🗊Презентация Repeated games. (Lecture 6)

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Repeated games. (Lecture 6), слайд №1Repeated games. (Lecture 6), слайд №2Repeated games. (Lecture 6), слайд №3Repeated games. (Lecture 6), слайд №4Repeated games. (Lecture 6), слайд №5Repeated games. (Lecture 6), слайд №6Repeated games. (Lecture 6), слайд №7Repeated games. (Lecture 6), слайд №8Repeated games. (Lecture 6), слайд №9Repeated games. (Lecture 6), слайд №10Repeated games. (Lecture 6), слайд №11Repeated games. (Lecture 6), слайд №12Repeated games. (Lecture 6), слайд №13Repeated games. (Lecture 6), слайд №14Repeated games. (Lecture 6), слайд №15Repeated games. (Lecture 6), слайд №16Repeated games. (Lecture 6), слайд №17Repeated games. (Lecture 6), слайд №18Repeated games. (Lecture 6), слайд №19Repeated games. (Lecture 6), слайд №20Repeated games. (Lecture 6), слайд №21Repeated games. (Lecture 6), слайд №22Repeated games. (Lecture 6), слайд №23Repeated games. (Lecture 6), слайд №24Repeated games. (Lecture 6), слайд №25Repeated games. (Lecture 6), слайд №26Repeated games. (Lecture 6), слайд №27Repeated games. (Lecture 6), слайд №28Repeated games. (Lecture 6), слайд №29Repeated games. (Lecture 6), слайд №30Repeated games. (Lecture 6), слайд №31Repeated games. (Lecture 6), слайд №32Repeated games. (Lecture 6), слайд №33Repeated games. (Lecture 6), слайд №34Repeated games. (Lecture 6), слайд №35
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LECTURE 6 REPEATED GAMES
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LECTURE 6 REPEATED GAMES

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Introduction Lectures 1-5: One-shot games The game is played just once, then the interaction ends. Players have a short term horizon, they are opportunistic, and are unlikely to cooperate (e.g. prisoner’s dilemma). Firms, individuals, governments often interact over long periods of time Oligopoly Trade partners
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Introduction Lectures 1-5: One-shot games The game is played just once, then the interaction ends. Players have a short term horizon, they are opportunistic, and are unlikely to cooperate (e.g. prisoner’s dilemma). Firms, individuals, governments often interact over long periods of time Oligopoly Trade partners

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Introduction Players may behave differently when a game is repeated. They are less opportunistic and prioritize the long-run payoffs, sometimes at the expense of short-term payoffs. Types of repeated games: Finitely repeated: the game is played for a finite and known number of rounds, e.g. 2 rounds/repetitions. Infinitely: the game is repeated infinitely. Indefinitely repeated: the game is repeated for an unknown number of times. The interaction will eventually end, but players don’t know when.
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Introduction Players may behave differently when a game is repeated. They are less opportunistic and prioritize the long-run payoffs, sometimes at the expense of short-term payoffs. Types of repeated games: Finitely repeated: the game is played for a finite and known number of rounds, e.g. 2 rounds/repetitions. Infinitely: the game is repeated infinitely. Indefinitely repeated: the game is repeated for an unknown number of times. The interaction will eventually end, but players don’t know when.

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A model of price competition Two firms compete in prices. The NE is to set low prices to gain market shares. They could obtain a higher payoff by cooperating (Prisoner’s dilemma situation)
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A model of price competition Two firms compete in prices. The NE is to set low prices to gain market shares. They could obtain a higher payoff by cooperating (Prisoner’s dilemma situation)

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A model of price competition The equilibrium that arises from using dominant strategies is worse for every player than cooperation. Why does defection occur? No fear of punishment Short term or myopic play What if the game is played “repeatedly” for several periods? The incentive to cooperate may outweigh the incentive to defect.
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A model of price competition The equilibrium that arises from using dominant strategies is worse for every player than cooperation. Why does defection occur? No fear of punishment Short term or myopic play What if the game is played “repeatedly” for several periods? The incentive to cooperate may outweigh the incentive to defect.

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Finite repetition Games where players play the same game for a certain finite number of times. The game is played n times, and n is known in advance. Nash Equilibrium: Each player will defect in the very last period Since both know that both will defect in the last period, they also defect in the before last period. etc…until they defect in the first period
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Finite repetition Games where players play the same game for a certain finite number of times. The game is played n times, and n is known in advance. Nash Equilibrium: Each player will defect in the very last period Since both know that both will defect in the last period, they also defect in the before last period. etc…until they defect in the first period

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Finite repetition When a one-shot game with a unique PSNE is repeated a finite number of times, repetition does not affect the equilibrium outcome. The dominant strategy of defecting will still prevail. BUT…finitely repeated games are relatively rare; how often do we really know for certain when a game will end? We routinely play many games that are indefinitely repeated (no known end), or infinitely repeated games.
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Finite repetition When a one-shot game with a unique PSNE is repeated a finite number of times, repetition does not affect the equilibrium outcome. The dominant strategy of defecting will still prevail. BUT…finitely repeated games are relatively rare; how often do we really know for certain when a game will end? We routinely play many games that are indefinitely repeated (no known end), or infinitely repeated games.

