AN ABSTRACT OF THE THESIS OF

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2 AN ABSTRACT OF THE THESIS OF Leslie M. McDonald for the degree of Master of Science in Mathematics presented on March, 205. Title: robabilities of Voting aradoxes with Three Candidates Abstract approved: Mina E. Ossiander ardoxes in voting has been an interest of voting theorists since the 800 s when Condorcet demonstrated the key example of a voting paradox: voters with individually transitive rankings produce an election outcome which is not transitive. With Arrow s Impossibility Theorem, the hope of finding a fair voting method which accurately reflected society s preferences seemed unworkable. Recent results, however, have shown that paradoxes are unlikely under certain assumptions. In this paper, we corroborate results found by Gehrelin for the probabilities of paradoxes, but also give results which indicate paradoxes are extremely likely under the right conditions. We use simulations to show there can be many situations where paradoxes can arise, dependent upon the variability of voters preferences, which echo Saari s statements on the topic.

3 c Copyright by Leslie M. McDonald March, 205 All Rights Reserved

4 robabilities of Voting aradoxes with Three Candidates by Leslie M. McDonald A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science resented March, 205 Commencement June 205

5 Master of Science thesis of Leslie M. McDonald presented on March, 205. AROVED: Major rofessor, representing Mathematics Chair of the Department of Mathematics Dean of the Graduate School I understand that my thesis will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request. Leslie M. McDonald, Author

6 ACKNOWLEDGEMENTS Academic I would like to thank my friends and professors at the University of South Alabama for urging me to pursue graduate mathematics and thesis work. Much thanks also goes to my advisor Mina Ossiander for her excellent suggestions and her patience. Last, but not least, thanks to all my mathematical friends at Oregon State University for helping me out along the way and generally being superb mathematicians. ersonal I thank my parents for creating opportunities for me. My Mother has ever been my biggest supporter. I would not be where I am without her encouragement and guidance, and my Father s natural diligence in his endeavors inspires me to continue. Thanks to the rest of the world for thinking I must be a genius to be doing mathematics.

7 TABLE OF CONTENTS age Introduction. History Basic Definitions and Setting Recent Results and Methods Notation and Methods 2. Key Definitions Central Limit Theorem Approximations Simulation methods Simple Examples The probability that A wins by the pairwise method Condorcet s Example Single-peaked profiles Results 28. Main Analyses lurality Winner is the airwise Loser lurality Winner is Borda Loser Summary of Results Conclusion Further Work Appendix 56 Bibliography 66

8 Figure LIST OF FIGURES age 2. Limiting behavior in simulation when p k = 6 for all k robability of A being the pairwise winner under conditions p +p 2 +p 4 = 2 and p + p 2 + p = Limiting behavior in simulation when p k = 6 for all k Limiting behavior in simulation when p = p 2 = 0.20, p = p 4 = 0.05, p 5 = 0.5, and p 6 = Limiting behavior in simulation when p = p 2 = 0.67, p = p 4 = 0.05, p 5 = 0.284, and p 6 = Limiting behavior in simulation when p = p 2 = 6, p = p 4 = 20, and p 5 = p 6 = Limiting behavior in simulation when p = p 2 = 0.2, p = p 4 = p 5 = 0., and p 6 = Limiting behavior in simulation when p = p 2 = 0.2, p = p 4 = 0, and p 5 = p 6 = Limiting behavior in simulation when p = p 2 = 0.2, p = p 4 = 0.08, and p 5 = p 6 = Limiting behavior in simulation when p = p 2 = 0.2, p = p 4 = 0.07, and p 5 = p 6 = Limiting behavior in simulation when p = p 2 = 0.2, p = p 4 = 0.078, and p 5 = p 6 = Limiting behavior in simulation when p k = 6 for all k Limiting behavior in simulation when p = 0.2, p 2 = 0.5, p = 0., p 4 = 0.09, p 5 = 0.24, and p 6 = Limiting behavior in simulation when p = 0.28, p 2 = 0.07, p = 0., p 4 = 0.09, p 5 = 0.24, and p 6 = Limiting behavior in simulation when p = 0.29, p 2 = 0.06, p = 0., p 4 = 0.09, p 5 = 0.24, and p 6 =

9 Figure LIST OF FIGURES (Continued) age.4 Limiting behavior in simulation when p = p = 2, p 2 = p 5 = 4, and p 4 = p 6 = Limiting behavior in simulation when p = p = p 4 = p 6 = 5 6 and p 2 = p 5 = Limiting behavior in simulation when p = p 2 = 0.2, p = p 4 = 0.05, and p 5 = p 6 = 0.25, candidate A is the lurality winner and Borda loser Limiting behavior in simulation when p = p 2 = 0.2, p = p 4 = 0.05, and p 5 = p 6 = 0.25, candidate A is the lurality winner and airwise loser... 54

10 Table LIST OF TABLES age. Borda s example Calculating the pairwise tallies from Borda s example Summary of results from Gehrlein et al Condorcet s example lurality winner is the airwise loser, limiting probabilities for different constraint categories lurality winner is the airwise loser, limiting probabilities for different constraint categories lurality winner is the airwise loser, limiting probabilities for different constraint categories lurality winner is the Borda loser, limiting probabilities for different constraint categories lurality winner is the Borda loser, limiting probabilities for different constraint categories lurality winner is the Borda loser, limiting probabilities for different constraint categories lurality winner is the Borda loser, limiting probabilities for different constraint categories lurality winner is the Borda loser, limiting probabilities for different constraint categories

11 Chapter Introduction. History The analysis of how a group of individuals preferences are combined to create a collective decision is called social choice theory. Social choice theory is predominantly comprised of welfare economics and voting theory. The main question of social choice theorists in studying voting is whether there exists a voting method which accurately reflects the preferences of the voters. Mathematical analysis of this question began during the French Revolution, most notably with the writings of Jean-Charles de Borda, Marie Jean Antoine Nicholas Caritat (Marquis de Condorcet), and ierre-simon Marquis de Laplace. Condorcet introduced the method of pairwise voting and a criterion for choosing the winning candidate: The candidate who wins against any other candidate head to head should be the societal choice. He also discovered a troublesome effect that can occur with pairwise voting, that candidates can be ranked cyclically. For example, with three candidates {A, B, C}, there are situations where pairwise voting tallies could produce the ranking that puts A B, then B C, but also C A. Another voting strategy was suggested by Borda. Voters rank the candidates and tallies are calculated by assigning points to a candidate based on how a voter ranks them, then these points are totalled for each candidate. So for our three candidates {A, B, C}, if a voter has preference B A C, then B will get point, A will get half a point, and C will get no points. Both Condorcet and Laplace, among others, felt that the Borda rule showed promise as a fair voting method. Indeed, Borda s method of voting was used by the Academy of Sciences in France from 784 until 800. It was discontinued at the behest of new member Napolean Bonaparte []. Other mathematicians such as E. J. Nanson, Francis Galton, and the well-known fiction writer Rev. C. L. Dodgson, proposed other methods and mathematical analyses of voting theory. However, the first successful axiomatization of voting theory came from Kenneth Arrow. The well-known Arrow s Theorem (see Section.2 Theorem.) gives the surprising result that any voting method which satisfies his axioms must be a dictatorship. That is, Arrow s axioms are not consistent with each other, that is to say, there is no fair voting method which satisfies all of them [].

12 2 Alternatives to Arrow s axioms have been proposed most of which are kinds of restrictions on the voter profiles (a profile is a collection of all the voters preferences). Duncan Black showed that if the voter profile satisfies a condition known as single-peakedness, then all of Arrow s axioms are satisfied by simple majority rule []. Amaryta Sen gives a similar statement about value-restricted profiles [24]. Most recently, D. G. Saari proposed relaxing the condition of Independence of Irrelevant Alternatives to Intensity of Binary Independence. He shows that the Borda rule then satisfies these new axioms and is the only rule which does so. Saari also showed that the Borda rule satisfies Arrow s original axioms when a preference profile has a certain subset of voters removed. [2] In addition to Arrow s disturbing result, we also have paradoxes in voting. Borda illustrated a voting situation where the pairwise rule ranking C B A is totally reversed by the majority rule ranking A B C (see Table.). Condorcet gave an example where no positional voting method (such as the Borda rule and the plurality rule) elects the pairwise method winner [, 25]. Innumerable other examples can be created, as Saari shows in [2]. Studying voting paradoxes has been of interest since the beginnings of voting theory, but powerful mathematical tools were not used on the problem until the late 970 s. Such people as Richard Niemi, eter Fishburn, and William Gehrlein utilized probability, combinatorics, and computer algorithms. Their main results give probabilities for finding paradoxes, often with conditioning on the measure of single-peakedness and other similar parameters [20, 7, 4]. Work along this vein continues to recent years and is used extensively in this paper. In addition, we consider the vector space approach developed by Saari which illuminates underlying structure and completely explains paradoxes between voting methods [2]. Typically, Condorcet s pairwise criterion was used as a measuring stick against other voting methods to show how well they represented the voters preferences. Arguments made by Saari and others point out flaws with this pairwise voting method and present evidence for the Borda rule as the best measure of preferences [2]. Saari identifies subspaces of the space of profiles (the space of all possible ways a set of voters can have voting preferences) which are responsible for all discrepancies between voting rules. By using this vector space approach, an intuitive understanding of these paradoxes is brought to light. He also gives a number of theorems on the wild variability that can arise in profiles, of which the Borda rule is mostly exempt, further supporting the Borda rule [2, 2]. Recent results show that paradoxes are unlikely to show up when the number of candidates is small and the number of voters is large. However, as the number of candidates, or

13 alternative choices, gets bigger, with ten being considered big, discrepancies among different voting methods becomes more of an issue. Thus, it would seem that large elections like that of the US president are probably exempt from these specific paradoxes due to the large number of voters (but of course are susceptible to other types of manipulation). In many instances where a choice must be made amongst several alternatives by a small number of individuals, such as a university choosing a new faculty member or the judging of a sports competition, we should be wary of pardoxes. In these situations we see many complicated and varied voting methods which we should view with suspicion, and which, some would argue, should be replaced with the Borda rule. Our results indicate that paradoxes can occur with high probability with only three candidates if certain conditions occur..2 Basic Definitions and Setting Individuals in a group faced with making a choice amongst a set of alternatives are called voters, and those alternatives are called the candidates (other social choice situations can be analogously defined). A voter s transitive ranking of the candidates is called a voter preference, and the set of all the voters preferences is called the voter profile or also the voting situation. The voting method is a map from the voter profile to a voting tally or voting count. The voting method specifies how to assign points to each candidate based on a voter s preference, then the tally is found by summing up each candidate s votes, expressed as a vector ( A, B, C ). From the voting tally we get the ranking (more generally called the societal outcome) by simply listing the candidates from largest to smallest number of votes obtained. Table.: Borda s example Voter preference number of voters Voter preference number of voters A B C 0 C A B 0 A C B 5 B C A 4 B A C 0 C B A For example, take Borda s own example with three candidates shown in Table.. The Borda method (also called the Borda count) assigns one point to a candidate every time they are ranked first and 2 point every time they are ranked second. Using the Borda count, the voting tally is ( A, B, C ) = (5, 5.5, 7.5) which gives the ranking C B A. The lurality method assigns one point to a candidate every time they are ranked first.

