Probabilities of consistent election outcomes with majorities based on difference in support

Group Decision and Negotiation manuscript No. (will be inserted by the editor) Probabilities of consistent election outcomes with majorities based on difference in support Received: date / Accepted: date Abstract Computer simulations are used to evaluate the likelihood of consistent outcomes under the class of majorities based on difference in support. These majorities require certain consensus in collective preferences to declare an alternative as the winner. More precisely, individuals show preference intensities in the unit interval among each pair of alternatives and it is required that the winner alternative obtains a difference in the sum of the intensities with respect to the loser alternative. This difference is a real number located between 0 and the total number of voters. We introduce the values of the required threshold for which majorities based on difference in support lead to transitive and triple-acyclic collective decisions with a probability of 1. Our results improve the previous theoretical ones since they require softer thresholds to reach consistent collective decisions. Keywords: Computer simulations; Majorities based on difference in support; Probability; Transitivity; Triple-acyclicity. 1 Introduction Many consistent conditions such as the transitivity are imposed as minimal requirement concerning any social choice rule used to aggregate individual preferences into a collective outcome. Transitivity represents the idea that, for an individual or collective preference, when alternative x1is preferred to x2 and x2is preferred to x3then x1is preferred to x3. The literature related to this notion dates back 200 years to the great work of the Marquis de Condorcet (1785) which states that voters with individual transitive rankings can produce an election outcome which is not transitive if one chooses pairwise majority voting as aggregation. This result gave rise to numerous studies and promoted the probabilistic study of the occurrence of voting paradoxes and their consequences under different aggregation rules. 2 The probability of voting paradoxes has been the subject of a whole strand of literature. Under this approach, it is assumed an a priori probability model to describe the individual preferences, derived the conditions under which the paradox appears and reached probabilities through combinatoric calculus. In this context, stand out the studies about majority rule (Fishburn and Gehrlein 1980; Gehrlein 1983; Gehrlein and Fishburn 1976), supermajority rules (Balasko and Cr`es 1997; Tovey 1997) or scoring rules (Cervone et al 2005; Gehrlein and Fishburn 1980, 1981, 1983), among others. To circumvent the impossibility of having analytical results, other papers have undertaken a methodology using computer simulations of elections. Specifically, the study of the cyclical and intransitive collective decisions under the simple majority rule are carried out in Campbell and Tullock (1965); DeMeyer and Plott (1970); Jones et al (1995); Klahr (1966), among others. Voting rules based on the majority principle are the most studied in the literature and that principle remains one of the most widely used voting rules in real life. The majority rule, and other classic voting systems, have the advantage of being simple to use and easy to understand by the voters but they have major drawbacks. One of them is based on the idea that the preferences expressed by the voters are assumed to be dichotomous (also called crisp or ordinal), i.e. individuals can only declare if an alternative is preferred to another, or if they are indifferent. As the Nobel laureate Amartya Sen (1970) has noted, “voters opinions can be misrepresented when the preferences are dichotomous in the sense that the collective decision does not take into consideration the intensities of each individual preference”. The importance of considering intensities in the individual preferences has also been noted in Cook and Kress (1985); Meek (1975); Nurmi (1981, 2008); Tanino (1984), and Morales (1797) (see translation in McLean and Urken (1995)), among others. Reciprocal preferences have been introduced in the literature in order to deal with preference intensities. This framework allows individuals to show preference intensities among each pair of alternatives by a real number in the unit interval [0,1]. It is clear that the information contained in reciprocal preferences is much richer than the case of dichotomous preferences. Indeed, for each voter, a reciprocal preference between two alternatives x1and x2 expresses the degree by which the voter prefers x1to x2. More precisely, the closer the intensity is to 1, the more x1is preferred to x2, and the closer the intensity is to 0, the more x2is preferred to x1. In addition, an intensity of 0.5 stands for the exact indifference between the two alternatives x1and x2. Following this approach, when we extend the set of alternatives from two to three alternatives, we assume that each individual reciprocal preference fulfil some properties of transitivity in order to guarantee individual rationality of each reciprocal preference. Considering preference intensities, majorities based on difference in support f Mkhave been introduced in the literature as a possible aggregation system. These majorities require certain consensus in collective preferences to declare an alternative as the winner and depend on the idea of a threshold. More precisely, these aggregation