On the geometric mean method for incomplete pairwise comparisons

When creating the ranking based on the pairwise comparisons very often, we face difficulties in completing all the results of direct comparisons. In this case, the solution is to use the ranking method based on the incomplete PC matrix. The article presents the extension of the well known geometric mean method for incomplete PC matrices. The description of the methods is accompanied by theoretical considerations showing the existence of the solution and the optimality of the proposed approach.


Introduction
The ability to compare has accompanied mankind for centuries. When comparing products in a convenience store, choosing dishes in a restaurant, or selecting a gas station with the most attractive fuel price, people are trying to make the best choice. During the process of selecting the best option, the available alternatives are compared in pairs. This observation underlies many decision-making methods. The first use of pair comparison as a formal basis for the decision procedure is attributed to the XIII-century mathematician Ramon Llull who proposed a binary electoral system [9]. His method over time was forgotten and rediscovered in a similar form by Condorcet [10]. Although both Llull and Condorcet treated comparisons as binary, i.e. the result of comparisons can be either a win or loss (for a given alternative), Thurstone proposed the use of pairwise comparisons (PC) in a more generalized, quantitative way [34]. Since then, the result of the single pairwise comparison can be identified with a real positive number where the values greater than 1 mean the degree to which the first alternative won, and in the same way, the values smaller than 1 mean the degree to which the second alternative won. Llull's electoral system was in some form reinvented by Copeland [9,11].
After performing appropriate comparisons, their results are subject to further calculations, resulting in the final ranking of considered objects. It is easy to compute that the set consisting of n alternatives allows n(n − 1)/2 comparisons to be made. Thus, for five alternatives, experts need to perform ten comparisons, for six -fifteen, and so on. The number of necessary collations increases with the square of the number of alternatives. Since pairwise judgments are very often made by experts, making a large number of paired comparisons can be difficult and expensive. This observation encouraged experts to search for ranking methods using a reduced number of pairwise comparisons. These studies have resulted in several methods, including Harker's eigenvalue based approach [18], the logarithmic least square (LLS) method for incomplete PC matrices [4,5], spanning-tree approach [28] or missing values estimation [14].
In this study, we propose a direct extension of the popular geometric mean (GM) method for incomplete PC matrices. The method proposed by Harker served as the starting point of our procedure. The modified method, like the original GM method, is equivalent to the LLS method for incomplete PC matrices. Hence, similarly to the LLS method, it minimizes the logarithmic least square error of the ranking.
The paper is composed of 7 sections including an introduction and summary. The notion of an incomplete PC matrix and the indispensable amount of definitions are introduced in (Sec. 2). Section 3 briefly presents the existing priority deriving methods for incomplete PC matrices with particular emphasis on those which the presented solution is based on. The next part, Section 4, presents the modified GM method for incomplete PC matrices. It is followed by an illustrative example (Sec. 5). The penultimate Section 6 addresses the problems of optimality and the existence of a solution. A brief summary is provided in (Sec. 7).

