Mean Field Games with Ergodic cost for Discrete Time Markov Processes

TL;DR

This paper studies mean field games with ergodic cost in discrete time Markov processes, proving existence of equilibria and their convergence from finite N-player games to the mean field limit.

Contribution

It establishes the existence of mean field game equilibria and demonstrates convergence of N-player Nash equilibria to the mean field solution in a general setting.

Findings

Existence of mean field game equilibrium under certain conditions

N-player Nash equilibria converge to the mean field game solution as N increases

Analysis conducted in a general -compact Polish space setting

Abstract

We consider mean field games with ergodic cost in the framework of a general discrete time controlled Markov processes. The state space of the processes is given by a general $\sigma$-compact Polish space. Under certain conditions, we show the existence of a mean field game equilibrium. We also study the $N$-person game where the players interacts with each other via their empirical measure. We show that the $N$-person game has Nash equilibrium and as $N$ tends to infinity the equilibria converge to a mean field game solution.