It is well known from the Folk Theorem that infinitely repeated games admit a multitude of equilibria. This paper demonstrates that in some types of games, the Folk Theorem form of multiplicity is an artifact of the standard representation which assumes perfect synchronization in the timing of actions between the players. We define here a more general family of repeated settings called renewal games. Specifically, a renewal game is a setting in which a stage game is repeated in continuous time, and at certain stochastic points in time determined by an arbitrary renewal process some set of players may be called upon to make a move. A stationary, ergodic Markov process determines who moves at each decision node. We restrict attention in this paper to a natural subclass of renewal games called asynchronously repeated games, in which no two individuals can change their actions simultaneously. Special cases include the alternating move game, and the Poisson revision game. In the latter, each player adjusts his action independently at Poisson distributed times.
Our main result concerns asynchronously repeated games of pure coordination (where the pay­ offs of all players in the stage game are identical up to an affine transformation): given € > 0, if players are sufficiently patient then every Perfect equilibrium payoff comes within € of the Pareto dominant payoff. We also show that the "Folk wisdom" in the standard model that repetition always expands (weakly) the set of equilibrium payoffs is not true generally in asynchronously repeated games.