{"id":"c37a38e5-ae30-47da-9824-fb77daf0c941","arxiv_id":"2508.19682","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Cheaper verification is claimed to coarsen optimal public signals, but the paper's own equations imply full revelation is optimal and the main lemma is false.","lead":"This paper asks whether cheaper fact-checking makes a sender's public messages more or less informative. It claims less informative, but the model as written makes full disclosure optimal regardless of verification costs, and a key lemma used in the proof is false.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unconstrained optimum is full revelation, making Theorem 4.2's coarsening claim vacuous; Lemma 4.1's contraction inequality is also false.","rationale":"The reader's verdict identifies the two decisive flaws: the unconstrained optimum is full revelation, so the benchmark comparative static is vacuous, and Lemma 4.1's contraction inequality is false. My stress test confirms both from the manuscript text. Eq. (3.5) shows v ≤ −b² with equality only at the endpoints, so the concavification problem is degenerate: the sender's best response is to reveal the state fully regardless of F. The introduction and Theorem 4.2 claim a strict coarsening response to cheaper verification, which cannot happen when full revelation is already optimal for every F. This is not a matter of degree or an auxiliary assumption; it invalidates the paper's headline result. The additional failure of Lemma 4.1, with an explicit FOSD pair, further undermines the proof of Theorem 4.2 even if one attempted to modify the model to avoid the full-revelation degeneracy. The paper's Protocol B (Section 4.2) does impose EPIC constraints that may rule out full revelation for b>0, but the authors present Theorem 4.2 as the benchmark 'reverse comparative static' and the abstract/introduction emphasize the unconstrained result. Since the central claim as stated fails, a REJECT verdict is appropriate. I found no reason to soften the reader's judgment; the concerns are load-bearing and directly testable.","tokens_in":19920,"tokens_out":7393,"duration_ms":74741,"concrete_test":"Two checks. (1) Evaluate the unconstrained program (4.1) for uniform F(x)=x, π=1/2, any b>0. Show that Π=0.5δ₀+0.5δ₁ attains −b², while any experiment with positive mass on interior posteriors attains strictly less, so full revelation is the unique optimum independent of F. (2) Test Lemma 4.1 with the counterexample: set F(x)=x and let F' be piecewise linear with F'(0.09)=0.15, F'(0.25)=0.25, F'(1)=1 (extended continuously). Compute v(0.1;F')−v(0.5;F') and v(0.1;F)−v(0.5;F); the former exceeds the latter, contradicting the lemma's pointwise contraction.","verdict_should_be":"REJECT","load_bearing_attack":"The central comparative static rests on the unconstrained concavification problem in Section 4.1. But from Eq. (3.5), v(µ;F) = −b² − (1−λ(µ;F))²µ(1−µ) ≤ −b², with equality if and only if µ ∈ {0,1}. Therefore, for any prior π∈(0,1), the unique maximizer of EΠ[v(µ;F)] over Bayes-plausible Π is the fully revealing experiment (mass 1−π at 0 and π at 1). This optimum is independent of F, so no FOSD improvement of F can make the optimal experiment strictly less informative; the claimed reverse comparative static is vacuous. The authors' Theorem 4.2 asserts a strict coarsening response that cannot occur. Lemma 4.1, which the proof invokes, is also false in general: take F(x)=x and a piecewise-linear F' with F'(0.09)=0.15, F'(0.25)=0.25. At µ=0.1, s=0.09, the FOSD improvement raises v(0.1;F')−v(0.5;F') above v(0.1;F)−v(0.5;F), reversing the asserted inequality. Even if the lemma were repaired, the degeneracy of the objective would remain: full revelation dominates any interior experiment for every F.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a sender who commits to a public experiment before a continuum of receivers who can privately verify the state at heterogeneous costs. The sender's loss is quadratic in the distance between the aggregate action and θ+b, and verifiable reports must be truthful (EPIC). The main claim is a reverse comparative static: if the verification-cost distribution improves in FOSD, the sender's indirect value v(µ;F) becomes more concave and every optimal public experiment becomes strictly less Blackwell informative. After the benchmark, the paper imposes EPIC (Protocol A: hard evidence + silence; Protocol B: minimal soft layer) and extends the framework to falsification and repression. The appendix provides proofs and a constructive implementation.","tokens_in":20282,"tokens_out":8869,"duration_ms":88140,"significance":"The paper is clearly written and the EPIC-constrained concavification formulation is a natural way to model hard-evidence constraints in mass-audience persuasion. The closed-form uniform-cost special case is a useful pedagogical benchmark. However, the central comparative static is invalidated by the paper's own Eq. (3.5): the