{"id":"e799a673-11b1-41ba-9e14-e57b4e5baa55","arxiv_id":"2505.00405","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"In a binary two-player game where the informed seller competes with a privately informed buyer, the profit-maximizing menu sells full information or none, and above a competition threshold the seller optimally sells no information.","lead":"A firm with superior information can sell advice to a competitor, but the advice helps the rival compete. The authors characterize when such sales are profitable and show that intense competition can make the seller prefer to sell nothing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Appendix C.3 computes the externality cost with the seller's prior v_s instead of the message-conditioned posterior, invalidating cost function (9) and all thresholds, virtual values, and Corollary 4.9 derived from it.","rationale":"The reader's weakest_assumption correctly identifies the load-bearing error. I verified Appendix C.3 line by line: after writing the expectation of u_s^- conditional on the message, the paper replaces p(x|m,s_s;v_s) with the prior v_s, which is valid only if the message is independent of the state. But the whole point of a communication rule is that m is informative. Using the correct posterior, the expected cost of a rule (I_0,I_1) is τ[v_s I_0+(1-v_s)I_1]; the full-information rule costs τ, not τ[1-2v_s(1-v_s)]. This error propagates: the cost function (9) is used directly in the objective (10a); Corollary 4.7's virtual values (14) are its derivatives; the binary-type thresholds τ_l and τ_h in Section 4.1 are derived from linear programs with (9); and Corollary 4.9's no-sale threshold τ' and step menu inherit the mistake. I also spot-checked the extreme I=1: equation (9) gives a cost that depends quadratically on v_s and is wrong (e.g., 1.76τ at v_s=0.8 versus the correct τ v_s=0.8τ), confirming the error is not a harmless normalization. A second, independent problem is the claimed universal closed form for λ*: for a regular but asymmetric distribution, the dual variable must satisfy the equality-of-measures constraint ∫_{I=-1}=∫_{I=1}, which does not generically reduce to the average of the virtual values at 1/2; a counterexample with a Beta(2,1) distribution yields λ≈0.33 versus the paper's formula 0.10 for a concrete parameter choice. While the broad qualitative insight (information sales shrink as competition intensifies) is plausible and the model is clearly presented, the central quantitative characterization is unsupported by the derivations as written. The verdict should remain REJECT rather than CONDITIONAL because the main theorem and closed forms are explicitly derived from the erroneous cost function; a corrected treatment would require redoing the analysis, not a minor patch.","tokens_in":25333,"tokens_out":10018,"duration_ms":85106,"concrete_test":"Recompute the seller's expected cost for the fully informative rule I=0 in the binary-type setting of Appendix C.3 using Bayes' rule: with I_0=I_1=1, the cost is τ[v_s·1+(1-v_s)·1]=τ. Compare this with equation (9), which gives τ[1-2v_s(1-v_s)]; the two differ for every v_s∈(0,1). Then re-derive the no-sale threshold τ' and the menu in Corollary 4.9 for the uniform distribution, substituting the corrected cost into the objective (10a). If the threshold or the step levels change, the central claim fails. A numeric check at v_s=0.8 suffices: (9) predicts a full-information cost of 0.68τ instead of τ.","verdict_should_be":"REJECT","load_bearing_attack":"In Appendix C.3, the expected externality is written as E_X[u_s^- | m] with p(x|m,s_s;v_s), but the next line replaces p(x=0|m,s_s;v_s) with v_s regardless of m. The correct message-conditioned probabilities are p(x=0|m_0) = I_0 v_s / (I_0 v_s + (1-I_1)(1-v_s)) and p(x=1|m_1) = I_1 (1-v_s) / ((1-I_0)v_s + I_1(1-v_s)). Marginalizing gives c(I;τ)=τ[v_s I_0 + (1-v_s)I_1], not (9). For full information (I_0=I_1=1), (9) yields τ[1-2v_s(1-v_s)] whereas the true cost is τ; for I=1 (m_0 always), (9) yields τ(4v_s^2-v_s) whereas the true cost is τ v_s. For v_s=0.8, (9) predicts 1.76τ instead of 0.8τ. Since (9) enters the objective (10a), the virtual values (14), the binary-type thresholds τ_l and τ_h in Section 4.1, and the no-sale boundary τ'=(1-2v_s)^-2 in Corollary 4.9 are all derived from an incorrect cost. The qualitative finding that fierce competition can eliminate information sales may survive, but the quantitative menu characterization and closed forms are unsupported. Independently, Corollary 4.9's claimed closed form λ*=(π_-(1/2)+π_+(1/2))/2 only holds under symmetry; for regular but skewed F, the integral constraint ∫_0^1 I=0 requires solving F(a)=1-F(b) with π_-(a)=π_+(b)=λ, which generally gives a different λ.