Belief-Based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring

We analyze the infinitely repeated prisoners' dilemma with imperfect private monitoring and discounting. The main contribution of this paper is to construct ``belief-based'' strategies, where a player's continuation strategy is a function only of his beliefs. This simplifies the analysis considerably, and allows us to explicitly construct sequential equilibria for such games, thus enabling us to invoke the one-step deviation principle of dynamic programming. By doing so, we prove that one can approximate the efficient payoff in any prisoners' dilemma game provided that the monitoring is sufficiently accurate. Furthermore, for a class of prisoners' dilemma games, one can approximate every individually rational feasible payoff. These results require that monitoring be sufficiently accurate, but only require a uniform lower bound on the discount rate.