The (Mis)use of Information in Decentralised Markets

D. Carlos Akkar

REVIEW 2 major objections 5 minor 3 references

The (Mis)use of Information in Decentralised Markets

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that in a decentralised market, more buyers improve allocative efficiency only when buyers' signals can fully reveal high quality; otherwise surplus converges to the no-information level.

desk verdict A genuinely novel paper on information and competition in decentralized markets, but Theorem 1's proof has a load-bearing gap that the authors must fix before the central claim can be accepted. read the letter →

arxiv 2506.06848 v4 pith:TPVQY4N4 submitted 2025-06-07 econ.TH

D. Carlos Akkar This is my paper

classification econ.TH MSC 91B2691B44

keywords adverseselectiondecentralisedmarketscommonvalueallocativeefficiencyinformationaggregationsequentialsearchBlackwellinformativenesslikelihoodratio

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

A seller walks a single asset through a queue of buyers; each buyer gets a private signal about whether the asset is high or low quality, but no buyer knows how many earlier buyers rejected the seller. The paper asks whether adding more buyers, or giving each buyer a more informative signal, makes trading more efficient. The answer is that both channels feed adverse selection as well as information. Total surplus rises with the number of buyers only when a buyer's signal can sometimes prove the asset is high quality; then a large market approaches the full-information outcome. Without such a signal, a large market sinks to the no-information outcome, and better-informed buyers can reduce surplus whenever the new information lets a buyer approve a deal she would otherwise have rejected.

What carries the argument

The load-bearing object is the buyer's interim belief after being visited, $\psi = \rho \nu_H/(\rho \nu_H + (1-\rho)\nu_L)$, where $\nu_\theta = \frac{1}{n}\sum_{k=0}^{n-1} r_\theta^k$ and $r_\theta$ is the per-visit rejection probability under a strategy. This identity converts past rejections into a pessimistic prior, and posterior odds are interim odds times the signal's likelihood ratio $s/(1-s)$. The theorems turn on comparing that quantity with $c/(1-c)$, especially through the adverse-selection-irrelevance condition $\frac{\rho}{1-\rho}\left(\frac{r_H}{r_L}\right)^{n-1}\frac{s}{1-s} \ge \frac{c}{1-c}$. For general finite experiments, the paper decomposes any Blackwell improvement into local mean preserving spreads, labels them positive or negative overrides, and thereby signs the surplus change without tracking the full change in interim beliefs.

What would settle it

Fix a binary experiment with $s_H < 1$ and $\rho > c$; Theorem 1 predicts that under the most selective equilibrium, total surplus eventually decreases and converges to $\rho - c$ as $n$ grows, so computing that equilibrium for large $n$ and finding surplus instead increasing or failing to approach $\rho - c$ would refute the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1: under the most selective equilibrium, expected total surplus as the number of buyers grows eventually increases and converges to the full-information benchmark if and only if the experiment has an outcome that fully reveals High quality; otherwise it eventually decreases and converges to the no-information benchmark. The mechanism is that in a larger market a buyer is more likely to be visited only after several earlier rejections, so past refusals become a stronger signal of Low quality. Theorems 2 and 3 extend this to better information: a Blackwell improvement that makes acceptance harder (a negative override) raises surplus, while one that makes rejection harder (a positive override) lowers surplus unless adverse selection is irrelevant, meaning the buyer would want to trade even if all other buyers had rejected. The paper also shows that a surplus-maximising regulator will coarsen buyers' information into binary accept/reject recommendations, and when the reservation value weakly exceeds the prior, the optimum is the least selective such garbling under which adverse selection is irrelevant.

Load-bearing premise

The buyer has no information about how many other buyers the seller has already visited, and if buyers could see their place in the queue, past rejections would no longer drive the pessimistic inference on which all three theorems depend.

Editorial extensions

If this is right

In a large market, adding buyers is good for efficiency exactly when some signal fully reveals high quality; otherwise the market ends up at the no-information benchmark.
More accurate data in the hands of buyers can lower total surplus when it mainly rescues deals that would otherwise have been rejected; a regulator should restrict such positive overrides unless a buyer would trade even after everyone else rejected.
With binary signals, stronger good news always helps, while stronger bad news eventually hurts once it is strong enough to make buyers reject on low signals.
A surplus-maximising regulator can restrict attention to monotone binary garblings that produce incentive-compatible accept/reject recommendations; when $\rho \le c$, the optimum is the least selective garbling under which adverse selection is irrelevant.
The extension to buyer ultimatum offers shows that any equilibrium surplus level can be achieved with buyers offering either the seller's value $c$ or zero.

