A rank based mean field game in the strong formulation

TL;DR

This paper introduces a mean field game model with rank-dependent rewards under common noise, providing solutions, approximate Nash equilibria for finite players, and convergence rates.

Contribution

It presents a novel mean field game framework with rank-based rewards and common noise, including solution methods and convergence analysis.

Findings

Derived the mean field game solution with rank-dependent rewards

Established an approximate Nash equilibrium for finite players

Quantified the convergence rate to the mean field limit

Abstract

We discuss a natural game of competition and solve the corresponding mean field game with \emph{common noise} when agents' rewards are \emph{rank dependent}. We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence.