## Annals of Probability

- Ann. Probab.
- Volume 48, Number 1 (2020), 211-263.

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

François Delarue, Daniel Lacker, and Kavita Ramanan

#### 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 nonasymptotic 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.

#### Article information

**Source**

Ann. Probab., Volume 48, Number 1 (2020), 211-263.

**Dates**

Received: April 2018

Revised: February 2019

First available in Project Euclid: 25 March 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1585123327

**Digital Object Identifier**

doi:10.1214/19-AOP1359

**Mathematical Reviews number (MathSciNet)**

MR4079435

**Zentralblatt MATH identifier**

07206757

**Subjects**

Primary: 60F10: Large deviations 60E15: Inequalities; stochastic orderings 60H10: Stochastic ordinary differential equations [See also 34F05] 91A13: Games with infinitely many players 91A15: Stochastic games

Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Mean field games master equation McKean–Vlasov limit interacting particle systems common noise large deviation principle concentration of measure transport inequalities linear-quadratic systems systemic risk

#### Citation

Delarue, François; Lacker, Daniel; Ramanan, Kavita. From the master equation to mean field game limit theory: Large deviations and concentration of measure. Ann. Probab. 48 (2020), no. 1, 211--263. doi:10.1214/19-AOP1359. https://projecteuclid.org/euclid.aop/1585123327