The weaker player's option to exit as a source of bargaining power in bilateral bargaining with fixed costs

We study the strategic value of the proposer's option to exit in alternating-offer bargaining where players incur fixed costs per period. Despite the multiplicity of equilibria, the proposer's option to exit unambiguously improves the bargaining outcome of the player with the higher cost when she is a proposer. When her opponent with the lower cost rejects her offer, he has to give some share to her in the next period in order to prevent her from leaving the negotiation. This effect does not arise when the cost of bargaining comes from discounting, and there are no outside options.