On the probabilistic modeling of consistency for iterated positional election procedures

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Spring 2014 On the probabilistic modeling of consistency for iterated positional election procedures Mark A. Krines University of Iowa Copyright 2014 Mark A. Krines This dissertation is available at Iowa Research Online: Recommended Citation Krines, Mark A.. "On the probabilistic modeling of consistency for iterated positional election procedures." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Applied Mathematics Commons

2 ON THE PROBABILISTIC MODELING OF CONSISTENCY FOR ITERATED POSITIONAL ELECTION PROCEDURES by Mark A. Krines A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Applied Mathematical and Computational Sciences in the Graduate College of The University of Iowa May 2014 Thesis Supervisors: Associate Professor Douglas Dion Associate Professor Jonathan K. Hodge

3 Copyright by MARK A. KRINES 2014 All Rights Reserved

4 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Mark A. Krines has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Applied Mathematical and Computational Sciences at the May 2014 graduation. Thesis Committee: Douglas Dion, Thesis Supervisor Jonathan K. Hodge, Thesis Supervisor Weimin Han Joseph B. Lang David E. Stewart

5 ACKNOWLEDGEMENTS I would like to express my gratitude to those who have assisted me with my research and completing this dissertation. My immediate family, namely my parents Jerome and Sandra and my sister Amy, have been quite supportive throughout this process, especially during some of the more stressful periods of this project. I would also like to thank my two co-advisors, Dr. Douglas Dion and Dr. Jonathan K. Hodge. My research proceeds in a unique direction, one which has allowed me to establish an independent research program, and it would not have been possible without Dr. Dion agreeing to be my advisor, offering various literature to get me going in my research, and providing significant feedback on my efforts. It also would not have been possible without Dr. Hodge proofreading my ideas and mathematical writing while also providing invaluable advice and resources for contextualizing my work. Further, I appreciate Dr. William Gehrlein s willingness to discuss with me his research and other present research in this field. His feedback was especially helpful to me when my initial research efforts led me to his work and required me to expand the focus of my dissertation. Finally, I would like to thank Professors Weimin Han, Joseph B. Lang, and David E. Stewart for serving on my committee and comprehending my research even though it only indirectly connects to their areas of expertise. I would also like to express my gratitude for my mathematics professors at St. Norbert College, especially Professors John Frohliger, Richard Poss, and Larry Thorsen. Their efforts within the Mathematics Department were invaluable in preparii

6 ing me for graduate school and in shaping how I want to proceed in my career as a mathematician. Finally, I would like to thank all of my past teachers for their influence on my life and in developing my passion for pursuing a career in post-secondary education. I would not have been able to complete this thesis and embark on my career without all of their support and influence. iii

7 ABSTRACT A well-known fact about positional election procedures is that its ranking of m alternatives can change when some of the alternatives are removed from consideration even if the voters preferences remain constant. In particular, suppose we are given a positional procedure on each of 2, 3,..., m alternatives and a collective preference order for each distinct subset of the m alternatives. Saari [26] has established that with few exceptions, we can find a voter profile for which the collective preference order for each subset under the according positional procedure is the one given. However, Saari s results do not quantify the likelihood of finding such voter profiles. For small numbers of alternatives, William Gehrlein [19] developed a statistical model to explore the probabilities that particular collective preference orders on subsets of alternatives can occur for large electorates. One goal of this research is to determine whether changes in the collective preference order as alternatives are removed can be considered to be the norm or an outlier for positional procedures. This dissertation extends the research headed by Gehrlein in two directions. One, I generalize his statistical model to explore probabilities for iterated election procedures. Gehrlein s model previously produced results only for three alternatives and in limited cases for four alternatives. I have extended this model to produce results for up to five alternatives, including analysis of instant-runoff voting and runoff elections. Two, Gehrlein s model required specific conditions on the probability distribution of individual voter preferences across the population. I relax this assumption so that for any probability distribution of individual voter preferences across the population, I can explore the probability that a collective preference order is inconsistent with the outcomes when alternatives are removed. iv

8 These results provide a foundation for discussing the impact of removing alternatives on elections across all large electorates. I also apply these results to two recent United States elections wherein a third-party candidate received a significant share of the votes: the 1992 U.S. Presidential election and the 1998 Minnesota Gubernatorial election. Overall, my research will suggest that as the number of alternatives increases, the likelihood of finding changes in the collective preference order as alternatives are removed will approach one. v

9 TABLE OF CONTENTS LIST OF TABLES viii LIST OF FIGURES x CHAPTER 1 PRELIMINARIES Introduction Prior Research Outline of Later Chapters FOUNDATION Definitions and Assumptions Method RANKING-INVARIANCE UNDER THE IMPARTIAL CULTURE AS- SUMPTION Ranking-Invariance when m = Ranking-Invariance when m = Ranking-Invariance when m = Method Results Summary of Results Reducing Some of the Variates: Primary Elections WINNER-INVARIANCE UNDER THE IMPARTIAL CULTURE AS- SUMPTION Method Winner-Invariance for m = Winner-Invariance for m = Application to Instant-Runoff Voting Summary of Results RESULTS FOR ARBITRARY PROBABILITY DISTRIBUTIONS FOR M = Preliminaries vi

10 5.2 Ranking-Invariance Method Results Pairwise-Invariance Method Results Application to Recent Elections The 1992 U.S. Presidential Election The 1998 Minnesota Gubernatorial Election CONCLUSION APPENDIX A VARIATE VALUES FOR THE FIVE-ALTERNATIVE PREFERENCE ORDERS B SELECTED VALUES FOR P R C SELECTED VALUES FOR P P W REFERENCES vii

11 LIST OF TABLES Table 1.1 Committee preferences for their open position Alternate committee preferences for their open position Preference Order Assignment for p Values for Y i 1 and Y i Values for the Three Variables in the Three-Alternative Case Values for the Six Variables in the Four-Alternative Case P R m for Baldwin s Method and m Winner-Invariance Probabilities for IRV and Baldwin s Method Voter preferences for Example Values for Y i j for computing P R 123 (p, u) List of Expected Values for Y i j List of Covariances when Y i j and Y i k are zero List of Permutations for each Strict Collective Preference Order Values for Y i 4 and Y i List of Expected Values for Y i j List of Covariances when Y i j and Y i k are zero Times-Mirror Poll Results for Voters Second Choices Popular Vote Percentages for the 1992 U.S. Presidential Election List of Expected Values for the 1992 U.S. Presidential Election List of Adjusted Expected Values for the 1992 U.S. Presidential Election 141 viii

12 5.13 Popular Vote Percentages for the 1998 Minnesota Gubernatorial Election Lacy and Monson s Results for Voters Second Choices List of Expected Values for the 1998 Minnesota Gubernatorial Election. 148 A.1 Values for Variates A.2 Values for Variates B.1 Selected Values of P R 3 (u, p) C.1 Selected Values of P P W 3 (u, p) ix

13 LIST OF FIGURES Figure 3.1 P3 R (u) P4 R ((0, 0), u) for plurality rule on the first iteration P4 R ((1, 1), u) for antiplurality rule on the first iteration P R 4 (( 2, ) ) 1 3 3, u for the Borda count on the first iteration P4 R (t, 0) for plurality rule in the second iteration Slice of the graph of P4 R (t, 0) with t 2 = Slice of the graph of P4 R (t, 0) with t 3 = P4 R (t, 1) for antiplurality rule in the second iteration Slice of the graph of P4 R (t, 1) with t 2 = Slice of the graph of P4 R (t, 1) with t 3 = P4 R (t,.5) for the Borda count in the second iteration Slice of the graph of P4 R (t,.5) with t 2 = Slice of the graph of P4 R (t,.5) with t 3 = P4 R (t, u) for natural iteration Slice of the graph of P4 R (t, u) for natural iteration with t 2 = Slice of the graph of P4 R (t, u) for natural iteration with t 3 = Probability that plurality rule is ranking-invariant for u = Probability that plurality rule is ranking-invariant for u = Probability that plurality rule is ranking-invariant for u = Probability that the Borda count is ranking-invariant for u = x

14 3.21 Probability that the Borda count is ranking-invariant for u = Probability that the Borda count is ranking-invariant for u = Probability that antiplurality rule is ranking-invariant for u = Probability that antiplurality rule is ranking-invariant for u = Probability that antiplurality rule is ranking-invariant for u = Average Values for P R 5 under Natural Iteration The graph of P R 5 when t and u correspond to plurality rule The graph of P R 5 when t and u correspond to antiplurality rule The graph of P R 5 when t and u correspond to the Borda count The graph of P R 5 when t is antiplurality rule and u is plurality rule P R 5 when s and t are defined with symmetry P P R 4 (t) for various four-alternative positional procedures P P R 5 (s) for various five-alternative positional procedures P W 4 ((0, 0), u) for plurality rule on the first iteration P W 4 ((1, 1), u) for antiplurality rule on the first iteration P W 4 (( 2, ) ) 1 3 3, u for the Borda count on the first iteration Graph of P4 W (t, 0) for plurality rule in the second iteration Slice of the graph of P4 W (t, 0) with t 2 = Slice of the graph of P4 W (t, 0) with t 3 = Graph of P4 W (t, 1) for antiplurality rule in the second iteration Slice of the graph of P4 W (t, 1) with t 2 = Slice of the graph of P4 W (t, 1) with t 3 = xi

15 4.10 Graph of P W 4 (t,.5) for the Borda count in the second iteration Slice of the graph of P W 4 (t,.5) with t 2 = Slice of the graph of P W 4 (t,.5) with t 3 = Graph of P W 4 (t, u) for natural iteration Slice of the graph of P W 4 (t, u) for natural iteration with t 2 = Slice of the graph of P W 4 (t, u) for natural iteration with t 3 = Probability that plurality rule on five alternatives is winner-invariant for u = Probability that plurality rule on five alternatives is winner-invariant for u = Probability that plurality rule on five alternatives is winner-invariant for u = Probability that antiplurality rule on five alternatives is winner-invariant for u = Probability that antiplurality rule on five alternatives is winner-invariant for u = Probability that antiplurality rule on five alternatives is winner-invariant for u = Probability that the Borda count on five alternatives is winner-invariant for u = Probability that the Borda count on five alternatives is winner-invariant for u = Probability that the Borda count on five alternatives is winner-invariant for u = Average values for P W 5 under natural iteration The graph of P W 5 when t and u are plurality rule The graph of P W 5 when t and u are antiplurality rule xii

16 4.28 The graph of P5 W when t and u are the Borda count The graph of P5 W when t is antiplurality rule and u is plurality rule P5 W when s and t are defined with symmetry Average probabilities for ranking-invariance Average probabilities for pairwise-invariance xiii

17 1 CHAPTER 1 PRELIMINARIES 1.1 Introduction Suppose that a university committee must fill a position with one of four available candidates: Alice, Bob, Candace, and Dave. Their preferences are tallied with the distribution given in Table 1.1. Considering the presence of four candidates, the committee decides to use instant-runoff voting (IRV) to determine which candidate will fill the position. IRV is an iterated election procedure based on the repeated use of plurality rule. 1 At each iteration, the ballots are tallied according to plurality rule among the remaining alternatives, and if one candidate receives a majority of the votes, then that candidate is declared the winner. Otherwise, if no candidate receives a majority of the votes, then the candidate receiving the fewest votes is eliminated and the process repeats itself. Given our committee s preferences, on the first iteration, no candidate receives a majority of the votes (i.e. at least seven votes). As Candace receives only one vote, Candace is eliminated first. Among Alice, Bob, and Dave, Bob is eliminated second due to receiving only three votes while neither Alice nor Dave receive a majority of the votes. Finally, between Alice and Dave, Dave wins the IRV election by a 7:5 margin. IRV attempts to improve upon plurality rule through the use of iteration. The general argument is that by eliminating less-preferred candidates initially, we can be 1 Under plurality rule, each voter votes for one alternative. The alternative receiving the most votes wins. This is the most popular election procedure used in democratic societies.

