# Almost optimal manipulation of a pair of alternatives

arXiv: https://arxiv.org/abs/2304.03060 · HTML: https://mlsanity.com/paper/2304.03060/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 60 references that are in the map.
- First posted on arXiv 6 Apr 2023.

## Builds on (most cited first, 9 of 9)

- [On the monotonicity of the eigenvector method](https://mlsanity.com/paper/1902.10790/) (2019)
- [The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices](https://mlsanity.com/paper/1701.04265/) (2017)
- [Pairwise Comparisons Simplified](https://mlsanity.com/paper/1311.6863/) (2013)
- [Studying a set of properties of inconsistency indices for pairwise comparisons](https://mlsanity.com/paper/1507.08826/) (2015)
- [On the Derivation of Weights from Incomplete Pairwise Comparisons Matrices via Spanning Trees with Crisp and Fuzzy Confidence Levels](https://mlsanity.com/paper/2112.10253/) (2021)
- [Towards secure judgments aggregation in AHP](https://mlsanity.com/paper/2303.15099/) (2023)
- [Characterization of the row geometric mean ranking with a group consensus axiom](https://mlsanity.com/paper/1706.07256/) (2017)
- [Incomplete pairwise comparison matrices and weighting methods](https://mlsanity.com/paper/1507.00461/) (2015)
- [A gradient method for inconsistency reduction of pairwise comparisons matrices](https://mlsanity.com/paper/2108.04034/) (2021)

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

- [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)
- [Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process](https://mlsanity.com/paper/2608.21187/) (2026)

## Abstract (arXiv)

The role of an expert in the decision-making process is crucial, as the final recommendation depends on his disposition, clarity of mind, experience, and knowledge of the problem. However, the recommendation also depends on their honesty. But what if the expert is dishonest? Then, the answer on how difficult it is to manipulate in a given case becomes essential. In the presented work, we consider manipulation of a ranking obtained by comparing alternatives in pairs. More specifically, we propose an algorithm for finding an almost optimal way to swap the positions of two selected alternatives. Thanks to this, it is possible to determine how difficult such manipulation is in a given case. Theoretical considerations are illustrated by a practical example.