{"id":"8584f5b8-9b72-40fc-921b-212d9182158e","arxiv_id":"2505.16141","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A data-driven sender can learn a near-optimal decision-calibrated predictor without knowing the prior, but the proof as written has a critical Lagrangian error and the Bayesian benchmark is restricted by construction.","lead":"A machine-learned prediction can steer people's choices even when no one knows the true probabilities. The authors propose an algorithm for learning such persuasive predictions from data, but the proof contains a sign error in its central optimization step.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (3)'s negative-sign Lagrangian term s(E−γ) makes the dual unbounded at any E<γ (including perfectly calibrated E=0), so the minimax game does not enforce the calibration constraint; Lemma 3.2 and Theorem 3.1 fail as written, though a sign correction may repair the proof.","rationale":"The reader's weakest assumption is exactly where the paper breaks. Eq. (3) writes the Lagrangian penalty as λ_+(E−γ)+λ_−(−E+γ), which is not the Lagrangian of the constraint |E|≤γ. Because the s=− term gives positive weight to any E<γ, the dual is unbounded for all feasible interior predictors, and the bounded game adds a spurious penalty Cγ at E=0. This invalidates Lemma 3.2 and hence Theorem 3.1, the central algorithmic claim; Theorems 4.1 and 5.1 inherit the failure. I do not see a comparably decisive second issue: the Hedge/BRNR framework and uniform-convergence bounds are standard, and the quantal-response extension would plausibly follow once the Lagrangian is corrected. The Bayesian benchmark is indeed weaker than the abstract suggests because it is restricted to Π_H, but that is a framing overstatement rather than the main correctness break. The flaw appears to be a fixable sign error, but as submitted the paper's core guarantee is unproven; the reader's REJECT verdict should stand.","tokens_in":25967,"tokens_out":8394,"duration_ms":72506,"concrete_test":"Evaluate the bounded Lagrangian Eq. (4) at a perfectly calibrated predictor with E=0, for γ>0 and C>0: max_λ∈Λ L_D(f,λ) = −U(f)+Cγ, achieved at λ_−=C. Since the correct dual for a feasible f should equal −U(f), this value shows the minimax reduction is not the Lagrangian of Eq. (2). Then replace s(E−γ) by λ_+(E−γ)+λ_−(−E−γ) and re-derive Lemma 3.2; if the bound DecCE(f)≤γ+(1+2ε)/C now follows, the flaw is a fixable sign typo, while if it still fails, the minimax approach itself is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (3) defines the calibration penalty as Σ_{s∈{+,-}} λ_{s,i,j,a} s(E_{i,j,a}−γ), where E_{i,j,a} := E_{h∼f}E_D[(h_j−y_j)b_i(h(x),a)]. For s=−, the summand is λ_−(−E+γ). For any predictor with E<γ, including a perfectly calibrated predictor with E=0 and γ>0, this term is positive, so in Eq. (3) the unbounded max over λ_− drives L_D(f,λ) to +∞. The only way to avoid +∞ is to have E=γ exactly, not to satisfy |E|≤γ as required by DecCE(f)≤γ. The bounded game Eq. (4) inherits the problem: at E=0, max_λ∈Λ sets λ_−=C and adds Cγ to the objective, so an approximate equilibrium need not be an approximately feasible predictor. Lemma 3.2's proof also selects the violating sign by maximizing s(E−γ), so it classifies every E<γ as a violation. Since Lemma 3.2 is the bridge to Theorem 3.1, and Theorems 4.1 and 5.1 depend on it, the central guarantee is not established as written. The intended constraint |E|≤γ corresponds to λ_+(E−γ)+λ_−(−E−γ); replacing s(E−γ) with that expression would restore the standard Lagrangian relaxation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes 'persuasive prediction,' a data-driven variant of Bayesian persuasion in which a sender learns from samples a decision-calibrated predictor to influence a receiver. The main algorithmic claim (Theorem 3.1) is that for a finite hypothesis class H, PerDecCal, based on a Lagrangian minimax reformulation with an ERM oracle and Hedge, achieves near-optimal sender utility among γ-decision-calibrated predictors with sample complexity independent of the feature-space size |X|. Section 4 claims this matches a Bayesian persuasion benchmark in the single-receiver case, and Section 5 extends the approach to quantal responses and infinite hypothesis classes via covering numbers. The paper also contains no-regret guarantees for receivers and a finite-sample uniform-convergence analysis.","tokens_in":26212,"tokens_out":11077,"duration_ms":91950,"significance":"If established, the results would offer a novel connection between decision