An Anti-Folk Theorem for Large Repeated Games with Imperfect Monitoring

We study infinitely repeated games in settings of imperfect monitoring. We first prove a family of theorems that show that when the signals observed by the players satisfy a condition known as (ǫ, γ)-differential privacy, that the folk theorem has little bite: for values of ǫ and γ sufficiently small, for a fixed discount factor, any equilibrium of the repeated game involve players playing approximate equilibria of the stage game in every period. Next, we argue that in large games (n player games in which unilateral deviations by single players have only a small impact on the utility of other players), many monitoring settings naturally lead to signals that satisfy (ǫ, γ)-differential privacy, for ǫ and γ tending to zero as the number of players n grows large. We conclude that in such settings, the set of equilibria of the repeated game collapse to the set of equilibria of the stage game. Our results nest and generalize previous results of Green (1980), Sabourian (1990) and suggest that differential privacy is a natural measure of the “largeness” of a game. Further, techniques from the literature on differential privacy allow us to prove quantitative bounds, where the existing literature focuses on limiting results. We would like to especially thank Yu Awaya and Vijay Krishna for several helpful comments regarding the manuscript. Department of Economics, University of Pennsylvania. Email: [email protected]. Department of Computer and Information Science, University of Pennsylvania. Email: [email protected] Supported in part by NSF Grant CCF-1101389, and NSF CAREER grant, and a Google Research Grant. Center for Research on Computation and Society, Harvard University. Email: [email protected]