Axiomatizations of inconsistency indices for triads
This work addresses the foundational issue of inconsistency measurement in decision-making and ranking systems, but it is incremental as it builds upon existing axiomatic frameworks.
The paper tackles the problem of measuring inconsistency in pairwise comparison matrices by expanding a set of axioms for inconsistency indices, focusing on triads (matrices with three alternatives). It proves that the chosen axioms uniquely determine the inconsistency ranking for most indices on triads, which can help extend these results to larger matrices.
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.