Approximation of Mean Field Games to N-Player Stochastic Games, with Singular Controls

Abstract

This paper establishes that $N$-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.

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Cited by in corpus (4)

• Nonzero-sum stochastic games and mean-field games with impulse controls
• Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift
• MFGs for partially reversible investment
• Mean Field Games with Singular Controls