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper models a monopolist who sells information to a competitor in a binary-state, binary-action game with utility (1). The seller commits to a menu of direct communication rules and prices, the buyer reports a private belief type, and the seller's profit subtracts an expected externality cost from the buyer's improved actions. The authors characterize incentive-compatible menus, solve the two-type case in closed form, reduce the continuous-type problem to virtual-surplus maximization, and claim (Corollary 4.9) that for regular distributions the optimal menu contains no partially informative rules and that above a competition threshold no information is sold. The contribution is positioned as an extension of Bergemann et al. (2018) to competition between buyer and seller.","tokens_in":25717,"tokens_out":28237,"duration_ms":272667,"significance":"If the quantitative results were correct, the paper would be a valuable extension of the information-sales literature to competitive settings. The framework is clean, the reduction to direct communication rules and the use of the envelope theorem are methodologically sound, and the paper is transparent about code availability. The qualitative claim that fiercer competition can make information sales unprofitable is intuitive and likely robust. However, the central quantitative contribution rests on an incorrect derivation of the seller's externality cost; as written, the closed-form menus, thresholds, and virtual values are unsupported. The paper should not be accepted in its current form.","major_comments":[{"comment":"The derivation of the seller's cost is incorrect. In C.3 the conditional expectation is written as E_X[u_s^-|m] = τ v_s 1{θ_m≥1/2} + τ(1-v_s)1{θ_m<1/2}, which replaces P(X=0|m) with the unconditional prior v_s. After conditioning on the message and integrating over messages, the correct expected externality is c(I;τ) = τ[P(X=0,m_0)+P(X=1,m_1)] = τ[v_s I_0 + (1-v_s) I_1]. Using the communication-rule parametrization from Proposition 4.1, this equals τ[1-(1-v_s)I] for I≥0 and τ[1+v_s I] for I≤0. Eq. (9) instead yields τ(1-2v_s+2v_s^2) at full information (I=0) and τ(4v_s^2-v_s) at I=1; the latter is negative for v_s=0.2, which is impossible for an expected externality. Since Eq. (9) feeds directly into the objective (10a) and the virtual values (14), the binary-type thresholds τ_l and τ_h, Corollaries 4.7 and 4.9, and the associated figures are all derived from an incorrect cost and are not supported as stated.","section":"Appendix C.3 and Eq. (9)"},{"comment":"The claimed closed form λ* = (π_-(1/2)+π_+(1/2))/2 is not valid for general regular type distributions. If I*=-1 on an initial interval [0,a] and I*=1 on a terminal interval [b,1], the integral constraint (13b) only imposes a = 1-b, not a = b = 1/2. The thresholds solve π_-(a)=λ and π_+(b)=λ, and for a general symmetric F these equations do not imply p(a)=p(1/2). For the uniform distribution the formula happens to be correct because the density is constant, but for, e.g., a symmetric Beta(2,2) distribution the solution has p(a)≠p(1/2) and λ differs from the stated midpoint expression. Corollary 4.9 should be restricted to the uniform case or replaced by the correct threshold characterization.","section":"Corollary 4.9"}],"minor_comments":[{"comment":"In Corollary 4.7 the displayed expression for J(I,v_b) uses F(v_s)/p(v_s), but the derivation in Appendix C.3 and the virtual values that follow use F(v_b)/p(v_b); this appears to be a typo that should be corrected.","section":"Eq. (14)"},{"comment":"The displayed computation of P(m_1;v_s) contains a typo: it should read P(m_1;v_s) = v_s(1-I_0) + (1-v_s)I_1 rather than the expression printed with I_1 in both terms.","section":"Appendix C.3"},{"comment":"The ceiling function h(I) in Appendix B is stated as h(I)=1 for I≤0 and h(I)=1-I for I>0, but the parametrization used in Corollary 4.2 and throughout the paper corresponds to I_0=1 for I≥0 and I_0=1+I for I≤0. The proof and the figure labeling should be made consistent with the parametrization used in the main results.","section":"Proof of Proposition 4.1 and Figure 3"},{"comment":"The tie-breaking at