calibration and Bayesian persuasion, with a plausible oracle-efficient algorithm and finite-sample guarantees that do not depend on |X|. The manuscript contains substantial technical components: a minimax reformulation, uniform-convergence bounds, a Lipschitz covering-number analysis for smoothed responses, and regret guarantees for receivers. However, the central proof contains a sign error in the Lagrangian that invalidates Theorem 3.1 as written, and the 'Bayesian benchmark' in Section 4 is defined to coincide with the decision-calibrated optimum, so the headline claim of matching a fully informed Bayesian sender is not supported. These are load-bearing issues, so the contribution as stated is not established.","major_comments":[{"comment":"The Lagrangian term Σ_{s∈{+,-}} λ_{s,i,j,a_i} s(E_{i,j,a_i} - γ) is incorrect for the constraint |E_{i,j,a_i}| ≤ γ. For s = -1, the summand is λ_-(-E + γ), which is positive for every E < γ, including the perfectly calibrated value E = 0. Hence max_{λ ∈ R_+^{2Nmd}} L_D(f, λ) is infinite even for feasible predictors, and the minimax game (3)-(4) does not enforce the decision-calibration constraint. The correct relaxation is λ_+(E - γ) + λ_-(-E - γ).","section":"Eq. (3)"},{"comment":"The proof of Lemma 3.2 relies on the same erroneous sign. In the first case it asserts -E[u] = max_λ L(f̂, λ) for a feasible f̂, which is false because max_λ L(f̂, λ) is infinite whenever E < γ. In the second case, the argmax over s of s(E - γ) selects s = - with value γ - E for any feasible E ∈ (-γ, γ), so the proof treats every feasible predictor as a violation and derives an artificial bound C(γ - E) ≤ 1 + 2ε. Consequently the DecCE bound in Lemma 3.2, and hence Theorem 3.1, is not established as written.","section":"Lemma 3.2"},{"comment":"The update in line 6, c_t(λ_{s,i,j,a_i}) = λ_{s,i,j,a_i} s(E - γ), together with the ERM loss ℓ_λ in Definition 3.1, makes PerDecCal optimize a different objective from Eq. (2): the s = - coordinate receives positive cost whenever E < γ, so the Hedge weights push the predictor away from the feasible region rather than enforce |E| ≤ γ. A correction to the Lagrangian must be accompanied by a corresponding change to the algorithm's cost and loss functions; the algorithm as printed does not solve the stated constrained problem.","section":"Algorithm 1"},{"comment":"The 'Bayesian benchmark' Π_H is defined as the set of signaling schemes induced by the very class FDCAL(H) of perfectly decision-calibrated predictors over H. Therefore BayesOPT(μ_D, Π_H) = OPT(H, D, 0) by construction, and Theorem 4.1 is a corollary of Theorem 3.1 that renames the same optimization problem. The abstract's claim that the method 'matches the utility of a Bayesian sender who has full knowledge of the underlying prior distribution' is not supported: the comparison is only against the restricted class of schemes induced by H, not against the full Bayesian persuasion value.","section":"Definition 4.2 / Theorem 4.1"},{"comment":"Definition 2.1 defines the strict best response as arg min of the receiver's utility v_i(a'_i, h(x)), but the no-regret proof of Theorem 2.1 and the informal Eq. (1) require the receiver to maximize utility; with the printed arg min the inequality in the proof of Theorem 2.1 has the wrong direction. Example 5.1 follows the arg-min convention, so the paper is internally inconsistent about the direction of receiver preferences. The definition should be corrected to arg max (or the utility convention should be renamed to a cost).","section":"Definition 2.1"}],"minor_comments":[{"comment":"The proof chooses C = 2/ε without stating this in Theorem 3.1 or Algorithm 1; the theorem should specify the dual bound C as an input, or the guarantee should be stated for this choice.","section":"Theorem 3.1 proof"},{"comment":"The bound contains a term sqrt(ln|4H|/δ / 2n) with an unexplained factor of 4; the proof says the budget is split to δ/2, but the displayed expression should be derived explicitly.","section":"Lemma 3.3"},{"comment":"The swap regret bound is printed as ln m + 1 / η while Theorem C.3 gives (ln|A| + 1) / η; the notation should be reconciled.","section":"Theorem 5.1"},{"comment":"The last sentence refers to 'receivers who play η-quantal response' and cites Theorem 2.1, but Theorem 2.1 is for strict best responses; the reference should be to Theorem C.3.","section":"Appendix D, proof of Theorem 3.1"},{"comment":"The proof should explicitly justify the equality of events {f'(x) = f'_a} and {b(h(x), a) = 1}; this is intuitive because f' takes the value