Games with incomplete information in continuous time and for continuous types

TL;DR

This paper analyzes a continuous-time, two-player zero-sum game with incomplete information and a continuous set of payoffs, characterizing its value function through measure-valued processes and Hamilton-Jacobi equations.

Contribution

It generalizes previous finite-payoff models to continuous payoffs, providing a new characterization of the value function as a viscosity solution.

Findings

Value function characterized as a viscosity solution of a Hamilton-Jacobi equation

Equivalent formulation using measure-valued processes

Extension from finite to continuous payoff sets

Abstract

We consider a two-player zero-sum game with integral payoff and with incomplete information on one side, where the payoff is chosen among a continuous set of possible payoffs. We prove that the value function of this game is solution of an auxiliary optimization problem over a set of measure-valued processes. Then we use this equivalent formulation to characterize the value function as the viscosity solution of a special type of a Hamilton-Jacobi equation. This paper generalizes the results of a previous work of the authors, where only a finite number of possible payoffs is considered.