Delegated Information Provision

This paper models a designer's optimal strategy for restricting an experimenter's choice of information experiments to counteract persuasive incentives, demonstrating that when preferences are S-shaped, the best approach involves a "double censorship" mechanism that outperforms full delegation by balancing information provision with persuasion constraints.

The Gist

Imagine you are a Judge (the Decision Maker) trying to decide whether to sentence a defendant to prison or set them free. You don't know the truth about the defendant's guilt; you only have a hunch based on past cases.

Enter the Prosecutor (the Experimenter). They have access to all the evidence. They can run tests, gather data, and present it to you.

Here is the problem: The Prosecutor wants you to convict the defendant, regardless of the actual truth. They are a "persuader." If they can choose how to present the evidence, they will cherry-pick the worst parts of the story and hide the good parts to make the defendant look guilty.

Now, imagine a third person: the Legislator (the Designer). The Legislator wants the Judge to make the correct decision based on the truth, not just be tricked by the Prosecutor. The Legislator can't force the Prosecutor to tell the truth, but they can set the rules of the game. They can say, "You are only allowed to present evidence in these specific formats."

This paper asks: What is the best set of rules the Legislator should write to stop the Prosecutor from lying, while still getting useful information?

The Core Conflict: The "Garbage" Problem

The paper highlights a tricky situation. Even if the Legislator says, "You must show me a full, detailed forensic report," the Prosecutor can still garble it. They can take that detailed report, throw away the parts that help the defendant, and only show you the scary parts.

So, the Legislator can't force the Prosecutor to be honest. They can only set an upper limit on how much information the Prosecutor can show. The Prosecutor will then choose the most deceptive version of that limit.

The Old Way: "Full Delegation" (The Status Quo)

If the Legislator does nothing and lets the Prosecutor choose any evidence they want (Full Delegation), the Prosecutor will use a strategy called "Upper Censorship."

• How it works: The Prosecutor reveals the truth for low-level crimes (e.g., "This guy is definitely innocent"). But for high-level crimes (e.g., "This guy might be a serial killer"), they lump everything together into one scary bucket.
• The Result: You know the innocent are innocent, but you can't tell the difference between a "maybe guilty" person and a "definitely guilty" person. The Prosecutor pools all the high-risk cases together to maximize the chance of a conviction.

The New Discovery: "Double Censorship"

The authors of this paper discovered that the Legislator can do better than doing nothing. They can impose a specific rule called Double Censorship.

Think of the states of the world (the defendant's actual guilt level) as a line from 0 (Innocent) to 100 (Guilty).

1. The Old Way (Full Delegation):

   • 0 to 40: Revealed clearly.
   • 40 to 100: All lumped together into one "Guilty" bucket.
   • Problem: You lose a lot of detail in the middle and high ranges.

2. The New Way (Double Censorship):

   • 0 to 20: Revealed clearly (Innocent).
   • 20 to 60: Lumped into an "Intermediate" bucket (Maybe Guilty).
   • 60 to 100: Lumped into a "High" bucket (Very Guilty).

Why is this better?
It sounds like you are hiding more information (by creating that middle "Maybe" bucket), but it actually helps the Judge.

Here is the magic trick: By forcing the Prosecutor to reveal the "Maybe Guilty" cases separately, you discipline them.

• If the Prosecutor tries to hide the "Very Guilty" cases by lumping them with the "Maybe Guilty" ones, the Judge will realize the "Maybe" bucket is too scary and might acquit.
• To avoid this, the Prosecutor is forced to be more honest about the very worst cases. They have to separate the "Very Guilty" from the "Maybe Guilty."

The Trade-off:
You sacrifice a little bit of detail in the middle (the "Maybe" bucket) to gain a huge amount of clarity at the top (the "Very Guilty" bucket). The paper proves that this trade-off always results in a better outcome for the Judge than letting the Prosecutor run wild.

A Creative Analogy: The "Grading Scale"

Imagine a teacher (The Prosecutor) who wants to give a student (The Judge) a bad grade to get them fired. The student's actual performance is a number between 0 and 100.

• No Rules (Full Delegation): The teacher says, "If you scored below 50, I'll tell you exactly what you got. If you scored above 50, I'll just say 'Fail'." The teacher hides the difference between a 51 and a 99.
• The Legislator's Rule (Double Censorship): The Legislator says, "You must report scores in three categories: 'Pass' (0-40), 'Warning' (41-80), and 'Fail' (81-100)."
  • The teacher must separate the 85s from the 45s.
  • The teacher can't hide the 85s inside the 45s anymore.
  • Even though the teacher can still hide the difference between 41 and 79 (the "Warning" zone), the Judge now knows for sure that anyone in the "Fail" zone is a 90+ student.