rules suggest that alternative x1is preferred Title Suppressed Due to Excessive Length 3 to alternative x2if and only if the sum of the intensities for x1exceeds the aggregated intensity of x2by a threshold kgiven by a real number located between 0 and the total number of voters. In other words, a higher value of kmeans that a stronger preference for x1over x2is needed in order to x1to be declared socially preferred to x2. Conversely, a low value of kmeans that a weak preference for x1over x2is needed to declare x1socially preferred to x2. The axiomatic characterization of majorities based on difference in support is introduced in Garc´ıa-Lapresta and Llamazares (2010). Notice that, for other kinds of operators which can be used for the aggregation of reciprocal preferences and their characterizations, the reader is referred to Garc´ıa-Lapresta and Llamazares (2000); Llamazares (2004, 2007); Llamazares and Garc´ıa-Lapresta (2003, 2008), among others. This paper is devoted to analyze the probabilities of consistent collective decisions over the class of majorities based on difference in support. We specifically calculate the probabilities of transitive and triple-acyclic strict collective preferences and the corresponding ones of transitive weak collective preferences for these majorities. To calculate these probabilities, we apply the Monte-Carlo simulation methodology inspired by the studies in Campbell and Tullock (1965); DeMeyer and Plott (1970); Jones et al (1995); Klahr (1966), among others. Specifically, we generate the individual reciprocal preference relations for the case of three alternatives. Each individual intensity of preference is understood as a continuous random 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 majority based on difference in support. We derive the resultant collective ordering of alternatives and evaluate its consistency. Finally, we iterate that procedure to estimate desired probabilities as the number of consistent orderings over the total number of simulated collective orderings. The objective of this paper is twofold. First, the methodology proposed here allows us to hypothesize about a relationship between the type of individual preference intensities and the likelihood of consistent collective decisions. Intuitively, we expected to find softer conditions to have consistent collective decisions with probability 1 as individuals are more rational, i.e. the strongest the assumed individual transitivity condition is. As we will see, our results refute that idea. Second, we compare our results on probabilities with the theoretical ones about the consistency of the collective preferences under majorities based on difference in support in Llamazares et al (2013) and Llamazares and P´erez-Asurmendi (2015). In these articles, necessary and sufficient conditions for which these majorities provide transitive and tripleacyclic collective decisions have been introduced. The algebraic tools used in the proofs lead to several impossibility results. Specifically, for the case of the transitivity of the collective decisions they proved that it can be always found an example of intransitive collective preferences when the individual transitivity condition is weak. In the case of assuming stronger conditions to individual 4 preferences, the collective preference is transitive only when individual preferences show a higher level of unanimity. Although they found less demanding conditions in their results about the triple-acyclicity of the collective decision, it is also required a high level of similarity in individual preferences in order to find consistent collective decisions. Under the probabilistic approach followed here, the required conditions for having consistent collective decisions are much softer than there promoting the use of these types of majorities whenever the consistency of the collective decision is a concern for the society or the policy maker. The paper is organized as follows. Section 2 describes the theoretical framework followed in this paper and introduces majorities based on difference in support. Section 3 is devoted to the method of simulation. Sections 4 and 5 discuss the results and Section 6 concludes. 2 Preliminaries Consider a set of alternatives X={x1, x2, x3}in an election with mindividuals. Let Sbe a binary relation on X, i.e. a subset of the Cartesian product X×X. In what follows, xiSxjstands for (xi, xj)∈S, i.e. when xiis in the relation Swith xj.S−1is the inverse relation of Sdefined by xiS−1xj⇔xjSxj and Scis the complement relation of Sdefined by xiScxj⇔ ¬(xiSxj). Given two binary relations Sand T, the intersection of Sand Tis also a binary relation defined by xi(S∩T)xj⇔(xiSxj∧xiTxj). A binary relation Son X is 1. reflexive if ∀x∈X, xSx, 2. symmetric if ∀xi, xj∈X, xiSxj⇒xjSxi, 3. asymmetric if ∀xi, xj∈X, xiSxj⇒ ¬(xjSxi), 4. antisymmetric if ∀xi, xj∈X, (xiSxj∧xjSxi)⇒xi=xj, 5. complete if ∀xi, xj∈X, xiSxj∨xjSxi, 6. transitive if ∀xi, xj, xl∈X, (xiSxj∧xjSxl)⇒xiSxl, 7. triple-acyclic if ∀xi, xj, xl∈X, (xiSxj∧xjSxl)⇒ ¬(xlSxi). Aweak preference Ris a complete binary relation on the set of alternatives X. The strict preference Passociated with Ris the asymmetric binary relation on Xdefined by P= (R−1)cand the corresponding indifference relation Iis the reflexive and symmetric binary relation on Xdefined 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 antisymmetric. From definitions above it is well known that any weak ordering implies 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. 