From pairwise comparisons to consistency with respect to a group operation and Koczkodaj's metric
The problem of pairwise comparisons is known for ages. It is probably one of the oldest techniques that were used by people who would like to make some choices. It refers to comparison of two or more objects or their properties in order to make a right decision. If there are more than two objects the comparisons can create a reciprocal \(n \times n\) matrix \(A=[i,j]\) where there are real (positive) numbers that express preferences. It is expected that between these preferences there is a consistency that is expressed as: for all \(i, j, k \in {1, \ldots, n}\), the following equality holds: \(a_{i,k}\cdot a_{k, j} =a_{i, j}\). But human's choices and judgements are not always consistent and this property is not ensured. There are a lot scientific papers about the problem of inconsistency and its reduction. There are also two important measures about this property. One is related to Saaty's approach and the second one is based on Koczkodaj's proposal. The paper consists of 7 Sections and after necessary introduction to the whole problem shown in Section 1, Sections 2--5 give mathematical preliminaries and background to the concept proposed in the paper. The most important part of this paper is Section 6 related to Koczkodaj's metric on \(\mathbb R_+\). It shows that Koczkodaj's metric \(\rho\) is not totally bounded, is Cantor complete and is uniformly equivalent with the metric \(\rho^*\). But in contradiction to what is stated in this paper it can be shown that only torsion free abelian groups are orderable and PC matrix entries cannot be generalized to any algebraic group as it is stipulated in this paper.
scientific article; zbMATH DE number 7073984
scientific article