Private Monitoring, Likelihood Ratio Condition, and the Folk Theorem

This paper investigates infinitely repeated prisoner-dilemma games where the discount factor is less than but close to 1. We assume that players not only imperfectly but also privately monitor their opponents’ choices of action. We provide a likelihood ratio condition under which there exist Nash equilibrium payoff vectors better than the one-shot Nash equilibrium payoff vector. We show that an efficient payoff vector is approximated by a Nash equilibrium payoff vector if the minimum likelihood ratio is zero. We show the full folk theorem when players’ private signals are independent and a stronger version of this zero likelihood ratio condition is satisfied. In contrast with previous works, this paper assumes that monitoring is neither almost perfect nor almost public, and there exist no public signals, no announcement of publicly observed messages, and no public randomization devices. We also investigate machine games, and we newly require that players always play a Nash equilibrium irrespective of their initial states of machine, i.e., they play according to a uniform equilibrium. We show that there exists the unique payoff vector sustained by a uniform equilibrium, i.e., the unique uniformly sustainable payoff vector, which Pareto-dominates all other uniformly sustainable payoff vectors. This is in contrast with the multiplicity of Pareto-undominated perfect equilibrium payoff vectors. We show also that this Pareto-dominant uniformly sustainable payoff is efficient if and only if the minimum likelihood ratio is zero.