📈 economics

Persuasion under the Threat of Verification

This paper demonstrates that the impact of cheaper private verification on a sender's public communication strategy depends critically on the sender's own cost of generating public information: while low public costs lead to more informative experiments and reduced verification, high public costs cause the sender to coarsen information and shift the burden of fact-finding onto receivers.

Original authors: Georgy Lukyanov, Samuel Safaryan

Published 2026-07-28

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Georgy Lukyanov, Samuel Safaryan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to convince a crowd to do something, like buying a new gadget or voting for a candidate. In the world of economics, this is called persuasion. Usually, we assume the person doing the convincing (the "sender") has all the power: they can show you a picture, tell a story, or run an ad, and you just believe them. But in the real world, we aren't that gullible. We have fact-checkers. We might read a review, hire an expert, or do our own research before making a decision. This paper lives in the intersection of information design (how to best present facts) and costly verification (the price we pay to check if those facts are true).

The big question the authors ask is: What happens when fact-checking gets cheaper? If it becomes easier and cheaper for people to verify the truth, does the person trying to persuade them get scared and start telling the truth? Or does it make them lazy, letting the audience do all the heavy lifting of finding the truth while the sender just gives vague, unhelpful hints? It turns out, the answer isn't a simple "yes" or "no." It depends entirely on how much it costs the sender to produce their own information in the first place.

The Story of the Shifty Salesperson and the Busy Crowd

Let's meet our characters. There is a Sender (think of a salesperson or a politician) who wants everyone to choose "Action A" (buy the product, vote for the candidate), no matter what the reality is. Then there is a Crowd of receivers. Some in the crowd are cheap to convince; others are expensive. Everyone in the crowd has a secret "verification cost"—a price tag on how much effort or money it takes them to find out the real truth.

Before the crowd decides, the Sender commits to a Public Experiment. This is like a pre-announced test or a rating system. The Sender pays a fee to run this test. The test gives a public signal (a score or a grade) that updates everyone's belief about the truth. After seeing the public signal, the crowd members decide: "Is this signal trustworthy enough, or should I pay my own cost to check the truth myself?"

If the public signal is weak, the crowd checks. If the signal is super clear, the crowd trusts it and saves their money. The Sender wants to design the test to get the crowd to choose "Action A" while paying the least amount of money for the test itself.

The Great Phase Reversal: Two Different Worlds

The paper discovers a fascinating "phase reversal," which is like a switch that flips the rules of the game depending on how expensive it is for the Sender to run their own tests.

Scenario 1: The Sender Has Cheap Tests (The "Discipline" Zone)
Imagine the Sender can run a high-quality, super-accurate test for very little money. In this world, if fact-checking becomes cheaper for the crowd, the Sender gets disciplined.

What happens: The Sender thinks, "Oh no, if I give a vague signal, the crowd will easily check me and catch me lying!" So, to avoid being checked, the Sender decides to buy a sharper, more informative test. They make their public signal so crystal clear and decisive that the crowd doesn't need to check anymore.
The Result: Paradoxically, as fact-checking gets cheaper, the actual amount of checking by the crowd goes down. The Sender's fear of being caught forces them to be so honest and clear that the crowd feels safe trusting them without doing the extra work. The public information becomes more informative, and the crowd saves money.

Scenario 2: The Sender Has Expensive Tests (The "Substitution" Zone)
Now, imagine the Sender's tests are incredibly expensive to run. Maybe they need expensive lab equipment or a team of analysts.

What happens: If fact-checking becomes cheaper for the crowd, the Sender thinks, "Phew! It's too expensive for me to make a perfect signal. But hey, if the crowd can check the truth cheaply now, I don't need to try so hard!"
The Result: The Sender stops trying to be clear. They might even give up and just give a vague, uninformative signal (a "pool" where everyone gets the same boring message). They let the crowd do the fact-checking. As fact-checking gets cheaper, the public signal actually becomes less informative (or stays the same), and the crowd ends up checking more because the Sender's signal is so weak. The Sender effectively shifts the burden of finding the truth onto the audience.

