Theoretical properties of the eigenvector method

TL;DR

This paper analyzes the theoretical limitations of the eigenvector method for deriving weights from pairwise comparison matrices, highlighting issues like asymmetry, group incoherence, and efficiency, with illustrative examples and open questions.

Contribution

It provides a comprehensive theoretical critique of the eigenvector method, identifying five key issues and proposing directions for future research.

Findings

Identifies right-left asymmetry in eigenvector calculations

Shows group incoherence can lead to inconsistent rankings

Demonstrates eigenvector may lack Pareto efficiency

Abstract

A classical proposal to derive weights from a pairwise comparison matrix is the right eigenvector. The literature has identified some potential weaknesses of this method in previous decades. This chapter discusses five of these issues. First, right-left asymmetry emerges because of the difference between the right and inverse left eigenvectors. Second, group incoherence for choice means that, in group decision-making problems, the ranking given by the aggregated individual weight vectors is not guaranteed to coincide with the ranking derived from the aggregated pairwise comparison matrix. Third, the ranking based on the right eigenvector may depend on the intensity of the preferences, represented by taking a positive power of all comparisons. Fourth, both the ranking position and the normalised weight of an object might change counter-intuitively after modifying a particular comparison. Fifth, the right eigenvector is not necessarily Pareto efficient: a dominating weight vector that approximates each pairwise comparison at least as well, with an improvement in at least one position, could exist. All violations of the theoretical properties are highlighted by illustrative examples. We also present several open questions in order to inspire future research.