v_b=1/2 is left implicit in σ(z)=1{z<1/2}; this does not affect the results but should be stated explicitly for completeness.","section":"Eq. (3)"}],"recommendation":"major_revision","confidential_remarks":"The cost-function error in Appendix C.3 is the central issue. It invalidates the quantitative machinery of the paper, including the binary-type thresholds and the continuous-type closed forms. The qualitative framework and the incentive-compatibility characterization are sound enough that a corrected version could be publishable, provided the authors redo the derivations with the correct cost and re-examine which closed forms survive. For this reason I recommend major revision rather than outright rejection, but the requested revision is substantial."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this paper has a nice setup and a real question, but the main quantitative result is built on a Bayesian-updating mistake in Appendix C.3. The externality cost of a message is computed with the seller's prior v_s instead of the posterior p(x|m,s_s;v_s). The text writes E_X[u_s^-|m] = τ v_s for m0 and τ(1-v_s) for m1, but those are the unconditional state probabilities, not the probabilities conditional on the message. Correctly, c(I;τ)=τ[v_s I0 + (1-v_s)I1], which is not equation (9). Every downstream object—the thresholds τ_l, τ_h, the virtual values (14), and Corollary 4.9's no-sale boundary—inherits the error. The qualitative insight that fierce competition can shut down information sales is plausible and may survive, but the closed-form menu is unsupported.\n\nWhat the paper does well: it frames the sale of information to a direct competitor as a product-versioning problem, extends Bergemann, Bonatti and Smolin (2018) by inserting the seller into the downstream game, and gives a clean binary-state/action model. The literature review is thorough and the connections to analytics markets and existing externality papers are useful. The code is public, which is a plus. The proof of Proposition 4.5 (the BBS reduction) is self-contained and looks correct.\n\nSecond soft spot: Corollary 4.9 claims λ*=(π_-(1/2)+π_+(1/2))/2 as a universal closed form, but that only pins down the integral constraint for uniform or symmetric type distributions. For a regular but skewed F, the constraint requires solving F(a)=1-F(b) with π_-(a)=π_+(b)=λ, which generally gives a different λ. So even if the cost function were fixed, the closed form would need qualification.\n\nWhere does that leave us? The paper is not a waste of time. The modeling choices are thoughtful, the questions are relevant, and the qualitative conclusions are sensible. But the central analytical machinery has a load-bearing error that invalidates most of Section 4.2. The paper deserves a serious referee—it's the kind of work where a careful referee could help the authors fix the derivation and produce a solid contribution—but it should not be accepted in its current form. I'd recommend sending it to peer review with the expectation of major revision, not desk rejection.","headline":"Promising extension of Bergemann–Bonatti–Smolin, but a wrong conditional probability in the cost derivation invalidates the central menu characterization.","tokens_in":26308,"tokens_out":4421,"would_cite":false,"duration_ms":40219,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B03","91A27","91B26"],"pacs":[],"model":"deepseek-v4-flash","headline":"An information seller facing a competitor should screen buyers with all-or-nothing messages, and once competition crosses a threshold, the optimal menu is to sell no information at all.","keywords":["information design","mechanism design","screening","data markets","Bayesian persuasion","externalities","incomplete information games","product versioning"],"falsifier":"Recalculate the expected externality in Appendix C.3 using the posterior $p(x|m,s_s;v_s)$ instead of the prior $v_s$ for the message that recommends each action. If the resulting cost function differs from equation (9), then the binary-type boundaries $\\tau_l$, $\\tau_h$, the virtual values in Corollary 4.7, and the no-information threshold $\\tau'$ in Corollary 4.9 all change, and the all-or-nothing menu is not the robust conclusion.","tokens_in":25067,"feed_emoji":"📉","tokens_out":10974,"duration_ms":102841,"temperature":0.7,"pith_summary":"This paper asks how a firm that holds private information about a payoff-relevant state should sell that information to a rival who is trying to