18 2 Table 1.1: Committee preferences for their open position. First Second Third Last Frequency Alice Bob Candace Dave 5 Dave Bob Candace Alice 4 Bob Candace Dave Alice 2 Candace Bob Dave Alice 1 ensured that the winner of the election will be preferred by a majority of the voters. This is in contrast to plurality rule, wherein a candidate can be elected with far less than a majority of the votes and could be elected even if a majority of the voters rank that candidate last. It is expected, then, that IRV and plurality rule will differ in their outcomes in some proportion of elections. Our committee s election is one such example; while Dave wins the IRV election, it can be readily inferred from Table 1.1 that Alice wins the plurality election with five votes. However, it should be noted that neither plurality rule nor IRV appears to represent the committee s preferences effectively. Every member of the committee ranks Bob first or second, so it is likely that Bob is the most satisfying candidate across the committee. Although the above example illustrates that IRV and plurality rule can produce differing winners, we can also find examples wherein not only the winners agree, but the ranking at each iteration of IRV agrees with the ranking produced by plurality rule. Suppose instead that the preferences of the committee are those given in Table 1.2. The plurality ranking has Alice first, Bob second, Candance third, and Dave fourth. For IRV, the first iteration produces the same results, but as Alice did

19 3 not receive a majority of the votes, Dave is eliminated from consideration. Dave s supporter then votes for Candace, and the second-iteration plurality ranking among Alice, Bob, and Candace has Alice first, Bob second, and Candace third the same as the first iteration. As Alice still did not receive a majority of the votes, Candace is eliminated from consideration. Between Alice and Bob, Alice wins the election with a 7:5 margin the same ranking as in the plurality case. Table 1.2: Alternate committee preferences for their open position. First Second Third Last Frequency Alice Bob Candace Dave 5 Bob Dave Candace Alice 4 Candace Alice Bob Dave 2 Dave Candace Bob Alice 1 Of primary relevance to this research, IRV is a particular example of an iterated procedure: an election procedure carried out over multiple stages wherein some subset of the candidates is eliminated after each stage until a winner is chosen. IRV historically dates to 1871 with its invention by William Robert Ware. IRV is used internationally in elections involving three or more candidates, most notably in Australia where all of their elections follow IRV (and its generalized form, Single Transferable Voting, for multiple-winner elections). 2 IRV has been gaining traction 2 Technically, Single Transferable Voting is older than IRV, having been created in 1821 by Thomas Wright Hill.

20 4 in the United States in recent years, with various municipalities such as San Francisco, California, and St. Paul, Minnesota, adopting IRV for their elections. On the other hand, Burlington, Vermont, adopted IRV in 2006 before returning to their prior method following a controversial 2009 mayoral election wherein the IRV winner did not match the plurality winner or the Condorcet winner. 3 A general discussion of IRV can be found in Farrell [12], and a historical outline of IRV s implementation in Australia can be found in Farrell and Mcallister [11]. Iterated election procedures can be generalized beyond IRV in multiple fashions. Many democratic societies use some form of a runoff to narrow a field of candidates immediately to two alternatives before conducting a majority rule election. Most local and state governments in the United States follow this method by using plurality rule to reduce an election field to two candidates (usually with one candidate in each of the major parties). More notably, France uses a runoff system to narrow its field of presidential candidates to two candidates [31]. Their 2002 election garnered international attention when their initial round of balloting split the vote between major-party candidates and allowed Jean-Marie Le Pen, an extremist candidate, to advance to the second stage. Although Le Pen received about 16.9% of the vote among sixteen candidates, he only increased his total to 17.8% in his landslide loss to Jacques Chirac. Accordingly, we should be aware of the disastrous consequences that our choice of election procedure can have on the election results and on our society. 3 A Condorcet winner in an election is an alternative which defeats every other alternative in a pairwise comparison using majority rule. A Condorcet loser is defined similarly.

21 5 We can also generalize IRV by not using plurality rule at each iteration, instead using some other election procedure which produces a complete ranking of the alternatives. One example of this involves positional procedures. Generally speaking, positional procedures assign a fixed number of points to each alternative based on its ranking on each individual voter s ballot, with alternatives ranked higher on the ballot receiving more points. The alternatives are then ranked according to the number of points they each received, in decreasing order. It is worth noting that plurality rule is a special case of a positional procedure wherein only each voter s top alternative receives any points. To create an iterated positional procedure, we could initially rank all of the alternatives via the positional procedure, eliminate one or more of the lowest-ranked alternatives, and repeat the process using another positional procedure for the new number of alternatives. 4 By doing so, we can analyze how iterated positional procedures compare to their single-stage counterparts. In particular, by analyzing iterated positional procedures, the work presented in this dissertation will analyze the impact of removing alternatives on the collective preference order. With their mathematical structure, positional procedures readily lend themselves to linear algebraic, combinatorial, and statistical analysis. For our research, we will create statistical models which compute the probability of observing consistent preferences between an iterated positional procedure and its single-stage counterpart. For these statistical models, we will assume that elections are equivalent 4 Although the positional procedure on the smaller set of alternatives might be intuitively expected to be based on the original positional procedure, this is not necessary in order to have an iterated positional procedure.