The Big Takeaway

The paper's main conclusion is counter-intuitive: Sometimes, hiding a little bit of information is the best way to get the truth.

By restricting the Prosecutor's ability to "smooth over" the details in the middle, the Legislator forces them to be more honest about the extremes. The optimal strategy isn't to demand more information; it's to demand a different shape of information that aligns the Prosecutor's incentives with the truth.

In short: To stop a liar from lying about the worst things, you have to force them to tell the truth about the "okay" things, even if it means you learn slightly less about the "okay" things. It's a strategic sacrifice that pays off in the end.

Technical Summary

Here is a detailed technical summary of the paper "Delegated Information Provision" by Francesco Bilotta, Christoph Carnehl, and Justus Preusser.

1. Problem Statement

The paper addresses a principal-agent problem involving three parties: a Designer, an Experimenter, and a Decision Maker (DM).

• The Setting: The Designer relies on the Experimenter to generate information (an experiment) about a state $\omega \in [0,1]$ to inform the DM's decision.
• The Conflict: The Experimenter has a state-independent preference for the DM to take a specific action (e.g., "approve"). Consequently, the Experimenter has an incentive to persuade the DM rather than truthfully reveal information.
• The Constraint: The Designer cannot force the Experimenter to run a specific experiment. Instead, the Designer can only restrict the set of admissible experiments (imposing a Blackwell-upper bound).
• The Moral Hazard: Even with restrictions, the Experimenter can garble (obfuscate) any admissible experiment before it reaches the DM. Thus, the Designer can only impose an upper bound on informativeness, not a lower bound.
• The Goal: The Designer, who has strictly convex preferences over the DM's posterior mean (valuing information), seeks to design a delegation set (a restriction) that maximizes their expected utility, anticipating the Experimenter's strategic garbling.

2. Methodology and Framework

Model Setup

• State & Outside Option: The state $\omega$ is drawn from a prior $H$. The DM has a private outside option $r$ drawn from $G$. The DM acts if their posterior mean $m \geq r$.
• Payoffs:
  • Experimenter: Maximizes the probability of action, yielding an interim payoff of $G(m)$. Under Assumption 1, $G$ is S-shaped (convex on $[0, r_0]$, concave on $[r_0, 1]$).
  • Designer: Has a strictly convex payoff $u_D(m)$ in the posterior mean.
• Delegation as MPC: An experiment is represented by the CDF of posterior means, $F$. The set of all experiments is the set of Mean-Preserving Contractions (MPC) of the prior $H$, denoted $MPC(H)$. A restriction $\bar{F}$ limits the Experimenter to choosing $F \in MPC(\bar{F})$.

Key Concepts

1. Incentive Compatibility (IC): An experiment $F$ is IC if it is a best reply to itself. That is, if the Designer sets $F$ as the restriction, the Experimenter optimally chooses $F$ rather than garbling it to a less informative experiment.
2. Maximal Incentive-Compatible (MIC): An IC experiment $F$ is maximal if there is no other IC experiment $\tilde{F}$ such that $F$ is a proper MPC of $\tilde{F}$.
   • Intuition: Since the Designer values information, they will never choose a non-maximal restriction because a more informative IC restriction would always yield a higher payoff.
3. Double Censorship: A specific structure of experiments characterized by two thresholds and two atoms.

3. Key Contributions and Results

A. Characterization of Optimal Restrictions (The S-Shaped Case)

The paper's primary contribution is the full characterization of MIC experiments when the Experimenter's payoff is S-shaped.

• Theorem 1 (Double Censorship): All MIC experiments take the form of Double Censorship.

  • Structure: There exist two thresholds $s$ and $t$ ($s \leq t$) and two atoms $x$ and $y$.
    1. Low States ($[0, s)$): Fully revealed (follows the prior $H$).
    2. Intermediate States ($[s, t]$): Pooled into a single atom $x = E[\omega | \omega \in [s, t]]$.
    3. High States ($[t, 1]$): Pooled into a single atom $y = E[\omega | \omega \in [t, 1]]$.
  • Incentive Condition: The atoms $(x, y)$ must satisfy a tangency condition: $G(x) + g(y)(y - x) = G(y)$. This ensures the Experimenter has no incentive to further garble the high atom.
  • Maximality Condition: The thresholds must satisfy $s \leq x^* \leq t$, where $x^*$ is the threshold of the full-delegation (unrestricted) solution.

