I consider a bargaining game with two types of players – rational and stubborn. Rational players choose demands at each point in time. Stubborn players are restricted to choose from the set of “insistent” strategies that always make the same demand and never accept anything less. However, their initial choice of demand is unrestricted. I characterize the equilibria of this game. I show that while pooling equilibria exist, fully separating equilibria do not. Relative to the case with exogenous behavioral types, strong behavioral predictions emerge: in the limit, players randomize over at most two demands. However, unlike in a world with exogenous types, there is Folk-theorem-like payoff multiplicity.