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REVIEW 2 major objections 4 minor 113 references

An order statistic becomes more accurate when the sample grows if and only if the matching cumulative hazard is log-supermodular.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-14 11:45 UTC pith:FY4IFVLE

load-bearing objection Clean, checkable theory that generalizes order-statistic accuracy to every rank and unifies auction and voting aggregation under standard MLR. the 2 major comments →

arxiv 2607.10435 v1 pith:FY4IFVLE submitted 2026-07-11 econ.TH

Information Comparison of Order Statistics, with Applications to Auctions and Voting

Alfredo Di Tillio , Marco Ottaviani , Peter N. S{\o}rensen This is my paper
classification econ.TH
keywords accuracycomparison of experimentsorder statisticsauctionsinformation aggregationstrategic votingdata selectionlog-supermodularity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper compares how much an order statistic reveals about an unknown state when one more independent draw is added to the sample. It proves that the k-th highest of n+1 draws is Lehmann-more accurate than the k-th highest of n exactly when the cumulative reverse hazard is log-supermodular, and that the symmetric lower-tail comparison is governed by log-supermodularity of the cumulative hazard. Global reversals of these comparisons are exceptional: they force the signals, up to increasing transforms, to be exponential or reflected-exponential location experiments. In large samples, middle ranks become fully informative under identification alone, while fixed extreme ranks need unbounded-informativeness tail conditions; when full learning fails, residual noise is a location experiment that becomes Blackwell richer as the fixed extreme rank moves toward the center. The same order-statistic comparisons unify information aggregation for multi-unit auction prices and give a direct reading of how jury size and quota rules affect the accuracy of strategic voting.

Core claim

Under the monotone likelihood ratio property, more maximal selection raises Lehmann accuracy of the k-th highest order statistic if and only if the cumulative reverse hazard is log-supermodular, and more minimal selection raises accuracy of the k-th lowest if and only if the cumulative hazard is log-supermodular. Reversals at every node force exponential (or reflected-exponential) location form up to increasing transforms. Middle order statistics have asymptotic full accuracy under identification alone; bounded ranks require the matching unbounded-informativeness tail condition, and when that fails the residual limit experiments become Blackwell more informative as the extreme rank becomes m

What carries the argument

Lehmann accuracy applied to order statistics of conditionally i.i.d. signals, with log-supermodularity of the cumulative reverse hazard R = −log F and the cumulative hazard H = −log(1−F) as the exact conditions that orient the finite-sample “accordion” of adjacent rank comparisons.

Load-bearing premise

Higher signal realizations must be stronger evidence for higher states (monotone likelihood ratio); without that ranking the accuracy order need not control the monotone decision problems that justify the auction and voting conclusions.

What would settle it

Exhibit a continuous MLR family whose cumulative reverse hazard fails log-supermodularity, yet the k-th highest of n+1 is still Lehmann more accurate than the k-th highest of n for some interior k and n; or check whether any global reversal of maximal selection fails to reduce, after increasing transforms, to exponential location noise.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Holding the number of objects fixed, adding a bidder improves multi-unit auction price informativeness exactly when the cumulative reverse hazard is log-supermodular.
  • Raising both bidders and objects by one improves price informativeness exactly when the cumulative hazard is log-supermodular.
  • Proportional-quota rules make large juries asymptotically correct under identification alone; finite-quota rules (including unanimity) require the matching unbounded-informativeness tail condition.
  • Blocks of consecutive order statistics become more accurate under log-supermodularity of the hazard rate and reverse hazard rate.
  • When extreme ranks do not fully learn the state, more central fixed ranks leave strictly less residual noise in the limit location experiment.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same accordion can rank platform rules that publish only intermediate quantiles or trimmed blocks rather than full samples.
  • Boundary cases with log-linear reverse hazard (Gumbel noise) make the sample maximum equally accurate for every n, offering a design target when size-invariance at the extreme is desired.
  • Committee selection that strips both tails (as under peremptory challenge) inherits the block conditions, so hazard-rate log-supermodularity becomes the relevant design check.
  • Fixed-n horizontal comparisons of adjacent ranks, treated only briefly here, are likely the binding constraint for practical quota design even when vertical sample-size comparisons are global.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper compares the Lehmann accuracy of order statistics X_{k,n} from conditionally i.i.d. draws with MLR densities as sample size grows. Theorems 1 and 1* give necessary and sufficient conditions: more minimal selection (X_{k,n} to X_{k+1,n+1}) raises accuracy iff the cumulative hazard H is log-supermodular, and more maximal selection (X_{k,n} to X_{k,n+1}) raises accuracy iff the cumulative reverse hazard R is log-supermodular. Theorems 2 and 2* show that global reversals force (up to increasing transforms) exponential or reflected-exponential location form. Theorem 3 characterizes asymptotic full accuracy: middle ranks need only identification; bounded ranks need the corresponding unbounded-informativeness tail condition. Theorems 4–5* describe residual location limits and Blackwell rankings of more central extreme ranks. Multidimensional block comparisons (Theorem 6) require log-supermodularity of hazard rates. Applications unify auction price aggregation and recast strategic voting as order-statistic experiments.

