On the Mean-Field limit of diffusive games through the master equation: $L^{\infty}$ estimates and extreme value behavior
Erhan Bayraktar, Nikolaos Kolliopoulos
TL;DR
We study the mean-field limit of an $N$-player diffusive game with drift interactions under Nash equilibrium, using the Master equation to connect finite-$N$ dynamics to a limiting McKean–Vlasov system. The paper proves an $L^{\infty}$-norm error bound of order $\mathcal{O}(1/N)$ between the finite-$N$ Nash drifts and their mean-field approximations, strengthening prior $L^1$ results and enabling sharp uniform error control. It also establishes an Extreme Value Theory result for the top order statistics of the Nash states: for fixed $t$, the top $k$ states, after appropriate centering and scaling $(X_t^{i_k,N} - b_t^N)/a_t^N$, converge in law to the EVT limit with index $\gamma_t$, with the Gumbel case $\gamma_t=0$ recovered in common Gaussian-like settings. The work further discusses extensions to state-dependent volatility via a diffusion transformation, providing a framework for tail-risk analysis and more robust numerical simulations of large stochastic differential games.
Abstract
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.