Bargaining, Reputation and Equilibrium Selection in Repeated Games with Contracts1

Consider a two-person intertemporal bargaining problem in which players choose actions and o¤ers each period and collect payo¤s (as function of that period’s actions) while bargaining proceeds. This can alternatively be viewed as an in…nitely-repeated game wherein players can o¤er one another enforceable contracts that govern play for the rest of the game. Theory is silent regarding 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 speci…c 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 initally 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 twostage games. In any perfect Bayesian equilibrium, she can guarantee herself virtually her NBWT payo¤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 payo¤s 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 payo¤s-as-you-go.