Significance. If correct, the paper supplies a clean, unified comparative-statics toolkit for information aggregation under data selection. The finite-sample accordion comparisons (Theorems 1–2*) are sharp and novel beyond the boundary maxima/minima treated in Di Tillio et al. (2021); the large-sample results recover and extend Wilson–Milgrom and Pesendorfer–Swinkels double-largeness logic while adding residual-limit Blackwell rankings; the voting application cleanly links strategic pivotality to order statistics. Strengths include complete Appendix proofs using standard Beta/Gamma and star-order tools, necessity as well as sufficiency for the one-dimensional claims, and explicit dual statements via reflection. The work is of clear interest to mechanism design, information economics, and political economy.

major comments (2)
  1. Theorem 6 states only sufficient conditions (log-supermodularity of hazard rates h and r) for block accuracy, while Theorems 1/1* are if-and-only-if. The text notes that the rate conditions are stronger than the cumulative conditions because of endogenous truncations, but does not discuss whether they are close to necessary or whether counter-examples exist under mere log-supermodularity of H or R. A short remark or example clarifying the gap would strengthen the multidimensional claim that is used for committee selection and multi-bid auctions.
  2. Section 6 applies the results to affiliated-value multi-unit auctions by asserting that equilibrium bids are strictly increasing, so prices are monotone transforms of order statistics. The maintained setup is conditional i.i.d. signals given θ (Section 2). Affiliation of values is invoked via Milgrom–Weber, but the paper never states the precise joint distribution of (values, signals) under which the MLR and conditional-independence assumptions remain compatible with affiliation. A one-paragraph clarification of the maintained information structure would remove any ambiguity about the scope of Propositions 1–2.
minor comments (4)
  1. Figure 1 and the two panels of Figure 2 are described in the text but would benefit from explicit arrow labels (e.g., “H log-SM” vs “R log-SM”) so that the accordion orientations are immediately readable.
  2. The companion paper Di Tillio et al. (2026) is cited for local accuracy and equilibrium stability in voting; a single sentence in Section 7 stating what is deliberately left to the companion would help readers who encounter only the present manuscript.
  3. Notation for lower/upper classes (L, U) and the normalizations W^L_{r,n}, Y^U_{r,n} is introduced densely in Section 4.2; a short display summarizing the objects would improve readability.
  4. Typos: “V oting” with a space appears in the title and section headings; “HEUROPE” in the acknowledgments should be “Horizon Europe” or similar.

Circularity Check

1 steps flagged

No significant circularity: finite-sample and large-sample order-statistic comparisons are derived from Lehmann accuracy applied to explicit cdfs under MLR, not from fitted parameters or tautological self-definition.

specific steps
  1. self citation load bearing [Section 5 (Multidimensional Experiments), Theorem 6 and surrounding text; also Introduction and Section 2 referencing Di Tillio et al. (2021)]
    "Exploiting the multidimensional accuracy framework of Di Tillio et al. (2021), we present the counterpart of our one-dimensional results for vectors of selected data. ... Theorems 1 and 1* generalize the comparison in Theorem 2 of Di Tillio et al. (2021)."

