A characterization of the Logarithmic Least Squares Method

We provide an axiomatic characterization of the Logarithmic Least Squares Method (sometimes called row geometric mean), used for deriving a preference vector from a pairwise comparison matrix. This procedure is shown to be the only one satisfying two properties, correctness in the consistent case, which requires the reproduction of the inducing vector for any consistent matrix, and invariance to a specific transformation on a triad, that is, the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3-cycle by a positive scalar.


Introduction
Pairwise comparisons are a fundamental tool in many decision-analysis methods such as the Analytic Hierarchy Process (AHP) (Saaty, 1980). However, in real-world applications the judgements of decision-makers may be inconsistent: for example, alternative A is two times better than alternative B, alternative B is three times better than alternative C, but alternative A is not six times better than alternative C. Inconsistency can also be an inherent feature of the data, for example, on the field of sport (Csató, 2013;Bozóki et al., 2016;Chao et al., 2018).
Therefore, a lot of methods have been suggested in the literature for deriving preference values from pairwise comparison matrices. In such cases, it seems to be fruitful to follow an axiomatic approach: the introduction and justification of reasonable properties may help to narrow the set of appropriate weighting methods and reveal some crucial features of them. The most important contribution of similar analyses can be an axiomatic characterization, that is, when a set of properties uniquely determine a preference vector.
Characterization of different methods is a standard tool in social choice theory, for instance, in the case of the Shapley value in game theory (see e.g. Shapley (1953); van den Brink and Pintér (2015)), or for the Hirsch index in scientometrics (see e.g. Woeginger (2008); Bouyssou and Marchant (2014)). This approach has been applied recently for inconsistency indices of pairwise comparison matrices (Csató, 2018a,b). Fichtner (1984), presumably the first work on the axiomatizations of weighting methods, characterized the Logarithmic Least Squares Method (Rabinowitz, 1976;Williams, 1980, 1985;De Graan, 1980) by using four requirements, correctness in the consistent case, comparison order invariance, smoothness and power invariance. Fichtner (1986) showed that substituting power invariance with rank preservation leads to the Eigenvector Method suggested by Saaty (1980). From this set of axioms, correctness in the consistent case and comparison order invariance are almost impossible to debate. However, according to Bryson (1995), there exists a goal-programming method satisfying power invariance and a slightly modified version of smoothness, which possesses the additional property that the presence of a single outlier cannot prevent the identification of the correct priority vector. While Fichtner (1984) introduces smoothness in terms of differentiable functions and continuous derivatives, the interpretation of Bryson (1995) -a small change in the input does not lead to a large change in the output -seems to be more natural for us. Cook and Kress (1988) approached the problem by focusing on distance measures in order to get another goal programming method on an axiomatic basis.
Smoothness and power invariance can be entirely left out from the characterization of the Logarithmic Least Squares Method. Barzilai et al. (1987) exchange them for a consistency-like axiom by considering two procedures: (1) some pairwise comparison matrices are aggregated to one matrix and the solution is computed for this matrix; (2) the priorities are derived separately for each matrix and combined by the geometric mean; which are required to result in the same preference vector. We think it is not a simple condition immediately to adopt. Barzilai (1997) managed to replace this axiom and comparison order invariance with essentially demanding that each individual weight is a function of the entries in the corresponding row of the pairwise comparison matrix only. Joining to Dijkstra (2013), we are also somewhat uncomfortable with this premise.
To summarize, the problem of weight derivation does not seem to be finally settled by the axiomatic approach. Consequently, it may not be futile to provide another characterization of the Logarithmic Least Squares Method, which hopefully highlights some new aspects of the procedure. This is the main aim of the current paper.
Presenting an axiomatic characterization does not mean that we accept all properties involved as wholly justified and unquestionable or we reject the axioms proposed by previous works. To consider an example from a related topic, although most axiomatic analysis of inconsistency (Brunelli and Fedrizzi, 2011;Brunelli, 2016Brunelli, , 2017Brunelli and Fedrizzi, 2015;Cavallo and D'Apuzzo, 2012;Koczkodaj and Szwarc, 2014;Koczkodaj and Szybowski, 2015) look for well-motivated axioms that should be satisfied by any reasonable measure, Csató (2018b) does not deal with the appropriate motivation of his axioms, the issue to be investigated is only how they can narrow the set of inconsistency indices. This paper strictly follows the latter direction, therefore, we only say that if one agrees with our axioms, then geometric mean remains the only choice.
The study is structured as follows. Section 2 presents some definitions on the field of pairwise comparison matrices. Two properties of weighting methods are defined in Section 3, which will provide the characterization of the Logarithmic Least Squares Method in Section 4. Section 5 summarizes our findings.

