[Paper Review] A Geometric Framework for the Inconsistency in Pairwise Comparisons

Waldemar W. Koczkodaj, Jean-Pierre Magnot|arXiv (Cornell University)|Jan 23, 2016

Data Management and Algorithms6 references3 citations

TL;DR

This paper introduces a geometric framework for analyzing inconsistency in pairwise comparison matrices by generalizing them to non-abelian Lie groups, providing a necessary and sufficient consistency criterion and a method to find the nearest consistent matrix within the group structure. The approach uses connections to a simplex for geometric interpretation, enhancing approximate reasoning through inconsistency reduction.

ABSTRACT

In this study, a pairwise comparison matrix is generalized to the case when coefficients create Lie group $G$, non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons matrix (extended to the class of group $G$) are proposed. A geometric interpretation of pairwise comparisons matrices in terms of connections to a simplex is given. Approximate reasoning is more effective when inconsistency in data is reduced.

Motivation & Objective

• To generalize pairwise comparison matrices from real numbers to elements of a non-abelian Lie group G.
• To establish a necessary and sufficient condition for consistency in the generalized group-based framework.
• To develop criteria for computing the nearest consistent matrix within the group G.
• To provide a geometric interpretation of pairwise comparison matrices via connections to a simplex.
• To improve approximate reasoning by reducing inconsistency in decision-making data.

Proposed method

• Generalize pairwise comparison matrices by replacing scalar entries with elements of a Lie group G.
• Define consistency using group-theoretic conditions that generalize the classical transitivity property.
• Formulate the problem of finding the nearest consistent matrix as a minimization over the group G.
• Use geometric structures—specifically connections on a simplex—to interpret the matrix entries and their consistency.
• Apply differential geometry tools to analyze the curvature and holonomy of the connection, linking inconsistency to geometric deviation.
• Derive conditions under which a matrix in G is consistent, based on path independence of group products.

Experimental results

Research questions

• RQ1 What is the necessary and sufficient condition for consistency in pairwise comparison matrices when entries belong to a non-abelian Lie group G?
• RQ2 How can one compute the nearest consistent matrix to a given inconsistent matrix in the group G?
• RQ3 What geometric structure underlies the representation of pairwise comparison matrices in the generalized group setting?
• RQ4 How does inconsistency in the matrix relate to curvature or holonomy in the associated geometric connection?
• RQ5 To what extent does reducing inconsistency improve the effectiveness of approximate reasoning?

Key findings

• A necessary and sufficient condition for consistency in the group-based framework is that the product of group elements along any closed loop in the comparison graph equals the identity element.
• The problem of finding the nearest consistent matrix in G is well-defined and can be approached via optimization on the Lie group manifold.
• The geometric interpretation via connections to a simplex provides a natural way to visualize and quantify inconsistency as curvature in the connection.
• Inconsistency in the matrix corresponds to non-trivial holonomy around closed loops, offering a differential-geometric measure of inconsistency.
• Reducing inconsistency through nearest consistent matrix approximation leads to more effective approximate reasoning in decision-making processes.