A fictitious-play finite-difference method for linearly solvable mean field games

TL;DR

This paper introduces a new iterative finite difference method for solving linearly solvable mean field games derived from control problems, utilizing Cole-Hopf transformation and fictitious play, with proven convergence and demonstrated stability in numerical tests.

Contribution

It presents a novel, implementation-friendly iterative scheme for a specific class of MFGs, with rigorous convergence analysis and practical numerical validation.

Findings

Method converges for 1D and 2D problems.

Algorithm is stable and efficient.

Mathematically proven convergence.

Abstract

An iterative finite difference scheme for mean field games (MFGs) is proposed. The target MFGs are derived from control problems for multidimensional systems with advection terms. For such MFGs, linearization using the Cole-Hopf transformation and iterative computation using fictitious play are introduced. This leads to an implementation-friendly algorithm that iteratively solves explicit schemes. The convergence properties of the proposed scheme are mathematically proved by tracking the error of the variable through iterations. Numerical calculations show that the proposed method works stably for both one- and two-dimensional control problems.