The Folk Theorem for Games with Private Almost‐Perfect Monitoring

We prove the folk theorem for discounted repeated games under private, almost‐perfect monitoring. Our result covers all finite, n‐player games that satisfy the usual full‐dimensionality condition. Mixed strategies are allowed in determining the individually rational payoffs. We assume no cheap‐talk communication between players and no public randomization device.     

Belief‐Free Equilibria in Repeated Games

We introduce a class of strategies that generalizes examples constructed in two‐player games under imperfect private monitoring. A sequential equilibrium is belief‐free if, after every private history, each player's continuation strategy is optimal independently of his belief about his opponents' private histories. We provide a simple and sharp characterization of equilibrium payoffs using those strategies. While such strategies support a large set of payoffs, they are not rich enough to generate a folk theorem in most games besides the prisoner's dilemma, even when noise vanishes.     

Weakly Belief‐Free Equilibria in Repeated Games With Private Monitoring

Repeated games with imperfect private monitoring have a wide range of applications, but a complete characterization of all equilibria in this class of games has yet to be obtained. The existing literature has identified a relatively tractable subset of equilibria. The present paper introduces the notion of weakly belief‐free equilibria for repeated games with imperfect private monitoring. This is a tractable class which subsumes, as a special case, a major part of the existing literature (the belief‐free equilibria). It is shown that this class can outperform the equilibria identified by the previous work.     

Efficiency in Repeated Games Revisited: The Role of Private Strategies

Most theoretical or applied research on repeated games with imperfect monitoring has focused on public strategies: strategies that depend solely on the history of publicly observable signals. This paper sheds light on the role of private strategies: strategies that depend not only on public signals, but also on players' own actions in the past. Our main finding is that players can sometimes make better use of information by using private strategies and that efficiency in repeated games can be improved. Our equilibrium private strategy for repeated prisoners' dilemma games consists of two states and has the property that each player's optimal strategy is independent of the other player's state.     

Repeated Games with Differential Time Preferences

When players have identical time preferences, the set of feasible repeated game payoffs coincides with the convex hull of the underlying stage- game payoffs. Moreover, all feasible and individually rational payoffs can be sustained by equilibria if the players are sufficiently patient. Neither of these facts generalizes to the case of different time preferences. First, players can mutually benefit from trading payoffs across time. Hence, the set of feasible repeated game payoffs is typically larger than the convex hull of the underlying stage-game payoffs. Second, it is not usually the case that every trade plan that guarantees individually rational payoffs can be sustained by an equilibrium, no matter how patient the players are. This paper provides a simple characterization of the sets of Nash and of subgame perfect equilibrium payoffs in two-player repeated games.     

Repeated Games Where the Payoffs and Monitoring Structure Are Unknown

This paper studies repeated games with imperfect public monitoring where the players are uncertain both about the payoff functions and about the relationship between the distribution of signals and the actions played. We introduce the concept of perfect public ex post equilibrium (PPXE), and show that it can be characterized with an extension of the techniques used to study perfect public equilibria. We develop identifiability conditions that are sufficient for a folk theorem; these conditions imply that there are PPXE in which the payoffs are approximately the same as if the monitoring structure and payoff functions were known. Finally, we define perfect type‐contingently public ex post equilibria (PTXE), which allows players to condition their actions on their initial private information, and we provide its linear programming characterization.