Following the risk-taking model of Seel and Strack, n players decide when to stop privately observed Brownian motions with drift and absorption at zero. They are then ranked according to their level of stopping and paid a rank-dependent reward. We study the optimal reward design where a principal is interested in the average performance and the performance at a given rank. While the former can be related to reward inequality in the Lorenz sense, the latter can have a surprising shape. Next, I will present the mean-field version of this problem. A particular feature of this game is to be tractable without necessarily being smooth, which turns out to offer a cautionary tale. We show that the mean field equilibrium induces n-player ε-Nash equilibria for any continuous reward function— but not for discontinuous ones. We also analyze the quality of the mean field design (for maximizing the median performance) 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. (Joint work with M. Nutz)