22 6 to sampling with replacement from a large population exhibiting a predetermined distribution of individual preference orders. By doing so, we can utilize the Central Limit Theorem in determining our probabilities. These probabilities can then be interpreted in discussing the extent to which removing alternatives can influence the outcome of the election. In the following chapters, I will analyze positional procedures from two perspectives using statistical models. For the first perspective, for up to five alternatives, we can consider the impact of iterated elections which advance a subset of the candidates at each iteration until a winner is determined. We will compare positional procedures to their iterated versions in finding the probability that their outcomes agree. For this perspective, we will assume that the individual voters are equally likely to exhibit any strict ranking of the alternatives. This is called the Impartial Culture (IC) assumption. In the second perspective, we will relax the IC assumption to allow for any initial probability distribution of the individual voters preferences. Considering the additional dimensions that relaxing the IC assumption adds to the model, we will focus on the three-alternative case for this perspective. With three alternatives, we can explore the probabilities of obtaining the same winner when an alternative is dropped from the election and obtaining consistency between the positional procedure s outcome and the pairwise rankings between each pair of alternatives. So we can better discuss the prior research in this area, I will intuitively define some of the terminology used in the next section. These terms will be formally defined in Section 2.1. Alternatives will be denoted by x 1, x 2, etc., with comparisons

23 7 between two alternatives x i and x j being denoted by x i x j (strict preference) and x i x j (indifference). A voter profile q is the collection of preferences for the voting population. An election procedure assigns a collective preference order to each possible voter profile. 1.2 Prior Research Modern research in social choice theory dates back to the eighteenth century with the proposal of the Borda count and the Condorcet method, by Jean Charles de Borda and Marie-Jean-Antoine-Nicolas de Caritat Condorcet, respectively. At the crux of their debate was whether elections on three or more alternatives should be considered as single comparison involving all of the alternatives or as a series of pairwise comparisons. As the Borda count assigns points to each alternative based on the number of other alternatives ranked below it, the Borda count determines a collective preference ranking from a single comparison. The Condorcet method generalizes the idea of majority rule which is always effective in the two-alternative case to the case of three or more alternatives, seeking to nominate a winner whenever there exists a candidate which is preferred by a majority of voters to each other candidate (called a Condorcet winner). These methods contain flaws, however. The Borda count is susceptible to manipulation by the voters. If a voter ranks x 2 x 1 x 3 and the voter perceives the election to be highly contested between x 1 and x 2, the voter might obtain a better outcome (for herself) if she instead casts the ballot x 2 x 3 x 1, effectively burying x 1 in her ballot in an effort to promote x 2. On the

24 8 other hand, the Condorcet method need not always elect a winner or even produce a transitive collective preference order. If three individuals vote in an election, with one individual each having the preferences x 1 x 2 x 3, x 2 x 3 x 1, and x 3 x 1 x 2, then the Condorcet method produces the pairwise outcomes x 1 x 2, x 2 x 3, and x 3 x 1, each by a 2:1 margin. These pairwise outcomes, however, form a cycle (called the Condorcet cycle) in which no alternative can be clearly considered the winner. In the two centuries following Borda and Condorcet, research in social choice theory was relatively dormant. Social choice theory reawakened in the 1950s when Kenneth Arrow published his Impossibility Theorem [2], which states that with at least two voters and at least three alternatives, the only election procedure which satisfies a particular set of reasonable axioms is a dictatorship. Arrow designed these axioms in an effort to find an election procedure that was democratic and would avoid counterintuitive election outcomes. As he defined some axioms and found some procedures that satisfied those axioms, he would then find some example that led to unsatisfactory results. He would add another axiom to the list, but instead of eventually obtaining a satisfactory election procedure, his axioms led to a proof that, intuitively speaking, all election procedures are flawed in some way. Although differing sources vary on the names of the axioms, I list Arrow s axioms as follows: universality, monotonicity, citizen sovereignty, and independence of irrelevant alternatives (IIA). Universality supposes that voters may submit any possible transitive preference ranking of the alternatives, and every possible collection of the individual preference orders must yield a transitive collective preference ranking. Monotonic-

25 9 ity establishes the intuitive expectation that if a voter increases his support for a particular alternative x i, then x i cannot be ranked lower in the collective preference order as a result. Citizen sovereignty states that every possible collective preference order must be attainable by some voter profile. Last, IIA assumes that the collective preference ranking between alternatives x i and x j must depend only on the individual voters preferences between x i and x j. That is, alternative x k (for k i, j) can have no bearing on the election procedure s ranking of x i and x j. Arrow established that any election procedure satisfying these four axioms must be dictatorial contrary to our desires for a democratic election procedure. Many research directions within social choice theory spawned as a result of Arrow s Impossibility Theorem. Some researchers proceeded with axiomatic approaches in creating and analyzing election procedures. One example of this involves weakening one of Arrow s axioms. In particular, Steven Brams and Peter Fishburn [6] publicized approval voting (AV) in 1978, wherein voters express either approval or disapproval for each candidate when casting their ballots. AV satisfies all of Arrow s axioms except universality; AV fails universality because voters cannot cast a strict ranking between every pair of alternatives. In particular, a voter must express indifference between any pair of approved (or disapproved) candidates. A second example emanates from Duncan Black [5], which considers single-peaked voter preferences. For voters to have single-peaked preferences, we must first be able to align voter preferences along a single dimension (e.g. the real line). Then each voter must have an ideal alternative and rank the rest of her preferences in decreasing order of proximity

26 10 to her ideal alternative. This assumption again limits universality, but Black uses it to establish the median voter theorem, which discusses some conditions (including single-peaked preferences) under which majority rule will select the outcome preferred by the median voter along a spectrum of options. A second direction emanating from Arrow s Impossibility Theorem is the microeconomic, game theoretic direction. These researchers focus on having only a few voters and modeling their decisions as a game. Collective choice is then equated with Nash equilibria. Further information about game theory can be found in Osborne [25] and Mas-Colell [10]. Mas-Colell also discusses further connections between game theory and collective choice. This includes the topic of strategic voting with regard to the Gibbard-Satterthwaite Theorem, which states that with at least three alternatives, all election procedures satisfy at least one of the following conditions: The procedure is dictatorial; some candidate can never win under the procedure; or the procedure is susceptible to strategic voting. Some positive results related to the economics of social choice theory are contained in Austen-Smith and Banks [3, 4]. A third direction analyzes the mathematical structure of election procedures in an effort to provide further insight into why and how frequently certain paradoxes occur. Donald Saari has many significant contributions to the structural understanding of election procedures especially with regard to positional procedures. A primary theme emanating through Saari s research is that election procedures cannot be considered as pairwise comparisons of alternatives due to the expectation that individuals must have transitive preferences. In particular, if an individual has the preferences

