Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls

We consider a general class of nonzero-sum N-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a veri?cation approach and provide su?cient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to in?nity, that is, a suitable mean-?eld game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N-player game, with $\mathrm{ϵ}=O\frac{1}{\sqrt{N}}$. As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player’s optimal strategy, with sensitivity analysis of the model parameters.