Moral Hazard in Delegated Bayesian Persuasion

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.

• The paper demonstrates that first-best signal design is implementable only under strict local and global affine alignment conditions.
• It reformulates the principal-agent problem via strong duality into a tractable concavification framework, using entropy-based costs for explicit analysis.
• The study quantifies moral hazard’s impact by showing a 28% reduction in posterior dispersion in binary experiments relative to first-best outcomes.

Moral Hazard in Delegated Bayesian Persuasion: A Detailed Analysis

Problem Setting and Theoretical Foundations

This paper analyzes a generalization of Bayesian persuasion, focusing on environments where the principal delegates the design and execution of information acquisition (signal structure) to an intermediary subject to moral hazard and convex costs. The canonical Bayesian persuasion model [Kamenica and Gentzkow, 2011] presumes that the sender (principal) directly chooses the signal; however, in many real-world applications—such as regulatory consulting, credit rating, and scientific peer review—the information provider (intermediary) privately selects the experiment (signal structure), which is costly. The intermediary’s effort is non-contractible, and the principal can only incentivize via outcome-dependent transfers, resulting in a principal-agent problem with classic moral hazard.

The intermediary’s cost function is convex and lower semicontinuous, enabling flexible modeling. Notably, the paper provides explicit results for entropy-based costs, inspired by rational inattention theory (e.g., [Sims 2003], [Caplin & Dean 2013]). The principal’s objective is to maximize her expected payoff by designing a transfer schedule over receiver actions, anticipating the intermediary’s best-response choice of experiment within the incentive and participation constraints.

First-Best Implementation and Alignment Conditions

The primary theoretical contribution is a sharp characterization of first-best implementability under delegation with moral hazard. The paper formalizes two alignment conditions for implementability:

• Necessary Condition—Local Affine Alignment: For the first-best experiment to be implementable, the difference between the mediator's and principal's payoff indices at each posterior must be equal to the transfer at the recommended action plus a constant. This stringent structure on transfer schedules emerges from quasi-linearity and the agent’s problem geometry.
• Sufficient Condition—Global Affine Alignment: If the intermediary’s and principal’s reduced-form payoff indices across all beliefs are globally affine (i.e., proportional), then first-best outcomes can always be implemented via an appropriate transfer schedule. This knife-edge condition only holds in special cases, such as when agent and principal value posteriors identically up to scale and shift.

These alignment conditions set exact boundaries on where first-best is achievable; outside these, first-best implementation generally fails. In generic environments with non-aligned payoffs, moral hazard precludes first-best and mandates second-best solutions.

Virtual Bayesian Persuasion and Second-Best Characterization

When first-best is unattainable, the principal’s problem admits a "virtual Bayesian persuasion" representation. By strong Lagrangian duality, the second-best outcome is characterized as standard Bayesian persuasion (a concavification over distributions of posteriors) but applied to a distorted objective:

$$V^{\mathrm{dist}}(\mu; t, \gamma) = V_P(\mu; t) - \gamma V_M(\mu; t)$$

Here, $\gamma$ is a scalar shadow price capturing the marginal cost of meeting the agent’s participation constraint. The distortion reduces the curvature ("flattens") of the effective payoff index wherever the agent values posterior dispersion more than the principal.

This reduction collapses a bilevel optimization (principal chooses transfers anticipating agent’s experiment choice) to a tractable single-level program, leveraging standard concavification techniques. The optimal transfer schedule and experiment are explicitly characterized, particularly in two-state environments.

Posterior Compression and Explicit Results for Entropy Costs

A pivotal practical implication is the compression of posterior dispersion under moral hazard. For entropy-based costs, the effect is transparent: delegation does not restrict the feasible information structures but compresses the informativeness (reduces the posterior spread) by distorting the payoff index. This is shown both analytically and through a fully-worked numerical example.

For a binary-state, binary-action setup:

• First-best solution: The optimal experiment is a two-posterior signal, maximizing the spread subject to Bayes plausibility and cost.
• Second-best with moral hazard: The solution remains a two-posterior experiment, but the spread is strictly reduced. The paper provides closed-form formulas for the posterior endpoints. The optimal transfer schedule is solved as a triangular system connecting the transfer gap, participation constraint, and shadow price.

Strong Numerical Result

In a calibrated example with typical parameters (binary state, binary action, Shannon entropy cost), the spread of posteriors under delegation with moral hazard is reduced by approximately 28% compared to the first-best. This substantial compression highlights the quantitative effect of agency frictions in practical scenarios.

Connections to Prior Literature

The analysis situates itself among studies of persuasion under costly or delegated information acquisition [Gentzkow & Kamenica 2014, Whitmeyer & Zhang 2023]. Distinctively, the present framework embeds costly, rationally inattentive information acquisition into the principal-agent architecture, with agency and convex costs endogenously shaping informational outcomes. It also draws an explicit analogy between the Lagrangian distortion here and Myerson’s virtual surplus in mechanism design—wherein hidden action (moral hazard) induces a scalar distortion, contrasting with hidden information (adverse selection) that creates richer, type-indexed distortions.

Implications and Future Developments

Theoretical Implications

This framework reveals that moral hazard in delegated persuasion operates entirely through geometric distortion of the persuasion objective, rather than directly constraining feasible experiments. This leads to systematically coarser (less informative) information provision, independent of signal structure feasibility constraints.

Notably, the scalar shadow price $\gamma$ serves as a sufficient statistic for the cost of delegation. This tractability may facilitate empirical identification in settings with observable posterior distributions and transfers, such as credit ratings or third-party evaluations.

Practical Implications

The findings are directly relevant for:

• Regulators and Firms: When outsourcing information production, even optimal contracts cannot generally avoid information coarsening unless objective alignment is exact.
• Platform Design: Delegated recommendation and scientific review processes should account for moral hazard-induced information compression.
• Empirical Analysis: The model provides explicit comparative statics and closed-form benchmarks for the degree of information loss under moral hazard.

Future Research

Directions for future work include:

• Extending the analysis to multi-dimensional state/action spaces with more complex alignment conditions
• Studying repeated and dynamic versions of delegated persuasion
• Empirically estimating the shadow price of information distortion in real organizations

Conclusion

This paper rigorously establishes that, in delegated Bayesian persuasion with moral hazard and convex information costs, first-best outcomes are rarely implementable outside knife-edge alignment. The optimal solution in the presence of agency frictions is a tractable virtual Bayesian persuasion problem with a scalar-distorted payoff index. The informativeness of the optimal experiment is strictly compressed relative to the first-best, as quantified in both general theory and explicit entropy-based examples. The results directly inform the design of contracts and institutions in settings where information production is costly and delegated, and establish a scalable methodology for analyzing such problems (2604.10006).