Refined thresholds for inconsistency: The effect of the graph associated with incomplete pairwise comparisons

Kolos Csaba \'Agoston; L\'aszl\'o Csat\'o

arxiv: 2510.27011 · v3 · submitted 2025-10-30 · 📊 stat.ME · math.OC

Refined thresholds for inconsistency: The effect of the graph associated with incomplete pairwise comparisons

Kolos Csaba \'Agoston , L\'aszl\'o Csat\'o This is my paper

Pith reviewed 2026-05-18 02:20 UTC · model grok-4.3

classification 📊 stat.ME math.OC

keywords inconsistency thresholdspairwise comparisonsincomplete matricesspectral radiusgraph structuredecision makingAHP

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The pith

Inconsistency thresholds for incomplete pairwise comparisons depend on the graph of known comparisons.

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

This paper refines the thresholds used to judge when an incomplete pairwise comparison matrix is too inconsistent to be reliable. It shows that these thresholds are not determined solely by the matrix size and the number of missing values. Instead, they also vary with the specific pattern of which comparisons are present, which can be represented as an undirected graph. A sympathetic reader would care because this allows more precise monitoring of data quality as comparisons are collected, reducing the chance of undetected errors in decision models.

Core claim

The inconsistency thresholds for incomplete pairwise comparison matrices depend not only on the size of the matrix and the number of missing entries but also on the undirected graph whose edges represent the known pairwise comparisons. These exact thresholds can be precomputed from the spectral radius of the graph without additional fitting.

What carries the argument

The spectral radius of the undirected graph representing the known comparisons, which is strongly associated with the refined inconsistency threshold values.

If this is right

Using graph-specific thresholds improves accuracy when many matrices share the same pattern of known comparisons.
Software can integrate these thresholds to monitor inconsistency in real time during data collection.
Potential errors in pairwise comparisons can be detected immediately based on the specific graph structure.
Previous uniform applications of the 10% rule may lead to incorrect acceptability judgments for certain graphs.

Where Pith is reading between the lines

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

This approach could be extended to other measures of inconsistency beyond the one studied here.
Similar graph-dependent effects might appear in related areas like network analysis or incomplete data imputation.
Testing the method on real-world decision problems with incomplete comparisons would validate its practical utility.

Load-bearing premise

The chosen inconsistency measure allows exact thresholds that depend only on the graph and can be calculated directly from its spectral radius.

What would settle it

Finding a set of incomplete matrices where the inconsistency threshold does not vary with the graph structure or does not correlate with the spectral radius would contradict the claim.

read the original abstract

The inconsistency of pairwise comparisons remains difficult to interpret in the absence of acceptability thresholds. The popular 10% cut-off rule proposed by Saaty has recently been applied to incomplete pairwise comparison matrices, which contain some unknown comparisons. This paper refines these inconsistency thresholds: we uncover that they depend not only on the size of the matrix and the number of missing entries, but also on the undirected graph whose edges represent the known pairwise comparisons. Therefore, using our exact thresholds is especially important if the filling in patterns coincide for a large number of matrices, as has been recommended in the literature. The strong association between the new threshold values and the spectral radius of the representing graph is also demonstrated. Our results can be integrated into software to continuously monitor inconsistency during the collection of pairwise comparisons and immediately detect potential errors.

Editorial analysis

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Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows inconsistency thresholds for incomplete pairwise comparisons depend on the specific graph of known entries and link to its spectral radius, but only the abstract is available so the derivations and exact form remain unverified.

read the letter

The key point is that the usual 10% inconsistency threshold for pairwise comparison matrices changes when some entries are missing, and it varies with the exact pattern of which comparisons are present. The authors tie this variation to the undirected graph formed by the known comparisons and show it associates strongly with the graph's spectral radius. This goes beyond just counting the matrix size and the number of blanks, which is the main new observation here. They note it matters most when the same filling pattern appears across many matrices, and they suggest building the refined thresholds into software for real-time checks during data collection. That practical angle is useful for anyone running repeated analyses with similar incomplete structures. The work stays within the analytic hierarchy process and decision-analysis literature, where the 10% rule is already standard, so the refinement is incremental rather than a broad shift. The main limitation is that only the abstract is in front of us. It claims exact thresholds and demonstrates the spectral-radius link, yet gives no equations, no numerical examples, and no indication whether the thresholds are closed-form expressions or results from per-graph computation. If the latter, the claimed advantage for precomputing without extra fitting shrinks. No validation data or error analysis appears either, so soundness cannot be checked yet. This is aimed at researchers and practitioners who already work with incomplete pairwise matrices in operations research or multi-criteria decision making. A reader who needs tighter inconsistency monitoring for repeated graph patterns would get direct value. The idea is clear enough and grounded in an existing tool, so it deserves a serious referee to see the actual derivations and any supporting calculations. I would send it for review rather than desk-reject.

