Numerical approximation of Nash equilibria for a class of non-cooperative differential games

TL;DR

This paper introduces a numerical method for approximating Nash equilibria in multi-player non-cooperative differential games with a specific structure, using Hamilton-Jacobi equations and fixed point iteration.

Contribution

It develops a novel numerical scheme based on Dynamic Programming and fixed point iteration for solving infinite horizon differential games with Hamilton-Jacobi systems.

Findings

Numerical solutions for two-player games in 1D and 2D are demonstrated.

The scheme exhibits desirable convergence and stability properties.

Some features and properties of the approximation method are discussed.

Abstract

In this paper we propose a numerical method to obtain an approximation of Nash equilibria for multi-player non-cooperative games with a special structure. We consider the infinite horizon problem in a case which leads to a system of Hamilton-Jacobi equations. The numerical method is based on the Dynamic Programming Principle for every equation and on a global fixed point iteration. We present the numerical solutions of some two-player games in one and two dimensions. The paper has an experimental nature, but some features and properties of the approximation scheme are discussed.