Engodeneity of Alternating Offers in a Bargaining Game

We investigate an infinite horizon two-person simultaneous offer bargaining game of incomplete information with discounted playoffs. In each period, each player chooses to give in or hold out. The game continues until at least one of the players chooses to give in, at which point agreement has been reached and the game terminates, with an agreement benefit accruing to each player, and a cost to the player (or players) that give in. Players have privately known agreement benefits. 'Low benefit players have a weakly dominant strategy to hold out forever; high benefit players would be better off giving in if they knew their opponent was planning to hold out forever. For any discount factor there is a unique Nash equilibrium in which the two players alternate in their willingness to give in, if the players' priors about each others type are sufficiently asymmetric. Second, for almost all priors, this is the unique equilibrium if the discount factor is close enough to one