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On strict ranking by pairwise comparisons

Jean-Pierre Magnot

TL;DR

This work investigates how to obtain a strict, tie-free ranking from pairwise comparison matrices used in AHP. It introduces the $\\mathcal{R}$-condition to guarantee a strict ranking and examines how standard consistencization can disrupt this property, even when starting from imperfect data. The authors characterize ranking loci, demonstrate instability under projection-based consistency procedures with 3×3 counterexamples, and propose a new functional $\Phi$ to enforce both consistency and the $\\mathcal{R}$-condition, suggesting a gradient-based optimization pathway. The results provide a framework for generating consistent PC matrices that induce a strict ranking and highlight open questions in the mathematical analysis of the proposed minimization problem with practical implications for decision-making processes relying on PC matrices.

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.

On strict ranking by pairwise comparisons

TL;DR

This work investigates how to obtain a strict, tie-free ranking from pairwise comparison matrices used in AHP. It introduces the -condition to guarantee a strict ranking and examines how standard consistencization can disrupt this property, even when starting from imperfect data. The authors characterize ranking loci, demonstrate instability under projection-based consistency procedures with 3×3 counterexamples, and propose a new functional to enforce both consistency and the -condition, suggesting a gradient-based optimization pathway. The results provide a framework for generating consistent PC matrices that induce a strict ranking and highlight open questions in the mathematical analysis of the proposed minimization problem with practical implications for decision-making processes relying on PC matrices.

Abstract

We attack the problem of getting a strict ranking (i.e. a ranking without equally ranked items) of items from a pairwise comparisons matrix. Basic structures are described, a first heuristical approach based on a condition, the 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.
Paper Structure (9 sections, 6 theorems, 40 equations)

This paper contains 9 sections, 6 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From multiplication to addition: the basic viewpoint for making consistent a PC matrix
  4. Multiplication versus addition: the viewpoint of Lie theory and quantum gravity
  5. Strict ranking, related sets of weights and pairwise comparisons
  6. Ranking loci in $\mathcal{R}-$condition and non-consistent ranking
  7. Unstability of $\mathcal{R}PC_n$ under consistencization procedures
  8. On a consistencization procedure leading to strict ranking
  9. Acknowledgements:

Key Result

Theorem 3.3

There is a bijection between $\mathfrak{S}_n$ and the admissible loci in $\mathcal{R}PC_n.$

Theorems & Definitions (12)

  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Remark 4.1
  • Theorem 4.2
  • Definition 4.3
  • ...and 2 more