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Title: Mean Field Contest with Singularity
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. Funding: M. Nutz is supported by an Alfred P. Sloan Fellowship and the Division of Mathematical Sciences of the National Science Foundation [Grants DMS-1812661 and DMS-2106056]. Y. Zhang is supported in part by the Natural Sciences and Engineering Research Council of Canada [NSERC Discovery Grant RGPIN-2020-06290].  more » « less
Award ID(s):
2106056 1812661
NSF-PAR ID:
10427731
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Mathematics of Operations Research
Volume:
48
Issue:
2
ISSN:
0364-765X
Page Range / eLocation ID:
1095 to 1118
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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