arxiv: 2604.10006 · v2 · submitted 2026-04-11 · 💰 econ.TH

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Moral Hazard in Delegated Bayesian Persuasion

Wilfried Youmbi Fotso , Xun Chen

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Pith reviewed 2026-05-10 16:19 UTC · model grok-4.3

classification 💰 econ.TH

keywords delegated Bayesian persuasionmoral hazardinformation designpayoff alignmentshadow pricevirtual persuasionconvex costsentropy costs

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The pith

Moral hazard in delegated Bayesian persuasion reduces to alignment conditions on payoff indices, or a single shadow price when those fail.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a setting where a principal hires an intermediary to choose an information experiment but the intermediary bears private convex costs of doing so. It derives conditions under which the principal can still obtain the unconstrained optimal experiment through outcome-contingent transfers. A local condition on the support of the target posteriors is necessary for first-best implementation, while a global affine alignment of the two parties' payoff indices is sufficient. When alignment fails, the problem collapses to a virtual Bayesian persuasion task in which the principal's objective is concavified after being scaled by a constant shadow price that captures the entire agency friction. Under entropy costs this distortion compresses posterior spread whenever the intermediary's payoffs differ across the actions it recommends, and closed forms exist for binary cases.

Core claim

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

What carries the argument

Pair of alignment conditions (local support necessity plus global affine sufficiency) on payoff indices, together with the virtual Bayesian persuasion representation that distorts the principal's objective by one scalar shadow price.

If this is right

Whenever the local support and global affine alignments hold, the principal obtains the unconstrained optimal experiment.
When alignment fails, the second-best solution is the Bayesian-persuasion optimum of the principal's objective after uniform scaling by the shadow price.
Under entropy costs, moral hazard strictly reduces posterior dispersion if the intermediary's payoffs differ across recommended actions.
Explicit closed-form posteriors, mixing probabilities, and transfer schedules are available for all binary environments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The single-scalar reduction suggests that many delegation-of-information problems may admit low-dimensional contract solutions rather than fully flexible mechanisms.
The same virtual-representation technique could be tested in dynamic or multi-intermediary versions of the model.
Regulators or firms could estimate the shadow price from observed information choices to approximate optimal delegation rules without solving the full mechanism-design problem.

Load-bearing premise

The intermediary privately chooses the experiment subject to convex costs, and the principal can commit to outcome-contingent transfers without further restrictions on the contract space.

What would settle it

Compute the second-best experiment for a concrete binary-action environment with non-aligned payoffs and entropy costs, then check whether it exactly maximizes the concavified objective under the scalar shadow price derived from the incentive constraints.

Figures

Figures reproduced from arXiv: 2604.10006 by Wilfried Youmbi Fotso, Xun Chen.

**Figure 2.** Figure 2: Geometric characterization of the second-best experiment (Proposition [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** Figure 3: Effective objective Φ(µ) = W(µ) + H(µ) for the first-best (solid blue, slope 1 above µ¯) and second-best (dashed red, slope 0.8 above µ¯) under Shannon costs with µ0 = 0.45, µ¯ = 0.5, Dv = 1 2 , and γ SB = 0.4. Both curves share the same left piece H(µ); the second-best right piece is shallower by γ SBDv = 0.2. Dotted lines are the supporting chords; filled circles mark first-best posteriors and squares ma… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean virtual-representation result for handling moral hazard when delegating experiment choice in Bayesian persuasion.

read the letter

Here's the quick take: the paper gives a clean way to handle delegation in Bayesian persuasion when the intermediary has private costs for choosing the experiment. They characterize first-best implementability with a local support condition that's necessary and a global affine alignment that's sufficient. When first-best can't be reached, the second-best looks like standard Bayesian persuasion but with the principal's payoffs adjusted by one scalar shadow price that captures all the agency friction. Under entropy costs they even get closed forms for binary actions. What stands out is how they keep the concavification structure intact while adding the moral hazard layer. The shadow price reduction is useful because it turns a complicated contract design problem into something familiar. The partial characterization of the intermediate region between local and global conditions shows they didn't gloss over the gaps. The assumptions are standard for this literature: convex costs and full commitment to transfers. That keeps things tractable but means the results apply best when the principal has strong contracting power. I didn't see any obvious holes in the logic from the abstract and stress test, though the full proofs would need checking for whether the local condition really pins down the support without additional restrictions. This is worth a serious look for anyone in information economics or contract theory. It extends existing tools without overclaiming. A referee could tighten the intermediate region or test robustness to other cost functions, but the main results look solid enough to publish after revision. I'd bring it to a reading group to discuss the alignment conditions and how they compare to standard mechanism design.

