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Mean Field Contest with Singularity

Author

Listed:
  • Marcel Nutz

    (Departments of Statistics and Mathematics, Columbia University, New York, New York 10027)

  • Yuchong Zhang

    (Department of Statistical Sciences, University of Toronto, Toronto, Ontario M5G1Z5, Canada)

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 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.

Suggested Citation

  • Marcel Nutz & Yuchong Zhang, 2023. "Mean Field Contest with Singularity," Mathematics of Operations Research, INFORMS, vol. 48(2), pages 1095-1118, May.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:2:p:1095-1118
    DOI: 10.1287/moor.2022.1297
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