{"id":"80cc7978-46b4-4bc5-8d5f-ed262f252677","arxiv_id":"2506.06848","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a sequential-search market, total surplus reaches the full-information level with more buyers only if a signal can fully reveal good quality; otherwise it collapses to the no-information level, and better-informed buyers reduce surplus unless adverse selection is irrelevant.","lead":"An economic theory paper shows that in a market where a seller visits buyers one by one until someone buys, adding more buyers or giving them better information can either improve or destroy allocative efficiency. The results identify precisely when more information helps in opaque markets such as over-the-counter trading and credit, with implications for regulation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's proof is not rigorous: the 'eventually decreasing' step uses an uncontrolled approximation, and the case split mishandles the no-information benchmark for ρ≤c. The central convergence claim is therefore not established as written.","rationale":"The reader's verdict CONDITIONAL is appropriate. The most load-bearing concern is not the queue-position assumption (a model feature) but the internal rigor of Theorem 1's proof. The proof of the sm<1 case asserts eventual monotonicity through an uncontrolled approximation and treats the no-information benchmark incorrectly for ρ≤c. Because Theorem 1 is the paper's central claim, this is a genuine correctness risk. However, the flaw may be repairable: the Cauchy property of the equilibrium strategies likely permits a standard epsilon-delta argument to close the gap, and the L_m = ρ/(1−ρ) cancellation for threshold strategies suggests the case split can be tied to ρ versus c. The reader already flagged a 'misstatement about the no-information benchmark' in the Theorem 1 proof, so there is partial overlap with the reader's rationale, but the reader's stated weakest_assumption is the queue-position assumption, which I do not share as the primary concern. The proposed computational test can uncover a concrete counterexample if the theorem is false; if it passes, the proof still needs a rigorous rewrite, but the conditional verdict already calls for that. Thus the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":43216,"tokens_out":16214,"duration_ms":166428,"concrete_test":"Run a computational search over binary experiments with parameters ρ>c and s_H<1 (e.g., grid ρ∈{0.55,0.6,0.7}, c∈{0.4,0.5}, s_L∈{0.1,0.3,0.4}, s_H∈{0.6,0.8,0.9}), computing the most selective equilibrium for n=1,...,100 by fixed-point iteration. Check whether the total-surplus sequence is eventually decreasing and converges to ρ−c. Any violation is a counterexample to Theorem 1; if no violation appears across the grid, the gap is likely a proof technicality rather than a false claim. To directly test the proof step, also record max_m |Π_m − Π_{m+1}| and the error from replacing rθ(ˆσ_m) by rθ(ˆσ_N) to see whether the approximation in the proof can be made rigorous for a chosen N.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is Theorem 1: with no fully-revealing signal the surplus sequence under the most selective equilibrium is eventually decreasing and converges to the no-information benchmark. The proof in Section 9.3, Case 2, contains a load-bearing gap. After showing the equilibrium rejection probabilities are Cauchy, the proof writes an inequality with '≈' to conclude Π_m > Π_{m+1} for all sufficiently large m, using a fixed N in place of m and m+1. The approximation error is never bounded, and the strict inequality between the fixed-N terms may be arbitrarily small relative to the error introduced by replacing m by N. Thus 'eventually decreasing' does not follow from the displayed argument. In addition, the proof asserts in the L_m < c/(1-c) case that 'Π_n = Π∅ = 0', which is only correct when ρ≤c; if ρ>c, the no-information benchmark is ρ−c>0, and the proof does not explain why this case cannot arise. Theorem 1 is the paper's first main result, so this gap is load-bearing: without a rigorous proof of eventual monotonicity and the correct limit, the paper's answer to 'does more competition improve efficiency?' is unproven. The modeling assumption that buyers do not observe their queue position is a deliberate feature, not a flaw; the vulnerability is internal to the proof of the central theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":43475,"tokens_out":14218,"duration_ms":137843,"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":[{"comment":"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":"Section 9.3, proof of Theorem 1, Case 2"},{"comment":"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.","section":"Section 9.3, proof of Theorem 1, subcase L_m < c/(1-c)"}],"minor_comments":[{"comment":"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":"Section 