Persuasion is an exceedingly difficult task. A leading cause of this difficulty is the misalignment of preferences, which is studied extensively by the literature on persuasion games. However, the difficulty of communication also has a first order effect on  outcomes and welfare of agents. Motivated by this observation, we study a model of Bayesian persuasion in which the communication between the sender and the receiver is constrained. We limit the cardinality of the signal space to be less than the cardinality of the action space and the state space . This limits the sender's ability of making arbitrarily many action recommendations . We prove the existence of a solution to the sender's utility maximization problem and characterize its properties . In solving this problem, we develop a novel approach for solving Bayesian persuasion problems, which can be applied to a wide range of settings. We characterize the sender's willingness to pay for an additional signal as a function of the prior belief, which we interpret as the value of precise communication. We show that increased precision might not be always welfare improving by showing that the receiver might prefer coarse communication  .
Persuasion is an exceedingly difficult task. A leading cause of this difficulty is the misalignment of preferences, which is studied extensively by the literature on persuasion games. However, the difficulty of communication also has a first order effect on the outcomes and welfare of agents. Motivated by this observation, we study a model of Bayesian Persuasion in which the communication between the sender and the receiver is constrained. This is done by allowing the cardinality of the signal space to be less than the cardinality of the action space and the state space , which limits the number of action recommendations that the sender can make. Existence of a maximum to the sender's problem is proven and its properties are characterized. This generalizes the standard Bayesian Persuasion framework, in which existence results rely on the assumption of rich signal spaces. We analyze the sender's willingness to pay for an additional signal as a function of the prior belief, which can be interpreted as the value of precise communication. We provide an upper bound for this value which applies to all finite persuasion games. While increased precision is always better for the sender, we show that the receiver might prefer coarse communication . We show this by analyzing a game of advice seeking, where the receiver has the ability to choose the size of the signal space .
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