unconstrained optimum is full revelation for every F, and Lemma 4.1, which the proof invokes, is false. As a result, the claimed reverse comparative static — the paper's main contribution — does not follow from the model as specified.","major_comments":[{"comment":"Eq. (3.5) shows v(µ;F)=−b²−(1−λ(µ;F))²µ(1−µ)≤−b², with equality only at µ∈{0,1} for any F with λ(µ;F)<1 on (0,1). Therefore, for any prior π, the fully revealing public experiment achieves the upper bound in problem (4.1), and any distribution with interior support is strictly worse. The unconstrained optimal experiment is the same full-revelation distribution for every F, so Theorem 4.2's claimed strict coarsening is impossible. Under EPIC with ε>0, the optimal policy is δ0=δ1=1 for every F; informativeness does not respond to F.","section":"Section 4.1, Eq. (3.5), Theorem 4.2"},{"comment":"The asserted mean-preserving contraction inequality is false. Let F(x)=x on [0,1], and let F' be a cdf with F'(0.09)=0.15 and F'(0.25)=0.25 (a FOSD improvement in the sense of cheaper verification). At µ=0.1, using Eq. (3.5) with b=0, p(µ;F)=(1−F(µ(1−µ)))²µ(1−µ), we have p(0.1;F)=0.074529, p(0.5;F)=0.140625, so v(0.1;F)−v(0.5;F)=0.066096. For F', p(0.1;F')=0.065025, p(0.5;F')=0.140625, so v(0.1;F')−v(0.5;F')=0.0756>0.066096, reversing the inequality. The lemma also mischaracterizes v: at b=0, v(0)=v(1)=0>v(1/2)=−(1−F(1/4))²/4, so 1/2 is a local minimum, not a global maximum. Since Theorem 4.2's proof and Theorem 4.4 rely on this lemma, the coarsening conclusion lacks support.","section":"Lemma 4.1 / Lemma A.2"}],"minor_comments":[{"comment":"The statement that a FOSD decrease in F makes v(µ;F) 'more concave in µ' is asserted without proof and is false as stated; see the counterexample in the major comments.","section":"Section 3.4"},{"comment":"The paragraph after Proposition 4.3 says the informativeness effect is a priori ambiguous, but in the actual model with the quadratic objective the maximal-disclosure policy dominates any non-degenerate protocol, so the ambiguity is not present.","section":"Section 4.2, Protocol A"},{"comment":"The continuous-state extension is stated without proof ('details are omitted for brevity'). This unsupported claim goes beyond the binary-state analysis and should be either proved or clearly labeled as conjectural.","section":"Section A.8 and Section 6.5"}],"recommendation":"reject","confidential_remarks":"The central result is contradicted by the model's own Eq. (3.5): the unconstrained optimum is full revelation independent of F, making the main comparative static vacuous. The additional failure of Lemma 4.1 is independent. These are load-bearing defects that cannot be repaired within the current framework; the authors would need to change the sender's objective or introduce a non-trivial cost of public information for the claimed reverse comparative static to have content."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The setup is genuinely novel: a mass audience that self-selects into costly verification, combined with an ex-post truthfulness constraint, is a nice idea that isn't in the cited literature. The verification cutoff and the aggregate-action derivation are clean, and the falsification/repression extension could be interesting in a different model. But the central result collapses under inspection of the paper's own value function.\n\nEq. (3.5) gives v(µ;F) = −b² − (1−λ)²µ(1−µ). Since (1−λ)²µ(1−µ) ≥ 0, v is at most −b², with equality only at µ=0 and µ=1. Therefore the sender's expected payoff is maximized by the fully revealing experiment, regardless of F. Any interior posterior reduces the payoff. So the optimum is full revelation for every verification-cost distribution, and Theorem 4.2's claimed strict coarsening under cheaper verification is vacuous—the optimal experiment doesn't change at all. The paper's description of v as 'single-peaked with a global maximum at µ∈{0,1/2,1}' is also wrong; v is actually higher at the endpoints than at 1/2.\n\nThe second leg is Lemma 4.1, which asserts that a FOSD improvement in verification costs makes v more concave in a pointwise sense. The proof only shows that (1−λ)²µ(1−µ) pointwise decreases, which does not imply the stated contraction around 1/2. A concrete counterexample exists: take F uniform and a piecewise-linear F' with F'(0.09)=0.15 and F'(0.25)=0.25. This F' is a FOSD improvement (cheaper costs), but at µ=0.1 the inequality in Lemma 4.1 is reversed. So the lemma is false, and the proof of Theorem 4.2 relies on it.\n\nThere is also a mismatch with the abstract: it promises an explicit counterexample showing that FOSD improvements alone don't sign the response, but no such counterexample appears in the main text.