From the master equation to mean field game limit theory: A central limit theorem

TL;DR

This paper proves a central limit theorem for mean field games, describing the fluctuations around the law of large numbers limit as solutions to a linear stochastic PDE, using the master equation and McKean-Vlasov systems.

Contribution

It establishes a functional CLT for MFGs under additional assumptions, linking the master equation to the fluctuations via a stochastic PDE, and introduces a new multidimensional CLT for McKean-Vlasov systems.

Findings

Proved a CLT characterizing fluctuations in MFGs as solutions to a stochastic PDE.

Derived a new multidimensional CLT for McKean-Vlasov systems.

Applied methodology to a linear-quadratic example, explicitly solving the stochastic PDE.

Abstract

Mean field games (MFGs) describe the limit, as $n$ tends to infinity, of stochastic differential games with $n$ players interacting with one another through their common empirical distribution. Under suitable smoothness assumptions that guarantee uniqueness of the MFG equilibrium, a form of law of large of numbers (LLN), also known as propagation of chaos, has been established to show that the MFG equilibrium arises as the limit of the sequence of empirical measures of the $n$-player game Nash equilibria, including the case when player dynamics are driven by both idiosyncratic and common sources of noise. The proof of convergence relies on the so-called master equation for the value function of the MFG, a partial differential equation on the space of probability measures. In this work, under additional assumptions, we establish a functional central limit theorem (CLT) that characterizes the limiting fluctuations around the LLN limit as the unique solution of a linear stochastic PDE. The key idea is to use the solution to the master equation to construct an associated McKean-Vlasov interacting $n$-particle system that is sufficiently close to the Nash equilibrium dynamics of the $n$-player game for large $n$. We then derive the CLT for the latter from the CLT for the former. Along the way, we obtain a new multidimensional CLT for McKean-Vlasov systems. We also illustrate the broader applicability of our methodology by applying it to establish a CLT for a specific linear-quadratic example that does not satisfy our main assumptions, and we explicitly solve the resulting stochastic PDE in this case.