{"id":"549f08ae-c967-41e5-b68a-425c075ed87a","arxiv_id":"2608.04332","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The discriminatory auction price is Lehmann more informative than the uniform-price auction price whenever k/n is large enough relative to a cumulative-score threshold determined by the signal distribution.","lead":"This paper asks which auction format reveals more about an asset's true value to an outside observer: a discriminatory auction, whose public price reflects the highest bidder signal, or a uniform-price auction, whose price reflects the (k+1)th highest signal. It gives conditions, depending on how informative high and low bidder signals are and on the ratio of objects to bidders, under which the highest signal is statistically more informative than the lower one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The most load-bearing point is the unproved existence of strictly increasing symmetric equilibria in the multi-unit auction formats; without it, auction prices need not invert to the order statistics the theorem compares.","rationale":"The paper's central mathematical contribution is Theorem 6.1, and I traced the proof in Appendix B.1 plus Lemma A.3: the derivative identities for derivative of log F^(1)_v and derivative of log F^(k+1)_v are correct, and the sufficient condition follows from the bounds on S_v and b/B <= 1. The only asserted but unproved ingredient inside the theorem is the monotonicity of S_v; this is a known consequence of MLRP via reverse hazard rate dominance, so it is a minor gap. The more serious concern is the bridge from order statistics to auction prices. Section 2's claim about equilibrium existence is genuinely load-bearing: the price in the uniform-price auction equals beta(X_(k+1)) only if a strictly increasing symmetric equilibrium exists and is the one played, and likewise for the discriminatory price and X_(1). The paper gives neither a proof nor a precise citation for the multi-unit common-value case. This is not an internal inconsistency in the order-statistic theorem, but it is a correctness risk for the advertised auction-pricing conclusion. The reader's CONDITIONAL verdict already captures this concern, and I do not see a reason to strengthen to REJECT because the equilibrium existence is standard and likely provable under the stated assumptions. A single check confirming strict monotonicity of the candidate equilibria would settle the matter.","tokens_in":18969,"tokens_out":21878,"duration_ms":212340,"concrete_test":"Derive the symmetric equilibrium for the uniform-price auction with k units and unit demand under strict MLRP using the standard candidate beta(s) = E[V | S_i = s, Y_k = s], where Y_k is the kth-highest rival signal, and verify beta'(s) > 0 on the signal support. For the discriminatory auction, solve the symmetric first-order differential equation for the bid function and verify a strictly increasing solution exists (for example, with n=3, k=1 and an explicit MLRP family). If either verification fails, Section 2's reduction is invalid; if both succeed, the conditional verdict stands and the gap is only a missing citation or proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2 asserts, with no proof and no direct multi-unit citation, that strict MLRP guarantees strictly increasing symmetric equilibria in both the discriminatory and uniform-price auctions. The entire auction-price interpretation of Theorem 6.1 depends on this reduction: the outside observer inverts the public price through the known equilibrium bid function to recover X_(1) or X_(k+1). If such an equilibrium does not exist, or if the selected equilibrium is not strictly increasing (uniform-price auctions can admit flat low-price equilibria), the price is not a monotone function of the relevant order statistic, and the comparison of X_(1) and X_(k+1) does not apply to the auctions. The order-statistic theorem is internally coherent, but the advertised claim about 'informational content of auction prices' is only as strong as this asserted reduction. A secondary gap is the unproved monotonicity of the cumulative score S_v used in the proof of Theorem 6.1; it is true under MLRP but should be shown.