On Discrete-Time Approximations to Infinite Horizon Differential Games

In this paper we study a discrete-time semidiscretization and a fully discretization (discrete-time, discrete-state) of an infinite time horizon noncooperative $N$-player differential game. We prove that as either the discretization time step or both time step and mesh size parameters approach zero the discrete value function approximates the value function of the differential game. Furthermore, the discrete Nash equilibrium is an $\epsilon$-Nash equilibrium for the continuous-time differential game both in the discrete-time and fully discrete cases.