Polarizing Persuasion

This paper considers Bayesian persuasion between a sender and two receivers. The sender's payoff is a function of the receivers' beliefs on the binary payoff relevant state. All agents share a common prior about this state. But, we assume disagreement about a payoff irrelevant state, a binary variable that enters no utility functions. If the sender's payoff is differentiable and strictly monotone, then the sender never fully conceals. A convex payoff is insufficient for full revelation, but guarantees that every signal rules out two of the four states. We measure polarization by the sender's expectation of the absolute difference between the receivers posterior beliefs on the payoff relevant state, and solve for the maximum polarization across all message services. With linear payoff functions, the sender chooses a message service that achieves a significant fraction of this maximum. The sender's payoff is strictly increasing in the prior disagreement between the receivers. Given extreme prior disagreement between the receivers, we explicitly solve for the optimal message service when the sender has monotone payoffs in two general cases: bi-concave and bi-convex preferences. In both cases, the optimal message services induce polarization equal to the common prior on the sender's least preferred payoff relevant state.