Folk theorems in a bargaining game with endogenous protocol

Two players bargain to select a utility allocation in some set X⊂R+2. Bargaining takes place in infinite discrete time, where each period t is divided into two sub-periods. In the first sub-period, the players play a simultaneous-move game to determine that period’s proposer, and bargaining takes place in the second sub-period. Rejection triggers a one-period delay and move to t+ 1. For every x∈X∩R++2, there exists a cutoff δ(x) < 1 , such that if at least one player has a discount factor above δ(x) , then for every y∈ X that satisfies y≥ x there exists a subgame perfect equilibrium with immediate agreement on y. The equilibrium is supported by “dictatorial threats.” These threats can be dispensed with if X is the unit simplex and the target-vector is Pareto efficient. The results can be modified in a way that allows for arbitrarily long delays in equilibrium.