Approximation of N-player stochastic games with singular controls by mean field games

TL;DR

This paper demonstrates that N-player stochastic games with singular controls can be effectively approximated by mean field games, extending classical results to include discontinuous control strategies.

Contribution

It introduces a framework for approximating N-player games with singular controls using mean field games, including cases with discontinuous controls of bounded velocity or finite variation.

Findings

Optimal control in MFGs approximates N-player Nash equilibria with error $O(1/\sqrt{N})$.

Approximation holds for both bounded velocity and finite variation singular controls.

Extends classical MFG approximation results to discontinuous control settings.

Abstract

This paper establishes that a class of $N$-player stochastic games with singular controls, either of bounded velocity or of finite variation, can both be approximated by mean field games (MFGs) with singular controls of bounded velocity. More specifically, it shows (i) the optimal control to an MFG with singular controls of a bounded velocity $\theta$ is shown to be an $\epsilon_N$-NE to an $N$-player game with singular controls of the bounded velocity, with $\epsilon_N = O(\frac{1}{\sqrt{N}})$, and (ii) the optimal control to this MFG is an $(\epsilon_N + \epsilon_{\theta})$-NE to an $N$-player game with singular controls of finite variation, where $\epsilon_{\theta}$ is an error term that depends on $\theta$. This work generalizes the classical result on approximation $N$-player games by MFGs, by allowing for discontinuous controls.