On the coincidence of optimal completions for small pairwise comparison matrices with missing entries

TL;DR

This paper investigates how different inconsistency indices influence the optimal completion of incomplete pairwise comparison matrices, revealing that for matrices up to order four, they lead to the same solutions, but diverge for larger matrices.

Contribution

It proves that Saaty's and the geometric inconsistency indices produce identical optimal completions for matrices up to order four, highlighting differences for larger matrices.

Findings

Optimal completions coincide for matrices up to order four.

Differences in optimal completions emerge for matrices of order five or more.

The relationship between inconsistency indices and completion methods is clarified.

Abstract

Incomplete pairwise comparison matrices contain some missing judgements. A natural approach to estimate these values is provided by minimising a reasonable measure of inconsistency after unknown entries are replaced by variables. Two widely used inconsistency indices for this purpose are Saaty's inconsistency index and the geometric inconsistency index, which are closely related to the eigenvector and the logarithmic least squares priority deriving methods, respectively. The two measures are proven to imply the same optimal filling for incomplete pairwise comparison matrices up to order four but not necessarily for order at least five.