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CORE CONCEPT Published: December 18, 2017

Game Theory and Cooperation: How Putting Others First Can Help Everyone Authors

David McAdams

Young Reviewers

Katherine Johnson Scholar Sisters

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INTRODUCTION: GAMES, GAMES EVERYWHERE

No man is an island, entire of itself. (John Donne, seventeenth-

century English poet)

When people speak of “games,” they are usually talking about

amusements, like checkers and Monopoly, or sports, like football or

basketball. But in the branch of mathematics known as “game theory,”

we speak of games in a much broader sense.

De�nition: Game

A situation is a “game” whenever (i) more than one person is making a

decision and (ii) people’s decisions impact one another.

ABSTRACT

Leo Durocher, a famous baseball player from the 1930s, once

said that “Nice guys �nish last.” That may be true in baseball,

but it is not true in life more generally. In this article, we use

ideas from the branch of mathematics known as “game

theory” to study a situation known as the “Prisoners’ Dilemma

(PD),” which sheds light on why people often fail to work well

together. Game theory shows that people who are kind and

trustworthy have a strategic advantage, as they can “change

the game” to escape the PD and make everyone better o�,

including themselves. So, truly, “nice guys �nish �rst.”

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Just about everything we do in life is a game in the game-theory

sense. At home, at school, everywhere we go, and just about

everything we do, we are playing games. Do not believe me? Think

back to the very beginning of your day, when you woke up. How

quickly you got out of bed impacted your parents—and how early

they woke you up impacted you—so that was a game! Think about

what happened next, throughout the rest of the day. From the

bathroom to the breakfast table, in the classroom and on the

playground, you make choices that impact others as well as yourself.

You’re playing games! Knowing about game theory can help you

improve your experience in those situations—not just to “win,” but to

improve your relationships and have a happier life.

Game theory is used to study how people are likely to behave in

strategic situations, with applications in economics, political science,

business strategy, law, entrepreneurship, and military science, to name

just a few. Just as physics describes how planets revolve around the

sun, game theory seeks to describe how people make decisions in

games. Mathematics is useful in game theory as a tool to analyze

players’ motivations and to predict outcomes. Sometimes, as in the

next example, the predictions that game theory makes can be quite

unexpected.

EXAMPLE: EATING OUT WITH FRIENDS

Eight friends have decided to go out together for lunch at a burger

restaurant. They will split the bill equally. There are two items on the

menu: (i) a regular burger that costs $4 and (ii) a deluxe burger that

costs $8. Each friend feels that eating a regular burger is worth $5

while eating a deluxe burger is worth $6. Note that a regular burger is

worth more than it costs ($5 > $4) while a deluxe burger is worth less

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than it costs ($6 < $8). Deluxe burgers are obviously a bad deal—you

have to pay $8 for something that is only worth $6 to you—but game

theory predicts that, when eating together, all eight friends will

splurge on deluxe burgers. Why?

To understand this unexpected outcome, it is helpful to think about

the game from a mathematical point of view. Let D be the number of

friends who order deluxe and let R be the number who order regular.

(D and R are integers from 0 to 8, with R + D = 8, because there are

eight friends in total.) Since deluxe burgers cost $8 and regular ones

cost $4, the total bill is 8D + 4R. Since R = 8 − D, we can rewrite this as 8D + 4 (8 − D). Multiplying this out, we get 8D + 32 − 4D which can be simpli�ed to

Total bill=32+4D

Each friend’s individual bill is 1/8th of the total bill. Since the total bill is

32 + 4D, each friend pays 32/8 + 4D/8 which can be simpli�ed to

Individual bill=4+D/2.

When someone upgrades their order from “regular” to “deluxe,” they

get an extra dollar of value from the burger (it’s now worth $6 to them

instead of $5). However, they only pay an extra 50 cents, since the

additional $4 cost for the deluxe burger is split eight ways. Each

individual, therefore, has a motivation to order a deluxe burger even

though, when everyone does this, they all wind up paying $8 for

something that is only worth $6 to them. (What would happen if one

person in the group decided to order a regular burger? The total bill

would fall from $64 to $60, causing everyone’s individual bill to fall

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from $8 to $7.50. So, the person who ordered the regular burger

would have to pay $7.50 for something that is only worth $5 to them,

even worse than paying $8 for something that is only worth $6.)

This seems strange but, in fact, this turns out to be an example of the

most famous and most well-studied strategic situation in game theory

—the game known as the “Prisoners’ Dilemma (PD).”