corresponding to the chosen action, but it is not stated.","section":"Lemma 4.2"}],"recommendation":"reject","confidential_remarks":"The sign error in the Lagrangian is pervasive: it appears in Eqs. (3)-(4), Lemma 3.2, Definition 3.1, and Algorithm 1, so the main optimization result is not sound as submitted. The Bayesian-benchmark claim in Section 4 is definitional rather than substantive. These issues could potentially be repaired, but they require substantial reworking of the proofs and algorithm, and the overclaim in the abstract would need to be removed. Given the current state, I do not see a path to acceptance without major changes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is worth taking seriously: a prior-free, data-driven sender who learns a decision-calibrated predictor, with sample complexity independent of the feature space and an oracle-efficient minimax algorithm, is a genuinely useful extension of the persuasive-calibration program. The quantal-response extension and the covering-number analysis for infinite hypothesis classes are also solid contributions. The paper does real work and engages honestly with the prior literature. I would not desk-reject this.\n\nThe problem is that the central theorem is not established as printed. The stress-test note is right: Eq. (3) writes the calibration penalty as s(E − γ), and for s = −1 that is −E + γ, which is positive for any E < γ, including a perfectly calibrated predictor with E = 0. The max over λ then drives the Lagrangian to +∞, so the bounded game in Eq. (4) does not enforce |E| ≤ γ. Lemma 3.2's proof selects the violating sign by maximizing s(E − γ), which classifies every E < γ as a violation. Since Lemma 3.2 is the bridge to Theorem 3.1, and Theorems 4.1 and 5.1 depend on it, the main guarantee fails as written. The fix looks straightforward — replace s(E − γ) with the standard λ₊(E − γ) + λ₋(−E − γ) — and I suspect the rest of the analysis would go through. But as submitted, the paper does not support its headline claim.\n\nThere is also a smaller issue the reader flagged correctly: the abstract's claim that the method 'matches the utility of a Bayesian sender who has full knowledge of the prior' overstates what Theorem 4.1 proves. The benchmark is restricted to signaling schemes induced by the same predictor class H, so the equality of optima holds substantially by construction. That is a framing flaw, not a technical one, but it matters for how the result is advertised.\n\nThe Hedge description in Algorithm 1 is ambiguous and the proof of Lemma 4.2 is skimpier than I would like, but those are minor relative to the sign error.\n\nWho gets value from this? Researchers working on calibration-based persuasion, omniprediction, and prior-free information design. The conceptual bridge between decision calibration and Bayesian persuasion is worth their attention even before the proof is fixed. The paper should go to peer review, with the sign error and the benchmark framing flagged for the authors. If the sign correction works and the authors tone down the abstract, this could be a useful contribution.","headline":"The paper has a real idea and a repairable-looking sign error, but as written the central guarantee does not go through.","tokens_in":26818,"tokens_out":1350,"would_cite":false,"duration_ms":13694,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A finite-data sender can learn a decision-calibrated predictor that matches the utility of a Bayesian sender who knows the prior.","keywords":["Bayesian persuasion","decision calibration","persuasive prediction","prior-free persuasion","minimax equilibrium","no-regret learning","swap regret","quantal response"],"falsifier":"Directly substitute a perfectly calibrated constant predictor, say $h(x) = \\mathbb{E}[Y]$, into Eq. (3) with $\\gamma > 0$ and a Lagrange multiplier placed on a negative-sign coordinate; the calibration term is $\\lambda\\gamma > 0$, so the Lagrangian value exceeds the negative sender utility, contradicting the claimed equivalence with the constrained problem and the proof of Lemma 3.2.","tokens_in":25674,"feed_emoji":"🎯","tokens_out":11643,"duration_ms":92049,"temperature":0.7,"pith_summary":"Bayesian persuasion normally presupposes that sender and receiver share an exact prior, which is unlearnable in high-dimensional or infinite state spaces. The paper introduces persuasive prediction, where the sender learns a predictor from past data and receivers best respond to it, and adopts decision calibration as the behavioral link: a prediction is credible