Mean Field Contest with Singularity

• 成果类型: Article
• 署名作者: Nutz, Marcel; Zhang, Yuchong
• 署名单位: Columbia University; University of Toronto
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.1297
• 发表日期: 2023
• 页码: 1095-1118
• 关键词: n-player games; CONVERGENCE; incentives; equilibria

摘要

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 epsilon-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.