From the master equation to mean field game limit theory: Large deviations and concentration of measure

TL;DR

This paper investigates the convergence, concentration, and large deviations of Nash equilibria in large symmetric stochastic differential games, using the master equation and mean field game theory, with and without common noise.

Contribution

It introduces new concentration bounds and large deviation principles for mean field game limits, extending previous results to cases with common noise and specific examples.

Findings

Concentration bounds for Nash equilibrium empirical measures without common noise.

Weak and full large deviation principles established for systems with and without common noise.

Methodology adaptable to specific examples lacking initial assumptions.

Abstract

We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with non-asymptotic concentration bounds for the Wasserstein distance between the $n$-player Nash equilibrium empirical measure and the mean field equilibrium. We also show that the sequence of Nash equilibrium empirical measures satisfies a weak large deviation principle, which can be strengthened to a full large deviation principle only in the absence of common noise. For both sets of results, we first use the master equation, an infinite-dimensional partial differential equation that characterizes the value function of the mean field game, to construct an associated McKean-Vlasov interacting $n$-particle system that is exponentially close to the Nash equilibrium dynamics of the $n$-player game for large $n$, by refining estimates obtained in our companion paper. Then we establish a weak large deviation principle for McKean-Vlasov systems in the presence of common noise. In the absence of common noise, we upgrade this to a full large deviation principle and obtain new concentration estimates for McKean-Vlasov systems. Finally, in two specific examples that do not satisfy the assumptions of our main theorems, we show how to adapt our methodology to establish large deviations and concentration results.