27 11 x 1 x 2 and x 2 x 3, then she is forced to have the preference x 1 x 3. Pairwisebased election procedures fail to account for this expectation of individual rationality, which is a primary reason why pairwise-based election procedures can change ranking when removing some of the alternatives from consideration. In particular, Saari rejects the IIA axiom due to the expected impact of the preferences individuals must have based on the transitivity of their preferences. Another primary theme of Saari s contributions involves the geometric and linear algebraic representations of election procedures especially for positional procedures. He has established an intuitive method for decomposing voter profiles into basis elements. In particular, he defines the basis elements according to particular properties that combinations of voters might exhibit. For example, we could have voters whose ballots directly oppose each other (i.e. reversed preference orders such as x 1 x 2 x 3 and x 3 x 2 x 1 ), and we could have voters who ballots combine to form a Condorcet cycle (e.g. x 1 x 2 x 3, x 2 x 3 x 1, and x 3 x 1 x 2 ). By writing voter profiles in terms of basis elements, Saari [27] describes how voter profiles operate under three alternatives. He also describes how some voting paradoxes can occur and presents an algorithm for creating voter profiles which fail IIA under a particular positional procedure. Saari [28, 29] extends this work to the case of more than three alternatives, but with the increased dimensions (i.e. m! basis elements for m alternatives), some of the practicality is diminished. With regard to positional procedures, Saari [26] provides some insight into what can happen when alternatives are removed from consideration. Suppose there

28 12 are m alternatives. For each k = 2, 3,..., m, fix a positional procedure on k alternatives. 5 By analyzing the space of voter profiles, Saari determines that for most combinations of positional procedures, any possible ranking of the alternatives for any subset of the m alternatives can be obtained by some profile. The only major exception to this result is when the Borda count is used for each k. This is because the Borda count satisfies the Condorcet loser criterion, meaning that a candidate who loses every pairwise election cannot win the election over all m alternatives. For example, on three alternatives, we cannot simultaneously obtain the three-alternative outcome x 1 x 2 x 3 and the pairwise outcomes x 2 x 1 and x 3 x 1. Thus, for the Borda count, some combinations of rankings cannot be obtained. Generally, Saari s results in this direction establish that the breadth of voting paradoxes is much greater than we would prefer. Although Saari provides significant results on what can happen when an alternative is removed, he does not discuss the extent to which these voting paradoxes can occur. Although almost any combination of preference rankings can be obtained on subsets of the m alternatives, we have little evidence toward whether or not such combinations are likely to occur in democratic elections. In particular, if there is a small probability that a particular election procedure will produce greatly inconsistent combinations of preferences on the subsets of the m alternatives, then we would have pragmatic evidence in support of using such an election procedure. Alternatively, if 5 The results which follow still hold if one only considers a subset of the values k = 2, 3,..., m.

29 13 we can determine the probability that an election procedure will have consistent preference rankings on subsets of the m alternatives, then we could determine whether or not that election procedure should be used for practical elections either by our own standards or in comparison to other election procedures. Such evidence could be helpful in guiding us toward using election procedures which better represent voters preferences while minimizing ambiguous outcomes. William Gehrlein has completed significant research on this question by analyzing Condorcet winners for any numbers of alternatives and the consistency of positional procedures on subsets of three and four alternatives. He has also organized the current research in these directions in two textbooks, one focused on Condorcetrelated paradoxes [16] and another on the consistency of voting procedures on subsets of alternatives [19]. His discussion of the current research involves the analysis of topics such as Borda s Paradox, Condorcet s Paradox, and the role of voter abstention utilizing experimental, combinatorial, and statistical approaches. Although the experimental and combinatorial approaches can provide some insight into the relationship between election procedures and particular properties, I will focus on the statistical approach (with positional procedures) as my research expands upon this direction. When attempting to use statistical methods to calculate the probability that an election under a particular positional procedure will satisfy some desired property, there exist some challenges in setting up an appropriate statistical model. The general method for this approach is to assume that an election on m alternatives involving n voters is a random sampling of n objects with replacement such that each of the

30 14 distinct possible m! objects (i.e. preference orders) has a fixed probability of being chosen. However, even with just three alternatives and assuming strict, transitive individual preference orders, there are six possible options for each voter. Thus, the probability space for the distribution of individual preference orders lies in R 5. To analyze the entire space at once can be daunting, and for some of the initial methods, researchers were only able to analyze particular subsets of the probability space. They accordingly utilized particular assumptions about the voting population so that their models could provide some results. These assumptions, discussed more in the following paragraphs, are Impartial Anonymous Culture (IAC), Impartial Culture (IC), and Dual Culture (DC). They only cover a small portion of the probability space (which has measure zero), and in an effort to cover a larger portion, some researchers discuss the homogeneity of voters preferences as a measure of how agreeable the voter s preferences are given a probability distribution. As described initially by Kiyoshi Kuga and Hiroaki Nagatani [23] and formalized by Fishburn and Gehrlein [13], the Impartial Anonymous Culture assumption supposes that each distinct voter profile on n voters is equally likely. Restricting our attention to just the possible voter profiles on n voters allows for combinatorial analysis. In particular, for a given property, it can be determined whether or not each voter profile satisfies the property. The voter profiles which satisfy the property can then be counted and divided by the number of possible voter profiles to calculate the probability that a random voter profile will satisfy the property. To perform these computations more efficiently, each voter profile can be assigned to a particular lattice

