A non-asymptotic approach to stochastic differential games with many players under semi-monotonicity

TL;DR

This work treats $N$-player stochastic differential games with semi-monotone costs and weak inter-player interactions in a non-asymptotic, dimension-free framework. It develops dimension-free bounds for both the Nash (PDE/FBSDE) and Pontryagin systems, enabling precise comparisons between closed-loop, open-loop, and distributed equilibria without resorting to master equations. The authors establish two main monotone regimes (displacement and Lasry–Lions), derive Lipschitz and $L^2$ bounds for decoupling fields, and deduce explicit gaps between equilibrium notions, along with universality results for mean-field game limits—even for dense networks—via forward–backward propagation of chaos. They also explore a joint vanishing viscosity and large-$N$ limit in the displacement setting and discuss extensions and open questions, including sharpness of the weak-interaction conditions and common-noise scenarios. Overall, the paper provides a robust, non-asymptotic bridge between finite $N$-player games and mean-field descriptions, with quantitative control that is robust to network structure.

Abstract

We consider stochastic differential games with a large number of players, with the aim of quantifying the gap between closed-loop, open-loop and distributed equilibria. We show that, under two different semi-monotonicity conditions, the equilibrium trajectories are close when the interactions between the players are weak. Our approach is non-asymptotic in nature, in the sense that it does not make use of any a priori identification of a limiting model, like in mean field game (MFG) theory. The main technical step is to derive bounds on solutions to systems of PDE/FBSDE characterizing the equilibria that are independent of the number of players. When specialized to the mean field setting, our estimates yield quantitative convergence results for both open-loop and closed-loop equilibria without any use of the master equation. In fact, our main bounds hold for games in which interactions are much sparser than those of MFGs, and so we can also obtain some "universality" results for MFGs, in which we show that games governed by dense enough networks converge to the usual MFG limit. Finally, we use our estimates to study a joint vanishing viscosity and large population limit in the setting of displacement monotone games without idiosyncratic noise.

Theorems & Definitions (73)

• Remark 2.1
• Remark 2.2
• Remark 2.4
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Remark 2.9
• Remark 2.11
• Theorem 2.13
• Theorem 2.14