14 4 Using this method the voting tally is then ( A, B, C ) = (5, 4, ) which gives the ranking A B C, the total reverse of the Borda ranking! Thus, one is led to the question, which voting method truly reflects the voters prefernces? The Borda count and the plurality count are each a kind of positional voting method. For three candidates, any positional procedure can be expressed as a vector (, λ, 0) = w λ which specifies that for each voter s preference, the first ranked candidate gets one point, the second ranked candidate gets 0 λ points, and the lowest ranked candidate gets zero points. Thus, the Borda count is represented by the vector w /2 and the plurality count is represented by w 0. Negative plurality count is represented by w, and is equivalent to assigning one point to the lowest ranked candidate, then ranking candidates based on the smallest number of points. Table.2: Calculating the pairwise tallies from Borda s example preference no. of voters preference no. of voters preference no. of voters A B 5 A C 5 B C 4 B A 7 C A 7 C B 8 B A C A C B The other voting method which features prominently in voting theory is the pairwise method. The winner of the pairwise method is the candidate who wins in every head to head contest with every other candidate. The pairwise loser is found similarly. The ranking is then found by comparing the rest of the two candidate subset plurality elections. For example, the pairwise ranking from Borda s example is found by comparing the plurality tallies for all six 2-candidate subsets, see Table.2. We see that the pairwise winner is C and the pairwise loser is A. The rankings C B and B A tell us that B is middle ranked. Clearly, the pairwise method is not a positional method, but it is the standard in the field for measuring whether a voting method is satisfactory, often dubbed the Condorcet criterion. The following famous theorem combines common sense restrictions on voters preferences and conditions that should define a voting method which accurately reflects voters preferences as a whole. The theorem, however, gives a rather unsatisfactory conclusion. It basically says that no fair voting method satisfies all of the conditions. Theorem.. Arrow s Impossibility Theorem[, 2] If the voting method used in an election with or more candidates satisfies the following

15 5 conditions, then it must be a dictatorship. societal outcome. That is, one voter s preference decides the. Each voter s rankings of the candidates form a complete, strict, transitive ranking. 2. There are no restrictions on how the voters can rank the candidates.. If all voters share the same ranking of a pair of candidates, then that should be the societal outcome. 4. The societal ranking of a particular pair of candidates depends only on how the voters rank that pair and not how they rank the other candidates. Condition is called individual rationality and means that a voter does not have a cyclic preference ordering such as A B C A. Condition 2 reflects freedom of choice, desirable in any democratic society. Condition is called areto and is equivalent to saying that if we have a new election and a certain individual ranks a candidate higher, with all other voters voting the same way as in the old election, the new social outcome should not then have that candidate ranked lower; the societal outcome should either not change or should also rank that candidate higher. Condition 4 is called the Independence of Irrelevant Alternatives. This condition reflects the belief that the pairwise voting method is the preferred method, that is, if a method satisfies this condition, it will give the same ranking as the pairwise method. The theorem gives the irritating result that no fair voting method satisfies all of these conditions because there is some voter whose preference determines the ranking. If that one voter changes their preference, then the whole ranking is changed to match it. Subsequent to this result, many believed the endeavor of finding a fair voting method was useless. Take the much quoted W. H. Riker: the choice of a positional voting method is subjective. And this quote from D. G. Saari [2]: A naïve, false belief is that the choice of an election procedure does not matter because the outcome is essentially the same no matter which method is used, no matter what candidates are added or dropped. While this belief identifies a tacit but major goal, it is so stringent that it cannot be satisfied even with an unanimity profile. (The three candidate unanimity plurality outcome differs from the BC outcome which differs from the outcome where each voter votes

16 6 for two candidates.) Indeed, there is no reason to expect these conditions ever to be satisfied. In the very least, only a winner could be reliably found, but not a full ranking. The findings of Gehrlein et al [8, 4, 7], though, seem to indicate that paradoxes are hard to come by in large elections, and when conditioned upon measures of group coherence, are even rarer. Duncan Black showed that, for three candidates, if the voters preferences are single-peaked, then the majority rule satisfies Theorem. s four conditions []. A profile is single-peaked if there exists some candidate who is never ranked last. This means that all candidates are evaluated along some common scale, for example in modern politics there is the scale of conservatism vs. liberalism []. This implies that the ranking will be transitive, that is, it will not be cyclic (see Fact 2.). Other conditions of group coherence such as single-troughed and perfectly polarizing also ensure that cyclic pairwise rankings do not occur. A profile is single-troughed when some candidate is never ranked first. A profile is perfectly polarizing if there is some candidate who is never ranked second place. These conditions guarantee that there is a pairwise winner and pairwise loser. However, paradoxes can still occur where the Borda winner(loser) is not the pairwise winner(loser), even though the Borda count must rank the pairwise winner above the pairwise loser [9] (see Fact.2). Such discrepancies between the positional methods and the pairwise method are due to a specific subspace of the space of all voter profiles. D. G. Saari constructs the profile decomposition based on which parts of the profile space are affected by different voting methods. He shows that by removing a certain subspace of the profile, Borda count will satisfy all the conditions of Arrow s Theorem. D. G. Saari also showed [2] that the Borda count satisfies the following modified version of Arrow s theorem: Theorem.2. Arrow s Modified Theorem If the Independence of Irrelevant Alternatives condition in Arrow s Theorem is modified to the Intensity of Binary Independence condition, The societal ranking for a pair only depends on how voters rank that pair AND the intensity of that ranking, i.e. how many rankings are between the pair. then the Borda Count is the only positional voting method which satisfies the conditions. Here, the condition of Independence of Irrelevant Alternatives is strengthened to require that voting methods consider the information lost by using the pairwise voting method.

