Folk theorems in a bargaining game with endogenous protocol

Two players bargain to select a utility allocation in some set $$X\subset {\mathbb {R}}_+^2$$ 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 $$t+1$$ t + 1 . For every $$x\in X\cap {\mathbb {R}}^2_{++}$$ x ∈ X ∩ R + + 2 , there exists a cutoff $$\delta (x)