Microscopic Derivation of Mean Field Game Models

TL;DR

This paper derives mean field game PDE systems from microscopic agent dynamics without measure derivatives, providing new insights into the scaling behavior of large agent systems, especially in financial models.

Contribution

It introduces a novel microscopic derivation of mean field game models from deterministic agent dynamics, avoiding measure derivatives and exploring new scaling limits.

Findings

Derived mean field PDE systems from agent ODEs.

Analyzed the system's behavior as the number of agents grows.

Applied results to agent-based financial market models.

Abstract

Mean field game theory studies the behavior of a large number of interacting individuals in a game theoretic setting and has received a lot of attention in the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In this work, we derive mean field game partial differential equation systems from deterministic microscopic agent dynamics. The dynamics are given by a particular class of ordinary differential equations, for which an optimal strategy can be computed (Bressan, Milan Journal of Mathematics, 2011). We use the concept of Nash equilibria and apply the dynamic programming principle to derive the mean field limit equations and we study the scaling behavior of the system as the number of agents tends to infinity and find several mean field game limits. Especially we avoid in our derivation the notion of measure derivatives. Novel scales are motivated by an example of an agent-based financial market model.