Download This PaperPersuasion with Coarse Communication
49 Pages Posted: 18 Mar 2020 Last revised: 12 Jun 2026
Date Written: June 12, 2026
Abstract
In many expert–decision maker settings, information is richer than the language used to convey it. Motivated by this communication friction, we study Bayesian persuasion when the sender is constrained to use k messages. We show that the sender's value is given by a $k$-point analogue of concavification, which we call k-concavification. An optimal information structure can be chosen with affinely independent posterior support, allowing the problem to be reduced to a lower-dimensional persuasion problem and then solved by standard concavification. We derive a tight bound on the value of communication capacity that applies to general persuasion games: the gain from a (k+1)st message is at most 2/(k-1) times the value attainable with k messages. Finally, we solve a class of belief-threshold games in which the receiver chooses between a safe default and several risky actions, the sender gets zero from the default and the same positive payoff from any risky action, and a risky action is taken only when the corresponding posterior probability exceeds a threshold. We characterize the optimal coarse information structure, derive comparative statics in the prior and the threshold, and extend the analysis to heterogeneous thresholds and heterogeneous sender values across risky actions.
Keywords: Bayesian Persuasion; Information Design; Coarse Communication, Information Design, Coarse Communication, Concavification
JEL Classification: D82, D83
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