An anti-folk theorem for finite past equilibria in repeated games with private monitoring

We prove an anti-folk theorem for repeated games with private monitoring. We assume that the strategies have a finite past (they are measurable with respect to finite partitions of past histories), that each period players' preferences over actions are modified by smooth idiosyncratic shocks, and that the monitoring is sufficiently connected. In all repeated game equilibria, each period play is an equilibrium of the stage game. When the monitoring is approximately connected and equilibrium strategies have a uniformly bounded past, then each period play is an approximate equilibrium of the stage game.