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：在缺乏可接受性阈值的情况下，成对比较的不一致性仍难以解释。Saaty提出的经典10%截断规则最近被应用于包含未知比较项的不完全成对比较矩阵。本文细化了这些不一致性阈值：我们发现阈值不仅取决于矩阵规模和缺失项数量，还与表示已知成对比较关系的无向图结构有关。因此，当大量矩阵具有相同的填充模式时（正如文献所建议的），采用我们提出的精确阈值尤为重要。研究同时证明了新阈值与表示图谱半径之间的强关联性。我们的成果可集成至软件中，用于在成对比较数据采集过程中持续监测不一致性并即时检测潜在错误。