Mean Field Contest with Singularity

29 Pages Posted: 9 Mar 2021

See all articles by Marcel Nutz

Marcel Nutz

Columbia University

Yuchong Zhang

University of Toronto - Department of Statistics

Date Written: March 6, 2021


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 ε-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.

Keywords: Mean Field Game, Stochastic Contest, Optimal Contract, Stackelberg Game

Suggested Citation

Nutz, Marcel and Zhang, Yuchong, Mean Field Contest with Singularity (March 6, 2021). Available at SSRN: or

Marcel Nutz

Columbia University ( email )

Yuchong Zhang (Contact Author)

University of Toronto - Department of Statistics ( email )

700 University Ave.
Toronto, Ontario M5S 1Z5

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