Referee Report

0 major / 2 minor

Summary. The paper studies moral hazard in delegated Bayesian persuasion, where a principal uses outcome-contingent transfers to incentivize an intermediary who privately chooses an experiment subject to convex costs. It characterizes first-best implementability via a necessary local alignment condition on the support of the target experiment and a sufficient global affine alignment condition on the payoff indices. The gap between these is shown to be non-empty with a partial characterization provided. When first-best is unattainable, the principal's problem reduces to a virtual Bayesian persuasion problem in which the second-best experiment maximizes the concavified objective with the principal's payoff index distorted by a single scalar shadow price summarizing the agency friction. Explicit closed-form solutions for posteriors, mixing weights, and transfers are derived for binary environments under entropy costs.

Significance. If the characterizations and virtual representation hold, this provides a clean and tractable framework for incorporating agency frictions into Bayesian persuasion models. The separation of local necessary and global sufficient conditions, combined with the single-shadow-price distortion for the second-best, offers a useful reduction that parallels standard mechanism-design techniques. The explicit solutions under entropy costs are a particular strength, as they deliver falsifiable predictions on posterior compression and could support applications in delegated information design.

minor comments (2)

The abstract and introduction refer to a 'partial characterization of the intermediate region' between the local and global alignment conditions; the main text should include at least one concrete example or proposition that illustrates the boundary of this region.
Notation for the principal's and intermediary's payoff indices (and their concavifications) should be introduced with a single consistent table or display early in the paper to aid readability across sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and insightful summary of our paper, as well as the recommendation for minor revision. The report raises no specific major comments or points requiring clarification, so we have no revisions to propose at this stage.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The paper's core results—necessary local support condition and sufficient global affine alignment for first-best implementability, plus the virtual Bayesian persuasion reduction for the second-best via an endogenous scalar shadow price—are derived directly from the model primitives (principal's and intermediary's payoff indices, convex experiment costs, and unrestricted outcome-contingent transfers) without reducing to fitted inputs, self-definitional loops, or load-bearing self-citations. The shadow price emerges as the Lagrange multiplier enforcing the intermediary's incentive constraint in the principal's optimization program, preserving independence from the target experiment; explicit binary solutions under entropy costs follow by direct substitution into the concavified objective. No step equates a claimed prediction or first-principles result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from Bayesian persuasion and mechanism design; no new entities are postulated and the shadow price is derived rather than introduced as a free parameter.

axioms (2)

standard math Players are Bayesian rational with a common prior.
Implicit background assumption for all Bayesian persuasion models.
domain assumption The intermediary's cost function for choosing an experiment is convex.
Stated in the model setup to guarantee well-behaved optimization and implementability results.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

Arieli, I., Babichenko, Y., and Smorodinsky, R. (2022). Mediators in persuasion problems.Theoretical Economics, 17(1):1–36. Bergemann, D. and Morris, S. (2016). Bayes correlated equilibrium and the compar- ison of information structures in games.Theoretical Economics, 11(2):487–522. Bharadwaj, A. and Iyer, G. (2025). Contracting for persuasion.Working pap...

2022
[2]

and Skreta, V

Doval, L. and Skreta, V. (2024). Optimal mechanism for the sale of a durable good. Theoretical Economics, 19(2):865–915. Dworczak, P. and Martini, G. (2019). The simple economics of optimal persuasion. Journal of Political Economy, 127:1993–2048. Gentzkow, M. and Kamenica, E. (2014). Costly persuasion.American Economic Review Papers and Proceedings, 104(5...

work page arXiv 2024