Federal Reserve Bank of

This paper develops recursive methods for considering stationary sequential equilibria in games with private actions and signals (private monitoring games), where play is assumed not only to continue forever into the future, but is also assumed to have occurred forever into the past. In particular, we develop set-based methods for constructing all pure strategy equilibria of the subclass of strategies where each players’ action, but not his beliefs, depends only on what he observed in the previous k periods. We then connect equilibria of games with infinite histories to the more traditional games with a start date. We show that each k-history dependent sequential equilibrium in a game with infinite history defines a simple correlated equilibrium for a game with a starting date. Further, for any correlation device which signals fictitious private k-period histories, we derive setbased necessary and sufficient conditions for this correlation device to deliver a correlated equilibrium of the game with a start date when coupled with a k-history dependent sequential equilibrium in the infinite history game. Since sequential equilibria are examples of correlated equilibria (with degenerate signaling devices) these methods deliver simple necessary and sufficient conditions for constructing all k-history dependent pure strategy sequential equilibria of private monitoring games with a start date. ∗The authors thank Narayana Kocherlakota, Peter DeMarzo, David Levine, George Mailath, Stephen Morris, Larry Samuelson, seminar participants at the Federal Reserve Bank of Minneapolis and the 2006 meetings of the Society for Economic Dynamics for helpful comments as well as the excellent research assistance of Roozbeh Hosseini. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.