Beliefs and Private Monitoring ∗

This paper develops new recursive, set based methods for studying repeated games with private monitoring. For any finite state strategy profile, we find necessary and sufficient conditions for whether there exists a distribution over initial states such that the strategy, together with this distribution, form a correlated sequential equilibrium. Also, for any given correlation device for determining initial states (including degenerate cases where players’ initial states are common knowledge), we provide necessary and sufficient conditions for the correlation device and strategy to be a correlated sequential equilibrium, or in the case of a degenerate correlation device, for the strategy to be a sequential equilibrium. We also consider several applications. In these, we show that the methods are computationally feasible, show how to construct and verify equilibria in a secret price-setting game and show that with private monitoring, stick-and-carrot strategies cannot support equilibrium in a classic repeated duopoly game. ∗The authors thank V. Bhaskar, Peter DeMarzo, Glenn Ellison, Larry Jones, Narayana Kocherlakota, David Levine, George Mailath, Stephen Morris, Ichiro Obara, Larry Samuelson, Itai Sher, Ofer Zeitouni, seminar participants at the Federal Reserve Bank of Minneapolis, the Harvard/MIT joint theory seminar, Stanford University, Iowa State University, Princeton University, the University of Chicago, the University of Minnesota, University College London, the London School of Economics and the 2006 meetings of the Society for Economic Dynamics and three anonymous referees for helpful comments as well as the excellent research assistance of Songzi Du, Kenichi Fukushima and Roozbeh Hosseini. Financial assistance from National Science Foundation Grant # 0721090 is gratefully acknowledged. 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.