    The multidimensional accuracy definition and the special-case maximum comparison are taken from the authors' prior work. This is framework continuity rather than a circular reduction of the new scalar Theorems 1–5, which are proved independently from the definition of Lehmann accuracy and the cdfs of order statistics. The self-citation is therefore minor and not load-bearing for the paper's strongest claims.

full rationale

The paper's strongest claims (Theorems 1/1*, 2/2*, 3–5) are self-contained mathematical comparisons. Accuracy is defined via Lehmann's order on the explicit cdfs F_{k,n} of order statistics; necessity and sufficiency for log-supermodularity of H (resp. R) reduce the adjacent comparison to a star-order property of exponential order statistics (Lemma 1) plus monotonicity of the ratio map T induced by log-supermodularity, with exceptional global reversals forcing the exponential-location form via boundary minima/maxima. Large-sample AFA and residual location limits follow from standard Beta/Gamma extreme-value arguments under the maintained MLR and identification. Self-citations to Di Tillio et al. (2021) supply the multidimensional accuracy notion and the special-case maximum comparison that is generalized here, and (2026) is a companion on local voting comparisons; neither is load-bearing for the new scalar theorems, which stand independently under stated assumptions. There is no fitted input renamed as prediction, no uniqueness theorem imported to forbid alternatives, and no renaming of a known empirical pattern as a first-principles derivation. Score 1 reflects only minor, non-load-bearing self-citation for framework continuity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 2 invented entities

The paper is a pure comparative-statics theory paper. Load-bearing content is standard probability/statistics plus the domain MLR assumption that makes Lehmann accuracy the right order for monotone decision problems. No free parameters are fitted. Invented entities are definitional (accordion geometry, lower/upper classes) rather than new physical objects.

axioms (5)
  • domain assumption Monotone likelihood ratio property (MLR) of the family f(x|θ)
    Section 2, display (MLR). Required so that accuracy comparisons translate into value rankings for IDO/monotone decision problems used in auctions and voting.
  • domain assumption Conditional i.i.d. draws with absolutely continuous F(·|θ) positive on interior support; F continuous in θ
    Section 2 regularity. Standard for order-statistic densities and continuous comparative statics.
  • domain assumption Identification (ID): distinct states induce distinct distributions F(·|θ)
    Section 4.1. Necessary for asymptotic full accuracy statements; paired with MLR to obtain strict stochastic dominance.
  • standard math Lehmann (1988) accuracy order as the informativeness criterion for IDO preferences
    Section 2. Standard comparison of experiments; linked to Quah-Strulovici IDO.
  • standard math Extreme-value and Beta-Gamma identities for order-statistic limits
    Used in Theorems 4–5 proofs (Leadbetter et al.; beta-gamma identity). Standard probability tools.
invented entities (2)
  • Accordion of adjacent order-statistic comparisons (more maximal vs more minimal selection) no independent evidence
    purpose: Organizes the finite-sample geometry of adding one draw at every rank
    Definitional diagram (Figure 1); not an ontological postulate, but a new organizing device for the comparisons.
  • Lower classes / upper classes of states (nonseparability partitions) no independent evidence
    purpose: Describe residual information when unbounded informativeness fails in a tail
    Defined from the failure of (LI)/(UI) via limits of F ratios; standard equivalence-class construction, not a new physical entity.

pith-pipeline@v1.1.0-grok45 · 37015 in / 2767 out tokens · 25800 ms · 2026-07-14T11:45:10.095007+00:00 · methodology

0 comments
read the original abstract

We compare the informativeness of order statistics in a sample of conditionally independent draws from a distribution \(F(x\mid\theta)\) as the sample size n increases. The k-th highest of n+1 draws is more accurate than the k-th highest of n if and only if the cumulative reverse hazard -\log F(x\mid\theta) is log-supermodular. Symmetrically, the k-th lowest is more accurate if and only if the cumulative hazard -\log(1-F(x\mid\theta)) is log-supermodular. Reversals are exceptional, occurring only for experiments that are, up to increasing transformations, exponential location experiments. In large samples, middle order statistics are asymptotically fully informative, while bounded lower and upper ranks require unbounded informativeness tail conditions. When full learning fails, bounded ranks converge to location experiments, and more central ranks are Blackwell more informative. Extending the analysis from scalar order statistics to blocks of selected data, we obtain multidimensional comparisons under log-supermodularity of hazard rates. The results unify and extend information-aggregation results in auctions and provide a new order-statistic approach to strategic voting.

Figures

Figures reproduced from arXiv: 2607.10435 by Alfredo Di Tillio, Marco Ottaviani, Peter N. S{\o}rensen.

Figure 1
Figure 1. Figure 1: The accordion when both more minimal and more maximal selection increase accuracy. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: more minimal selection increases and more maximal selection decreases accuracy. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

discussion (0)

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