Dimension reduction techniques in deterministic mean field games

TL;DR

This paper explores dimension reduction methods in mean field games, providing conditions for simplification in finite and infinite state spaces, and compares solutions with and without noise for better numerical modeling.

Contribution

It introduces general conditions for reducing mean field game equations to lower dimensions in both finite and infinite state spaces.

Findings

Reduction conditions are established for finite and infinite state spaces.

Reduced models approximate original equations effectively in small-noise regimes.

Comparison shows consistency between solutions with noise and their reduced counterparts.

Abstract

We present examples of equations arising in the theory of mean field games that can be reduced to a system in smaller dimensions. Such examples come up in certain applications, and they can be used as modeling tools to numerically approximate more complicated problems. General conditions that bring about reduction phenomena are presented in both the finite and infinite state-space cases. We also compare solutions of equations with noise with their reduced versions in a small-noise expansion.