Bounding Equilibrium Payoffs in Repeated Games With Private Monitoring

We provide a simple sufficient condition for the existence of a recursive upper bound on (the Pareto frontier of) the sequential equilibrium payoff set at a fixed discount factor in two-player repeated games with imperfect private monitoring. The bounding set is the sequential equilibrium payoff set with perfect monitoring and a mediator. We show that this bounding set admits a simple recursive characterization, which nonetheless necessarily involves the use of private strategies. Under our condition, this set describes precisely those payoff vectors that arise in equilibrium for some private monitoring structure if either nonstationary monitoring or communication is allowed.