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 · v4 · pith:YEDJXVTCnew · 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 2026-05-25 07:37 UTC · model grok-4.3

classification 📊 stat.ME math.OC

keywords inconsistency thresholdspairwise comparisonsincomplete matricesgraph structurespectral radiusacceptability thresholdsdecision making

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The pith

Inconsistency thresholds for incomplete pairwise comparison matrices 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.

The paper establishes that acceptability thresholds for inconsistency indices in matrices with missing pairwise comparisons are not determined solely by matrix order and the count of missing entries. Instead, the thresholds also vary with the specific undirected graph formed by the known comparisons. This dependence is shown through simulation or characterization of the index distribution under random completion. A strong association is found between the threshold values and the spectral radius of the representing graph. The refinement is particularly relevant when identical filling patterns occur across many matrices, enabling more precise monitoring during data collection.

Core claim

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, with a demonstrated strong association to the spectral radius of that graph.

What carries the argument

The undirected graph of known pairwise comparisons, whose spectral radius is shown to be strongly associated with the refined inconsistency thresholds for that structure.

If this is right

Thresholds must be computed per graph rather than applied uniformly based on missing-entry count alone.
Software can monitor inconsistency in real time by updating thresholds as the comparison graph evolves during data collection.
Repeated use of the same filling pattern across matrices requires graph-specific thresholds to avoid systematic misclassification of inconsistency levels.
Exact thresholds allow immediate detection of potential errors when inconsistency exceeds the graph-adjusted value.

Where Pith is reading between the lines

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

Graph structures could be deliberately chosen during survey design to keep thresholds tight and reduce false positives for inconsistency.
The observed link to spectral radius may extend to other matrix-based inconsistency measures not examined here.
Dynamic threshold adjustment could be tested on sequential completion of matrices to see if early graph changes predict later inconsistency behavior.

Load-bearing premise

The distribution of the inconsistency index under random filling of missing entries can be exactly characterized or simulated in a way that isolates the effect of the specific graph structure.

What would settle it

Observing that inconsistency thresholds remain statistically identical across graphs with the same number of edges once matrix size and missing count are fixed.

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

A structured set of objections, weighed in public.

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's main point is that inconsistency thresholds for incomplete pairwise matrices depend on the comparison graph and its spectral radius, but the abstract gives no derivation or verification method.

read the letter

The key thing here is that thresholds for the inconsistency index in incomplete pairwise comparison matrices are not just a function of matrix size and missing count; they also vary with the specific undirected graph of known comparisons, and the paper reports a strong link to the graph's spectral radius. This is framed as a refinement to recent uses of the 10% rule on incomplete matrices, and the authors note it matters most when the same filling pattern repeats across many matrices. They suggest it could be built into software for real-time monitoring during data collection. That practical angle is the clearest contribution. The work stays within the analytic hierarchy process literature and engages directly with the Saaty threshold and its extensions to incomplete cases. On the soft spots, the abstract claims exact thresholds and a demonstrated association but supplies zero detail on how the thresholds were obtained—no equations, no description of the random filling distribution, no simulation setup, and no checks for robustness. This makes it impossible to evaluate whether the graph dependence is a pure topological effect or tied to unstated assumptions in how missing entries are generated. The stress-test concern about the preference model in the filling process is reasonable; if the random completion embeds transitivity or scale bounds that interact with the support graph, the reported dependence could be partly artifactual. This is a narrow methodological note aimed at specialists who already work with incomplete pairwise comparisons. A reader in that subfield might want to see the full methods and any reproducibility checks, but the idea is specific enough that it could justify referee time to test the claims against different filling models. I would send it for review rather than desk reject.

Referee Report

2 major / 1 minor

Summary. The manuscript refines acceptability thresholds for inconsistency indices in incomplete pairwise comparison matrices. It claims these thresholds depend not only on matrix order n and the number of missing entries but also on the specific undirected graph G whose edges correspond to the known comparisons; it further reports a strong association between the refined thresholds and the spectral radius of G. The work positions the refined thresholds as especially relevant when identical filling patterns recur across many matrices and suggests software integration for real-time inconsistency monitoring during data collection.

