A Differential Game with Symmetric Incomplete Information on Probabilistic Initial Condition and with Signal Revelation

TL;DR

This paper studies a two-player zero-sum differential game with symmetric incomplete information on initial positions, where players receive signals revealing the state upon hitting a target, and proves the existence and characterization of the game's value.

Contribution

It introduces a framework for analyzing differential games with symmetric incomplete information and signal revelation, establishing the existence and uniqueness of the value via viscosity solutions.

Findings

Value of the game exists under signal-dependent strategies.

The extended value function solves a Hamilton-Jacobi-Isaacs equation.

The value function is unique and satisfies boundary conditions.

Abstract

In this paper, we investigate the existence and characterization of the value for a two-player zero-sum differential game with symmetric incomplete information on a continuum of initial positions and with signal revelation. Before the game starts, the initial position is chosen randomly according to a probability measure with compact support, and neither player is informed of the chosen initial position. However, they observe a public signal revealing the current state as soon as the trajectory of the dynamics hits a target set. We prove that, under a suitable notion of signal-dependent strategies, the value of the game exists, and the extended value function of the game is the unique viscosity solution of an associated Hamilton-Jacobi-Isaacs equation that satisfies a boundary condition.