# On the monotonicity of the eigenvector method

arXiv: https://arxiv.org/abs/1902.10790 · HTML: https://mlsanity.com/paper/1902.10790/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2019.

## Built on by (most cited first, 7 of 7)

- [Inconsistency thresholds for incomplete pairwise comparison matrices](https://mlsanity.com/paper/2102.10558/) (2021)
- [Towards secure judgments aggregation in AHP](https://mlsanity.com/paper/2303.15099/) (2023)
- [Almost optimal manipulation of a pair of alternatives](https://mlsanity.com/paper/2304.03060/) (2023)
- [Manipulation of individual judgments in the quantitative pairwise comparisons method](https://mlsanity.com/paper/2211.01809/) (2022)
- [Establishing a leader in a pairwise comparisons method](https://mlsanity.com/paper/2403.14885/) (2024)
- [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)

## Abstract (arXiv)

Pairwise comparisons are used in a wide variety of decision situations where the importance of alternatives should be measured on a numerical scale. One popular method to derive the priorities is based on the right eigenvector of a multiplicative pairwise comparison matrix. We consider two monotonicity axioms in this setting. First, increasing an arbitrary entry of a pairwise comparison matrix is not allowed to result in a counter-intuitive rank reversal, that is, the favoured alternative in the corresponding row cannot be ranked lower than any other alternative if this was not the case before the change (rank monotonicity). Second, the same modification should not decrease the normalised weight of the favoured alternative (weight monotonicity). Both properties are satisfied by the geometric mean method but violated by the eigenvector method. The axioms do not uniquely determine the geometric mean. The relationship between the two monotonicity properties and the Saaty inconsistency index are investigated for the eigenvector method via simulations. Even though their violation turns out not to be a usual problem even for heavily inconsistent matrices, all decision-makers should be informed about the possible occurrence of such unexpected consequences of increasing a matrix entry.