This paper studies the properties of an inconsistency index of a pairwise comparison matrix under the assumption that the index is defined as a norm-induced distance from the nearest consistent matrix. Under additive representation of preferences, it is proved that an inconsistency index defined in this way is a seminorm in the linear space of skew-symmetric matrices and several relevant properties hold. In particular, this linear space can be partitioned into equivalence classes, where each class is an affine subspace and all the matrices in the same class share a common value of the inconsistency index. The paper extends in a more general framework some results due, respectively, to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices satisfy a set of six characterizing properties previously introduced, as well as an upper bound property for group preference aggregation.

本文假设成对比较矩阵的不一致性指标的性质是将其定义为从最近的一致矩阵到范数的距离，则研究该属性。在偏好的加性表示下，证明以这种方式定义的不一致指数是偏对称矩阵的线性空间中的一个半范数，并且具有几个相关的性质。特别地，可以将该线性空间划分为等价类，其中每个类都是一个仿射子空间，并且同一类中的所有矩阵共享不一致索引的公共值。本文在更一般的框架内扩展了一些结果，分别应归功于Crawford和Barzilai。还证明了基于规范的不一致性指数满足了先前介绍的六个特征属性的集合，