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

TL;DR

This paper introduces a non-asymptotic method for analyzing stochastic differential games with many players, providing bounds on equilibria that do not rely on mean field limits and applying to sparse interactions.

Contribution

It develops a non-asymptotic framework for stochastic differential games under semi-monotonicity, avoiding mean field assumptions and applicable to sparse interaction networks.

Findings

Bounds on equilibrium solutions independent of number of players

Quantitative convergence results for open-loop and closed-loop equilibria

Universality results showing convergence of sparse games to MFG limits

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.