The Folk Theorem with Private Monitoring and Uniform Sustainability

This paper investigates infinitely repeated prisoner-dilemma games where the discount factor is less than but close to 1. We assume that monitoring is truly imperfect and truly private, there exist no public signals and no public randomization devices, and players cannot communicate and use only pure strategies. We show that implicit collusion can be sustained by Nash equilibria under a mild condition. We show that the Folk Theorem holds when players' private signals are conditionally independent. These results are permissive, because we require no conditions concerning the accuracy of private signals such as the zero likelihood ratio condition. We also investigate the situation in which players play a Nash equilibrium of a machine game irrespective of their initial states, i.e., they play a uniform equilibrium. We show that there exists a unique payoff vector sustained by a uniform equilibrium, i.e., a unique uniformly sustainable payoff vector, which Pareto-dominates all other uniformly sustainable payoff vectors. We characterize this payoff vector by using the values of the minimum likelihood ratio. We show that this payoff vector is efficient if and only if the zero likelihood ratio condition is satisfied. These positive results hold even if each player has limited knowledge on her opponent's private signal structure. Keywords: Repeated Prisoner-Dilemma Games, Private Monitoring, Conditional Independence, Folk Theorem, Uniform Sustainability, Zero Likelihood Ratio Condition, Limited Knowledge.