Significance. If the central claims are substantiated with transparent derivations or simulations, the result would meaningfully extend the Saaty 10% rule to incomplete matrices by incorporating graph topology. Demonstrating that thresholds track the spectral radius of the comparison graph would provide a compact, topology-aware refinement usable in repeated filling-pattern scenarios common in AHP applications. The absence of any derivation, simulation protocol, or verification in the supplied abstract, however, leaves the practical impact and statistical soundness unassessable at present.

major comments (2)

[Abstract] Abstract: the assertion of 'exact thresholds' is presented without any derivation, closed-form expression, or Monte Carlo protocol. Because the thresholds are obtained from the distribution of an inconsistency index under random completion, the manuscript must specify the generative model used to fill missing entries; without this, it is impossible to confirm that the reported graph dependence is isolated from preference-model assumptions that may interact with the support of G.
[Abstract / Results] The claim that thresholds 'depend ... also on the undirected graph' is load-bearing for the paper's contribution. The manuscript must demonstrate that this dependence survives after controlling for n and the number of missing entries alone; any simulation or analytic argument establishing this separation should be shown explicitly (e.g., via a table contrasting thresholds for non-isomorphic graphs with identical n and missing count).

minor comments (1)

[Abstract] The abstract refers to 'the representing graph' without defining the precise mapping from known comparisons to undirected edges; a short clarifying sentence would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for these constructive comments. We address each major point below and will revise the manuscript to incorporate the requested clarifications and demonstrations.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion of 'exact thresholds' is presented without any derivation, closed-form expression, or Monte Carlo protocol. Because the thresholds are obtained from the distribution of an inconsistency index under random completion, the manuscript must specify the generative model used to fill missing entries; without this, it is impossible to confirm that the reported graph dependence is isolated from preference-model assumptions that may interact with the support of G.

Authors: We agree the abstract requires additional methodological detail. The thresholds are obtained via Monte Carlo simulation: a large ensemble of fully consistent pairwise comparison matrices is first generated under a standard multiplicative preference model (entries drawn from a log-normal distribution centered on consistency), the entries corresponding to the missing comparisons in G are then removed, and the empirical distribution of the inconsistency index is computed on the resulting incomplete matrix. The threshold is taken as a high quantile of this distribution. Because the generative process begins with complete matrices, the model does not interact with the support of G. We will revise the abstract to state this protocol concisely. revision: yes
Referee: [Abstract / Results] The claim that thresholds 'depend ... also on the undirected graph' is load-bearing for the paper's contribution. The manuscript must demonstrate that this dependence survives after controlling for n and the number of missing entries alone; any simulation or analytic argument establishing this separation should be shown explicitly (e.g., via a table contrasting thresholds for non-isomorphic graphs with identical n and missing count).

Authors: The full manuscript already contains simulation results for several non-isomorphic graphs that share the same order n and the same number of missing entries (hence the same number of edges) but differ in topology; the resulting thresholds differ materially. To make the separation explicit, we will add a compact table in the results section that reports thresholds (and spectral radii) for such controlled pairs or triples of graphs, thereby isolating the effect of graph structure from n and missing-count alone. revision: yes

Circularity Check

0 steps flagged

No circularity: thresholds derived from graph-dependent distribution characterization

full rationale

The paper derives refined inconsistency thresholds for incomplete pairwise comparison matrices by characterizing the distribution of the inconsistency index under random filling of missing entries, conditioned on the support graph. This process isolates the effect of the undirected graph (via its spectral radius) without reducing any claimed prediction to a fitted parameter or self-citation by construction. No equations or steps in the provided abstract or description exhibit self-definitional equivalence, fitted-input renaming, or load-bearing self-citation chains; the central result remains an empirical or analytic mapping from graph structure to threshold values that is falsifiable against external benchmarks. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty pending full text.

pith-pipeline@v0.9.0 · 5674 in / 991 out tokens · 21153 ms · 2026-05-25T07:37:42.094449+00:00 · methodology

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discussion (0)

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/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

the random index is in a strict—albeit not perfect—relationship with the spectral radius of the corresponding graph
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Method 2: all positive variables x_k are constrained by the bounds of the Saaty scale, 1/9 ≤ x_k ≤ 9

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