Almost optimal manipulation of a pair of alternatives

The role of an expert in the decision-making process is crucial, as the ﬁnal recommendation depends on his disposition, clarity of mind, experience, and knowledge of the problem. However, the recommendation also depends on their honesty. But what if the expert is dishonest? Then, the answer on how diﬃcult it is to manipulate in a given case becomes essential. In the presented work, we consider manipulation of a ranking obtained by comparing alternatives in pairs. More speciﬁcally, we propose an algorithm for ﬁnding an almost optimal way to swap the positions of two selected alternatives. Thanks to this, it is possible to determine how diﬃcult such manipulation is in a given case. Theoretical considerations are illustrated by a practical example.


Introduction
The popularity of the pairwise comparison methods in the field of multicriteria decision analysis is largely due to their simplicity.It is easier for a decision maker to compare two objects at the same time, as opposed to comparing larger groups.Although the first systematic use of pairwise comparison is attributed to Ramon Llull [6], thirteenth-century alchemist and mathematician, it can be assumed that also prehistoric people used this method in practice.In the beginning, people were interested in qualitative comparisons.Over time, however, this method gained a quantitative character.The twentieth-century precursor of the quantitative use of pairwise comparison was Thurstone, who harnessed this method to compare social values [43].Continuation of studies on the pairwise comparison method [44,11,35] resulted the seminal work written by Saaty [40].In this article Saaty proposed the Analytic Hierarchy Process (AHP) -a new multiple-criteria decision-making method.Thanks to the popularity of AHP, the pairwise comparison method has become one of the most frequently used decision-making techniques.Its numerous variants and extensions have found application in economy [37], consumer research [14], management [31,36,45], construction [10], military science [18], education and science [32,21], chemical engineering [9], oil industry [17] and others.The method is constantly developing and inspires researchers who conduct work on the inconsistency of the paired comparisons [2,3,5,24,28], incompleteness of decision data [13,8,30,29], priority calculations [34,4,25,27,19], representation of uncertain knowledge [48,1,38] as well as new methods based on the pairwise comparisons principle [38,39,23,26].
Popularity of the decision-making methods also makes them vulnerable to attacks and manipulations.This problem has been studied by several researchers including Yager [46,47] who considered strategic preferential manipulations, Dong et al. [12], who addressed manipulation in the group decision-making or Sasaki [41], on strategic manipulation in the context of in group decisions with pairwise comparisons.Some aspects of decision manipulation in the context of electoral systems are presented in [16,15,42].
In the presented work, we want to take a step towards determining the degree of difficulty of manipulating in the pairwise comparison method.For this purpose, we will propose an algorithm for calculating the closest approximation of the pairwise comparison matrix (PCM), which equates the priorities of two selected alternatives.We apply a similar technique of orthogonal projections to that used in [20,22].The difference between the initial matrix and the modified matrix shows the degree of difficulty of a given manipulation.Although the reasoning is done for additive matrices, the obtained result is also valid for multiplicative matrices.
The article consists of four sections.Introduction (Sec. 1) and Preliminaries (Sec.2) present the state of research and introduce basic concepts and definitions in the field of the pairwise comparison method.The third section, Towards optimal manipulation of a pair of alternatives, defines the procedure to construct a manipulated pairwise comparisons matrix.It also contains a method for determining the difficulty of manipulation.The article ends with Conclusion (Sec.4), summarizing the achieved results.

Multiplicative pairwise comparisons matrices
Let us assume that we want perform pairwise comparison of a finite set E = {e 1 , . . ., e n } of alternatives.The comparisons can be expressed in a pairwise comparisons matrix (PCM) M = [m ij ] with positive elements satisfying the reciprocity condition for i, j ∈ {1, . . ., n}.
Given a single PCM M , one of the more popular procedures to assign a positive weight w k to each alternative e k (k ∈ {1, . . ., n}), showing its position in a ranking, is the Geometric Mean Method (GMM), introduced in [7].Then, the formula for w k can be calculated as the geometric mean of the k-th row elements: If we want to standardize the resulting weight vector, we divide each coordinate by the sum of all of them: Let us denote the set of all PCMs by M.

Additive pairwise comparisons matrices
The family M is not a linear space.However, we can easily transform every multiplicative PCM M into an additive one using the following map: is a linear space of additive PCMs.Obviously, we can define the map Since ϕ and µ are mutually reverse, from now on we will consider only the additive case in order to use the algebraic structure of A.
If we treat an additive PCM A as the image of M ∈ M by the map ϕ, we can also obtain the vector of weights v by use of the logarithmic mapping: By applying (2) we get so the k-th coordinate of A can be calculated as the arithmetic mean of the k-th row of A.

Towards optimal manipulation of a pair of alternatives
Let us start with a very simple example.
Example 1.Consider a family of additive pairwise comparisons matrices If we take ε = 1 n and ε = − 1 n (n ∈ N), we obtain two PCMs, whose weight vectors are It implies that the order of alternatives is (a 1 , a 3 , a 2 ) in the first case and (a 2 , a 3 , a 1 ) in the second case.
Since the standard Frobenious distance of the matrices is they can be arbitrarily close.
Example 1 shows that it is impossible to find the PCM closest to a given one such that the ranking positions of two given alternatives induced by both matrices will be reversed.
However, it is possible to find a matrix closest to a given one such that their positions are the same.