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Infinite Repetition What if the interaction never ends? No final period, so no rollback. Players may be using history-dependent strategies, i.e. trigger/contingent strategies: e.g. cooperate as long as the rivals do Upon observing a defection: immediately revert to a period of punishment (i.e. defect) of specified length.
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Infinite Repetition What if the interaction never ends? No final period, so no rollback. Players may be using history-dependent strategies, i.e. trigger/contingent strategies: e.g. cooperate as long as the rivals do Upon observing a defection: immediately revert to a period of punishment (i.e. defect) of specified length.

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Trigger Strategies Tit-for-tat (TFT): choose the action chosen by the other player last period
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Trigger Strategies Tit-for-tat (TFT): choose the action chosen by the other player last period

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Trigger Strategies Grim strategy: cooperate until the other player defects, then if he defects punish him by defecting until the end of the game
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Trigger Strategies Grim strategy: cooperate until the other player defects, then if he defects punish him by defecting until the end of the game

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Trigger Strategies Tit-for-Tat is most forgiving shortest memory proportional credible but lacks deterrence
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Trigger Strategies Tit-for-Tat is most forgiving shortest memory proportional credible but lacks deterrence

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Repeated games. (Lecture 6), слайд №12
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Infinite repetition and defection Is it worth defecting? Consider Firm1. Cooperation: Firm 1 defects: gain 36 (360-324) If Firm 2 plays TFT, it will also defect next period:
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Infinite repetition and defection Is it worth defecting? Consider Firm1. Cooperation: Firm 1 defects: gain 36 (360-324) If Firm 2 plays TFT, it will also defect next period:

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Infinite repetition and defection If Firm 1 keeps defecting: If Firm 1 reverts back to cooperation: If defection, trade-off defection - return to cooperation
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Infinite repetition and defection If Firm 1 keeps defecting: If Firm 1 reverts back to cooperation: If defection, trade-off defection - return to cooperation

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Discounting future payoffs Recall from the analysis of bargaining that players discount future payoffs. The discount factor is δ= 1/(1+r), with δ < 1. r is the interest rate Invest $1 today  get $(1+r) next year Want $1 next year  invest $1/(1+r) today For example, if r=0.25, then δ =0.8, i.e. a player values $1 received one period in the future as being equivalent to $0.80 right now.
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Discounting future payoffs Recall from the analysis of bargaining that players discount future payoffs. The discount factor is δ= 1/(1+r), with δ < 1. r is the interest rate Invest $1 today  get $(1+r) next year Want $1 next year  invest $1/(1+r) today For example, if r=0.25, then δ =0.8, i.e. a player values $1 received one period in the future as being equivalent to $0.80 right now.

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Discounting future payoffs Considering an infinitely repeated game, suppose that an outcome of this game is that a player receives $1 in every future play (round) of the game, starting from next period. Present value of $1 every period (starting from next period):
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Discounting future payoffs Considering an infinitely repeated game, suppose that an outcome of this game is that a player receives $1 in every future play (round) of the game, starting from next period. Present value of $1 every period (starting from next period):

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Defection? Defecting once vs. always cooperate against a TFT player. Gain 36 in period 1; Lose 108 in period 2. Defect if: Defecting forever vs. always cooperate against a TFT player. Gain 36 in period 1; Lose 36 every period ever after. Defect if:
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Defection? Defecting once vs. always cooperate against a TFT player. Gain 36 in period 1; Lose 108 in period 2. Defect if: Defecting forever vs. always cooperate against a TFT player. Gain 36 in period 1; Lose 36 every period ever after. Defect if:

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Defection? When r is high (r>minimum{1,2}, i.e. r>1 in this example), cooperation cannot be sustained. When future payoffs are heavily discounted, present gains outweigh future losses. Cooperation is sustainable only if r<1, i.e. if future payoffs are not too heavily discounted. Lesson: Infinite repetition increases the possibilities of cooperation, but r has to be low enough.
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Defection? When r is high (r>minimum{1,2}, i.e. r>1 in this example), cooperation cannot be sustained. When future payoffs are heavily discounted, present gains outweigh future losses. Cooperation is sustainable only if r<1, i.e. if future payoffs are not too heavily discounted. Lesson: Infinite repetition increases the possibilities of cooperation, but r has to be low enough.