Reading between the lines

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

If buyers could observe how many previous buyers rejected the seller, the adverse-selection channel would largely disappear, so a testable extension of the paper's logic is that revealing queue position in markets like housing or over-the-counter trading should blunt the surplus decline from more buyers.
The paper's policy principle is that data restrictions should be designed against the deepest adverse selection, not the average; this suggests banning data that upgrades previously rejected borrowers can raise total surplus even when it reduces lending.
The condition that a signal fully reveal High quality also appears in large common-value auctions, which suggests the real friction is the opacity of the trading history rather than competition itself; making histories observable may be an alternative to coarsening buyers' information.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The paper is worth reading: Theorems 2 and 3, which characterize when better information helps or hurts via local mean preserving spreads, are genuinely new and the binary-signal specialization in Theorem 2 is sharp. The regulator section is a thoughtful extension. But the proof of Theorem 1, the first main result, is not rigorous as written. The 'eventually decreasing' step in Case 2 (Section 9.3) uses an inequality with '≈' to replace m and m+1 with a fixed N, and never bounds the approximation error. The strict inequality between the fixed-N terms could be smaller than the error introduced. So the claimed monotonicity is not established. There's also a slip in the same proof: when L_m < c/(1-c) the text concludes Π_n = Π∅ = 0, which is only true when ρ ≤ c. If ρ > c, Π∅ = ρ − c > 0 and the case split is not handled. These are load-bearing because Theorem 1 is the paper's central answer to 'does more competition improve efficiency?'.

The rest is in better shape. The local spread argument for Theorems 2 and 3 is clever and seems correct; the proof of Proposition 3 is clumsy (the inequality chain is tautological as written), but the underlying contradiction argument is salvageable. The modeling assumption that buyers don't observe their queue position is a deliberate feature rather than a flaw; it's the source of the adverse selection, and the paper is transparent about it.

The paper is for people working on information aggregation, search, and OTC markets. It deserves a serious referee: the ideas are important and likely correct, but the current version's central theorem is unproven. I'd send it to review with a request for a rigorous proof of Theorem 1, and also for a clean statement of the ρ≤c versus ρ>c cases.

Referee Report

2 major / 5 minor

Summary. The paper studies a decentralised market in which a seller with a common-value asset sequentially visits n buyers, each of whom receives a private signal about the asset's quality. Buyers do not observe how many previous buyers rejected the seller, creating adverse selection. The paper asks whether allocative efficiency improves with (i) more buyers or (ii) better-informed buyers. The main theoretical results are: Theorem 1, which states that in the most selective equilibrium total surplus is eventually increasing and converges to the full-information benchmark if and only if the signal experiment has an outcome fully revealing High quality (sm=1); otherwise it is eventually decreasing and converges to the no-information benchmark max{0,ρ-c}. Theorems 2 and 3 characterize the effect of Blackwell-improved information: for binary signals, stronger good news always raises surplus while stronger bad news eventually lowers it; for general finite signals, negative overrides raise surplus and positive overrides lower it unless adverse selection is irrelevant. The paper also studies regulator-optimal coarsening of buyers' information and an extension with ultimatum offers by buyers.

Significance. If the results are correct, the paper makes a valuable contribution to the literature on information aggregation in decentralized markets. It provides a sharp, falsifiable condition (unbounded likelihood ratios) for when more competition improves or worsens efficiency, and it offers novel comparative statics on buyer informativeness, with direct policy implications for credit markets and OTC markets. The paper is well-motivated and connects to classic work by Wilson, Milgrom, and Lauermann-Wolinsky. The analytic approach is mostly self-contained and uses standard tools (Kakutani's fixed point theorem, Blackwell's theorem, monotone comparative statics). However, the proof of the central theorem (Theorem 1) contains serious gaps, as detailed below, so the main results are not yet established as rigorously as the journal would require.

major comments (2)