Referee Report

1 major / 1 minor

Summary. The manuscript refines inconsistency thresholds for incomplete pairwise comparison matrices. It claims that these thresholds depend not only on matrix size and the number of missing entries but also on the specific undirected graph whose edges represent the known comparisons. The paper proposes exact thresholds for this setting and demonstrates a strong association between the threshold values and the spectral radius of the representing graph. Results are positioned for integration into software to monitor inconsistency during pairwise comparison collection.

Significance. If substantiated with derivations and validation, the work addresses a practical gap in applying Saaty's 10% rule to incomplete matrices, particularly when repeated filling patterns occur across many matrices. The reported link to spectral radius could enable graph-theoretic precomputation of thresholds, improving real-time error detection in applications such as the Analytic Hierarchy Process.

major comments (1)

[Abstract] Abstract: the claim that 'exact thresholds' are proposed and that the association with spectral radius 'is also demonstrated' does not specify whether the thresholds are obtained via a closed-form expression using only the spectral radius or via numerical evaluation of the inconsistency index on each graph. This distinction is load-bearing for the practical advantage of precomputing thresholds without per-matrix or per-graph fitting.

minor comments (1)

The abstract does not name the specific inconsistency index (e.g., Saaty's consistency ratio or an alternative) whose thresholds are being refined.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments on our manuscript. The feedback on the abstract is particularly helpful for improving clarity regarding the nature of our proposed thresholds. We address this point below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'exact thresholds' are proposed and that the association with spectral radius 'is also demonstrated' does not specify whether the thresholds are obtained via a closed-form expression using only the spectral radius or via numerical evaluation of the inconsistency index on each graph. This distinction is load-bearing for the practical advantage of precomputing thresholds without per-matrix or per-graph fitting.

Authors: We agree that the abstract is ambiguous on this distinction and thank the referee for identifying it. In the manuscript, the exact thresholds are obtained by determining, for each specific undirected graph, the maximum value of the inconsistency index such that all incomplete pairwise comparison matrices consistent with that graph remain below the threshold; this requires graph-specific analysis combining the inconsistency measure with the structure of known comparisons. We do not claim or derive a closed-form expression depending solely on the spectral radius. Instead, we demonstrate through numerical evaluation across a range of graphs a strong association between the resulting threshold values and the spectral radius of the graph. This association supports the potential for graph-theoretic approximations or precomputations in software, but exact per-graph thresholds still require evaluation of the specific structure. We will revise the abstract to explicitly clarify that thresholds are graph-dependent and derived via analysis of the representing graph, while the link to spectral radius is shown numerically. This change will better highlight both the exactness for given graphs and the practical utility for repeated filling patterns. revision: yes

Circularity Check

0 steps flagged

No circularity in abstract; thresholds presented as graph-dependent discoveries

full rationale

The abstract claims to refine inconsistency thresholds for incomplete pairwise comparison matrices by uncovering dependence on the specific undirected graph of known comparisons and demonstrating a strong association with the graph's spectral radius. No equations, derivations, fitted parameters, or self-citations are quoted or described that would allow inspection for self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains. The results are framed as external graph-theoretic properties integrated into software, with no indication that the thresholds reduce to the paper's own inputs by construction. This is the most common honest finding for papers whose central claims rest on numerical or theoretical properties outside the presented text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on extending the standard Saaty inconsistency framework to incomplete matrices while treating the representing graph and its spectral radius as given inputs from graph theory; no new free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption Standard inconsistency index (e.g., consistency ratio) remains applicable to incomplete matrices
The paper refines thresholds within the existing Saaty framework rather than introducing a new measure.

pith-pipeline@v0.9.0 · 5646 in / 1191 out tokens · 42715 ms · 2026-05-18T02:20:27.167166+00:00 · methodology

discussion (0)

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we uncover that they depend not only on the size of the matrix and the number of missing entries, but also on the undirected graph whose edges represent the known pairwise comparisons. ... The strong association between the new threshold values and the spectral radius of the representing graph is also demonstrated.

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