9.3, proof of Theorem 1"},{"comment":"The proof uses both Z and A to denote the same set of score profiles; please standardize the notation.","section":"Section 9.2, proof of Lemma 11"},{"comment":"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":"Section 9.3, proof of Lemma 7"},{"comment":"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":"Section 3.2, proof of Proposition 3"},{"comment":"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.","section":"Section 5.1, Figure 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely and interesting question, and the main results, if correct, would constitute a solid contribution. The writing is clear and the literature review is thorough. However, the proof of Theorem 1—the paper's first main result—contains load-bearing gaps: an incorrect surplus expression and an unvalidated approximation step, plus an unjustified invocation of Π∅=0 in a subcase. These issues need to be resolved with a rigorous argument before the paper is publishable. I recommend major revision and suggest the author rework the proof of Theorem 1, possibly using monotonicity of the rejection probabilities in a more careful way. The remaining results and the policy section are promising, but they rely on the same technical framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe 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?'.\n\nThe 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.\n\nThe 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.","headline":"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.","tokens_in":44007,"tokens_out":2724,"would_cite":true,"duration_ms":25585,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B26","91B44"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["adverse selection","decentralised markets","common value","allocative efficiency","information aggregation","sequential search","Blackwell informativeness","likelihood ratio"],"falsifier":"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.","tokens_in":1699,"feed_emoji":"📉","tokens_out":1831,"duration_ms":76095,"temperature":0.7,"pith_summary":"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.","feed_headline":"More buyers help only when a signal can prove high quality","feed_subtitle":"More data can backfire by feeding adverse selection in opaque markets.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the classic large-auction result that the winning bid reveals common value when likelihood ratios are unbounded; Theorem 1 uses the same condition in a decentralised market.","marker":"Wilson (1977)"},{"why":"Provides the general 'distinguishability' condition that reduces to unbounded likelihood ratios in the binary-value case, the exact condition Theorem 1 relies on.","marker":"Milgrom (1979)"},{"why":"Closest model of a decentralised market with common values; Theorem 1 contrasts the partially informative trade outcome with their full-or-no revelation result.","marker":"Lauermann and Wolinsky (2016)"},{"why":"Baseline model with identical structure but trade always efficient; Theorem 1 strengthens the conclusion when trading with a Low quality seller is inefficient.","marker":"Zhu (2012)"},{"why":"Studies how an additional bidder affects allocative efficiency in a common-value auction; the paper compares the sufficient condition there with the necessary and sufficient condition here.","marker":"Riordan (1993)"},{"why":"Supplies the ranking of binary experiments and the Blackwell theorem used in Lemma 12 and the binary-signal analysis.","marker":"Blackwell and Girshick (1954)"},{"why":"Provides the local mean preserving spread decomposition of Blackwell improvements, which is the technical device behind Theorem 3.","marker":"Müller and Stoyan (n.d.)"}],"fun_headline_variants":["Larger markets only improve efficiency if signals can fully reveal high quality","More buyers backfire when no signal proves high quality","Sharper info on rejected trades reduces surplus; on accepted trades, it rises","Why more information can backfire in decentralised markets"],"cache_read_input_tokens":46080,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Larger markets only improve efficiency if signals can fully reveal high quality","More buyers backfire when no signal proves high quality","Sharper info on rejected trades reduces surplus; on accepted trades, it rises","Why more information can backfire in decentralised markets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000415,"raw_usage":{"total_tokens":2113,"prompt_tokens":886,"completion_tokens":1227,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":1156}},"tokens_in":502,"tokens_out":1227,"duration_ms":13084,"temperature":1.0,"reasoning_tokens":1156,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:48:46.763664+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}