Differential games with asymmetric information and without Isaacs condition

TL;DR

This paper studies a two-player zero-sum differential game with asymmetric information and no Isaacs condition, establishing the existence of a value using random strategies and characterizing it via a viscosity solution of a Hamilton-Jacobi-Isaacs equation.

Contribution

It introduces a framework for differential games without the Isaacs condition, proving the existence of a value with asymmetric information using novel random strategies.

Findings

Existence of a game value in mixed strategies.

Characterization of the value as a viscosity solution.

No Isaacs condition required for the analysis.

Abstract

We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs condition. The dynamics is an ordinary differential equation parametrised by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. Moreover, the value function can be characterised in term of the unique viscosity solution in some dual sense of a Hamilton-Jacobi-Isaacs equation. Here we do not suppose the Isaacs condition which is usually assumed in differential games.