Inconsistency thresholds for incomplete pairwise comparison matrices

TL;DR

This paper extends inconsistency thresholds for pairwise comparison matrices to incomplete data, providing a practical statistical criterion for decision-making with missing comparisons.

Contribution

It generalizes Saaty's inconsistency rule to incomplete matrices by minimizing the maximal eigenvalue, accounting for missing data.

Findings

Inconsistency depends on matrix size and missing elements

Random matrix inconsistency varies nearly linearly with missing elements

Results can be integrated into decision-making software

Abstract

Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices for similar incomplete data sets and no reasonable measure has an associated threshold. This paper generalises the famous rule of thumb for the acceptable level of inconsistency, proposed by Saaty, to incomplete pairwise comparison matrices. The extension is based on choosing the missing elements such that the maximal eigenvalue of the incomplete matrix is minimised. Consequently, the well-established values of the random index cannot be adopted: the inconsistency of random matrices is found to be the function of matrix size and the number of missing elements, with a nearly linear dependence in the case of the latter variable. Our results can be directly built into decision-making software and used by practitioners as a statistical criterion for accepting or rejecting an incomplete pairwise comparison matrix.