THE PRISONERS’ DILEMMA

Princeton mathematician and early game theorist1 Albert Tucker came

up with the story of the PD in 1950, to demonstrate an unexpected

lesson of game theory—that sometimes doing what is best for

ourselves individually can make everyone worse o�.

Story of the PD

The police have arrested two criminals—“Prisoner #1 (P1)” and

“Prisoner #2 (P2)”—on charges that carry a prison term of up to 5

years. But the police strongly suspect that the criminals also

committed a worse crime, armed robbery, which carries a prison term

of up to 20 years. The police interrogator places the prisoners in

separate cells and tells each prisoner, “It’s time for you to confess to

the robbery. How long you stay in prison depends on who confesses.

If only you confess, I will let you walk free today. Otherwise, you will

spend 5 years behind bars if neither of you confesses, 10 years if both

confess, or 20 years if you are the only one not to confess.” What

should we expect the prisoners to do?

Figure 1 shows how long P1 will spend in jail, depending on who

chooses to confess. Notice that, no matter what P2 does, P1 is better

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o� confessing himself. (If P2 confesses, confessing allows P1 to spend

10 years rather than 20 years in jail. If P2 does not confess, confessing

allows P1 to go free rather than spending 5 years in jail. Either way, P1

has an individual motivation to confess.) Of course, this logic also

applies to P2. Game theory, therefore, predicts that both prisoners will

confess and both spend 10 years in jail—a “lose–lose outcome” since

they could have both spent only 5 years in jail if they had both stayed

quiet.

Figure 1 - Jail time for Prisoner #1 in the Prisoners’ Dilemma.

This “payo� matrix” shows how much time Prisoner #1 will spend in jail, depending

on who confesses. There are four boxes, corresponding to the four possible

outcomes of the game. (Prisoner #1’s choices are shown in the rows and Prisoner

#2’s are shown in the columns.) Note that Prisoner #1 has an individual motivation

to confess—illustrated by the red arrows—whether Prisoner #2 confesses or not.

THE PD IN REAL LIFE

You will probably never be interrogated by the police, but many

situations that we face in life are essentially the same as the PD, from

a game-theory point of view. To make this point as clearly as possible,

it is helpful to de�ne what I mean by “Prisoners’ Dilemma” more

precisely. That way, we can more easily identify when a real-world

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situation is a PD, even when no prisoners are involved. To do so, I �rst

need to de�ne another important game-theory concept: “dominant

strategy.”

De�nition: Dominant Strategy

A “dominant strategy” for a player is an action that gives that player

the highest possible payo�, no matter what actions others choose.

De�nition: Prisoners’ Dilemma

A game is a “Prisoners’ Dilemma” whenever (i) all players have a

dominant strategy and (ii) all players are better o� if they all choose

not to play their dominant strategies.

This de�nition of the PD does not say anything about how many

players are in the game. In the classic story, there are only two

prisoners, but this is unimportant. The same basic game could have

been played with any number of prisoners. For instance, the “Eating

Out with Friends” game we considered earlier is a PD with eight

players, with dominant strategy “order a deluxe burger” (rather than

“confess”) and collectively bad outcome “everyone overpays for their

meal” (rather than “everyone spends more time in jail”). Here, are two

other examples of many-player PDs that we encounter in everyday

life.

Example: Littering

Every weekend, picnickers at a local park decide whether to throw

away their trash. Individually speaking, each person is better o�

leaving their trash on the ground—it is a hassle to throw it away—but,

if everyone were to do that, the park would be a mess. In this PD

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played among picnickers, each player’s dominant strategy is to leave

their trash on the ground, leading to a collectively bad outcome in

which the park is a mess.

Example: Standing up to a Bully

There is a bully who picks on all the kids in your class. When the bully

is picking on someone else, will you stand up in defense of the other

kid or will you stand by and do nothing? Individually speaking, you

and every other kid in class prefer not to stand up to the bully—he

might start going after you instead—but, if everyone were to do that,

the bully will never stop. In this PD played among kids, each player’s

dominant strategy is not to stand up to the bully, leading to a

collectively bad outcome in which the bully is never stopped.

ESCAPING THE PD: THE POWER OF HELPING OTHERS

Do unto others what you want them to do unto you. (The Golden

Rule)

In the PD, everyone gets a worse outcome when they do what is best

for themselves individually than if they each make a personal sacri�ce

to help others. One way to escape the PD is, therefore, to change

players’ motivations so that they want to help others. For instance, in

the littering example, a neighborhood association could put up

posters urging picnickers to “Be A Good Neighbor: Throw Out Your

Trash.” By emphasizing how “good neighbors” throw out their trash,

such posters could cause people to start cleaning up after themselves

because they want to feel like “good neighbors”—and because they

want to avoid being thought of as “bad neighbors.” As long as these

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new feelings are strong enough to persuade everyone to clean up,

everyone will be better o� than before, because the park will be clean

and people will feel good about themselves and their neighbors for

behaving well.

