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CHAPTER 10 Game Theory: Inside Oligopoly McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.

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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

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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

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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

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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

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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

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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. 10-7 Simultaneous-Move, One-Shot Games

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Normal-Form Game 10-8 Simultaneous-Move, One-Shot Games Player A Player B StrategyLeftRight Up10, 2015, 8 Down-10, 710, 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”

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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. 10-9 Simultaneous-Move, One-Shot Games

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Dominant Strategy 10-10 Simultaneous-Move, One-Shot Games Player A Player B StrategyLeftRight Up10, 2015, 8 Down-10, 710, 10 Player A has a dominant strategy: Up Player B has no dominant strategy Player A Player B StrategyLeftRight Up10, 2015, 8 Down-10, 710, 10

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Secure Strategy Simultaneous-Move, One-Shot Games Player A Player B Strategy LeftRight Up10, 2015, 8 Down-10, 710, 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 LeftRight Up10, 2015, 8 Down-10, 710, 10 Player A Player B StrategyLeftRight Up10, 2015, 8 Down-10, 710, 10 10-11

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Nash Equilibrium Strategy Simultaneous-Move, One-Shot Games Player A Player B StrategyLeftRight Up10, 2015, 8 Down-10, 710, 10 A Nash equilibrium results when Player A’s plays “Up” and Player B plays “Left” Player A Player B Strategy LeftRight Up10, 2015, 8 Down-10, 710, 10 10-12

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Application of One-Shot Games: Pricing Decisions Simultaneous-Move, One-Shot Games Firm A Firm B Strategy Low priceHigh price Low price0, 050, -10 High price-10, 5010, 10 A Nash equilibrium results when both players charge “Low price” Firm A Firm B Strategy Low priceHigh price Low price0, 050, -10 High price-10, 5010, 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

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Application of One-Shot Games: Coordination Decisions Simultaneous-Move, One-Shot Games Firm A Firm B Strategy 120-Volt Outlets90-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 Outlets90-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 Outlets90-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 communication2) government set standard 10-14

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Application of One-Shot Games: Monitoring Employees Simultaneous-Move, One-Shot Games Manager Worker Strategy MonitorDon’t Monitor Monitor -1, 11, -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. 10-15

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Application of One-Shot Games: Nash Bargaining Simultaneous-Move, One-Shot Games Management Union Strategy 050100 00, 00, 500, 100 5050, 050, 50-1, -1 100100, 0-1, -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 10-16

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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

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Present Value Analysis Review Infinitely Repeated Games 10-18

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Supporting Collusion with Trigger Strategies Infinitely Repeated Games Firm A Firm B Strategy Low priceHigh price Low price0, 050, -40 High price-40, 5010, 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. 10-19

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Supporting Collusion with Trigger Strategies Infinitely Repeated Games Firm A Firm B Strategy Low priceHigh price Low price0, 050, -40 High price-40, 5010, 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. 10-20

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Trigger Strategy Conditions to Support Collusion 10-21 Infinitely Repeated Games

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Supporting Collusion with Trigger Strategies 10-22 Infinitely Repeated Games Firm A Firm B Strategy Low priceHigh price Low price 0, 050, -40 High price -40, 5010, 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.

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Supporting Collusion with Trigger Strategies 10-23 Infinitely Repeated Games Firm A Firm B Strategy Low priceHigh price Low price 0, 050, -40 High price -40, 5010, 10 Q: What are firm A’s profits if it does not cheat on the collusive agreement?

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Supporting Collusion with Trigger Strategies 10-24 Infinitely Repeated Games Firm A Firm B Strategy Low priceHigh price Low price 0, 050, -40 High price -40, 5010, 10 Q: Does an equilibrium result where the firms charge the high price in each period?

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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.

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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

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Theory 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. 10-27

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Games with Uncertain Final Period Finitely Repeated Games Firm A Firm B Strategy Low priceHigh price Low price 0, 050, -40 High price -40, 5010, 10 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: 10-28

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Repeated Games with a Known Final Period: End-of-Period Problem Finitely Repeated Games Firm A Firm B Strategy Low priceHigh price Low price 0, 050, -40 High price -40, 5010, 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. 10-29

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Theory Multistage 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

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Theory: Sequential-Move Game in Extension Form Multistage Games B B A Up Down Decision node denoting the beginning of the game Player B’s decision nodes Player A payoffPlayer 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 10-31

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Equilibrium Characterization Multistage Games B B A Up 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! 10-32

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Subgame Perfect Equilibrium Multistage 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

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Equilibrium Characterization Multistage Games B B A Up Down Subgame Perfect Equilibrium Player A: Up Player B: Up, if player A chooses Up, and Down if Player A chooses Down 10-34

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Application of Multistage Games: The Entry Game 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. 10-35

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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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