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 outcomes. Title Suppressed Due to Excessive Length 5 Table 1 Possible social outcomes in a three-alternative election. 1. x1Px2x2Px3x1Px310. x1Ix2x2P x3x1P x319. x1Ix2x2P x3x1Ix3 2. x1Px3x3Px2x1Px211. x2Ix3x3P x1x2P x120. x1P x2x2P x3x1Ix3 3. x2Px1x1Px3x2Px312. x1Ix3x3P x2x1P x221. x3P x1x1P x2x2Ix3 4. x2Px3x3Px1x2Px113. x1Ix2x2Ix3x1Ix322. x2P x3x3P x1x1Ix2 5. x3Px1x1Px2x3Px214. x1Px2x2Ix3x1Ix323. x3P x2x1P x3x1Ix2 6. x3Px2x2Px1x3Px115. x1Ix2x2Ix3x1P x324. x2P x1x1P x3x2Ix3 7. x1Px2x2Ix3x1P x316. x1Ix2x2Ix3x3P x125. x3P x2x2P x1x1Ix3 8. x2Px1x1Ix3x2P x317. x1Ix3x3P x2x1Ix226. x1P x2x2P x3x3P x1 9. x3Px1x1Ix2x3P x218. x2P x1x1Ix3x2Ix327. x2P x1x1P x3x3P x2 Our interest focuses on the frequency of consistent social outcomes given the 27 possible outcomes above. We distinguish among three cases of consistent outcomes; the case of weak orderings corresponding to the first thirteen outcomes, 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 outcomes. 2.1 Individual preferences We consider that individuals compare the alternatives on Xby pairs and declare their preferences by means of values rp ij which belong to the unit interval [0,1] with the following interpretation: rp ij >0.5 indicates that the individual p prefers the alternative xito the alternative xj, the more the nearer is the value of rp ij to 1 that represents the maximum degree of preference for xiover xj; conversely, rp ij <0.5, means that individual pprefers alternative xjto xi, the more the nearer is the value of rp ij to 0 that represents the maximum degree of preference for xjover xi; finally, rp ij = 0.5 stands for the indifference between xiand xjfor individual p. The reciprocity of these preferences is described by the condition rp ij +rp ji = 1. As noted in the Introduction, to avoid the possibility of having incoherent individual preferences, we need to assume some kind of rationality condition. But, in this framework, several concepts could be taken to ensure such rationality requirement (see, among others, Dasgupta and Deb 1996; Dubois and Prade 1980; Garc´ıa-Lapresta and Meneses 2005; Zadeh 1971). Here, we consider the following transitivity conditions for reciprocal preference relations. Definition 1 We say that individual pis 1. 0.5–transitive if ∀i, j, l ∈ {1,2,3} (rp ij >0.5∧rp jl >0.5) ⇒rp il >0.5, 2. min–transitive if ∀i, j, l ∈ {1,2,3} (rp ij >0.5∧rp jl >0.5) ⇒rp il ≥min{rp ij, rp jl}, 6 3. am–transitive if ∀i, j, l ∈ {1,2,3} (rp ij >0.5∧rp jl >0.5) ⇒rp il ≥rp ij +rp jl/2, 4. max–transitive if ∀i, j, l ∈ {1,2,3} (rp ij >0.5∧rp jl >0.5) ⇒rp il ≥max{rp ij, rp jl}. The preferences of each individual over the alternatives in X={x1, x2, x3} can be represented using a 3 ×3 matrix Rp=Ärp ijäas follows: Rp=Ö0.5rp 12 rp 13 1−rp 12 0.5rp 23 1−rp 13 1−rp 23 0.5è.(1) Individual preferences are collected in a vector where each vector-element represents the preferences of an individual. Assuming mindividuals1a profile of reciprocal preferences is a vector (R1, . . . , Rm)∈ R(X)m, where R(X) the set of all reciprocal preference relations. 2.2 Majorities based on difference in support In Garc´ıa-Lapresta and Llamazares (2010), majorities based on difference in support were introduced in the framework 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 Pkover 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. Formal definition for these majorities is as follows. Definition 2 (Majorities based on difference in support or f Mkmajorities Garc´ıa-Lapresta and Llamazares (2010)) Given k∈[0, m), the majority based on difference in support or f Mkmajority is the mapping f Mk:R(X)m−→ P(X) defined by f Mk(R1, . . . , Rm) = Pk, where xiPkxj⇔ m X p=1 rp ij > m X p=1 rp ji +k. (2) Using the fact that rp ij = 1 −rp ji for each voter p, (2) is equivalent to: xiPkxj⇔Pm p=1 rp ij m>0.5 + k 2m.(3) 1To calculate the probabilities presented here, mtakes the following values: 3, 4, 5, 10, 100, 1,000 and 100,000. Title Suppressed Due to Excessive Length 7 The indifference relation associated with Pkis defined by: xiIkxj⇔      m X p=1 rp ij − m X p=1 rp ji      ≤k. (4) Putting the relation rp ij = 1 −rp ji into (4), one obtains: xiIkxj⇔    Pm p=1 rp ij m−0.5    ≤k 2m.(5) Example 1 Let RIand RII be the following reciprocal preference relations over the alternatives on X={x1, x2, x3}. RI=Ñ0.5 1 0.9 0 0.5 0.6 0.1 0.4 0.5é, RII =Ñ0.5 0.8 1 0.2 0.5 0.7 0 0.3 0.5é. Consider the profile (R1, R2, R3, R4, R5) where Rp=®RIif p= 1,2,3, RII if p= 4,5. Assuming a required difference in support kequal to 1.75 and applying the corresponding f M1.75 majority we have 5 X p=1 rp 12 = 4.6> 5 X p=1 rp 21 + 1.75 = 0.4+1.75 ⇒x1P1.75 x2,       5 X p=1 rp 23 − 5 X p=1 rp 32      =|3.2−1.8| ≤ 1.75 ⇒x2I1.75 x3, 5 X p=1 rp 13 = 4.7> 5 X p=1 rp 31 + 1.75 = 0.3+1.75 ⇒x1P1.75 x3. As noticed in the Introduction, in the case of crisp preferences, given a pair of alternatives, individuals declare if they prefer an alternative to another one or if they are indifferent between them. In other words, the values of rp ij are restricted to the set of discrete values {0,0.5,1}. If rp ij = 1, individual p prefers alternative xito alternative xj, whereas if rp ij = 0, individual pprefers xjto xi. If rp ij = 0.5, individual pis indifferent between both alternatives. In this framework, the concept of majorities based on difference of votes was introduced in Garc´ıa-Lapresta and Llamazares (2001) and was later axiomatically characterized in Llamazares (2006), and subsequently in Houy (2007). Under these