[Section 9.3, proof of Theorem 1, Case 2] The argument that the sequence of total surplus is eventually decreasing relies on a displayed chain of inequalities using '≈' after establishing that the rejection probabilities rθ(σ̂_n;E) are Cauchy. This step is not rigorous. First, the total surplus formula is miswritten: the first term should be ρ(1-c)[1 - r_H(σ̂_m)^m], not ρ(1-c)[1 - r_H(σ̂_m)]^m as printed. Second, even with the correct formula, replacing r_H(σ̂_m) by r_H(σ̂_N) inside the term [1 - r_H^m] is not justified by Cauchy convergence alone, because the exponent m can amplify small differences in r_H; the error introduced by this replacement is not bounded. Hence the strict inequality Π_m > Π_{m+1} for all sufficiently large m is not established by the displayed argument.
[Section 9.3, proof of Theorem 1, subcase L_m < c/(1-c)] In the subcase where L_m < c/(1-c), the proof asserts that the most selective equilibrium is eventually more selective than σ_m, and concludes 'Π_n(σ̂;E) = Π∅ = 0' for all sufficiently large n. However, Π∅ = max{0,ρ-c}, so the conclusion 'Π∅ = 0' requires ρ ≤ c. The proof does not explain why ρ > c cannot arise in this subcase. A careful reader can verify that, when s_m < 1, the limit L_m equals ρ/(1-ρ), so L_m < c/(1-c) indeed implies ρ < c; but this identity is not derived or stated in the proof. As written, the 'otherwise' branch of Theorem 1 is incomplete because the zero-benchmark conclusion is unsupported for ρ > c.

minor comments (5)

[Section 9.3, proof of Theorem 1] The total surplus expression is written inconsistently: the first term appears as [1-r_H]^m while the second appears as [1-r_L^m]. The correct expression is ρ(1-c)[1-r_H^n] - c(1-ρ)[1-r_L^n].
[Section 9.2, proof of Lemma 11] The proof uses both Z and A to denote the same set of score profiles; please standardize the notation.
[Section 9.3, proof of Lemma 7] The proof states 'Since r*_H ≥ r_H and r*_L = r_L, efficiency is higher under σ*', but the preceding derivation showed r*_H ≤ r_H. The inequality sign appears to be reversed; please check and correct.
[Section 3.2, proof of Proposition 3] The proof of the lower bound is hard to follow; the chain with conditional expectations and the decomposition of the no-information benchmark could be written more clearly.
[Section 5.1, Figure 5.1] Figure 5.1 is referenced in the text but does not appear in the provided manuscript; the figures should be included or the reference adjusted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analysis is self-contained and derives its results from explicit model primitives and standard external theorems.

full rationale

The paper is a purely theoretical derivation with no parameters fitted to data and no load-bearing self-citations. Theorem 1 is proved from primitive objects—buyers' rejection probabilities rθ(σ;E), interim beliefs ψ, and the independently defined benchmarks Πf=ρ(1−c) and Π∅=max{0,ρ−c}—with the benchmark bounds established in Proposition 3 by a deviation argument rather than assumed. Theorems 2 and 3 derive comparative statics using the external local-mean-preserving-spread decomposition (Remark 1, citing Müller and Stoyan) and prove the relevant thresholds algebraically; Definition 4 names the condition 'adverse selection is σ-irrelevant' but does not presuppose the theorem's conclusion—it is exactly the condition that falls out of the marginal-reject calculation in the proof. All cited external results (Kakutani, Blackwell, Müller–Stoyan, Shaked–Shanthikumar) are standard and machine-independent. The proof of Theorem 1 contains a rigor concern in the '≈' step of Case 2, but that is a completeness/correctness issue, not a circularity: the derivation does not reduce to its own inputs by construction, nor does it rely on a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The model's parameters (prior ρ, reservation value c, signal distributions pθ) are primitives, not fitted. The main results rest on standard game-theoretic fixed-point arguments and on the domain assumption that buyers do not observe their place in the seller's search order. No new particles, forces, or entities are introduced.

assumptions (5)

domain assumption Finite signal space S with conditional distributions pθ
Section 2 restricts to finite signals, which excludes infinite-signal settings where likelihood ratios can be unbounded without a fully-revealing signal. This assumption is used throughout, especially in Theorems 1 and 3.
domain assumption Buyers do not observe their position in the seller's search order
Section 2 states that the buyer receives no information about how many others the seller previously visited. This creates the adverse selection channel and is essential for the over-ride results.
domain assumption Seller's reservation value c is fixed and known; the seller cannot bargain
Section 2 fixes c in [0,1] and assumes the seller visits sequentially and sells at this value unless extended in Section 7. This is a modeling simplification that drives the buyer surplus calculations.
standard math Kakutani fixed point theorem for equilibrium existence
Invoked in the proof of Proposition 1 in Section 9.3 to prove non-emptiness of the equilibrium set. This is a standard mathematical result.
standard math Blackwell's theorem and local mean preserving spread decomposition
Used in Section 5.2 and Remark 1 to characterize Blackwell informativeness via local spreads. This is standard decision theory.