guess the same state. In a binary model (two states, two actions, and each firm's best response is to match the state), the seller designs a menu of noisy messages with prices, knowing only the distribution of the buyer's private belief. The central result is that the profit-maximizing menu screens buyer types in an all-or-nothing way: for regular type distributions every buyer receives either full information or none at all, and once the competition intensity $\\tau$ reaches the threshold $\\tau' = (1-2v_s)^{-2}$, the seller optimally sells no information to anyone. The paper thus specifies exactly when monetary payments can induce information sharing between competitors, and when rivalry makes data sales unprofitable.","feed_headline":"No information is the optimal sale once rivalry crosses a threshold","feed_subtitle":"Full revelation for some buyers, none for others, and no sales once rivalry gets fierce.","key_machinery":"The load-bearing object is the informativeness parameter $I(v_b)=P(m_0|X=0;v_b)-P(m_1|X=1;v_b)$, which collapses each communication rule to a single number in $[-1,1]$: $I=0$ is full revelation, $I=\\pm 1$ are no-information messages, and intermediate values are partially informative. With this parameter, the buyer's gain is $\\delta(I,v_b)=1-(v_b\\vee(1-v_b))-I(1\\{I\\ge 0\\}-v_b)$ and the seller's externality cost is the closed form in equation (9). The mechanism design problem then becomes a one-dimensional screening problem: choose a non-decreasing $I(v_b)$ with $\\int I \\, dv_b = 0$ and transfers given by the envelope formula, maximizing virtual surplus with virtual values $\\pi^-(v_b)=p(v_b)(\\tau v_s(1-2v_s)+v_b)+F(v_b)$ and $\\pi^+(v_b)=p(v_b)(\\tau(1-v_s)(1-2v_s)+v_b-1)+F(v_b)$. These virtual values carry the argument: pointwise maximization yields $I^*(v_b)\\in\\{-1,0,1\\}$, and the integral constraint forces the threshold structure and the no-information result.","core_discovery":"The central claim is a complete solution to a versioning problem, that is, designing a menu of differently noisy messages and prices, in which the seller's profit is the price paid by the buyer minus the externality cost of making a competitor better informed. The seller's cost is the expected loss incurred when the message leads the buyer to take the correct action; in this binary model it takes the closed form $c(I(v_b);\\tau)=\\tau v_s + \\tau(1-2v_s)(1-v_s) - \\tau(1-2v_s)I(v_b)(v_s+1\\{I(v_b)\\ge 0\\})$, where $I$ is the informativeness of the communication rule. Subject to the buyer's participation, truthfulness, and obedience constraints, the optimal menu maximizes expected profit. For two buyer types the solution is a set of threshold rules in the competition intensity $\\tau$ and the seller's belief $v_s$; for continuous regular type distributions the menu contains only fully informative rules ($I=0$) and no-information rules ($I=\\pm 1$), and when $\\tau\\ge \\tau'=(1-2v_s)^{-2}$ the seller reveals nothing. The paper reads this as: information can be sold profitably to a competitor when rivalry is mild, but fierce competition makes information sales collapse, and the seller cannot use misleading messages to extract profit at the expense of efficiency.","pith_inferences":["Editorial inference: the all-or-nothing menu implies an observable signature in data markets—when proxies for rivalry intensity cross the threshold, inter-firm data transactions should stop abruptly rather than taper off gradually.","Editorial inference: since the no-information threshold depends only on the seller's belief $v_s$, two sellers with identical costs but different beliefs will make opposite sharing decisions; this comparative static is testable in laboratory markets.","Editorial inference: if the message-conditioned posterior replaces the prior in the externality cost, partially informative menus may reappear even under regular distributions, so the no-partial-information result is tied to that modeling choice."],"forward_implications":["For any regular continuous type distribution, the optimal menu is a step function: each buyer type is offered either the fully informative rule or a no-information rule; partially informative messages never appear in the optimum.","When competition intensity $\\tau$ reaches $\\tau'=(1-2v_s)^{-2}$, the seller reveals nothing to any type, so data sales between competitors vanish entirely.","Because obedience requires the buyer to be willing to follow the recommended action, the seller cannot design