majorities, an alternative, say xi, is declared the winner if the number of individuals who prefer that alternative, to the other one, say xj, 8 exceeds the number of individuals who prefer xjto xiin a difference of votes, fixed before the election process. With mindividuals, that difference could take any integer value in {0, . . . , m −1}. It is clear that these majorities are located between simple majority rule when the difference of votes is zero and unanimity when the difference of votes is the total number of individuals m minus one. Assuming weak or linear individual orderings and using the well-known Impartial Anonymous Culture (IAC) condition (Gehrlein and Fishburn 1976), calculations of the probabilities of consistent outcomes (transitivity and tripleacyclicity) under majorities based on difference in votes have been conducted in Diss and P´erez-Asurmendi (2015). The IAC condition assumes that all voting situations are drawn independently and uniformly given that each voting situation indicates a specific combination of the number of voters associated with each order of the alternatives. The objective of Diss and P´erez-Asurmendi (2015) was twofold. First, the needed thresholds which guarantee that the probability of consistent outcomes is close to 1 have been found. Second, the authors have set forth the impact of weak orderings and linear orderings for the individual preferences on the probability of consistent outcomes. It is shown for instance that in the cases of transitive and triple-acyclic strict preferences, the probabilities are higher considering weak than linear orderings. In the present paper, we extend the study in Diss and P´erez-Asurmendi (2015) to the majorities based on difference in support f Mk. However, as long as the intensities of preference between each pair of alternatives can take any value in the continuous interval [0,1], the IAC model can not be applied. Our difficulty is that there is not any equivalent model to IAC in the framework of reciprocal preferences since the set of voting situations is not numerable. In other words, the probabilistic analysis carried out in Diss and P´erez-Asurmendi (2015) turns impossible to study in the case of majorities based on difference in support f Mk. Consequently, we perform a computer simulation to estimate these probabilities. Our simulation method will follow the same spirit as the Impartial Culture (IC) condition (Guilbaud 1952), a well-known model which considers the set of all preference profiles as a sample space and where a voter preference profile identifies the specific preference ranking that each voter has for the candidates. Notice that individual voter’s preferences are not anonymous under IC condition while they are under IAC assumption. 3 Simulation method In this section, we detail the simulation method used in this paper in order to provide the probabilities of reaching consistent collective decisions under f Mk majorities. Our probabilities are estimated as the proportion of the number of consistent outcomes in the simulation over the total number of simulated Title Suppressed Due to Excessive Length 9 outcomes. We generate 100,000 outcomes to guarantee our results with a confidence level of 99% and a sampling error of less than a 0.0041%2. In the following, we describe the methodology applied in the simulations to estimate the probability for the considered three types of consistent collective decisions under f Mkmajorities, i.e. transitive weak preferences, transitive and triple-acyclic strict preferences. We follow that scheme taking into account each type of individual transitive reciprocal relations, i.e. 0.5–transitive, min– transitive, am–transitive and max–transitive reciprocal preference relations. Notice that the matrix in (1) representing a reciprocal preference relation is determined by the vector composed of the intensities r12, r23 and r13. 1. At the beginning of the evaluation, we randomly generate mvectors representing the reciprocal preferences of the mindividuals. For each individual, each component of the triplet (r12, r23, r13) is drawn from an uniform distribution on [0; 1]. Such vectors are built bearing in mind one of the considered transitivity conditions for reciprocal preference relations. In other words, if the resulting individual reciprocal preference (r12, r23, r13) is not transitive for the function under consideration (i.e. 0.5–transitivity, min–transitivity, am–transitivity and max–transitivity), it is rejected. This first step ends when exactly mindividual transitive reciprocal preferences are generated. 2. We compute the sum of the individuals’ intensities of preference over each pair of alternatives relative to the number of voters mthrough a vector S= (S12 m,S23 m,S13 m) where Sij = m P p=1 rp ij. 3. Having in mind the conditions in equations (3) and (5) and the value of the threshold a=k 2m, the collective decision is evaluated over each pair of alternatives in the vector S. 4. The collective decision in Sis classified following the cases of possible collective outcomes displayed in Table 1. For instance, the case 27 corresponds to S12 m<0.5−a,S23 m<0.5−a, and S13 m>0.5 + a. Indeed, if the collective decision is one of the cases 26 or 27, the strict preference Pkis not triple-acyclic. If it is one of the cases from 19 to 27, the strict preference Pkis not transitive. Finally, if it is one of the cases from 14 to 27, the weak preference Rkis not transitive. 2Assuming a proportion of consistent outcomes Pon the population of a 50%, the proportion pin a random sample of size n≥30 for a confidence level of 99%, diverges from the one of the population in an error of less than : Prob(|P−p| ≤ )≥0.99. Taken into account that the sample proportion pis distributed as NÄP, pP(1 −P)/nä, the sampling error is as follows: =zα/2pP(1 −P)/n. In our case, n= 100,000 and the corresponding percentile of the normal distribution for a confidence level of 99% is zα/2= 2.57. Thus, ≤0.00407. 16 2. The transitivity of the strict preference can not be ensured for any threshold of support kless than mif the reciprocal preference relations are less demanding than am–transitive ones. 3. The strict preference is transitive for any threshold of support such that k∈ [m−1, m) if the reciprocal preference relations are at least am–transitive ones. On the other hand, the case of triple-acyclic strict preferences is analysed in Llamazares and P´erez-Asurmendi (2015) with the following results: 1. The triple-acyclicity of the strict preference, in the case of 0.5–transitive reciprocal preference relations, can be guaranteed if the threshold of support kis located in b2m/3c, mwhere bacstands for the integer part of a. 2. The triple-acyclicity of the strict preference, in the case of min–transitive and max–transitive reciprocal preference relations, can be guaranteed if the threshold of support kbelongs to [m/3, m). 3. In the case of max–transitive reciprocal preference relations, it conjectures that strict preference relations are triple-acyclic if the threshold kbelongs to b2m/3c/2, m. The probabilistic results setting here complement the above theoretical ones by the following reasons. First, thresholds with associated probabilities of consistent strict preferences equal to 1 are found for all the considered types of transitive reciprocal preference relations. Second, reasonable thresholds are required to certify the consistency of the strict preference with a probability value of 1 in those cases where theoretical results asked a very high threshold to guarantee such consistency. Third, the conjecture about the needed thresholds in the case of max–transitive reciprocal preference relations seems to be true. Specifically, in the case of transitive strict preferences with 0.5–transitive and min–transitive reciprocal preference relations, the probabilities achieve the value of 1 for the considered values of m(see Table 9) whereas as it is said before, the theoretical result asserts that no threshold guarantees the transitivity of the strict preference for such types of reciprocal preference relations. Table 9 Thresholds ksuch that the simulated probability of Pkbeing transitive is equal to 1 with 0.5– and min–transitive reciprocal preference relations. m= 3 m= 4 m= 5 m= 10 m= 100 m= 1,000 k(0.5–trans.) 2.58 2.97 3.24 5.17 14.51 47.79 k(min–trans.) 1.96 2.49 2.61 3.47 12.51 35.24 In the cases of am–transitive and max–transitive reciprocal preference relations, the thresholds that provide a probability value of transitive strict preference relations equal to 1 are lower than the ones that guarantee the transitivity of the strict preference in the theoretical framework. Table 10 shows Title Suppressed Due to Excessive Length 17 the theoretical minimum threshold required, i. e. m−1, and the thresholds that provide a simulated probability value equal to 1. To illustrate, look at the case of m= 1,000. The theoretical result asserts that the threshold khas to belong to [999,1,000). By contrast, a probability value of 1 is achieved with a threshold of 33.10 in the case of am-transitive reciprocal preference relations and of 26.10 in the case of max–transitive reciprocal preference relations. Table 10 Theoretical threshold kvs. thresholds ksuch that the simulated probability of Pk being transitive is equal to 1 with am– and max–transitive reciprocal preference relations. m= 3 m= 4 m= 5 m= 10 m= 100 m= 1,000 m−1 2 3 4 9 99 999 k(am–trans.) 1.56 1.85 2.07 3.10 9.17 33.10 k(max–trans.) 1.51 1.66 1.93 2.78 9.72 26.10 In the case of triple-acyclic strict preferences, the thresholds to reach a probability value of 1 again are much lower than the ones required in the theoretical setting. Tables 11, 12 and 13 illustrate that fact in the cases of 0.5– and min–transitive reciprocal preference relations. In these cases we set forth the minimum theoretical threshold required and the thresholds that provide a simulated probability value equal to 1. Table 11 Theoretical threshold kvs. thresholds ksuch that the simulated probability of Pkbeing triple-acyclic is equal to 1 with 0.5–transitive reciprocal preference relations. m= 3 m= 4 m= 5 m= 10 m= 100 m= 1,000 b2m/3c2 2 3 3 66 666 k1.32 1.50 1.91 2.65 8.29 26.60 Table 12 Theoretical threshold kvs. thresholds ksuch that the simulated probability of Pkbeing triple-acyclic is equal to 1 with min–transitive reciprocal preference relations. m= 3 m= 4 m= 5 m= 10 m= 100 m= 1,000 m/3 1 1.Û3 1.Û6 3.Û3 33.Û3 333.Û3 k0.67 0.88 0.90 1.40 4.56 14.99 The case of max–transitive reciprocal preference relations deserves a special attention. In Llamazares and P´erez-Asurmendi (2015), authors conjecture that the needed threshold to guarantee the triple-acyclicity of the collective 18 strict preference when individuals are endowed with max–transitive reciprocal relations is lower or equal3than in the case of individual min–transitive reciprocal relations. Unfortunately, they could not provide a formal proof for that assertation. Comparing the thresholds ksuch that the simulated probability of Pkis triple-acylic with a probability of 1 from Tables 12 and 13, we can corroborate that conjecture. The thresholds in the case of max–transitive reciprocal preference relations are smaller or equal than the ones in min–transitive case. Table 13 Theoretical threshold kvs. thresholds ksuch that the simulated probability of Pkbeing triple-acyclic is equal to 1 with max–transitive reciprocal preference relations. m= 3 m= 4 m= 5 m= 10 m= 100 m= 1,000 b2m/3c/2, m1 1 1.5 3 33 333 k0.67 0.72 0.75 0.99 3.58 9.95 A quite natural question is to wonder if the consideration of reciprocal preference relations has an impact on the probability of consistent decisions when compared with the case of crisp preferences, i.e. when voters only declare if they prefer an alternative to another one or if they are indifferent between both alternatives. To highlight this idea, we consider two cases: First, we focus on voters who show linear preference orderings on the candidates. In other words, rp ij ∈ {0,1} for each pair of alternatives xiand xjand each voter p. Second, we also consider the case where individuals are endowed with weak preference orderings and declare their preferences by means of values rp ij ∈ {0,0.5,1}. If rp ij = 0.5, individual pis indifferent between both alternatives. Notice that in the first case, these majorities are equivalent to supermajorities (see for a formal proof of that Diss and P´erez-Asurmendi (2015)); in the second case, these voting rules are majorities based on difference of votes. For that aim, we find the thresholds which guarantee a probability equal to 1 of Pkbeing triple-acyclic (Table 14), Pkbeing transitive (Table 15), and Rkbeing transitive (Table 16) taking into account individual linear and weak orderings and compare them with the corresponding ones when considering individual reciprocal preference relations. 3Notice that m/3 = b(2 ·100)/3c/2 when mis multiple of three. Title Suppressed Due to Excessive Length 19 Table 14 Crisp preferences versus reciprocal preferences: Thresholds ksuch that the simulated probability of Pkbeing triple-acyclic is equal to 1. m↓ Crisp preferences Reciprocal preferences rp ij ∈ {0,1}rp ij ∈ {0,0.5,1}0.5–trans. min–trans. am–trans. max–trans. 3 1 1 1.32 0.67 0.56 0.67 4 0 1 1.50 0.88 0.67 0.72 5 1 1 1.91 0.90 0.83 0.75 10 2 2 2.65 1.40 1.11 0.99 100 10 8 8.29 4.56 3.34 3.58 1000 36 30 26.60 14.99 11.68 9.95 Table 15 Crisp preferences versus reciprocal preferences: Thresholds ksuch that the simulated probability of Pkbeing transitive is equal to 1. m↓ Crisp preferences Reciprocal preferences rp ij ∈ {0,1}rp ij ∈ {0,0.5,1}0.5–trans. min–trans. am–trans. max–trans. 3 1 1 2.58 1.96 1.56 1.51 4 2 2 2.97 2.49 1.85 1.66 5 3 3 3.24 2.61 2.07 1.93 10 6 7 5.17 3.47 3.10 2.78 100 26 21 14.51 12.51 9.17 9.72 1000 78 89 47.79 35.24 33.10 26.10 Table 16 Crisp preferences versus reciprocal preferences: Thresholds ksuch that the simulated probability of Rkbeing transitive is equal to 1. m↓ Crisp preferences Reciprocal preferences rp ij ∈ {0,1}rp ij ∈ {0,0.5,1}0.5–trans. min–trans. am–trans. max–trans. 3 - - 2.97 2.96 2.97 2.98 4 - - 3.81 3.84 3.91 3.84 5 - - 4.70 4.80 4.91 4.72 10 - - 7.95 8.32 7.48 7.56 100 48 43 26.40 27.23 27.90 25.47 1000 147 172 84.83 95.32 83.01 80.00 We can deduce the following facts: - For big electorates, required thresholds are lower when considering reciprocal preference relations than in the case of crisp preferences with independence of the analysed collective consistent decision (see the case of m= 1,000 in Tables 14, 15, and 16). 20 - In the case of the probability of Rkbeing transitive equal to 1 (Table 16), we find thresholds to reach that probability when considering reciprocal preference relations for all the considered number of individuals whilst in the case of crisp preferences we cannot provide some of them. Notice that, even for a threshold equal to k=m−1, the probability of transitive Rkis non null when considering crisp preferences. For instance, for m= 10 and a threshold k= 9, the probability of transitive Rkis equal to 0.9944 and 0.9993 for rp ij ∈ {0,1}and rp ij ∈ {0,0.5,1}, respectively. - In the case of the probability of Pkbeing triple-acyclic equal to 1 (Table 14), we find lower thresholds for min–transitive, am–transitive and max–transitive reciprocal preference relations than for crisp preferences. - The case of the probability of Pkbeing transitive equal to 1 (Table 15) is more complex to analyse. On the one hand, the thresholds are lower when considering crisp preferences and m= 3. On the other hand, reciprocal preferences provide lower thresholds for bigger electorates than that. Specifically, greater than m= 3 when considering am–transitive and max– transitive reciprocal preferences relations, greater than m= 4 in the case of min–transitive reciprocal preference relations and greater than m= 10 when regarding 0.5–transitive reciprocal preference relations. Therefore, we can ensure that considering reciprocal preference relations has an unequivocal effect on the needed thresholds to reach a probability of collective consistent decisions equal to 1 when considering big electorates. Finally, we set forth the probabilities of consistent collective decisions with crisp preferences and the ones with reciprocal preferences for a given number of voters mand a given threshold k. Now our goal is to know if the likelihood of consistent collective decisions is higher or smaller when the voters show their preferences reciprocally than in the case where they are endowed with linear or weak orderings. In Table 17, we set the probabilities when m= 3 and m= 1,000 with the aim of distinguishing between the cases of small and big electorates. Title Suppressed Due to Excessive Length 21 Table 17 The impact of the individual type of preferences in the probability. The examples of m= 3 and m= 1,000. m= 3 m= 1,000 k= 0 k= 1 k= 2 k= 3 k= 4 k= 10 rp ij ∈ {0,1} Rktransitive 0.9511 0.6772 0.6908 0.8739 0.9004 0.8047 Pktransitive 0.9511 1 1 0.8907 0.9160 0.9137 Pktriple-acyclic 0.9511 1 1 0.9492 0.9643 0.9897 rp ij ∈ {0,0.5,1} Rktransitive 0.8736 0.6943 0.7827 0.9399 0.9339 0.8893 Pktransitive 0.9599 1 1 0.9442 0.9411 0.9272 Pktriple-acyclic 0.9944 1 1 0.9727 0.9761 0.9910 0.5–trans. Rktransitive 0.8835 0.5438 0.8883 0.7816 0.7479 0.5794 Pktransitive 0.8835 0.9587 0.9994 0.8267 0.8241 0.8767 Pktriple-acyclic 0.8835 0.9997 1 0.9328 0.9477 0.9905 min–trans. Rktransitive 0.9563 0.6551 0.8978 0.8802 0.8534 0.7092 Pktransitive 0.9563 0.9950 1 0.9245 0.9262 0.9659 Pktriple-acyclic 0.9563 1 1 0.9827 0.9886 0.9996 am–trans. Rktransitive 0.9751 0.7036 0.8913 0.9113 0.8883 0.7640 Pktransitive 0.9751 0.9988 1 0.9536 0.9572 0.9857 Pktriple-acyclic 0.9751 1 1 0.9923 0.9959 1.0000 max–trans. Rktransitive 0.9777 0.7102 0.9015 0.9184 0.8971 0.7726 Pktransitive 0.9777 0.9993 1 0.9609 0.9657 0.9904 Pktriple-acyclic 0.9777 1 1 0.9945 0.9970 1 For a small electorate (m= 3) we have found the following: - For k= 0, the probabilities of transitive Rkare greater when considering min–transitive, am–transitive and max–transitive reciprocal preference relations than when considering crisp and 0.5–transitive reciprocal preferences. The probabilities of transitive Pkare also greater bearing in mind am–transitive and max–transitive reciprocal preference relations than any other considering preferences. Nevertheless, the probability of triple-acyclic Pkis greater in the case of weak orderings than in the other possible cases. - For k= 1, the probabilities of transitive Rkare greater when regarding am– transitive and max–transitive reciprocal preference relations than when considering crisp preferences and these last ones are also greater than when taking into account min–transitive and 0.5–transitive reciprocal preference relations. The probabilities of transitive Pkare greater when regarding crisp preferences than when considering reciprocal preference relations. In the case of the probabilities of triple-acyclic Pk, crisp preferences and reciprocal preference relations provide same probabilities with the exception of the case of 0.5–reciprocal preference relations attached with a lower probability. - For k= 2, the probabilities of consistent outcomes are greater or equal when considering reciprocal preference relations than when regarding crisp preferences with the exception of the probability of transitive Pkwhen having 0.5 reciprocal preference relations. For a big electorate (m= 1,000) we have the following: 22 - For k= 3 the lowest probabilities of consistent outcomes are reached when considering 0.5–transitive reciprocal preference relations. The probabilities of transitive Rkare greater when considering weak orderings than max– transitive, am–transitive, min–transitive reciprocal preference relations and those ones are also greater than when regarding linear orderings. The probabilities of transitive Pkare greater when considering am–transitive and max–transitive reciprocal relations than when regarding weak orderings. These last ones are greater than when considering min–transitive reciprocal relations which in turn are also greater than when regarding linear orderings. In the case of the probabilities of triple-acylcic Pk, the greatest ones corresponds to max–transitive, am–transitive and min–transitive reciprocal preference relations followed by the ones attached to crisp preferences. - For k= 4 the probabilities of transitive Rkare greater when considering crisp than reciprocal preference relations. The probabilities of transitive and triple-acyclic Pkbehave in the same way than in the case of k= 3 analysed above. - For k= 10 the probabilities of transitive Rkare greater when considering crisp than reciprocal preference relations. The probabilities of transitive Pkare greater when considering max–transitive, am–transitive and min– transitive reciprocal preference relations than when considering crisp preferences. The lowest probabilities are reached when regarding 0.5–transitive reciprocal preference relations. The probabilities of triple-acylcic Pkare greater when considering reciprocal preference relations than when regarding crisp preferences. Therefore, we can not draw a clear conclusion about the relationship between the probabilities and the considered type of individual preferences. For instance, 0.5–transitive reciprocal preference relations give rise to the lowest probabilities of consistent outcomes except in the cases of triple-acyclic Pk with k= 2 for small electorates and with k= 10 for big electorates, respectively. For big electorates weak orderings provide greater probabilities of transitive Rkthan the remaining considered individual preferences. Instead, in the case of small electorates, am–transitive and max–transitive reciprocal preference relations promote the greatest probabilities. Analysing the case of the probabilities of transitive Pkconsidering big electorates, the greatest ones are reached when regarding max–transitive and am–transitive reciprocal preference relations. That is also the case for m= 3 when considering k= 0 but not for the other considered values of k(notice that in the case of k= 1 crisp preferences give rise to the highest probabilities and in the case of k= 2 the probabilities are the same for all the considered individual preferences with the exception of 0.5–transitive reciprocal preference relations). Finally, when regarding the probabilities of triple-acyclic Pkwe found that the greatest probabilities are attached to weak orderings when m= 3 whereas Title Suppressed Due to Excessive Length 23 to max–transitive, am–transitive and min–transitive reciprocal preference relations when m= 1,000. 