31 15 point within a polytope, and as one method of counting the lattice points within a polytope, Ehrhart polynomials can be used [8]. General results emanating from the IAC condition are discussed by Gehrlein [16, 19]. One concern about the IAC assumption is that it treats unanimity profiles as equally likely to occur as voter profiles with relatively equally-distributed preferences. For example, with four voters, there is only one way the four voters can cast ballots so that the voter profile is unanimously x 1 x 2 x 3. However, suppose that the voter profile is such that eactly one voter has each of the preference orders x 1 x 2 x 3, x 1 x 3 x 2, x 2 x 1 x 3, and x 2 x 3 x 1. Then there are 24 ways for the four voters to cast their ballots. From this perspective, it would seem more likely for the latter profile to occur in practice. Thus, to consider both profiles as equally likely appears to misrepresent the probability that each profile will actually occur. If we must make an assumption about the voter preferences, we should take into account the notion that voters are more likely to have preferences which disagree to some extent than agree unanimously. By comparison to IAC, as first proposed by Gerhlein, the Impartial Culture assumption supposes that each voter is equally likely to have any strict preference order. The probability of a voter profile q occurring is thus dependent on the number of ways the voters could cast their ballots to produce q. The IC assumption implies that for any sample voter profile having n voters, no alternative has any inherent advantage over any other alternative both in pairwise ranking and in sequential ranking in each preference order. That is, the expected collective preference ranking between

32 16 any subset of the alternatives under any positional procedure is complete indifference. From this expected neutrality between alternatives, a statistical model involving the Central Limit Theorem can be developed to determine the limiting probabilities for sample voter profiles under positional procedures having various properties. This model, which will be discussed in Section 2.2, is the initial motivation for my research. Due to each voter profile q on n voters being weighted by the number of distinct orderings of the voters which can produce q, the methods involving Ehrhart polynomials do not apply in the IC case. In limited situations, exact formulas have been derived for the probability that a voter profile on n (finite) voters satisfies a given property. When comparing the probabilities between IAC and IC in the limiting case, IAC has been shown to imply some dependence between the individual voters preferences whereas IC implies independence [19]. Also discussed first by Gehrlein, the IC assumption can be generalized to the Dual Culture assumption by assuming only that no alternative has any advantage over any other alternative for just the pairwise comparisons. In particular, DC assumes the following about the population: For each preference order, if the probability that any individual voter has the preference order is p 0, then the probability that any individual voter has the reversed preference order is also p 0. For example, if there is a one-fourth probability that a voter has the preferences x 1 x 2 x 3, then there is also a one-fourth probability that a voter has the preferences x 3 x 2 x 1. Generally speaking, the DC assumption proposes that the population is equally split

33 17 on any pair of reversed preference orders. Clearly, the IC assumption is a special case of the DC assumption, and some results which focus on pairwise comparisons, such as Condorcet s paradox and the prevalence of a Condorcet winner in an election, can be explored under DC. However, when attempting to extend the research to positional procedures, the current models in the literature generally fail because they require the expected collective preference ranking to be complete indifference. In particular, while DC implies indifference for any pairwise comparison, DC generally does not imply indifference among subsets of three or more alternatives. 6 Example Suppose there is a one-half probability that a voter has the preference x 1 x 2 x 3. Then there s also a one-half probability that a voter has the preference x 3 x 2 x 1. The probability that a voter has any other preference order is zero. Suppose that election procedure is plurality rule. Then the expected value for any sample election is that x 1 and x 3 each receive half of the vote while x 2 receives no votes. The expected preference order is thus x 1 x 3 x 2. Accordingly, the current statistical models published in the research are unable to produce results for elections sampled from this population. In an effort to discuss arbitrary probability distributions of individual preference orders, R. Abrams [1] developed a measure of homogeneity for three-alternative 6 The one exception is the Borda count, and some analysis of the Borda count under DC with regard to electing a Condorcet winner when a Condorcet winner exists was completed by Gehrlein [15].

34 18 elections. He defined H (p), the homogeneity of the probability vector p, as follows: H (p) = 6 p 2 i. i=1 H (p) is maximized at one when the population unanimously chooses one profile, and H (p) is minimized at 1 6 under the IC condition. In general, larger values for H (p) indicate greater agreement between the voters. The intuition which follows from this homogeneity measure is that paradoxes are less likely to occur with higher values of H (p). However, although experimental research supports this intuition, there is currently no mathematical proof for this. Further, although homogeneity allows us to compare all probability distributions of individual voter preferences, homogeneity does not discuss anything about the actual prevalence of certain properties in sample elections drawn from various probability distributions. Part of my research will define a statistical model that is capable of discussing the intended purpose of homogeneity. It will allow us to analyze arbitrary probability distributions of individual preference orders on three alternatives. Given an arbitrary probability distribution p of individual preference orders, the model will produce the probabilities that a sample election drawn from p will satisty two distinct properties: one, that we obtain the same ranking on the top two alternatives after removing the third-place candidate, and, two, obtaining pairwise results which agree with the initial ranking. Other properties could be studied at a later time, such as properties related to Condorcet winners. Overall, such research provides insight into the likelihood that

35 19 a large election with given preliminary poll results and a fixed election procedure will produce an outcome which passes certain properties. Although Fishburn and Gehrlein do not discuss their research with the context of iterated positional procedures, some of their results under the IC assumption in the three- and four-alternative cases apply. In the three-alternative case, they explore what happens when any alternative is removed from consideration [18]. They determine the probability that any positional procedure agrees with the pairwise outcome on the two remaining alternatives. In the context of iterated positional procedures, if we focus on the removal of the last-place alternative, this is equivalent to computing the probabilities for a two-iteration positional procedure on three alternatives which removes just the last alternative after the first iteration. In the same paper as the three-alternative case, Fishburn and Gehrlein discuss some results for the fouralternative case. They explore the cases of removing any one or two alternatives after the first iteration, but they do not discuss any results for three-iteration positional procedures which remove one alternative after each iteration. In the case of removing two alternatives, which can correspond to a runoff election when the two last-place alternatives are removed, Gehrlein writes out the exact forms for these probabilities in his 2011 book [19], but he only discusses the method for obtaining them in the original paper [18]. Nonetheless, in Chapters 3 and 4, I will reproduce Fishburn and Gehrlein s results, explore the probabilities for three-iteration elections on four alternatives, and expand the research to five alternatives.

