Efficiency in a Repeated Prisoners' Dilemma with Imperfect Private Monitoring

We study the repeated two-player Prisoners' Dilemma with imperfect private monitoring and no communication. Letting the discount factor go to one and holding the monitoring structure fixed, we achieve asymptotic efficiency. Unlike previous works on private monitoring, which have confined attention to signals that are either almost perfect or conditionally independent, we allow for both imperfect and correlated signals but assume that they are sufficiently private, i.e. private actions are more informative than private signals about the opponent's signals. Interestingly, for the game we study, even the existing literature that allows communication has not yet yielded efficiency.