Abstract
How does an expert's ability persuade change with the availability of messages? We study games of Bayesian persuasion the sender is unable to fully describe every state of the world or recommend all possible actions. We characterize the set of attainable payoffs. Sender always does worse with coarse communication and values additional signals. We show that there exists an upper bound on the marginal value of a signal for the sender. In a special class of games, the marginal value of a signal is increasing when the receiver is difficult to persuade. We show that an additional signal does not directly translate into more information and the receiver might prefer coarse communication. Finally, we study the geometric properties of optimal information structures. Using these properties, we show that the sender's optimization problem can be solved by searching within a finite set.
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