This paper examines repeated implementation of a social choice function (SCF) with infinitely lived agents whose preferences are determined randomly in each period. An SCF is repeatedly implementable in Nash equilibrium if there exists a sequence of (possibly history-dependent) mechanisms such that its Nash equilibrium set is nonempty and every equilibrium outcome path results in the desired social choice at every possible history of past play and realizations of uncertainty. We show, with minor qualifications, that in the complete information environment an SCF is repeatedly implementable in Nash equilibrium if and only if it is efficient. We also discuss several extensions of our analysis.

We study the role of incomplete information and outside options in determining bargaining postures and surplus division in repeated bargaining between a long-run player and a sequence of short-run players. The outside option is not only a disagreement point but reveals information privately held by the long-run player. In equilibrium, the uninformed short-run players' offers do not always respond to changes in reputation and the informed long-run player's payoffs are discontinuous. The long-run player invokes inefficient random outside options repeatedly in order to build reputation to a level where the subsequent short-run players succumb to his extraction of a larger payoff, but he also runs the risk of losing reputation and relinquishing bargaining power. We investigate equilibrium properties when the discount factor goes to 1 and when the informativeness of outside option diffuses. In both cases, bargaining outcomes become more inefficient and the limit reputation building probabilities are interior.