$N$-player games and mean field games of moderate interactions

TL;DR

This paper investigates the behavior of large populations of interacting players using mean field game theory, establishing approximate Nash equilibria and deriving a law of large numbers for empirical processes.

Contribution

It introduces a rigorous derivation of mean field games with moderate interactions and proves a law of large numbers for empirical processes, providing a foundation for approximate Nash equilibria.

Findings

Law of large numbers for empirical processes

Approximate Nash equilibria for large finite games

Characterization of solutions via verification argument

Abstract

We study the asymptotic organization among many optimizing individuals interacting in a suitable "moderate" way. We justify this limiting game by proving that its solution provides approximate Nash equilibria for large but finite player games. This proof depends upon the derivation of a law of large numbers for the empirical processes in the limit as the number of players tends to infinity. Because it is of independent interest, we prove this result in full detail. We characterize the solutions of the limiting game via a verification argument.