Long Range Games

TL;DR

This paper introduces a novel class of long range games derived as a limit of finite N-player games on geometrical structures, establishing existence, uniqueness, and equilibrium properties, with applications to various interaction mechanisms.

Contribution

It develops a new asymptotic framework called long range games, extending mean field games to non-permutation invariant structures and proving key equilibrium properties.

Findings

Existence of at least one equilibrium in long range games

Uniqueness of equilibrium under monotonicity conditions

Feedback strategies form quasi-Nash equilibria for finite N-player games

Abstract

We consider $N$-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 {\em long range game}. We prove that this asymptotic scheme satisfies the following key properties: a) the long range game admits al 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 $N$-player games. We finally show that this scheme includes several examples of interaction mechanisms, in particular Kac-type interactions and interactions on generalized Erd\"{o}s-Renyi graphs.