Nash bargaining solution under externalities

• We define a Nash bargaining solution (NBS) for partition function games.
• We define a bargaining game where the rejecter of a proposal stochastically exits.
• We show that the NBS is supported by any efficient stationary equilibrium.
• We provide a necessary and sufficient condition for such an equilibrium to exist.

We define a Nash bargaining solution (NBS) of partition function games. Based on a partition function game, we define an extensive game, which is a propose–respond sequential bargaining game where the rejecter of a proposal exits from the game with some positive probability. We show that the NBS is supported as the expected payoff profile of any stationary subgame perfect equilibrium (SSPE) of the extensive game such that in any subgame, a coalition of all active players forms immediately. We provide a necessary and sufficient condition for such an SSPE to exist. Moreover, we consider extensions to the cases of nontransferable utilities, time discounting and multiple-coalition formation.