Continuous time finite state mean field games

TL;DR

This paper develops a mean field model for symmetric continuous-time finite state games with many players, proving convergence to equilibrium and analyzing the relation to finite-player games.

Contribution

It introduces a new mean field model for finite state games, proves convergence to equilibrium, and establishes the rate of convergence from finite-player to mean field models.

Findings

Proved convergence to stationary solutions in the mean field limit.

Established the rate of convergence as the number of players increases.

Provided examples for potential mean field games.

Abstract

In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study the $N+1$-player problem, which the mean field model attempts to approximate. Our main result is the convergence as $N\to \infty$ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.