A general class of simple majority decision rules based on linguistic opinions

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1 A general class of simple majority decision rules based on linguistic opinions José Luis García-Lapresta Dept. de Economía Aplicada (Matemáticas), Universidad de Valladolid Avda. Valle de Esgueva 6, Valladolid, Spain Abstract In this paper we have introduced a class of decision rules related to simple majority, by considering individual intensities of preference. These intensities will be shown by means of linguistic labels. In order to compare the amount of opinion obtained by each alternative, we have considered the total ordered monoid generated by the sums of the original labels, according to an addition and an ordering. In this general framework different sets of linguistic labels can be employed and these sets can be represented by means of diverse mathematical objects. Moreover, on these mathematical representations of linguistic labels several orderings can be considered. Thus, flexibility is an important feature of this new class of group decision making procedures. Some examples of putting in practice the simple majority decision rules based on linguistic labels are provided, and the main properties of these voting systems are analyzed. It is worth emphasizing that these properties are satisfied for any total ordered monoid, regardless of the mathematical representation of linguistic labels or the ordering used to compare collective opinions. Keywords: Simple majority, Preferences, Linguistic labels, Ordered monoids, Fuzzy numbers. 1 Introduction Simple majority is one of the most usual decision rules in practice. According to this voting procedure, x defeats y when the number of voters who prefer x to y is greater than the number of voters who prefer y to x. The 1998 Nobel Prize Laureate A.K. Sen asserts in Sen [18, p. 162]:... the method of majority decision takes no account of intensities of preference, and it is certainly arguable that what matters is not merely the number who prefer x to y and the number who prefer y to x, but also by how much each prefers one alternative to the other. This idea had already been considered in the 18th Century by the Spanish mathematician J.I. Morales, who in Morales [16] states: opinion is not something that can be quantified but rather something which has to be weighed (see English translation in McLean Urken [15, p. 204]), or:... majority opinion... is something which is independent of any fixed number of votes or, which is the same, it has a varying relationship with this figure (see English translation in McLean Urken [15, p. 214]). On the other hand, Sen [18, p. 163] states: intensities of preference and relative measures of well-being are difficult to handle in an interpersonal context, and while our value judgments may make use of these concepts, they are not easy to put together and operate on. In this paper we consider the possibility of handling intensities of preference in the interpersonal framework of simple majority. In order to do this, we have considered that individuals tend to 1

2 express their preferences in a linguistic manner rather than in precise numerical values (see Zadeh [20], Herrera Herrera-Viedma Verdegay [9] and Bordogna Fedrizzi Pasi [1], among others). We generalize the classical simple majority voting rule by considering linguistic labels, in order to allow the agents to show intensities of preference in the pair-wise comparisons of alternatives. In this framework, labels need to be added up, and their sums need to be ordered. For this reason we have considered the totally ordered commutative monoid generated by the original linguistic labels according to an additive operation and an ordering. Linguistic simple majority decision rules have been designed similarly to the classical simple majority: an alternative defeats another when the sum of linguistic labels showing preference for the first alternative over the second is greater than the sum of linguistic labels showing preference for the second alternative over the first. For a fuzzy logic approach to fuzzy majorities based on calculus of linguistically quantified propositions, see Kacprzyk [11] and Chiclana Herrera Herrera-Viedma Poyatos [2], among others. May [14] characterizes the classical simple majority decision rule through four independent axioms: decisiveness, egalitarianism, neutrality and positive responsiveness. In this paper we generalize May s properties in the framework of linguistic preferences, and we have proven the new properties for the class of simple majority decision rules based on linguistic labels. The paper is organized as follows. In Section 2 we formalize the classical simple majority voting rule appropriately, in order to define the class of simple majority decision rules based on linguistic labels. In Section 3 we introduce linguistic labels, the ordered monoid generated by them, and a semantics with three mathematical representations through real numbers, intervals and triangular fuzzy numbers. In Section 4 the class of linguistic simple majority decision rules is defined, three examples are provided and some properties are established. 2 Classical simple majority Suppose m agents, with m 3, who have to show their preferences between the alternatives x and y. According to May [14] and Fishburn [6], we can use an index D for distinguish among the three possible cases of preference and indifference between x and y: 1, if x is preferred to y, D = 0, if x is indifferent to y, 1, if y is preferred to x. Now we take d = D + 1, and then 2 1, if x is preferred to y, d = 0.5, if x is indifferent to y, 0, if y is preferred to x. According to this interpretation, the opinion of the voter k between x and y is denoted by d k {0, 0.5, 1}. A profile of crisp preferences is a vector (d 1,..., d m ) {0, 0.5, 1} m that contains the opinion of the m voters between x and y. 2

3 Definition 1. A decision rule is a mapping where for each profile (d 1,..., d m ) {0, 0.5, 1} m F : {0, 0.5, 1} m {0, 0.5, 1}, F (d 1,..., d m ) = 1 means that x defeats y F (d 1,..., d m ) = 0.5 means that x and y tie F (d 1,..., d m ) = 0 means that y defeats x. Now we present several equivalent expressions of the classical simple majority decision rule. In this section we consider the classical negation on [0, 1], N(d) = 1 d. Given a profile (d 1,..., d m ) {0, 0.5, 1} m, in the simple majority decision rule 1. x defeats y card {k d k = 1} > card {k d k = 0} card {k d k > 0.5} > card {k d k < 0.5} d k > d k >0.5 d k < x and y tie card {k d k = 1} = card {k d k = 0} card {k d k > 0.5} = card {k d k < 0.5} d k = d k >0.5 d k >0.5 d k < y defeats x card {k d k = 1} < card {k d k = 0} card {k d k > 0.5} < card {k d k < 0.5} d k < d k <0.5 N(d k ). N(d k ). N(d k ). Taking into account the above expressions, now we present the particular description of the classical simple majority which will be the starting point for designing the simple majority decision rules based on linguistic labels. Definition 2. The simple majority decision rule, F S : {0, 0.5, 1} m {0, 0.5, 1}, is defined by F S (d 1,..., d m ) = 1, if d k > N(d k ), 0.5, if 0, if d k >0.5 d k >0.5 d k <0.5 d k = N(d k ), d k < d k <0.5 d k >0.5 d k <0.5 N(d k ), for every profile (d 1,..., d m ) {0, 0.5, 1} m of crisp opinions between x and y. 3

4 3 Linguistic labels Linguistic labels are introduced to extend simple majority decision rule to cases where voters show intensities of preference linguistically. Let L = {l 0, l 1,..., l s } be a set of linguistic labels, with s 2, ranked by a linear order: l 0 < l 1 < < l s. We consider that individuals use these labels for declaring their preferences between x and y. Suppose that there is an intermediate label representing indifference, and the rest of labels are defined around it symmetrically. The number of labels, s + 1, will be odd and, consequently, l s/2 is the central label. Now let N : L L be the negation operator defined by N(l i ) = l s i for all i {0, 1,..., s}. Obviously, N satisfies N(l s/2 ) = l s/2, N(N(l i )) = l i and l i < l j N(l j ) < N(l i ), for all i, j {0, 1,..., s}. In what follows, we will consider the following scheme for preferences of voter k: d k = l s, if x is definitely preferred to y l s/2 < d k < l s, if x is somewhat preferred to y d k = l s/2, if x is indifferent to y l 0 < d k < l s/2, if y is somewhat preferred to x d k = l 0, if y is definitely preferred to x. Linguistic labels can be represented mainly by real numbers, intervals and triangular and trapezoidal fuzzy numbers. Concrete representations of linguistic labels by fuzzy numbers can be found in Zadeh [20], Marimin Umano Hatono Tamura [13], Herrera Herrera-Viedma [8] and García-Lapresta Lazzari Martínez-Panero [7], among others. On the other hand, linguistic labels can be managed symbolically by means of the linguistic OWA operators introduced in Herrera Herrera-Viedma Verdegay [10]. We note that the conventional case of crisp preferences can be included in the previous model: L = {l 0, l 1, l 2 }, where l 0 = 0, l 1 = 0.5 and l 2 = 1. A profile of linguistic preferences is a vector (d 1,..., d m ) L m that contains the voters linguistic opinions between x and y. 3.1 The ordered monoid generated by L To define the simple majority decision rules based on linguistic labels, we consider the commutative monoid ( L, +) generated by L by means of all possible sums of labels of L with an associative and commutative operation + on L, where l 0 is the neutral element: 1. L L 2. l + l L, for all l, l L 4

5 3. l + (l + l ) = (l + l ) + l, for all l, l, l L 4. l + l = l + l, for all l, l L 5. l + l 0 = l, for all l L. We also consider a total order on L compatible with the original order on L: 6. l l, for all l L 7. (l l and l l) l = l, for all l, l L 8. (l l and l l ) l l, for all l, l, l L 9. l l or l l, for all l, l L 10. l 0 < l 1 < < l s, where < is the strict order associated with (l < l if l l and l l, for all l, l L ). Moreover, we suppose 11. l l l + l l + l, for all l, l, l L. Then, ( L, +, ) is a totally ordered commutative monoid. We note that for each l L, there exist integers λ 0, λ 1,..., λ s 0, not necessarily unique, such s that l = λ k l k, where λ k l k is the addition of λ k times the label l k and 0l k = l 0. k=0 3.2 A semantics Let L = {l 0, l 1, l 2, l 3, l 4, l 5, l 6 } be the set of 7 linguistic labels defined in Table 1. Table 1: A semantics with 7 terms. Label l 0 l 1 l 2 l 3 l 4 l 5 l 6 Meaning y is definitely preferred to x y is highly preferred to x y is slightly preferred to x x and y are indifferent x is slightly preferred to y x is highly preferred to y x is definitely preferred to y 5

6 L can be represented in different mathematical ways. Delgado Verdegay Vila [4] propose using only trapezoidal fuzzy numbers for representing linguistic labels. The main argument for avoiding sophisticated fuzzy sets in the representation of linguistic variables is that linguistic assessments are just approximate assessments. Moreover, trapezoidal fuzzy numbers include, as particular cases, three models very used in the literature for representing linguistic labels: real numbers, intervals and triangular fuzzy numbers. Given 4 real numbers a, b, c, d such that a b c d, the trapezoidal fuzzy number l = (a, b, c, d) is defined by its membership function µ l : IR [0, 1], where µ l (x) = 0, if x < a or x > d, x a, if a < x < b, b a 1, if b < x < c, and d x d c, if c < x < d 1. µ l (a) = µ l (d) = 0 and µ l (b) = µ l (c) = 1, if a < b < c < d, 2. µ l (a) = µ l (b) = µ l (c) = 1 and µ l (d) = 0, if a = b = c < d, 3. µ l (a) = µ l (b) = µ l (c) = µ l (d) = 1, if a = b < c = d, 4. µ l (a) = 0 and µ l (b) = µ l (c) = µ l (d) = 1, if a < b = c = d, 5. µ l (a) = µ l (b) = µ l (c) = µ l (d) = 1, if a = b = c = d. Such as we have mentioned above, real numbers, intervals and triangular fuzzy numbers can be represented through trapezoidal fuzzy numbers: (a, a, a, a) is the real number a; the interval [a, b] is (a, a, b, b); and (a, b, b, c) is the triangular fuzzy number (a, b, c). Table 2: Representation of linguistic labels. Label Real number Interval Triangular fuzzy number l 0 0 [0, 0] (0, 0, 0) l [0, 0.2] (0, 0.2, 0.4) l [0.2, 0.4] (0.1, 0.3, 0.5) l [0.4, 0.6] (0.4, 0.5, 0.6) l [0.6, 0.8] (0.5, 0.7, 0.9) l [0.8, 1] (0.6, 0.8, 1) l 6 1 [1, 1] (1, 1, 1) 6

7 In Table 2 we consider three concrete representations of the linguistic labels by means of real numbers, intervals and triangular fuzzy numbers. The examples of the next section are based on this table. 4 Simple majorities based on linguistic labels Definition 3. A decision rule based on L is a mapping where for each profile (d 1,..., d m ) L m F : L m {0, 0.5, 1}, F (d 1,..., d m ) = 1 means that x defeats y F (d 1,..., d m ) = 0.5 means that x and y tie F (d 1,..., d m ) = 0 means that y defeats x. According to Definition 2, we now extend the simple majority decision rule naturally to the framework of gradual preferences based on linguistic labels. In this generalization the role of the addition and the order on L is meaningful. Definition 4. The simple majority decision rule based on L, F LS : L m {0, 0.5, 1}, is defined by F LS (d 1,..., d m ) = 1, if d k > N(d k ), d k >l s/2 d k <l s/2 0.5, if d k = N(d k ), d k >l s/2 d k <l s/2 0, if d k < N(d k ), d k >l s/2 d k <l s/2 for every profile (d 1,..., d m ) L m of linguistic opinions between x and y. By convention, given (d 1,..., d m ) L m 1. If d k l s/2 for all k {1,..., m}, then 2. If d k l s/2 for all k {1,..., m}, then d k = l 0. d k >l s/2 d k = l 0. d k <l s/2 7

8 4.1 Example 1 Consider the set of linguistic labels and the semantics given in 3.2 (see Tables 1 and 2). Suppose 7 voters whose preferences of x over y are contained in the profile (d 1, d 2, d 3, d 4, d 5, d 6, d 7 ) = (l 4, l 0, l 1, l 4, l 4, l 0, l 4 ). To obtain the outcome, it is necessary to compare d k >l 3 d k = 4l 4 and d k <l 3 N(d k ) = 2l 6 + l 5. Addition of real numbers, of intervals and of triangular fuzzy numbers is conventional. Analogously, the order on the real line is conventional, as well. However, several class of orderings on the sets of intervals and triangular fuzzy numbers can be used. Suppose the following ordering of intervals, based on Delgado Vila Voxman [5] with an attitude against uncertainty: a + b > a + b [a, b] > [a, b ] or a + b = a + b and b a < b a. The following triangular fuzzy number ordering is based on Yao Wu [19] and used in García- Lapresta Lazzari Martínez-Panero [7]. Given two triangular fuzzy numbers l = (a, b, c) and l = (a, b, c ): l > l a + 2b + c > a + 2b + c or a + 2b + c = a + 2b + c and c > c or a + 2b + c = a + 2b + c, c = c and a > a. Consequently, we have introduced three totally ordered commutative monoids based on the same set of linguistic labels, by considering three different representations of the linguistic labels and their additions and orderings. Now we show that the vote result depends on these monoids. 1. Real numbers: x and y tie, since d k = 2.8 = N(d k ). d k >l 3 d k <l 3 2. Intervals: y defeats x, since d k = [2.4, 3.2] < [2.8, 3] = N(d k ). d k >l 3 d k <l 3 3. Triangular fuzzy numbers: x defeats y, since d k = (2, 2.8, 3.6) > (2.6, 2.8, 3) = N(d k ). d k >l 3 d k <l 3 Then, the linguistic label representation has been relevant in the vote result. These differences depend not only on the mathematical objects used in the representation, but on the concrete 8

9 assignments. For instance, if the linguistic labels of L were represented by the real numbers l 0 = 0, l 1 = 0.15, l 2 = 0.35, l 3 = 0.5, l 4 = 0.65, l 5 = 0.85 and l 6 = 1, then y would defeats x: d k = 2.6 < 2.85 = N(d k ). d k >l 3 d k <l Example 2 Now suppose that seven friends are planning to go to dinner. Three of them have a small inclination towards eating shellfish to pizza, one prefers highly eating shellfish to pizza and the other three prefer without a shadow of doubt eating pizza to shellfish. If these friends use the classical simple majority, they will go to a shellfish restaurant, because four of them prefer eating shellfish to pizza and only three prefer eating pizza to shellfish. But, if intensities of preference are to be taken into account, we can ask the question: do they go to eat pizza? Taking into account the preferences of these friends of shellfish over pizza, contained in the profile to obtain the result it is necessary to compare (d 1, d 2, d 3, d 4, d 5, d 6, d 7 ) = (l 4, l 4, l 4, l 5, l 0, l 0, l 0 ), d k >l 3 d k = 3l 4 + l 5 and d k <l 3 N(d k ) = 3l 6. According to 3.2 (see Tables 1 and 2) and the orderings considered in the previous example, we show that these friends should go to eat pizza. 1. Real numbers: 2. Intervals: d k = 2.9 < 3 = N(d k ). d k >l 3 d k <l 3 d k = [2.6, 3.4] < [3, 3] = N(d k ). d k >l 3 d k <l 3 3. Triangular fuzzy numbers: d k = (2.1, 2.9, 3.7) < (3, 3, 3) = N(d k ). d k >l 3 d k <l 3 Consequently, in this example the information provided by the individual intensities of preference has produced a different decision (shellfish) to that where only a truncate information about individual preferences is considered (pizza). 4.3 Example 3 We can extend linguistic simple majorities to more than two alternatives. Again m agents, m 3, have to show their preferences but now over alternatives in X = {x 1,..., x n }, where n > 2. With r k ij we denote the linguistic preference between x i and x j for the agent k. So r k ij L, rk ii = l s/2 and r k ji = N(rk ij ) for all x i, x j X. Linguistic preferences of agent k over alternatives are included in 9

10 the matrix M k = r k 11 r k 12 r k 1n r k 21 r k 22 r k 2n r k n1 r k n2 r k nn Suppose 5 agents whose preferences over x 1, x 2, x 3 and x 4 are given by the matrices M 1 = l 3 l 1 l 1 l 4 l 5 l 3 l 4 l 6 l 5 l 2 l 3 l 5 l 2 l 0 l 1 l 3 M 4 =, M 2 = l 3 l 4 l 1 l 1 l 2 l 3 l 0 l 0 l 5 l 6 l 3 l 2 l 5 l 6 l 4 l 3 l 3 l 1 l 0 l 2 l 5 l 3 l 1 l 5 l 6 l 5 l 3 l 6 l 4 l 1 l 0 l 3, M 5 =., M 3 = l 3 l 4 l 0 l 2 l 2 l 3 l 0 l 2 l 6 l 6 l 3 l 6 l 4 l 4 l 0 l 3 l 3 l 6 l 5 l 5 l 0 l 3 l 0 l 0 l 1 l 6 l 3 l 2 l 1 l 6 l 4 l 3 We note that for all the three representation of linguistic labels considered in Table 2 we have:., x 3 defeats x 1 : r k 31 = 2l 5 + 2l 6 > l 5 = N(r k 31). x 3 defeats x 2 : r k 31 >l 3 r k 32 = l 5 + 3l 6 > l 4 > r k 31 <l 3 N(r k 32). x 3 defeats x 4 : r32 k >l 3 r32 k <l 3 r34 k = l 5 + 2l 6 > 2l 4 = N(r k 34). x 4 defeats x 1 : r34 k >l 3 r34 k <l 3 r41 k = 2l 4 + l 5 > l 4 + l 5 = N(r k 41). x 4 defeats x 2 : r41 k >l 3 r41 k <l 3 r42 k = l 4 + 2l 6 > l 5 + l 6 = N(r42). k x 1 defeats x 2 : r42 k >l 3 r12 k = 2l 4 + l 6 > 2l 5 = r12 k >l 3 r42 k <l 3 N(r12). k r k 12 <l 3 Then, the linguistic simple majority decision rule ranks the alternatives in the following order x 3, x 4, x 1, x 2, and x 3 is the Condorcet winner (defeats all the other alternatives). It is well-known that classical simple majority can produce cycles when more than two alternatives are compared by the agents (voting paradox). Analogously, linguistic simple majorities could generate inconsistencies. If inconsistencies appear, the Copeland method [3] can be used in order to obtain collective coherent decisions (for more details, see Levin Nalebuff [12]). Another possibility consists in use other procedures such as the AHP (see for instance Saaty [17]) or a fuzzy Borda count (see García- Lapresta Lazzari Martínez-Panero [7]). 10

11 4.4 Properties Now we introduce anonymity, neutrality and positive responsiveness in the framework of decision rules over a set of linguistic labels. These notions are natural generalizations of those given by May [14] for ordinary preferences. Anonymity means that the collective decision depends on only the set of individual intensities of preference, but not on which individuals have these preferences. Neutrality means that if all the voters reverse their preferences, then the collective decision is also reversed. Positive responsiveness means that if two alternatives are tied and, in a second voting, one individual changes their vote favorably to one of the alternatives, then this alternative will be the winner. Definition 5. Let F : L m {0, 0.5, 1} be a decision rule based on L. 1. F is anonymous if and only if for all bijection σ : {1,..., m} {1,..., m} and all profile (d 1,..., d m ) L m : F (d σ(1),..., d σ(m) ) = F (d 1,..., d m ). 2. F is neutral if and only if for all profile (d 1,..., d m ) L m : F (N(d 1 ),..., N(d m )) = 1 F (d 1,..., d m ). 3. F is positive responsive if and only if for all pair of profiles (d 1,..., d m ), (d 1,..., d m) L m such that d j > d j for some j {1,..., m}, and d i = d i for every i {1,..., m} \ {j}: F (d 1,..., d m ) 0.5 F (d 1,..., d m) = 1. Now we justify that every simple majority decision rule based on whatever set of linguistic labels satisfies the above properties, independently of the mathematical representation of the linguistic labels and the addition and the ordering considered in the nonoid. Proposition. responsive. The simple majority decision rule based on L is anonymous, neutral and positive Proof: Since addition of labels is associative and commutative, and the result of the decision rule only depends on the number of individuals who prefer an alternative to another for each linguistic label, F LS is anonymous. In order to prove neutrality, let (d 1,..., d m ) L m be a profile of linguistic preferences. We have: F (N(d 1,..., d m )) = 1 d k <l s/2 N(d k ) > N(d k )>l s/2 N(d k ) > d k >l s/2 d k F (d 1,..., d m ) = 0. N(d k )<l s/2 N(N(d k )) Analogously, we obtain: F (N(d 1,..., d m )) = 0.5 F (d 1,..., d m ) =

12 and F (N(d 1,..., d m )) = 0 F (d 1,..., d m ) = 1. Consequently, F (N(d 1,..., d m )) = 1 F (d 1,..., d m ). In order to prove that F LS is positive responsive, let (d 1,..., d m ), (d 1,..., d m) L m be two profiles such that d i = d i whenever i j and d j > d j for some j {1,..., m}. Suppose F LS (d 1,..., d m ) 0.5, i.e., d k N(d k ). Three cases are possible: d k >l s/2 d k <l s/2 1. d j > d j l s/2 : 2. d j > l s/2 > d j : 3. l s/2 d j > d j: d k > d k N(d k ) = N(d k). d k >l s/2 d k >l s/2 d k <l s/2 d k <l s/2 d k > d k N(d k ) > N(d k). d k >l s/2 d k >l s/2 d k <l s/2 d k <l s/2 d k = d k N(d k ) > N(d k). d k >l s/2 d k >l s/2 d k <l s/2 d k <l s/2 Then, in all the cases we have F LS (d 1,..., d m) = 1. Concluding remarks Classical simple majority only considers the number of individuals who prefer an alternative to another. So, voters can only show if an alternative is preferred to another, but they can t declare intensities of preference. All kinds of preference modalities are identified and voters opinions are misrepresented. Consequently, the outcome might not be reflected properly in terms of the voters opinion. For this reason we have introduced a generalization of the classical simple majority, where voters can reveal their preferences among alternatives in a linguistic manner, such as it happens in real life. Analogously to the classical voting procedure, an alternative defeats another when the amount of opinion obtained is bigger. Now this amount is not measured by counting number of individuals, but adding up intensities of preference. Since these intensities are given by means of linguistic labels, we have considered the monoid generated by the original labels. In order to compare the total amount of opinion obtained by each alternative, we have taken into account a total order in the monoid. This new voting procedure is very flexible: different linguistic labels can be used; these labels can be represented by means of a wide class of mathematical objects; and several orderings can be employed. Depending on the choice of the labels, their semantics and orderings, the outcome of the voting might be different. This apparent drawback is the consequence of the flexibility. Each real voting needs an accurate election of these objects. In order to avoid possible collective inconsistencies (voting paradox) classical simple majority is usually used to compare between only two alternatives. Nevertheless, in 4.3 we have considered 12

13 the possibility of using linguistic simple majorities for more than two alternatives. If cycles appear, then several kinds of procedures can be used in order to obtain coherent collective decisions. On the other hand, voters could falsify their preferences by declaring insincere intensities of preference, in order to obtain an outcome close to their desires. But in the classical simple majority, the information that voters have to show is unrealistic: all intensities of preference are identified. The advantages are clear. Moreover, the properties used by May [14] to characterize the classical simple majority remain. Acknowledgements The financial support of the Junta de Castilla y León (Consejería de Educación y Cultura, Proyecto VA057/02), the Spanish Ministerio de Ciencia y Tecnología, Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (I+D+I) (Proyecto BEC ) and ERDF are gratefully acknowledged References [1] Bordogna, G. Fedrizzi, M. Pasi, G. (1997). A linguistic modelling of consensus in group decision making based on OWA operators. IEEE Transactions on Systems, Man, and Cybernetics, 27, pp [2] Chiclana, F. Herrera, F. Herrera-Viedma, E. Poyatos, M.C. (1996). A classification method of alternatives for multiple preference ordering criteria based on fuzzy majority. The Journal of Fuzzy Mathematics, 4, pp [3] Copeland, A.H. (1951). A reasonable social welfare function. Mimeo. University of Michigan Seminar on Applications of Mathematics to the Social Sciences. [4] Delgado, M. Verdegay, J.L. Vila, M.A. (1988). Ranking fuzzy numbers using fuzzy relations. Fuzzy Sets and Systems, 26, pp [5] Delgado, M. Vila, M.A. Voxman, W. (1998). On a canonical representation of fuzzy numbers. Fuzzy Sets and Systems, 93, pp [6] Fishburn, P.C. (1973). The Theory of Social Choice. Princeton University Press, Princeton. [7] García-Lapresta, J.L. Lazzari, L.L. Martínez-Panero, M. (2001). A group decision making method using fuzzy triangular numbers. In: Zopounidis, C., Pardalos, P.M. and Baourakis, G. (eds.), Fuzzy Sets in Management, Economics and Marketing. World Scientific, Singapore, pp [8] Herrera, F. Herrera-Viedma, E. (2000). Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets and Systems, 115, pp [9] Herrera, F. Herrera-Viedma, E. Verdegay, J.L. (1995). A sequential selection process in group decision making with linguistic assessment. Information Sciences, 85, pp

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