On the monotonicity of the eigenvector method

László Csató, Dóra Gréta Petróczy

Published 2019-02-27Version 1

Pairwise comparisons are used in a wide variety of decision situations when the importance of different alternatives should be measured by numerical weights. One popular method to derive these priorities is based on the right eigenvector of a multiplicative pairwise comparison matrix. We introduce an axiom called monotonicity: increasing an arbitrary entry of a pairwise comparison matrix should increase the weight of the favoured alternative (which is in the corresponding row) by the greatest factor and should decrease the weight of the favoured alternative (which is in the corresponding column) by the greatest factor. It is proved that the eigenvector method violates this natural requirement. We also investigate the relationship between non-monotonicity and the Saaty inconsistency index. It turns out that the violation of monotonicity is not a problem in the case of nearly consistent matrices. On the other hand, the eigenvector method remains a dubious choice for inherently inconsistent large matrices such as the ones that emerge in sports applications.