Axiomatizations of inconsistency indices for triads

TL;DR

This paper expands the axiomatic foundation for inconsistency indices in triads, providing a unique characterization that enhances understanding of inconsistency measurement in pairwise comparison matrices.

Contribution

It introduces two new properties to the axioms and proves that these, along with existing ones, uniquely determine the inconsistency ranking for triads.

Findings

A set of axioms characterizes the inconsistency ranking for triads.

Two new properties are added to the axiomatic framework.

Results can inform inconsistency measurement in larger matrices.

Abstract

Pairwise comparison matrices often exhibit inconsistency, therefore many indices have been suggested to measure their deviation from a consistent matrix. A set of axioms has been proposed recently that is required to be satisfied by any reasonable inconsistency index. This set seems to be not exhaustive as illustrated by an example, hence it is expanded by adding two new properties. All axioms are considered on the set of triads, pairwise comparison matrices with three alternatives, which is the simplest case of inconsistency. We choose the logically independent properties and prove that they characterize, that is, uniquely determine the inconsistency ranking induced by most inconsistency indices that coincide on this restricted domain. Since triads play a prominent role in a number of inconsistency indices, our results can also contribute to the measurement of inconsistency for pairwise comparison matrices with more than three alternatives.