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This paper concerns the two‐stage game introduced in Nash (1953). It formalizes a suggestion made (but not pursued) by Nash regarding equilibrium selection in that game, and hence offers an arguably more solid foundation for the “Nash bargaining with endogenous threats” solution. Analogous reasoning is then applied to an infinite horizon game to provide equilibrium selection in two‐person repeated games with contracts. In this setting, issues about enforcement of threats are much less problematic than in Nash's static setting. The analysis can be extended to stochastic games with contracts.

Consider a two-person intertemporal bargaining problem in which players choose actions and offers each period, and collect payoffs (as a function of that period's actions) while bargaining proceeds. This can alternatively be viewed as an infinitely repeated game wherein players can offer one another enforceable contracts that govern play for the rest of the game. Theory is silent with regard to how the surplus is likely to be split, because a folk theorem applies. Perturbing such a game with a rich set of behavioral types for each player yields a specific asymptotic prediction for how the surplus will be divided, as the perturbation probabilities approach zero. Behavioral types may follow nonstationary strategies and respond to the opponent's play. In equilibrium, rational players initially choose a behavioral type to imitate and a war of attrition ensues. How much should a player try to get and how should she behave while waiting for the resolution of bargaining? In both respects she should build her strategy around the advice given by the "Nash bargaining with threats" (NBWT) theory developed for two-stage games. In any perfect Bayesian equilibrium, she can guarantee herself virtually her NBWT payoff by imitating a behavioral type with the following simple strategy: in every period, ask for (and accept nothing less than) that player's NBWT share and, while waiting for the other side to concede, take the action Nash recommends as a threat in his two-stage game. The results suggest that there are forces at work in some dynamic games that favor certain payoffs over all others. This is in stark contrast to the classic folk theorems, to the further folk theorems established for repeated games with two-sided reputational perturbations, and to the permissive results obtained in the literature on bargaining with payoffs as you go. Copyright The Econometric Society 2007.

This paper investigates pure strategy sequential equilibria of repeated games with imperfect monitoring. The approach emphasizes the equilibrium value set and the static optimization problems embedded in extremal equilibria. A succession of propositions, central among which is "self-generation, " allow properties of constrained efficient supergame equilibria to be deduced from the solutions of the static problems. The authors show that the latter include solutions having a "bang-bang" property; this affords a significant simplification of the equilibria that need be considered. These results apply to a broad class of asymmetric games, thereby generalizing their earlier work on optimal cartel equilibria. Copyright 1990 by The Econometric Society.

In a repeated partnership game with imperfect monitoring, we distinguish among the effects of (1) reducing the interest rate, (2) shortening the period over which actions are held fixed, and (3) shortening the lag with which accumulated information is reported.All three changes are equivalent in games with perfect monitoring.With imperfect monitoring, reducing the interest rate always increases the possibilities for cooperation, but the other two changes always have the reverse effect when the interest rate is small.