Geometry of efficient weight vectors

Abstract

Pairwise comparison matrices and the weight vectors obtained from them are important concepts in multi-criteria decision making. A weight vector calculated from a pairwise comparison matrix is called Pareto efficient if the approximation of the matrix elements by the weight ratios cannot be improved for any element of the matrix without worsening it for another element. The aim of this paper is to show the geometrical properties of the set of Pareto efficient weight vectors for $4\times 4$ pairwise comparison matrices. We prove that the set of efficient weight vectors is a union of three tetrahedra, each determined by four weight vectors calculated from an incomplete submatrix of the pairwise comparison matrix that can be represented by a path spanning tree graph. It is shown that with suitable rearrangements the orientations of the $4$-cycles in the Blanquero-Carrizosa-Conde graphs for the efficient weight vectors are determined as well. The special cases (double perturbed, simple perturbed, consistent matrices) are discussed in the online appendices.