Borda count versus approval voting: A fuzzy approach

Transcription

1 Public Choice 112: , Kluwer Academic Publishers. Printed in the Netherlands. 167 Borda count versus approval voting: A fuzzy approach JOSÉ LUIS GARCÍA-LAPRESTA & MIGUEL MARTÍNEZ-PANERO Departamento de Economía Aplicada, Universidad de Valladolid, Valladolid, Spain; lapresta@cpd.uva.es Accepted 6 March 2001 Abstract. In this paper we consider a fuzzy variant of the Borda count taking into account agents intensities of preference. This fuzzy Borda count is obtained by means of score gradation and normalization processes from its original pattern. The advantages of the Borda count hold, and are even improved, providing an appropriate scheme in collective decision making. In addition, both classic and fuzzy Borda counts are related to approval voting, establishing a unified framework from distinct points of view. 1. Introduction Among the great variety of methods in collective decision making, in this paper we shall focus our attention on a variant of the Borda count, the vote-counting scheme introduced in 1770 by Jean Charles de Borda. 1 This engineer and navy officer denounced in a Memory read in the French Academy of Sciences that the usual collective decision procedures (such as plurality rule) only considered the most preferred alternative for each agent, ignoring the rest. Taking into account this partial information, the final output could not faithfully reflect the agents preferences. Borda showed examples with this fault, and then advocated the following method: each agent ranks all the alternatives, and gives integer marks to each of them: the highest score, which coincides with the number of alternatives, to the most preferred; one point less to the next alternative; and so on, in a descent manner, till the least preferred is reached, which is given only one point. This Borda count was chosen by the French Academy of Sciences to select its members and, from that time on, it has been both criticized and praised. Some advantages and drawbacks of the method can be shown according to procedural and epistemic arguments. This paper has been partially supported by the Consejería de Educación y Cultura de la Junta de Castilla y León (project VA09/98). The authors want to acknowledge the valuable comments of an anonymous referee.

2 168 Procedural arguments. The Spanish enlightened mathematician Morales (1797) already considered the Borda count as an straightforward and representative procedure, taking into account the information from the entire preference rankings of all the voters. 2 However, it was said that the Borda count did not respect the agents freedom of scoring and did not reflect the candidates merit with accuracy. This drawback was partially refuted in Morales (1805: 18 30), showing that, if the scale of values 0,1,2..., is used instead of 1,2,3..., to score the alternatives from the least to the most preferred, then the total score of a fixed alternative is the number of alternatives evaluated worse than it. This fact had already been pointed out by Borda and commented by Condorcet to ensure that the range of values used by the Borda count was not arbitrary (see McLean and Urken (1995: 81 89)). On the other hand, Condorcet censured the manipulability of this method 3 and was aware of the following fact: an alternative which defeats all others by simple majority in pairwise comparisons 4 might not be selected if the Borda count is used (see Black (1958: 156)). More recently, the non-fulfillment of the independence of irrelevant alternatives principle considered by Arrow (1963) has been added to the above-mentioned drawbacks. 5 Epistemic arguments. Dummett (1998: 290) has argued that the Borda count is the best tool for reaching the decision most likely to be correct when the object is to reconcile different judgements about effective means to a common aim, and the most equitable method of determining a resultant of divergent desires. 6 However, following the maximum likelihood estimation approach of Condorcet (1785), Young (1988) has shown that the voters competence level influences who the most probable winner in pairwise comparisons is. Assuming that the best candidate exists and that each voter will choose it with probability p, the candidate with the highest probability of being the best is the Borda winner only when p is sufficiently near 0.5. From a theoretical point of view, several characterizations of the Borda count have been obtained by Young (1974), Hansson and Sahlquist (1976), Nitzan and Rubinstein (1981), and Debord (1992), among others. Although Nitzan and Rubinstein (1981) consider non-transitive preferences over the alternatives, in this paper transitivity will be a coherence assumption needed to ensure reasonable Borda scores, as we shall show. The Borda rule has been re-examined many times, inspiring other collective decision making procedures. In 1882, Nanson (see McLean and Urken (1995: )) introduced a sort of Borda count with eliminations. Copeland (1951) considered Borda-type scores for each alternative, which were the

3 169 number of alternatives better than that marked minus the number of alternatives worse than it (see Saari and Merlin (1996)). Black (1958: 66) proposed a hybrid procedure consisting in the choice of the Condorcet winner, if it exists. Otherwise, Borda count should be used. Kemeny (1959) formuled a rule defined with arguments similar to those of the Borda rule (see Young, 1995: 61). Later on, Sen (1977) considered the so-called narrow and broad Borda count in connection with the independence of irrelevant alternatives principle. In addition, Dummett (1998) introduced the revised and adjusted Borda count to penalize the manipulability of the classic Borda count. In this paper the classic Borda count is considered and extended in a natural way to non-strict orderings, where the agents can declare indifference among distinct alternatives. This method will even be generalized by considering fuzzy preferences for the agents and not necessarily integer 7 scores over the alternatives. We note that Marchant (1996a, 1996b, 2000) has used this idea to design a fuzzy variant of the Borda count. However, this author deals with this procedure especially in connection with the PROMETHEE method in multi-criteria decision making. In the present paper, we have essentially come to a similar fuzzy Borda count, but in a collective decision making framework. On the other hand, our approach focuses on the underlying coherence hypotheses to ensure individual pairwise comparisons consistent with the marks given to the alternatives. In addition, the Borda count will be related to another collective decision making procedure with less tradition, but widely studied in recent years: approval voting. 8 This expression was first considered by Weber (1977) for a method whose fundamental impulse is due to Brams and Fishburn. With this procedure, agents can approve of as many alternatives as they wish; that is, as coined by the last mentioned authors: one man, n votes. Brams and Fishburn (1978, 1983) have also pointed out the main advantages of approval voting: more flexibility than other single vote methods, inhibition of abstention, legitimacy to the outcome, eminent practicability, etc. Other studies by the same authors show the difficulty to manipulate approval voting. There are some drawbacks of approval voting pointed out again by Brams and Fishburn (1983): loss of preference gradations allowed by several methods (such as Borda does); agents ambivalence; possibility of inequity among agents, depending on the number of approved alternatives, etc. On the other hand, Saari (1995: 181) has noted that approval voting shares undesirable properties as plurality voting (where agents choose only one alternative) and antiplurality voting (where agents choose necessarily all the alternatives except one of them). From a theoretical point of view, characterizations of approval voting have been given by Fishburn (1978), Sertel (1988) and Baigent and Xu (1991). And

4 170 different versions of approval voting have been proposed by Brams (1990), who consider a coinstrained approval voting by means of determining the number of approved alternatives from different categories of agents, and Ylmaz (1999) who considers an alternative voting method with three categories rather than two. On the other hand, Weber (1995) has revisited the subject of approval voting in connection with other procedures, the Borda count among others. As the title of this paper suggests, we shall also consider the Borda count in connection with approval voting, and a variant of the last method based on the gradual acceptance of the alternatives by the agents is also presented. The paper is organized as follows. In Section 2 we formalize the classic (discrete) Borda count. In Section 3 a fuzzy Borda count is defined. These Borda counts are connected with approval voting in Section 4. Finally, in Section 5, the results obtained are commented and a utilitarian view to the scores used is exposed. 2. The discrete Borda count At first, the Borda count was designed to score the alternatives sequentially, assuming the agents preferences to be strict orderings. This can be formalized and generalized in a natural way as follows. Let P 1,P 2,..., P m be the preference relations (binary asymmetric relations 9 ) of m agents over n alternatives x 1,x 2,...,x n. Each agent gives a mark to each alternative, according to the number of alternatives worse than it: r k (x i ) = #{x j x i P k x j } is the score given by agent k to the alternative x i. The same result is obtained by considering, for each agent k, its preference matrix: r k 11 rk rk 1n r k 21 rk rk 2n , r k n1 rk n2... rk nn where r k ij = { 1, if xi P k x j, 0, otherwise. In this way, the agent k gives the alternative x i the score:

5 171 r k (x i ) = n j = 1 x i P k x j r k ij We note that the possible score range is contained in the set of values {0, 1,...,,n 1}. Some of the upper values might not be reached if there is indifference (absence of preference) among distinct alternatives. However, if the orderings were strict, the above-mentioned set would exactly be ranged. With these individual counts a collective one is obtained: m r(x i ) = r k (x i ). k=1 So, the collective preference relation P B is defined by: x i P B x j r(x i )> r(x j ), which is negatively transitive. 10 Thus, the highest scored alternatives (maximals for P B ) will be chosen. Nevertheless, it is reasonable for the Borda count to require the fulfillment of the following property of monotonicity: when two alternatives are compared by an agent, the highest scored must be the preferred one. Definition 1. The count r k is monotonic if and only if for all pair of alternatives x i, x j X. x i P k x j r k (x i )>r k (x j ), Remark 1. It is easily seen that if the preference relation P k is transitive, then the count r k is monotonic. However, if P k is not transitive, then the count r k is not necessarily monotonic, as it is showed in the following example. Suppose a preference relation P k over 4 alternatives, explicitly given by: x 1 P k x 2, x 2 P k x 3, x 3 P k x 4, x 2 P k x 4.Thenr k (x 1 ) = 1 < r k (x 2 ) = 2, although, as pointed out, x 1 P k x A fuzzy Borda count In the classic Borda count the agents only consider which alternatives are preferred to the others. In this Section we present a natural extension of the classic Borda count by allowing the agents to show numerically how much some alternatives are preferred to the others, 11 evaluating their preference

6 172 intensities from 0 to 1. This is possible by considering fuzzy binary relations instead of ordinary binary relations. The following pre-requisites are introduced to formalize the fuzzy Borda counts. A fuzzy subset A of X is defined through its membership function, µ A : X [0, 1], whereµ A (x) is the membership grade of x to A. If µ A (x) = 1, then x belongs absolutely to A; if µ A (x) = 0, then x does not belong to A; and if 0 <µ A (x) <1, then x partly belongs to A (the closer µ A (x) is to 1, the more x belongs to A). Thus, the notion of fuzzy subset generalizes the concept of ordinary subset: a fuzzy subset A of X is ordinary if µ A (X) {0, 1}. According to the notion of fuzzy subset, a fuzzy binary relation on X is a fuzzy subset of X X. We shall consider a finite set of alternatives, X ={x 1,...,x n }, over each one of the m agents show their preferences by means of fuzzy binary relations R 1,...,R m.letµ R k : X X [0, 1] the membership function of R k, k = 1,...,m. The number r k ij = µ R k(x i, x j ) [0, 1] has been interpreted in the literature in two main ways (see García-Lapresta and Llamazares (2000: 675) for references). For some authors r k ij is understood as the degree of certainty or confidence with which agent k prefers (strictly or weakly) x i to x j. For other authors, r k ij denotes the intensity with which agent k prefers x i to x j. Our paper is based on this second viewpoint. Because of normalization reasons, for each agent, the total capacity to prefer between pairs of alternatives is considered to be unitary. It is also plausible to admit that however greater the intensity r ij with which x i is preferred to x j, the lower intensity r ji with which x j is preferred to x i will be. These considerations are contemplated in the following axiom of reciprocity, which will be assumed in our arguments. 12 Definition 2. A fuzzy binary relation R over X is reciprocal if and only if it is verified r ij + r ji = 1foralli, j {1,...,n}. Given the reciprocal fuzzy binary relation R k of the agent k, for each α [0, 1) an ordinary binary relation over X can be defined by x i k x j r k ij > α. However, k is an ordinary preference relation (i.e., asymmetric) only if α 0.5. In this case, the indifference relation associated with k is defined by x i k x j neither x i k x j nor x j k x i, i.e., r k ij α and r k ji α. By reciprocity, both conditions are equivalent to 1 α r k ij α. Then, for every pair of alternatives x i, x j, X one and only one of the following statements holds: x i k x j (r k ij >α),x i k x j (1 α r k ij α),x j k x i (r k ji >α, i.e., r k ij < 1 α). Among all possible thresholds α [0.5, 1) which ensure k to be an ordinary preference relation, in what follows we shall consider only the smaller threshold, α = 0.5. This election is supported by two considerations:

7 173 simplicity and the fact that, by reciprocity of R k, r k ij > r k ji is equivalent to r k ij > 0.5. Thus, choosing α = 0.5 todefine k, this ordinary binary relation show those alternatives somehow preferred by the agent k. Definition 3. For each agent k {1,...,m}, the reciprocal fuzzy binary relation R k induces an ordinary preference relation over X defined by x i k x j r k ij > 0.5, for all pair of alternatives x i, x j X. We note that for every pair of alternatives x i, x j X, one and only one of the following statements holds: x i k x j (r k ij > 0.5), x i k x j (r k ij = 0.5), x j k x i (r k ij < 0.5). Given R k, the reciprocal fuzzy preference relation of the agent k over X, we can consider the matrix of preference intensities: r k 11 rk rk 1n r k 21 rk rk 2n , r k n1 rk n2... rk nn where the symmetric entries respect the main diagonal add up to 1. For each agent k {1,...,m}, we shall define a new count to evaluate each alternative according to its preference intensities. In the discrete case, the value assigned by the individual count is the number of alternatives considered worse than it for an ordinary preference relation P k.now, analogously, this role will be played by the ordinary preference relation k induced by the reciprocal fuzzy binary relation R k. So, the introduced individual count will add up the preference intensities among the alternative and those considered worse according to k. Taking into account Definition 3, agent k gives the alternative x i the value n n r k (x i ) = r k ij = r k ij, j = 1 j = 1 x i k x j r k ij > 0.5 which coincides with the sum of the entries greater than 0.5 in the row i in the above matrix. Again, with these individual counts a collective one is obtained: m r(x i ) = r k (x i ). k=1

8 174 Now, the collective preference relation P FB is defined by x i P FB x j r(x i )> r(x j ), which is negatively transitive. Thus, the highest scored alternatives (maximals for P FB ) will be chosen. As we mentioned above, it could be possible to consider other Borda counts by choosing α>0.5 in the definition of k : r k (x i ) = n j = 1 x i k x j r k ij = n j = 1 r k ij >α r k ij. This voting procedure would require a greater support on the alternatives by the agents than the Borda count associated to α = 0.5 (only intensities greater than α would be taken into account). We note that if α>0.5, r k ij > r k ji and x i k x j would be possible simultaneously. We have to point out that the discrete Borda count gives individual scores belonging to the set {0, 1,...,n 1}, while the fuzzy Borda count takes its values in the interval [0, n 1], all intermediate values being possible. In this fuzzy framework we also consider a monotoniciy property, as in the discrete case: when two alternatives are compared by an agent, the highest score must be given to the preferred alternative according to k. Definition 4. The count r k is monotonic if and only if x i k x j r k (x i )>r k (x j ), for all pair of alternatives x i, x j X. According to the discrete case, one can believe that the count r k is monotonic if the ordinary preference relation k is transitive. However, as we shall show, this assumption is not sufficient. All the hypotheses of fuzzy coherence related below ensure the transitivity of k, but only the so-called weak max-max transitivity in the agents individual preferences implies the monotonicity of the count r k. Definition 5. Let properties: be a binary operation in [0, 1] with the following Commutativity: a b = b aforalla, b [0, 1]. Non-decreasing in each component: (a a and b b ) a b a b, for all a, a, b, b [0, 1]. Super-idempotency: a a aforalla [0, 1].

9 175 A reciprocal fuzzy binary relation R k over X is weak 13 max- transitive if and only if it is verified: (x i k x j and x j k x l ) r k il rk ij rk jl, for all x i, x j, x 1 X. Among the operations used in the literature to modelize the agents rationality, the following ones, proposed by Zadeh (1971) and Bezdek and Harris (1978), among others, are considered: a 1 b = min{a, b}, a 2 b = a + b 2, a 3 b = max{a, b}. Definition 6. T i (X) denotes the set of reciprocal fuzzy binary relations verifying weak max- i transitivity, for i = 1, 2, 3. Remark 2. Taking into account that for all a, b [0, 1]] it is verified min{a, b} a + b 2 then max{a, b} and (a > 0.5, b > 0.5) min{a, b} > 0.5, R k T 3 (X) R k T 2 (X) R k T 1 (X) k transitive. In the following result we show that, under the coherence assumption related to T 3 (X), the agents will assign scores to the alternatives in a monotonic way. We note that when a fuzzy relation R k belongs to T 3 (X), drops in the levels of preference are not allowed if agent k compares linked alternatives x i, x j, x l, each one subject to being strictly preferred to the next (according to k ). More specifically, we mean that r k il, the intensity with which k prefers x i to x l, ought to be greater than or equal to both r k ij and r k jl, the preference intensities of k for x i over x j,andforx i over x l, respectively (if these last two values are supposed to be greater than 0.5). Proposition. If R k T 3 (X), then the count r k is monotonic. Proof. For each i {1,...,n} consider the set P(i) = {l r k il > 0.5} {1,...,n}. Thus r k (x i ) = r k il and r k (x j ) = r k jl. Suppose x i k x j, l P(i) l P(j) i.e., r k ij > 0.5. Then P(j) P(i) is verified: if l P(j), thenr k jl > 0.5 By hypothesis, r k ij > 0.5; consequently r k il max{r k ij, rk jl } > 0.5 andsol P(i). What is more, this inclusion is strict, because r k ij > 0.5 implies j P(i),

10 176 while j / P(j) since by reciprocity r k jj = 0.5. Now, if l P(j) P(i), then r k il max {rk ij, rk jl } rk jl and, consequently, r k(x i )>r k (x j ). Remark 3. If R k T 1 (X) or R k T 2 (X), thenr k is not necessarily monotonic, as is shown in the following example. Let R k be the reciprocal fuzzy binary relation over X ={x 1, x 2, x 3, x 4, x 5 } whose matrix is: It is easy to verify R k T 2 (X) and, taking into account Remark 3, also R k T 1 (X). Note that x 1 k x 2, because r k 12 = 0.52 > 0.5. Nevertheless, r k (x 1 ) = = 2.95 < = 3 = r k (x 2 ). A notable advantage of the fuzzy Borda count with respect to the classic Borda count is the agents possibility to show their preferences more faithfully. So, if an agent prefers x i to x j with a weak level of intensity, for instance 0.6, in the classic Borda count that agent must declare the extreme level of intensity 1, the same value as if the agent preferred x i to x j absolutely. No kind of intermediate intensity can be shown, only 0 or 1. However, in the fuzzy Borda count every agent can declare the real intensity of preference between each pair of alternatives. These aspects appear in the following example. Example. Suppose two agents whose preference intensities over a set of two alternatives X ={x 1, x 2 } are given by the matrices associated to R 1 and R , , respectively. We note that R 1, R 2 T 3 (X). If these agents had to show their preferences in a crisp manner in order to give classic Borda scores, they would only take into account those alternatives somehow preferred to the others, i.e., with a intensity of preference greater than 0.5 in pairwise comparisons. That is, they switch from their fuzzy preference relations R k to the ordinary associated preference relations k, k = 1, 2. Thus, each level of intensity greater than 0.5 becomes 1, and each value less than or equal to 0.5 becomes 0, and the matrices above would change to

11 , We note that with the classic Borda count we obtain r(x 1 ) = 2, r(x 2 ) = 3, r(x 3 ) = 1, and the second alternative would be the winner. However, with the matrices of preference intensities the fuzzy Borda count gives r(x 1 ) = 1.9, r(x 2 ) = 1.8, r(x 3 ) = 0.6, and the first alternative will be the winner. There is no paradox in the discordance: the second way takes into account more accurate information and the result may differ with respect to the binary data provided by the classic Borda count. As commented in Section 1, the use of integers to score the alternatives by means of the Borda count was already criticized in the XVIII th century. Since then, such marks have been partially justified in several ways, but it seems to us that the main adverse argument remains: the lack of freedom to score the alternatives according to the real agents preference intensities. Thus, with the fuzzy Borda method, the principal advance would be the gradation allowed the agents in their scores, and hence, a more faithful adherence to their true estimations. There is one more step in terms of representation: not only are all the alternatives considered in agents preferences (this already occurs with the Borda count with respect to plurality rule); now, while maintaining this equitable quality, the alternatives are taken into account in a more accurate way than in the classic Borda method. We note that the fuzzy Borda count shares the same drawbacks as the discrete one: non-fulfillment of the independence of irrelevant alternatives principle and manipulability. In this last aspect, the above arguments about gradation of the scores have a clear incidence in more extensive way than the one pointed out by Morales (1797: 42) (see McLean and Urken (1995: )). Thus, when agents hide their true preferences in order to favor one alternative against other, in the majority rules they do not vote for this alternative (in score terms, they switch from 1 to 0); in the classic Borda count they can reduce their individual scores (for example from m to m 1 or m 2, etc.) in a descent integer manner. However, in the fuzzy Borda count they are not constrained in their assessments (true or false) by the modality of the scores.

12 Approval voting: From binary to gradual assessments Up to now the agents scores given by means of Borda counts are a consequence of their pairwise preferences (crisp or fuzzy) over the alternatives. Another way in collective decision making is that suggested by the approval voting procedure. In this method, we consider that the agents assess at the alternatives one by one, using what we call approval counts, inacrispor fuzzy manner. In the first one, each alternative is approved or disapproved absolutely by each agent. In the second manner, the individual decision is more precise, estimating the agent s level of resolution. In both cases the individual agent s results are added up together in order to obtain the final decision. These fuzzy approval counts are formalized below. Again, it is considered a finite set of alternatives X ={x 1,...,x n } to be evaluated by each of the m agents. With the discrete approval count, each agent accepts each alternative or not, as many as wanted. This can be formalized by defining the binary index: { r k 1, if the alternative xi is approved by the agent k, i = 0, if the alternative x i is disapproved by the agent k. r 1 r r m 1 Thus, the matrix r 1 2 r r m is defined, and each alternative is given r 1 n r 2 n... r m n the sum of individual scores: r(x i ) = m r k i is the total score obtained by the k=1 alternative x i, which coincides with the sum of the entries of the i-th row. As in the Borda count, this scoring defines a collective preference relation x i P A x j r(x i )>r(x j ), which is negatively transitive. The highest scored alternatives (maximals for P A ) will be chosen. The fuzzy approval count is defined in a parallel manner. Each agent estimates the level of acceptance over each alternative as follows. Let r k i [0, 1] be the approval estimation given by the agent k to the alternative x i,and dispose these indices as entries in the scoring matrix r 1 1 r r m 1 r 1 2 r r m r 1 n r 2 n... r m n. Again, we can define, for each alternative x i, the sum of individual scores: r(x i ) = m is the total score obtained by x i, which coincides with the r k i k=1

13 179 sum of the values of the i-th row. Thus, a collective preference relation can be defined by x i P FA x j r(x i )>r(x j ), which is once more negatively transitive. The highest scored alternatives (maximals for P FA ) will be chosen. In order to compare fuzzy Borda count and fuzzy approval voting, we can associate each agent k with a fuzzy binary relation and a fuzzy subset, respectively: In the fuzzy Borda count, µ R k : X X [0, 1], the membership function of the fuzzy binary relation R k of agent k, assigns each pair of alternatives (x i, x j ) the index r k ij = µ R k(x i, x j ), the degree of preference with which the agent k prefers x i to x j. In the fuzzy approval voting, µ Ak : X [0, 1], the membership function of the fuzzy subset A k of approved alternatives in X for agent k, assigns each alternative x i the index r k i = µ Ak (x i ), the degree of acceptance of x i for the agent k. We have to point out the following fact about the ranges of the marks used. While the discrete approval count gives total scores belonging to the set {0, 1,...,m}, the fuzzy approval count does so in the interval [0, m], all the intermediate values being reachable. And then, the last mentioned decision making procedure also provides more flexibility to the agents. As a result, the existence of ties is less likely in the maximum total scores (which are maximals according to the collective preference relation defined in each case) if fuzzy counts are used instead of the discrete ones. Nevertheless, if it happens, procedures to break the ties must be established. The mentioned comment is also applicable to the fuzzy Borda count with respect to the discrete Borda count. 5. Concluding remarks The two voting procedures introduced in the paper, fuzzy Borda count and fuzzy approval voting, use more complex information than classic Borda count and approval voting. While in the classic procedures only 0 s and 1 s should be added as individual marks in pairwise comparisons or individual assessments, in the fuzzy procedures any number from 0 to 1 could appear. In the fuzzy approval voting each voter assigns a value from 0 to 1 to each alternative. Adding up the values that voters give to the alternatives, we obtain the winner(s). Consequently, no essential differences between putting in practice original and fuzzy approval voting appear. In the fuzzy Borda count each voter has to show the intensities of preference between all the pairs of alternatives by means of values from 0 to 1. In practice, every agent must first compare each pair of alternatives showing

14 180 Table 1. Possible marks for n alternatives, m agents Count Borda Approval Discrete Individual {0,...,n 1} Individual {0, 1} step = 1 step = 1 Total {0,...,m(n 1)} Total {0,...,m} step = 1 step = 1 Fuzzy Individual [0, n 1] Individual [0, 1] no step no step Total [0, m(n 1)] Total [0, m] no step no step preference for one over the other; otherwise, the agent is indifferent to both alternatives (in this case, the agent gives 0.5 points to each alternative). If the agent prefers one alternative to the other, then the agent must show the intensity of preference (a number greater than 0.5 and less than or equal to 1). When all the information provided by the agents is collected in the corresponding matrices, then the total score of an alternative is obtained adding up the values greater than 0.5 in the entries of the corresponding row of the agents matrices. The alternative(s) with the highest score will be the winner(s). All this process can be implemented with computer assistance. This is the case of MACBETH, a decision procedure developed in a similar framework to the fuzzy Borda count (see Bana e Costa and Vansnick (1997)). We note that MACBETH and its software consider linguistic labels instead of numbers in the assessments of pairwise comparisons. In Table 1 possible ranges of values appear, individual as well as total, of the considered counts. It is possible to unify the amplitude of the individual scores in Borda and approval counts by dividing these individual scores by n 1. So, the scale changes, but not the ordering, and then the scores range from 0 to 1 with step 1 n 1 in the discrete case, and continuously in the interval [0, 1] in the fuzzy one. When these normalized individual scores are added up, total possible values are obtained from 0 to m with the same step as in the discrete case, and again continuously in the interval [0, m] in the fuzzy one. It is also feasible to unify the variation amplitude of the individual and total scores, so that both of them will vary from 0 to 1. To do this, we normalize again by dividing the total scores by the number of agents, m. That is

15 181 to say, individual scores are aggregated in a collective one as an arithmetic mean. 14 In this way, Table 2 is obtained. Table 2. Possible marks for n alternatives, m agents Count Normalized Normalized Borda approval Discrete Individual {0,...,1} Individual {0, 1} step = m 1 1 step = 1 Total {0,...,1} Total {0,...,1} step = 1 m(n 1) step = m 1 Fuzzy Individual [0, 1] Individual [0, 1] no step no step Total [0, 1] Total [0, 1] no step no step One can observe by examining Table 2 that discrete counts are particular cases of the fuzzy ones, where marks are scaled in the unit interval. As for the fuzzy Borda and approval counts, the marks obtained are the same, although corresponding to distinct conceptions in each case, as pointed out earlier. The marks used in the counts appearing in this paper admit a utilitariancardinal perspective, as pointed out by Black (1976), Sugden (1981) and recently by Marchant (2000). The total score obtained by the alternatives can be understood as defined by a collective utility function U : X [0, 1], such that U(x i ) = u 1(x 1 ) + u 2 (x i ) + +u m (x i ), m where u j (x i ) [0, 1] are the normalized scores which are aggregated through the arithmetic mean, as indicated. Notes 1. Historical background can be found in Black (1958: ) and McLean and Urken (1995: 94 97). These last two authors have also indicated that Nicolaus Cusanus (XV th c.) foresaw the idea of the Borda count (see McLean and Urken (1995: 19 23)). 2. Black (1958, 1976), Mueller (1979) and Straffin Jr. (1980), among others, have noted that the Borda count chooses the alternative which stands highest on average in the agents preference orderings.

