  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
$\varepsilon$-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.