when it is unbiased conditioned on the receiver's best response. The central result is that with a sample count growing polynomially in the number of receiver actions and outcome dimensions, but independent of the size of the context space, the PerDecCal algorithm finds a randomized predictor that is nearly decision-calibrated and nearly maximizes sender utility among all such predictors. In the single-receiver case this matches the utility of a fully informed Bayesian sender restricted to the same predictor class.","feed_headline":"One predictor can persuade as well as a Bayesian who knows the prior","feed_subtitle":"Decision calibration lets a sender with finite samples reach near-optimal utility, whatever the context space.","key_machinery":"The load-bearing object is decision calibration: a randomized predictor $f \\in \\Delta(H)$ is $\\epsilon$-decision-calibrated when, for every receiver $i$, outcome coordinate $j$, and action $a$, the expectation $|\\mathbb{E}[(y_j - h(x)_j) \\cdot b_i(h(x),a)]|$ is at most $\\epsilon$, where $b_i$ is the receiver's strict best response. The argument converts the sender's constrained optimization problem into a zero-sum game via Lagrange multipliers: the min player chooses $f$, the max player chooses bounded dual variables $\\lambda$, and the payoff is the negative sender utility plus calibration-violation penalties. PerDecCal solves this game by alternating an ERM oracle (best response for the min player) with the Hedge algorithm (no-regret for the max player), and uniform-convergence bounds transfer the empirical equilibrium to the true distribution. The bridge to Bayesian persuasion is a pair of lemmas: calibrated predictors correspond exactly to signaling schemes over posterior means, and any perfectly decision-calibrated predictor can be post-processed into a perfectly calibrated predictor with the same sender utility, which is what makes the Bayesian-benchmark comparison possible.","core_discovery":"The paper's central claim is that decision calibration is enough to make persuasion prior-free. Formally, for a finite predictor class $H$, PerDecCal returns a predictor $\\hat{f}$ with $\\mathrm{DecCE}(\\hat{f}) \\le \\gamma + \\epsilon$ and $\\mathbb{E}[u(a,y)b(h(x),a)] \\ge \\mathrm{OPT}(H,D,\\gamma) - \\epsilon$, with probability $1-\\delta$, using $n = O(\\log(|H|Ndm/\\delta)/\\epsilon^4)$ samples; the bound does not depend on $|X|$. The proof routes through a Lagrangian reformulation of the constrained optimization problem as a zero-sum game between a predictor and a dual player, and shows that best-response-vs-no-regret dynamics converge to an approximate equilibrium. The authors further show that in the single-receiver case the attained utility is at least $\\mathrm{BayesOPT}(\\mu_D, \\Pi_H) - \\epsilon$, so the data-driven sender matches a Bayesian sender with full knowledge of $D$ who is confined to signaling schemes induced by $H$. For infinite $H$ and quantal-responding receivers, an analogous algorithm achieves the same type of guarantee with sample complexity governed by a covering number.","pith_inferences":["Beyond the paper: the same Lagrangian-minimax template could turn any constraint expressible as a family of linear inequalities on a predictor, such as multi-calibration or fairness constraints, into a finite-sample optimization problem solved by no-regret dynamics.","Beyond the paper: the benchmark-matching result suggests decision calibration is not merely a proxy but an operational realization of the common-prior assumption, so the choice of calibration notion determines which rational receiver behavior is being assumed in prior-free persuasion.","Beyond the paper: replacing the exact ERM oracle with an approximate oracle, such as SGD-trained neural networks, and measuring the gap in utility and calibration would provide a direct empirical test of how much the oracle-efficiency guarantee degrades in practice.","Beyond the paper: varying the inverse temperature $\\eta$ in the quantal-response extension interpolates between strict no-regret behavior and the Bayesian best-response limit, offering a way to experimentally separate decision-calibrated persuasion from classical prior-based persuasion."],"forward_implications":["With $O(\\log(|H|Ndm/\\delta)/\\epsilon^4)$ samples the sender can produce a $\\gamma+\\epsilon$-decision-calibrated predictor that is $\\epsilon$-optimal in sender utility, no matter how large $|X|$ is.","Receivers who strict best respond to the output predictor incur at most $2mL(\\gamma+\\epsilon)$ swap regret, so