Two numerical approaches to stationary mean-field games

TL;DR

This paper explores two numerical algorithms for solving stationary mean-field games, demonstrating their effectiveness across various models including periodic, congestion, and higher-dimensional cases.

Contribution

It introduces and compares a gradient-flow method and a monotonicity-based approach for stationary MFGs, expanding computational tools in this area.

Findings

Both methods successfully solve diverse MFG models.

The gradient-flow approach leverages variational principles.

The monotonicity method exploits structural properties of MFG.

Abstract

Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient-flow method based on the variational characterization of certain MFG. The second one uses monotonicity properties of MFG. We illustrate our methods with various examples, including one-dimensional periodic MFG, congestion problems, and higher-dimensional models.