On Discrete-Time Approximations to Infinite Horizon Differential Games

TL;DR

This paper analyzes how discrete-time approximations of infinite horizon differential games converge to the continuous game, establishing the accuracy of discrete models and equilibria.

Contribution

It proves convergence of discrete value functions to the continuous one and shows discrete Nash equilibria are approximate equilibria for the continuous game.

Findings

Discrete value functions approximate the continuous value function as discretization parameters approach zero.

Discrete Nash equilibria are $ ext{epsilon}$-Nash equilibria for the continuous game.

Convergence holds for both semi-discretization and full discretization cases.

Abstract

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.