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

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.

💡 Research Summary

The paper tackles a long‑standing problem in Analytic Hierarchy Process (AHP) applications: how to interpret the inconsistency index (CI) of an incomplete pairwise comparison matrix when no clear acceptability threshold is available. The classic 10 % rule (CI/RI ≤ 0.1) was originally devised for complete matrices and relies on the Random Index (RI), the average CI of matrices whose upper‑triangular entries are drawn uniformly from the Saaty scale. Recent work (Agoston & Csató, 2022) extended this idea to incomplete matrices by making RI a function of the number of alternatives (n) and the number of missing comparisons (m). However, that approach assumes the missing entries are placed at random, which is rarely true in practice where the pattern of missing comparisons (the “filling‑in pattern”) is often fixed across many decision makers.

The authors propose a fundamentally richer model. They represent an incomplete matrix by an undirected graph G = (V,E): vertices correspond to alternatives, and an edge exists if the corresponding comparison is known. Missing comparisons are simply absent edges. The key insight is that the structure of G, especially its spectral radius ρ(G) (the largest absolute eigenvalue of its adjacency matrix), strongly influences the average inconsistency of random completions. Consequently, the appropriate RI – and thus the admissible CI threshold – should be computed for each specific graph, not merely for each (n,m) pair.

Methodologically, the study proceeds as follows. For a fixed n, m and a chosen connected graph G, the authors generate one million random incomplete matrices. Each known comparison is drawn independently with probability 1/17 for each of the 17 Saaty scale values (1/9,…,1/9,…,9). The missing entries are treated as positive variables x_k constrained to the interval