The mathematical equivalence of the “spanning tree” and row geometric mean preference vectors and its implications for preference analysis

Abstract

Article:Lundy, M, Siraj, S orcid.org/0000-0002-7962-9930 and Greco, S (2017) The mathematical equivalence of the "spanning tree" and row geometric mean preference vectors and its implications for preference analysis. Pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker (DM), and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the DM's judgements. While the most widely used method is the eigenvector method, the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation. In this paper, we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach. This is an important nding due to the fact that it identies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector, and yet, in its entirety, the spanning tree method has more to oer than the row geometric mean method, in that, it is inherently applicable to incomplete sets of pairwise comparison judgements, and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the DM's judgements.

The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices

“…The arithmetic (geometric) mean of weight vectors calculated from all spanning trees was proved to be (logarithmic) least squares optimal. The proof of the complete case [39] cannot be extended to the incomplete case, due to that the incomplete (L)LS optimal solution does not have an explicit formula. However, the implicit formula (2) was still applicable to operations with spanning trees.…”

“…The numerical answers are collected into a multiplicative pairwise comparison matrix A = [a ij ] i,j=1...n fulfilling reciprocity, i.e., a ij = 1/a ji . A pairwise comparison matrix can be complete, as in the Analytic Hierarchy Process (AHP) [45], or incomplete [7,13,23,31,37,40,39,42,46,47,48,51,56]. A complete multiplicative pairwise comparison matrix A = [a ij ] is called consistent if cardinal transitivity, i.e., a ij a jk = a ik holds for all i, j, k. Otherwise, the matrix is inconsistent, and several inconsistency indices have been proposed, see [9,11,40,45].…”

“…The most natural candidates for the aggregation of weight vectors calculated from all spanning trees are the arithmetic [47,48,53,54] and the geometric means [39,55].…”

Characterization of the Row Geometric Mean Ranking with a Group Consensus Axiom

“…Perhaps it is not only a coincidence that row geometric mean has a number of other favourable properties (see, e.g. Barzilai et al (1987); Barzilai (1997); Dijkstra (2013); Csató (2015); Lundy et al (2017); Csató (2018c)).…”

Inconsistency indices for incomplete pairwise comparisons matrices

“…There are many other priority deriving methods [31,36,44]. In general, all of them lead to the same ranking vector unless the set of paired comparisons is inconsistent.…”