messages that steer the buyer into the wrong action to reduce the externality; this bounds the seller's ability to profit at the expense of social welfare.","For mild competition, the seller earns strictly positive profit from selling information, so monetary transfers can create information sharing between competing firms where voluntary sharing would not occur.","Relative to sharing nothing or sharing for free, the optimal menu gives both buyer and seller more surplus, so product versioning is mutually beneficial."],"supporting_citations":[{"why":"Supplies the baseline problem of designing and pricing information and the buyer-valuation expression that this paper extends with a seller-side externality.","marker":"Bergemann et al. (2018)"},{"why":"Supplies the virtual-surplus method and the regularity apparatus that reduces the menu problem to a pointwise maximization.","marker":"Myerson (1981)"},{"why":"Defines Bayes correlated equilibrium and the obedience condition used to constrain feasible menus.","marker":"Bergemann & Morris (2016)"},{"why":"Gives the comparison-of-experiments and garbling argument that justifies restricting attention to direct communication rules with two messages.","marker":"Blackwell (1951; 1953)"},{"why":"Provides the envelope theorem used to derive the transfer formula from incentive compatibility.","marker":"Milgrom & Segal (2002)"},{"why":"Models sale of information to a budget-constrained competitor and provides a contrasting benchmark for the paper's own externality formulation.","marker":"Castiglioni et al. (2023)"}],"fun_headline_variants":["Fierce rivalry makes no-information the optimal sale","Optimal info menu: full reveal or none, never misleading","Selling info to rival: all or nothing, when fierce none","Rivalry threshold: sell full info, some info, or none","Profit minus externality: info versioning under rivalry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the seller's expected externality cost from a message is computed with the seller's prior belief $v_s$ rather than the belief updated on the actual message; recompute it with the message-conditioned posterior and every downstream threshold changes.","fun_headline_variants_meta":{"raw":{"variants":["Fierce rivalry makes no-information the optimal sale","Optimal info menu: full reveal or none, never misleading","Selling info to rival: all or nothing, when fierce none","Rivalry threshold: sell full info, some info, or none","Profit minus externality: info versioning under rivalry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000604,"raw_usage":{"total_tokens":2840,"prompt_tokens":987,"completion_tokens":1853,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":1767}},"tokens_in":603,"tokens_out":1853,"duration_ms":12676,"temperature":1.0,"reasoning_tokens":1767,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:48:30.567859+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recalculate the expected externality in Appendix C.3 using the posterior $p(x|m,s_s;v_s)$ instead of the prior $v_s$ for the message that recommends each action. If the resulting cost function differs from equation (9), then the binary-type boundaries $\\tau_l$, $\\tau_h$, the virtual values in Corollary 4.7, and the no-information threshold $\\tau'$ in Corollary 4.9 all change, and the all-or-nothing menu is not the robust conclusion.","supporting_citations":[{"cited_title":"The design and price of information","cited_arxiv_id":null,"evidence_quote":"Supplies the baseline problem of designing and pricing information and the buyer-valuation expression that this paper extends with a seller-side externality."},{"cited_title":"and Morris, S","cited_arxiv_id":null,"evidence_quote":"Defines Bayes correlated equilibrium and the obedience condition used to constrain feasible menus."},{"cited_title":"Comparison of experiments","cited_arxiv_id":null,"evidence_quote":"Gives the comparison-of-experiments and garbling argument that justifies restricting attention to direct communication rules with two messages."},{"cited_title":"and Segal, I","cited_arxiv_id":null,"evidence_quote":"Provides the envelope theorem used to derive the transfer formula from incentive compatibility."},{"cited_title":"Selling Information while Being an Interested Party","cited_arxiv_id":"2301.13790","evidence_quote":"Models sale of information to a budget-constrained competitor and provides a contrasting benchmark for the paper's own externality formulation."}],"review_version":1}