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Selling Information in Games with Externalities

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Promising extension of Bergemann–Bonatti–Smolin, but a wrong conditional probability in the cost derivation invalidates the central menu characterization. read the letter →

arxiv 2505.00405 v1 pith:K375BM26 submitted 2025-05-01 cs.GT

Thomas Falconer , Anubhav Ratha , Jalal Kazempour , Pierre Pinson , Maryam Kamgarpour This is my paper · ORCID
classification cs.GT MSC 91B0391A2791B26
keywords informationdesignmechanismscreeningdatamarketsBayesianpersuasionexternalitiesincompletegamesproductversioning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Appendix C.3 and Eq. (9)] 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.
  2. [Corollary 4.9] 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.
minor comments (4)
  1. [Eq. (14)] 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.
  2. [Appendix C.3] 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.
  3. [Proof of Proposition 4.1 and Figure 3] 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.
  4. [Eq. (3)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained, with self-citations appearing only in related work.

full rationale

The paper's core derivation is not circular. The buyer's gain δ(I,vb) is derived in Corollary 4.2 from the dominant-strategy characterization (3), the direct-communication-rule reduction (Proposition 3.2), and the ceiling result (Proposition 4.1); it is not imported as an input. The seller's cost function (9) is derived in Appendix C.3 from the seller's utility (1) and the message structure; even if that derivation is mathematically flawed (it appears to condition on the seller's prior vs rather than a message-conditioned posterior, and the printed coefficient of 1{I(vb)≥0} differs from the paper's own intermediate algebra), this is a correctness or modeling defect, not a circular equivalence: the cost is not defined in terms of the menu it is used to predict. Proposition 4.5 is attributed to Bergemann et al. (2018), but the paper supplies its own proof in Appendices C.1 and C.2, so the result is not supported by an unverified citation. The self-citations (Falconer et al. 2024, 2025; Pinson et al. 2022) occur only in the related-work discussion of analytics markets and play no role in the mechanism design theorems. The closed-form λ* in Corollary 4.9, λ* = (π−(1/2)+π+(1/2))/2, is an analytic claim about the dual variable; if it is invalid for skewed regular distributions, that is a mathematical correctness issue, not a reduction of the conclusion to its assumptions. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in by citation. The main characterization (Corollary 4.9) is a consequence of the model's equations, so the derivation is self-contained against the external benchmark of Bergemann et al. (2018).

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the exogenous model parameters tau and v_s, the independence and dominant-strategy structure of the information game, the regularity of the type distribution, and on a specific (and in the paper mishandled) computation of the expected externality cost. No new entities are postulated. The buyer's gain function and IC/IR reduction are imported from Bergemann et al. (2018).

free parameters (2)
  • competition intensity tau
    Exogenous parameter encoding how much the seller loses when the buyer also chooses the correct action. Not fitted; figures vary it.
  • seller's belief v_s
    Exogenous posterior P(X=0) of the seller. Not fitted; figures vary it. The full-information cost and all thresholds depend on it.
assumptions (4)
  • domain assumption Private signals are independent across players conditional on the state: p(s_b, s_s|x) = p(s_b|x)p(s_s|x).
    Invoked in Section 2; it lets the buyer's valuation be independent of the seller's action and keeps the posterior within the Bernoulli family.
  • ad hoc to paper The cost of a message is computed with the seller's prior belief v_s instead of the posterior p(x|m,s_s;v_s).
    This is the step in Appendix C.3 that produces equation (9). It is the load-bearing fragility of the paper's quantitative results.
  • domain assumption Regularity: virtual values pi-(v_b) and pi+(v_b) are non-decreasing in v_b.
    Assumed in Section 4.2 before Corollary 4.7; the authors acknowledge that bimodal type distributions (most buyers well informed) violate it and leave ironing to future work.
  • standard math Boundary types v_b in {0,1} place zero value on information, giving Delta(0)=Delta(1)=0 in the envelope proof.
    Used in the proof of Proposition 4.5 to derive the integral constraint integral_0^1 I(v_b) dv_b = 0 from the envelope theorem.

