Equilibrium Characterization of Repeated Games with Private Monitoring

Abstract

This paper studies sequential equilibria of repeated games with private monitoring. We recover a recursive structure by means of stochastic filtering. The set of complete, infinite-length outcome paths forms a filtered space, over which the filtrations represent the players' information sets. Finite-period histories are setwise identical to the filtered space, so that players face continuation games strategically identical to the original game. A player's first-order beliefs of her opponents' private histories along with her finite-period histories are sufficient statistics for continuation of play, and the value of continuation is a function of the two. In a pure-strategy sequential equilibrium profile, each player has a partition of her belief manifold. Over each submanifold, value functions, which indicate continuation values, are maximized when the histories and beliefs are consistent, that is, when the beliefs are reachable from initial beliefs through the histories. The mixed-strategy equilibria are the belief-free equilibria in the literature. Private beliefs are irrelevant in action choices and state transitions as all value functions are identical. However, beliefs affect the continuation values. We are able to characterize the pure-strategy equilibrium sets when the actions and signals are finite.