Long Range Games

Francesca Albertini; Paolo Dai Pra

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arXiv · 2410.02822

Long Range Games

Francesca Albertini·Paolo Dai Pra

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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.

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Francesca Albertini, Paolo Dai Pra. 2024-09-27. Long Range Games. https://arxiv.org/abs/2410.02822

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