Finite State Mean Field Games with Wright–Fisher Common Noise as Limits of N -Player Weighted Games

Erhan Bayraktar, Alekos Cecchin (), Asaf Cohen () and François Delarue ()
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Alekos Cecchin: Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau, France
Asaf Cohen: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
François Delarue: Université Côte d’Azur, CNRS, Laboratoire J.A. Dieudonné, 06108 Nice, France

Mathematics of Operations Research, 2022, vol. 47, issue 4, 2840-2890

Abstract: Forcing finite state mean field games by a relevant form of common noise is a subtle issue, which has been addressed only recently. Among others, one possible way is to subject the simplex valued dynamics of an equilibrium by a so-called Wright–Fisher noise, very much in the spirit of stochastic models in population genetics. A key feature is that such a random forcing preserves the structure of the simplex, which is nothing but, in this setting, the probability space over the state space of the game. The purpose of this article is, hence, to elucidate the finite-player version and, accordingly, prove that N -player equilibria indeed converge toward the solution of such a kind of Wright–Fisher mean field game. Whereas part of the analysis is made easier by the fact that the corresponding master equation has already been proved to be uniquely solvable under the presence of the common noise, it becomes however more subtle than in the standard setting because the mean field interaction between the players now occurs through a weighted empirical measure. In other words, each player carries its own weight, which, hence, may differ from 1 / N and which, most of all, evolves with the common noise.

Keywords: Primary: 91A13; 91A15; 35K65; mean field games; diffusion approximation; convergence problem; Wright–Fischer common noise (search for similar items in EconPapers)
Date: 2022
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