17 7 In using the pairwise method, voter rationality is lost because several voting profiles can give the same pairwise tallies. Consider the following profile where 2 voters have the irrational preference of A B C A and one voter has the irrational preference of A C B A (consequently, these are the only cyclic preferences for three candidates). By looking at the 2 candidate subsets, we arrive at the following pairwise preference profile A B = 2 B C = 2 C A = 2 A C = B A = C B = from which we can recombine the pairwise orderings to obtain the rational profile A B C = B C A = C A B =. This profile, of course, gives a positional method tally of a complete tie while the pairwise ranking is the cyclic A B C A.. Recent Results and Methods From the literature, we consider two major avenues of analysis. The first is the calculation of probabilities of finding paradoxes, conditioned upon certain profile restrictions or assumptions. William Gehrlein has written extensively in this area [, 0, 9]. The second is the characterization of the structure of voting profiles with respect to the voting methods, spearheaded by Donald Saari [2]. It is the aim of this exposition to corroborate Gehrlein s numerical findings and use simulations to explore frequencies of paradoxes for finite numbers of voters. We will see a connection between Saari s profile decomposition and the probabilities of finding paradoxes. We start by defining the assumptions on the voter profiles. We assume that voters have strictly transitive preferences on the candidates {A, B, C}. Thus, the set of preferences is {A B C, A C B, B A C, C A B, B C A, C B A}. The number of voters is n and the distribution of these n voters amongst the 6 preferences can be given in a number of ways. The model we use here is the multinomial distribution where n k is the number of voters with the kth preference and the probability that a voter has the kth preference is p k. A voting profile is obtained by making n sequential random assignments of the possible preference rankings to voters according to the p k probabilities.

18 8 The probability of a profile p = (n, n 2, n, n 4, n 5, n 6 ) is then ( ) ( ) 6 A B C = n, A C B = n 2, B A C = n, n = C A B = n 4, B C A = n 5, C B A = n 6 n n 2 n 6 When p k = /6 for all k, the above distribution is called Impartial Culture (IC). This model represents complete independence between the voters preferences [4, 4]. Another model for the voter profiles is that of Impartial Anonymous Culture (IAC) where each voting profile has equal probability of being observed. Thus, ( p) = / ( ) n+5 5 for any profile p. In this model, there is a degree of dependence between the voters preferences. k= p n k k. It is termed anonymous because unlike the IC assumption, if we were to change the preferences of some voters, the probability of the profile would be the same. The multinomial counts the different ways n non-anonymous voters can be arranged, which is precisely why the multinomial coefficient shows up in the distribution function. For both assumptions about voting preferences, as n, all candidates will tie in head to head comparisons. Thus more voter profiles will have a small degree of group cohesiveness, and thus conditioning upon IC or IAC will overestimate any paradoxical behavior[4]. However, Gehrlein shows that the probability of a profile being single-peaked reaches a finite value, 5 6, as n but for odd n (for even n, there is a similar limiting value) []. Gehrlein also uses the voter model of IAC*, where all profiles which are consistent with single-peaked preferences are equally likely. There are n(n + )(n + 5)/2 of these profiles. The ratio of IAC* to IAC profiles is (60n)/((n + 4)(n + )(n + 2)), so the proportion of profiles which are single-peaked goes to zero as n []. references which are single-peaked can be said to be unidimensional [20]. For three candidates, this is equivalent to saying that there is some candidate who is never ranked last. This creates a continuum in one dimension on which all voters rank the three candidates. Unidimensional rankings therefore prevent cyclic pairwise outcomes and paradoxes [2,, ]. Measuring how close a profile is to being single-peaked, single-troughed, or polarizing correlates with the probability of not getting certain paradoxes. Gehrlein defines these three measures of group coherence, b, t, and m, respectivley single-peaked, single-troughed, and

19 9 polarizing: ( ) B C A + C B A, A C B + C A B, b = min A B C + B A C ( ) A B C + A C B, B A C + B C A, t = min C B A + C A B ( ) B A C + C A B, A B C + C B A, m = min. A C B + B C A We say that a candidate who is never ranked last is the positively unifying candidate and the candidate who is never ranked first is the negatively unifying candidate. The candidate who is never ranked in the middle is the polarizing candidate since this candidate is either first or last in everyone s preferences []. In a series of papers, W. V. Gehrlein and others provide closed form representations and limiting representations for various probabilities involving voting paradoxes. He uses a brute force algorithm to calculate conditional probabilities for so called Borda paradoxes conditioned upon b, t, or m, and IC, IAC, or IAC*. The general conclusion is that the probabilities are small as n increases (less than 5% in most cases), and that probabilities become smaller as b, t, or m tend toward zero. A summary of some of these results is given in Table. at the end of this section. The following bizarre theorem from Saari [2] guarantees the existence of paradoxical profiles but gives no indication of their frequency. Theorem.. Ranking and dropping candidates Let N represent the number of candidates and suppose there are at least candidates. Rank the candidates in any way and select a positional election method to tally the ballots. There are N ways to drop one candidate. For each of these, rank the remaining (N ) candidates in any way and select a positional election method to tally the ballots. Continue for each subset of or more candidates.

20 0 For each pair of candidates assign a ranking and use the plurality method to tally the ballots. For almost all choices of positional voting methods, there exists a profile with the above choices for rankings and methods at each stage. The only method which does not guarantee this is the Borda count. Saari s profile decomposition breaks the space of all profiles into orthogonal subspaces in the sense that the rankings of certain voting methods are affected by one subspace and not another. In Section 2. we explain the decomposition in more detail. The following theorem gives a general sense of how these subspaces behave. Theorem.4. rofile decomposition [22] All profiles can be expressed as a linear combination of vectors from the following subspaces: All voting methods yield a complete tie for profiles from the Kernel. The Basic subspace gives identical rankings for all positional methods and the pairwise method. Every positional method gives a zero tie on the Condorcet subspace, while the pairwise method gives the cyclic ranking A B C A. On Reversal profiles, different positional methods give different rankings but the pairwise method and Borda count give a zero tie. The Condorcet portion of a profile is completely responsible for all discrepancies between positional methods and the pairwise method. The Reversal portion of a profile is responsible for all differences between different positional methods. Necessary and sufficient conditions for agreement between the various methods are given in [22] and are found by simple algebraic manipulations. Saari also gives illuminating geometric interpretations of these subspaces and their actions. To summarize, we give the following theorem. Theorem.5. Ranking possibilities for three candidates [22] Choose any ranking of the three candidates and any ranking for the pairs. If the Borda ranking is not equal to any of the positional rankings, then there is a profile where the pairwise rankings and the positional rankings are as described. Only the Borda ranking must be related to the pairwise rankings.

