Moral Hazard in Delegated Bayesian Persuasion

We study delegated Bayesian persuasion: a principal incentivizes an intermediary to design information via outcome-contingent transfers, while the intermediary privately chooses the experiment subject to convex costs. We characterize first-best implementability through a pair of alignment conditions on the principal's and intermediary's payoff indices. A local condition on the support of the target experiment is necessary; a global affine alignment condition is sufficient. We show that the gap between them is non-empty and provide a partial characterization of the intermediate region. When the first-best is unattainable, the principal's problem admits a virtual Bayesian persuasion representation: the second-best experiment maximizes the same concavified objective as the first-best, with the principal's payoff index distorted by a single scalar shadow price that summarizes the entire agency friction. Under entropy costs, moral hazard compresses posterior dispersion whenever the intermediary's utility differs across the actions it recommends. Explicit closed-form solutions for posteriors, mixing weights, and the optimal transfer schedule are derived for binary environments.