Finite state mean field games with Wright Fisher common noise as limits of $N$-player weighted games

TL;DR

This paper investigates finite state mean field games influenced by Wright-Fisher common noise, demonstrating that equilibria of N-player games converge to the mean field limit with this specific stochastic structure.

Contribution

It establishes the convergence of N-player equilibria to Wright-Fisher mean field games with weighted interactions, extending the understanding of common noise effects in finite state models.

Findings

Proves convergence of N-player equilibria to the mean field limit.

Shows the master equation remains uniquely solvable with Wright-Fisher noise.

Analyzes the impact of weighted empirical measures on the game dynamics.

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, to prove that $N$-player equilibria indeed converge towards the solution of such a kind of Wright-Fisher mean field game. Whilst 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.