Inconsistency thresholds revisited: The effect of the graph associated with incomplete pairwise comparisons

TL;DR

This work targets the difficulty of interpreting inconsistency in incomplete pairwise comparison matrices by showing that Saaty-like thresholds depend on the graph of known comparisons. By fixing a connected graph $G=(V,E)$ and solving a graph-constrained eigenvalue-minimisation problem, the authors generate $10^6$ random incomplete matrices and compute the average inconsistency index $CI$, revealing a strong link between the random index $RI$ and the graph's spectral radius $\rho(G)$. The main contributions are graph-dependent, exact inconsistency thresholds for incomplete matrices and evidence that neglecting the graph structure can misclassify many matrices under the traditional 0.1 rule, which has practical implications for live inconsistency monitoring during data collection. The results advocate integrating graph-aware thresholds into decision-support software to improve error detection and data-quality control in hierarchical decision analyses.

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 revises 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.

Figures (6)

• Figure 1: The graph representation of the pairwise comparison matrix $\mathbf{A}$ in Example \ref{['Examp1']}
• Figure 2: Spectral radius and random index for incomplete pairwise comparison matrices with $m=2$ missing entries
• Figure 3: Possible graph representations of incomplete pairwise comparison matrices of size five with at least two and at most five missing entries Notes: Dotted lines indicate the known comparisons, thick red lines indicate the unknown comparisons. Graph $G_{i,j}$ has $i$ missing edges. The vectors show the degree distribution of the associated graph. See Table \ref{['Table_A1']} for the spectral radii and random indices of these incomplete pairwise comparison matrices.
• Figure 4: Possible graph representations of incomplete pairwise comparison matrices of size six with six missing entries Notes: Dotted lines indicate the known comparisons, thick red lines indicate the unknown comparisons. The vectors show the degree distribution of the associated graph. See Table \ref{['Table_A2']} for the spectral radii and random indices of these incomplete pairwise comparison matrices.
• Figure 5: Spectral radius and random index for incomplete matrices of size five and six

Theorems & Definitions (1)

• Example 1