6 Conclusion In this paper, we estimate through simulations the probabilities of consistent preferences under majorities based on difference in support: the probability of having transitive weak preferences Rk, the probability of having transitive strict preferences Pk, and the probability of having triple-acyclic strict preferences Pk. We start our simulations by assuming four types of transitivity conditions for individual reciprocal preferences: 0.5-transitivity, min-transitivity, am-transitivity, and max-transitivity. This paper contains new contributions to the literature of reciprocal preferences. First, we have found the thresholds which guarantee that the probability of consistent outcomes is equal to 1. These thresholds are much lower than the ones required in the theoretical setting. In fact, our results show a dramatic difference between these two thresholds: For instance, in some cases, the simulated threshold represents less than a 3% of the total number of voters whereas the theoretical one reaches the 67%. Given that the needed thresholds to reach consistent decisions with probability equal to 1 are not too demanding, the implementation of majorities based in difference of support have sense whenever the flexibility attached to individual preferences by the consideration of reciprocal preference relations and the consistency of the collective decision is a concern for the policy maker or for the society as a whole. Second, we have set forth the impact of the threshold kon the probability of consistent outcomes for a given number of voters. The probability of having transitive weak preferences Rkas well as the probability of having transitive strict preferences Pkexhibit a similar behaviour: for the lowest values of the threshold k, it decreases as the value of the threshold kincreases whereas for higher values of k, it increases with the value of the threshold k. However, the probability of having triple-acyclic strict preferences Pkincreases when the value of the threshold kdoes, for any number of voters. Third, we have compared the needed thresholds to reach a probability of collective consistent decisions equal to 1 regarding reciprocal preference relations with the ones considering crisp preferences and concluded that the thresholds are significantly lower in the case of reciprocal preference relations than in the case of crisp preferences. Finally, we have studied if the consideration of reciprocal preference relations instead of crisp preferences has an impact in the probability of collective consistent decisions. We have found that the probability depends on the concrete considered individual preference relation and on the size of the electorate. 24 Appendix Table A.1 Probabilities of transitive Rkfor 0.5–transitive reciprocal preference relations. m→3 4 5 10 100 1,000 100,000 k↓ 0 0.8835 0.8781 0.8763 0.8751 0.8706 0.8705 0.8728 2.97 1 0.9836 0.9485 0.7630 0.5927 0.7826 0.8655 3.81 1 0.9964 0.9065 0.5481 0.7544 0.8633 4.70 1 0.9748 0.5312 0.7248 0.8609 7.95 1 0.6729 0.6266 0.8518 26.40 1 0.7012 0.7954 84.83 1 0.6156 95.32 0.5913 Table A.2 Probabilities of transitive Rkfor min–transitive reciprocal preference relations. m→3 4 5 10 100 1,000 100,000 k↓ 0 0.9563 0.9513 0.9505 0.9487 0.9468 0.9476 0.9455 2.96 1 0.9831 0.9528 0.8015 0.7220 0.8814 0.9400 3.84 1 0.9966 0.9205 0.6776 0.8576 0.9383 4.80 1 0.9809 0.6552 0.8307 0.9366 8.32 1 0.7559 0.7432 0.9291 27.23 1 0.7702 0.8831 95.32 1 0.7189 Table A.3 Probabilities of transitive Rkfor am–transitive reciprocal preference relations. m→3 4 5 10 100 1,000 100,000 k↓ 0 0.9751 0.9710 0.9712 0.9694 0.9676 0.9670 0.9692 2.97 1 0.9812 0.9492 0.8087 0.7738 0.9117 0.9650 3.91 1 0.9971 0.9225 0.7295 0.8904 0.9636 4.91 1 0.9813 0.7043 0.8683 0.9622 7.48 1 0.7370 0.8112 0.9582 27.90 1 0.7873 0.9180 83.01 1 0.7967 95.32 0.7731 Title Suppressed Due to Excessive Length 25 Table A.4 Probabilities of transitive Rkfor max–transitive reciprocal preference relations. m→3 4 5 10 100 1,000 100,000 k↓ 0 0.9777 0.9766 0.9751 0.9741 0.9727 0.9726 0.9744 2.98 1 0.9842 0.9561 0.8219 0.7867 0.9189 0.9708 3.84 1 0.9964 0.9241 0.7450 0.9003 0.9696 4.72 1 0.9782 0.7188 0.8805 0.9681 7.56 1 0.7495 0.8190 0.9637 25.47 1 0.7720 0.9298 80.00 1 0.8030 95.32 0.7744 Table A.5 Probabilities of transitive Pkfor 0.5–transitive reciprocal preference relations. m→3 4 5 10 100 1,000 100,000 k↓ 0 0.8835 0.8781 0.8763 0.8751 0.8706 0.8705 0.8728 2.58 1 0.9999 0.9995 0.9916 0.8512 0.8287 0.8668 2.97 1 1.0000 0.9970 0.8661 0.8268 0.8659 3.24 1 0.9985 0.8759 0.8257 0.8654 5.17 1 0.9465 0.8267 0.8611 14.51 1 0.9295 0.8440 47.79 1 0.8293 Table A.6 Probabilities of transitive Pkfor min–transitive reciprocal preference relations. m→3 4 5 10 100 1,000 100,000 k↓ 0 0.9563 0.9513 0.9505 0.9487 0.9468 0.9476 0.9455 1.96 1 0.9999 0.9997 0.9968 0.9377 0.9271 0.9421 2.49 1 1.0000 0.9993 0.9514 0.9255 0.9414 2.61 1 0.9995 0.9533 0.9254 0.9413 3.47 1 0.9704 0.9248 0.9398 12.51 1 0.9793 0.9291 35.24 1 0.9251 47.79 0.9311