Analysis of natural cardinal ranking vectors for pairwise comparisons and the universal efficiency of the Perron geometric mean

TL;DR

This paper compares various methods for deriving ranking vectors from pairwise comparison matrices, demonstrating that the geometric mean of the Perron vectors is universally efficient and outperforming other approaches.

Contribution

It introduces and empirically evaluates several alternative ranking vectors, proving the universal efficiency of the geometric mean of Perron vectors.

Findings

The geometric mean of Perron vectors is universally efficient.

The Gass and Rapcsák (2004) vector performs better than others.

All eight vectors are efficient for matrices derived from consistent matrices with modifications.

Abstract

In models using pair-wise (ratio) comparisons among alternatives, a cardinal ranking vector should be deduced from a reciprocal matrix. The right Perron eigenvector (RP) was traditionally used, though several other options have emerged. We consider some alternatives, mostly new, namely the entry-wise reciprocal of the left Perron vector (LP), the left singular vector (LS), the entry-wise reciprocal of the right singular vector (RS), the arithmetic and geometric means of RP and LP (AP and GP), and of LS and RS (AS and GS). (The ranking vector AS was proposed by Gass and Rapcs\'ak (2004)). All 8 of these vectors produce the natural vector in the consistent case. We compare them empirically, in terms of efficiency, for random matrices, as a function of the number of alternatives. It turns out that the vector GP is universily efficient, and this fact is proven. The vector GS performs better that the remaining 6 vectors. We show that, for reciprocal matrices obtained from consistent matrices by modifying one column and the corresponding row, all 8 vectors are efficient. Moreover, the cone generated by the columns is efficient.