On strict ranking by pairwise comparisons

Jean-Pierre Magnot

On strict ranking by pairwise comparisons

T0 review · 2 major / 0 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read A minimization problem produces consistent pairwise comparison matrices that induce strict rankings.

desk verdict The paper sketches an R-condition heuristic and a minimization idea for strict rankings from pairwise matrices but supplies no formulation, solvability argument, or example to support the central claim. read the letter →

arxiv 2501.14738 v2 submitted 2024-12-11 cs.IT math.IT

Jean-Pierre Magnot This is my paper

classification cs.ITmath.IT

keywords pairwisecomparisonsstrictrankingconsistentmatrixminimizationproblemR-conditionproceduretransitive

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

The paper tackles deriving a strict ranking of n items, meaning one with no ties, directly from a matrix of pairwise comparisons. It begins by describing basic structures and offering a heuristic method that works when the matrix satisfies the R-condition. After showing where this heuristic falls short, the work shifts to a minimization problem defined over a larger family of matrices. When the minimization succeeds, the output matrix is consistent and its induced ranking is strict.

What carries the argument

The minimization problem that adjusts an input pairwise comparison matrix to a nearby consistent matrix whose ranking has no ties.

What would settle it

A concrete pairwise comparison matrix outside the R-condition class for which every solution of the minimization problem either fails to be consistent or produces at least one tie in the ranking.

Watch

Extended reading notes

Core claim

For pairwise comparison matrices that lie outside the R-condition class, a minimization problem can be stated whose solution yields a consistent matrix from which a strict ranking of the n items follows.

Load-bearing premise

The minimization problem always possesses a solution that is both consistent and induces a strict ranking.

Editorial extensions

If this is right

Strict rankings become available for matrices that the R-condition heuristic cannot handle.
The output matrix satisfies the consistency property required for transitive rankings.
The procedure applies to any input matrix for which the minimization can be solved numerically.

Reading between the lines

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

The method could be tested on empirical preference data from voting or consumer surveys to measure how often the minimization succeeds.
It suggests that ranking tasks can be recast as optimization problems over matrix space rather than purely combinatorial searches.
If the minimization is convex, standard solvers would guarantee global solutions and therefore reliable strict rankings.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The punchline is that this work identifies the problem of strict ranking from pairwise comparisons, proposes an R-condition as a heuristic, notes its limits, and ends by stating a minimization problem that supposedly handles more cases. Nothing beyond that is shown. The abstract gives no explicit objective or constraints for the minimization, no existence or uniqueness result, and no numerical check that a solution, if found, actually produces a consistent transitive matrix with no ties. The stress-test concern holds up on the available text: the claim is conditional on solving the problem, yet the paper offers no reason to believe solutions exist or work as advertised outside the R-condition class. What the paper does reasonably is frame the basic structures and point out where the heuristic stops. That part is clear and direct. The rest stays at the level of suggestion without the supporting math or verification that would make the minimization usable. No derivations, no error bounds, no comparison to existing ranking methods appear. This leaves the contribution looking incremental at best and unproven at worst. The paper is aimed at people already working on preference aggregation or decision theory who might want another heuristic to try, but a reader would have to supply the missing pieces themselves. It is not ready for serious refereeing because the load-bearing claim rests on an unshown assumption rather than evidence or proof. I would not bring it to a reading group or cite it as is.

Referee Report

2 major / 0 minor

Summary. The manuscript addresses deriving strict (tie-free) rankings from pairwise comparison matrices. It describes basic structures, proposes a heuristic based on the R-condition, analyzes its limits, and concludes by formulating a minimization problem applicable to a wider class of matrices; the central claim is that solving this problem yields consistent pairwise comparisons that induce a strict ranking.

Significance. If the minimization problem is shown to be solvable with solutions that are guaranteed to be consistent and strict, the work could provide a useful extension beyond the R-condition class for ranking applications. The identification of limitations in the initial heuristic is a positive step, but the absence of any existence argument, explicit formulation details, or verification data means the significance cannot yet be assessed as realized.

major comments (2)

[Abstract and minimization-problem section] Abstract (final sentence) and the section introducing the minimization problem: the claim that solving the minimization 'produces consistent pairwise comparisons that produce a strict ranking' for matrices outside the R-condition class rests on an unshown solvability assumption. No existence proof, uniqueness result, or numerical example is supplied showing that a minimizer exists, is attained, and forces both transitivity (consistency) and absence of ties (strictness). This is load-bearing for the extension beyond the R-condition.
[Minimization-problem section] No derivations, error analysis, or validation data appear for the minimization problem (as noted in the abstract's conditional phrasing). Without these, it is impossible to confirm that the feasible set is nonempty or that attained solutions satisfy the required properties for the claimed wider class.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's constructive comments. We address the major concerns point by point and indicate the planned revisions to strengthen the manuscript.

read point-by-point responses

Referee: Abstract (final sentence) and the section introducing the minimization problem: the claim that solving the minimization 'produces consistent pairwise comparisons that produce a strict ranking' for matrices outside the R-condition class rests on an unshown solvability assumption. No existence proof, uniqueness result, or numerical example is supplied showing that a minimizer exists, is attained, and forces both transitivity (consistency) and absence of ties (strictness). This is load-bearing for the extension beyond the R-condition.

