Approachability, viability and differential games in incomplete information.

Summary Our dissertation has three parts, all of which are related to the issue of lack of information in game theory. First, we study the notion of Blackwell's approximability in repeated games with vector payments by using techniques developed in the framework of qualitative differential games. Indeed, we reformulate the sufficient condition of approximability of a closed set (B-set) by the notion of a discriminative domain for an appropriate qualitative differential game. By introducing an auxiliary repeated game, we prove that a closed set is *-approachable (i.e., deterministically approachable) if and only if it contains a nonempty B-set. One of our main results is to establish the links between the behavioral strategies in repeated games and the non-anticipatory strategies in the approximability game. We also study a discounted infinite horizon differential game with lack of information on both sides. For this we follow the model introduced by Cardaliaguet and extend it to the infinite horizon framework. We first obtain a principle of dynamic subprogramming. Then we prove that the upper and lower value functions are respectively sub-solutions and over-solutions in the dual sense of the associated Hamilton Jacobi equation. Using a comparison principle we prove the uniqueness of the solution in the dual sense and thus the existence of the value. In the last chapter, we study a control system with probabilistic information about the initial state and extend the viability and invariance theorems to the Wasserstein space of probability measures. As an application we consider a Mayer-type optimal control problem where the state of the system is known according to a probability law. Following Frankowska's epigraphic approach we characterize the function as a unique proximal episolution of a Hamilton-Jacobi type equation.