An Antifolk Theorem for Large Repeated Games

In this article, we study infinitely repeated games in settings of imperfect monitoring. We first prove a family of theorems showing that when the signals observed by the players satisfy a condition known as (ϵ, γ)-differential privacy, the folk theorem has little bite: for values of ϵ and γ sufficiently small, for a fixed discount factor, any equilibrium of the repeated game involves 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 (ϵ, γ)-differential privacy for ϵ 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 collapses to the set of equilibria of the stage game.

Our results nest and generalize previous results of Green [1980] and Sabourian [1990], suggesting 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.