Authors: We acknowledge that the manuscript does not include an existence proof or numerical examples for the minimization problem. The abstract uses conditional language ('If solved') precisely because these aspects are not demonstrated. To address this, the revised version will include a discussion of the problem's feasibility and at least one concrete numerical example illustrating that a minimizer can be attained and yields a consistent strict ranking for a matrix outside the R-condition class. revision: yes
Referee: No derivations, error analysis, or validation data appear for the minimization problem (as noted in the abstract's conditional phrasing). Without these, it is impossible to confirm that the feasible set is nonempty or that attained solutions satisfy the required properties for the claimed wider class.

Authors: The current manuscript prioritizes the formulation of the minimization problem over its full analysis. We agree that additional material is needed to validate the approach. In revision, we will add derivations for the objective and constraints, a brief error analysis, and validation through examples showing nonempty feasible sets and attainment of solutions with the desired properties. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: minimization problem introduced as external proposal without self-referential reduction

full rationale

The paper's derivation chain consists of describing basic structures, proposing an R-condition heuristic, noting its limits, and then stating a minimization problem for wider applicability. The abstract and provided text present this minimization conditionally ('If solved...') without any equations, fitted parameters, self-citations, or uniqueness theorems that reduce the output to the input by construction. No load-bearing step equates a claimed prediction or result to a prior definition or fit within the paper itself; the approach is offered as an independent method to be solved externally. This is the common case of a self-contained proposal with no detected circularity.

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

Only the abstract is available; no free parameters, axioms, or invented entities are identifiable.

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Cite this review

Pith. "Pith review of On strict ranking by pairwise comparisons." pith.science (2026). https://pith.science/paper/2501.14738

@misc{pith2026250114738,
  author       = {Pith},
  title        = {Pith review of: On strict ranking by pairwise comparisons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2501.14738}},
  note         = {Machine review of arXiv:2501.14738}
}

abstract

We attack the problem of getting a strict ranking (i.e. a ranking without equally ranked items) of $n$ items from a pairwise comparisons matrix. Basic structures are described, a first heuristical approach based on a condition, the $\mathcal{R}-$condition, is proposed. Analyzing the limits of this ranking procedure, we finish with a minimization problem which can be applied to a wider class of pairwise comparisons matrices. If solved, it produces consistent pairwise comparisons that produce a strict ranking.

Discussion (0). Sign in to comment.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We finish with a minimization problem which can be applied to a wider class of pairwise comparisons matrices. If solved, it produces consistent pairwise comparisons that produce a strict ranking.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Φ((ai,j)) = (∏ ii(A) / ((log ai,j)^2 + (ii(A))^{n²/2})) · ∑ ((log ai,j)^4 + 1)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.

[1] [1]

Information Sciences, 615 103-117 (2022)

Bartl, D.; Ramík, J.; A new algorithm for computing prior ity vector of pairwise comparisons matrix with fuzzy elements. Information Sciences, 615 103-117 (2022)

work page 2022

[2] [2]

Benford, F.; The law of anomalous numbers. Proc. Am. Philos. Soc. 78 no 4, 551-572 (1938)

work page 1938

[3] [3]

Bozóki, S.; Rapcsák, T.; On Saaty’s and Koczkodaj’s inco nsistencies of pairwise comparison matrices; J. Glob. Optim. 42 no.2 (2008) 157-175

work page 2008

[4] [4]

Social Choice and Welfare 40 no2, 317-328 (2011)

Colomer, J-M.; Ramon Llull: from Ars electionis to socia l choice theory. Social Choice and Welfare 40 no2, 317-328 (2011)

work page 2011

[5] [5]

Journal of Mathematical Psychology , 29 387–405 (1985) 15

Crawford, R.; Williams, C.; A note on the analysis of subj ective judgement matrices. Journal of Mathematical Psychology , 29 387–405 (1985) 15

work page 1985

[6] [6]

Darko, A.; Chan, A

A. Darko, A.; Chan, A. P. C.; Ameyaw, E. E.; Owusu, E. K.; Pä rn, E.; D. J. Edwards, D. J.; Review of application of analytic hi erarchy process (AHP) in construction. International Journal of Construction Management 19 no5, 436-452 (2019)

work page 2019

[7] [7]

The six blin d men and the elephant

Ellingsen, A.; Lundholm, D.; Magnot, J-P.; “The six blin d men and the elephant”: an interdisciplinary selection of measurem ent features. in: Kielanowski, Piotr (ed.) et al., Geometric methods in ph ysics XL, workshop, Bialowieza, Poland, June 20?25, 2023. Cham: Birk hauser. Trends Math., 275-307 (2024)

work page 2023

[8] [8]

Fadell, E.R.; Husseini, S.Y.; Geometry and topology of conﬁguration spaces Springer, Berlin (2001)

work page 2001