Fictitious Play via Finite Differences for Mean Field Games with Optimal Stopping

TL;DR

This paper introduces a generalized fictitious play algorithm with finite difference schemes to compute mixed equilibria in mean field games with optimal stopping, addressing challenges posed by obstacle and Fokker-Planck equations.

Contribution

It proposes a novel iterative method for solving OSMFGs that converges to mixed strategy equilibria and incorporates efficient numerical schemes.

Findings

Algorithm effectively computes mixed equilibria

Convergence is rigorously justified

Numerical experiments demonstrate robustness and efficiency

Abstract

This paper considers mean field games with optimal stopping time (OSMFGs) where agents make optimal exit decisions, the coupled obstacle and Fokker-Planck equations in such models pose challenges versus classic MFGs. This paper proposes a generalized fictitious play algorithm that computes OSMFG mixed equilibria by iteratively solving pure strategy systems, i.e. approximating mixed strategies through averaging pure strategies according to a certain updating rule. The generalized fictitious play allows for a broad family of learning rates and the convergence to the mixed strategy equilibrium can be rigorously justified. The algorithm also incorporates efficient finite difference schemes of the pure strategy system, and numerical experiments demonstrate the effectiveness of the proposed method in robustly and efficiently computing mixed equilibria for OSMFGs.