The "Inverse-U" Surprise

One of the coolest findings is what happens to the number of people actually checking the facts in the "Cheap Sender" world.

At first, as fact-checking gets cheaper, more people start checking because it's affordable.
But once the Sender gets scared enough to upgrade their signal to be super-clear, the crowd stops checking!
So, the number of fact-checkers goes up, hits a peak, and then goes back down. It's an "inverse-U" shape. The threat of verification disciplines the Sender so effectively that the actual act of verifying becomes unnecessary.

What This Paper Rules Out

The authors are very careful to tell us what doesn't happen. You might think that if fact-checking gets cheaper (meaning the distribution of costs shifts down), the Sender will always be forced to be more honest. The paper proves this is false.

It is not true that cheaper verification always leads to better public information.
It is not true that the amount of public information and the amount of private checking always move in the same direction. Sometimes they move in opposite directions!
The paper also shows that you can't just look at the "average" cost of checking to predict the outcome. You need to know the specific shape of the costs and how the Sender's own costs interact with them. A simple "first-order" improvement in verification costs isn't enough to guarantee a specific result; the math gets tricky and depends on the curvature of the costs.

The Bottom Line

This paper uses rigorous math (specifically, a method called "concavification" and "Blackwell order" comparisons) to prove that the relationship between public information and private checking is a two-sided coin.

If the Sender can afford to be clear, cheaper checking makes them clearer (and checking drops).
If the Sender can't afford to be clear, cheaper checking makes them vague (and checking rises).

The authors show this with a specific benchmark model where costs are quadratic (a specific mathematical shape) and then prove that the logic holds even for more general, complex cost structures. They don't just guess; they derive exact formulas and prove that this "phase reversal" is a fundamental feature of how information works when the audience has the power to double-check.

So, the next time you see a politician or a company release a report, ask yourself: "Is their signal getting better because they are scared of my fact-checking, or are they getting worse because they know I'll do the work for them?" The answer depends entirely on how much it costs them to speak the truth.

Technical Summary: Persuasion under the Threat of Verification

Problem and Motivation
This paper investigates the strategic interaction between a sender who commits to a costly public experiment and a continuum of heterogeneous receivers who, after observing the public signal, may choose to privately verify the state at a cost. The central question is whether a reduction in the cost of private verification disciplines the sender into providing more informative public information, or whether it allows the sender to shift the informational burden onto the receivers, resulting in less informative public signals.

The model departs from standard Bayesian persuasion by introducing endogenous receiver verification. Unlike models where receivers are passive or where verification is a binary "hard evidence" constraint, here receivers possess a private cost of acquiring perfect state information. The sender's objective is to induce a specific action (action 1) regardless of the state, while receivers wish to match their action to the true state. The sender incurs a posterior-separable cost to produce public information, representing the resources required to install and operate a signal technology, distinct from any penalty for lying.

Methodology
The authors employ a standard ex-ante persuasion framework with a binary state $\theta \in \{0, 1\}$ and a binary action space. The timing is as follows:

The sender commits to a public experiment (a Markov kernel) before the state is realized.
The state is drawn, and the public signal generates a posterior belief $\mu$ .
Receivers, observing $\mu$ and knowing their private verification costs $c_i$ , decide whether to pay $c_i$ to learn the true state perfectly.
Receivers choose an action based on their information (verified or unverified).

The analysis relies on the concavification approach (Kamenica and Gentzkow, 2011) adapted for costly information (Gentzkow and Kamenica, 2014). The sender's problem is reduced to choosing a distribution of posteriors $\Pi$ that satisfies Bayes plausibility ( $\mathbb{E}[\mu] = \pi$ ) to maximize an adjusted interim payoff function. This payoff function, $v_{r,\kappa}(\mu)$ , incorporates the sender's expected aggregate action (which depends on the mass of verifiers at $\mu$ ) minus the cost of information generation.

