Some Notes on the Similarity of Priority Vectors Derived by the Eigenvalue Method and the Geometric Mean Method

TL;DR

This paper compares the eigenvalue and geometric mean methods for deriving priority vectors from pairwise comparison matrices, analyzing their similarities, differences, and the influence of matrix inconsistency.

Contribution

It introduces propositions on the (dis)similarity of rankings from both methods and explores their relationship with matrix inconsistency and rank correlation measures.

Findings

Rankings become more similar as matrix size increases.

Higher inconsistency leads to greater differences between methods.

Numerical examples and simulations support theoretical results.

Abstract

This paper examines the differences in ordinal rankings obtained from a pairwise comparison matrix using the eigenvalue method and the geometric mean method. First, we introduce several propositions on the (dis)similarity of both rankings concerning the matrix size and its inconsistency expressed by the Koczkodaj's inconsistency index. Further on, we examine the relationship between differences in both rankings and Kendall's rank correlation coefficient $\tau$ and Spearman's rank coefficient $\rho$. Apart from theoretical results, intuitive numerical examples and Monte Carlo simulations are also provided.