36 Outline of Later Chapters Chapter 2 will set forth the assumptions and method for our project. Chapter 3 will utilize this method for analyzing iterated positional procedures for up to five alternatives under the IC assumption, particularly from the perspective of obtaining the same ranking as the original positional procedure at each iteration. Chapter 4 will continue the discussion from Chapter 3 from the perspective of obtaining the same winner as the original positional procedure. Chapter 5 will utilize this method in the three-alternative case using any probability distribution of the individual voters preferences. It will establish the probabilities for obtaining the same ranking when the last-place alternative is removed from the election and for the collective preference order agreeing with all of the pairwise rankings. As well, it will apply the results to two recent United States elections which featured a strong third-party presence: the 1992 United States Presidential election in which Ross Perot earned 19% of the popular vote and the 1998 Minnesota Gubernatorial election in which former professional wrestler Jesse Ventura won the election.

37 21 CHAPTER 2 FOUNDATION 2.1 Definitions and Assumptions Let X be a set of m alternatives in an election. By default, the alternatives X are named x 1, x 2,..., x m, although on occasion, it will be natural to substitute A for x 1, B for x 2, C for x 3, etc. Let N be a finite set of n individual voters, where for all i = 1, 2,..., n, individual i has a total order i of preference over X under the ordering assumptions of completeness and transitivity. In particular, completeness states that for any two alternatives x j and x k in X, either x j i x k or x k i x j. Transitivity requires that for any three alternatives x j, x k, and x l, if x j i x k and x k i x l, then x j i x l. Definition Let x i, x j X. When x i x j and x j x i, the preference between x i and x j is strict and denoted by x i x j. When x i x j and x j x i, the preference between x i and x j is indifferent and denoted by x i x j. In my research, it will be useful to denote all preferences by and. By doing so, one may represent any preference order as a single sequence of preferences (e.g. x 1 x 3 x 4 x 2 ). I will denote by R the set of all possible preference orders on X. Unless stated otherwise, I will assume that all individual preferences are strict. In this setting, P will denote the set of all possible strict preference orders on X. It is obvious that P R. Definition Let X and n be given. A voter profile q is a set of n individual

38 22 preference orders over X, where for i = 1, 2,..., n, i is the individual preference order for the i th voter. Generally, q R n, but if strict individual preferences are assumed, then q P n. Let Q denote the set of all possible profiles (from all possible sizes of n) over X. Given a profile q, a common objective of social choice theory is to associate with q a collective preference order, namely a total ordering of the alternatives in X which represents the preferences expressed by q. Although I assume by default that all individual preference orders are strict, I will never assume that collective preference orders are strict. That is, the entire set of possible collective preference orders is equal to R. Further, in denoting a collective preference order as a sequence of alternatives, I will explicitly use and/or. Definition An election procedure is any function ψ : Q R which assigns a collective preference order to each q Q. An election is defined as the application of ψ to a given voter profile q. Generally speaking, an election procedure could provide almost any assignment of collective preference orders and need not make any practical sense. Considering the motivation of finding a collective preference order which best represents a profile, researchers have defined some properties which election procedures should have in order to be taken seriously. These include nondictatorship, anonymity, neutrality, and unanimity. Briefly, nondictatorship is the idea that no individual s preferences are identical to the collective preference order determined by ψ for every profile q Q. Anonymity implies that if any two individuals preferences are interchanged, then the

39 23 collective preference order stays the same. Neutrality states that if any two alternatives x i and x j are swapped in all individual preference orders, then the collective preference order also swaps x i and x j. Lastly, unanimity requires that for all pairs of alternatives x i, x j X, if all individuals exhibit the preference x i x j, then the collective preference order also exhibits the preference x i x j. We will assume that all election procedures are anonymous and neutral. By doing so, we can define some notation for voter profiles. Consider a profile q. Organize all of the possible strict individual preference orders in lexicographic order (see Example 2.1.4). The number of voters exhibiting each possible preference order can be written as an m!-tuple, where for i = 1, 2,..., m, the i th entry of the m-tuple corresponds to the number of voters exhibiting the i th preference order in the lexicographic ordering of preference orders. 1 Example Suppose m = 3 and q comprises two voters with preferences x 1 x 3 x 2, three voters with preferences x 2 x 1 x 3, and one voter with preferences x 3 x 2 x 1. Then q is represented by the six-tuple (0, 2, 3, 0, 0, 1). Note that the entries of q correspond to the preference orders x 1 x 2 x 3, x 1 x 3 x 2, x 2 x 1 x 3, x 2 x 3 x 1, x 3 x 1 x 2, and x 3 x 2 x 1, respectively, which is the lexicographic ordering of the six strict preference orders on three alternatives. In the following definitions, I rigorously define a particular class of election 1 D. Saari [27] uses a different convention for a profile on three voters. His convention is based on the geometry of individual preference orders going around the Saari Triangle. However, such a convention is not similarly suggested for higher values of n, so I have adopted the lexicographic order as my convention.

