeable property rights are established and enforced, then negotiation between
the parties a↵ected by the externality will lead to a socially optimal outcome no matter who
initially owns the property rights. For example, in our earlier scenario of the smoker in the
restaurant, all that was necessary for social optimality was to establish and enforce the right
to smoke or the right to have smoke-free air. Then trade between the parties would result in
a socially optimal allocation. Of course, who owns the right will a↵ect how well o↵ each of
the parties is in the resulting equilibrium and they will certainly disagree about who should
initially own the right. But, no matter who it is given to, smoking will occur if and only if it
2For an accessible discussion of the issues involved in designing an optimal mechanism, see the Nobel Prize Committee’s Scientific Background statement in support of the 2007 Nobel Prize in Economics [329], which was awarded for work in mechanism design.
780 CHAPTER 24. PROPERTY RIGHTS
is socially optimal. The same idea applies to pollution emission permits. If property rights
are clearly established with someone as the initial owner then trade will lead to optimality.
Again the initial allocation of permits will make some better o↵ and some worse o↵, and it
is sure to be politically contentious [81].
The one qualification that is necessary in Coase’s argument (that initial ownership is
irrelevant) is that it ignores transaction costs and simply assumes that negotiation beginning
from any assignment of property rights will lead to an e�cient outcome. As we noted in the
smoking example this is not plausible when many individuals are involved in the negotiation.
Similarly, in the case of pollution, establishing marketable pollution rights is more likely to
minimize transaction costs and lead to socially optimal outcomes.
24.2 The Tragedy of the Commons
In a 1968 article in Science, entitled “The Tragedy of the Commons” [205], Garrett Hardin
o↵ered a compelling story about the inevitable “tragedy” of commonly shared resources. In
his story there is a village commons on which any herdsman can freely graze his cattle.3
Hardin noted that inevitably the commons will be overused to the detriment of all the
villagers. He then argued that establishing property rights would solve the problem. These
property rights could be privately held, the commons could be sold to some individual, or
they could be publicly held. However, if the village continues to own the commons it must
carefully limit the use of the commons if we are to have a socially optimal allocation.
A Model for the Commons. Let’s construct a simple example to see how Hardin’s story
plays out. Suppose that there is a village with N people, for some large number N , and each
villager owns one cow. If a fraction x of this population of N cows grazes on the commons,
then the revenue generated per cow is equal to f (x) for some function f (·). Hardin noted that the fewer cows there are on the commons, the more grass there is per cow, and so the
greater revenue per cow. That is, the function f (·) is decreasing. Let’s suppose for example that f (x) = c � x for some number c < 1. This means that the revenue per cow remains positive until x reaches c, at which point it becomes zero: increasing the fraction of cattle
using the commons beyond x = c will cause a negative revenue per cow due to the crowding
of the commons.
So the total revenue generated, if the commons is used by an x fraction of all cattle,
is equal to f (x) · xN , which in our example is (cx � x2)N . Figure 24.1 shows the curve y = (cx � x2)N . If the goal of using the commons is to generate revenue from the cattle that graze there, then the socially optimal fraction of cattle on the commons is the value x⇤
3The term “commons” comes from the common use of a village green in Europe. Many old villages still have them, although generally not for use by cattle any more.
24.2. THE TRAGEDY OF THE COMMONS 781
0
Total Revenue
Fraction of population
using commons cc/2
Optimal utilization
Over-utilization
Figure 24.1: In the Tragedy of the Commons, a freely shared resource can easily be overused unless some form of property rights are established.
that maximizes the function f (x) · xN ; in our case, as we see from the figure, this maximum is achieved midway between the two points where the curve crosses the x-axis, at x⇤ = c/2.
Thus the maximum revenue is
f (x⇤) · x⇤N = ⇣ c �
c
2
⌘ · ⇣ c
2
⌘ N =
c2N
4 .
We analyzed a similar function in Chapter 17 to describe the maximum price that users
would pay for a good with network e↵ects when an x fraction of the population is using it.
In that setting there is also an externality, since each individual’s willingness to pay for the
good is a↵ected by the total number of users. There are key di↵erences, however, between
the setting of network e↵ects and the issues involving the commons. With network e↵ects,
each additional user had a positive e↵ect on the people already using the product, due to
the positive externality. Here, on the other hand, increasing the population on the commons
has a negative e↵ect on each individual already there, due to the crowding. Revenue is
maximized when the increased crowding due to extra cattle is traded o↵ against the greater
revenue from having more cattle using the commons — and this maximum occurs at a
fraction x = c/2 of the population in the case of our simple example.
