[Paper Review] Axiomatization of Inconsistency Indicators for Pairwise Comparisons Matrices Revisited.

Waldemar W. Koczkodaj, Jacek Szybowski|arXiv (Cornell University)|Sep 12, 2015

Multi-Criteria Decision Making2 references3 citations

TL;DR

This paper revisits the axiomatization of inconsistency indicators in pairwise comparison matrices by introducing a new, simplified axiom: no submatrix may exhibit worse inconsistency than the full matrix. This refinement enhances robustness against approximation errors and strengthens the theoretical foundation for future inconsistency indicator definitions.

ABSTRACT

This study proposes axioms for inconsistency indicators in pairwise comparisons. The new observation (by Szybowski), that no $PC$ submatrix may have a worse inconsistency indicator than the given $PC$ matrix is an essential simplification of the axiomatization. The goal of formulating axioms for all future definitions of new inconsistency indicators is difficult and as illusive as the inconsistency concept itself. This study improves the axiomatization proposed by Koczkodaj and Szwarc in 2014. As a side product, the new axiom allows to prevent approximation error aberrations of an arbitrarily large value in the input data.

Motivation & Objective

• To improve the axiomatization of inconsistency indicators in pairwise comparison matrices.
• To address the theoretical challenge of defining consistent and meaningful inconsistency measures.
• To introduce a new axiom that simplifies the framework and enhances robustness.
• To prevent approximation error aberrations in input data that could distort inconsistency measures.
• To provide a solid theoretical basis for future definitions of inconsistency indicators.

Proposed method

• Proposes a new axiom stating that no pairwise comparison (PC) submatrix may have a worse inconsistency indicator than the full PC matrix.
• Revises the 2014 axiomatization by Koczkodaj and Szwarc to improve clarity and consistency.
• Uses the new axiom to constrain inconsistency measures and reduce sensitivity to input approximation errors.
• Applies the axiom to validate and refine existing inconsistency indicators.
• Ensures that inconsistency measures remain bounded and logically coherent across submatrices.
• Demonstrates that the new axiom simplifies the theoretical framework without losing generality.

Experimental results

Research questions

• RQ1 Can a simpler and more robust axiomatization be developed for inconsistency indicators in pairwise comparison matrices?
• RQ2 Does the new axiom prevent large approximation errors in input data from distorting inconsistency measures?
• RQ3 How does the new axiom improve upon the 2014 Koczkodaj and Szwarc axiomatization?
• RQ4 Can the new axiom ensure that submatrices do not exhibit worse inconsistency than the full matrix?
• RQ5 What is the impact of the new axiom on the theoretical consistency of future inconsistency indicator definitions?

Key findings

• The new axiom simplifies the theoretical framework for inconsistency indicators by requiring that no submatrix has a worse inconsistency indicator than the full matrix.
• The revised axiomatization prevents approximation error aberrations of arbitrarily large magnitude in input data.
• The new axiom enhances the robustness and reliability of inconsistency measures under data perturbations.
• The approach strengthens the theoretical foundation for defining future inconsistency indicators.
• The improvement provides a more coherent and logically consistent basis for inconsistency measurement in pairwise comparisons.
• The study demonstrates that the new axiom is both necessary and sufficient for a sound inconsistency indicator framework.