Learning sets in state dependent signalling game forms: A characterization

We consider repeated game forms with incomplete information and state dependent signalling structure. We study the information a player can learn about the state of nature through a communication procedure, in a way robust to unilateral deviations. More precisely, we say that player i can distinguish state ω from state ω′ if there exists a strategy profile such that if ω is the true state and at most one player deviates from the profile, then player i will know after a finite number of stages that ω′ is not the true state. The learning set of player i at ω is the set of states that player i cannot distinguish from ω. It corresponds to the information player i can learn, in a way immune against unilateral deviations, if the state is ω and players communicate through the game form for a finite number of stages. This notion was introduced in a joint paper with T. Tomala. We provide here a general characterization of the learning sets.