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Cite this review

Pith. "Pith review of The (Mis)use of Information in Decentralised Markets." pith.science (2026). https://pith.science/paper/TPVQY4N4

@misc{pith2026250606848,
  author       = {Pith},
  title        = {Pith review of: The (Mis)use of Information in Decentralised Markets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPVQY4N4}},
  note         = {Machine review of arXiv:2506.06848}
}

read the original abstract

A seller offers an asset in a decentralised market. Buyers have private signals about their common value. I study whether the market becomes allocatively more efficient with (i) more buyers, (ii) better-informed buyers. Both increase the information available about buyers' common value, but also the adverse selection each buyer faces. With more buyers, trade surplus eventually increases and converges to the full-information upper bound if and only if the likelihood ratios of buyers' signals are unbounded from above. Otherwise, it eventually decreases and converges to the no-information lower bound. With better information about trades buyers would have accepted, trade surplus increases. With better information about trades they would have rejected, trade surplus decreases--unless adverse selection is irrelevant. For binary signals, a sharper characterisation emerges: stronger good news increase total surplus, but stronger bad news eventually decrease it.

Figures

Figures reproduced from arXiv: 2506.06848 by the authors.

**Figure 5.1.** Theorem 2 illustrated To understand the intuition behind Theorem 2, let us start from the case of stronger good news. Instead of the experiment E, buyers now observe the outcome of E ′ = (S ′ , p′ L , p′ H), which delivers stronger good news than E, s ′ H > sH, but the same strength of bad news as E, sL = s ′ L . Brushing equilibrium considerations aside, simply assume that, under both experiments, a buyer accepts u… view at source ↗

**Figure 5.2.** Blackwell Improvements of a Binary Signal [PITH_FULL_IMAGE:figures/full_fig_p019_5_2.png] view at source ↗

**Figure 5.3.** 35 Restricting to local spreads is without loss for finite experiments36—every Blackwell improvement, and a fortiori, ordinary mean preserving spread can be constructed through a finite number of local spreads. Remark 1. [Müller and Stoyan (n.d.), Theorem 1.5.29] An experiment E ′ is Blackwell more informative than another, E, if and only if there is a finite sequence of experiments E1, E2, ..., Ek such that E1 = E… view at source ↗

Figures from the paper (2 more)

**Figure 8.** Figure 8: b: Probability that [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]

**Figure 8.** Figure 8: c: Probability that some buyer trades when θ = H [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]

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Works this paper leans on

3 extracted references · 2 canonical work pages

[1]

Searching for Approval

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work page doi:10.3982/ecta18554 1970
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work page arXiv 2013
[3309]

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doi: 10.1257/jep.22.2.217. Bikhchandani, Sushil, David Hirshleifer, and Ivo Welch (1992). “A Theory of Fads, Fashion, Cus- tom, and Cultural Change as Informational Cascades”. In:Journal of Political Economy100.5, pp. 992–1026. issn: 00223808, 1537534X.doi: 10.1086/261849. Blackwell, David (1953). “Equivalent Comparisons of Experiments”. In:The Annals of ...

arXiv 1992

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Searching for Approval

Agarwal, Sumit et al. (July 2024). “Searching for Approval”. In:Econometrica 92.4, pp. 1195–1231. doi: 10.3982/ECTA18554. Akerlof, George A. (1970). “The Market for "Lemons": Quality Uncertainty and the Market Mech- anism”. In:The Quarterly Journal of Economics84.3, pp. 488–500.issn: 00335533, 15314650. doi: 10.2307/1879431. BCBS (Dec. 2017).High-level su...

work page doi:10.3982/ecta18554 1970

[2] [5]

Biased social learning

Elsevier B.V, pp. 813–886.isbn: 9780444595331. doi: 10.1016/B978-0-444-59531-7.00013-2 . Herrera, Helios and Johannes Hörner (2013). “Biased social learning”. In:Games and Economic Behavior 80.C, pp. 131–146.doi: 10.1016/j.geb.2012.12.006. Kay, John (2024). “HBOS Collapse Shows Danger of the Winner’s Curse”. In:Financial Times. url: https://www.ft.com/con...

work page arXiv 2013

[3] [3309]

A Theory of Fads, Fashion, Cus- tom, and Cultural Change as Informational Cascades

doi: 10.1257/jep.22.2.217. Bikhchandani, Sushil, David Hirshleifer, and Ivo Welch (1992). “A Theory of Fads, Fashion, Cus- tom, and Cultural Change as Informational Cascades”. In:Journal of Political Economy100.5, pp. 992–1026. issn: 00223808, 1537534X.doi: 10.1086/261849. Blackwell, David (1953). “Equivalent Comparisons of Experiments”. In:The Annals of ...

arXiv 1992