Asymptotic Analysis of Mean Field Games with Small Common Noise

TL;DR

This paper develops an approximation method for Nash equilibria in mean field games with small common noise, using linear stochastic equations to characterize the first order correction to the no-noise solution.

Contribution

It introduces a first order approximation for MFGs with small common noise via linear stochastic equations, extending the no-noise MFG solution to noisy settings.

Findings

The approximate strategy achieves an order  Nash equilibrium.

The first order correction is characterized by a Gaussian process.

The method provides a practical way to handle small common noise in MFGs.

Abstract

In this paper, we consider a mean field game (MFG) model perturbed by small common noise. Our goal is to give an approximation of the Nash equilibrium strategy of this game using a solution from the original no common noise MFG whose solution can be obtained through a coupled system of partial differential equations. We characterize the first order approximation via linear mean-field forward-backward stochastic differential equations whose solution is a centered Gaussian process with respect to the common noise. The first order approximate strategy can be described as follows: at time $t \in [0,T]$, applying the original MFG optimal strategy for a sub game over $[t,T]$ with the initial being the current state and distribution. We then show that this strategy gives an approximate Nash equilibrium of order $\epsilon^2$.