The tie spaces
Fix i, j ∈ {1, . . ., n}.Let us define the subspace A ij of all additive PCMs which induce the ranking such that alternatives i and j are equal: We will call such a linear space a tie space.
Proof.Each reciprocal additive matrix is uniquely defined by n 2 −n 2 independent numbers a qr , 1 ≤ q < r ≤ n above the main diagonal.The equation and to Finally, ( 3) is equivalent to This way we express a jn by the rest of the elements above the main diagonal in the i-th and j-th row and column, which decreases the dimension of the space by one.Now let us define a basis for the tie space A ij .With no loss of generality we can assume that i < j < n.Let Proof.The number of all elements above the main diagonal is n 2 −n 2 .There are: • n − j elements in the j-th row.
Thus, the total number of Z ij elements equals At first, for each (q, r) ∈ Z ij let us define C qr ∈ A, whose elements are given by Next, we define the elements of additive PCMs D p for p ∈ {1, . . ., i − 1}, E p for p ∈ {1, . . ., j − 1}, F p for p ∈ {i + 1, . . ., j − 1, j + 1, . . ., n} and G p for p ∈ {j + 1, . . ., n − 1} by formulas: Proof.Summing up the numbers of the consecutive matrices, we get the number Theorem 5. A family of matrices Proof.By Propositions 2 and 4, the cardinality of B is equal to the dimension of A ij , so it is enough to show that each matrix A ∈ A ij is generated by matrices from B.
For this purpose, let us define a matrix H as a linear combination of matrices from B: It is straightforward (as all but one addends of the sum are zeros) that for (q, r) ∈ {(j, n), (n, j)} we have Likewise, The last equality follows from (4).By analogy, h nj = a nj , so A = H, which completes the proof.
Let us redefine B as a family of matrices as follows: We set {B p } as matrices {C qr } (q,r)∈Zij , ordered lexicografically, i.e.

Orthogonalization
It is a well-known fact of linear algebra that any basis of an inner product space can be easily transformed into an orthogonal basis by a standard Gram-Schmidt process.
In particular, if we apply that to the basis B 1 , . . ., B n 2 −n 2 −1 of the A ij vector space equipped with a standard Frobenius inner product •, • then we obtain a pairwise orthogonal basis as follows: Example 7. Let us consider matrices B 1 , . . ., B 9 from Example 6.We will apply the Gram-Schmidt process to obtain an orthogonal basis H 1 , . . ., H 9 of A 23 : .

The best approximation of a PCM equating two alternatives
Consider an additive PCM A. In order to find its projection A ′ onto the subspace A ij we present A ′ as a linear combination of the orthogonal basis vectors We will look for the factors Then, ∀C ∈ A ij , A − A ′ , C F = 0, which is equivalent to the system of linear equations: Thus, the PCM A ′ which generates a ranking equating the i-th and j-th alternatives and which is the closest to A can be calculated from the formula Example 8. Let us consider the PCM The weights in a ranking vector obtained as the arithmetic means of row elements of A are w = (0.2, 1.2, −1.8, −3.4,3.8) T .
In order to find the PCM closest to A which generates a ranking equating the second and the third alternative, we take the orthogonal basis H 1 , . . ., H 9 described in Ex. 7. Next, we calculate the coefficients in (11): Finally, we obtain the orthogonal projection of A onto A 23 : The corresponding vector of weights is Let us notice that the weights of the second and third alternatives are actually equal.What's more, the weights of the rest of alternatives have not changed.The common weight of the second and the third alternative in w ′ is the arithmetic mean of the corresponding weights in w.
It appears that the remark above is true regardless of the dimension of the PCM and of the choice of the two alternatives whose weights are equalized, i.e the following theorem is true: (2) Proof.Let us assume, without the loss of generality, that i < j.Note that

On the other hand, A
which proves (15).

3.4
Measuring the ease of manipulation 3.4.Measuring the ease of manipulation Manipulation carried out by an expert carries the risk of detection.The consequences for the expert may vary, from loss of trust of the person ordering the ranking to criminal sanctions.Of course, the smaller the difference between the correct matrix and the manipulated one, the lower the chances of dishonest answers being detected, and thus the easier the manipulation.Therefore, the distance between the individual elements of the matrix can be considered as the indicator of the ease of manipulation.
It can be seen that both matrices differ in the rows and columns corresponding to the replaced alternatives.For example, the absolute difference of matrices A (12) and A ′ (13) given as Thus, A ′′ has 14 elements different than 0. In general, such a matrix can have 4n − 6 non-zero elements.Therefore, the Ease of Manipulation Index (EMI) takes the form of the average difference between elements of the original matrix and the manipulated one, i.e.

EMI(A, A
In the case of A (12) and A ′ (13), the value of EMI(A, A ′ ) = 1.785.

Conclusion and summary
In the presented work, we introduce a method to find a closest approximation of a PCM which equates two given alternatives (let's say, the i-th and the jth one).We also prove that the weights of all of the other alternatives do not change, while the new weights of the equated alternatives are equal to the arithmetic mean of the original ones.
Example 1 shows that it is impossible to find the best approximation of a PCM such that the positions of the i-th and the j-th alternatives in a ranking are reversed.However, if two alternatives have the same ranking, we may slightly change the element a ij in order to tip the scales of victory in favor of one of them.The resulting matrix will satisfy the manipulation condition.
We also proposed an Ease of Manipulation Index (EMI), which allows to determine the average difficulty of switching the positions of two alternatives.The analysis of A ′′ resulting from the difference between the original and manipulated PCMs can also indicate pairwise comparisons where manipulation is particularly difficult (the distance between a ij and its counterpart a ′ ij is exceptionally large).
An alternative gradient method presented in [33] is a possible subject of the future research.Another possible generalization could be incomplete PCMs considered for example in [29].
Consider three cases: (a) If k < i, we add equations (2 k ) and (3 k ) and we get S = 0. (b) If i < k < j, we subtract equation (4 k ) from (3 k ) and we get S = 0. (c) If k > j, we add equations (4 k ) and (5 k ) and we get S = 0.