On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games

This paper provides a dual characterization of the limit set of perfect public equilibrium payoffs in stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg, Levine and Maskin (1994), Kandori and Matsushima (1998) and Hörner, Sugaya, Takahashi and Vieille (2011) obtain. As a second corollary, in the context of repeated games, it follows that this limit set of payoffs is a polytope (a bounded polyhedron) when attention is restricted to equilibria in pure strategies. We provide a two-player game in which this limit set is not a polytope when mixed strategies are considered.Stochastic games, Repeated games, Folk theorem

On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games

This paper provides a dual characterization of the limit set of perfect public equilibrium payoffs in stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg, Levine and Maskin (1994), Kandori and Matsushima (1998) and Hörner, Sugaya, Takahashi and Vieille (2011) obtain. As a second corollary, in the context of repeated games, it follows that this limit set of payoffs is a polytope (a bounded polyhedron) when attention is restricted to equilibria in pure strategies. We provide a two-player game in which this limit set is not a polytope when mixed strategies are considered

A Complete Characterization of Infinitely Repeated Two-Player Games having Computable Strategies with no Computable Best Response under Limit-of-Means Payoff

It is well-known that for infinitely repeated games, there are computable strategies that have best responses, but no computable best responses. These results were originally proved for either specific games (e.g., Prisoner's dilemma), or for classes of games satisfying certain conditions not known to be both necessary and sufficient. We derive a complete characterization in the form of simple necessary and sufficient conditions for the existence of a computable strategy without a computable best response under limit-of-means payoff. We further refine the characterization by requiring the strategy profiles to be Nash equilibria or subgame-perfect equilibria, and we show how the characterizations entail that it is efficiently decidable whether an infinitely repeated game has a computable strategy without a computable best response

Subgame Perfect Equilibria in Continuous-Time Repeated Games

This paper considers subgame perfect equilibria of continuous-time repeated games with perfect monitoring when immediate reactions to deviations are allowed. The set of subgame perfect equilibrium payoffs is shown to be a fixed-point of a set-valued operator introduced in the paper. For a large class of discrete time games the closure of this set corresponds to the limit payoffs of  when the discount factors converge to one. It is shown that in the continuous-time setup pure strategies are sufficient for obtaining all equilibrium payoffs supported by the players' minimax values. Moreover, the equilibrium payoff set is convex and satisfies monotone comparative statics when the ratios of players' discount rates increase.</p