This is what happens if we are able to choose a quantity of cattle on the commons so as
to maximize the total revenue. But what happens if the commons is freely available to all
the villagers? In this case, villagers will add cattle to the commons as long as the revenue
from doing so is positive. This will eventually lead to the revenue per cow being driven to
782 CHAPTER 24. PROPERTY RIGHTS
zero (as long as there are enough cows available, which in our example is true). To see why
this must happen, note that if there is presently a fraction x of cattle on the commons and
f (x) > 0, then there are villagers whose cows are not currently using the commons, and it
would be in the interest of any one of these villagers to add his cow to the commons in order
to reap even a small bit of positive revenue. This can stop only when the fraction of cattle
on the commons reaches the number x̄ for which f (x̄) = 0, which in our example is x̄ = c.
At that point, there will be twice as many cows on the commons as is socially optimal and
the total revenue from the collective use of the commons is x̄ · f (x̄)N = 0. This is Hardin’s tragedy. The village owns a resource which is clearly valuable, but because its use is not
restricted, everyone who uses it receives zero reward from using it.
Avoiding the Tragedy. A further “tragic” aspect to this loss of social optimality is that
the village could easily solve the problem. There are two obvious ways to do it and several
variations on each approach.
One approach is for the village to continue to own the commons jointly but somehow
limit the quantity of cattle to the socially optimal number. This could be done either by
charging a price for grazing on the commons, or by simply setting the fraction of cattle that
are allowed on the commons at the optimal value x⇤, which in our example is equal to c/2.
If the village charges a price, the optimal price to charge is c/2 per cow. To see why this
is optimal, note that a villager will add his cow to the commons if and only if the revenue
from grazing a cow on the commons is greater than the price. Thus, in an equilibrium, the
revenue from grazing a cow on the commons must be equal to the price. So the equilibrium
will be the value of x which solves f (x) = c/2 and this is x = c/2. Alternatively, the village
could sell grazing rights for x⇤N cows. Again, the maximum price per cow the village could
charge for such grazing rights is c/2 per cow, as we just found above. Either variation on
this method results in the socially optimal use of the commons and a revenue to the village
of c2N/4.
Instead of owning it jointly, the village could sell the commons to a large owner of livestock
who has many cows. This livestock owner who buys the commons would put x⇤N cows on
the commons too as this is the number of cows that maximizes his revenue. The maximum
price that the village could sell the commons for is the revenue that the buyer would receive
from optimal use of the commons, which is again c2N/4. So in either approach — joint
ownership by the village with an appropriate price charged per cow for grazing; or outright
sale to an individual — the village receives a revenue of c2N/4, and the commons is used by
the optimal fraction of cattle.
In Hardin’s village commons example all that is needed to avoid the tragedy is to establish
a property right. It can either be a property right held by some form of government which
optimally limits use of the resource or it can be a privately held property right. Just as
24.3. INTELLECTUAL PROPERTY 783
with Coase’s argument it does not matter (for social optimality) who owns the property;
all that matters is that someone owns it. Hardin uses this story to argue why too much
pollution is produced, why national parks are overused (if there are no admission charges or
limits on the number of users), why there is over-fishing, and, more controversially, to argue
that over-population and thus overuse of the world’s resources is inevitable. In each of these
examples there is an externality as the actions of any one firm or individual a↵ect others,
and if there are no property rights, there is no compensation, and thus no reason to use the
resource optimally.
24.3 Intellectual Property
The property that we discussed in our exploration of Coase’s Theorem or Hardin’s tragedy
existed independently of investment by individuals or firms. The air in the restaurant, and
the air or water that might be polluted by the power plant, were there regardless of what
property rights society creates. Similarly, once the village green is established, it exists
independently of any actions by the villagers. But there is a di↵erence between the green
and the air or water; and, as we will see there is yet another di↵erence between either of
these examples and intellectual property.
To this point in our analysis we have focused on e�cient use of resources without regard
to where theses resources came from. For air and water this seems reasonable. These natural
resources are not created by any human e↵ort and no e↵ort is needed to make them useful.
The village green is also a natural resource, but human e↵ort can reasonably make it more or
less valuable. Cutting the grass regularly, removing weeds, and applying chemical or natural
fertilizer can all make the green more productive. What property rights are established will
matter for whether these activities are carried out. If no one owns the green, then no one
can fully benefit from undertaking these costly but productivity-enhancing investments, and
it seems likely that they will not occur. Solving the property rights problem by assigning
the property rights to someone solves the problem of ine�cient use of the resource and it
also solves the problem of a lack of incentives for productivity-enhancing investment. If
an individual owns the green, then that individual reaps the reward of investment in the
green, and this provides the individual with the proper incentive to make exactly those
investments that are worthwhile. The individual will undertake any investment in the green
that generates more revenue than the investment costs. Similarly, if the village owns the
green jointly, but sells rights to use it, the village also has the proper incentives to maintain
the green. So assigning property rights to the village green is even more important for social
optimality than our earlier discussion indicated.