• Comparison with Full Delegation:

  • Under Full Delegation (no restriction), the Experimenter chooses Upper Censorship: revealing states below $x^*$ and pooling all states above $x^*$ into a single high atom $y^*$. This is a special case of Double Censorship where $s = t = x^*$.
  • The Trade-off: MIC experiments are Blackwell-incomparable to each other and to the full-delegation outcome. To reduce the pooling of the highest states (improving information at the top), the Designer must expand the pooling of intermediate states (worsening information in the middle).

B. Optimality of Non-Trivial Restrictions

• Theorem 2 (Full Delegation is Suboptimal): The Designer strictly prefers a non-trivial Double Censorship restriction over Full Delegation.
  • Mechanism: By introducing a small intermediate pooling region (moving from $s=t=x^*$ to $s < t$), the Designer disciplines the Experimenter.
  • First-Order Gain vs. Second-Order Loss:
    • Gain: Reducing the size of the top pooling region (moving the top atom $y$ closer to the true state) yields a first-order benefit because the Designer values information in the high states.
    • Loss: Creating the intermediate pooling region causes a second-order loss because the states pooled are close to the revelation threshold.
  • Result: The first-order gain from better information in high states strictly outweighs the second-order loss from intermediate pooling.

C. Generalization Beyond S-Shaped Preferences

• Section 6.2: The authors extend the analysis to general Experimenter payoffs.
  • Conditions for Profitable Restrictions: Full delegation is suboptimal if:
    1. The Experimenter has locally strict curvature preferences (payoff is strictly convex or concave in local intervals).
    2. The full-delegation outcome admits incomplete full revelation (the Experimenter reveals some states but pools others, rather than revealing everything or nothing).
  • Counter-examples: If the DM has no private information (step-function payoff) or if the prior is such that the optimal persuasion is a binary experiment supported on the peaks of an M-shaped payoff, full delegation may remain optimal.

D. Connection to Extreme Points

• Theorem 3: Every MIC experiment is an extreme point of the set $MPC(H)$.
• Significance: This links the novel incentive-constrained problem to the standard literature on information design (Kleiner et al., 2021), which solves unconstrained persuasion via extreme points (bi-pooling experiments). The authors show that the incentive-compatibility constraint effectively selects a specific subset of these extreme points.

4. Technical Mechanisms and Proofs

• Price-Theoretic Approach: The authors utilize the duality approach of Dworczak and Martini (2019). An experiment $F$ is IC if and only if there exists a convex "price" function $p \geq G$ such that the support of $F$ lies where $p(m) = G(m)$.
• Local Perturbation: The proof that full delegation is suboptimal (Theorem 2) relies on a local perturbation argument. The authors calculate the derivative of the Designer's utility with respect to the top atom $y$ at the full-delegation point $y^*$. They show this derivative is strictly positive, driven by the strict convexity of the Designer's utility function.
• Construction of MIC: The proof of Theorem 1 constructs the Double Censorship experiment by rotating the Integrated CDF (ICDF) of a candidate experiment until it becomes tangent to the prior $H$ at a new threshold, ensuring maximality while maintaining the IC condition.

5. Significance and Implications

1. New Mechanism for Control: The paper demonstrates that a principal can improve outcomes not by forcing more information, but by reshaping the information structure. By deliberately inducing "noise" (pooling) in intermediate states, the principal can prevent the agent from "censoring" the most valuable (high) states.
2. Blackwell-Incomparability: It highlights that optimal restrictions are not simply "less informative" versions of the full-delegation outcome. They are structurally different (incomparable) information structures that reallocate informativeness across the state space.
3. Practical Applications: The framework models real-world scenarios such as:
   • Regulation: Limiting the types of tests firms can run.
   • Law: Restricting the evidence prosecutors can present.
   • Privacy: Designing privacy-preserving signals where certain sets of states cannot be distinguished.
4. Theoretical Advancement: It bridges the gap between Bayesian Persuasion (where the sender chooses the experiment) and Information Design with constraints, showing how incentive compatibility constraints interact with the geometry of the set of experiments ($MPC(H)$).

In summary, the paper provides a rigorous characterization of how a designer can optimally constrain a strategic information provider. The key insight is that strategic obfuscation can be curbed by accepting local information loss in exchange for global gains, specifically by transforming the "upper censorship" of the unconstrained equilibrium into a "double censorship" structure.