Refined thresholds for inconsistency: The effect of the graph associated with incomplete pairwise comparisons

TL;DR

This paper refines inconsistency thresholds for incomplete pairwise comparison matrices, showing they depend on matrix size, missing entries, and the associated graph's properties, especially its spectral radius.

Contribution

It introduces exact inconsistency thresholds considering the graph structure, improving the interpretation of incomplete pairwise comparisons over the traditional 10% rule.

Findings

Thresholds depend on matrix size, missing entries, and graph structure.

Spectral radius of the comparison graph correlates with inconsistency thresholds.

Results enable software to monitor inconsistency during data collection.

Abstract

The inconsistency of pairwise comparisons remains difficult to interpret in the absence of acceptability thresholds. The popular 10% cut-off rule proposed by Saaty has recently been applied to incomplete pairwise comparison matrices, which contain some unknown comparisons. This paper refines these inconsistency thresholds: we uncover that they depend not only on the size of the matrix and the number of missing entries, but also on the undirected graph whose edges represent the known pairwise comparisons. Therefore, using our exact thresholds is especially important if the filling in patterns coincide for a large number of matrices, as has been recommended in the literature. The strong association between the new threshold values and the spectral radius of the representing graph is also demonstrated. Our results can be integrated into software to continuously monitor inconsistency during the collection of pairwise comparisons and immediately detect potential errors.