Preliminaries
Assume that n alternatives should be measured with respect to a given criterion on the basis of pairwise comparisons such that a ij is an assessment of the relative importance of alternative i with respect to alternative j.
Let R n + and R n×n + denote the set of positive (with all elements greater than zero) vectors of size n and matrices of size n × n, respectively.
Any pairwise comparison matrix is well-defined by its elements above the diagonal since we discuss only multiplicative pairwise comparison matrices with the reciprocal property throughout the paper. Let A n×n be the set of pairwise comparison matrices of size n × n.
A pairwise comparison matrix A ∈ A n×n is called consistent if a ik = a ij a jk for all 1 ≤ i, j, k ≤ n. Otherwise, it is said to be inconsistent. Any pairwise comparison matrix is allowed to be inconsistent unless its consistency is explicitly stated. A weighting method associates a weight vector to each pairwise comparison matrix. Several weighting methods have been suggested in the literature, see Choo and Wedley (2004) for an overview. This paper discusses two of them, which are among the most popular.

Definition 2.2. Weight vector:
Definition 2.4. Eigenvector Method (EM) (Saaty, 1980) where λ max denotes the maximal eigenvalue, also known as principal or Perron eigenvalue, of matrix A. Definition 2.5. Logarithmic Least Squares Method (LLSM) Williams, 1980, 1985;De Graan, 1980): The Logarithmic Least Squares Method is the function A → w LLSM (A) such that the weight vector w LLSM (A) is the optimal solution of the problem: (1) LLSM is sometimes called (row) geometric mean because the solution of (1) can be computed as

Axioms
In this section, two properties of weighting methods will be discussed.
CR requires the reproduction of the inducing vector for any consistent pairwise comparison matrix. It was introduced by Fichtner (1984) under the name correct result in the consistent case and was used by Fichtner (1986), Barzilai et al. (1987) and Barzilai (1997), among others. Definition 3.1. Transformation of triad improvement: Let A ∈ A n×n be a pairwise comparison matrix and 1 ≤ i, j, k ≤ n be three different alternatives. A transformation of triad improvement on (i, j, k) -on the triad determined by the three alternatives i, j, and k -by α provides the pairwise comparison matrixÂ ∈ A n×n such thatâ ij = αa ij (â ji = a ji /α),â jk = αa jk (â kj = a kj /α),â ki = αa ki (â ik = a ik /α) andâ ℓm = a ℓm for all other elements.
This transformation changes three elements of a pairwise comparison matrix along a 3-cycle. It can produce local consistency: the choice α = 3 a ik /(a ij a jk ) leads toâ ijâjk = α 2 a ij a jk = a ik /α =â ik . Naturally, this process modifies all values of the triad, while maybe two of the comparisons are accurate and one contains all the inaccuracy. However, if no further information is available, then the assumption behind our triad improvement seems to be reasonable. IT I means that the weights of alternatives are not influenced by transformations of triad improvement. It has been inspired by the axiom independence of circuits in Bouyssou (1992).
A motivation for independence of triad improvement can be the following. Consider a sport competition where player i has defeated player j, player j has defeated player k, while player k has defeated player i, and suppose that the three wins are equivalent. Then the final ranking is not allowed to change if the margins of victories are modified by the same amount. In particular, the three results can be reversed (j beats i, k beats j, and i beats k), or all comparisons can become a draw. IT I is practically a generalization of this idea. Proof. Take two pairwise comparison matrices A,Â ∈ A n×n such thatÂ is obtained from A through a transformation of triad improvement, namely, they are identical except Consequently, the product of row elements does not change, so w LLSM (A) = w LLSM (Â) according to (2).  (Crawford and Williams, 1985).