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper compares the informational content, in the Lehmann sense, of the highest order statistic X_(1) and the (k+1)-th highest order statistic X_(k+1) of n i.i.d. signals drawn from a common-value experiment {F_v}. It motivates this comparison by a multi-unit auction model in which the discriminatory auction's publicly observed highest winning bid reveals X_(1), while the uniform-price auction's clearing price reveals X_(k+1). The main theorem (Theorem 6.1) gives a sufficient condition, k/n >= 1 - S_v(omega_v)/S_v(alpha_v), where S_v = -partial_v log F_v, under which X_(1) Lehmann dominates X_(k+1); the condition specializes to k/n >= 1 - rho(1)/rho(0) in scale families and to k/n >= 1 - r(0)/r(-infinity) in location families. Section 6.2 and the appendices establish qualitatively necessary conditions: unbounded likelihood ratios, no conclusive left-endpoint signals, and liminf k_n/n > 0. Section 7 applies the ranking to jury voting.","tokens_in":19091,"tokens_out":11405,"duration_ms":117216,"significance":"If the equilibrium-reduction step is supplied, the paper offers a clean, prior-free ranking of two auction formats in terms of the Lehmann informativeness of their public prices. The paper goes beyond local indices by using a global comparison criterion, covers fixed n rather than only asymptotics, and complements the vertical comparisons of Di Tillio, Ottaviani, and Sorensen with horizontal comparisons across order statistics. The explicit examples and the jury-voting application are useful, and the three qualitative necessity results in Section 6.2 are a genuine strength. The main derivations in Appendices A through C are coherent, and I found no algebraic error in the central inequality of Theorem 6.1 and its special cases. However, the advertised auction interpretation rests on an unproved equilibrium-existence and monotonicity assertion, and one key monotonicity fact used in the proof of Theorem 6.1 is asserted without proof. Once those points are filled, the order-statistic results themselves appear sound.","major_comments":[{"comment":"The assertion that strict MLRP guarantees symmetric, strictly increasing equilibria in both the discriminatory and the uniform-price auctions is stated without proof and without a direct citation; footnote 5 merely says that weaker conditions suffice. This assertion is load-bearing: Section 2.1 inverts the observed price through the equilibrium bid function to recover X_(1) or X_(k+1), and the entire auction interpretation of Theorem 6.1 depends on that inversion. If no strictly increasing equilibrium exists, or if a flat low-price equilibrium of the uniform-price auction is selected, the public price need not be a monotone function of the relevant order statistic, and the comparison of X_(1) and X_(k+1) would not apply to the auctions. The authors should provide a proof or a precise citation for the multi-unit existence and monotonicity result, and they should state explicitly which equilibrium is being selected.","section":"Section 2 (equilibrium reduction)"},{"comment":"The proof of Theorem 6.1 uses the claim that S_v(x) is non-increasing in x, introduced in Section 6.1 and repeated in Appendix B.1 as 'easy to verify,' and Proposition 6.3 uses the strict version S'_v(y) < 0. This monotonicity is central because it is what allows inequality (13) to be bounded by the endpoint values S_v(omega_v) and S_v(alpha_v). The fact is true under MLRP via reverse hazard rate dominance, but as written it is unproved. Please add a lemma with a proof (or a standard reference) for the non-increasing property and a clear statement of the strict version needed in the proof of Proposition 6.3.","section":"Appendix B.1 and Section 6.1"}],"minor_comments":[{"comment":"The informal proof of Proposition 6.3 uses the notation L({l,m}) without defining it in the body; the formal proof in Appendix C.3 is correct but dense, and a one-sentence definition would improve readability.","section":"Section 6.2.3"},{"comment":"The statement that 'the unanimity rule is welfare superior to the k voting rule' can confuse readers because the paper also says that the unanimity rule has k=0; the condition in Proposition 7.1 is imposed on the alternative rule's k, not on the unanimity rule's k. Please clarify this in the text.","section":"Section 7"},{"comment":"The reference 'Hollander, F. (2000)' should be 'den Hollander, F. (2000)' for the large-deviations monograph.