The Equilibrium-Value Convergence for the Multiple-Partners Game

Abstract

We study the multiple-partners game (Sotomayor, 1992), the simplest many-to-many generalization of the assignment game. Our main result is that the Shapley value of a replicated multiple-partners game converges to a competitive equilibrium payoff when the number of replicas tends to infinity. Furthermore, the result also holds for a large subclass of semivalues since we prove that they converge to the same value as the replica becomes large. In the proof of our theorem, we use properties of the "multiple-partners game with types," where several agents are of each type. We show, in particular, that every competitive equilibrium outcome of a "large” game with types satisfies equal treatment of equals and equal treatment of partnerships.

Keywords: Assignment game, Shapley value, Replica, Semivalues