16 About this great weakness of voting procedures in connection with the Borda count, see Black (1976), Ludwin (1978), Sen (1984) and Dummet (1998), among others. 4. This alternative, if it exists, is called the Condorcet winner. 5. Again, Condorcet anticipated Arrow s independence of irrelevant alternatives principle. See McLean (1995) and McLean and Urken (1995: 32 34). 6. This viewpoint of Dummet can be supported by the distance-based consensus ideas of disagreement minimization appearing in Cook and Seiford (1982). On the other hand, Biswas (1994) considers efficiency and consistency reasons related to the Borda count. 7. We have just pointed out the arguments of Morales and Borda to justify integer value scale of the Borda count as the number of times each alternative is preferred pairwise to each other. Laplace, on his own, obtained this spectrum of scores by means of probabilistictype arguments based on the agents latent estimates, recently re-formulated by Tanguiane (1991: 80 and ff.) and Tangian (2000), who already suggest the fuzzy counts introduced in this paper. Laplace s arguments also appear in Basset Jr. and Perski (1999). 8. The system used to elect the Venetian Dux anticipates approval voting. See Lines (1986) and McLean and Urken (1995: 22). 9. A binary relation P is asymmetric if x i Px j implies not x j Px i. 10. An ordinary preference relation P is negatively transitive if, not being verified x i Px j nor x j Px k, then does not satisfy x i Px k. This definition, widely used in the literature of preference modeling, is equivalent to other more intuitive statements. So, an asymmetric binary relation P is negatively transitive if and only if R, the weak preference relation associated with P (x i Rx j if not x j Px i ) is transitive. And both statements are equivalent to the transitivity of P and I, the indifference relation associated with P and R (x i Ix j can be defined by no x i Px j nor x j Px i, or equivalently by x i Rx j and x j Rx i ). 11. This idea appears implicitly in Morales (1797, 1805) and in Condorcet s comments to the Memory of Borda (see McLean and Urken (1995: 81 89)). 12. To justify the axiom of reciprocity see, among others, Bezdek, Spillman and Spillman (1978), Nurmi (1981) and García-Lapresta and Llamazares (2000). 13. Max- transitivity for R k is initially defined by demanding r k il r k ij rk jl. Weak (or restricted ) conditions are considered by Tanino (1984) and Dasgupta and Deb (1996), among others, when certain additional hypotheses are required. In our case, these are to consider preference intensities in Definition 5 greater than About the arithmetic mean as an aggregation procedure, see Chichilnisky and Heal (1983), Le Breton and Uriarte (1990), Candeal, Induráin and Uriarte (1992), Candeal and Induráin (1995), Quesada (2000) and García-Lapresta and Llamazares (2000). References Arrow, K.J. (1963). Social choice and individual values. 2nd edition. New York: Wiley. Baigent, N. and Xu, Y. (1991). Independent necessary and sufficient conditions for approval voting. Mathematical Social Sciences 21: Bana e Costa A. and Vansnick J.C. (1997). The MACBETH approach: Basic ideas, software, and an application. In N. Meskens and M. Roubens (Eds.), Advances in decision analysis, Dordrecht: Kluwer Academic Publishers. Basset Jr., G.W. and Perski, J. (1999). Robust voting. Public Choice 99: Bezdek, J.C. and Harris, J.D. (1978). Fuzzy partitions and relations: An axiomatic basis for clustering. Fuzzy Sets and Systems 1:

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