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to the auction-pricing claim is the unproved existence of strictly increasing symmetric equilibria in the multi-unit formats. This is likely fixable by a precise citation or a short proof, but if it cannot be supplied, the paper could be reframed as an order-statistics comparison result, with the auction application presented as a conditional corollary; the order-statistics results are interesting enough to stand on their own. I would also ask the editor to ensure that the boundary with Di Tillio et al. (2026) is stated precisely, since both papers use Lehmann comparisons of order statistics in auction and voting applications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the theorem is right and the auction packaging has a hole in it. If you read this as a paper about comparing the highest order statistic with the (k+1)th under Lehmann dominance, it is a solid piece of economic theory. If you read it as a paper about what auction prices reveal, it leans on an equilibrium existence claim that is asserted, not proved.\n\nWhat is actually new: for fixed n, comparing X_(1) to X_(k+1) horizontally is a different question from Di Tillio, Ottaviani, and Sorensen's vertical comparisons (fix rank, vary n). The cumulative-score threshold in Theorem 6.1 is a clean sufficient condition, and it reduces to the scale-family condition in Theorem 4.1 and the location-family condition in Theorem 5.1. The necessity results in Section 6.2 are also worth having: they show you cannot drop conclusive high signals, inconclusive low signals, or a positive k/n ratio without losing the ranking. The large-deviation arguments in Appendix C are coherent, and the jury-voting application is a useful byproduct.\n\nThe soft spots are exactly where the reader put them. Section 2 asserts that strict MLRP guarantees strictly increasing symmetric equilibria in both the discriminatory and uniform-price auctions, and calls them straightforward generalizations of Milgrom-Weber (1982). That is not a proof, and it is not a direct citation. Multi-unit common-value auctions are not a trivial corollary of the single-object case, and uniform-price auctions can have flat, non-monotone equilibria. The entire reduction of prices to order statistics depends on the observer knowing that the equilibrium is strictly increasing. This should be either proved in an appendix or cited to a theorem that actually covers these formats. It is a load-bearing gap, but it is fixable.\n\nThe secondary gap is smaller. The monotonicity of S_v(x) in x is asserted in Section 6.1 and \"easy to verify\" in Appendix B.1. It does follow from reverse hazard rate dominance under MLRP, but the paper should show the two-line argument. Minor.\n\nThe citation pattern looks fair. The distinction from Di Tillio et al. and from the information-aggregation literature is drawn honestly. No fitted parameters, no circularity.\n\nWho benefits: auction theorists and anyone working on Lehmann comparisons of order statistics. The paper deserves a serious referee; it should be sent out with a request to close the equilibrium existence gap and prove the monotonicity claim. If those are done, it is publishable. I would take it to reading group and would cite it for the horizontal comparison result.","headline":"Genuinely new horizontal order-statistic comparison under Lehmann dominance; the auction interpretation rests on an unproved equilibrium existence claim that should be fixed before publication.","tokens_in":19673,"tokens_out":4248,"would_cite":true,"duration_ms":39702,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62B15","62G30","91B26","91B12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a discriminatory auction's publicly reported top price is Lehmann more informative about a common value than a uniform-price auction's clearing price once the object-to-bidder ratio clears a threshold set by the…","keywords":["common value auctions","price informativeness","Lehmann dominance","order statistics","discriminatory auction","uniform-price auction","cumulative score function","jury voting"],"falsifier":"A direct check would fix $n=3$, $k=1$ and a strict-MLRP family such as the scale family $F_v(x) = (x/v)^\\alpha$ on $[0,v]$, compute the two order-statistic CDFs, and test whether every pair $(x,y)$ with $F_v^{(1)}(x) = F_v^{(2)}(y)$ satisfies $-\\partial_v \\log F_v^{(1)}(x) \\geq -\\partial_v \\log F_v^{(2)}(y)$ whenever $k/n$ is above the paper's