Geometric interpretation of efficient weight vectors

Pairwise comparison matrices (PCMs) are frequently used in different multicriteria decision making problems. A weight vector is said to be efficient if no other weight vector is at least as good in estimating the elements of the PCM, and strictly better in at least one position. Understanding the efficient weight vectors is crucial to determine the appropriate weight calculation technique for a given problem. In this paper we study the set of efficient weight vectors for three and four dimensions (alternatives) from a geometric viewpoint, which is a complementary to the algebraic approach used in the literature. Besides providing well-interpretable demonstrations, we also draw attention to the particular role of weight vectors calculated from spanning trees. Weight vectors corresponding to line graphs are vertices of the (polyhedral, but usually nonconvex) set of efficient weight vectors, while weight vectors corresponding to other spanning trees are also on the boundary.