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Games of unknown length Interactions don’t last forever: Suppose there is a probability p<1 that the interaction will continue next period  Indefinitely repeated games. present value of 1 tomorrow is Future losses are discounted more heavily than in infinitely repeated games, because they may not even materialize. Cooperation is more difficult to sustain when p<1 than when p=1.
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Games of unknown length Interactions don’t last forever: Suppose there is a probability p<1 that the interaction will continue next period  Indefinitely repeated games. present value of 1 tomorrow is Future losses are discounted more heavily than in infinitely repeated games, because they may not even materialize. Cooperation is more difficult to sustain when p<1 than when p=1.

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Games of unknown length The effective rate of return R is the rate of return used to discount future payoffs when p<1. R is such that: i.e. the discount factor δ is lower when p<1. R>r, and future payoffs are more heavily discounted, which decreases the possibilities of cooperation.
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Games of unknown length The effective rate of return R is the rate of return used to discount future payoffs when p<1. R is such that: i.e. the discount factor δ is lower when p<1. R>r, and future payoffs are more heavily discounted, which decreases the possibilities of cooperation.

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Games of unknown length We found that the condition for defecting against a TFT player is: e.g. suppose that r=0.05  no defection Now assume that there is each period a 10% chance that the game stops: p=0.90.  R=0.16 (still <1, hence no defection) If instead p=0.5, then R=1.1, and there is defection (1.1>minimum{1,2}).
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Games of unknown length We found that the condition for defecting against a TFT player is: e.g. suppose that r=0.05  no defection Now assume that there is each period a 10% chance that the game stops: p=0.90.  R=0.16 (still <1, hence no defection) If instead p=0.5, then R=1.1, and there is defection (1.1>minimum{1,2}).

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Example with asymmetric payoffs
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Example with asymmetric payoffs

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Example with asymmetric payoffs Firm 1: no change Defect once better than cooperate if: Defect forever better than cooperate if:
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Example with asymmetric payoffs Firm 1: no change Defect once better than cooperate if: Defect forever better than cooperate if:

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Example with asymmetric payoffs Firm 2: Defect once better than cooperate if: Defect forever better than cooperate if: Cooperation may not be stable when r>0.66
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Example with asymmetric payoffs Firm 2: Defect once better than cooperate if: Defect forever better than cooperate if: Cooperation may not be stable when r>0.66

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Experimental evidence from a prisoner’s dilemma game From Duffy and Ochs (2009), Games and Economic Behavior. Initially 30% of players cooperate, and this increase to 80% with more repetitions. Trust between players increases over time and fewer of them defect.
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Experimental evidence from a prisoner’s dilemma game From Duffy and Ochs (2009), Games and Economic Behavior. Initially 30% of players cooperate, and this increase to 80% with more repetitions. Trust between players increases over time and fewer of them defect.

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The Axelrod Experiment: Assessing trigger strategies Axelrod (1980s) invited selected specialists to enter strategies for cooperation games in a round-robin computer tournament. Strategies specified for 200 rounds. TFT obtained the highest overall score in the tournament. Why did TFT win? TFT's can adapt to opponents. It resists exploitation by defecting strategies but reciprocates cooperation. Programs that defect suffer against TFT programs. Programs that never defect lost against programs that defect.
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The Axelrod Experiment: Assessing trigger strategies Axelrod (1980s) invited selected specialists to enter strategies for cooperation games in a round-robin computer tournament. Strategies specified for 200 rounds. TFT obtained the highest overall score in the tournament. Why did TFT win? TFT's can adapt to opponents. It resists exploitation by defecting strategies but reciprocates cooperation. Programs that defect suffer against TFT programs. Programs that never defect lost against programs that defect.

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The Axelrod Experiment: Assessing trigger strategies In another experiment, some “players” were programmed to defect, some to cooperate, some to play trigger strategies such as TFT and grim. The programs that do well “reproduce” themselves and gain in population. The losing programs lose population. After 1000 rounds, TFT accounted for 70% of the population. TFT does well against itself and other cooperative strategies. Defecting strategies fare badly when their own kind spreads, and against TFT.
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The Axelrod Experiment: Assessing trigger strategies In another experiment, some “players” were programmed to defect, some to cooperate, some to play trigger strategies such as TFT and grim. The programs that do well “reproduce” themselves and gain in population. The losing programs lose population. After 1000 rounds, TFT accounted for 70% of the population. TFT does well against itself and other cooperative strategies. Defecting strategies fare badly when their own kind spreads, and against TFT.

