On the Convergence to Nash Bargaining Solution for Endogenous Bargaining Protocols

We consider non-cooperative multilateral bargaining games with endogenous bargaining protocols.Under an endogenous protocol, the probability with which a player becomes the proposer in a roundof bargaining depends on the identity of the player who previously rejected. An important exampleis the frequently studied rejector-becomes-proposer protocol. We focus on subgame perfectequilibria in stationary strategies which are shown to exist and to be efficient. Equilibriumproposals do not depend on the probability to proposeconditional on the rejection by another player, though equilibrium acceptance sets do depend onthese probabilities. Next we consider the limit, as the bargaining friction vanishes. In case noplayer has a positive probability to propose conditional on his rejection, each player receiveshis utopia payoff conditional on being recognized and equilibrium payoffs are in general Paretoinefficient. Otherwise, equilibrium proposals of all players converge to a weighted NashBargaining Solution, where the weights are determined by the probability to propose conditional ona rejection.