21 The last part of this theorem is referring to the fact that the Borda count must rank the pairwise winner above the pairwise loser. Fact.6. Borda count always ranks the pairwise winner above the pairwise loser. roof. Suppose without loss of generality that the pairwise ranking is A B C. We must then have that ABC + ACB + CAB > BAC + BCA + CBA ABC + BAC + BCA > ACB + CAB + CBA that is, A B i.e. B C ABC + ACB + BAC > CAB + BCA + CBA and this is A C. On the other hand, the Borda tallies are A s tally = ABC + ACB + ( BAC + CAB ) 2 B s tally = BAC + BCA + ( ABC + CBA ) 2 Recall that C s tally = CAB + CBA + ( ACB + BCA ). 2 n = ABC, n = BAC, n 5 = BCA, n 2 = ACB, n 4 = CAB, n 6 = CBA. If we sum the inequalities which come from the pairwise conditions for A B and A C, and sum the inequalities which arise from the conditions B C and A C we get 2(n + n 2 ) + n + n 4 > 2(n 5 + n 6 ) + n + n 4 n + n 2 > n 5 + n 6 2(n + n ) + n 2 + n 5 > 2(n 4 + n 6 ) + n 2 + n 5 n + n > n 4 + n 6. Now add the resulting inequalities, 2n + n 2 + n > n 4 + n 5 + 2n 6 2n + 2n 2 + n + n 4 > 2n 4 + 2n 6 + n 2 + n 4, which is equivalent to saying that the Borda count ranks A over C.

22 Table.: Summary of results from Gehrlein et al Voter distribution Event Result/Range Method used Reference airwise winner exists rb= mulitvariate normal [5] Borda winner = lurality winner rb= multivariate normal [5] airwise loser = lurality winner rb=0.084 multivariate normal [5] IC Borda winner = airwise winner rb= multivariate normal [5] lurality winner = airwise winner rb= multivariate normal [5] airwise creates a cyclic outcome, given b [ n, 0] rb: 0.25 to 0 computer [20] λ-rule winner = airwise loser, given one exists λ: 0 to /2 to, multivariate normal [4] n rb: 0.07 to 0 to 0.07 IAC IAC* airwise winner exists, odd n rb= 5(n+)2 6(n+2)(n+4) computer [2] λ-rule winner = airwise loser, n rb= (2λ ) (2 9λ 2λ 2 ) 405(λ ) polyhedra volume [4] λ-rule winner = airwise winner, given one exists λ: 0,, /2, EUIA2 [] n rb: 9/5, 68/08, 2/5 all λ-rule winners = airwise winner EUIA2 [] n given on exists rb=47/ λ-rule winner = airwise winner, given one exists λ: 0,, /2, EUIA2 [] n rb: /6, /4, /2 all λ-rule winners = airwise winner EUIA2 [] n given on exists rb=/8 2

23 Chapter 2 Notation and Methods In the subsequent analyses, the set of candidates is {A, B, C}. We assume the distribution of voter profiles follows a multinomial distribution as explained in Section. and that voters preferences are strictly transitive. The set of preferences is S = {A B C, A C B, B A C, C A B, B C A, C B A} which is the sample space. The set of voters preferences is denoted as {v i } n i=, where vi S. We will shorten a preference in S by leaving off the s, for example we will write ACB in place of A C B. The following describes the various voting methods in terms of sums of binomial random variables and calculates expectation, variance, and covariance for the general case. By presenting the basic components of our probabilistic events in this way, it is easily seen how one can use the Central Limit Theorem (see Theorem 2.) to obtain limiting probabilities which serve as approximations of the likelihoods of voting paradoxes. 2. Key Definitions n A voting sum will be denoted as S n (E) = E (v i ), where E S. The characteristic i= function E (v i ) takes values from the sample space and is equal to when v i E and 0 when v i / E. Based on our assumptions about the voting profiles, S n (E) is a binomial random variable with mean n (E) and variance n (E)( (E)). All of the different positional voting method outcomes and the pairwise method can be defined in terms of S n (E). For example, the plurality sums are A = n {ABC,ACB} (v i ), B = i= n {BAC,BCA} (v i ), C = i= n {CAB,CBA} (v i ). i= Then, the plurality tally is the vector ( A, B, C ). The ranking is given by listing the candidates in order of largest to smallest of the K, for K {A, B, C}.

