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

• We refine inconsistency thresholds for incomplete pairwise comparison matrices • They are shown to depend not only on the number of alternatives and missing entries • They are affected by the undirected graph representing the known matrix entries • The new thresholds are strongly associated with the spectral radius of the graph • Our results can be integrated into software to continuously monitor inconsistency 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.