Mean Field Contest with Singularity

TL;DR

This paper develops a mean field game model where players stop Brownian motions and are ranked for rewards, establishing equilibrium properties, analyzing a principal's reward design, and revealing singularities as the number of players grows.

Contribution

It introduces a novel mean field game with ranking-based stopping times, derives the equilibrium and its limits, and analyzes optimal reward design and its limitations in large populations.

Findings

Unique mean field equilibrium exists and is the limit of n-player games.

Optimal reward distribution for the median player is derived in closed form.

Mean field design quality deteriorates as the number of players increases due to asymptotic singularities.

Abstract

We formulate a mean field game where each player stops a privately observed Brownian motion with absorption. Players are ranked according to their level of stopping and rewarded as a function of their relative rank. There is a unique mean field equilibrium and it is shown to be the limit of associated $n$-player games. Conversely, the mean field strategy induces $n$-player $\varepsilon$-Nash equilibria for any continuous reward function -- but not for discontinuous ones. In a second part, we study the problem of a principal who can choose how to distribute a reward budget over the ranks and aims to maximize the performance of the median player. The optimal reward design (contract) is found in closed form, complementing the merely partial results available in the $n$-player case. We then analyze the quality of the mean field design when used as a proxy for the optimizer in the $n$-player game. Surprisingly, the quality deteriorates dramatically as $n$ grows. We explain this with an asymptotic singularity in the induced $n$-player equilibrium distributions.