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

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 $θ$ is shown to be an $ε_N$-NE to an $N$-player game with singular controls of the bounded velocity, with $ε_N = O(\frac{1}{\sqrt{N}})$, and (ii) the optimal control to this MFG is an $(ε_N + ε_θ)$-NE to an $N$-player game with singular controls of finite variation, where $ε_θ$ is an error term that depends on $θ$. This work generalizes the classical result on approximation $N$-player games by MFGs, by allowing for discontinuous controls.

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