24 4 We also define the negative plurality sums and the middle sums, N K and M K, respectively. N A = n {BCA,CBA} (v i ), N B = i= n {ACB,CAB} (v i ), N C = i= n {ABC,BAC} (v i ) i= and M A = n {BAC,CAB} (v i ), M B = i= n {ABC,CBA} (v i ), M C = i= n {ACB,BCA} (v i ). i= The Borda method sum, B K, is then a combination of the plurality and middle sums, with λ = 2 : B K = K + 2 M K. Recall, p k is the probability that a voter has preference k. Specifically we have, p = (v i = A B C) p 2 = (v i = A C B) p = (v i = B A C) p 4 = (v i = C A B) p 5 = (v i = B C A) p 6 = (v i = C B A). For each of the six voting preferences on three candidates, Gehrlein uses n k for the total number of voters who hold that preference: n = n {ABC} (v i ), n 2 = i= n {ACB} (v i ), n = i= n {BAC} (v i ), i= n n n n 4 = {CAB} (v i ), n 5 = {BCA} (v i ), n 6 = {CBA} (v i ). i= i= i= These n k are each binomial random variables and will be the fundamental random variables that we use in this analysis. We restate the voting sums in terms of these n k, A = n {ABC,ACB} (v i ) = n + n 2, B = i= n {BAC,BCA} (v i ) = n + n 5, i= C = n {CAB,CBA} (v i ) = n 4 + n 6. i=

25 5 are the plurality sums. The Borda sums are just B A = A + 2 M A = n + n (n + n 4 ), B B = B + 2 M B = n + n (n + n 6 ), B C = C + 2 M C = n 4 + n (n 2 + n 5 ). For a general voting sum S n (F ) for F S, we can calculate the expectation, variance, and covariance: [ n ] E[S n (F )] = E F (v i ) = i= n E [ F (v i ) ] = ne [ F (v ) ] = n [F ], and since S n (F ) is a binomial random variable with mean n [F ], where i= Var[S n (F )] = E [ (S n (F )) 2] ( E[S n (F )] ) 2 = n [F ]( [F ]); Cov [ S n (F ), S n (G) ] [ n = Cov F (v i ), so that we get i= n i= ] G (v i ) = n n i= j= [ ] { [F G] [F ] [G] if i = j Cov F (v i ), G (v j ) = 0 if i j Cov [ S n (F ), S n (G) ] = [ ] Cov F (v i ), G (v j ) n Cov [ F (v i ), G (v i ) ] = ncov [ F (v ), G (v ) ] i= = n (E [ F (v ) G (v ) ] E [ F (v ) ] E [ G (v ) ]) ( = n E [ F G (v ) ] ) [F ] [G] = n ( [F G] [F ] [G] ). As an example, take F = A is ranked first place. Then S n (F ) = n i= {ABC,ACB}(v i ) = n i= {ABC}(v i ) + n i= {ACB}(v i ) counts the number of times A is first place in all of

26 6 the voters preferences. We calculate the expectation to be E [ S n ({ABC, ACB}) ] = E [ n n {ABC} (v i )+ {ACB} (v i ) ] = i= i= n n E[ {ABC} (v i )]+ E[ {ACB} (v i )] i= i= n n = (v i = ABC) + (v i = ACB) = i= i= and the variance is then n n p + p 2 = n(p + p 2 ), i= i= Var [ S n ({ABC, ACB}) ] = n [{ABC, ACB}]( [{ABC, ACB}]) = n(p +p 2 )( (p +p 2 )). Next we explain Saari s profile decomposition. The space of all voter profiles where preferences are strictly transitive can be viewed as a n! dimensional vector subspace in R n!, denoted n = {(n, n 2, n, n 4, n 5, n 6 ) : 6 k= n k = n, n k Z}. Basis vectors of this space are called profile differentials. The Kernel is spanned by the profile K = (,,,,, ) for n = candidates. For n, there is a space called the Universal Kernel, UK n, of dimension n! 2 n (n 2) 2 in which K n = (,,..., ) R n! is contained. All the profiles of this space result in complete ties for all positional methods and the pairwise method. It is interesting to note that for n 5, the dimension of this space is more than half the dimension of n, and approaches the dimension of n as n. In essence this is an illustration of the law of large numbers since profiles which have no effect on election outcomes become the largest part of the profile space [2]. The Basic subspace of profiles gives the same tally for any positional procedure and the pairwise ranking agrees with the positional ranking. For n =, this is a 2-dimensional subspace spanned by any two of the profile differentials {B A, B B, B C } where B A + B B + B C = (0, 0, 0, 0, 0, 0, ), B A = (,, 0, 0,, ), B B = (0,,,,, 0), B C = (, 0,,, 0, ). The Condorcet subspace is spanned by the profile differential C = (,,,,, ). For n, this subspace has dimension 2 (n )! and the Basic subspace has dimension (n ). For all positional methods, the tallies are all zero ties, and for the pairwise method, the ranking is the cyclic A B C A.

27 7 The last subspace is the 2-dimensional Reversal subspace (for n this space is (n )-dimensional). The profiles of this space result in a Borda count zero tie but positional procedures other than the Borda rule have non-zero tallies which differ for each rule. These profiles are R A = (,, 2, 2,, ), R B = ( 2,,,,, 2), R C = (, 2,,, 2, ), and also R A + R B + R C = 0. We shall write a decomposition of a profile p = (n, n 2, n, n 4, n 5, n 6 ) as p = ab A + bb B + r A R A + r B R B + γc + kk and the vector v = (a, b, r A, r B, γ, k) as the vector of decomposition coefficients. Saari gives a way to find all the coefficients for the decomposition. The matrix T is It is the inverse of the matrix (B A, B B, R A, R B, C, K), where the columns are the basis vectors expressed in the canonical basis of R 6. We can do matrix multiplication to get T p = v, that is, we can find the decomposition coefficients from multiplying the profile by T [22]. Conversely, we can get the decomposition from the coefficients simply by using the definition of the profile decomposition vectors. 2.2 Central Limit Theorem Approximations Theorem 2.. Central Limit Theorem in R m [2] For each i, let X i = (X i,..., X im ) T be an independent random vector, where all X i have the same distribution. Suppose that E[X 2 ik ] < for k m; let the vector of means be c = (c,..., c m ) where c k = E[X ik ], and let the covariance matrix be Σ = [σ ij ] where σ ij = E[(X i c i )(X j c j )]. ut S n = n i= X i.