784 CHAPTER 24. PROPERTY RIGHTS
Rivalrous and Non-Rivalrous Goods. Perhaps surprisingly given our focus to this
point on the desirability of assigning and enforcing property rights, there are commodities
for which assigning property rights can lead to less e�cient use than would occur if there
were no ownership at all. Consider, for example, the result of a creative process. This could
be a book, a song, a new computer program, a new variety of crop, a new drug to treat
cancer, or a new manufacturing process to produce batteries. The use by one individual or
firm of the outcome of the creative process (not the physical object, if one exists, but the
idea that is captured in the physical object) does not a↵ect the ability of anyone else to use
it.4 The same process for making a battery or making a drug, for example, can be used by
an unlimited number of people. Everyone can listen to a song downloaded from the Internet,
or read a book on-line, without a↵ecting the ability of anyone else to listen to the song or to
read the book. In contrast, the can of Diet Coke that one person drinks cannot be consumed
by anyone else. The grass on the village green that one villager’s cow consumes cannot be
consumed by another cow. Goods such that use or consumption by one user precludes use or
consumption by any other potential user are called rivalrous goods; goods that can be used
or consumed over and over are called non-rivalrous goods.
For non-rivalrous goods, establishing property rights can interfere with e�cient use of
the goods. The owner of the good can charge for use of the good, and any non-zero price
may result in some potential users not purchasing the good, and thus not using it (or at
least not using it legally). This results in an ine�ciency as there is no cost to society in
allowing everyone to use the good, so prohibiting anyone from using it is wasteful. This
contrasts with the socially optimal use of the village green in which it was necessary to limit
the use of the good (the green) in order to achieve social optimality. In that case the good
was rivalrous. But for non-rivalrous goods, Hardin’s ideas do not apply.
This is, of course, not the end of the story about property rights for non-rivalrous goods.
We also have to ask where the good came from and whether it would exist without property
rights. If the creator of a non-rivalrous good does not own it, then the creator’s ability to
profit from his creation is limited. Even so the creator may receive some benefit from his
creation. Humans discovered many useful ideas (using fire to cook meat, for example) long
before there was any legal protection for the creators of the results of these ideas. These
early inventors received a direct benefit from their own use of their inventions. Plato wrote
and Mozart created music with little or no protection for their works. They too received
benefits both directly and from users of their work. So some creative activity will exist
and creators will receive some benefits from creation without property rights that protect
intellectual property. But it is not obvious that the socially optimal amount of creative
activity will occur without these property rights, nor is it obvious what form those rights
4Use by many people may of course a↵ect how profitable it is to use the result of the creative process, but it does not a↵ect the feasibility of use by others.
24.3. INTELLECTUAL PROPERTY 785
should take in order to generate the optimal amount of creative activity.
The relevant question is the tradeo↵ between providing incentives for creative activity
and allowing e�cient use of the creation once it exists. No one will have the creation unless
the creator releases it, so the creator at least can benefit from the first use of a valuable
good. If there are no protections then eventually the good will become publicly available
and the creator’s ability to charge for it will vanish. Thus, without property rights, the
financial reward for creative activity may be small, and in the modern economy with the
advent of fast and inexpensive copying and communication that ability may be very small
indeed. Instituting property rights to creative activity increases the creator’s incentives, but
it does so at the cost of ine�cient use of creation once it exists.
Copyrights. Rather than attempt to determine how this balancing should occur in the
abstract we will look at several examples. First, let’s consider the case of books, songs, plays,
television shows and movies. In the U.S. all of these creations are covered by the copyright
law (currently the Copyright Act of 1976) which gives the creator of the work the exclusive
right to copy the work, to distribute it, to modify it, or in the case of songs, plays, television
shows and movies to perform the work. This right lasts for 70 years beyond the lifetime of
the creator of the work. The owner of the copyright has the right to transfer this right to
others.
It is important to be careful about what use of copyrighted works is allowed, and what
uses are illegal without permission of the holder of the copyright. First, the copyright holder’s
exclusive right to copy the work does not actually prohibit all copying. The doctrine of fair
use has evolved over time to permit limited copying for non-commercial use of parts of a
copyrighted work. For example, it is permissible to quote a copyrighted work in a review
of the work, in a scholarly article, or in a classroom. The doctrine of fair use is outlined
in Section 107 of Title 17 of the Copyright Act of 1976 (the entire law can be found at
http://www.copyright.gov/title17/). This law does not define exactly what is or is not fair
use; instead whether a use of copyrighted work is fair or not is to be determined case by case
with the intent of the copier a critical factor in making the determination. Second, copyright
law does not prohibit the owner of a copy of a work from reselling that copy. (This is distinct
from making another copy and selling the new copy, which is prohibited.) So although it is
illegal to copy a book or CD and transfer that copy to someone else, selling a legally obtained
copy of the book or CD is permitted.
