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1 GROUPE D ANALYSE ET DE THÉORIE ÉCONOMIQUE LYON - ST ÉTIENNE WP 1402 Consistent collective decisions under ajorities based on differences Mostapha Diss, Patrizia Pérez- Asurendi January 2014 Docuents de travail Working Papers

2 GATE Groupe d Analyse et de Théorie Éconoique Lyon- St Étienne 93, chein des Mouilles Ecully France Tel. +33 (0) Fax +33 (0) , rue Basse des Rives Saint- Etienne cedex 02 France Tel. +33 (0) Fax. +33 (0) Messagerie électronique / Eail : gate@gate.cnrs.fr Téléchargeent / Download : Publications / Working Papers

3 Consistent collective decisions under ajorities based on differences Mostapha Diss Patrizia Pérez-Asurendi This version: January 2014 Abstract The ain criticis to the aggregation of individual preferences under ajority rules refers to the possibility of reaching inconsistent collective decisions fro the election process. In these cases, the collective preference includes cycles and even could prevent the election of any alternative as the collective choice. The likelihood of consistent outcoes under two classes of ajority rules constitutes the ai of this paper. Specifically, we focus on ajority rules that require certain consensus in individual preferences to declare an alternative as the winner. In the case of ajorities based on difference of votes, such requireent asks to the winner alternative to obtain a difference in votes with respect to the loser alternative taken into account that individuals are endowed with weak preference orderings. Sae requireent is asked to the restriction of these rules to individual linear preferences, whereas in the case of ajorities based on difference in support, the requireent has to do with the difference in the su of the intensities for the alternatives in contest. Keywords Majorities based on difference of votes Majorities based on difference in support Probability Transitivity Triple-acyclicity. M. Diss Université de Lyon, Lyon, F-69007, France; CNRS, GATE Lyon Saint-Etienne, Ecully, F , France; Université Jean Monnet, Saint-Etienne, F-42000, France. E-ail: diss@gate.cnrs.fr Corresponding author: P. Pérez-Asurendi Grupo de investigación PRESAD, Universidad de Valladolid, Avda. Valle de Esgueva 6, 47011, Valladolid, Spain & Grupo de investigación SEED, Universidad Pública de Navarra, Capus de Arrosadia, 31006, Paplona, Spain. E-ail: patrizia.perez@eco.uva.es

4 2 Diss, Pérez-Asurendi 1 Introduction Since Condorcet (1785) introduced the Voting Paradox, it is well known that the aggregation of transitive individual preferences under siple ajority rule could lead to inconsistent collective preferences. Recalling the classical exaple, consider a three-alternative election with alternatives x 1, x 2, x 3 and three individuals endowed with the following rankings x 1 x 2 x 3, x 2 x 3 x 1 and x 3 x 1 x 2, where, for instance, x 1 x 2 x 3 eans that x 1 is preferred to x 2, x 2 is preferred to x 3 and x 1 is preferred to x 3. For each pair of alternatives, each individual casts a vote for her/his preferred alternative following just assued orderings. Adding up these votes, alternatives x 1, x 2 and x 3 defeat x 2, x 3 and x 1 respectively, by two votes to one. In that voting situation, there is a cycle on the ordering induced by the strict collective preference. In such a case, that preference fails on transitivity and on triple-acyclicity given the requireents of such conditions. To illustrate, assue that alternative x 1 defeats x 2 and x 2 defeats x 3 ; x 1 defeats x 3 whenever the strict collective preference is transitive whereas x 3 does not defeat x 1 whenever the strict collective preference is triple-acyclic. Consider now the following voting process outcoe: x 1 defeats x 2 and it is indifferent to x 3, and x 2 is also indifferent to x 3. In this case, the weak collective preference fails on consistency. Notice that the strict preference associated with that weak preference behaves right but the indifference relation associated with the weak preference fails on transitivity. The idea that the Voting Paradox certainly could not be very likely to ever be observed in realistic situation stated by Gehrlein and Fishburn (1976), prootes the probabilistic study of the occurrence of that paradox and of their consequences under different aggregation rules. In several studies, it is assued an a priori probability odel to estiate the likelihood of different voting situations, derived the conditions under which the paradox or the effects of that appear and reached probabilities through cobinatoric calculus. In this context, stand out the studies about siple ajority rule (Gehrlein and Fishburn 1976; Fishburn and Gehrlein 1980; Gehrlein 1983), superajority rules (Balasko and Crès 1997; Tovey 1997) or scoring rules (Gehrlein and Fishburn 1980, 1981, 1983; Cervone et al 2005), aong others. In other ones, these probabilities are calculated following the Montecarlo siulation ethodology. Specifically, the study of the cyclical and intransitive collective decisions under the siple ajority rule are carried out in Capbell and Tullock (1965), Klahr (1966), DeMeyer and Plott (1970) and Jones et al (1995). This paper is devoted to analyze and copare the probabilities of consistent collective decisions over three alternatives for two different classes of ajorities rules: ajorities based on difference of votes (García-Lapresta and Llaazares 2001; Llaazares 2006; Houy 2007) and ajorities based on difference in support (García-Lapresta and Llaazares 2010). Given two alternatives, these ajorities based on differences focus on requiring to an alternative, to be declared the winner, to reach a nuber of votes or a support that exceeds the nuber of votes or the support for the other alternative in a quantity fixed before the voting process. Therefore, the difference between these two classes

