Bargaining in Stationary Networks

We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain over a unit surplus. Players that reach agreement are replaced by new players at the same positions in the network. We prove that for each discount factor all equilibria are payoff equivalent. The equilibrium payoffs and the set of equilibrium agreement links converge as players become patient. Several new concepts– mutually estranged sets, partners, and shortage ratios–provide insights into the relative strengths of the positions in the network. We develop a procedure to determine the limit equilibrium payoffs by iteratively applying the following results. Limit payoffs are lowest for the players in the largest mutually estranged set that minimizes the shortage ratio, and highest for the corresponding partners. In equilibrium, for high discount factors, the partners act as an oligopoly for the estranged players. In the limit, surplus within the induced oligopoly subnetwork is divided according to the shortage ratio. We characterize equitable networks, stable networks, and non-discriminatory buyer-seller networks. The results extend to heterogeneous discount factors and general matching technologies.