Stationarity and Chaos in Infinitely Repeated Games of Incomplete Information

Consider an incomplete information game in which the players first learn their own types, and then infinitely often play the same normal form game with the same opponents. After each play, the players observe their own payoff and the action of their opponents. The payoff for a strategy n-tuple in the infinitely repeated game is the discounted present value of the infinite stream of payoffs generated by the strategy. This paper studies Bayesian learning in such a setting. Kalai and Lehrer [1991] and Jordan [1991] have shown that Bayesian equilibria to such games exist and eventually look like Nash equilibria to the infinitely repeated full information game with the correct types. However, due to folk theorems for complete information games, this still leaves the class of equilibria for such games to be quite large. In order to refine the set of equilibria, we impose a restriction on the equilibrium strategies of the players which requires stationarity with respect to the profile of current beliefs: if the same profile of beliefs is reached at two different points in time, the players must choose the same behavioral strategy at both points in time. This set, called the belief stationary equilibria, is a subset of the Bayesian Nash equilibria. We compute a belief stationary equilibrium in an example. The equilibria that result can have elements of chaotic behavior. The equilibrium path of beliefs when types are not revealed can be chaotic, and small changes in initial beliefs can result in large changes in equilibrium actions.