# An Overview and Comparison of Axiomatization Structures Regarding Inconsistency Indices' Properties in Pairwise Comparisons Methods

arXiv: https://arxiv.org/abs/2408.13297 · HTML: https://mlsanity.com/paper/2408.13297/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 87 references that are in the map.
- First posted on arXiv 23 Aug 2024.

## Builds on (most cited first, 15 of 15)

- [An application of incomplete pairwise comparison matrices for ranking top tennis players](https://mlsanity.com/paper/1611.00538/) (2016)
- [On the monotonicity of the eigenvector method](https://mlsanity.com/paper/1902.10790/) (2019)
- [On Axiomatization of Inconsistency Indicators for Pairwise Comparisons](https://mlsanity.com/paper/1307.6272/) (2013)
- [Inconsistency thresholds for incomplete pairwise comparison matrices](https://mlsanity.com/paper/2102.10558/) (2021)
- [Studying a set of properties of inconsistency indices for pairwise comparisons](https://mlsanity.com/paper/1507.08826/) (2015)
- [Axiomatizations of inconsistency indices for triads](https://mlsanity.com/paper/1801.03355/) (2018)
- [Axiomatic properties of inconsistency indices for pairwise comparisons](https://mlsanity.com/paper/1306.6852/) (2013)
- [Characterization of the row geometric mean ranking with a group consensus axiom](https://mlsanity.com/paper/1706.07256/) (2017)
- [Notes on discrepancy in the pairwise comparisons method](https://mlsanity.com/paper/1312.2986/) (2013)
- [Recent advances on inconsistency indices for pairwise comparisons - a commentary](https://mlsanity.com/paper/1503.08289/) (2015)
- [New results on inconsistency indices and their relationship with the quality of priority vector estimation](https://mlsanity.com/paper/1505.06573/) (2015)
- [Characterization of an inconsistency ranking for pairwise comparison matrices](https://mlsanity.com/paper/1610.07388/) (2016)
- [On Inconsistency Indices and Inconsistency Axioms in Pairwise Comparisons](https://mlsanity.com/paper/1703.05204/) (2017)
- [A characterization of the Logarithmic Least Squares Method](https://mlsanity.com/paper/1704.05321/) (2017)
- [Entropy production rate as a criterion for inconsistency in decision theory](https://mlsanity.com/paper/1801.01733/) (2018)

## Abstract (arXiv)

Mathematical analysis of the analytic hierarchy process (AHP) led to the development of a mathematical function, usually called the inconsistency index, which has the center role in measuring the inconsistency of the judgements in AHP. Inconsistency index is a mathematical function which maps every pairwise comparison matrix (PCM) into a real number. An inconsistency index can be considered more trustworthy when it satisfies a set of suitable properties. Therefore, the research community has been trying to postulate a set of desirable rules (axioms, properties) for inconsistency indices. Subsequently, many axiomatic frameworks for these functions have been suggested independently, however, the literature on the topic is fragmented and missing a broader framework. Therefore, the objective of this article is twofold. Firstly, we provide a comprehensive review of the advancements in the axiomatization of inconsistency indices' properties during the last decade. Secondly, we provide a comparison and discussion of the aforementioned axiomatic structures along with directions of the future research.