Copyright law grants a monopoly (single seller) to the holder of the copyright. Generally
monopolies are harmful as the monopolist artificially restricts the use of the good by setting
a price above the socially optimal price. For copyrighted works the socially optimal price
would clearly be zero if this price had no e↵ect on the creation of the work. The extent to
which the protection a↵orded by a copyright is necessary to provide su�cient incentives to
786 CHAPTER 24. PROPERTY RIGHTS
produce goods that are currently copyrighted is unclear. Some authors, such as Boldrin and
Levine [66] for example, argue that copyrights should not exist at all as they believe that
copyrights are not necessary for innovation and they prevent e�cient use of works once they
are created. The more common view is that copyrights are a necessary evil: they do prevent
e�cient use, but without them the amount of resources devoted to creative activities would
be ine�ciently small.
Patents. Next, let’s consider inventions such as a new drug, a new manufacturing process,
or a new piece of computer hardware. The inventor can apply to the United States Patent
and Trademark O�ce for a patent on the invention, and if the patent is granted, the inventor
has the right to exclude others from using the invention for a fixed period of time, usually
20 years. The web site of the U.S. Patent and Trademark O�ce5 provides a description of
the law.
The economic role of patents is much like that of copyrights. They increase the reward
to inventive activity at the expense of ine�cient use of the patented invention once it is
created. However, patents are di↵erent from copyrights in several ways. First, granting
of copyrights for original work is automatic; the creator only needs to indicate that the
work is copyrighted. Patents are obtained by filing an application with the U.S. Patent
and Trademark O�ce which reviews the application for originality of the work. Second,
enforcement of copyrights and patents is generally left to the holder of the copyright or
patent. The primary exception to this is copyright infringement of songs and movies over
the Internet, and the creation of devices or procedures that can be used to circumvent
digital rights management (technologies designed to prevent copying). These activities have
been partially criminalized by the Digital Millennium Copyright Act of 1998. Third, the
investment in research and development needed to create many patentable commodities is
very large relative to the investment needed to create most works of art. The pharmaceutical
industry, for example, spends large sums on R&D and it seems likely that this would not
occur without the ability to patent discoveries. So the case for restrictive and strongly
enforced patent law is more compelling than that for copyright law. As with many other
aspects of property rights, the trade-o↵s here are complex, and they remain the subject of
active consideration.
24.4 Exercises
1. Consider an airport that is trying to sell exclusive rights to operate a wireless access
network in its terminals. Depending on what proportion of all travelers use the net-
work, it can become congested and result in a low-quality experience for everyone.
5http://www.uspto.gov/main/patents.htm
24.4. EXERCISES 787
Specifically, suppose for simplicity that there are N travelers at any given time during
normal hours in the airport, and if an x fraction of them attempt to use the network
concurrently, then the payo↵ to each of them will be 1 2
� x. (We can view this payo↵ as the amount they’d be willing to pay for the service.)
(a) When the airport sells the rights to operate the network to a third-party access
provider, this provider will make back what it pays for the rights by charging
travelers in the airport a fee to connect to the networks. How much should the
airport expect to be able to sell the rights for, how much will the third-party
access provider in turn charge to travelers, and what will be the resulting sum of
payo↵s to all travelers? Explain.
(b) Suppose instead that the airport were to let people use the service for free. What
would be the sum of payo↵s to all travelers in this case? Explain.
2. Consider the set-up from Exercise 1, but now let’s change the scenario a bit to suppose
that there are two populations among the travelers through the airport, with members
of one population valuing the wireless access service more than members of the other.
In particular, suppose that travelers of Type 1 receive a payo↵ of 1 2
� x when an x fraction of all travelers use the service. (Here, the x fraction is based on the total
usage by travelers of both types, since travelers of both types contribute to the shared
congestion.) Travelers of Type 2 receive twice this payo↵, 1 � 2x, when an x fraction of all travelers use the service. Notice that both payo↵s become 0 when x reaches 1
2
,
since at that point the high congestion makes the service useless for everyone.
As in Exercise 1, the airport is going to sell the right to operate the network to an
access provider, who will then charge a single price to all travelers (regardless of which
type they are).
(a) Suppose that the airport and the access provider know that half of all travelers
are of Type 1 and half are of Type 2. How much can the airport expect to sell
the rights for, and how much will the access provider charge?
(b) Now let’s consider a variation: suppose instead that only 5% of all travelers are
of Type 2, and the rest are of Type 1. Again, both the airport and the access
provider know this. Now how much can the airport expect to sell the rights for,
and how much will the access provider charge?
788 CHAPTER 24. PROPERTY RIGHTS
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