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 {omega} from state {omega}' if there exists a strategy profile such that if {omega} 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 {omega}' is not the true state. The learning set of player i at {omega} is the set of states that player i cannot distinguish from {omega}. It corresponds to the information player i can learn, in a way immune against unilateral deviations, if the state is {omega} and players communicate through the game form for a finite number of stages. This notion was introduced in a joint paper with T. Tomala (Renault and Tomala 2000). We provide here a general characterization of the learning sets.