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Cite this review

Pith. "Pith review of Selling Information in Games with Externalities." pith.science (2026). https://pith.science/paper/K375BM26

@misc{pith2026250500405,
  author       = {Pith},
  title        = {Pith review of: Selling Information in Games with Externalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K375BM26}},
  note         = {Machine review of arXiv:2505.00405}
}
read the original abstract

A competitive market is modeled as a game of incomplete information. One player observes some payoff-relevant state and can sell (possibly noisy) messages thereof to the other, whose willingness to pay is contingent on their own beliefs. We frame the decision of what information to sell, and at what price, as a product versioning problem. The optimal menu screens buyer types to maximize profit, which is the payment minus the externality induced by selling information to a competitor, that is, the cost of refining a competitor's beliefs. For a class of games with binary actions and states, we derive the following insights: (i) payments are necessary to provide incentives for information sharing amongst competing firms; (ii) the optimal menu benefits both the buyer and the seller; (iii) the seller cannot steer the buyer's actions at the expense of social welfare; (iv) as such, as competition grows fiercer it can be optimal to sell no information at all.

Figures

Figures reproduced from arXiv: 2505.00405 by the authors.

Figure 1
Figure 1. Existing frameworks for sharing information. The blue, red and green arrows indicate monetary transactions, strategic actions, and information flows between the firms and the market, respectively. Whilst in each framework, Firm B shares information with Firm A, the setups differ as follows: in (a) information is shared freely between competitors, in (b) information is purchased yet Firm B is a third party that doesn… view at source ↗
Figure 2
Figure 2. One-dimensional informativeness. and P(m1|X = 0; b)b ≤ P(m1|X = 1; b)(1 − b), (5) thereby inducing the respective actions.5 Lastly, given the parametrization in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Feasible region of I0(b) and I1(b). The ceiling of the feasible region is highlighted in red. The Buyer’s Gain. With the dominant strategy in (3), the expected value of the nonnegative term of the buyer’s utility function before any information sharing reduces to EX [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Gain as a function and buyer types, as described in (8), for I = 0 (solid), I = 1/2 (dashed) and I = −1/2 (dotted). This expression matches that for the buyer’s valuation in Bergemann et al. (2018), as dominant strategies ensure that selling information to a competitor…
Figure 5
Figure 5. Figure 5: A congruent binary type distribution. Without additional information, both types will take the same action, that is, σ(v l b ) = σ(v h b ) = 0. the mechanism design problem in (10) reduces to: max I k,tk ϕ(t h − c h ) + (1 − ϕ)(t l − c l ) (11a) s.t. I k ∈ R[−1,1], tk …
Figure 6
Figure 6. Figure 6: Decision boundaries τ h (dashed) and τ l (solid) for congruent types. vb 0 1/6 1/2 2/3 1 v h b v l b [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: A noncongruent binary type distribution. Without addi￾tional information, types take different actions, that is, σ(v l b ) = 1 and σ(v h b ) = 0. τ , the seller may offer less information to both types. There￾fore, for noncongruent types, the problem in (11) reduces to…
Figure 8
Figure 8. Figure 8: Decision boundaries τ h (dashed) and τ l (solid) for non￾congruent types. violating incentive constraints. The information offered to v l b depends on the relative precision (i.e., distance from 1/2) of the types. As the low type’s prior becomes less informa￾tive, appr…
Figure 9
Figure 9. Figure 9: Virtual surplus with vb = 2/5 for varying levels of competition. of Myerson (1981), we can instead maximize virtual surplus pointwise, so the menu is optimal if there exists λ ∗ with I ∗ (vb) = arg max I∈I I(vb) (π(I, vb) − λ ∗ ). (15) Proposition 4.8. For regular type…
Figure 10
Figure 10. Figure 10: Both negative (solid) and positive (dashed) virtual values, with vs = 4/5, for varying levels of competition. The primal solution, I ∗ (vb), is indicated by the blue vertical lines, with only types within the enclosed interval offered the fully informative communicati…
Figure 11
Figure 11. Figure 11: Expected profit from either strategic versioning (solid), offering full information to all types (dashed) or sharing no infor￾mation at all (dashdot) virtual values after re-introducing the integral constraint. To satisfy constraint (13b), the dual variable λ ∗ must a…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.