Private Bayesian Persuasion

We consider a multi-receiver Bayesian persuasion problem where an informed sender tries to persuade a group of receivers to adopt a certain product. The sender is allowed to commit to a signaling policy where she sends a private signal to every receiver. The utility of the sender is a function of the subset of adopters and the realized state. We first consider a setting with a binary state space and no payoff externalities among receivers. We characterize an optimal signaling policy and the maximal revenue to the sender for two different types of utility functions: supermodular, and anonymous submodular. In particular, for supermodular utilities we show that the optimal policy correlates positive recommendation to adopt the product as much as possible. We generalize these results to the case of a nonbinary state space. The result for supermodular utilities is generalized to the case where receivers have payoff externalities. We also provide a necessary and sufficient condition under which public and conditionally independent signaling policies are optimal.

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