\n\nThe flaws are load-bearing, not cosmetic. The model's objective makes full revelation the unique optimum, so the entire comparative static and the phase-reversal narrative are unsupported. The framework might be salvageable with a different sender objective (e.g., one that rewards belief dispersion), but as written the paper should be rejected. A referee would quickly identify these issues, so the paper does deserve a serious referee to document them, but it should not be published without major revision.","headline":"The paper's central comparative static is vacuous because its own Eq. (3.5) makes full revelation optimal, and the key lemma is false.","tokens_in":20705,"tokens_out":8189,"would_cite":false,"duration_ms":80062,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Cheaper fact-checking makes the optimal public signal less informative, the paper argues.","keywords":["Bayesian persuasion","information design","verifiable evidence","costly verification","public signals","Blackwell informativeness","falsification","repression"],"falsifier":"Test the paper's key concavity lemma by choosing a distribution F′ that first-order stochastically dominates F and evaluating v(µ;F′)−v(1/2;F′) against v(µ;F)−v(1/2;F) at an interior belief such as µ=0.1. The paper's conclusion that optimal experiments become Blackwell-less-informative requires this inequality at every µ; a single reversal—for example with F uniform and F′ piecewise linear with F′(0.09)=0.15, F′(0.25)=0.25—would falsify the lemma. One could also directly compute the optimal experiment under each F and check whether the spread of the posterior distribution shrinks.","tokens_in":19803,"feed_emoji":"🔍","tokens_out":9218,"duration_ms":93404,"temperature":0.7,"pith_summary":"This paper studies a sender who commits to a public experiment before a mass audience decides whether to pay a private cost to verify the state. It tries to establish a reverse comparative static: when verification becomes cheaper in the population, the sender's optimal public signal becomes strictly less informative, because extreme claims invite more scrutiny. The mechanism is a constrained concavification problem: folding receivers' endogenous verification into the sender's payoff makes the indirect value more concave in the public belief, so the optimal experiment coarsens. With an ex-post truthfulness constraint on verifiable evidence, cheaper verification also shifts the sender toward ex-post manipulation—upward falsification first, then fixed-cost repression—once thresholds are crossed. If true, the result connects transparency, message precision, and repression in one framework.","feed_headline":"Cheaper fact-checking makes public signals noisier","feed_subtitle":"A sender facing well-verified audiences waters down messages, then shifts to manipulation and repression.","key_machinery":"The carrying object is the sender's indirect payoff at a public posterior, v(µ;F) = −(b² + (1−F(µ(1−µ)))²µ(1−µ)), where F(µ(1−µ)) is the mass of receivers who verify because the private benefit of verification equals the posterior variance. The argument works by showing that a first-order stochastic improvement in F makes v(·;F) pointwise more concave, then applying Bayesian-persuasion concavification: the optimal experiment is a Bayes-plausible distribution over posteriors maximizing E[v(µ;F)], and more concavity pulls the supporting chord inward. The ex-post truthfulness constraint enters through the indifference condition (1−F(µ_s(1−µ_s)))µ_s = 2b, which determines the only interior poste","core_discovery":"The paper's central claim is that in a public-signal persuasion game where receivers can verify at heterogeneous costs and the sender must be truthful about verifiable claims, a fall in verification costs makes the sender's indirect payoff more concave in the public posterior and thereby makes every optimal public experiment less informative in the Blackwell order. The authors derive this through a concavification argument: the sender maximizes the expected value of v(µ;F) over Bayes-plausible posterior distributions, subject to an ex-post implementability constraint that pins down the silence posterior by (1−F(µ_s(1−µ_s)))µ_s = 2b. Under a protocol with hard evidence plus a minimal soft lab","pith_inferences":["The same concavification logic implies that the coarsening response should be strongest for receivers whose beliefs sit near 1/2, where the verification benefit µ(1−µ) peaks; this is a testable cross-sectional prediction the paper does not spell out.","When falsification is unconstrained, the persuasion margin is predicted to be neutral—only manipulation scales—so observed message coarsening alongside a verification-cost shock can be used to detect whether falsification capacity binds.","The threshold structure of repression implies a temporal ordering—noisier messages and rising manipulation before a discrete onset of violence—which event studies of censorship breaks or fact-checking rollouts could trace.","A dynamic extension of the model would predict that sustained declines in verification costs produce gradual coarsening and falsification punctuated by repression spikes once accumulated gaps clear the fixed