Ranking stability under uniform preference intensification in pairwise comparison matrices

L.A. Calvo, J. Mazurek

Operations Research Perspectives Vol. 17, pp. 100406

Summary:

This paper examines ranking stability in multiplicative pairwise comparison matrices under uniform preference intensification, represented by the entrywise power transformation A → A^(k) = [α^k_ij]. The associated invariance requirement is known as scale invariance; here, we study its failure at the level of the induced ranking, referred to as intensity-of-preference rank reversal. We combine theoretical observations, illustrative examples, and Monte Carlo experiments to analyse how this phenomenon depends on matrix order, inconsistency, and the priority derivation method. The row geometric mean method is used as the known scale-invariant benchmark, since its ranking is preserved under uniform intensification. In contrast, the eigenvector method and several other commonly used procedures may change the induced ranking, including the top-ranked alternative. The simulations indicate that such instability becomes more frequent as the number of compared objects increases, persists even among matrices satisfying conventional consistency-ratio thresholds, and differs substantially across priority derivation methods. These results show that robustness to uniform preference intensification is distinct from consistency screening and should be considered separately when evaluating priority derivation methods.

Spanish layman's summary:

El estudio muestra que intensificar uniformemente las preferencias en matrices de comparación por pares puede alterar los rankings, especialmente con más alternativas. La inestabilidad depende del método usado y no queda resuelta solo con controlar la consistencia, por lo que la robustez debe evaluarse aparte.

English layman's summary:

The study shows that uniformly intensifying preferences in pairwise comparison matrices can change rankings, especially as the number of alternatives grows. This instability depends on the priority method and is not captured by consistency checks alone, so robustness should be assessed separately.

Keywords: Eigenvector method; Geometric mean method; Pairwise comparison matrices; Rank reversal; Uniform preference intensification

JCR-JIF Impact Factor and WoS quartile: 3,300 - Q2 (2025)

DOI reference: https://doi.org/10.1016/j.orp.2026.100406

Published on paper: December 2026.

Published on-line: July 2026.

Citation:
L.A. Calvo, J. Mazurek, "Ranking stability under uniform preference intensification in pairwise comparison matrices", Operations Research Perspectives, Vol. 17, pp. 100406, December 2026. [Online: July 2026] doi: 10.1016/j.orp.2026.100406