Coasian Bargaining with an Arriving Outside Option

We consider Coasian bargaining problems where the buyer has an outside option arriving at a stochastic time. We study both observable outside option models and unobservable outside option models. In both models, we show that a Coasian equilibrium exists if (1) the arrival of the outside option is public, or (2) the arrival of the outside option is private but the arrival probability is small enough. (1) the seller makes multiple rounds of offers, and (2) the Coase conjecture holds for an arbitrarily large arrival rate of the outside option. The result also applies to the time-varying outside option model. This exercise helps us to understand the sharp difference between Board and Pycia (2013), where the buyer's outside option is always available, and the standard Coasian bargaining literature, where the buyer has no outside option.