Repeated games with lack of information on one side: The dual differential approach

We introduce the dual differential game of a repeated game with lack of information on one side as the natural continuous time version of the dual game introduced by De Meyer. A traditional way to study the value of differential games is through discrete time approximations. Here, we follow the opposite approach: we identify the limit value of a repeated game in discrete time as the value of a differential game. Namely, we use the recursive structure for the finitely repeated version of the dual game to construct a differential game for which the upper values of the uniform discretization satisfy precisely the same property. The value of the dual differential game exists and is the unique viscosity solution of a first-order derivative equation with a limit condition. We identify the solution by translating viscosity properties in the primal.