CHAPTER 10 Game Theory: Inside Oligopoly McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights
CHAPTER 10 Game Theory: Inside Oligopoly McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
Dec 15, 2015
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CHAPTER 10
Game Theory: Inside Oligopoly
McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Outline• Overview of games and strategic thinking• Simultaneous-move, one-shot games
– Theory– Application of one-shot games
• Infinitely repeated games– Theory– Factors affecting collusion in pricing games– Application of infinitely repeated games
• Finitely repeated games– Games with an uncertain final period– Games with a known final period: the end-of-period problem
• Multistage games– Theory– Applications of multistage games
10-2
Chapter Overview
Introduction
• Chapter 9 examined market environments when only a few firms compete in a market, and determined that the actions of one firm will impact its rivals. As a consequence, a manager must consider the impact of her behavior on her rivals.
• This chapter focuses on additional manager decisions that arise in the presence of interdependence. The general tool developed to analyze strategic thinking is called game theory.
10-3
Chapter Overview
Game Theory Framework• 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.
10-4
Overview of Games and Strategic Thinking
Order of Decisions in Games• Simultaneous-move game– Game in which each player makes decisions
without the knowledge of the other players’ decisions.
• Sequential-move game– Game in which one player makes a move after
observing the other player’s move.
10-5
Overview of Games and Strategic Thinking
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.
10-6
Overview of Games and Strategic Thinking
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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Simultaneous-Move, One-Shot Games
Normal-Form Game
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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”
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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Simultaneous-Move, One-Shot Games
Dominant Strategy
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 has a dominant strategy: UpPlayer B has no dominant strategy
Player A
Player B
Strategy Left Right
Up 10, 20 15, 8
Down -10 , 7 10, 10
Secure StrategySimultaneous-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
10-11
Nash Equilibrium StrategySimultaneous-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
10-12
Application of One-Shot Games: Pricing Decisions
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.
10-13
Application of One-Shot Games: Coordination Decisions
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
10-14
Application of One-Shot Games: Monitoring Employees
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 randomizesover two or more available actions in order to keep rivals from being able to predict his or her actions.
10-15
Application of One-Shot Games: Nash Bargaining
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 50Equilibrium strategy 3: Management chooses 0, Union chooses 100
10-16
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.
Infinitely Repeated Games
10-17
Present Value Analysis Review• When a firm earns the same profit, , in each
period over an infinite time horizon, the present value of the firm is:
Infinitely Repeated Games
10-18
Supporting Collusion with Trigger StrategiesInfinitely 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 agame and in which some particular past action “triggers” a differentaction by a player.
10-19
Supporting Collusion with Trigger StrategiesInfinitely 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, providedneither 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 conditionsunder which neither firm has an incentive to cheat on the collusive outcome.
10-20
Trigger Strategy Conditions to Support Collusion
10-21
Infinitely Repeated Games
• 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.
Supporting Collusion with Trigger Strategies
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.
Supporting Collusion with Trigger Strategies
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:
Supporting Collusion with Trigger Strategies
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 pricein 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.
Factors Affecting Collusion in Pricing Games
10-25
Infinitely Repeated 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.
Factors Affecting Collusion in Pricing Games
Infinitely Repeated Games
• Number of firms in the market• Firm size• History of the market• Punishment mechanisms
10-26
TheoryFinitely 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.
10-27
Games with Uncertain Final PeriodFinitely 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:
Π 𝐴h𝐶 𝑒𝑎𝑡=50 ≤
10𝜃
=Π 𝐴𝐶𝑜𝑜𝑝
10-28
Repeated Games with a Known Final Period: End-of-Period Problem
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 timesand there is only one Nash equilibrium, then collusion cannot work.
The only equilibrium is the single-shot, simultaneous-move Nashequilibrium; in the game above, both firms charge low price.
10-29
TheoryMultistage Games
• 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.
10-30
Theory: Sequential-Move Game in Extension Form
Multistage Games
B
B
A
(10,15)
(5,5)(0,0)
(6,20)
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:
UpDown
Up, if player A plays Down and Down, if player A plays DownUp, if player A plays Up and Down, if player A plays Up
10-31
Equilibrium CharacterizationMultistage Games
B
B
A
(10,15)
(5,5)(0,0)
(6,20)
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 DownIs 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!
10-32
Subgame Perfect EquilibriumMultistage Games
• 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.
10-33
Equilibrium CharacterizationMultistage Games
B
B
A
(10,15)
(5,5)(0,0)
(6,20)
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
10-34
Application of Multistage Games: The Entry Game
Multistage Games
B
A
(−1,1)
(5,5)
(0,10)
In
Hard
Soft
Out
Nash Equilibrium I: Player A: Out Player B: Hard, if player A chooses InNon-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.
10-35
Conclusion• Firms operating in a perfectly competitive market
take the market price as given.– Produce output where P = MC.– Firms may earn profits or losses in the short run.– … but, in the long run, entry or exit forces profits to zero.
• A monopoly firm, in contrast, can earn persistent profits provided that source of monopoly power is not eliminated.
• A monopolistically competitive firm can earn profits in the short run, but entry by competing brands will erode these profits over time.
10-36
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