# Probabilistic approach to finite state mean field games

## TL;DR

This paper introduces a probabilistic framework for finite state mean field games using stochastic differential equations driven by Poisson measures, establishing existence, approximation, and uniqueness results.

## Contribution

It develops a probabilistic representation for finite state mean field games, proving existence of solutions and their role as approximate Nash equilibria for large N-player games.

## Key findings

- Existence of solutions in relaxed controls
- Mean field solutions form approximate Nash equilibria with error rate 1/√N
- Uniqueness under small time horizon or monotonicity

## Abstract

We study mean field games and corresponding $N$-player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric $\epsilon_N$-Nash equilibria for the $N$-player game, both in open-loop and in feedback strategies (not relaxed), with $\epsilon_N\leq \frac{\text{constant}}{\sqrt{N}}$. Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity.