Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC - Coimbra

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1 Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC - Coimbra Claude Lamboray Luis C. Dias Pairwise support maximization methods to exploit valued outranking relations in ranking problems No ISSN: Instituto de Engenharia de Sistemas e Computadores de Coimbra INESC - Coimbra Rua Antero de Quental, 199; Coimbra; Portugal

2 Pairwise support maximization methods to exploit valued outranking relations in ranking problems Claude Lamboray ULB/University of Luxembourg CODE (Computer and Decision Engineering), C.P.210/01, B-1050 Brussels, Belgium Luis C. Dias INESC Coimbra and Faculdade de Economia da Universidade de Coimbra Av Dias da Silva 165, Coimbra, Portugal. Abstract Many outranking methods involve two steps. First, the alternatives are compared pairwise to build a valued outranking relation; this relation is then exploited to derive a recommendation for the decision maker. We present three related exploitation models providing three types of solutions, depending on the decision maker s request: a linear order, a weak order, or a weak partial order. Each solution is evaluated by computing the support for the pairwise comparisons it implies. This comparison may be based on the minimum support present in the solution (prudence principle), the lexicographically minimum support, or the sum of the supports. Relations between these models and some of their properties are discussed and illustrated using data sets found in the literature. 1 Introduction Outranking methods [9, 19, 18] usually involve 2 steps. First, the alternatives are compared pairwise in order to build an outranking relation. This relation can be crisp, such as in ELECTRE I, or valued such as in ELECTRE III. In a second step, this outranking relation is exploited in order to come up with a recommendation for the decision maker. In this paper, we focus on this exploitation step in ranking problems. A major issue with the outranking approach is that the pairwise preference information can contradict the purpose of deriving a global level ranking. There 1

3 can be preference cycles which prevent the construction of a transitive ranking on the alternatives that would completely respect the initial information. Indifference thresholds sometimes used in the pairwise comparison computations are another possible source of intransitivity. The possibility of the outranking relation suggesting an incomparability or an indifference between two alternative further adds to the complexity of the problem. All this motivates the need for reasonable exploitation methods. Despite these difficulties, various exploitation methods have been proposed and studied (see for instance [15], [22] [16], [8], [7], [12],[13]). None of these techniques seems to be objectively preferable to another one. Most of them are inspired by or are at least closely related to voting rules. For instance, it is well known that ranking alternatives by flows, as it is done in the PROMETHEE method [14], is closely related to Borda s rule [5]. The aim of this research is to investigate how the prudence principle initially proposed by Arrow and Raynaud [1] can be applied to exploit a valued outranking relation in order to build a ranking. Intuitively, the prudence principle stipulates that the weakest support should be maximized, considering the preferences between pairs of alternatives implied in a ranking. In its original context of social choice theory, the prudence principle applied to binary relations satisfying the constant-sum property (S ij + S ji = S xy + S yx, i, j, x, y, i j, x y), an assumption that we drop in this paper. The original idea of Arrow and Raynaud ia also extended by presenting different models which follow their principle and which, depending on the decision problem, either constructs a linear order, a weak order or a partial weak order. We consider that among the results maximizing the weakest support, we can break existing ties following the same principle in a lexicographical way [11]. We also consider models not based on Arrow and Raynaud s idea, but rather on maximizing the sum of the supports, in what could be regarded as an extension of Kemeny s voting method to a general outranking relation, and to weak or partial weak orders. This paper is organized as follows. Section 2 briefly introduces some notation and definitions about preference structures. The definition of support for a conclusion about a pair of alternatives and support for a preference structure are presented in Section 3, along with exploitation models to derive a linear order, a weak order or a partial weak order from a valued outranking relation. These models are complemented in Section 4 by introducing two variants that can be applied to derive linear orders. The relations between all the models introduced are discussed in Section 5, and some of the models properties are the subject of Section 6. Section 7 presents some illustrative examples, and Section 8 presents some concluding remarks. 2

4 2 Preference structures Let A = {a 1, a 2,..., a n } be a set of n alternatives. Let R be a binary relation over A. We denote by a i Ra j the fact that the pair (a i, a j ) belongs to the relation R. A binary relation R may or not satisfy each of the following properties: R is symmetric if a i, a j A, a i Ra j a j Ra j. R is asymmetric if a i, a j A, a i Ra j not(a j Ra j ). R is reflexive if a i A, a i Ra i. R is irreflexive if a i A, not(a i Ra i ). R is complete if a i, a j A, a i Ra j or a j Ra i. R is transitive if a i, a j, a k A, a i Ra j and a j Ra k a i Ra k. Following the terminology presented in [17], we say that a preference structure is a triplet (P, I, J), where P is the strict preference relation, I, the indifference relation and J, the incomparability relation. A preference structure (P, I, J) must verify the following properties: P is asymmetric. I is reflexive and symmetric. J is irreflexive and symmetric. P I J is complete. A preference structure is a linear order if P is complete (i.e., I = {(a i, a i ) : a i A, J = ) and transitive. We denote by LO the set of all linear orders on A. A preference structure is a weak order if P I is complete (J = ) and P I is transitive. We denote by WO the set of all the weak orders on A. A preference structure is a partial weak order if P I is transitive. We denote by PWO the set of all the partial weak orders on A. It is clear that LO WO PWO. 3 Pairwise support exploitation models In this section, we present our main exploitation models. We suppose from now on that a valued outranking relation has been previously built. For all a i, a j A, we denote S ij the value which indicates to what extent a i is as least as good as a j. We denote S the matrix such that i, j, the entry of row i and column j corresponds to S ij. We suppose that: 3

5 1. a i, a j A, S ij [0, 1]. 2. a i, a j A, if S ij > 0.5, then this means that "a i is as least as good as a j " is more credible than "a i is not as least as good as a j ". 3. a i, a j A, if S ij < 0.5, then this means that "a i is not as least as good as a j " is more credible than "a i is as least as good as a j ". 4. a i, a j, a k, a l A, if S ij > S kl, then this means that "a i is as least as good as a j " is more credible than "a k is as least as good as a l ". Such a valued outranking relation can be computed as in the ELECTRE methods [18, 19, 9]. It usually requires fixing parameters such as weights and preference thresholds which model the preferences of the decision maker. As shown by Bouyssou [6], no structural properties can be made on such a valued outranking relation in the sense that any relation S can be obtained by choosing a suitable evaluation table of the alternatives with the right set of parameters. The key idea of our exploitation models is to evaluate the support for preference, indifference or incomparability between two alternatives based on the initial valued outranking relation. More particularly, a i, a j A, we denote S ij >, S= ij and S? ij the support for preference, indifference or incomparability between alternative a i and alternative a j. By relying on the bi-polar credibility calculus [2], we define these supports as follows: a i, a j A, a i a j, S > ij = min(s ij, 1 S ji ) S = ij = min(s ij, S ji ) S? ij = min(1 S ij, 1 S ji ) Accordingly, the support for a preference of a i over a j is high if it is highly credible that a i is as least as good as a j and not much credible that a j is as least as good as a i. If there is a high credibility that a i is at least as good as a j and vice-versa, then there is a high support for considering a i indifferent to a j. Finally, a i and a j are considered incomparable with a high support if there is not much credibility to consider any of them at least as good as the other. The bi-polar calculus seems to provide an appropriate way of computing pairwise supports. However, other possibilities could be considered. For instance, in Section 4.2, we introduce a dual version of these supports. If the valued outranking relation is such that a i, a j, a i a j, S ij + S ji = 1 (satisfying the constant-sum property), then it is easy to see that S > ij = S ij and that S = ij = S? ij. 4

6 Although Electre-type methods traditionally assume the valued outranking degree to be in the [0,1] interval, with 0.5 as midpoint, other possibilities could be considered. For instance, Bisdorff [4] suggests to work in a [-1,1] interval, with 0 as undetermined mid-point. Following the bi-polar credibility calculus, the pairwise supports for preference, indifference and incomparability have to be adapted accordingly in the sense that the negation 1 S ij has to be replaced with S ij. Given a preference structure (P, I, J), we denote σ {i,j} (P, I, J) the evaluation of the support of pair {a i, a j } in the preference structure (P, I, J) defined as follows: a i, a j A, a i a j, σ {i,j} (P, I, J) = S ij > S ji > S ij = S ij? if a i P a j if a j P a i if a i Ia j if a i Ja j Since there are n(n 1) 2 different pairs, there are n(n 1) 2 such pairwise support evaluations for each preference structure. We denote σ(p, I, J) an n(n 1) 2 - dimensional vector containing these pairwise supports. Each pairwise support is well-defined because, (P,I,J) being a preference structure, exactly one of the 4 possibilities is satisfied. We want to evaluate which of two preference structures (P, I, J) and ( ˆP, Î, Ĵ) translates better the information contained in the initial valued outranking relation. To do so, we introduce a binary relation between the n(n 1) 2 dimensional support vector for (P, I, J) and the n(n 1) 2 dimensional support vector for ( P, Ĩ, J). If σ( P, Ĩ, J) σ( ˆP, Î, Ĵ), then (P, I, J) translates the information contained in S at least as good as ( ˆP, Î, Ĵ). With the purpose of finding the preference structure that best translates the information is S, we need the relation to be transitive and complete. We now present three possibilities of defining for comparing two preference structures. 1. We denote σ (1) (P, I, J), the smallest pairwise support. σ (1) (P, I, J) = min σ {i,j}(p, I, J). {i,j}:i j The relation min compares two preference structures (P, I, J) and ( P, Ĩ, J) based on the weakest pairwise supports: σ(p, I, J) min σ( P, Ĩ, J) σ (1) (P, I, J) σ (1) (P, I, J). 5

7 2. More generally, we denote σ (k) (P, I, J), the k th smallest pairwise support. The relation lex (a refinement of min ) compares two preference structures (P, I, J) and ( P, Ĩ, J) lexicographically as follows: σ(p, I, J) lex σ( P, Ĩ, J) σ (i) (P, I, J) = σ (i) n(n 1) (P, I, J), i = 1,..., 2 { n(n 1) σ or t : (i) (P, I, J) = σ (i) (P, I, J), i < t 2 σ (t) (P, I, J) > σ (t) (P, I, J) 3. We can also define a relation based on a sum, although such an operator should not be used unless the pairwise supports have a cardinal nature (i.e., measured according to an interval scale). σ(p, I, J) sum σ( P, Ĩ, J) n n i=1 j=i+1 σ {i,j} (P, I, J) n n i=1 j=i+1 σ {i,j} ( P, Ĩ, J). We are now ready to formalize our exploitation models, for situations in which the decision maker requests the solution to be a linear order, a weak order or a partial weak order. If, in the tradition of outranking methods (Roy et al. s ELECTRE II, II, and IV; Brans and Vincke s PROMETHEE I), the decision maker accepts the solution to be a partial weak order, then we look for the following set of solutions: PWO (S) = {(P, I, J) PWO : ( ˆP, Î, Ĵ) PWO : σ(p, I, J) σ( ˆP, Î, Ĵ)}. We denote this exploitation model "PWO " since 1) we are looking for partial weak orders and 2) we compare two preference structures using the relation. In some situations, exploitation model PWO may not be satisfactory for the decision maker because the partial weak order contains too many incomparabilities. In such cases we can restrict the set of solutions to the set of weak orders. Model WO can thus be defined as follows: WO (S) = {(P, I, J) WO : ( ˆP, Î, Ĵ) WO : σ(p, I, J) σ( ˆP, Î, Ĵ)}. If the decision maker requests the solution not to contain any indifference, then exploitation model LO can be considered: LO (S) = {(P, I, J) LO : ( ˆP, Î, Ĵ) LO : σ(p, I, J) σ( ˆP, Î, Ĵ)}. If a i, a j A, a i a j, we have that S ij + S ji = 1, then model LO min is equivalent to the set of prudent orders as defined by Arrow and Raynaud [1]. In fact, a prudent order has been initially defined in a constant-sum context as a linear order such that the smallest S ij of the ordered pairs belonging to that 6

8 linear order is maximal (see also Lamboray [10]). We illustrate the exploitation models on the following example. There are 3 alternatives a, b and c with the following valued outranking relation: a b c a b c This leads to the following pairwise supports: S > S = S? (S > ) 1 (a, b) (b, c) (a, c) The data suggests that the most supported relation between a and b is ap b, whereas the most supported relation between b and c is bjc and the most supported relation between a and c is aic. However, this does not yield a linear order, or a weak order, or even a partial weak order, since ap b together with aic would by transitivity imply that cp b, which has a support of When looking for a linear order, the supports that are taken into account are those that concern preference: 0.59 if ap b vs if bp a; 0.12 for bp c vs if cp b; 0.40 if ap c vs if cp a. If the chosen criterion is to maximize the minimum support (prudent order principle), using relation min, then only pair {b, c} is determinant, and any linear extension of cp b will maximize the minimum support. A possible refinement is to use relation lex to make a distinction between these ex-aequo extensions. Then, ap cp b will maximize the second worst support (0.40), avoiding the low support of stating cp a (0.24). The support for ap cp b is then, lexicographically, (0.20, 0.40, 0.59). This linear order is also the best one when sum is used (sum=1.19, meaning an average support of approx. 0.40) If the decision maker accepted a weak order, then the support can increase by considering aic instead of ap c, maintaining both a and c preferred to b. Indeed, the weak order aicp b is the best according to lex and sum. The supports have now increased to, lexicographically, (0.20, 0.59, 0.60). If the decision maker accepted a partial weak order, then we must have bjc to avoid a support of 0.20 or lower, and then the best options would be to make ap b and ap c. The supports would again increase to, lexicographically, (0.40, 0.59, 0.80), and this would be the best partial weak order according to lex (and also sum ). There is naturally a trade-off between the type of ranking (from a complete ranking without ties to a partial weak order) and the support with respect to the original outranking relation information. 7

9 4 Two other models for linear orders In this section, we present two other exploitation models which can be used when the solution is required to be a linear order. The first model is based on the notion of a relevant pair. The second model looks at the dual of the supports. 4.1 Relevant pairs model When both S ij and S ji are small, then this may suggest an incomparability between a i and a j. Similarly, when both S ij and S ji are large, then this may suggest an indifference between a i and a j. In such cases, the use of model LO min (or its refinement LO lex ) results in a strong influence from such pairs, since the credibility for a preference will be very low (possibly the minimum). Furthermore, the resulting minimum support may be considered as very low. As an example, let us look at the following valued outranking relation with three alternatives a 1, a 2 and a 3. a b c a b c This example suggests that a is preferred to b and that b is preferred to c, and so abc could be a potentially interesting solution if one requires the result to be a weak or linear order. However, model LO lex leads to the solution cab. In fact, the pair {a, c} discards all the solutions where a is preferred to c or a is indifferent to c, since S a,c > = 0.09 < S c,a > = 0.10, although none of these two supports is very convincing. In order to overcome this limitation, we will introduce the notion of an irrelevant pair. We say that {a i, a j } is an irrelevant pair if S ij < 0.5 and S ji < 0.5 or S ij > 0.5 and S ji > 0.5. If the solution is a linear order, then a pair is irrelevant if it suggests an incomparability or an indifference (i.e., if incomparability or indifference has higher support than a preference in either direction), since a linear order cannot handle incomparability or indifference. A pair which is not irrelevant is said to be relevant. One possible variant is to evaluate only pairs which are relevant. In other words, σ {i,j} (P, I, J) will only be defined if {a i, a j } is relevant. Consequently, the number of pairs which are evaluated may be less than n(n 1) 2 and is equal to the number of relevant pairs, given the initial valued outranking relation. We denote σ rp (P, I, J) the pairwise support evaluation vector restricted to the set of relevant pairs. We denote this model LO rp : LO rp (S) = {(P, I, J) LO : ( ˆP, Î, Ĵ) LO : σrp (P, I, J) σ rp ( ˆP, Î, Ĵ)}. 8

10 In the example at the beginning of this section, the solution found by model is abc (note that pair (a, c) would be considered irrelevant). LO rp lex 4.2 Dual model Instead of looking at the support that a i is preferred to a j, one may look at the support that a j is not preferred to a i. According to the bi-polar credibility calculus, the credibility that a j is not preferred to a i is equal to 1 min(s ji, 1 S ji ), which is equal to max(s ij, 1 S ji ). Hence, we can define pairwise supports as follows: a i, a j A, S >d ij = max(s ij, 1 S ji ) We denote σ d (P, I, J) the pairwise dual support evaluation vector. We denote the model by LO d : LO d (S) = {(P, I, J) LO : ( ˆP, Î, Ĵ) LO : σd (P, I, J) σ d ( ˆP, Î, Ĵ)}. In the example used to present the relevant pairs model, the solution found by model LO d is abc, according to the dual supports presented in the table below. As in the relevant pairs model, the strong incomparability between a and c does not penalize placing ap c (or the reverse). a b c a b c Relationships between the models In this section we analyze some relationships between the models that we have introduced. 5.1 Solution consistency between the PWO, WO and LO models The solutions obtained in models PWO, WO and LO can be consistent: if a solution to the partial weak order model is a weak order (or a linear order), then it will also be a solution to the weak order model (or the linear order model, resp.); if a solution to the weak order model is a linear order, then it will also be a solution to the linear order model. This property holds for lex, min, and sum. 9

11 Proposition 1 Let { lex, min, sum }. 1. For any valued outranking relation S, let (P, I, J) PWO (S). If (P, I, J) WO, then (P, I, J) WO (S). If (P, I, J) LO, then (P, I, J) WO (S) and (P, I, J) LO (S). 2. For any valued outranking relation S, let (P, I, J) WO (S). If (P, I, J) LO, then (P, I, J) LO (S). Proof: Let (P, I, J) be a solution obtained in model PWO. Then: ( ˆP, Î, Ĵ) PWO : σ(p, I, J) σ( ˆP, Î, Ĵ). Since WO PWO, this implies that ( ˆP, Î, Ĵ) WO : σ(p, I, J) σ( ˆP, Î, Ĵ). If (P, I, J) WO, this means that (P, I, J) will be obtained in model WO. Since LO WO, this also implies that ( ˆP, Î, Ĵ) LO : σ(p, I, J) σ( ˆP, Î, Ĵ). If (P, I, J) LO, this means that (P, I, J) will be obtained in model LO. The proof of part 2 of the property is similar. Such solution consistency is not verified between model PWO and model LO d. Consider the following example: a b c a b c We have that abc is a solution obtained in model PWO lex, and consequently, this solution is also obtained in model PWO min. However, the dual supports of abc are 0.78 (for pair {a, b}), 0.87 (for pair {b, c}) and 0.40 (for pair {a, c}), whereas the dual supports of bca are 0.87 (for pair {b, c}), 0.60 (for pair {c, a}) and 0.41 (for pair {b, a}). In fact bca is the unique solution obtained in model LO d lex and in model LO d min. Similarly, solution consistency is violated between models PWO lex and lex. Consider the following counterexample with four alternatives a, b, c LO rp and d: a b c d a b c d

12 On the one hand, the linear order bcda is obtained in model PWO lex. On the other hand, {b, d} being an irrelevant pair, cdab is the unique linear order found in model LO rp lex. 5.2 Sum-based original and dual pairwise support models We will show that if we rely on the relation sum to compare two preference structures, then the dual model is equivalent to the original model. Proposition 2 S, LO sum (S) = LO d sum (S) Proof: First we show that a i, a j A, Let ij = S ij S ji. We then have: S > ij S> ji = S>d ij S >d ji. (1) min(s ij, 1 S ji ) min(s ji, 1 S ij ) = ( ij + S ji S ij ) + min(s ij, 1 S ji ) min(s ji, 1 S ij ) = ij + min(s ij S ij, 1 S ji S ij ) min(s ji S ji, 1 S ij S ji ) = ij + min(0, 1 S ji S ij ) min(0, 1 S ji S ij ) = ij In the same manner, we can check that: max(s ij, 1 S ji ) max(s ji, 1 S ij) ) = ij This proves equation 1. Consequently, a i, a j A, there must exist a constant c {i,j} such that: S ij >d = S ij > + c {i,j} and S ji >d = S ji > + c {i,j}. Let (P, I, J) be a solution found in model LO sum. This means that: ( P, Ĩ, J) LO, S ij > ( P, Ĩ, J) LO, ( P, Ĩ, J) LO, (a i,a j ) P (a i,a j ) P (a i,a j) P (a i,a j ) P S > ij (S > ij + c {i,j}) S >d ij (a i,a j) P (a i,a j) P S >d ij (S > ij + c {i,j}) This means that (P, I, J) is also a solution found in model LO d sum. 11

13 5.3 Sum-based and leximin-based original pairwise support models There exists a particular link between the solutions obtained with leximin-based models and the solutions obtained with sum-based models. In fact, either every solution obtained under the leximin-based models is also obtained by the sum-based models, or no solution obtained under the leximin-based models is obtained by the sum-based models. This idea is formalized in the next proposition. Proposition 3 S, PWO lex (S) PWO sum (S) or PWO lex (S) PWO sum (S) =. S, WO lex (S) WO sum (S) or WO lex (S) WO sum (S) =. S, LO lex (S) LO sum (S) or LO lex (S) LO sum (S) =. S, LO rp lex (S) LO rp sum (S) or LO rp lex (S) LO rp sum (S) =. S, LO d lex (S) LO d sum (S) or LO d lex (S) LO d sum (S) =. Proof: Either PWO lex (S) PWO sum (S) =, in which case the proof is complete. Otherwise, let (P b, I b, J b ) PWO lex (S) PWO sum (S). We then need to show that if (P, I, J) PWO lex (S), then (P, I, J) PWO sum (S). Since (P b, I b, J b ) PWO sum (S), we know that: ( P, Ĩ, J) PWO, n n i=1 j=i+1 σ {i,j} (P b, I b, J b ) n n i=1 j=i+1 σ {i,j} ( P, Ĩ, J). Since (P b, I b, J b ) PWO lex (S) and (P, I, J) PWO lex (S), we also know that σ(p b, I b, J b ) lex σ(p, I, J), where lex denotes the symmetric part of lex. Consequently we have that: n n i=1 j=i+1 σ {i,j} (P b, I b, J b ) = n n i=1 j=i+1 σ {i,j} (P, I, J). We can thus conclude that: ( P, Ĩ, J) PWO, σ {i,j} (P, I, J) i,j i,j σ {i,j} ( P, Ĩ, J). Hence we have that (P, I, J) PWO sum (S). The proof for the other models is similar. 12

14 5.4 Leximin and min based models Since lex is a refinement of min, it is easy to see that every solution which is obtained in the leximin based models is also obtained in the min based models. Proposition 4 S, LO lex (S) LO min (S) WO lex (S) WO min (S) PWO lex (S) PWO min (S) LO d lex (S) LO d min (S) LO lex (S) LO rp/ rp min (S) A direct corollary of this proposition is that if the solution obtained in a minbased models is unique, it must also be the unique solution of the corresponding leximin-based model. 5.5 Relevant pairs models We show that the relevant pairs model can be transformed into the original model if we use the operators lex or sum. This can be useful from a practical point of view. If we have an algorithm which solves the original models, we have at the same time an algorithm which solves the relevant pairs models. Proposition 5 For any valued outranking relation S, let S be defined as follows: a i, a j A, Then and { S ij Sij if {a = i, a j } is a relevant pair 0.5 otherwise LO rp lex (S) = LO lex (S ). LO rp sum (S) = LO sum (S ). Proof: We denote σ(p, I, J) (resp. σ (P, I, J)) the evaluations of the support of (P, I, J) when referring to S (resp. to S ). We denote IR the set of irrelevant pairs and by R the set of relevant pairs. If S ij = S ji = 0.5, then (P, I, J) LO, σ {i,j} (P, I, J) =

15 We have: (P, I, J) LO rp sum (S) ( P, Ĩ, J) LO, ( P, Ĩ, J) LO, ( P, Ĩ, J) LO, ( P, Ĩ, J) LO, (P, I, J) LO sum (S ) {a i,a j } R {a i,a j } R {a i,a j } R {a i,a j } R {a i,a j } R {a i,a j} R IR σ {i,j} (P, I, J) {a i,a j } R σ {i,j} (P, I, J) IR σ {i,j} ( P, Ĩ, J) IR σ {i,j} (P, I, J) + σ {i,j} ( P, Ĩ, J) + σ {i,j} (P ) {a i,a j } IR {a i,a j } IR {a i,a j} R IR σ {i,j} ( P, Ĩ, J) σ {i,j} (P, I, J) σ {i,j} ( P, Ĩ, J) σ {i,j} ( P, Ĩ, J) This proves the proposition for the sum-based models. We denote (σ(p, I, J)) R the vector σ(p, I, J) restricted to the relevant pairs. We denote lex the usual leximin relation defined between two vectors of the same length. We then have: (P, I, J) LO rp lex (S) ( P, Ĩ, J) LO, (σ(p, I, J)) R lex (σ( P, Ĩ, J)) R ( P, Ĩ, J) LO, ((σ(p, I, J)) R, 0.5,..., 0.5) lex ((σ( P, Ĩ, J)) R, 0.5,..., 0.5) ( P, Ĩ, J) LO, σ (P, I, J) lex σ ( P, Ĩ, J) (P, I, J) LO lex (S ) This proves the proposition for the leximin-based models. 6 Properties In this section we analyze more in detail some properties of the exploitation models. 6.1 Respect of data For any valued outranking relation S, we define the following preference structure ( P S, ĨS, J S ) which translates the pairwise relations contained in S that are 14

16 more credible than their negation: a i PS a j if S ij 0.5 and S ji 0.5, with at least one of the two inequalities being strict. a i Ĩ S a j if S ij 0.5 and S ji 0.5. a i JS a j if S ij < 0.5 and S ji < 0.5. Alternatively, some of our results are based on the following similar preference structure denoted by (P S, I S, J S ): a i P S a j if S ij > 0.5 and S ji < 0.5. a i I S a j if S ij > 0.5 and S ji > 0.5. a i J S a j if S ij < 0.5 and S ji < 0.5. It is easy to see that P S P S, I S ĨS and J S = J S. Unlike the binary relation P S Ĩ S JS, P S I S J S is not necessarily complete for every S. However, if P S I S J S is complete, then it must be identical to P S Ĩ S JS. Property 1 Let { lex, min, sum }. Let (P, I, J) be a preference structure obtained in model PWO (or model WO, or model LO ), i.e., (P, I, J) is optimal for the chosen model. It then holds that: σ (1) (P, I, J) > 0.5 (P, I, J) = (P S, I S, J S ). Proof: a) σ (1) (P, I, J) > 0.5 (P, I, J) = (P S, I S, J S ). Let us suppose by contradiction that (P, I, J) (P S, I S, J S ). Hence the two preference structures differ for at least one pair. For every pair {a k, a l } where (P, I, J) differs from (P S, I S, J S ), one of the following three situations must occur: a k P S a l and not a k P a l. We then have, by definition of P S, that S kl > 0.5 and S lk < 0.5, and it is easy to check that S > kl > 0.5 > max{s= kl, S? kl, S> lk }. Hence, σ {k,l} (P S, I S, J S ) > σ {k,l} (P, I, J) (the support for this pair under (P S, I S, J S ) is greater than the support under (P, I, J)). a k I S a l and not a k Ia l. We then have, by definition of I S, that S kl > 0.5 and S lk > 0.5, and it is easy to check that S = kl > 0.5 > max{s> kl, S? kl, S> lk }. Again, σ {k,l} (P S, I S, J S ) > σ {k,l} (P, I, J). a k J S a l and not a k Ja l. We then have, by definition of J S, that S kl < 0.5 and S lk < 0.5, and it is easy to check that S? kl > 0.5 > max{s> kl, S= kl, S> lk )}. Again, σ {k,l} (P S, I S, J S ) > σ {k,l} (P, I, J). Thus, for every pair where (P, I, J) differed from (P S, I S, J S ), the support σ {k,l} (P, I, J) would be lower than σ {k,l} (P S, I S, J S ), and for the pairs where the structures would not differ the support would be the same. As a consequence, (P, I, J) would have a support strictly lower than (P S, I S, J S ) regardless of the comparison relation used ( lex, min, or sum ). This is a contradiction since 15

17 we supposed that (P, I, J) had optimal support. Therefore, (P, I, J) cannot differ from (P S, I S, J S ). The same reasoning applies when restricting the analysis to linear orders or weak orders. b) σ (1) (P, I, J) > 0.5 (P, I, J) = (P S, I S, J S ). Obvious: if (P S, I S, J S ) is optimal for model PWO, model WO or model LO, then its worst support will be higher than 0.5, since all the pairs would have support higher than 0.5, by definition of P S, I S, and J S. According to Property 1, if the weakest relation in a preference structure has support higher than 0.5, then we have no local contradiction. In other words, local contradictions may appear on a global level only in cases where there exists no good solution, i.e., there exists no preference structure with support higher than 0.5 for all the relations it comprises. In voting theory, this phenomenon is known as Condorcet s paradox: by considering those ordered pairs for which there is a majority, a cyclic relation can appear. Property 2 Let { lex, min, sum }. Let (P, I, J) be a preference structure obtained in model PWO (or model WO, or model LO ), i.e., (P, I, J) is optimal for the chosen model. It then holds that: σ (1) (P, I, J) 0.5 ( P S, ĨS, J S ) is (also) optimal. Proof: a) σ (1) (P, I, J) 0.5 ( P S, ĨS, J S ) is optimal. Let us suppose by contradiction that (P, I, J) is optimal whereas ( P S, ĨS, J S ) is not optimal, which implies that (P, I, J) ( P S, ĨS, J S ). This is possible only, whatever the choice of the comparison relation ( lex, min, or sum ), if there exists at least one pair {a k, a l } where the two preference structures differ and such that σ {k,l} (P, I, J) > σ {k,l} ( P S, ĨS, J S ). Supposing this is true, then for such a pair {a k, a l } one of the following three situations must occur: a k PS a l and not a k P a l. We then have, by definition of P S, that S kl 0.5 and S lk 0.5, and it is easy to check that S > kl 0.5 max{s= kl, S? kl, S> lk }. Hence, σ {k,l} ( P S, ĨS, J S ) σ {k,l} (P, I, J). a k Ĩ S a l and not a k Ia l. We then have, by definition of I S, that S kl 0.5 and S lk 0.5, and it is easy to check that S = kl 0.5 max{s> kl, S? kl, S> lk }. Again, σ {k,l} ( P S, ĨS, J S ) σ {k,l} (P, I, J). a k JS a l and not a k Ja l. We then have, by definition of J S, that S kl < 0.5 and S lk < 0.5, and it is easy to check that S? kl > 0.5 max{s> kl, S= kl, S> lk )}. Hence, σ {k,l} ( P S, ĨS, J S ) > σ {k,l} (P, I, J). 16

18 Regardless of which of the three situations would occur, we would reach a contradiction. Therefore, ( P S, ĨS, J S ) coincides with (P, I, J) or it is another ex-aequo optimal preference structure. The same reasoning applies when restricting the analysis to linear orders or weak orders. b) σ (1) (P, I, J) 0.5 ( P S, ĨS, J S ) is optimal. If ( P S, ĨS, J S ) is optimal for model PWO, model WO or model LO, then its worst support will be at least 0.5, since all the pairs would have support 0.5 or higher, by definition of P S, ĨS, and J S. If the comparison relation is lex or min, then the fact that (P, I, J) is optimal solution implies that: σ (1) (P, I, J) σ (1) ( P S, ĨS, J S ) 0.5. If the comparison relation is sum, then the fact that (P, I, J) is an optimal solution implies that: σ {i,j} ( P S, ĨS, J S ). {i,j} σ {i,j} (P, I, J) = {i,j} Let us suppose by contradiction that σ (1) (P, I, J) < 0.5. Since σ (1) ( P S, ĨS, J S ) 0.5, and since the sum of the supports for both (P, I, J) and (P S, I S, J S ) is the same, there must exist at least one pair {a k, a l } such that σ {k,l} (P, I, J) > σ {k,l} ( P S, ĨS, J S ). This is only possible if the pair {a k, a l } is different in (P, I, J), and ( P S, ĨS, J S ), otherwise the support would be the same. Consequently, for such a pair {a k, a l } one of the following three situations must occur: a k PS a l and not a k P a l. We then have, by definition of P S, that S kl 0.5 and S lk 0.5, and it is easy to check that S > kl 0.5 max{s= kl, S? kl, S> lk }. Hence, σ {k,l} ( P S, ĨS, J S ) σ {k,l} (P, I, J). a k Ĩ S a l and not a k Ia l. We then have, by definition of ĨS, that S kl 0.5 and S lk 0.5, and it is easy to check that S = kl 0.5 max{s> kl, S? kl, S> lk }. Again, σ {k,l} ( P S, ĨS, J S ) σ {k,l} (P, I, J). a k JS a l and not a k Ja l. We then have, by definition of J S, that S kl < 0.5 and S lk < 0.5, and it is easy to check that S? kl > 0.5 max{s> kl, S= kl, S> lk )}. Hence, σ {k,l} ( P S, ĨS, J S ) > σ {k,l} (P, I, J). Regardless of which of the three situations would occur, we would reach a contradiction, since we also have that σ {k,l} (P, I, J) > σ {k,l} ( P S, ĨS, J S ). This proves that, when using as a comparison relation sum, we must have that σ (1) (P, I, J)

19 In fact, (P S, I S, J S ) translates pairwise the information contained in the valued outranking relation. Usually, such pairwise preference information is contradicted on a global level, e.g., if P S contains a cycle it is impossible to define linear (or weak) order not contradicting P S. However, if this pairwise information remains consistent on a global level, then exactly this preference structure is obtained in our exploitation models. Hence our models verify what Vincke [22] calls "respect of data", as shown in the following properties 3 and 4. Property 3 Let { lex, min, sum }. 1. a) If (P S, I S, J S ) PWO, then PWO (S) = {(P S, I S, J S )}. b) If (P S, I S, J S ) WO, then WO (S) = {(P S, I S, J S )}. c) If (P S, I S, J S ) LO, then LO (S) = {(P S, I S, J S )}. 2. If (P S, I S, J S ) LO, then LO rp (S) = {(P S, I S, J S )}. 3. If (P S, I S, J S ) LO, then LO d (S) = {(P S, I S, J S )}. Proof: 1. If a i P S a j, then this means that S ij > 0.5 and S ji < 0.5. Hence it is easy to check that: S > ij > 0.5 > max{s= ij, S? ij, S> ji }. If a i I S a j, then this means that S ij > 0.5 and S ji > 0.5. Hence it is easy to check that: S = ij > 0.5 > max{s> ij, S? ij, S> ji }. If a i J S a j, then this means that S ij < 0.5 and S ji < 0.5. Hence it is easy to check that: S? ij > 0.5 > max{s> ij, S= ij, S> ji }. Let us now consider a partial weak order ( ˆP, Î, Ĵ) different from (P S, I S, J S ). Hence the two preference structures differ for at least one pair {a k, a l }. From the definition of (P S, I S, J S ), it is obvious that one (and only one) of these relations will hold for any pair of alternatives. Hence, for any pair {a i, a j }, changing that relation would lead to decreased support. Any other preference structure ( ˆP, Î, Ĵ) different from (P S, I S, J S ) will thus have decreased support for the pairs where it differs from (P S, I S, J S ). If { lex, min, sum }, we can check that: ( ˆP, Î, Ĵ) PWO, ( ˆP, Î, Ĵ) (P S, I S, J S ), σ(p S, I S, J S ) σ( ˆP, Î, Ĵ), where denotes the asymmetric part of. Hence (P S, I S, J S ) is the unique solution obtained in model PWO, which proves 1a). The same reasoning can be used to prove 1b) and 1c). 2. If (P S, I S, J S ) LO, then this implies that J S = and that I S is reduced to the identity pairs. Hence there does not exists any pair {a i, a j } such that S ij < 0.5 and S ji < 0.5 or such that S ij > 0.5 and S ji > 0.5. Hence 18

20 there does not exist any irrelevant pair and so model LO rp is equivalent to model LO. We can deduce from point 1 that if (P S, I S, J S ) LO, then this is the unique solution obtained by model PWO, which proves If a i P S a j, then this means that S ij > 0.5 and S ji < 0.5. Hence it is easy to check that: S ij >d > 0.5 > S ji >d. Let us now consider a linear order ( ˆP, Î, Ĵ) different from (P S, I S, J S ). Hence there exists at least one pair {a k, a l } such that a k P S a l and a l ˆP ak. For any pair {a i, a j }, changing the relation from a i P S a j to a j ˆP ai would lead to decreased support. Any other linear order ( ˆP, Î, Ĵ) different from (P S, I S, J S ) will thus have decreased support for the pairs where it differs from (P S, I S, J S ). If { lex, min, sum }, we can check that: ( ˆP, Î, Ĵ) LO, ( ˆP, Î, Ĵ) (P S, I S, J S ), σ d (P S, I S, J S ) σ d ( ˆP, Î, Ĵ), where denotes the asymmetric part of. Hence (P S, I S, J S ) is the unique solution obtained in model LO d, which proves 3. We can also state a similar property based on the relation ( P S, ĨS, J S ). Property 4 Let { lex, min, sum }. 1. a) If ( P S, ĨS, J S ) PWO, then ( P S, ĨS, J S ) PWO (S). b) If ( P S, ĨS, J S ) WO, then ( P S, ĨS, J S ) WO (S). c) If ( P S, ĨS, J S ) LO, then ( P S, ĨS, J S ) LO (S). 2. If ( P S, ĨS, J S ) LO, then ( P S, ĨS, J S ) LO rp (S). 3. If ( P S, ĨS, J S ) LO, then ( P S, ĨS, J S ) LO d (S). Proof: 1. If a i PS a j, then this means that S ij 0.5 and S ji 0.5. Hence it is easy to check that: S > ij 0.5 max{s= ij, S? ij, S> ji }. If a i Ĩ S a j, then this means that S ij 0.5 and S ji 0.5. Hence it is easy to check that: S = ij 0.5 max{s> ij, S? ij, S> ji }. If a i JS a j, then this means that S ij < 0.5 and S ji < 0.5. Hence it is easy to check that: S? ij > 0.5 > max{s> ij, S= ij, S> ji }. Let us now consider a partial weak order ( ˆP, Î, Ĵ) different from ( P S, ĨS, J S ). Hence the two preference structures differ for at least one pair {a k, a l }. From the definition of ( P S, ĨS, J S ), it is obvious that one (and only one) of these relations will hold for any pair of alternatives. Hence, for any pair 19

21 {a i, a j }, changing that relation will not lead to increased support. Any preference structure ( ˆP, Î, Ĵ) will not have increased support for the pairs where it differs from ( P S, ĨS, J S ). If { lex, min, sum }, we can check that: ( ˆP, Î, Ĵ) PWO, σ( P S, ĨS, J S ) σ( ˆP, Î, Ĵ). Hence ( P S, ĨS, J S ) is a solution obtained in model PWO, which proves 1a). The same reasoning can be used to prove 1b) and 1c). 2. If ( P S, ĨS, J S ) LO then this implies that J S = and that ĨS is reduced to the identity pairs. Hence there does not exists any pair {a i, a j } such that S ij < 0.5 and S ji < 0.5 or such that S ij > 0.5 and S ji > 0.5. Hence there does not exist any irrelevant pair and so model LO rp is equivalent to model LO. We can deduce from point 1 that if ( P S, ĨS, J S ) LO, then this is the unique solution obtained by model PWO, which proves If a i PS a j, then this means that S ij 0.5 and S ji 0.5. Hence it is easy to check that: S ij >d 0.5 S ji >d. Let us now consider a linear order ( ˆP, Î, Ĵ) different from ( P S, ĨS, J S ). Hence there exists at least one pair {a k, a l } such that a k P S a l and a l ˆP ak. For any pair {a i, a j }, changing the relation from a i PS a j to a j ˆP ai will not lead to increased support. Any other linear order ( ˆP, Î, Ĵ) different from ( P S, ĨS, J S ) will not have increased support for the pairs where it differs from ( P S, ĨS, J S ). If { lex, min, sum }, we can check that: ( ˆP, Î, Ĵ) LO, σd ( P S, ĨS, J S ) σ d ( ˆP, Î, Ĵ). Hence ( P S, ĨS, J S ) is a solution obtained in model LO d, which proves 3. Let us now assume that (P S, I S, J S ) is a partial weak order, but not necessarily a linear order. According to Property 3, (P S, I S, J S ) is the unique solution obtained in model PWO min. The next property then shows that the strict preference P S will not be contradicted by the solutions obtained in models LO rp min and LO d min (resp. in the refined models LO rp lex and LO d lex ). The proof is based on the fact that P S I S J S is complete. Property 5 Let { lex, min }. Let us suppose that (P S, I S, J S ) PWO. 1. Let (P, I, J) LO rp (S). Then P S P. 2. Let (P, I, J) LO d (S). Then P S P. 20

22 Proof: 1. Let us suppose by contradiction that there exists a pair {a k, a l } such that a k P S a l, but not(a k P a l ). The completeness of P implies that a l P a k. Since a k P S a l, we furthermore know that S kl > 0.5 and S lk < 0.5. It is easy to check that consequently S lk > < 0.5. Hence: Consequently: σ {k,l} (P, I, J) < 0.5. σ (1) (P, I, J) < 0.5. Let us now consider any asymmetric and complete relation P which extends P S. We know that if a i P aj, then not(a j P ai ), which implies that not(a j P S a i ), which implies that a i I S aj or a i J S a j or a i P S a j. If a i I S a j, then S ij > 0.5 and S ji > 0.5 and so pair {a i, a j } is irrelevant. If a i J S a j, then S ij < 0.5 and S ij < 0.5 and so pair {a i, a j } is irrelevant. If a i P S a j, then S ij > 0.5 and S ji < 0.5 and so a i, a j A: σ {i,j} ( P, I, J) > 0.5. Consequently, when considering only relevant pairs, we know that: σ (1) ( P, I, J) > 0.5. This implies that σ rp ( P, I, J) σ rp (P, I, J), which is a contradiction since we supposed that (P, I, J) is obtained in model LO rp. 2. Let us suppose by contradiction that there exists a pair {a k, a l } such that a k P S a l, but not(a k P a l ). The completeness of P implies that a l P a k. Since a k P S a l, we furthermore know that S kl > 0.5 and S lk < 0.5. It is easy to check that consequently S lk >d < 0.5. Hence we have: Consequently: σ{k,l} d (P, I, J) < 0.5. (σ d ) (1) (P, I, J) < 0.5. Let us now consider any asymmetric and complete relation P which extends P S. We know that if a i P aj, then not(a j P ai ), which implies that not(a j P S a i ), which implies that either a i P S a j or a i I S aj or a i J S a j. If a i P s a j, then S ij > 0.5 and S ji < 0.5 and so it is easy to check that S >d ij > 0.5. If a i I s a j, then S ij > 0.5 and S ij > 0.5 and so it is easy to check that S ij >d > 0.5. We can thus conclude that a i, a j A, Consequently: σ d {i,j} ( P, I, J) > 0.5. (σ d ) (1) ( P, I, J) > 0.5. This implies that σ d ( P, I, J) σ d (P, I, J), which is a contradiction since we supposed that (P, I, J) is obtained in model LO d. 21

