On the coincidence of optimal completions for small pairwise comparison matrices with missing entries
László Csató, Kolos Csaba Ágoston, Sándor Bozóki
TL;DR
The two widely used inconsistency indices, Saaty's inconsistency index and the geometric inconsistency index, 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.
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.