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The Axelrod Experiment: Assessing trigger strategies According to Axelrod, TFT follow the following rules: “Don’t be envious, don’t be the first to defect, reciprocate both cooperation and defection, don’t be too clever.” Folk theorem: two TFT strategies are best replies for each other (i.e. it is a Nash Equilibrium). However, other Nash equilibria also exist, and may involve defecting strategies.
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The Axelrod Experiment: Assessing trigger strategies According to Axelrod, TFT follow the following rules: “Don’t be envious, don’t be the first to defect, reciprocate both cooperation and defection, don’t be too clever.” Folk theorem: two TFT strategies are best replies for each other (i.e. it is a Nash Equilibrium). However, other Nash equilibria also exist, and may involve defecting strategies.

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Repeated games. (Lecture 6), слайд №29
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Cournot in repeated games In a one-shot Cournot game, the unique NE is that producers defect rather than cooperate. Cooperation yields higher payoff, but is not stable. Cartels do form, and governments may have to intervene to prevent cartel formation. Some cartels are unstable, but some are stable.
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Cournot in repeated games In a one-shot Cournot game, the unique NE is that producers defect rather than cooperate. Cooperation yields higher payoff, but is not stable. Cartels do form, and governments may have to intervene to prevent cartel formation. Some cartels are unstable, but some are stable.

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Cournot in repeated games How to reconcile the Cournot model with the fact that many cartels are formed? Repetition increases the possibilities of cooperation, provided that producers attach sufficient weight on future payoffs (low r). “Short-termism” makes cartels less stable.
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Cournot in repeated games How to reconcile the Cournot model with the fact that many cartels are formed? Repetition increases the possibilities of cooperation, provided that producers attach sufficient weight on future payoffs (low r). “Short-termism” makes cartels less stable.

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Cournot in repeated games High p also helps. Cartels are more likely to be stable in “static” industries, where producers know that they will have a very long-term relationship. e.g. OPEC. The list of oil exporting countries is unlikely to change much over the next decades. In “dynamic” industries, where market shares quickly change, collusion is less stable.
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Cournot in repeated games High p also helps. Cartels are more likely to be stable in “static” industries, where producers know that they will have a very long-term relationship. e.g. OPEC. The list of oil exporting countries is unlikely to change much over the next decades. In “dynamic” industries, where market shares quickly change, collusion is less stable.

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Other factors affecting the possibilities of collusion I The more complex the negotiations, the greater the costs of cooperation (and create a cartel) It is easier to form a cartel when… Few producers are involved. 77% of cartels have six or fewer firms (Connor, 2003) The market is highly concentrated. Cartel members usually control 90%+ of the industry sales (Connor, 2003) Producers have a nearly identical product. If the products are different it is difficult to spot cheating because different products naturally have different prices
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Other factors affecting the possibilities of collusion I The more complex the negotiations, the greater the costs of cooperation (and create a cartel) It is easier to form a cartel when… Few producers are involved. 77% of cartels have six or fewer firms (Connor, 2003) The market is highly concentrated. Cartel members usually control 90%+ of the industry sales (Connor, 2003) Producers have a nearly identical product. If the products are different it is difficult to spot cheating because different products naturally have different prices

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Other factors affecting the possibilities of collusion II The incentive to defect from the cartel are larger when there are many producers. Consider an industry with N producers. π is the monopoly profit. Profit if all producers cooperate: π /N Profit if one defects: become a monopolist and get π Profit if is being punished: 0 As the number of producers rises, the gain from defection increases: π - π /N increases with N. With a high number of producers, the incentives to defect are strong.
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Other factors affecting the possibilities of collusion II The incentive to defect from the cartel are larger when there are many producers. Consider an industry with N producers. π is the monopoly profit. Profit if all producers cooperate: π /N Profit if one defects: become a monopolist and get π Profit if is being punished: 0 As the number of producers rises, the gain from defection increases: π - π /N increases with N. With a high number of producers, the incentives to defect are strong.

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Summary One-shot games: defection in equilibrium. Having a finite number of repetitions does not increase the possibilities of defection. Infinite repetitions can induce players to cooperate, but r has to be low enough. Players may use trigger strategies, and experiments suggest that TFT is a strong strategy. In indefinitely repeated games, a low p is associated with reduced possibilities of cooperation.
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Summary One-shot games: defection in equilibrium. Having a finite number of repetitions does not increase the possibilities of defection. Infinite repetitions can induce players to cooperate, but r has to be low enough. Players may use trigger strategies, and experiments suggest that TFT is a strong strategy. In indefinitely repeated games, a low p is associated with reduced possibilities of cooperation.

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