ESCAPING THE PD: THE POWER OF PROMISES

You scratch my back and I’ll scratch yours. (Old saying, believed

to have originated among English sailors during the seventeenth

century)

Back in the seventeenth century, English sailors caught drunk on duty

faced severe punishment. They were tied to the ship’s mast and

beaten with a whip (known as a “cat o’ nine tails”) by another crew

member. To avoid serious injury, sailors struck deals to whip each

other lightly (merely “scratching” each other’s backs). So long as

everyone agreed to the deal, everyone could then be sure to receive a

light whipping when their time for punishment came.

“You scratch my back and I’ll scratch yours” is an example of a

promise—a commitment to do something that helps another person

if that other person �rst does something to help you. A promise can

also help players escape the PD.

The Prisoner’s Promise

If you don’t confess, I promise that I also won’t confess. However,

if you do confess, then so will I.

Suppose that P2 makes this promise and P1 believes it. P1’s choice is

then between 10 years in jail (if both confess) or 5 years in jail (if

neither confesses), and he will choose not to confess. PD solved! See

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Figure 2. But what if P1 does not believe P2? If P1 feels that P2 will

break his promise, P1 will confess and both will go to jail for 10 years.

Still locked in the dilemma!

Figure 2

The Prisoner’s Promise allows the players to “escape” the PD. This �gure shows the

“game tree” of the PD when one player (Prisoner #1) moves �rst and the last-mover

(Prisoner #2) is able to commit to the Prisoner’s Promise. Everything shown in red is

for Prisoner #1, while everything in black is for Prisoner #2. The game proceeds

from left to right. The choice that each player makes at each “decision node” is

shown with a thick arrow, while the choice they do not make is shown with a thin

line. The predicted outcome—that neither confesses and both serve 5 years in jail—

is circled.

How can you make sure that others believe you when you make a

promise? The simplest and most powerful approach is to be an

honest and honorable person. If others know that “your word is your

bond” and that you care about your personal reputation, they will trust

you to do what you say. You can then fully use the power of promises,

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to escape the PD in your own life and to make yourself and everyone

around you better o�.

SUGGESTED FURTHER READING

This article discusses two ways to escape the PD—by either changing

what motivates players or by committing to a promise. However,

there are in fact �ve di�erent “escape routes” from the PD. For more

on this and other game-theory topics, see my book Game-Changer

and the other suggested readings listed below.

Books that provide easy-to-read introductions to game theory:

“Game-Changer: Game Theory and the Art of Transforming

Strategic Situations” by David McAdams, 2014

“Thinking Strategically: The Competitive Edge in Business, Politics,

and Everyday Life” by Avinash K. Dixit and Barry J. Nalebu�, 1991

Books that dig into important applications of game-theory ideas:

“Getting to YES: Negotiating Agreement Without Giving In” by

Roger Fisher and William Ury, 1981

“Who Gets What—And Why” by Alvin E. Roth (winner of the Nobel

Prize in Economics), 2015

My favorite game-theory textbook:

“Games of Strategy, fourth edition” by Avinash Dixit, Susan Skeath,

and David H. Reiley, Jr., 2015

CONFLICT OF INTEREST STATEMENT

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The author declares that the research was conducted in the absence

of any commercial or �nancial relationships that could be construed

as a potential con�ict of interest.

Footnote

[1] ↑ One of Tucker’s PhD students, John Nash, went on to win the Nobel Prize in Economics for his contributions to game theory. The Oscar-winning movie “A

Beautiful Mind” tells the story of Nash’s life during the years when he was Tucker’s

student at Princeton, with Russell Crowe starring as Nash.

Citation

McAdams D (2017) Game Theory and Cooperation: How Putting Others First Can

Help Everyone. Front. Young Minds. 5:66. doi: 10.3389/frym.2017.00066

Editor

Jonathan Montaño

Publishing dates

Submitted: July 25, 2017; Accepted: November 30, 2017; Published online:

December 18, 2017.

Copyright © 2017 McAdams

This is an open-access article distributed under the terms of the Creative Commons

Attribution License (CC BY). The use, distribution or reproduction in other forums is

permitted, provided the original author(s) and the copyright owner(s) are credited

and that the original publication in this journal is cited, in accordance with accepted

academic practice. No use, distribution or reproduction is permitted which does not

comply with these terms.

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