23 We can state a similar property, but working with ( P S, ĨS, J S ) instead of (P S, I S, J S ). Property 6 Let { lex, min }. Let us suppose that ( P S, ĨS, J S ) PWO. 1. There exists (P, I, J) LO rp (S). such that P S P. 2. There exists (P, I, J) LO d (S) such that P S P. Proof: 1. Let (P, I, J) LO rp (S). If P S P, then the proof is complete. Otherwise, there exists a pair {a k, a l } such that a k PS a l, but not(a k P a l ). The completeness of P implies that a l P a k. Since a k PS a l, we furthermore know that S kl 0.5 and S lk 0.5. It is easy to check that consequently S lk > 0.5. Hence: σ {k,l} (P, I, J) 0.5. Consequently: σ (1) (P, I, J) 0.5. Let us now consider any asymmetric and complete relation P which extends P S. We know that if a i P a j, then not(a j P a i ), which implies that not(a j PS a i ), which implies that a i Ĩ S a j or a i JS a j or a i PS a j. If a i Ĩ S a j, then S ij 0.5 and S ji 0.5 and so either pair {a i, a j } is irrelevant or S > ij = 0.5. If a i J S a j, then S ij < 0.5 and S ij < 0.5 and so pair {a i, a j } is irrelevant. If a i PS a j, then S ij 0.5 and S ji 0.5 which implies that S > ij 0.5 Consequently, we have a i, a j A such that {a i, a j } is a relevant pair: σ {i,j} (P, I, J) 0.5. Consequently, when considering only relevant pairs, we know that: σ (1) (P, I, J) 0.5. This implies that (P, I, J) (P, I, J). Since we assumed that (P, I, J) LO rp (S), we must also have that (P, I, J) LO rp (S). This completes the proof. 2. Let (P, I, J) LO d (S). If P S P, then the proof is complete. Otherwise, there exists a pair {a k, a l } such that a k PS a l, but not(a k P a l ). The completeness of P implies that a l P a k. Since a k PS a l, we furthermore know that S kl 0.5 and S lk 0.5. It is easy to check that consequently S lk >d 0.5. Hence: σ{k,l} d (P, I, J) 0.5. Consequently: (σ d ) (1) (P, I, J)

24 Let us now consider any asymmetric and complete relation P which extends P S. We know that if a i P a j, then not(a j P a i ) (because of the completeness of P ), which implies that not(a j PS a i ) (because P extends P S ), which implies that a i Ĩ S a j or a i JS a j or a i PS a j (because of the completeness of PS Ĩ s JS ). If a i Ĩ S a j, then S ij 0.5 and S ij 0.5 and so it is easy to check that S ij >d 0.5. If a i JS a j, then S ij < 0.5 and S ji < 0.5 and so it is easy to check that S ij >d > 0.5. If a i PS a j, then S ij 0.5 and S ji 0.5 and so it is easy to check that S ij >d 0.5. We can thus conclude that a i, a j A, Consequently, we know that: σ d {i,j} (P, I, J) 0.5. (σ d ) (1) (P, I, J) 0.5. This implies that σ d (P, I, J) σ d (P, I, J). Since we assumed that (P, I, J) LO d (S), we must also have that (P, I, J) LO d (S). This completes the proof. If (P S, I S, J S ) PWO, then such a pairwise consistency is not verified for models for models PWO min and LO min, respectively for models PWO lex and LO lex. Consider the following valued outranking relation with four alternatives a, b, c and d. and a b c d a b c d It is easy to see that (P S, I S, J S ) is a partial weak order, with P S = {(a, b), (a, c), (a, d), (c, d)} J S = {(b, c), (b, d)}. However, when restricting the set of solutions to weak orders or linear orders, then the linear order adbc (which contradicts cp S d) will be obtained with a smallest support of In fact, the linear orders abcd, acbd, acdb, adcb and abdc all have a smallest support of

25 6.2 Monotonicity Let (P, I, J) be a solution obtained in a particular model when exploiting a valued outranking relation S. We assume that a k P a l and define a new valued outranking relation S such that S kl > S kl and such that forall (i, j) (k, l), we have that S ij = S ij. We say that the model is increasing monotonic if there exists a solution (P, I, J ) obtained when exploiting the valued outranking relation S such that a k P I a l. is strongly increasing monotonic if every solution (P, I, J ) obtained when exploiting the valued outranking relation S is such that a k P I a l. In fact, increasing the valued outranking relation between a k and a l increases the support for both indifference and preference between a k and a l. Alternatively, let (P, I, J) be a solution obtained in a particular model when exploiting a valued outranking relation S. We assume that a k P a l and we define a new valued outranking relation S such that S lk < S lk and such that forall (i, j) (l, k), we have that S ij = S ij. We say that the model is decreasing monotonic if there exists a solution (P, I, J ) obtained when exploiting the valued outranking relation S such that a k P J a l. is strongly decreasing monotonic if every solution (P, I, J ) obtained when exploiting the valued outranking relation S is such that a k P J a l. In fact, decreasing the valued outranking relation between a l and a k increases the support for both incomparability and preference between a k and a l. We say that a model is (strongly) monotonic if it is both (strongly) increasing monotonic and (strongly) decreasing monotonic. A model which is strongly monotonic is also monotonic. The difference between strong monotonicity and monotonicity is that in the first case, changing the preference between a i and a j must discard all the solutions which "contradict" this change, whereas in the second case, changing the preference between a i and a j may discard some, but not necessarily all the solutions which "contradict" this change. 24

26 Property 7 1. Models PWO lex, WO lex, LO lex, PWO sum, WO sum, LO sum, LO d lex and LO d sum are strongly monotonic. 2. Models PWO min, WO min, LO min and LO d min are monotonic. 3. Models LO rp lex 4. Model LO rp min and LO rp sum is monotonic. are monotonic. Proof: We denote σ(p, I, J) the supports of the preference structure (P, I, J) under the valued outranking relation S and σ (P, I, J) the supports under the valued outranking relation S. 1. We assume that { lex, sum }. We denote the asymmetric part of. Let (P, I, J) PWO (S) and let us suppose that a k P a l. If S kl < S kl (in the case of increasing monotonicity) or if S lk > S lk (in the case of decreasing monotonicity), we always have that: σ {k,l} (P, I, J) > σ {k,l}(p, I, J). Since the valued outranking relation does not change for the other pairs, we have for every pair {a i, a j } different from the pair {a k, a l } that: σ {i,j} (P, I, J) = σ {i,j}(p, I, J). If { lex, sum }, we can check that: σ (P, I, J) σ(p, I, J). (*) We now analyze the case of increasing and decreasing monotonicity separately. Increasing monotonicity Let us now consider a preference structure (P, I, J ) with a l P J a k. Since S kl < S kl, we can check that: σ {k,l} (P, I, J ) < σ {k,l} (P, I, J ). Since S ij = S ij for every pair {a i, a j } different form pair {a k, a l }, we have that: σ {i,j} (P, I, J ) = σ {i,j} (P, I, J ). If we use { lex, sum }, we can conclude that: σ(p, I, J ) σ (P, I, J ). 25