Moral Hazard in Delegated Bayesian Persuasion

Original Abstract

We study Bayesian persuasion when information design is delegated to an intermediary who privately chooses the experiment subject to convex costs and would be incentivized by the principal via outcome-dependent transfers. We provide a sharp characterization of first-best implementability: implementing the first-best requires local affine alignment between the principal's and intermediary's reduced-form payoff indices on the posteriors induced by the target experiment, while a stronger global alignment condition guarantees implementability. Outside the global alignment condition, moral hazard typically prevents first-best implementation. We then characterize the second best: the principal's problem admits a virtual Bayesian persuasion representation in which the objective is distorted by a shadow cost proportional to the intermediary's valuation of posteriors. Under entropy costs, moral hazard compresses posterior dispersion relative to the first-best benchmark. In two-state environments with a binary-action receiver, the optimal second-best experiment has a tractable two-posterior form with explicit formulas for posterior endpoints and mixing weights, and the optimal transfer schedule is characterized in closed form as a triangular system in the shadow price, transfer gap, and participation constraint. A numerical example quantifies the compression: moral hazard reduces posterior spread by approximately 28 percent relative to first best under the baseline parameterization.