Mean-field ranking games with diffusion control

We consider a stochastic differential game, where each player continuously controls the diffusion intensity of her own state process. The players must all choose from the same diffusion rate interval \([\sigma _1, \sigma _2]\), and have individual random time horizons that are independently drawn from the same distribution. The players whose states at their respective time horizons are among the best \(p \in (0,1)\) of all terminal states receive a fixed prize. We show that in the mean field version of the game there exists an equilibrium, where the representative player chooses the maximal diffusion rate when the state is below a given threshold, and the minimal rate else. The symmetric n-fold tuple of this threshold strategy is an approximate Nash equilibrium of the n-player game. Finally, we show that the more time a player has at her disposal, the higher her chances of winning.