Informed communication equilibrium

We consider a privately informed principal selecting a communication device to influence players' actions. In contrast to standard information design, the principal cannot commit ex ante, and the device must elicit information from all players. We define an informed communication equilibrium (informed CE) as a perfect Bayesian equilibrium outcome of this informed communication design game. We show that the set of informed CE is the subset of communication equilibria (CE) that yield principal payoff vectors bounded below by an equilibrium payoff vector of the silent game, under some consistent interim beliefs. The ex ante optimal CE may fail to be an informed CE. We provide sufficient conditions for a CE to be an informed CE and identify classes of games in which CE and informed CE coincide.