Long Range Games
Abstract
We consider -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 -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.
Cite
@article{arxiv.2410.02822,
title = {Long Range Games},
author = {Francesca Albertini and Paolo Dai Pra},
journal= {arXiv preprint arXiv:2410.02822},
year = {2024}
}