Markov Perfect Equilibria in Discrete Finite-Player and Mean-Field Games

We study dynamic finite-player and mean-field stochastic games within the framework of Markov perfect equilibria (MPE). Our focus is on discrete time and space structures without monotonicity. Unlike their continuous-time analogues, discrete-time finite-player games generally do not admit unique MPE. However, we show that uniqueness is remarkably recovered when the time steps are sufficiently small, and we provide examples demonstrating the necessity of this assumption. This result, established without relying on any monotonicity conditions, underscores the importance of inertia in dynamic games. In both the finite-player and mean-field settings, we show that MPE correspond to solutions of the Nash-Lasry-Lions equation, which is known as the master equation in the mean-field case. We exploit this connection to establish the convergence of discrete-time finite-player games to their mean-field counterpart in short time. Finally, we prove the convergence of finite-player games to their continuous-time version on every time horizon.