Efficiency analysis for the Perron vector of a reciprocal matrix

TL;DR

This paper investigates the efficiency of the Perron vector in reciprocal matrices used for prioritization, showing how extensions can alter efficiency and characterizing specific cases, thus advancing understanding of ranking methods.

Contribution

It provides constructive methods to generate reciprocal matrices with efficient or inefficient Perron vectors and characterizes 4-by-4 cases, extending prior results.

Findings

Existence of reciprocal matrix extensions with efficient and inefficient Perron vectors.

Most extensions of a consistent matrix have efficient Perron vectors.

Characterization of 4-by-4 reciprocal matrices with inefficient Perron vectors.

Abstract

In prioritization schemes, based on pairwise comparisons, such as the Analytical Hierarchy Process, it is necessary to extract a cardinal ranking vector from a reciprocal matrix that is unlikely to be consistent. It is natural to choose such a vector only from efficient ones. One of the most used ranking methods employs the (right) Perron eigenvector of the reciprocal matrix as the vector of weights. It is known that the Perron vector may not be efficient. Here, we focus on extending arbitrary reciprocal matrices and show, constructively, that two different extensions of any fixed size always exist for which the Perron vector is inefficient and for which it is efficient, with the following exception. If B is consistent, any reciprocal matrix obtained from B by adding one row and one column has efficient Perron vector. As a consequence of our results, we obtain families of reciprocal matrices for which the Perron vector is inefficient. These include known classes of such matrices and many more. We also characterize the 4-by-4 reciprocal matrices with inefficient Perron vector. Some prior results are generalized or completed.