We consider an $N$-player game where the players control the drifts of their diffusive states which have no interaction in the noise terms. The aim of each player is to minimize the expected value of her cost, which is a function of the player's state and the empirical measure of the states of all the players. Our aim is to determine the $N \to \infty$ asymptotic behavior of the upper order statistics of the player's states under Nash equilibrium (the Nash states). For this purpose, we consider also a system of interacting diffusions which is constructed by using the Master PDE of the game and approximates the system of the Nash states, and we improve an $L^2$ estimate for the distance between the drifts of the two systems which has been used for establishing Central Limit Theorems and Large Deviations Principles for the Nash states in the past. By differentiating the Master PDE, we obtain that estimate also in $L^{\infty}$, which allows us to control the Radon-Nikodym derivative of a Girsanov transformation that connects the two systems. The latter allows us to reduce the problem to the case of $N$ uncontrolled diffusions with standard mean-field interaction in the drifts, which has been treated in a previous work.

翻译：我们考虑一个$N$人博弈，其中参与者控制其扩散状态的漂移项，且噪声项互不相关。每位参与者的目标是最小化其成本的期望值，该成本函数取决于参与者自身状态及所有参与者状态的实证测度。本文旨在确定纳什均衡下（即纳什状态）参与者状态的上阶统计量在$N \to \infty$时的渐近行为。为此，我们同时构建了一个通过博弈主方程构造的交互扩散系统，用以逼近纳什状态系统，并改进了用于估计两系统漂移项距离的$L^2$范数界——该估计曾用于建立纳什状态的中心极限定理与大偏差原理。通过对主方程进行微分，我们进一步得到了$L^{\infty}$范数下的估计，从而能够控制连接两系统的Girsanov变换的Radon-Nikodym导数。基于此，我们将原问题转化为具有标准均值场漂移交互的$N$个无控扩散系统问题，该情形已在先前工作中得到解决。