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**Game Theory: Inside Oligopoly**

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**

Chapter Overview Chapter Outline Overview of games and strategic thinking Simultaneous-move, one-shot games Theory Application of one-shot games Infinitely repeated games 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 Applications of multistage games

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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.

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**Overview of Games and Strategic Thinking**

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.

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**Order of Decisions in Games**

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.

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**Frequency of Interaction in Games**

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.

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**Theory Strategy Normal-form game**

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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**Normal-Form Game Simultaneous-Move, One-Shot Games Set of players**

Player B’s strategies Player A Player B Strategy Left Right Up 10, 20 15, 8 Down -10 , 7 10, 10 Player B’s possible payoffs from strategy “right” Player A’s strategies Player A’s possible payoffs from strategy “down”

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**Possible Strategies Dominant strategy Secure strategy**

Simultaneous-Move, One-Shot Games 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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**Dominant Strategy Player A has a dominant strategy: Up**

Simultaneous-Move, One-Shot Games Dominant Strategy 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 Player A has a dominant strategy: Up Player B has no dominant strategy

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**Simultaneous-Move, One-Shot Games**

Secure Strategy 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 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

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**Nash Equilibrium Strategy**

Simultaneous-Move, One-Shot Games Nash Equilibrium Strategy 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 A Nash equilibrium results when Player A’s plays “Up” and Player B plays “Left”

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**Application of One-Shot Games: Pricing Decisions**

Simultaneous-Move, One-Shot Games Application of One-Shot Games: Pricing Decisions Firm A Firm B Strategy Low price High price 0, 0 50, -10 -10 , 50 10, 10 Firm A Firm B Strategy Low price High price 0, 0 50, -10 -10 , 50 10, 10 A Nash equilibrium results when both players charge “Low price” Payoffs associated with the Nash equilibrium is inferior from the firms’ viewpoint compared to both “agreeing” to charge “High price”: hence, a dilemma.

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**Application of One-Shot Games: Coordination Decisions**

Simultaneous-Move, One-Shot Games Application of One-Shot Games: Coordination Decisions Firm A Firm B Strategy 120-Volt Outlets 90-Volt Outlets $100, $100 $0, $0 $0 , $0 Firm A Firm B Strategy 120-Volt Outlets 90-Volt Outlets $100, $100 $0, $0 $0 , $0 Firm A Firm B Strategy 120-Volt Outlets 90-Volt Outlets $100, $100 $0, $0 $0 , $0 There are two Nash equilibrium outcomes associated with this game: Equilibrium strategy 1: Both players choose 120-volt outlets Equilibrium strategy 2: Both players choose 90-volt outlets Ways to coordinate on one equilibrium: 1) permit player communication 2) government set standard

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**Application of One-Shot Games: Monitoring Employees**

Simultaneous-Move, One-Shot Games Application of One-Shot Games: Monitoring Employees Manager Worker Strategy Monitor 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.

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**Application of One-Shot Games: Nash Bargaining**

Simultaneous-Move, One-Shot Games Application of One-Shot Games: Nash Bargaining Management Union Strategy 50 100 0, 0 0, 50 0, 100 50 , 0 50, 50 -1, -1 100, 0 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

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**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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**Present Value Analysis Review**

Infinitely Repeated Games Present Value Analysis Review When a firm earns the same profit, 𝜋, in each period over an infinite time horizon, the present value of the firm is: 𝑃𝑉 𝐹𝑖𝑟𝑚 = 1+𝑖 𝑖 𝜋

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**Supporting Collusion with Trigger Strategies**

Infinitely Repeated Games Supporting Collusion with Trigger Strategies Firm A Firm B Strategy Low price High price 0, 0 50, -40 -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.

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**Supporting Collusion with Trigger Strategies**

Infinitely Repeated Games Supporting Collusion with Trigger Strategies Firm A Firm B Strategy Low price High price 0, 0 50, -40 -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.

