Persuasion with Coarse Communication

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.