On the Mean-Field limit of diffusive games through the master equation: $L^{\infty}$ estimates and extreme value behavior

We consider an $N$-player game where the states of the players evolve with time as Stochastic Differential Equations (SDEs) with interaction only in the drift terms. Each player controls the drift of the SDE satisfied by her state process, aiming to minimize the expected value of a cost that depends on the paths of the player's state and the empirical measure of the states of all the players until a terminal time. When $N \to \infty$, previous works have established Central Limit Theorems and Large Deviation Principles for the state processes when the game is in Nash Equilibrium (the Nash states), by using the Master Equation to construct approximations of those processes that evolve with time as SDEs with classical Mean-Field interaction. Staying in this framework, we improve an existing $L^{1}$ estimate for the total error of approximating all the Nash states to an $L^{\infty}$ one, and we also establish the $N \to \infty$ asymptotic behavior of the upper order statistics of the Nash states. The latter initiates the development of an Extreme Value Theory for Stochastic Differential Games.