# Axiomatizations of inconsistency indices for triads

arXiv: https://arxiv.org/abs/1801.03355 · HTML: https://mlsanity.com/paper/1801.03355/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2018.

## Built on by (most cited first, 3 of 3)

- [Detection of decision-making manipulation in the pairwise comparisons method](https://mlsanity.com/paper/2405.16693/) (2024)
- [An Overview and Comparison of Axiomatization Structures Regarding Inconsistency Indices' Properties in Pairwise Comparisons Methods](https://mlsanity.com/paper/2408.13297/) (2024)
- [Classifying Inconsistency in AHP Pairwise Comparison Matrices Using Machine Learning](https://mlsanity.com/paper/2505.06293/) (2025)

## Abstract (arXiv)

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.