the behavioral model is internally consistent.","In the single-receiver case the data-driven sender's utility matches the Bayesian-persuasion benchmark restricted to signaling schemes induced by $H$, even though the prior was never estimated.","When receivers use $\\eta$-quantal responses, the same near-optimal utility and smoothed-calibration guarantees hold for infinite hypothesis classes with bounded covering numbers.","The algorithm is oracle-efficient: it needs only one ERM call per iteration, so any class with a practical ERM oracle is computationally tractable."],"supporting_citations":[{"why":"Defines Bayesian persuasion and the commitment/signaling model the paper tries to replicate without a prior.","marker":"Kamenica and Gentzkow [2011]"},{"why":"Introduces decision calibration and its no-regret-by-type guarantee, which the paper adopts as the receiver-behavior model.","marker":"Zhao et al. [2021]"},{"why":"Provides the best-response-vs-no-regret convergence lemma used to solve the minimax game.","marker":"Freund and Schapire [1996]"},{"why":"Supplies the Hedge algorithm used for the dual player's no-regret updates.","marker":"Freund and Schapire [1997]"},{"why":"Supplies the folklore Lagrangian equivalence between the constrained problem and the minimax game.","marker":"Boyd and Vandenberghe [2004]"},{"why":"Establishes no-swap-regret properties of decision-calibrated forecasts in high dimensions, grounding the receiver-regret claim.","marker":"Noarov et al. [2023]"},{"why":"The closest prior work on selecting calibrated predictors for sender utility, which the paper extends to unknown priors and rich context spaces.","marker":"Feng and Tang [2025]"},{"why":"Connects calibrated forecasting to Bayesian persuasion, the bridge used in the Bayesian-benchmark comparison.","marker":"Jain and Perchet [2024]"},{"why":"Gives the quantal-response regret bound used in the infinite-hypothesis extension.","marker":"Roth and Shi [2024]"},{"why":"Introduces quantal response equilibrium, the stochastic receiver model used in Section 5.","marker":"McKelvey and Palfrey [1995]"}],"fun_headline_variants":["Decision calibration makes persuasion prior-free","No prior needed: calibrated predictors match Bayesian persuasion","Calibrated predictions reach Bayesian-level persuasion from samples","Persuasive prediction: calibration replaces knowledge of the prior","Prior-free persuasion via decision calibration"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Lagrangian in Eq. (3) correctly encodes the calibration constraint, but as printed the sign convention gives a perfectly calibrated predictor a positive dual penalty, so the central guarantee is not established unless that expression is a typo.","fun_headline_variants_meta":{"raw":{"variants":["Decision calibration makes persuasion prior-free","No prior needed: calibrated predictors match Bayesian persuasion","Calibrated predictions reach Bayesian-level persuasion from samples","Persuasive prediction: calibration replaces knowledge of the prior","Prior-free persuasion via decision calibration"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1487,"prompt_tokens":1081,"completion_tokens":406,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":697,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":697,"tokens_out":406,"duration_ms":3968,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:06:56.158069+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly substitute a perfectly calibrated constant predictor, say $h(x) = \\mathbb{E}[Y]$, into Eq. (3) with $\\gamma > 0$ and a Lagrange multiplier placed on a negative-sign coordinate; the calibration term is $\\lambda\\gamma > 0$, so the Lagrangian value exceeds the negative sender utility, contradicting the claimed equivalence with the constrained problem and the proof of Lemma 3.2.","supporting_citations":[{"cited_title":"Calibrating predictions to decisions: A novel approach to multi-class calibration","cited_arxiv_id":null,"evidence_quote":"Introduces decision calibration and its no-regret-by-type guarantee, which the paper adopts as the receiver-behavior model."},{"cited_title":"Game theory, on-line prediction and boosting","cited_arxiv_id":null,"evidence_quote":"Provides the best-response-vs-no-regret convergence lemma used to solve the minimax game."},{"cited_title":"A decision-theoretic generalization of on-line learning and an application to boosting","cited_arxiv_id":null,"evidence_quote":"Supplies the Hedge algorithm used for the dual player's no-regret updates."}],"review_version":1}