On the use of group theory to generalize elements of pairwise comparisons matrix: a cautionary note

TL;DR

This paper critically examines the limitations of applying group theory to pairwise comparisons, highlighting that certain groups cannot be used for ratios due to Levi's theorems, and provides counterexamples to illustrate these constraints.

Contribution

It clarifies the theoretical limitations of using non-torsion-free groups in pairwise comparison methods based on Levi's theorems.

Findings

Groups with torsion cannot be used for ratios in pairwise comparisons.

Counterexamples demonstrate the restrictions of group theory in this context.

The paper emphasizes the need for caution when applying group theory to pairwise comparison matrices.

Abstract

This paper examines the constricted use of group theory in the studies of pairwise comparisons. The presented approach is based on the application of the famous Levi Theorems of 1942 and 1943 for orderable groups. The theoretical foundation for multiplicative (ratio) pairwise comparisons has been provided. Counterexamples have been provided to support the theory. In our opinion, the scientific community must be made aware of the limitations of using the group theory in pairwise comparisons. Groups, which are not torsion free, cannot be used for ratios by Levi's theorems.