A differential game with a blind player

TL;DR

This paper analyzes a zero-sum differential game where one player has full information and observes the other's moves, while the other player is blind, establishing the game's value via a Hamilton-Jacobi equation on probability measures.

Contribution

It introduces a novel differential game with asymmetric information and characterizes its value as a viscosity solution of a Hamilton-Jacobi equation on probability spaces.

Findings

The game has a well-defined value under the given conditions.

The value is characterized as a unique viscosity solution.

The framework extends differential game theory to asymmetric information scenarios.

Abstract

We consider a zero sum differential game with lack of observation on one side. The initial state of the system is drawn at random according to some probability $\mu_0$ on $\R^N$. Player-I is informed of the initial position of state while player-II knows only $\mu_0$. Moreover Player-I observes Player-II's moves while Player-II is blind and has no further information. We prove that in this game with a terminal payoff the value exists and is characterized as the unique viscosity solution of some Hamilton-Jacobi equation on a space of probability measures.