A rank-based mean field game in the strong formulation

Erhan Bayraktar,Yuchong Zhang

doi:10.1214/16-ecp24

Abstract

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

Highlights

We discuss a natural game of competition and solve the corresponding mean field game with common noise when agents’ rewards are rank-dependent
Mean field games (MFGs), introduced independently by [8] and [6], provide a useful approximation for the finite player Nash equilibrium problems in which the players are coupled through their empirical distribution
In this paper we will consider a particular game in which the interaction of the players is through their ranks

Summary

Introduction

Mean field games (MFGs), introduced independently by [8] and [6], provide a useful approximation for the finite player Nash equilibrium problems in which the players are coupled through their empirical distribution. In this paper we will consider a particular game in which the interaction of the players is through their ranks. The rest of the paper is organized as follows: In Section 2 we introduce the N-player game in which the players are coupled through the reward function which is rank-based. We first find the mean field limit, discuss the uniqueness of the Nash equilibrium, and construct an approximate Nash equilibrium using the mean field limit

The N -player game

Mean field approximation when there is no common noise

Existence of a Nash equilibrium

Approximate Nash equilibrium of the N -player game

Mean field approximation when there is common noise