Monotone numerical methods for finite-state mean-field games

TL;DR

This paper introduces monotone numerical methods for finite-state mean-field games, leveraging a contraction flow approach to effectively solve the complex boundary value problems inherent in MFGs.

Contribution

The paper develops a novel numerical approach based on monotonicity that guarantees convergence to solutions of finite-state MFGs with challenging boundary conditions.

Findings

The methods successfully solve stationary and time-dependent MFGs.

Numerical experiments demonstrate convergence and accuracy.

Application to a paradigm-shift problem illustrates practical utility.

Abstract

Here, we develop numerical methods for finite-state mean-field games (MFGs) that satisfy a monotonicity condition. MFGs are determined by a system of differential equations with initial and terminal boundary conditions. These non-standard conditions are the main difficulty in the numerical approximation of solutions. Using the monotonicity condition, we build a flow that is a contraction and whose fixed points solve the MFG, both for stationary and time-dependent problems. We illustrate our methods in a MFG modeling the paradigm-shift problem.