"Repeated Games, Entry in The New Palgrave Dictionary of Economics, 2nd Edition"

This entry shows why self-interested agents manage to cooperate in a long-term relationship. When agents interact only once, they often have an incentive to deviate from cooperation. In a repeated interaction, however, any mutually beneficial outcome can be sustained in an equilibrium. This fact, known as the folk theorem, is explained under various information structures. This entry also compares repeated games with other means to achieve efficiency and briefly discuss the scope for potential applications.

"The Folk Theorem with Private Monitoring"

This paper investigates infinitely repeated prisoner-dilemma games, where the discount factor is less than but close to 1. We assume that monitoring is imperfect and private, and players' private signal structures satisfy the conditional independence. We require almost no conditions concerning the accuracy of private signals. We assume that there exist no public signals and no public randomization devices, and players cannot communicate and use only pure strategies. It is shown that the Folk Theorem holds in that every individually rational feasible payoff vector can be approximated by a sequential equilibrium payoff vector. Moreover, the Folk Theorem holds even if each player has no knowledge of her opponent's private signal structure.

A Robust Folk Theorem for the Prisoners' Dilemma

We prove the folk theorem for the Prisoner's dilemma using strategies that are robust to private monitoring. From this follows a limit folk theorem : when players are patient and monitoring is sufficiently accurate, (but private and possibly independent) any feasible individually rational payoff can be obtained in sequential equilibrium. The strategies used can be implemented by finite (randomizing) automata.

Folk theorems with Bounded Recall under(Almost) Perfect Monitoring

A strategy profile in a repeated game has bounded recall L if play under the profile after two distinct histories that agree in the last L periods is equal. Mailath and Morris (2002, 2006) proved that any strict equilibrium in bounded-recall strategies of a game with full support public monitoring is robust to all perturbations of the monitoring structure towards private monitoring (the case of almost-public monitoring), while strict equilibria in unbounded-recall strategies are typically not robust. We prove that the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict. The general result uses calendar time in an integral way in the construction of the strategy profile. If the players’ action spaces are sufficiently rich, then the strategy profile can be chosen to be independent of calendar time. Either result can then be used to prove a folk theorem for repeated games with almost-perfect almost-public monitoring.Repeated games, bounded recall strategies, folk theorem, imperfect monitoring

Folk Theorems with Bounded Recall under (Almost) Perfect Monitoring

A strategy profile in a repeated game has L bounded recall if play under the profile after two distinct histories that agree in the last L periods is equal. Mailath and Morris (2002, 2006) proved that any strict equilibrium in bounded-recall strategies of a game with full support public monitoring is robust to all perturbations of the monitoring structure towards private monitoring (the case of "almost-public monitoring"), while strict equilibria in unbounded-recall strategies are typically not robust. We prove that the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict. The general result uses calendar time in an integral way in the construction of the strategy profile. If the players' action spaces are sufficiently rich, then the strategy profile can be chosen to be independent of calendar time. Either result can then be used to prove a folk theorem for repeated games with almost-perfect almost-public monitoring.Repeated games, bounded recall strategies, folk theorem, imperfect monitoring

Folk Theorems with Bounded Recall under (Almost) Perfect Monitoring, Second Version

A strategy profile in a repeated game has bounded recall L if play under the profile after two distinct histories that agree in the last L periods is equal. Mailath and Morris (2002, 2006) proved that any strict equilibrium in bounded-recall strategies of a game with full support public monitoring is robust to all perturbations of the monitoring structure towards private monitoring (the case of almost-public monitoring), while strict equilibria in unbounded-recall strategies are typically not robust. We prove the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict. As a consequence, the perfect monitoring folk theorem is shown to be behaviorally robust under almost-perfect almost-public monitoring. That is, the same specification of behavior continues to be an equilibrium when the monitoring is perturbed from perfect to highly-correlated private.Repeated games, bounded recall strategies, folk theorem, imperfect monitoring

Folk Theorems with Bounded Recall under (Almost) Perfect Monitoring, Third Version

We prove the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict. As a consequence, the perfect monitoring folk theorem is shown to be behaviorally robust under almost-perfect almost-public monitoring. That is, the same specification of behavior continues to be an equilibrium when the monitoring is perturbed from perfect to highly-correlated private.Repeated games, bounded recall strategies, folk theorem,imperfect monitoring

"The Folk Theorem with Private Monitoring and Uniform Sustainability"

This paper investigates infinitely repeated prisoner-dilemma games where the discount factor is less than but close to 1. We assume that monitoring is truly imperfect and truly private, there exist no public signals and no public randomization devices, and players cannot communicate and use only pure strategies. We show that implicit collusion can be sustained by Nash equilibria under a mild condition. We show that the Folk Theorem holds when playersf private signals are conditionally independent. These results are permissive, because we require no conditions concerning the accuracy of private signals such as the zero likelihood ratio condition. We also investigate the situation in which players play a Nash equilibrium of a machine game irrespective of their initial states, i.e., they play a uniform equilibrium. We show that there exists a unique payoff vector sustained by a uniform equilibrium, i.e., a unique uniformly sustainable payoff vector, which Pareto-dominates all other uniformly sustainable payoff vectors. We characterize this payoff vector by using the values of the minimum likelihood ratio. We show that this payoff vector is efficient if and only if the zero likelihood ratio condition is satisfied. These positive results hold even if each player has limited knowledge on her opponentfs private signal structure. Keywords: Repeated Prisoner-Dilemma Games, Private Monitoring, Conditional Independence, Folk Theorem, Uniform Sustainability, Zero Likelihood Ratio Condition, Limited Knowledge.

"Repeated Games with Private Monitoring: Two Players"

We investigate two-player infinitely repeated games where the discount factor is less than but close to unity. Monitoring is private and players cannot communicate. We require no condition concerning the accuracy of players' monitoring technology. We show the folk theorem for the prisoners' dilemma with conditional independence. We also investigate more general games where players' private signals are correlated only through an unobservable macro shock. We show that efficiency is sustainable for generic private signal structures when the size of the set of private signals is sufficiently large. Finally, we show that cartel collusion is sustainable in price-setting duopoly.