5 Consistent collective decisions under ajorities based on differences 3 of ajorities restricts to the types of individual preferences considered. In the first case, individual preferences are understood as crisp preferences, i.e. given a pair of alternatives individuals declare if they prefer an alternative to another one or if they are indifferent between the. Here, we distinguish between the case where individuals are endowed with weak preferences and the case where individuals are endowed with linear orderings. In the second case, individual preferences are understood as reciprocal preference relations, i.e. given two alternatives, individuals declare the degree with which they prefer an alternative to another one, in other words their intensities of preference, by eans of nuerical values in the unit interval. Coing back to the consistent collective decisions analyzed here, we specifically calculate the probabilities of transitive and triple-acyclic strict collective preferences and the corresponding ones of transitive weak collective preferences for the three specifications of ajorities based on differences stated before, as the proportion of collective decisions that fulfill each of such consistency conditions over the total nuber of possible collective decisions. To calculate the probabilities of consistent outcoes under ajorities based on difference of votes taken into account weak and linear individual preferences respectively, we consider that each individual preference ordering is equiprobable by ebracing the Ipartial Anonyous Culture (IAC) condition (Gehrlein and Fishburn 1976) to describe the likelihood of the possible individual orderings. On the one hand, assuing weak or linear individual orderings jointly with the IAC condition allow to know the total nuber of possible collective preferences (again, Gehrlein and Fishburn 1976). On the other hand, the nuber of consistent profiles is calculated by eans of Ehrhart polynoials, a ethod recently introduced in the social choice literature by Wilson and Pritchard (2007) and Lepelley et al (2008) in order to estiate the probabilities of soe voting paradoxes under the IAC condition. In the case of the calculations of the probabilities of consistent outcoes under ajorities based on difference in support, we propose to apply the Montecarlo siulation ethodology to estiate such probabilities inspired by the studies in Capbell and Tullock (1965), Klahr (1966), DeMeyer and Plott (1970) and Jones et al (1995). Specifically, we generate the individual reciprocal preference relations for the case of three alternatives. Each individual intensity of preference is understood as a continuous rando variable in the unit interval consistently built with a specific transitivity condition over the individual s reciprocal preference relations. Then, we fix the required difference in support and aggregate these individual preferences with the corresponding ajority based on difference in support. We derive the resultant collective ordering of alternatives and evaluate its consistency. Finally, we iterate that procedure to estiate desired probabilities as the nuber of consistent orderings over the total nuber of siulated collective orderings. The ethodology proposed here allows us to hypothesize about a relationship between the type of individual preferences assued under each rule and the likelihood of inconsistent collective decisions. Moreover, we set forth our

6 4 Diss, Pérez-Asurendi results for ajorities based on difference of votes with previous ones on siple ajority (Gehrlein (1997) and Lepelley and Martin (2001)). We also are able to analyze the ipact of the types of transitivity conditions for individual reciprocal preferences on the probability of consistent collective decisions under ajorities based on difference in support. In addition, we copare our results on probabilities with soe theoretical ones about the consistency of the collective preferences under ajorities based on difference in support (Llaazares et al 2013; Llaazares and Pérez-Asurendi 2013). The paper is organized as follows. Section 2 describes the theoretical fraework followed in this paper and introduces ajorities based on difference of votes and ajorities based on difference in support. Sections 3 and 4 provide the results about the probability of consistent collective decisions under ajorities based on difference of votes with linear preferences and with weak preferences, respectively. Section 5 is devoted to the siulated probabilities in the case of ajorities based on difference in support. Section 6 discusses the results and concludes. 2 Preliinaries Consider a set of alternatives X = {x 1, x 2, x 3 } in an election with individuals. Let S be a binary relation on X, i.e. a subset of the cartesian product X X. In what follows, x i Sx j stands for (x i, x j ) S, i.e. when x i is in the relation S with x j. S 1 is the inverse relation of S defined by x i S 1 x j x j Sx j and S c is the copleent relation of S defined by x i S c x j (x i Sx j ). Given two binary relations S and T, the intersection of S and T is also a binary relation defined by x i (S T )x j (x i Sx j x i T x j ). A binary relation S on X is 1. reflexive if x X, xsx, 2. syetric if x i, x j X, x i Sx j x j Sx i, 3. asyetric if x i, x j X, x i Sx j (x j Sx i ), 4. antisyetric if x i, x j X, (x i Sx j x j Sx i ) x i = x j, 5. coplete if x i, x j X, x i Sx j x j Sx i, 6. transitive if x i, x j, x l X, (x i Sx j x j Sx l ) x i Sx l, 7. triple-acyclic if x i, x j, x l X, (x i Sx j x j Sx l ) (x l Sx i ). A weak preference R is a coplete binary relation on the set of alternatives X. The strict preference P associated with R is the asyetric binary relation on X defined by P = (R 1 ) c and the corresponding indifference relation I is the reflexive and syetric binary relation on X defined by I = R R 1. P(X) is the set of strict preferences. A weak ordering is a transitive weak preference whereas a linear ordering is also antisyetric. Fro definitions above it is well know that any weak ordering iplies a transitive strict preference relation and a transitive indifference relation. Moreover, any transitive strict preference is also a triple-acyclic preference relation. Notice that the converse is not true.

7 Consistent collective decisions under ajorities based on differences 5 Given that the social decision between two alternatives is given by either a strict preference relation or an indifference relation, and that three alternatives are in contest, we consider the 27 cases in Table 1 as possible social outcoes. Table 1 Possible social outcoes in a three alternative election. 1. x 1 P x 2 x 2 P x 3 x 1 P x x 1 P x 2 x 2 Ix 3 x 1 Ix 3 2. x 1 P x 3 x 3 P x 2 x 1 P x x 1 Ix 2 x 2 Ix 3 x 1 P x 3 3. x 2 P x 1 x 1 P x 3 x 2 P x x 1 Ix 2 x 2 Ix 3 x 3 P x 1 4. x 2 P x 3 x 3 P x 1 x 2 P x x 1 Ix 3 x 3 P x 2 x 1 Ix 2 5. x 3 P x 1 x 1 P x 2 x 3 P x x 2 P x 1 x 1 Ix 3 x 2 Ix 3 6. x 3 P x 2 x 2 P x 1 x 3 P x x 1 Ix 2 x 2 P x 3 x 1 Ix 3 7. x 1 P x 2 x 2 Ix 3 x 1 P x x 1 P x 2 x 2 P x 3 x 1 Ix 3 8. x 2 P x 1 x 1 Ix 3 x 2 P x x 3 P x 1 x 1 P x 2 x 2 Ix 3 9. x 3 P x 1 x 1 Ix 2 x 3 P x x 2 P x 3 x 3 P x 1 x 1 Ix x 1 Ix 2 x 2 P x 3 x 1 P x x 3 P x 2 x 1 P x 3 x 1 Ix x 2 Ix 3 x 3 P x 1 x 2 P x x 2 P x 1 x 1 P x 3 x 2 Ix x 1 Ix 3 x 3 P x 2 x 1 P x x 3 P x 2 x 2 P x 1 x 1 Ix x 1 Ix 2 x 2 Ix 3 x 1 Ix x 1 P x 2 x 2 P x 3 x 3 P x x 2 P x 1 x 1 P x 3 x 3 P x 2 Our interest focuses on the frequency of consistent social outcoes given the 27 possible outcoes above. We distinguish aong three cases of consistent outcoes; the case of weak orderings corresponding to the first thirteen outcoes, the case of transitive strict preferences corresponding to the first nineteen and the case of triple-acyclic strict preferences corresponding to the first twenty-fifth outcoes. 2.1 Individual preferences We consider that individuals copare the alternatives on X by pairs and declare their preferences by eans of values r p ij [0, 1], where rp ij > 0.5 eans that alternative x i is soewhat preferred to x j by the individual p, whereas r p ij < 0.5 signifies that alternative x j is soewhat preferred to x i by the individual p. At that point, we distinguish between the general case in which preferences are represented by reciprocal preferences and the particular case of that, referred to as crisp preferences. 1. Crisp preferences: the values of r p ij are restricted to the set of discrete values {0, 0.5, 1}. If r p ij = 1, individual p prefers alternative x i to alternative x j, whereas if r p ij = 0, individual p prefers x j to x i. If r p ij = 0.5, individual p is indifferent between both alternatives. Condition r p ij + rp ji = 1 guarantees that the preference of individual p is a weak preference. Moreover, the conditions (r p ij = 1 rp jl = 1) rp il = 1, (r p ij = 0.5 rp jl = 0.5) rp il = 0.5,

8 6 Diss, Pérez-Asurendi assure that the preference of individual p is a weak ordering. Individual linear orderings could also be represented in this fraework by including the following condition: r p ij {0, 1} i j. Thus, individuals could only be indifferent between an alternative and itself. 2. Reciprocal preferences: the values of r p ij belong to the unit interval [0, 1] with the following interpretation: r p ij > 0.5 indicates that the individual p prefers the alternative x i to the alternative x j, the ore the nearer is the value of r p ij to 1 that represents the axiu degree of preference for x i over x j ; conversely, r p ij < 0.5, eans that individual p prefers alternative x j to x i, the ore the nearer is the value of r p ij to 0 that represents the axiu degree of preference for x j over x i ; finally, r p ij = 0.5 stands for the indifference between x i and x j for individual p. The reciprocity of these preferences is described by the condition r p ij + rp ji = 1. To avoid the possibility of having incoherent individual preferences, we need to assue soe kind of rationality condition. But, in this fraework, several concepts could be taken to ensure such rationality requireent (see, aong others, Zadeh 1971; Dubois and Prade 1980; Dasgupta and Deb 1996; García- Lapresta and Meneses 2005). Here, we consider the following transitivity conditions for reciprocal preference relations. Definition 1 We say that individual p is (a) 0.5 transitive if i, j, l {1, 2, 3} (b) in transitive if i, j, l {1, 2, 3} (r p ij > 0.5 rp jl > 0.5) rp il > 0.5, (r p ij > 0.5 rp jl > 0.5) rp il in{rp ij, rp jl }, (c) a transitive if i, j, l {1, 2, 3} (r p ij > 0.5 rp jl > 0.5) rp il ( r p ij + rp jl) /2, (d) ax transitive if i, j, l {1, 2, 3} (r p ij > 0.5 rp jl > 0.5) rp il ax{rp ij, rp jl }. The preferences of each individual over the alternatives in X = {x 1, x 2, x 3 } can be represented using a 3 3 atrix R p = Ä rijä p as follows: Ö è 0.5 r p 12 r p 13 R p = 1 r p r p 23. (1) 1 r p 13 1 r p Individual preferences are collected in a vector where each vector-eleent represents the preferences of an individual. Assuing individuals 1 and taking into account the above distinction aong linear, weak and reciprocal preferences, a profile of linear orderings is a vector (R 1,..., R ) L(X), being 1 To calculate the probabilities presented here, takes the following values: 3, 4, 5, 10, 100, 1,000 and 100,000.

9 Consistent collective decisions under ajorities based on differences 7 L(X) the set of all linear orderings; a profile of weak orderings is a vector (R 1,..., R ) W(X), being W(X) the set of all weak orderings; and a profile of reciprocal preferences is a vector (R 1,..., R ) R(X), being R(X) the set of all reciprocal preference relations. 2.2 Majorities based on differences Given the nature of the three types of individual preferences exained in Subsection 2.1, we also consider three different specifications of ajorities based on differences relying on the types of individual preferences that we take into account in each case. In what follows, we refer to these ajorities as ajorities based on difference of votes with linear orderings, ajorities based on difference of votes with weak orderings and ajorities based on difference in support for reciprocal preference relations. The concept of ajorities based on difference of votes was introduced in García-Lapresta and Llaazares (2001) and was later axioatically characterized in Llaazares (2006), and subsequently in Houy (2007). These rules involve crisp preferences, i.e. given a pair of alternatives, individuals could declare their preference for one of the or their indifference between both alternatives. Under these ajorities, an alternative, say x i, is declared the winner if the nuber of individuals who prefer that alternative, to the other one, say x j, exceeds the nuber of individuals who prefer x j to x i in a difference of votes, fixed before the election process. Assuing individuals, that difference could take any integer value in {0,..., 1}. These ajorities are located between siple ajority rule where the difference of votes is zero and unaniity where the difference of votes is the total nuber of individuals inus one. Moreover, if the indifference state is ruled out fro individual preferences, these ajorities are equivalent to superajority rules. Taking into account the forer ajorities based on difference of votes, we introduce in Definition 2 the ajorities based on difference of votes with linear orderings and in Definition 3 the ajorities based on difference of votes with weak orderings. In what follows, the sybol # stands for the cardinality of a set.

10 8 Diss, Pérez-Asurendi Definition 2 (Majorities based on difference of votes with linear orderings or Mk L ajorities) Given k {0, 1,..., 1}, the ajority based on difference of votes with linear orderings or Mk L ajority is the apping M L k : L(X) P(X) defined by Mk L (R1,..., R ) = Pk L, where x i P L k x j #{p r p ij = 1} > #{p rp ji = 1} + k. The indifference relation associated with P L k x i I L k x j is as follows: #{p r p ij = 1} #{p rp ji = 1} k. Exaple 1 Let R I and R II be the following individual linear preference orderings over the alternatives on X = {x 1, x 2, x 3 }. R I = Ñ é Ñ é , R II = Consider the profile (R 1, R 2, R 3, R 4, R 5 ) where R p = R I if p = 1, 2, 3, R II if p = 4, 5. Assuing a required difference of votes k equal to 1 and applying the corresponding M1 L ajority we have #{p r p 12 = 1} #{p rp 21 = 1} = x 1 I L 1 x 2, #{p r p 23 = 1} = 5 > #{p rp 32 = 1} + 1 = x 2 P L 1 x 3, #{p r p 13 = 1} = 5 > #{p rp 31 = 1} + 1 = x 1 P L 1 x 3. Definition 3 (Majorities based on difference of votes with weak orderings or M k ajorities) Given k {0,..., 1}, the ajority based on difference of votes with weak orderings or M k ajority is the apping M k : W(X) P(X) defined by M k (R 1,..., R ) = P k, where x i P k x j #{p r p ij = 1} > #{p rp ji = 1} + k. The indifference relation associated with P k is as follows: x i I k x j #{p r p ij = 1} #{p rp ji = 1} k.

11 Consistent collective decisions under ajorities based on differences 9 Exaple 2 Let R I and R II be the following individual weak preference orderings over the alternatives on X = {x 1, x 2, x 3 }. Ñ é Ñ é R I = , R II = Consider the profile (R 1, R 2, R 3, R 4, R 5 ) where R p = R I if p = 1, 2, 3, R II if p = 4, 5. Assuing a required difference of votes k equal to 2 and applying the corresponding M 2 ajority we have #{p r p 12 = 1} = 3 > #{p rp 21 = 1} + 2 = x 1 P 2 x 2, #{p r p 23 = 1} #{p rp 32 = 1} = x 2 I 2 x 3, #{p r p 13 = 1} #{p rp 31 = 1} = x 1 I 2 x 3. In García-Lapresta and Llaazares (2010), ajorities based on differences of votes were extended to the fraework of reciprocal preferences allowing individuals to declare their degrees of preferences over pairs of alternatives. Majorities based on difference in support allow us to aggregate each profile of reciprocal preferences into a strict collective preference P k over the set of alternatives. Under these rules, the winner alternative is required to reach a support that exceeds the support for the other alternative in a quantity, fixed before the voting process. Foral definition for these ajorities is as follows. Definition 4 (Majorities based on difference in support or M k ajorities (García-Lapresta and Llaazares 2010)) Given k [0, ), the ajority based on difference in support or M k ajority is the apping M k : R(X) P(X) defined by M k (R 1,..., R ) = P k, where x i P k x j r p ij > + k. (2) p=1 p=1 r p ji The indifference relation associated with P k is defined by: x i I k x j r p ij r p ji k. (3) p=1 Notice that when considering crisp preferences, the expression in (2) goes for defining the ajorities based on difference of votes with linear and weak preference orderings. p=1

12 10 Diss, Pérez-Asurendi Exaple 3 Let R I and R II be the following reciprocal preference relations over the alternatives on X = {x 1, x 2, x 3 }. Ñ é Ñ é R I = , R II = Consider the profile (R 1, R 2, R 3, R 4, R 5 ) where R R p I if p = 1, 2, 3, = R II if p = 4, 5. Assuing a required difference in support k equal to 1.75 and applying the corresponding M 1.75 ajority we have 5 r p 12 = 4.6 > p=1 5 r p = x 1 P 1.75 x 2, p=1 5 5 r p 23 p=1 5 r p 13 = 4.7 > p=1 p=1 r p 32 = x 2 I 1.75 x 3, 5 r p = x 1 P 1.75 x 3. p=1 3 Probability of consistent collective decisions under ajorities based on difference of votes with linear orderings In this section the results about the probabilities of consistent collective decisions under Mk L ajorities are introduced under IAC assuption. Given that voters are endowed with coplete linear preference orderings, there are six possible preference orders that they ight have, x 1 x 2 x 3 ( 1 ) x 1 x 3 x 2 ( 2 ) x 2 x 1 x 3 ( 3 ) x 2 x 3 x 1 ( 4 ) x 3 x 1 x 2 ( 5 ) x 3 x 2 x 1 ( 6 ) where i is the nuber of voters with the associated linear preference ordering. In this fraework, a voting situation is a vector = ( 1, 2, 3, 4, 5, 6 ) 6 such that i =. As the IAC condition is assued, all possible voting i=1 situations are equally liked to be observed. Gehrlein and Fishburn (1976) showed that for agents and 3 alternatives, the total nuber of voting situations is given by the expression: ψ() = (4) ( + 1)( + 2)( + 3)( + 4)( + 5). (5) 120

13 Consistent collective decisions under ajorities based on differences Probabilities of triple-acyclic strict preferences under ajorities based on difference of votes with linear orderings To calculate the probability of triple-acyclic strict preferences under Mk L ajorities we focus on the cases fro 1 to 25 in Table 1. Specifically, we first cal- culate the probability of cyclic strict preferences, i.e. the probability of having preferences like the ones described in the cases 26 and 27 (see again Table 1). Thereafter, we obtain the probability of triple-acyclic cases as 1 inus the probability of cyclic strict preferences. Going deeper on the strict preference described in the case 26, we notice that for such preference to exist, the nubers of voters associated with the linear orderings described in (4) have to fulfil the following conditions: > k, > k and > k. In other words, the strict preference in the case 26 requires a voting situation = ( 1, 2, 3, 4, 5, 6 ) that fulfils the conditions given by the syste of inequalities below > k, > k, (x 1 Pk L x 2, x 2 Pk L x 3 and x 3 Pk L x > k, 1) i 0 for i {1,..., 6}, 1 k 0, =. Therefore, to calculate the probability of cyclic strict preferences, we need to solve the syste of linear inequalities derived fro conditions that the nubers of voters associated with the linear orderings in (4) have to hold for strict preferences like the ones in the cases 26 and 27 to exist. We copute the nuber of voting situations that fulfil these conditions by eans of the Paraeterized Barvinok s algorith (Verdoolaege et al (2004)) 2. Such algorith allows to quantify the nuber of integer solutions for systes of inequalities with paraeters. The connection of such algorith to Social Choice Theory was recently pointed out by Wilson and Pritchard (2007) and Lepelley et al (2008). Given the two paraeters and k, the nuber of voting situations for our syste is given by bivariate quasi polynoials in and k with 2- periodic coefficients eaning that such coefficients depend on the parity of the paraeters and k. Following the notation introduced in Lepelley et al (2008), we represent these coefficients by a list of 2 rational nubers enclosed in square brackets. To illustrate, assue the bracketed list [[a, b], [b, a] ] k. In the case of even k, the relevant list corresponds to [a, b]. The coefficient will be either a when is even or b when is odd. Accordingly, in the case 2 The free software to calculate the integer points under the Paraeterized Barvinok s algorith can be found in

14 12 Diss, Pérez-Asurendi of odd k, the relevant list is [b, a] and therefore, the coefficient will be either b when is even or a when is odd. Thus, the coefficient will be a when and k have the sae the parity and b otherwise. Notice that in the case of the cyclical strict preferences depicted in the cases 26 and 27 (Table 1), the nuber of solutions in the syste of inequalities derived fro the strict preference in the case 26 is the sae as in the syste derived fro the strict preference in the case 27 given the syetry of such cases. The progra indicates that the corresponding quasi polynoial for each of these cases is as follows: ( k ( [ ï 9 64, 63 ò ï, , 9 64 ( [ ï , 27 ò 128 [ ï 9 64, 63 ò ï 63, , 9 64 ( [ ï , 3 ò 64 [ ï 3 64, 21 ò ï, , 3 64 [ ï 1 20, [ ï 0, 81 ò ï 81, , 0 ò ] [ ï 0, 27 ò ï, , 0 ò ] ) ò [ ï , [ ï 1 ò 192, [ ï 0, ò [ ï 0, ò ï 27, 128, 0 ï 71, 1280, ò k ò k 3 + ] ò k + ] k 2 + k ò ] [ ï + 0, 27 ò 128 k, ï 364 ò ], k ò ] [ ï 2 + 0, 9 ò 64 k ò ] ) k + ï ò 1, 256, 0 ï 7, 384, ] ï 3, 128, 0 ò ï, 7 256, 0 ò ] k ] k k [ ï , 71 ò 3840 k ò ] k. ) k 4 + ï ò ] ) 27, 128, 0 k 2 + k ï, 9 ò ] 64, 0 + k ï, , 1 ò ] + 60 k In addition, the progra points out that this relation holds only if k ( 3)/3. Otherwise, the nuber of voting situations is zero.

15 Consistent collective decisions under ajorities based on differences 13 We siplify 3 the quasi polynoial above by considering different values of and k. Thus, it can be deduced that the nuber of voting situations corresponding to each of the strict preferences represented by the cases 26 and 27 in Table 1 is given by F 1 (, k ) if both and k are odd (or even) and by F 2 (, k ) if one of the paraeters ( or k ) is odd and the other one is even such that: F 1 (, k ) = Å ( 3 k 4) ( 3 k ) ( 3 k + 4) ( 3 k 2) ( 3 k + 2) F 2 (, k ) = 1 Å ( 3 k + 3) ( 3 k + 7) ( 3 k + 1) 3840 ã ( 3 k + 5) ( 3 k 1). ã. As a consequence of the above nuber of voting situations and taking into account the syetry of the strict preferences of the cases 26 and 27 in Table 1 and the total nuber of voting situations ψ() in (5), we introduce the probabilities of having triple-acyclic strict preferences under Mk L ajorities in the following result. Proposition 1 Consider a three candidate election with voters under M L k ajority rules where each individual vote consists of a linear preference ordering on the candidates. Assuing that all voting situations are equally likely (IAC), if k ( 3)/3, the probability of triple-acyclic strict preference is as follows: If both and k are odd (or even): 1 2F 1(, k ). ψ() If one of the paraeters ( or k ) is odd and the other one is even: 1 2F 2(, k ). ψ() Coputed values of this probability are listed in Table 2. The values of the difference of votes k correspond to the ones that provide a probability of triple-acyclic strict preferences equal to 1. As it is previously entioned, these probabilities indicate that the nuber of solutions in the systes of inequalities corresponding to the cyclic strict preferences (cases 26 and 27 in Table 1) are equal to zero when k > ( 3)/3. Soe interesting facts could be ephasized fro the probabilities in Table 2. 3 Such siplification is done with Maple software.

16 14 Diss, Pérez-Asurendi Table 2 Probability of triple-acyclic P L k , ,000 k ,332 1 First, M0 L ajority provide a probability of triple-acyclic strict preferences equal to 1 for the case of = 4 whereas a difference of votes equal to 1 is necessary in the case of = 3 and = 5 to achieve such probability. We conjecture that this odd result attends to the fact that the likelihood of having ties is greater in the case of an even nuber of individuals than in the case of an odd nuber of individuals when these voters are endowed with linear orderings. Second, the weight of the difference of votes necessary to achieve a probability of triple-acyclic strict preferences over the total nuber of votes equal to 1 increases with the nuber of individuals fro = 10 to = 100,000. In the case of = 10 required difference signifies a 20% of the value of, a 32% in the case of = 100, a 33.2% in the case of = 1,000 and a % in the case of = 100,000. Third, the probabilities do not reflect sall changes in the agnitude of the thresholds. See for instance, the probabilities attached to the cases = 10, = 100 and = 1,000 for the values of the difference k equal to 0 and 1 and for the case of = 100,000 in the case of k equal 0, 1 and 2. Finally, triple-acyclic strict preferences under Mk L ajorities can be guaranteed with a probability of 1 for not too deanding differences of votes. 3.2 Probabilities of transitive strict preferences under ajorities based on difference of votes with linear orderings To study the probability of transitive strict preferences under Mk L ajorities, we follow the sae ethodology as the one applied in Subsection 3.1 adding the cases 20, 21, 22, 23, 24 and 25 in Table 1 to the outcoes 26 and 27 analyzed in the case of triple-acyclic strict preferences. Once we calculate the probability of non transitive strict preferences collected in the cases fro 20 to 27, we deterine the probability of transitive strict preferences as 1 inus the previous probability. Ordinary preferences collected in cases fro 20 to 25 are siilar and hence, the nuber of integer solutions given by Barvinok s algorith is the sae in each of the six systes of inequalities representing these strict preferences.

17 Consistent collective decisions under ajorities based on differences 15 For these cases, two validity doains can be distinguished. On the one hand, if k ( 3)/3, the nuber of voting situations is given by G 1 (, k ) if both and k are odd (or even) and by G 2 (, k ) if one of the paraeters ( or k ) is odd and the other one is even such that: G 1 (, k ) = (k + 1) Ä 121 k k k k k k k 40 3 k k k ). G 2 (, k ) = k Ä 121 k k k k k k k k 840 k 360 k ). On the other hand, if ( 2)/3 k 2, the nuber of voting situations is given by G 3 (, k ) if both and k are odd (or even) and by G 4 (, k ) if one of the paraeters ( or k ) is odd and the other one is even such that: G 3 (, k ) = Å ( k 2) ( k + 4) ( k ) ( k + 6) ( k + 2) G 4 (, k ) = 1 Å ( k + 7) ( k + 3) ( k 1) 3840 ã ( k + 5) ( k + 1). Bearing in ind above nubers of voting situations, the results in Proposition 1 and the total nuber of voting situations ψ() in (5), we derive the probability of transitive strict preferences under Mk L ajorities as follows. ã. Proposition 2 Consider a three candidate election with voters under M L k ajority rule where each individual vote consists of a linear preference ordering on the candidates. Assuing that all voting situations are equally likely (IAC), the probability of transitive strict preferences is as follows: 1. If k ( 3)/3 If both and k are odd (or even): 1 2F 1(, k ) + 6G 1 (, k ). ψ() If one of the paraeters ( or k ) is odd and the other is even: 1 2F 2(, k ) + 6G 2 (, k ). ψ()

18 16 Diss, Pérez-Asurendi 2. If ( 2)/3 k 2 If both and k are odd (or even): 1 6G 3(, k ). ψ() If one of the paraeters ( or k ) is odd and the other is even: 1 6G 4(, k ). ψ() Coputed values of this probability are listed in Table 3. Going deeper on the, the weight of the required difference of votes k to guarantee a probability of transitive strict preferences equal to 1 with respect to the total nuber of individuals increases as the nuber of individuals does. To illustrate, in the case of = 3 the required k = 1 represents around a 33.33% of the value of whereas in the case of = 100,000 the required k represents a % of the value of. In fact, the required differences are very large for all the considered cases with the exception of = 3. Even for = 4, the difference signifies a 50% of the value of. For not too deanding differences, the probabilities increase in the cases of = 4, = 5 and = 10; this is not the case for = 100, = 1,000 and = 100,000 where asking reasonable differences of votes decreases the probability of transitive strict preferences. Table 3 Probability of transitive P L k , ,000 k ,998 1 Moreover, as in the case of the probabilities stated in Table 2, sall variations in the agnitude of the differences of votes do not change, at least in a significant way, the probabilities. To illustrate, look at the probabilities of = 100,000 with differences k equal to 0, 1, 2 and Probabilities of transitive weak preferences under ajorities based on difference of votes with linear orderings To derive the probability of transitive weak preferences under Mk L ajorities, we need to consider, in addition with the cases analyzed in Proposition 2, the

19 Consistent collective decisions under ajorities based on differences 17 cases fro 14 to 19 in Table 1. With that, we calculate the probability of non transitive weak preferences and therefore, the probability of transitive weak preferences is deterined as 1 inus the probability of non transitive weak preferences. By syetry arguents, the weak preferences represented in cases fro 14 to 19 are siilar and therefore the nuber of integer solutions of the six systes of inequalities corresponding to such cases is the sae. Using again the Barvinok s algorith, two validity doains can be considered. If k ( 2)/3, the nuber of voting situations inside each syste is given by H 1 (, k ) if both and k are odd (or even), and by H 2 (, k ) if one of the paraeters ( or k ) is odd and the other one is even such that: H 1 (, k ) = Å (k + 1) Ä 17 k 4 30 k 3 22 k k 2 28 k k k + 48 k 5 3 k 5 2 k ) ã. H 2 (, k ) = k Ä 17 k 4 30 k 3 90 k k k k k 45 2 k 100 k 30 k ). For the second validity doain, if ( 1)/3 k 1, this nuber is given by H 3 (, k ) if both and k are odd (or even) and by H 4 (, k ) if one of the paraeters ( or k ) is odd and the other one is even such that: H 3 (, k ) = 1 Å ( k + 2) ( k ) ( k + 4) 3840 Ä 29 k k + 94 k ä ã. H 4 (, k ) = 1 Å ( k + 1) ( k + 5) ( k + 3) 3840 Ä 29 k k + 36 k ä ã. Taking into consideration the intersections between the different validity doains and using the results in Propositions 1 and 2, the probability of transitive weak preferences is as follows. Proposition 3 Consider a three candidate election with voters under M L k ajority rule where each individual vote consists of a linear preference ordering on the candidates. Assuing that all voting situations are equally likely (IAC), the probability of transitive weak preferences is as follows: 1. If k ( 3)/3

20 18 Diss, Pérez-Asurendi If both and k are odd (or even): 1 2F 1(, k ) + 6G 1 (, k ) + 6H 1 (, k ). ψ() If one of the paraeters ( or k ) is odd and the other one is even: 1 2F 2(, k ) + 6G 2 (, k ) + 6H 2 (, k ). ψ() 2. If ( 1)/3 k 2 If both and k are odd (or even): 1 6G 3(, k ) + 6H 3 (, k ). ψ() If one of the paraeters ( or k ) is odd and the other one is even: 1 6G 4(, k ) + 6H 4 (, k ). ψ() 3. If k = ( 2)/3 Either both and k are odd or both are even: 1 6H 1(, k ) + 6G 3 (, k ). ψ() 4. If k = 1 One of the paraeters ( or k ) is odd and the other one is even: 1 6H 4(, k ). ψ() Analyzing the probabilities of transitive weak preferences displayed in Table 4, M0 L ajority provides the highest values for the probability of having transitive weak preferences for alost all the considered values of. In fact, any difference of votes can be asked to guarantee a probability value of 1. Only in the cases of = 1,000 and = 100,000 the probability arrives to the value of , i.e. the probability approxiates to the value of 1 without reaching it 4. Even so, in both cases the required difference of votes is extreely large. Specifically, it represents a 99.9% of the value of in the case of = 1,000 and a % in the case of = 100,000. Finally, as in the previous cases stated in Tables 2 and 3, the probabilities do not significantly change with sall variations of the agnitude of the difference in votes. On this, see for instance the cases of k equal 2 and 3 for equal 4, 10, 100, 1,000 and 100, Notice that this value also appears in Tables 5, 7, 12, 13, 14 and 15 with the sae eaning.

21 Consistent collective decisions under ajorities based on differences 19 Table 4 Probability of transitive R L k , ,000 k , Probabilities of consistent collective decisions under ajorities based on difference of votes with weak orderings In this section the results about the probabilities of consistent collective decisions under M k ajorities are introduced under the IAC assuption. Given that voters could be indifferent between the alternatives, we have to take into account the six linear preference orderings in (4), the six possible orderings that collect the partial indifference and the one that represents the coplete indifference aong three alternatives. Therefore, x 1 x 2 x 3 ( 1 ) x 1 x 3 x 2 ( 2 ) x 2 x 1 x 3 ( 3 ) x 2 x 3 x 1 ( 4 ) x 3 x 1 x 2 ( 5 ) x 3 x 2 x 1 ( 6 ) {x 1 x 2 }x 3 ( 7 ) {x 1 x 3 }x 2 ( 8 ) {x 2 x 3 }x 1 ( 9 ) x 1 {x 2 x 3 } ( 10 ) x 2 {x 1 x 3 } ( 11 ) x 3 {x 1 x 2 } ( 12 ) {x 1 x 2 x 3 } ( 13 ) where i represents the nuber of voters with the associated preference ordering and {x i x j } stands for the indifference between the alternatives x i and x j. As the IAC condition is assued, all possible voting situations are equally liked to be observed. For individuals and 3 alternatives, if the indifference between alternatives is allowed, the total nuber of voting situations is given by the expression: Ψ() = (6) ( + 1)( + 2) ( + 12). (7) 12! Using the sae approach applied in Section 3, the probability of consistent outcoes is calculated by eans of the coputation of the probability of inconsistent outcoes. As there, such probabilities are given by Ehrhart polynoials that provide the nuber of integer points inside the systes of inequalities that characterize each of the analyzed inconsistent outcoes. In the fraework of the preferences represented in (6), the coplexity of the conditions akes ipossible the derivation of a general atheatical

22 20 Diss, Pérez-Asurendi representation as the one provided in Section 3. This is because for each considered consistency condition, the nuber of validity doains and the length of the polynoials are greater than in the cases of Section 3. Fortunately, when the nuber of individuals and the threshold k are fixed, the probabilities can be calculated for the given nuber of voting situations. 4.1 Probabilities of triple-acyclic strict preferences under ajorities based on difference of votes with weak orderings In Table 5, the probabilities of triple-acyclic strict preferences under M k ajorities are displayed. The following facts can be pointed our fro these results. The probabilities of having triple-acyclic strict preferences when no difference of votes is required, i.e. when M 0 ajority is applied, reach very high values. Specifically, they are located between and For = 3, = 4 and = 5, the needed difference of votes to achieve a probability value of 1 equals 1. Therefore, it represents a one third of the value of in the case of = 3, a 25% in the case of = 4 and a 20% in the case of = 5. For the reaining considered values, the weight of the required differences represent around one third of the value of which eans that we can guarantee with a probability of 1 the triple-acyclicity of strict preferences under M k ajorities for reasonable values of the difference of votes. Table 5 Probability of triple-acyclic P k , ,000 k , Probabilities of transitive strict preferences under ajorities based on difference of votes with weak orderings In Table 6, the probabilities of transitive strict preferences under M k ajorities are presented. It is rearkable that to reach a probability of transitive strict preferences equal to 1, the weight of the required difference of votes k with respect to the

23 Consistent collective decisions under ajorities based on differences 21 Table 6 Probability of transitive P k , ,000 k ,998 1 total nuber of individuals increases as the nuber of individuals does. For instance, in the case of = 4, it represents a 50% of the value of whereas in the case of = 1,000 it does a 99.8% of the value of. Moreover, the required differences are too deanding for all the cases with the exception of the case of = 3 where it represents one third of the value of. 4.3 Probabilities of transitive weak preferences under ajorities based on difference of votes with weak orderings Probabilities of transitive weak preferences under M k ajorities defined for weak orderings are displayed in Table 7. Table 7 Probability of transitive R k , ,000 k , The probability value of 1 is alost achieved for the values of equal to 100, 1,000 and 100,000. In these cases, the required differences in votes k are so high that signify a 98% of the value of in the case of = 100, a 99.8% in the case of = 1,000 and a % in the case of = 100,000.

24 22 Diss, Pérez-Asurendi 5 Probabilities of consistent collective decisions under ajorities based on difference in support In this section, we provide the probabilities of reaching consistent collective decisions under M k ajorities for three alternatives. As long as the intensities of preference between each pair of alternatives can take any value in the continuous interval [0, 1], the IAC odel can not be applied to provide an a priori probability to each possible voting situation. Therefore the probabilistic analysis carried out in Sections 3 and 4 turns ipossible to study the case of M k ajorities. Consequently, we perfor a siulation with the software Matlab to estiate these probabilities as the proportion of the nuber of consistent outcoes in the siulation over the total nuber of siulated outcoes. We generate for each of these values 100,000 outcoes to guarantee our results with a confidence level of 99% and a sapling error of less than a % 5. Below, we describe the ethodology applied in the siulations to estiate the probability for the considered three types of consistent collective decisions under M k ajorities, i.e. transitive weak preferences, transitive and tripleacyclic strict preferences. We follow that schee taking into account each type of individual transitive reciprocal relations, i.e. 0.5 transitive, in transitive, a transitive and ax transitive reciprocal preference relations. Notice that the atrix in (1) representing a reciprocal preference relation is deterined by the vector coposed of the intensities r 12, r 23 and r We randoly generate vectors representing the transitive reciprocal preference relations of the individuals. Such vectors are built bearing in ind one of the considered transitivity conditions for reciprocal preference relations. 2. We copute the su of the individuals intensities of preference over each pair of alternatives through a vector S = (S 12, S 23, S 13 ) where S ij = r p ij. p=1 3. Having in ind the conditions in equations (2) and (3) and the value of k, the collective decision is evaluated over each pair of alternatives in the vector S. 5 Assuing a proportion of consistent outcoes P on the population of a 50%, the proportion p in a rando saple of size n 30 for a confidence level of 99%, diverges fro the one of the population in an error of less than ɛ: P rob( P p ɛ) Taken into account that the saple proportion p is distributed as N the sapling error ɛ is as follows: ɛ = z α/2 P (1 P )/n. Ä ä P, P (1 P )/n, In our case, n = 100,000 and the corresponding percentile of the noral distribution for a confidence level of 99% is z α/2 = Thus, ɛ

25 Consistent collective decisions under ajorities based on differences The collective decision in S is classified following the cases of possible collective outcoes displayed in Table 1. If it is one of the cases 26 or 27, the strict preference P k is not triple-acyclic. If it is one of the cases fro 19 to 27, the strict preference P k is not transitive. Finally, if it is one of the cases fro 14 to 27, the weak preference R k is not transitive. 5. This four steps are iterated 100,000 ties to obtain the nuber of inconsistent collective decisions. Specifically, the nuber of siulated outcoes in which the weak preference R k is not transitive, in which the strict preference P k is not transitive and in which P k is not triple-acyclic, respectively. 6. The nuber of each considered type of consistent social outcoes is coputed as the total nuber of siulated outcoes, i.e. 100,000, inus the nuber of inconsistent ones coputed in the previous step. 7. Finally, each of the desired probabilities, i.e. the probability of transitive R k and the probability of transitive and triple-acyclic P k, is calculated as the nuber of consistent outcoes over the total nuber of siulated outcoes. In the following, the siulated probabilities of consistent collective decisions under M k ajorities are listed in tables. 5.1 Probability of transitive weak preferences under ajorities based on difference in support Tables 8, 9, 10 and 11 provide the probabilities of transitive weak preferences when reciprocal preference relations fulfil 0.5 transitivity, in transitivity, a transitivity and ax transitivity, respectively. Table 8 Probabilities of transitive R k for 0.5 transitive reciprocal preference relations , ,000 k When the required difference in support equals zero, the harder the transitivity condition over the reciprocal preference relations is, the higher the probabilities of having transitive weak preferences are. To illustrate, notice that the probabilities vary in between and considering 0.5 transitive reciprocal preference relations, between and taking into account