Analysis of a Finite State Many Player Game Using its Master Equation

TL;DR

This paper studies a symmetric n-player stochastic game with finite states, demonstrating that players' value functions can be approximated by a master PDE and analyzing the fluctuations via stochastic differential equations.

Contribution

It introduces the master equation as an approximation tool for players' value functions and analyzes the fluctuations of the empirical measure in the game.

Findings

Value functions approximated by the master PDE.

Fluctuations governed by stochastic differential equations.

Proved regularity of the master equation.

Abstract

We consider an n-player symmetric stochastic game with weak interaction between the players. Time is continuous and the horizon and the number of states are finite. We show that the value function of each of the players can be approximated by the solution of a partial differential equation called the master equation. Moreover, we analyze the fluctuations of the empirical measure of the states of the players in the game and show that it is governed by a solution to a stochastic differential equation. Finally, we prove the regularity of the master equation, which is required for the above results.