A Folk Theorem for Repeated Sequential Games

Abstract

We study repeated sequential games where players may not move simultaneously in stage games. We introduce the concept of effective minimax for sequential games and establish a Folk theorem for repeated sequential games. The Folk theorem asserts that any feasible payoff vector where every player receives more than his effective minimax value in a sequential stage game can be supported by a subgame perfect equilibrium in the corresponding repeated sequential game when players are sufficiently patient. The results of this paper generalize those of Wen (1994), and of Fudenberg and Maskin (1986). The model of repeated sequential games and the concept of effective minimax provide an alternative view to the Anti-Folk theorem of Lagunoff and Matsui (1997) for asynchronously repeated pure coordination games. It has long been recognized that players behave differently in one-shot games and in the corresponding repeated games due to players' different objectives in short-term and long-term relationships. Repeat interactions allow players to respond to others' past actions in the future and a player must consider others' reactions in the future when making a decision. When all players evaluate their future sufficiently highly, repeated interactions enable the enforcement of almost all reasonable outcomes. This type of result is referred to as a Folk theorem, not only for repeated games but also for other situations. Intensive research makes repeated games one of the most mature subjects in game theory.' One seminal work is the infinitely repeated game model with discounting of Fudenberg and Maskin (1986). Their Folk theorem asserts that if the discounting is low enough then any feasible and strictly individually rational payoff vector of a stage game can be supported by a subgame perfect equilibrium in the corresponding infinitely repeated game, when the stage game either has only two players or satisfies the full dimensionality condition. In Fudenberg and Maskin's and many other repeated game models,2 stage games are modelled in normal form. It is interpreted that players choose their actions simultaneously in normal form games. What happens in repeated games if players do not move simultaneously in stage games? Fudenberg and Tirole (1991) first raised this question. Unfortunately, this question did not attract lot of attention at the time since it was commonly believed that a repeated game of any form of stage game should not differ too much from the repeated normal form representation of the stage game. Inspired by Fudenberg and Tirole's question, we study a class of repeated games, called repeated sequential games, in which players may not move simultaneously in stage games. We will introduce the concept of effective minimax for sequential games and establish a Folk theorem for repeated sequential games. Since normal form games are sequential games by definition, the effective minimax values and the Folk theorem in 1. Most game theory textbooks have chapters on repeated games. For a survey, see Aumann (1981), Friedman