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.

翻译：成对比较的不一致性在缺乏可接受阈值的情况下仍难以解释。Saaty提出的常用10%截断规则最近已被应用于包含未知比较项的不完整成对比较矩阵。本文对这些不一致性阈值进行了精化：我们发现这些阈值不仅取决于矩阵的规模与缺失条目数量，还取决于以已知成对比较为边的无向图。因此，当大量矩阵的填充模式一致时（正如文献中所推荐的），使用本文的精确阈值尤为重要。文中还证明了新阈值与表征图谱半径之间的强关联性。我们的研究结果可集成至软件中，用于在成对比较收集过程中持续监控不一致性并即时检测潜在错误。