Theoretical properties of the eigenvector method
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910072350703616 |
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| author | Bozóki, Sándor Csató, László |
| author_facet | Bozóki, Sándor Csató, László |
| contents | A classical proposal to derive weights from a pairwise comparison matrix is the right eigenvector. The literature has identified some potential weaknesses of this method in previous decades. This chapter discusses five of these issues. First, right-left asymmetry emerges because of the difference between the right and inverse left eigenvectors. Second, group incoherence for choice means that, in group decision-making problems, the ranking given by the aggregated individual weight vectors is not guaranteed to coincide with the ranking derived from the aggregated pairwise comparison matrix. Third, the ranking based on the right eigenvector may depend on the intensity of the preferences, represented by taking a positive power of all comparisons. Fourth, both the ranking position and the normalised weight of an object might change counter-intuitively after modifying a particular comparison. Fifth, the right eigenvector is not necessarily Pareto efficient: a dominating weight vector that approximates each pairwise comparison at least as well, with an improvement in at least one position, could exist. All violations of the theoretical properties are highlighted by illustrative examples. We also present several open questions in order to inspire future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Theoretical properties of the eigenvector method Bozóki, Sándor Csató, László Optimization and Control 90B50, 91B06 A classical proposal to derive weights from a pairwise comparison matrix is the right eigenvector. The literature has identified some potential weaknesses of this method in previous decades. This chapter discusses five of these issues. First, right-left asymmetry emerges because of the difference between the right and inverse left eigenvectors. Second, group incoherence for choice means that, in group decision-making problems, the ranking given by the aggregated individual weight vectors is not guaranteed to coincide with the ranking derived from the aggregated pairwise comparison matrix. Third, the ranking based on the right eigenvector may depend on the intensity of the preferences, represented by taking a positive power of all comparisons. Fourth, both the ranking position and the normalised weight of an object might change counter-intuitively after modifying a particular comparison. Fifth, the right eigenvector is not necessarily Pareto efficient: a dominating weight vector that approximates each pairwise comparison at least as well, with an improvement in at least one position, could exist. All violations of the theoretical properties are highlighted by illustrative examples. We also present several open questions in order to inspire future research. |
| title | Theoretical properties of the eigenvector method |
| topic | Optimization and Control 90B50, 91B06 |
| url | https://arxiv.org/abs/2603.24274 |