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Baye_9e_Chapter_101.pptxGame Theory: Inside Oligopoly
© 2017 by McGraw-Hill Education. All Rights Reserved. Authorized only for instructor use in the classroom. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
Chapter 10
Learning Objectives
Apply normal form and extensive form representations of games to formulate decisions in strategic environments that include pricing, advertising, coordination, bargaining, innovation, product quality, monitoring employees, and entry.
Distinguish among dominant, secure, Nash, mixed, and subgame perfect equilibrium strategies, and identify such strategies in various games.
Identify whether cooperative (collusive) outcomes may be supported as a Nash equilibrium in a repeated game, and explain the roles of trigger strategies, the interest rate, and the presence of an indefinite or uncertain final period in achieving such outcomes.
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2
Overview of Games and Strategic Thinking
Game theory is a general framework to aid decision making when agents’ payoffs depends on the actions taken by other players.
Games consist of the following components:
Players or agents who make decisions.
Planned actions of players, called strategies.
Payoff of players under different strategy scenarios.
A description of the order of play.
A description of the frequency of play or interaction.
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10-3
Overview of Games and Strategic Thinking
3
Order of Decisions in Games is Important
Simultaneous-move game
Game in which each player makes decisions without the knowledge of the other players’ decisions.
Bertrand duopoly game
Sequential-move game
Game in which one player makes a move after observing the other player’s move.
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10-4
Overview of Games and Strategic Thinking
4
Frequency of Interaction in Games
One-shot game
Game in which players interact to make decisions only once.
Repeated game
Game in which players interact to make decisions more than once.
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10-5
Overview of Games and Strategic Thinking
5
Simultaneous-Move, One-Shot Games: Theory
Strategy
Decision rule that describes the actions a player will take at each decision point.
Normal-form game
A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies.
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10-6
Simultaneous-Move, One-Shot Games
6
Normal-Form Game
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10-7
Simultaneous-Move, One-Shot Games
| Player A | Player B | ||
| Strategy | Left | Right | |
| Up | 10, 20 | 15, 8 | |
| Down | -10 , 7 | 10, 10 |
Set of players
Player A’s strategies
Player B’s strategies
Player A’s possible payoffs
from strategy “down”
Player B’s
possible
payoffs
from
strategy
“right”
7
Possible Strategies
Dominant strategy
A strategy that results in the highest payoff to a player regardless of the opponent’s action.
Secure strategy
A strategy that guarantees the highest payoff given the worst possible scenario.
Nash equilibrium strategy
A condition describing a set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other players’ strategies.
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10-8
Simultaneous-Move, One-Shot Games
8
Dominant Strategy
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10-9
Simultaneous-Move, One-Shot Games
| Player A | Player B | ||
| Strategy | Left | Right | |
| Up | 10, 20 | 15, 8 | |
| Down | -10 , 7 | 10, 10 |
Player A has a dominant strategy: Up
Player B has no dominant strategy
| Player A | Player B | ||
| Strategy | Left | Right | |
| Up | 10, 20 | 15, 8 | |
| Down | -10 , 7 | 10, 10 |
9
Secure Strategy
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10-10
Simultaneous-Move, One-Shot Games
| Player A | Player B | ||
| Strategy | Left | Right | |
| Up | 10, 20 | 15, 8 | |
| Down | -10 , 7 | 10, 10 |
Player A’s secure strategy: Up … guarantees at least a $10 payoff
Player B’s secure strategy: Right … guarantees at least an $8 payoff
| Player A | Player B | ||
| Strategy | Left | Right | |
| Up | 10, 20 | 15, 8 | |
| Down | -10 , 7 | 10, 10 |
| Player A | Player B | ||
| Strategy | Left | Right | |
| Up | 10, 20 | 15, 8 | |
| Down | -10 , 7 | 10, 10 |
10
Nash Equilibrium Strategy
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10-11
Simultaneous-Move, One-Shot Games
| Player A | Player B | ||
| Strategy | Left | Right | |
| Up | 10, 20 | 15, 8 | |
| Down | -10 , 7 | 10, 10 |
A Nash equilibrium results when Player A’s plays “Up”
and Player B plays “Left”
| Player A | Player B | ||
| Strategy | Left | Right | |
| Up | 10, 20 | 15, 8 | |
| Down | -10 , 7 | 10, 10 |
11
Application of One-Shot Games: Pricing Decisions
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10-12
Simultaneous-Move, One-Shot Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -10 | |
| High price | -10 , 50 | 10, 10 |
A Nash equilibrium results when both players charge “Low price”
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -10 | |
| High price | -10 , 50 | 10, 10 |
Payoffs associated with the Nash equilibrium is inferior from the
firms’ viewpoint compared to both “agreeing” to charge
“High price”: hence, a dilemma.
