Working Papers
By Year:
Paper #  Author  Title  

12043 
V. Bhaskar George J. Mailath Stephen Morris 
"A Foundation for Markov Equilibria in Infinite Horizon Perfect Information Games"  
We study perfect information games with an infinite horizon played by an arbitrary number of players. This class of games includes infinitely repeated perfect information games, repeated games with asynchronous moves, games with long and short run players, games with overlapping generations of players, and canonical noncooperative models of bargaining. We consider two restrictions on equilibria. An equilibrium is purifiable if close by behavior is consistent with equilibrium when agents' payoffs at each node are perturbed additively and independently. An equilibrium has bounded recall if there exists K such that at most one player's strategy depends on what happened more than K periods earlier. We show that only Markov equilibria have bounded memory and are purifiable. Thus if a game has at most one longrun player, all purifiable equilibria are Markov. Download Paper


12003 
V. Bhaskar George J. Mailath Stephen Morris 
“A Foundation for Markov Equilibria with Finite Social Memory”  
We study stochastic games with an infinite horizon and sequential moves played by an arbitrary number of players. We assume that social memory is finiteevery player, except possibly one, is finitely lived and cannot observe events that are sufficiently far back in the past. This class of games includes games between a longrun player and a sequence of shortrun players and games with overlapping generations of players. Indeed, any stochastic game with infinitely lived players can be reinterpreted as one with finitely lived players: Each finitelylived player is replaced by a successor, and receives the value of the successor's payoff. This value may arise from altruism, but the player also receives such a value if he can “sell” his position in a competitive market. In both cases, his objective will be to maximize infinite horizon payoffs, though his information on past events will be limited. An equilibrium is purifiable if closeby behavior is consistent with equilibrium when agents' payoffs in each period are perturbed additively and independently. We show that only Markov equilibria are purifiable when social memory is finite. Thus if a game has at most one longrun player, all purifiable equilibria are Markov. Download Paper


09029 
V. Bhaskar George J. Mailath Stephen Morris 
"A Foundation for Markov Equilibria in Infinite Horizon Perfect Information Games"  
We study perfect information games with an infinite horizon played by an arbitrary number of players. This class of games includes infinitely repeated perfect information games, repeated games with asynchronous moves, games with long and short run players, games with overlapping generations of players, and canonical noncooperative models of bargaining. We consider two restrictions on equilibria. An equilibrium is purifiable if close by behavior is consistent with equilibrium when agents' payoffs at each node are perturbed additively and independently. An equilibrium has bounded recall if there exists K such that at most one player's strategy depends on what happened more than K periods earlier. We show that only Markov equilibria have bounded memory and are purifiable. Thus if a game has at most one longrun player, all purifiable equilibria are Markov. Download Paper


07024 
V. Bhaskar George J. Mailath Stephen Morris 
"Purification in the InfinitelyRepeated Prisoners' Dilemma", Second Version  
This paper investigates the Harsanyi (1973)purifiability of mixed strategies in the repeated prisoners’ dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of oneperiod memory mixed strategy equilibria used by Ely and Välimäki (2002) in their study of the repeated prisoners’ dilemma with private monitoring. We find that any such strategy profile is not the limit of oneperiod memory quilibrium strategy profiles of the perturbed game, for almost all noise distributions. However, if we allow infinite memory strategies in the perturbed game, then any completelymixed equilibrium is purifiable. Download Paper


04004 
V. Bhaskar George J. Mailath Stephen Morris 
"Purification in the InfinitelyRepeated Prisoners' Dilemma"  
This paper investigates the Harsanyi (1973)purifiability of mixed strategies in the repeated prisoners' dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of a class of oneperiod memory mixed strategy equilibria used by Ely and Välimäki (2002) in their study of the repeated prisoners' dilemma with private monitoring. We find that the strategy profile is purifiable by perturbedgame finitememory strategies if and only if it is strongly symmetric, in the sense that after every history, both players play the same mixed action. Thus "most" strategy profiles are not purifiable by finite memory strategies. However, if we allow infinite memory strategies in the perturbed game, then any completelymixed equilibrium is purifiable. Download Paper