threshold; a violation would refute the theorem, and a separate numerical equilibrium computation would verify whether public prices actually invert to these order statistics.","tokens_in":18710,"feed_emoji":"📈","tokens_out":11110,"duration_ms":102971,"temperature":0.7,"pith_summary":"The paper asks which auction format better reveals an asset's unknown common value to someone who only sees the final price. In a discriminatory auction the published top bid reveals the highest private signal, while in a uniform-price auction the single clearing price reveals the (k+1)st highest signal, so the question reduces to comparing two order statistics. The central result is that the highest order statistic is Lehmann more informative—better for every decision maker whose preferred action rises with the state—than the (k+1)st whenever the ratio $k/n$ of objects to bidders clears a threshold set by the signal distribution's cumulative score function. Specialized to scale families, the condition becomes $k/n \\geq 1 - \\rho(1)/\\rho(0)$, which is met by common distributions once the object share is large enough. The same threshold condition is shown to be qualitatively necessary, and it carries over to jury voting, where unanimity beats less demanding rules under the same circumstances.","feed_headline":"Top bid beats clearing price past a ratio threshold","feed_subtitle":"The discriminatory top bid beats the uniform clearing price once k/n clears a score threshold.","key_machinery":"The load-bearing object is the cumulative score function $S_v(x) = -\\partial_v \\log F_v(x)$, the derivative with respect to the state $v$ of the negative log of the conditional CDF; MLRP makes $S_v$ non-increasing in $x$, and its endpoint values $S_v(\\alpha_v)$ and $S_v(\\omega_v)$ summarize how informative low and high signals are. Lehmann dominance is verified through the identity that $F$ dominates $G$ iff, whenever $F_v(x) = G_v(y)$, one has $-\\partial_v \\log F_v(x) \\geq -\\partial_v \\log G_v(y)$. The proof then uses the binomial expression for the (k+1)st order statistic's CDF, $F_v^{(k+1)}(x) = B(k; n, 1 - F_v(x))$, together with the identity $\\partial_q B(k;n,1-q) = (1/q)(n-k)b(k;n,1-q)$, to reduce the comparison to $n S_v(x) \\geq (n-k) S_v(y)$ times a binomial ratio, which is bounded once the endpoint ratio condition holds. The auction-to-order-statistic link is carried by strictly increasing equilibrium bid functions, which turn observed prices into the corresponding signal order statistics.","core_discovery":"The paper's central claim is Theorem 6.1: for any signal family $F_v$ satisfying the monotone likelihood ratio property, $X_{(1:n)}$ Lehmann dominates $X_{(k+1:n)}$ if for every $v$, $k/n \\geq 1 - S_v(\\omega_v)/S_v(\\alpha_v)$, with $S_v(x) = -\\partial_v \\log F_v(x)$. Because monotone equilibria let the discriminatory price be inverted to the highest signal and the uniform price to the (k+1)st signal, this order-statistic ranking is a ranking of the two auction formats. The proof starts from the equivalence that Lehmann dominance holds exactly when $-\\partial_v \\log F_v^{(1)}(x) \\geq -\\partial_v \\log F_v^{(k+1)}(y)$ at equal quantiles, substitutes the binomial formula for the (k+1)st order statistic, and uses monotonicity of $S_v$ to turn a pointwise inequality into the ratio threshold. The paper also establishes qualitative necessity: dominance can hold only if high signals are unboundedly informative (with the right endpoint of support increasing in large samples), the left endpoint of support is constant, and $k/n$ is bounded away from zero as $n$ grows.","pith_inferences":["The paper's proof does not depend on which bidder pays what, only on monotone invertibility of prices; a natural extension is to other sealed-bid formats or to markets where a regulator chooses which summary statistic, such as top, median, or average, to disclose, with a similar score threshold governing the comparison.","Because the necessary conditions show dominance fails when $k/n \\to 0$, the paper implies that in very thin markets the uniform-price format may carry more information; the authors mention this only as a contrast to information-aggregation results.","The jury application suggests a testable implication: if conclusive innocence signals are present and the acquittal threshold requires at least $k+1$ votes, unanimity should outperform supermajority