cost."],"forward_implications":["Cheaper fact-checking makes optimal public communication coarser: measured by the Blackwell order, the public experiment becomes less informative as the verification-cost distribution improves.","At the optimal message, a larger share of receivers chooses to verify, but the sender's response is to make bold claims rare, so realized verification can fall even as the underlying cost falls.","Under upward-only or capacity-limited falsification, cheaper verification increases the sender's ex-ante use of falsification in the unfavorable state while decreasing it in the favorable state.","There is a fixed-cost threshold for repression: cheaper verification expands the set of posteriors and states in which violence is worth buying.","A policy that lowers verification costs should therefore expect noisier messages, more manipulation, and—at thresholds—discrete repression, rather than a simple increase in transparency."],"supporting_citations":[{"why":"Supplies the concavification characterization of optimal experiments and Bayes-plausible posterior laws that the paper adapts.","marker":"Kamenica and Gentzkow (2011)"},{"why":"Defines the single-receiver costly-information benchmark whose deterrence result the paper's mass-audience coarsening claim reverses.","marker":"Matysková and Montes (2023)"},{"why":"Models costly state verification in persuasion, the basis for the cutoff rule λ(µ;F)=F(µ(1−µ)).","marker":"Yang (2024)"},{"why":"Shows how auxiliary feasibility constraints can be embedded in design problems, motivating the EPIC-filtered concavification.","marker":"Doval and Skreta (2022)"},{"why":"Provides the hard-evidence unraveling logic that underwrites the ex-post truthfulness constraint.","marker":"Grossman (1981)"},{"why":"Establishes the good-news/bad-news representation that underpins the verifiable-disclosure discipline.","marker":"Milgrom (1981)"},{"why":"Extends hard-evidence logic to mechanism design with commitment and robustness, informing the paper's implementability restriction.","marker":"Ben-Porath et al. (2019)"}],"fun_headline_variants":["Cheaper fact-checking blurs public signals","Low verification costs muddy public messages","Cheap private checks make public info vaguer","Verification cost drop lowers signal clarity","Cheaper truth-seeking, noisier public claims"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that a first-order improvement in the verification-cost distribution makes the sender's payoff more concave at every interior belief—and that an interior experiment can beat full revelation, since equation (3.5) alone would make the endpoints optimal.","fun_headline_variants_meta":{"raw":{"variants":["Cheaper fact-checking blurs public signals","Low verification costs muddy public messages","Cheap private checks make public info vaguer","Verification cost drop lowers signal clarity","Cheaper truth-seeking, noisier public claims"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000185,"raw_usage":{"total_tokens":1134,"prompt_tokens":694,"completion_tokens":440,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":373}},"tokens_in":438,"tokens_out":440,"duration_ms":5520,"temperature":1.0,"reasoning_tokens":373,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:36:59.874493+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the paper's key concavity lemma by choosing a distribution F′ that first-order stochastically dominates F and evaluating v(µ;F′)−v(1/2;F′) against v(µ;F)−v(1/2;F) at an interior belief such as µ=0.1. The paper's conclusion that optimal experiments become Blackwell-less-informative requires this inequality at every µ; a single reversal—for example with F uniform and F′ piecewise linear with F′(0.09)=0.15, F′(0.25)=0.25—would falsify the lemma. One could also directly compute the optimal experiment under each F and check whether the spread of the posterior distribution shrinks.","supporting_citations":[{"cited_title":"and Gentzkow, M","cited_arxiv_id":null,"evidence_quote":"Supplies the concavification characterization of optimal experiments and Bayes-plausible posterior laws that the paper adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Models costly state verification in persuasion, the basis for the cutoff rule λ(µ;F)=F(µ(1−µ))."},{"cited_title":"and Skreta, V","cited_arxiv_id":null,"evidence_quote":"Shows how auxiliary feasibility constraints can be embedded in design problems, motivating the EPIC-filtered concavification."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the hard-evidence unraveling logic that underwrites the ex-post truthfulness constraint."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the good-news/bad-news representation that underpins the verifiable-disclosure discipline."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends hard-evidence logic to mechanism design with commitment and robustness, informing the paper's implementability restriction."}],"review_version":1}