12
Application of One-Shot Games: Advertising and Quality Decisions
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10-13
Simultaneous-Move, One-Shot Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Advertise | $4, $4 | $2, $1 | |
| Don’t Advertise | $1 , $20 | $10, $10 |
A Nash equilibrium results when both firms “Advertise”
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Advertise | $4, $4 | $20, $1 | |
| Don’t Advertise | $1 , $20 | $10, $10 |
Collusion would not work because this is a one-shot game; if you
and your rival “agreed” not to advertise each of you would have an
incentive to cheat on the agreement.
13
Application of One-Shot Games: Coordination Decisions
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10-14
Simultaneous-Move, One-Shot Games
| Firm A | Firm B | ||
| Strategy | 120-Volt Outlets | 90-Volt Outlets | |
| 120-Volt Outlets | $100, $100 | $0, $0 | |
| 90-Volt Outlets | $0 , $0 | $100, $100 |
There are two Nash equilibrium outcomes associated with this game:
Equilibrium strategy 1: Both players choose 120-volt outlets
| Firm A | Firm B | ||
| Strategy | 120-Volt Outlets | 90-Volt Outlets | |
| 120-Volt Outlets | $100, $100 | $0, $0 | |
| 90-Volt Outlets | $0 , $0 | $100, $100 |
Equilibrium strategy 2: Both players choose 90-volt outlets
| Firm A | Firm B | ||
| Strategy | 120-Volt Outlets | 90-Volt Outlets | |
| 120-Volt Outlets | $100, $100 | $0, $0 | |
| 90-Volt Outlets | $0 , $0 | $100, $100 |
Ways to coordinate on one equilibrium:
1) permit player communication
2) government set standard
14
Application of One-Shot Games: Monitoring Employees
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10-15
Simultaneous-Move, One-Shot Games
| Manager | Worker | ||
| Strategy | Monitor | Don’t Monitor | |
| Monitor | -1, 1 | 1, -1 | |
| Don’t Monitor | 1, -1 | -1, 1 |
There are no Nash equilibrium outcomes associated with this game.
Q: How should the agents play this type of game?
A: Play a mixed (randomized) strategy, whereby a player randomizes
over two or more available actions in order to keep rivals from
being able to predict his or her actions.
15
Application of One-Shot Games: Nash Bargaining
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10-16
Simultaneous-Move, One-Shot Games
| Management | Union | |||
| Strategy | 0 | 50 | 100 | |
| 0 | 0, 0 | 0, 50 | 0, 100 | |
| 50 | 50 , 0 | 50, 50 | -1, -1 | |
| 100 | 100, 0 | -1, -1 | -1, -1 |
There three Nash equilibrium outcomes associated with this game:
Equilibrium strategy 1: Management chooses 100, union chooses 0
Equilibrium strategy 2: Both players choose 50
Equilibrium strategy 3: Management chooses 0, Union chooses 100
16
Infinitely Repeated Games: Theory
An infinitely repeated game is a game that is played over and over again forever, and in which players receive payoffs during each play of the game.
Disconnect between current decisions and future payoffs suggest that payoffs must be appropriately discounted.
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10-17
Infinitely Repeated Games
17
Review of Present Value
When a firm earns the same profit, , in each period over an infinite time horizon, the present value of the firm is:
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10-18
Infinitely Repeated Games
18
Supporting Collusion with Trigger Strategies
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10-19
Infinitely Repeated Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -40 | |
| High price | -40 , 50 | 10, 10 |
The Nash equilibrium to the one-shot, simultaneous-move
pricing game is: Low, Low
When this game is repeatedly played, it is possible for firms to
collude without fear of being cheated on using trigger strategies.
Trigger strategy: strategy that is contingent on the past play of a
game and in which some particular past action “triggers” a different
action by a player.
19
Supporting Collusion with Trigger Strategies
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10-20
Infinitely Repeated Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -40 | |
| High price | -40 , 50 | 10, 10 |
Trigger strategy example: Both firms charge the high price, provided
neither of us has ever “cheated” in the past (charge low price).
If one firm cheats by charging the low price, the other player will
punish the deviator by charging the low price forever after.
When both firms adopt such a trigger strategy, there are conditions
under which neither firm has an incentive to cheat on the collusive
outcome.
20
Sustaining Cooperative Outcomes with Trigger Strategies
Suppose a one-shot game is infinitely repeated and the interest rate is . Further, suppose the “cooperative” one-shot payoff to a player is , the maximum one-shot payoff if the player cheats on the collusive outcome is , the one-shot Nash equilibrium payoff is , and .
Then the cooperative (collusive) outcome can be sustained in the infinitely repeated game with the following trigger strategy: “Cooperate provided that no player has ever cheated in the past. If any player cheats, “punish” the player by choosing the one-shot Nash equilibrium strategy forever after.