rules in simulated signal environments; the paper does not run such simulations.","The threshold $1 - S_v(\\omega_v)/S_v(\\alpha_v)$ can in principle be estimated from data by fitting parametric signal distributions to bid sets, giving a practical decision rule for choosing or designing auction formats; this estimation route is not developed in the paper."],"forward_implications":["In scale families, the condition becomes $k/n \\geq 1 - \\rho(1)/\\rho(0)$, so, for example, the truncated normal (threshold about 0.29) and truncated Gumbel (0.22) yield dominance once the object share is high enough, while power distributions yield dominance for every $k$ and $n$.","In location families, the parallel condition $k/n \\geq 1 - r(0)/r(-\\infty)$ holds, extending the ranking to additive-error models.","Because $X_{(1)}$ dominates $X_{(k+1)}$, the discriminatory auction's published top price dominates the uniform auction's clearing price even if the discriminatory auction announces additional statistics such as average or median winning bids.","Under the same sufficient condition, the unanimity rule is welfare superior to any $k$-voting rule (including majority rule) in the jury model, and this holds for finite juries, not only in the large-$n$ limit.","The necessary conditions show that if $k/n$ vanishes as $n$ grows, the highest order statistic cannot dominate the (k+1)st, so the uniform-price format may be the more informative one in sufficiently thin markets."],"supporting_citations":[{"why":"Defines Lehmann dominance and supplies the derivative inequality that the paper uses to compare experiments.","marker":"Lehmann (1988)"},{"why":"Provides the symmetric monotone equilibrium strategies whose invertibility links auction prices to order statistics.","marker":"Milgrom and Weber (1982)"},{"why":"Gives the hazard-rate and reverse hazard-rate consequences of MLRP used in the sufficiency proof.","marker":"Shaked and Shanthikumar (2007)"},{"why":"Benchmark model with growing k and n; the paper contrasts its ratio condition with information-aggregation conditions.","marker":"Pesendorfer and Swinkels (1997)"},{"why":"Gives the mapping between order-statistic informativeness and voting rules used in the jury application.","marker":"Di Tillio et al. (2026)"},{"why":"Supplies conditions for a symmetric responsive equilibrium in jury voting, which the jury application relies on.","marker":"Duggan and Martinelli (2001)"}],"fun_headline_variants":["Auction top bid outshines clearing price past k/n threshold","k/n ratio decides if top bid beats uniform price","Top bid reveals more info above k/n threshold","Clearing price loses info race when k/n is high","Auction price informativeness flips at k/n"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The auction comparison collapses if either format fails to have a symmetric, strictly increasing equilibrium whose bid function the outside observer knows, so that the public price exactly reveals the highest or the (k+1)st highest signal; the paper states that strict MLRP guarantees this but gives no proof or direct multi-unit citation.","fun_headline_variants_meta":{"raw":{"variants":["Auction top bid outshines clearing price past k/n threshold","k/n ratio decides if top bid beats uniform price","Top bid reveals more info above k/n threshold","Clearing price loses info race when k/n is high","Auction price informativeness flips at k/n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000556,"raw_usage":{"total_tokens":2642,"prompt_tokens":937,"completion_tokens":1705,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":1626}},"tokens_in":553,"tokens_out":1705,"duration_ms":12866,"temperature":1.0,"reasoning_tokens":1626,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T19:35:17.047278+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check would fix $n=3$, $k=1$ and a strict-MLRP family such as the scale family $F_v(x) = (x/v)^\\alpha$ on $[0,v]$, compute the two order-statistic CDFs, and test whether every pair $(x,y)$ with $F_v^{(1)}(x) = F_v^{(2)}(y)$ satisfies $-\\partial_v \\log F_v^{(1)}(x) \\geq -\\partial_v \\log F_v^{(2)}(y)$ whenever $k/n$ is above the paper's threshold; a violation would refute the theorem, and a separate numerical equilibrium computation would verify whether public prices actually invert to these order statistics.","supporting_citations":[],"review_version":1}