Long Range Games

Abstract. We consider [Formula: see text]-player games, in continuous time, finite state space, and finite time horizon, on a geometrical structure possessing a macroscopic limit in a suitable sense. This geometrical structure breaks the permutation invariance property that gives rise to mean field games. The corresponding limit game is a variant of mean field games that we call a long range game. We prove that this asymptotic scheme satisfies the following key properties: (a) the long range game admits at least one equilibrium; (b) this equilibrium is unique under a suitable monotonicity condition; (c) the feedback corresponding to any equilibrium of the long range game is a quasi-Nash equilibrium for the [Formula: see text]-player games. We finally show that this scheme includes several examples of interaction mechanisms, in particular, Kac-type interactions and interactions on generalized Erdös–Renyi graphs.