Characterization of the Logarithmic Least Squares Method Theorem 4.1. The Logarithmic Least Squares Method is the unique weighting method satisfying correctness and independence of triad improvement.
Proof. LLSM satisfies both axioms according to Propositions 3.1 and 3.3. For uniqueness, consider an arbitrary pairwise comparison matrix A ∈ A n×n and a weighting method f : A n×n → R n , which meets correctness and independence of triad improvement. Denote by P i = n n k=1 a i,k the geometric mean of row elements for alternative i. In order to prove that f is the Logarithmic Least Squares Method, it is enough to show that f i (A)/f j (A) = P i /P j .
• If j > i + 1, then introduce the pairwise comparison matrix A (i,j−1) ∈ A n×n such that a (i,j−1) 1,i i,j−1 and a (i,j−1) k,ℓ := a (i,j) k,ℓ for all other elements, where α i,j−1 = P i / P j−1 a (i,j) i,j−1 . It can be checked that a 1 For the sake of simplicity, only the elements above the diagonal are indicated.
• If j = i+1 and i > 2, then define the pairwise comparison matrix A (i−1,n) ∈ A n×n such that a k,ℓ for all other elements, where α i−1,n = P i−1 / P n a (i,n) i−1,n . It can be checked that a However, a m,j = 1/a j,m due to the reciprocity condition and n m=1 a j,m = P n j , therefore a (2,3) It is clear that P 1 = 1/ ( n m=2 P m ) as the product of all elements of A gives one, which leads to a (2,3) 1,j = P 1 P j for all j ≥ 2. In other words, A (2,3) ∈ A n×n is a consistent pairwise comparison matrix such that a Consequently, f A (2,3) = w LLSM A (2,3) due to correctness. Weighting method f is independent of triad improvement, hence = 1, so the pairwise comparison matrix remains unchanged, A (2,3) = A (2,4) . It is still a consistent matrix, therefore any weighting method satisfying correctness and independence of triad improvement should give w LLSM (A) as the weight vector associated with the pairwise comparison matrix A.
Proposition 4.1. CR and IT I are logically independent axioms.
Proof. It is shown that there exist weighting methods, which satisfy one axiom, but do not meet the other: 1 CR: Eigenvector Method (see Propositions 3.1 and 3.2); 2 IT I: flat method such that f i (A) = 1/n for all 1 ≤ i ≤ n.

Conclusions
We have proved LLSM to be the unique weighting method among the procedures used to derive priorities from reciprocal pairwise comparison matrices, which is correct in the consistent case and invariant to a transformation called triad improvement. The somewhat surprising fact is that our algorithm used for triad improvement aims only to recover local consistency by focusing on a given triad without the consideration of other elements of the pairwise comparison matrix. Hence satisfaction of a local property fully determines a global weight vector.
Naturally, one can debate whether the axiom independence of triad improvement should be accepted, but, at least, it reveals an important aspect of the geometric mean, contributing to the long list of its favourable theoretical properties (Barzilai et al., 1987;Barzilai, 1997;Dijkstra, 2013;Lundy et al., 2017;Csató, 2018c). Furthermore, the violation of this property can be an argument against the Eigenvector Method, which has a number of other disadvantages, for example, the possibility of strong rank reversal in group decision-making that has been revealed recently (Pérez and Mokotoff, 2016;Csató, 2017).
Some directions of future research is also worth to mention. First, further axiomatic analysis and characterizations of weighting methods can help in a better understanding of them. Second, the Logarithmic Least Squares Methods has been extended to the incomplete case when certain elements of the pairwise comparison matrix are unknown (Bozóki et al., 2010). Axiomatization on this more general domain seems to be promising and within reach as revealed by Bozóki and Tsyganok (2017), although LLSM sometimes behaves strangely on this general domain (Csató and Rónyai, 2016).