Complete and incomplete additive/multiplicative pairwise comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known methods is proved in this paper. The arithmetic (geometric) mean of weight vectors, calculated from all spanning trees, is proved to be optimal to the (logarithmic) least squares problem, not only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the “spanning tree” and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197–208, but for incomplete matrices as well. Unlike the complete case, where an explicit formula, namely the row arithmetic/geometric mean of matrix elements, exists for the (logarithmic) least squares problem, the incomplete case requires a completely different and new proof. Finally, Kirchhoff's laws for the calculation of potentials in electric circuits is connected to our results.

Having a pairwise comparison matrix in a multi-attribute decision problem, two basic problems arise: how to compute the weight vector, and, how to associate an inconsistency index to the matrix. Two key concepts of the Analytic Hierarchy Process, the eigenvector method and inconsistency index are discussed. (In)efficiency is a well-known property in multiple objective optimization. We introduce a restriction of it in the paper. Given a pairwise comparison matrix, a weight vector is called internally inefficient if there exists another weight vector such that every pairwise ratio of the latter’s components is between the corresponding element of the pairwise comparison matrix and the pairwise ratio of the former’s components, and there exists at least one position, where the latter’s approximation is strictly better than the former’s. A class of pairwise comparison matrices with arbitrarily small inconsistency is provided such that the eigenvector method yields an internally inefficient weight vector. The paper is closed by another pairwise comparison matrix with an internally inefficient eigenvector and open questions.