The paper first establishes a benchmark with uniformly distributed verification costs and quadratic persuasion costs. It then generalizes the results to arbitrary symmetric convex persuasion costs and arbitrary continuous distributions of verification costs. The analysis utilizes tools from convex analysis, including supporting lines, Blackwell order comparisons, and comparative convexity (Curello and Sinander, 2025).

Key Results

1. The Phase Reversal (Discipline vs. Substitution)
The paper's primary finding is a "phase reversal" in the sender's response to cheaper verification, determined by the sender's own cost of producing public information ( $\kappa$ ):

Low-Cost Regime ( $\kappa < 2$ ): When public information is relatively cheap, a reduction in verification costs disciplines the sender. The sender responds by making the favorable posterior strictly more decisive (moving it further above the action threshold of $1/2$ ). Consequently, the optimal public experiment becomes strictly more informative in the Blackwell order. Paradoxically, as verification becomes cheaper, the realized mass of verifiers decreases because the public signal becomes so decisive that fewer receivers find it worthwhile to check. The relationship between public informativeness and realized verification is inverse-U-shaped.
High-Cost Regime ( $\kappa > 2$ ): When public information is expensive, the sender substitutes away from producing information. Cheaper verification leads the sender to tolerate more private fact-finding by the audience. The sender weakly coarsens the public experiment (moving the lower posterior closer to $1/2$ ) and may eventually resort to complete pooling. In this regime, public informativeness and realized verification move in opposite directions: as verification becomes cheaper, the public signal becomes less informative, and the mass of verifiers increases.

2. Generalization and Conditions
The paper demonstrates that this phase reversal is not an artifact of the quadratic benchmark.

General Costs: For general symmetric convex persuasion costs, the authors derive primitive curvature and supporting-line conditions that recover the two geometries (discipline and substitution). The discipline case requires the left branch of the payoff to be convex enough relative to the right branch, while the substitution case relies on the concavity of the left branch.
Arbitrary Verification Distributions: The authors show that a First-Order Stochastic Dominance (FOSD) reduction in verification costs is insufficient to sign the response of the optimal experiment. The relevant global condition is a "comparative convexity" order on the induced interim payoffs. They provide a counterexample where an FOSD improvement in verification costs leads to a more informative public experiment even in the high-cost regime, contradicting the intuition that cheaper verification always leads to substitution.

3. Welfare Implications

Receiver Welfare: In the low-cost regime, receiver welfare strictly increases as verification becomes cheaper, driven by both lower private costs and a more informative public signal. In the high-cost regime, the effect is ambiguous: the direct benefit of cheaper verification is offset by the endogenous contraction of the public signal.
Sender Welfare: In the low-cost regime, the sender's equilibrium value decreases as verification becomes cheaper, as the sender must incur higher information costs to maintain the discipline effect.

Significance and Claims
The paper claims to resolve the ambiguity regarding the effect of independent fact-checking on public information provision. It establishes that the direction of the effect depends entirely on the sender's internal cost structure. The authors emphasize that public informativeness and realized verification need not move together; they can move in opposite directions depending on the regime.

The paper modestly notes that while the binary environment isolates the mechanism cleanly, the results are derived under full commitment and specific cost structures. The authors state that the two-point support structure driving these results may not survive extensions to noisy verification, richer action spaces, or dynamic settings, as those would require ordering multidimensional mixtures of information rather than scalar posteriors. The work contributes to the literature on Bayesian persuasion, costly information, and endogenous verification by integrating the sender's production cost with the receiver's acquisition cost in a unified framework.

The Story of the Shifty Salesperson and the Busy Crowd

The Great Phase Reversal: Two Different Worlds

The "Inverse-U" Surprise

What This Paper Rules Out

The Bottom Line

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