Mean Field Game with Delay: a Toy Model

TL;DR

This paper introduces a simplified linear-quadratic mean field game model with delay, transforming it into an infinite-dimensional framework to analyze the master equation and approximate Nash equilibria.

Contribution

It presents a novel approach to handle delays in mean field games by lifting the problem into an infinite-dimensional space and deriving the master equation.

Findings

Derived the master equation for the delayed mean field game.

Computed explicit solutions to the master equation.

Showed the solution approximates Nash equilibria in finite games.

Abstract

We study a toy model of linear-quadratic mean field game with delay. We "lift" the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.