Differential Games with Incomplete Information on a Continuum of Initial Positions and without Isaacs Condition

This article deals with a two-player zero-sum differential game with infinitely many initial positions and without Isaacs condition. The structure of information is asymmetric: The first player has a private information on the initial position, while the second player knows only a probability distribution on the initial position. In the present model, we face two difficulties: First, the incomplete information structure does not reduce to a finite set (as in the famous Aumann–Maschler model for repeated games). Second, the game does not satisfy the Isaacs condition (crucially used in classical approaches to differential games). Therefore, we use tools from optimal transportation theory and stochastic control. Our main result shows that with a suitable concept of mixed strategies, there exists a value of the game with such random strategies. As a byproduct of our approach, we obtain the Lipschitz continuity of the random value with respect to the Wasserstein distance and we show the existence of a value in pure strategies in the specific case of an initial distribution without atoms. We also discuss an extension of our model when the asymmetric information concerns continuous scenarios.