Bargaining efficiency and the repeated prisoners' dilemma

The infinitely repeated prisoners' dilemma has a multiplicity of Pareto-unranked equilibria. This leads to a battle of the sexes problem of coordinating on a single efficient outcome. One natural method of achieving coordination is for the players to bargain over the set of possible equilibrium allocations. If players have different preferences over cooperative bargaining solutions, it is reasonable to imagin that agents randomize over their favorite choices. This paper asks the following question: do the players risk choosing an inefficient outcome by resorting to such randomizations? In general, randomizations over points in a convex set yields interior points. We show, however, that if the candidate solutions are the two most frequently used the Nash and Kalai-Smorodinsky solutions then for any prisoners'' dilemma, this procedure guarantees coordination of an efficient outcome.