# Inconsistency thresholds for incomplete pairwise comparison matrices

arXiv: https://arxiv.org/abs/2102.10558 · HTML: https://mlsanity.com/paper/2102.10558/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 41 references that are in the map.
- First posted on arXiv 21 Feb 2021.

## Builds on (most cited first, 6 of 6)

- [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 the ranking of a Swiss system chess team tournament](https://mlsanity.com/paper/1507.05045/) (2015)
- [Filling in pattern designs for incomplete pairwise comparison matrices: (quasi-)regular graphs with minimal diameter](https://mlsanity.com/paper/2006.01127/) (2020)
- [Revenue allocation in Formula One: a pairwise comparison approach](https://mlsanity.com/paper/1909.12931/) (2019)
- [On reducing inconsistency of pairwise comparison matrices below an acceptance threshold](https://mlsanity.com/paper/1311.0748/) (2013)

## Built on by (most cited first, 5 of 5)

- [Detection of decision-making manipulation in the pairwise comparisons method](https://mlsanity.com/paper/2405.16693/) (2024)
- [My part is bigger than yours -- assessment within a group of peers](https://mlsanity.com/paper/2407.01843/) (2024)
- [An Overview and Comparison of Axiomatization Structures Regarding Inconsistency Indices' Properties in Pairwise Comparisons Methods](https://mlsanity.com/paper/2408.13297/) (2024)
- [An experimental approach: The graph of graphs](https://mlsanity.com/paper/2508.17520/) (2025)
- [Machine Learning Model for Sparse PCM Completion](https://mlsanity.com/paper/2601.04366/) (2026)

## Abstract (arXiv)

Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices for similar incomplete data sets and no reasonable measure has an associated threshold. This paper generalises the famous rule of thumb for the acceptable level of inconsistency, proposed by Saaty, to incomplete pairwise comparison matrices. The extension is based on choosing the missing elements such that the maximal eigenvalue of the incomplete matrix is minimised. Consequently, the well-established values of the random index cannot be adopted: the inconsistency of random matrices is found to be the function of matrix size and the number of missing elements, with a nearly linear dependence in the case of the latter variable. Our results can be directly built into decision-making software and used by practitioners as a statistical criterion for accepting or rejecting an incomplete pairwise comparison matrix.