28 8 Then the distribution of the random vector n (S n nc) converges weakly to the centered multivariate normal distribution with covariance matrix Σ and density f Z (z) = (2π) m/2 Σ /2 e 2 zt Σ z. Each n k is a sum of Bernoulli random variables Ek, where E k is the kth preference, with mean p k. Applying Theorem 2. to the vector the multivariate standard normal distribution, n np n n 2 np 2 n. n 5 np 5 n D Z Z 5 ( nk np k n Z 2. ) T k=,...,5 gives convergence to where Z = (Z,..., Z 5 ) T is a multidimensional normal random variable with density f Z (z) = (2π) 5/2 Σ /2 e 2 zt Σ z, and where Σ = [E[( Ei p i )( Ej p j )]] i,j 5 is the covariance matrix p ( p ) p p 2 p p 5 p 2 p p 2 ( p 2 ) p 2 p p 5 p p 5 p 2 p 5 ( p 5 ) Note that we can calculate Z 6, since 6 k= n k = n and 6 k= p k = gives Z 6 = (Z + + Z 5 ). Subsequently, we give probabilities for two different paradoxes where a winner in one method is the loser in another method. In calculating the paradoxes, we find that changing the probabilities p k in the voter model yields different values for the probabilities of paradoxes. The following lemma will be used to calculate some of these probabilities. Lemma 2.2. A Bivariate Normal robability Let (X, X 2 ) T be a centered bivariate normal random vector with covariance matrix Σ =.

29 9 ( [σ i,j ] i,j=,2, σ,2 = σ 2,. Then (X > 0, X 2 > 0) = 4 + 2π sin σ2 σ σ 22 ). X V ar[x ] X roof. Let = X and 2 Var[X 2 ] = ax + by, where a, b R. We will require that Var[X] = Var[Y ] = and Cov[X, Y ] = 0 so that (X, Y ) is a bivariate normal random vector with mean zero and covariance matrix equal to the identity matrix. We need to find the values for a and b: [ ] X 2 = Var = Var[aX + by ] = a 2 + b 2 = a 2 + b 2 =. Var[X 2 ] [ ] X Cov[X, ax + by ] = Cov V ar[x ], X 2 Var[X 2 ] Also, we will need that So with these we get that = b = a 2 = = a = σ 2 σ σ 22, a b = = σ 2 σ σ 22 and Cov[X, ax + by ] = a σ2 2 σ σ 22 = σ 2 2 σ σ 22 σ2 2. σ σ 22 σ 2 2 σ σ 22. ( ) X (X > 0, X 2 > 0) = V ar[x ] > 0, X 2 Var[X 2 ] > 0 = = = ( X > 0, Y > 0 σ 2 2 σ σ 22 σ 2 2 a b x 2π e (x2 +y 2 )/2 dydx. ) ( X > 0, Y > a ) b Next, make a change of variables to polar coordinates. The integral becomes π/2 0 α 2π e r2 /2 rdrdθ

30 20 where α = tan ( σ 2 2 σ σ 22 σ 2 2 ) ( ) α = sin σ2. σ σ 22 Thus the integral is 0 α 2π e r2 /2 rdrdθ = = ( ( π 2π 2 sin σ )) 2 σ σ π e r2 /2 rdr π/2 α dθ = 4 + ( ) σ2 2π sin. σ σ Simulation methods Using MATLAB 20b version, we have written algorithms to calculate the frequency of two voting paradoxes. The first is where the lurality winner is the airwise loser and the second is where the lurality winner is the Borda loser, see Alorithm 2. and Algorithm 2.2 below. The MATLAB code is included in the Appendix. We use two versions of these algorithms. In one version, we do not count ties between candidates. For example, if candidates A and B are both the lurality winner but only candidate A is the airwise loser, we don t count this as a paradox. Similarly if two candidates are tied losers, or two candidates tie as lurality winners and airwise losers, etc, we do not count this. The other version is where we do count these tied situations as a voting paradox. These two approaches do not appear to be produce significantly different limiting behavior. Algorithm 2.. Finds the average number of times cadidate A is the lurality winner and airwise loser generate 0, 000 trials of random preference profiles based on the multinomial distribution with probabilities p k for i = to 0000 c = 0 counts whether a paradox occured calculate the lurality ranking and airwise ranking from the ith random preference profile using the definitions in Section 2.

31 2 test whether candidate A is the lurality winner and whether candidate A is the airwise loser if both these occur, c = c +, increase the count by one end for loop the average number of times a paradox occured is then c/000 Algorithm 2.2. Finds the average number of times cadidate A is the lurality winner and Borda loser generate 0, 000 trials of random preference profiles based on the multinomial distribution with probabilities p k for i = to 0000 c = 0 counts whether a paradox occured calculate the lurality ranking and Borda ranking from the ith random preference profile using the definitions in Section 2. test whether candidate A is the lurality winner and whether candidate A is the Borda loser if both these occur, c = c +, increase the count by one end for loop the average number of times a paradox occured is then c/000 To get the random preference profiles, we use MATLAB s function makedist to create the multinomial distribution using the six probabilities. Then using the function random, we can generate n preferences for the voters, from which we create the preference profile. It does this 0, 000 times, the number of trials we wish to produce. Apart from these two algorithms, we also make use of the function mvncdf to calculate the limiting probabilities. Suppose we want to calculate (Y > 0, Y 2 > 0, Y > 0, Y 4 > 0) where the centered and scaled normal variables {Y i } 4 i= have the covariance matrix Σ which we name Sigma in our MATLAB work space. Then, mvncdf([ ],[],Sigma)= (Y < 0, Y 2 < 0, Y < 0, Y 4 < 0) which then equals ( Y > 0, Y 2 > 0, Y > 0, Y 4 > 0) and this equals (Y > 0, Y 2 > 0, Y > 0, Y 4 > 0) since the variables are centered. We

32 22 shall use this function for the three dimensional and four dimensional normal distribution probability calculations. For bivariate and trivariate distributions, mvncdf uses adaptive quadrature on a transformation of the t density, based on methods developed by Drezner and Wesolowsky and by Genz. The default absolute error tolerance for these cases is e 8. For four or more dimensions, mvncdf uses a quasi-monte Carlo integration algorithm based on methods developed by Genz and Bretz, as described in the references. The default absolute error tolerance for these cases is e 4. [5, 6, 6, 7, 8] 2.4 Simple Examples The following examples give of flavor of the content of Chapter, showing how the preceding methods work to give probabilities and simulations. We also give and example of a particular profile decomposition as well as how to compute a profile decomposition. Lastly, we look at a single-peaked profile and see what its profile decomposition tells us The probability that A wins by the pairwise method The candidate A wins by the pairwise method whenever n + n 2 + n 4 > n + n 5 + n 6 and n +n 2 +n > n 4 +n 5 +n 6, or equivalently, whenever n +n 2 +n 4 > n 2 and n +n 2 +n > n 2. So then the probability of A being the pairwise winner is ( n +n 2 +n 4 > n 2, n +n 2 +n > n ) = 2 n np n + n 2 np 2 n + n 4 np 4 n > n np n + n 2 np 2 n + n np n > ( ) n 2 n n(p +p 2 +p 4 ) n n 2 n n(p +p 2 +p ) n (, ) = ( n np n + n 2 np 2 n + n 4 np 4 n > n( 2 (p + p 2 + p 4 )), n np n + n 2 np 2 n + n np n > n( 2 (p + p 2 + p )) ). Letting n we get convergence to the multivariate normal in R 2. Let Z +Z 2 +Z 4 = Y and Z + Z 2 + Z = Y 2. Then the vector (Y Y 2 ) T has covariance matrix Σ,

33 2 ( ) p + p 2 + p 4 (p + p 2 + p 4 ) 2 (p + p 2 )( (p + p 2 )) Σ = ( (p + p 4 )(p + p 2 ) p 4 p ) (p + p 2 )( (p + p 2 )) p + p 2 + p (p + p 2 + p ) 2. (p + p 4 )(p + p 2 ) p 4 p We get a positive probability whenever both p + p 2 + p 4 2 and p + p 2 + p 2. There will be four cases depending on whether we take equality or strict inequality. In the first case, if both p + p 2 + p 4 = 2 and p + p 2 + p = 2, then the covariance matrix can be expressed more simply as and the probability is (Y > 0, Y 2 > 0) = ( Σ = 4 4 p y =0 4 p y 2 =0 4 ), 2π Σ y t Σy e /2 2 dy. Figure 2.: Limiting behavior in simulation when p k = 6 for all k

34 24 Using Lemma 2.2, this integral reduces to 4 + 2π sin ( 4p ). When we let p = /6, we get 4 + 2π sin (/) 0.04 as given in [4]. Therefore, the probability of getting a pairwise winner is , whenever p k = /6 for all k. Figure 2. shows the simulation results for these values of p k. We see the convergence is practically immediate. In Figure 2.2 we see that the limiting probability goes to zero as p 2. When p is /2 then by the condition p + p 2 + p = /2, we get that p = p 2 = 0. This means that the probability that a random voter has preference ABC or ACB is zero, and so there is literally no chance that candidate A will get any first place votes, and thus cannot be the pairwise winner. Figure 2.2: robability of A being the pairwise winner under conditions p + p 2 + p 4 = 2 and p + p 2 + p = 2 Alternately, if p + p 2 + p 4 > 2 and p + p 2 + p = 2, we get the probability (Y >, Y 2 > 0) = y = y 2 =0 2π Σ where now the covariance matrix is ( ) 2 + p 4 p 4 2 (p + p 4 ) 4 2 (p, + p 4 ) 4 with the equivalent constraints y t Σy e /2 2 dy, p 4 > p, p + p 2 + p = 2, p 4 + p 5 + p 6 = 2.

35 25 This probability is just 2 since the centered normal distribution is symmetric about the origin. Thus, these conditions on the p k force A to be the pairwise winner half the time. As an illustration, if we maximize p 4 = p CAB at /2 and make p = p BAC close to /2, then p 5 = p 6 = 0 and also, there will be enough of a probability that A is ranked above B and C via the small but non-zero probabilities of p = p ABC and p 2 = p ACB. In addition, p 5 and p 6 are zero, which means there is no chance that A is last. One can deduce that the most likely ranking is then A B C. This result is analogous for the similar conditions p + p 2 + p 4 = 2, p + p 2 + p > 2 equivalent to p 4 < p, p + p 2 + p = 2, p 4 + p 5 + p 6 = 2, which give the probability (Y > 0, Y 2 > ) = 2. If we maximize p at /2 we can make similar arguments to find that the most likely ranking will be A C B. Now, suppose we let p 5 +p 6 = /2 under the constraints p +p 2 +p = 2 and p 4+p 5 +p 6 = 2 with either p 4 > p or p > p 4. Then we get that p 4 = p = 0 which implies then that p + p 2 = /2. In this situation, A is ranked first half of the time and last half of the time (so to speak). All of this makes sense because the p k are simply the frequency with which voters will have preference k and thus directly relate to the actual numbers of those voters with preference k, to within the variance of the random variables, which depends on the underlying assumption about the distribution of voter preferences Condorcet s Example Condorcet created the example where no positional procedure ranks the pairwise winner as the positional winner. The profile vector is p = (0,, 29, 0, 0, ), and the table below gives the profile in terms of the preferences. Using the matrix T to find the vector of coefficients [2], we get that p = 6 (68B A + 76B B 28R A 20R B + 9C + 8K). The coefficients on B A and B B will ensure a Borda rule ranking of B A C since 76 is greater than 68. The plurality ranking is also