The linear algebra of pairwise comparisons

摘要

In this paper, we start from the premise that pairwise comparisons between alternatives can be modeled by means of the additive representation of preferences. In this setting we study some algebraic properties of three sets: the set of pairwise comparison matrices, its subset of consistent ones and the orthogonal complement of the latter. The three sets are all vector spaces and we propose and interpret simple bases for each one. We prove that a convenient inner product can be found in the three cases such that the corresponding basis is orthonormal with respect to the considered inner product. In addition (i) we prove that the well-known method of the logarithmic least squares used to estimate the weight vector can be reinterpreted by referring to a basis for the set of consistent preferences and (ii) we interpret a transformation recently proposed by Csato. (C) 2019 Elsevier Inc. All rights reserved.