Linear Programming Fictitious Play algorithm for Mean Field Games with optimal stopping and absorption

TL;DR

This paper introduces a fictitious play algorithm based on linear programming to approximate mean field game dynamics involving optimal stopping and absorption, avoiding the direct computation of value functions.

Contribution

It develops a novel algorithm for mean field games that handles optimal stopping and absorption without solving complex value functions, with proven convergence in measure topology.

Findings

Algorithm converges in measure topology.

Numerical examples demonstrate effective approximation.

Handles mean field games with stopping and absorption.

Abstract

We develop the fictitious play algorithm in the context of the linear programming approach for mean field games of optimal stopping and mean field games with regular control and absorption. This algorithm allows to approximate the mean field game population dynamics without computing the value function by solving linear programming problems associated with the distributions of the players still in the game and their stopping times/controls. We show the convergence of the algorithm using the topology of convergence in measure in the space of subprobability measures, which is needed to deal with the lack of continuity of the flows of measures. Numerical examples are provided to illustrate the convergence of the algorithm.