I consider a model of reputational bargaining in which the stubborn type can choose their initial demand. There are two types of players: rational and stubborn. The game has two stages: a demand stage and a concession stage. Types can pool or separate in equilibrium for any fixed probability of facing a stubborn type. When the probability of facing a stubborn type is small, any feasible payoff can be achieved in equilibrium for either type. When the probability is large, there is either immediate agreement or long delays.
Anna Sanktjohanser (Thu,) studied this question.