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10-21
Infinitely Repeated Games
21
Supporting Collusion with Trigger Strategies In Action
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10-22
Infinitely Repeated Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -40 | |
| High price | -40 , 50 | 10, 10 |
Suppose firm A and B repeatedly play the game above, and the
interest rate is 40 percent. Firms agree to charge a high price in
each period, provided neither has cheated in the past.
Q: What are firm A’s profits if it cheats on the collusive agreement?
A: If firm B lives up to the collusive agreement but firm A cheats,
firm A will earn $50 today and zero forever after.
22
Supporting Collusion with Trigger Strategies in Action
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10-23
Infinitely Repeated Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -40 | |
| High price | -40 , 50 | 10, 10 |
Q: What are firm A’s profits if it does not cheat on the collusive
agreement?
A:
23
Supporting Collusion with Trigger Strategies in Action
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10-24
Infinitely Repeated Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -40 | |
| High price | -40 , 50 | 10, 10 |
Q: Does an equilibrium result where the firms charge the high price
in each period?
A: Since , the present value of firm A’s profits are higher
if A cheats on the collusive agreement. In equilibrium both firms
will charge low price and earn zero profit each period.
24
Factors Affecting Collusion in Pricing Games
Sustaining collusion via trigger strategies is easier when firms know:
who their rivals are, so they know whom to punish, if needed.
who their rival’s customers are, so they can “steal” those customers with lower prices.
when their rivals deviate, so they know when to begin punishment.
be able to successfully punish rival.
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10-25
Infinitely Repeated Games
25
Factors Affecting Collusion in Pricing Games
Number of firms in the market
Firm size
History of the market
Punishment mechanisms
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10-26
Infinitely Repeated Games
26
Finitely Repeated Games
Finitely repeated games are games in which a one-shot game is repeated a finite number of times.
Variations of finitely repeated games: games in which players
do not know when the game will end
know when the game will end
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10-27
Finitely Repeated Games
27
Games with an Uncertain Final Period
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10-28
Finitely Repeated Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -40 | |
| High price | -40 , 50 | 10, 10 |
Suppose the probability that the game will end after a given play is
, where .
An uncertain final period mirrors the analysis of infinitely repeated
games. Use the same trigger strategy.
No incentive to cheat on the collusive outcome associated with a
finitely repeated game with an unknown end point above, provided:
28
Repeated Games with a Known Final Period: End-of-Period Problem
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10-29
Finitely Repeated Games
| Firm A | Firm B | ||
| Strategy | Low price | High price | |
| Low price | 0, 0 | 50, -40 | |
| High price | -40 , 50 | 10, 10 |
When this game is repeated some known, finite number of times
and there is only one Nash equilibrium, then collusion cannot work.
The only equilibrium is the single-shot, simultaneous-move Nash
equilibrium; in the game above, both firms charge low price.
29
Applications of the End-of-Period Problem
Resignations and Quits
The “Snake-Oil” Salesman
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30
Finitely Repeated Games
Multistage Games: Theory
Multistage games differ from the previously examined games by examining the timing of decisions in games.
Players make sequential, rather than simultaneous, decisions.
Represented by an extensive-form game.
Extensive form game
A representation of a game that summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payoffs resulting from alternative strategies.
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10-31
Multistage Games
31
Theory: Sequential-Move Game in Extension Form
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10-32
Multistage Games
B
B
A
Up
Up
Up
Down
Down
Down
Decision node
denoting the
beginning of the
game
Player B’s decision nodes
Player A payoff
Player B payoff
Player A feasible strategies:
Player B feasible strategies:
Up
Down
Up, if player A plays Down and Down, if player A plays Down
Up, if player A plays Up and Down, if player A plays Up
32
Equilibrium Characterization
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10-33
Multistage Games
B
B
A
Up
Up
Up
Down
Down
Down
Nash Equilibrium
Player A: Down
Player B: Down, if player A chooses Up,
and Down if Player A chooses Down
Is this Nash equilibrium reasonable?
No! Player B’s strategy involves a non-credible threat since if A plays Up,
B’s best response is Up too!
33
Subgame Perfect Equilibrium
A condition describing a set of strategies that constitutes a Nash equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies.
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10-34
Multistage Games
34
Equilibrium Characterization
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10-35
Multistage Games
B
B
A
Up
Up
Up
Down
Down
Down
Subgame Perfect Equilibrium
Player A: Up
Player B: Up, if player A chooses Up,
and Down if Player A chooses Down
35
Application of Multistage Games: The Entry Game
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10-36
Multistage Games
B
A
In
Hard
Soft
Out
Nash Equilibrium I:
Player A: Out
Player B: Hard, if player A chooses In
Non-credible, threat since if A plays
In, B’s best response is Soft
Nash Equilibrium II:
Player A: In
Player B: Soft, if player A chooses In
Credible. This is subgame perfect equilibrium.
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