Preliminaries
The input data for the priority deriving procedure is a set of pairwise comparisons. Due to the convenience of calculations, it is usually presented in the form of a pairwise comparison (PC) matrix C = [c ij ] where a single entry c ij represents the results of comparisons of two alternatives a i and a j . Unless explicitly stated otherwise, we will assume that the set of alternatives A consists of n objects (options, alternatives), i.e. A = {a 1 , . . . , a n }. In C, not all entries may be specified. Such a matrix will be called incomplete or partial. The missing comparison will be denoted by ?. Let us define the PC matrix formally. Definition 1. An incomplete PC matrix for n alternatives is said to be the matrix C = [c ij ] such that c ij ∈ R n + ∪{?}, c ii = 1 for i, j = 1, . . . , n. Comparison of the i-th and j-th alternatives for which c ij =? is said to be missing.
A 4 × 4 PC matrix may look as follows: 1 The ranking procedure aims to assign numerical values corresponding to the strength of preferences to alternatives. In this way, each of the alternatives gains a certain weight (also called priority or importance) that determines its position in the ranking. The higher the weight, the higher the position. Let us denote this weight by the function w : A → R + . As elements of C are the results of paired comparisons between alternatives, then it is natural to expect 1 that c ij ≈ w(a i )/w(a j ). Since c ij represents a comparison of the i-th and jth alternatives and is interpreted as the ratio w(a i )/w(a j ), then c ji means the comparison of the j-th alternative versus the i-th alternative interpreted as the ratio w(a j )/w(a i ). For this reason, we will accept that c ij = 1/c ji . In this context, it will be convenient to use the reciprocity property defined as follows. In our further considerations, it will be convenient to interpret the set of comparisons as a graph. Very often, there is a directed graph where the direction of the edge indicates the winner of the given comparison. For the purpose of this article, however, we will use an undirected graph. This will allow us to represent the existence (or absence) of comparisons, without having to indicate the exact relationship between the two alternatives.
One of the often desirable properties of a graph is connectivity.
In terms of the PC matrix, the connectivity of vertices (identified with alternatives) is necessary to calculate the ranking [18]. This is quite an intuitive observation. If we assume that the graph consists of two subgraphs separated from each other, then there will be no relation (comparison) allowing the decision maker to determine how one subgraph is relative to the other. In this case, it is clear that it would be impossible to build a ranking of all alternatives. The matrices whose graphs are connected are irreducible 2 [30].
One of the frequently used properties of graph vertices is the vertex degree.
The concept of the vertex degree allows us to construct the degree matrix defined as follows.
The adjacency matrix is a frequently used representation of the graph [12].
Definition 7. An adjacency matrix of the graph The matrix that combines both previous matrices is the Laplacian matrix L(T C ).
The properties of the Laplacian matrix allow us to justify the existence of the solution of the method proposed in (Section 4).

Priority deriving methods for incomplete PC matrices
One of the first methods allowing the decision maker to calculate the ranking based on incomplete PC matrices was proposed by Harker [18]. In this approach, the author uses the eigenvalue method (EVM) proposed by Saaty [33]. According to EVM, the ranking is determined by the principal eigenvector of C understood as the solution of where λ max is the principal eigenvalue of C. Of course, EVM cannot be directly applied to an incomplete PC matrix. Thus Harker proposed the replacement of every missing c ij =? by the expression w(a i )/w(a j ). He argued that since c ij ≈ w(a i )/w(a j ) then the most natural replacement for c ij =? is just the ratio w(a i )/w(a j ). Thus, instead of solving (2), one has to deal with the following equation: where w is the weight vector and C * = [c * ij ] is the PC matrix such that Of course, (3) cannot be directly solved as C * contains a priori unknown values w(a i )/w(a j ). Fortunately, (3) is equivalent to the following linear equation system: Since B is an ordinary matrix, then (5) can be solved by using standard mathematical tools, including Excel Solver provided by Microsoft Inc. Harker proved that (5) has a solution and the principal eigenvector w is real and positive, which is a condition for the solution to be admissible.
Another approach to the problem of ranking for incomplete PC matrices has been proposed in Bozóki et al. [4]. Following Crawford and Williams and their Geometric Mean (GM) Method [13], the authors assume that the optimal solution: . . .
needs to minimize the distance between every pair c ij and the ratio w(a i )/w(a j ).
In the original work proposed for complete PC matrices, this condition takes the form of a square of the logarithms of these expressions, i.e.
Since the authors are interested in incomplete PC matrices, then the above condition takes the form: where the distance between missing entries and the corresponding ratios are just not taken into account. The authors prove [4,5] that solving the following problem: where L(T C ) is the Laplacian matrix of C, w = [ w(a 1 ), . . . , w(a n )] T is the logarithmized priority vector w i.e. w(a i ) = ln w(a i ) and provides the ranking vector minimizing S * (C). Hence, in order to receive the primary weight vector (6), it is enough to adopt w(a i ) = e w(ai) for i = 1, . . . , n.
Due to the form of S * (C), the method is called the Logarithmic Least Square (LLS) Method for incomplete PC matrices. In their work, Bozóki et al. also consider the algorithm for principal eigenvalue optimal completion. In [28], Lundy et al. point out that the ranking based on "spanning trees" of T C is equivalent to the GM method. In particular, they indicate that the method can be used for incomplete PC matrices. Later on, Bozóki and Tsyganok proved that the "spanning trees" method for incomplete PC matrices is equivalent to the LLS method for incomplete PC matrices [5].