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**Trigger Strategy Conditions to Support Collusion**

Infinitely Repeated Games Trigger Strategy Conditions to Support Collusion Suppose a one-shot game is infinitely repeated and the interest rate is 𝑖. Further, suppose the “cooperative” one-shot payoff to a player is 𝜋 𝐶𝑜𝑜𝑝 , the maximum one-shot payoff if the player cheats on the collusive outcome is 𝜋 𝐶ℎ𝑒𝑎𝑡 , the one-shot Nash equilibrium payoff is 𝜋 𝑁 , and 𝜋 𝐶ℎ𝑒𝑎𝑡 − 𝜋 𝐶𝑜𝑜𝑝 𝜋 𝐶𝑜𝑜𝑝 − 𝜋 𝑁 ≤ 1 𝑖 . 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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**Supporting Collusion with Trigger Strategies**

Infinitely Repeated Games Supporting Collusion with Trigger Strategies Firm A Firm B Strategy Low price High price 0, 0 50, -40 -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.

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**Supporting Collusion with Trigger Strategies**

Infinitely Repeated Games Supporting Collusion with Trigger Strategies Firm A Firm B Strategy Low price High price 0, 0 50, -40 -40 , 50 10, 10 Q: What are firm A’s profits if it does not cheat on the collusive agreement? A: …= =$35

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**Supporting Collusion with Trigger Strategies**

Infinitely Repeated Games Supporting Collusion with Trigger Strategies Firm A Firm B Strategy Low price High price 0, 0 50, -40 -40 , 50 10, 10 Q: Does an equilibrium result where the firms charge the high price in each period? A: Since $50>$35, 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.

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**Factors Affecting Collusion in Pricing Games**

Infinitely Repeated Games 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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**Factors Affecting Collusion in Pricing Games**

Infinitely Repeated Games Factors Affecting Collusion in Pricing Games Number of firms in the market Firm size History of the market Punishment mechanisms

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**Finitely Repeated Games**

Theory 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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**Games with Uncertain Final Period**

Finitely Repeated Games Games with Uncertain Final Period Firm A Firm B Strategy Low price High price 0, 0 50, -40 -40 , 50 10, 10 Suppose the probability that the game will end after a given play is 𝜃, where 0<𝜃<1. 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 𝐴 𝐶ℎ𝑒𝑎𝑡 =50≤ 10 𝜃 = Π 𝐴 𝐶𝑜𝑜𝑝

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**Repeated Games with a Known Final Period: End-of-Period Problem**

Finitely Repeated Games Repeated Games with a Known Final Period: End-of-Period Problem Firm A Firm B Strategy Low price High price 0, 0 50, -40 -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.

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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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**Theory: Sequential-Move Game in Extension Form**

Multistage Games Theory: Sequential-Move Game in Extension Form Player A payoff Player B payoff (10,15) Decision node denoting the beginning of the game Up B Up Down (5,5) A (0,0) Down Up Player A feasible strategies: B Up Down Down Player B’s decision nodes Player B feasible strategies: (6,20) 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

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**Equilibrium Characterization**

Multistage Games Equilibrium Characterization (10,15) Up B Up Down (5,5) A (0,0) Down Up Nash Equilibrium Player A: Down Player B: Down, if player A chooses Up, and Down if Player A chooses Down B Down (6,20) 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!

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**Subgame Perfect Equilibrium**

Multistage Games 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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**Equilibrium Characterization**

Multistage Games Equilibrium Characterization (10,15) Up B Up Down (5,5) A (0,0) Down Up B Subgame Perfect Equilibrium Player A: Up Player B: Up, if player A chooses Up, and Down if Player A chooses Down Down (6,20)

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**Application of Multistage Games: The Entry Game**

Nash Equilibrium I: Player A: Out Player B: Hard, if player A chooses In (−1,1) Hard Non-credible, threat since if A plays In, B’s best response is Soft B In Soft (5,5) A Nash Equilibrium II: Player A: In Player B: Soft, if player A chooses In Out Credible. This is subgame perfect equilibrium. (0,10)

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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.

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© 2008 Pearson Addison Wesley. All rights reserved Chapter Fourteen Game Theory.

© 2008 Pearson Addison Wesley. All rights reserved Chapter Fourteen Game Theory.

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