Convergence, Fluctuations and Large Deviations for finite state Mean Field Games via the Master Equation

TL;DR

This paper proves the convergence of finite state symmetric N-player differential games to a mean field game system using the Master Equation, and establishes related fluctuations and large deviations results.

Contribution

It introduces a novel convergence approach based on the Master Equation for finite state mean field games, including fluctuations and large deviations analysis.

Findings

Convergence of Nash equilibria and value functions to mean field limits

Establishment of a Central Limit Theorem for empirical measures

Derivation of a Large Deviation Principle for the system

Abstract

We show the convergence of finite state symmetric N-player differential games, where players control their transition rates from state to state, to a limiting dynamics given by a finite state Mean Field Game system made of two coupled forward-backward ODEs. We exploit the so-called Master Equation, which in this finite-dimensional framework is a first order PDE in the simplex of probability measures, obtaining the convergence of the feedback Nash equilibria, the value functions and the optimal trajectories. The convergence argument requires only the regularity of a solution to the Master equation. Moreover, we employ the convergence method to prove a Central Limit Theorem and a Large Deviation Principle for the evolution of the N-player empirical measures. The well-posedness and regularity of solution to the Master Equation are also studied.