Idea of the geometric mean method for incomplete PC matrices
Following Harker's method [18], the proposed solution assumes that C is irreducible (T C is connected) and the optimal completion of the incomplete PC matrix C is C * = [c * ij ] (4). Hence, every missing c ij =? in C is replaced by the ratio w(a i )/w(a j ) in C * . However, unlike in [18], C * is the subject of the GM method. Then, let us calculate the geometric mean of rows for C * , i.e.
One may observe that the above equation system is equivalent to n j=1 ln c * ij = n ln w(a i ) for i = 1, . . . , n.
Let us split the left side of the equation into two parts. One for the missing values in C, the other for the existing elements. It can be written in the matrix form as where By solving (10) we obtain the auxiliary vector w providing us the solution of the primary problem (9), i.e.
Of course, w can be the subject of scaling, so as the final ranking we may adopt αw where α = n i=1 e w(ai) −1 . In (Sec. 6.1) we show that M is nonsingular, hence, (10) and, as follows, (9) always has a unique solution.

Illustrative example
Let us consider the incomplete PC matrix C * given as It corresponds to the following equation system: .
Let us denote log w(a i ) df = w(a i ) and ln c ij df = c ij . Hence, the above equation system can be written down in the matrix form where the auxiliary matrix M is given as The constant term vector c and the vector w are as follows: Since the matrix M is non-singular, we may compute the auxiliary vector: 6. Properties of the method

Existence of a solution
One may observe that for the fixed incomplete and irreducible PC matrix C the auxiliary matrix M (11) can be written as: is the Laplacian matrix of T C , and J n is the n × n matrix, each of whose entries is 1. Let us prove that M is nonsingular. Proof. Let λ 1 , . . . , λ n be the eigenvalues of L(T C ) in the non increasing order, i.e. λ 1 ≥ . . . ≥ λ n and w 1 , . . . , w n are corresponding eigenvectors. Since T C is connected (as C is irreducible) then λ n−1 > 0 [29, p. 147]. On the other hand, for every i-th row of L(T C ) it holds that l ii = n j=1,i =j |l ij |, thus for w n = [1, . . . , 1] T we have L(T C )w n = 0, which implies that λ n = 0. In other words, all the eigenvalues except the smallest one are real and positive. L(T C ) is a symmetric, thus, all its eigenvectors are orthogonal. In particular this means that every w 1 , . . . , w n−1 is orthogonal to w n , i.e. w T i w n = 0 for i = 1, . . . , n. Let us consider the equation i.e. L(T C )w i + J n w i = λ i · w i , for i = 1, . . . , n For i = 1, . . . , n − 1 due to the orthogonality holds that w T i w n = 0 i.e.
It means that the eigenvalue 0 for the matrix L(T C ) has turned into n as the eigenvalue of M . Thus, the eigenvalues of M are n, λ 1 , λ 2 , . . . , λ n−1 . Since all the eigenvalues of M are positive, then M is nonsingular.
From the above theorem, it follows that (10) always has a unique solution.

Optimality
The GM method [13] is considered as optimal since it minimizes the logarithmic least square (LLS) condition S(C) (7) for some (complete) PC matrix C. It is natural to assume that the LLS condition for an incomplete PC matrix has to be limited to the existing entries. Hence, the LLS condition for the incomplete PC matrices S * (C) (8) has been formulated in a way that the missing values c ij =? are not taken into account [5,4,27].
The proposed method (Sec. 4) is, in fact, the GM method applied to the completed matrix C * . Therefore, due to Crawford's theorem, the vector w obtained as the solution of (9) minimizes S(C * ), where S(C * ) = Hence, it is optimal. This also means that it is equivalent to the LLS method for incomplete PC matrices, as defined in [4].

Summary
In this work, the GM method for incomplete PC matrices has been proposed. Its optimality and equivalence of the LLS method have been proven. The advantages of the presented solution are, on the one hand, optimality and, on the other hand, its relative simplicity. To compute the ranking, one needs to solve the linear equation system and perform appropriate logarithmic transformations on the matrix. Hence, with some reasonable effort, the ranking can be computed using even Microsoft's Excel with its embedded solver.
The proposed method meets the need for an efficient and straightforward ranking calculation procedure for incomplete matrices. It is based on the wellknown and trusted GM method initially defined for complete PC matrices. We hope that it will be a valuable addition to the existing solutions.

Acknowledgment
I would like to thank Prof. Ryszard Szwarc for his insightful remarks on the existence of the solution (Sec. 6.1). Special thanks are due to Ian Corkil for his editorial help. The research was supported by the National Science Centre, Poland, as part of project no. 2017/25/B/HS4/01617.