Universal Nash Equilibrium Strategies for Differential Games

TL;DR

This paper develops universal Nash equilibrium strategies for two-player nonzero-sum differential games, addressing cases with continuous and discontinuous value functions, and establishing the existence of discontinuous solutions.

Contribution

It introduces a construction of universal strategies applicable to both continuous and discontinuous value functions in differential games, and proves the existence of discontinuous value functions.

Findings

Discontinuous value functions exist in differential games.

Continuous value functions may not exist in general.

A smooth value function can fail to solve the Hamilton-Jacobi system.

Abstract

The paper is concerned with a two-player nonzero-sum differential game in the case when players are informed about the current position. We consider the game in control with guide strategies first proposed by Krasovskii and Subbotin. The construction of universal strategies is given both for the case of continuous and discontinuous value functions. The existence of a discontinuous value function is established. The continuous value function does not exist in the general case. In addition, we show the example of smooth value function not being a solution of the system of Hamilton--Jacobi equation.