Incomplete pairwise comparison matrices and weighting methods

TL;DR

This paper investigates how two common weighting methods, Eigenvector and Logarithmic Least Squares, perform on incomplete pairwise comparison matrices derived from directed acyclic graphs, revealing they violate linear order preservation.

Contribution

It demonstrates that both methods fail to satisfy the linear order preservation property on incomplete comparison data, with Eigenvector's ranking depending on the representation.

Findings

Eigenvector Method breaks linear order preservation.

Logarithmic Least Squares Method breaks linear order preservation.

Eigenvector ranking depends on the incomplete comparison representation.

Abstract

A special class of preferences, given by a directed acyclic graph, is considered. They are represented by incomplete pairwise comparison matrices as only partial information is available: for some pairs no comparison is given in the graph. A weighting method satisfies the linear order preservation property if it always results in a ranking such that an alternative directly preferred to another does not have a lower rank. We study whether two procedures, the Eigenvector Method and the Logarithmic Least Squares Method meet this axiom. Both weighting methods break linear order preservation, moreover, the ranking according to the Eigenvector Method depends on the incomplete pairwise comparison representation chosen.