40 24 procedures commonly known as positional procedures. The setup here will describe positional procedures in terms of a linear algebraic computation which produces a point total for each alternative. These point totals are then compared to produce a collective preference order. We will study positional procedures at length in the coming chapters. Definition Let the alternative set X be given. Let q be a voter profile on X. The matrix of alternative ranks (MAR) is an m-by-m matrix T such that for each i, j = 1, 2,..., m, (T i,j ) equals the number of voters in q who rank alternative x i in the j th position. The following example illustrates how T is computed from a voter profile on three alternatives. Example Let m = 3. Let q = (q 1, q 2, q 3, q 4, q 5, q 6 ) be given. Recalling the preference order associated with each entry of q, T is given by the following matrix. q 1 + q 2 q 3 + q 5 q 4 + q 6 T = q 3 + q 4 q 1 + q 6 q 2 + q 5 q 5 + q 6 q 2 + q 4 q 1 + q 3 Definition Let X be given. A voting vector is an m-tuple s = (s 1, s 2,..., s m ) such that s 1 > s m and s j s j+1 for j = 1, 2,... m If s 1 = s m, then s would contain identically one value. Any election under s would trivially produce a collective preference order of complete indifference.

41 25 It should be noted that voting vectors describe how many points each alternative receives when ranked in each possible position in an individual preference order. Definition Assume strict individual preference orders. Let the alternative set X and the voting vector s be given. A positional procedure F : Q R is an election procedure on X and s wherein for each q Q, the collective preference order for q is determined as follows: Compute y = T s. For i = 1, 2,..., m, the i th entry of y equals the number of points alternative x i received. For any pair of alternatives x i and x j, rank x i x j whenever y i > y j and x i x j whenever y i = y j. By connecting a positional procedure to its voting vector, we can identify any positional procedure on m alternatives as an s 1 : s 2 :... : s m method. However, in this form, positional procedures need not be unique. For example, on three alternatives, the methods 4 : 2 : 0 and 19 : 1 : 21 define positional procedures which output the same collective preference order for each q. To assist in analyzing positional procedures, a common convention is to normalize their respective voting vectors by forcing s 1 = 1, s m = 0, and the remaining s i -values to be somewhere in [0, 1] such that the sequence {s i } m i=1 is nonincreasing. Given an arbitrary positional procedure, this can be done by first translating the s i values additively so that s m = 0 and then dividing all s i -values by the resulting value for s 1. This allows every positional procedure on X to be uniquely identified by a single voting vector. It is assumed from here onward that all voting vectors are given in this normalized form. Example When s i = 0 for i = 2, 3,... m, the positional procedure is plurality

42 26 rule. When s i = 1 for i = 1, 2, 3,... m 1, the positional procedure is antiplurality rule. 3 The Borda count, usually identified by the voting vector m 1 : m 2 : m 3 :... : 1 : 0, is the positional procedure 1 : m 2 m 1 : m 3 m 1 :... : 1 m 1 : 0. Proposition The collective preference order determined in Definition is a total order. Proof. This proposition follows immediately by the completeness and transitivity of the relation. Because we will be using multiple positional procedures throughout our analysis, we will specify particular notation for the positional procedure on each fixed number of alternatives. When there are l m alternatives under consideration, and the positional procedure on l alternatives ranks x i above x j, we will denote such a ranking by x i l x j. A five-alternative positional procedure will be denoted by 1 : s 2 : s 3 : s 4 : 0. Collectively, the parameters s := {s 2, s 3, s 4 } provides a shorthand for a five-alternative positional procedure. On four alternatives, we will denote a positional procedure as 1 : t 2 : t 3 : 0, with t := {t 2, t 3 } as its shorthand. On three alternatives, we will denote a positional procedure as 1 : u : 0, referenced by u. When only two alternatives remain, there exists only one allowable positional procedure: 1 : 0. 3 Antiplurality rule can be intuitively defined by each voter casting a vote for her weakest preference. The candidates are then ranked in increasing order of votes, with the winner being the candidate who received the fewest votes. To translate this into our framework, we consider voters to vote for every candidate but one.

43 27 Although iterated positional procedures could remove any number of alternatives after each iteration, and the method we develop here can be used to analyze any of them, we will focus on two types of iterated positional procedures. The first is the runoff iterated positional procedure, which is a two stage-election procedure. On the first stage, a positional procedure is used to determine the top two alternatives. These two alternatives are compared via majority rule in the second stage of the election. The second is the maximal iterated positional procedure, which is an (m 1)-stage iteration procedure (similar to IRV) wherein the last-place candidate (under a given positional procedure) is removed at each stage of the election until majority rule determines a winner from the final two candidates. Although this is not discussed here, our methods can be applied to truncated forms of the (m 1)-stage iteration procedure wherein only some of the iterations are completed before a winner is decided from the remaining l > 2 alternatives. Example Baldwin s method is a particular example of a maximal iterated positional procedure wherein the Borda count is used at each iteration. Baldwin s method compares to instant-runoff voting, which is also a maximal iterated positional procedure, by substituting the Borda count for plurality rule at each iteration. 4 We next describe how our individual preference orders are considered at each iteration. Let a voter profile q be given. We begin the election by applying an 4 By definition, IRV is a maximal iterated positional procedure only when all of the iterations are completed. However, if x 1 is declard a winner among three or more alternatives under IRV, then x 1 will still win the election if the remaining iterations are completed. Thus, for our purposes, IRV is always a maximal iterated positional procedure.

44 28 m-alternative positional procedure to q to rank the alternatives. The last-ranked alternative(s) is then dropped from consideration. We next create the profile q which is obtained from q by blotting out the last-ranked alternative(s) from each individual s preferences. For example, when m = 5, k = 1, and x 5 is the last-place alternative, the individual preference orders of x 1 x 2 x 5 x 3 x 4 and x 5 x 1 x 2 x 3 x 4 are considered equally for the positional procedure on four alternatives after x 5 is removed namely as the preference order x 1 x 2 x 3 x 4. 5 After creating q, we apply an appropriate positional procedure to q to rank the alternatives. The lastranked alternative(s) is then dropped from consideration. The process repeats itself until the final iteration is completed. Definition Let F be an iterated positional procedure completed in K iterations. Let 1 : s 2 : s 3 :... : s m 1 : 0 be the positional procedure used in the first iteration. Let q be a voter profile. Then q is ranking-invariant with respect to F if for every k = 1, 2,..., K, for any two alternatives x i and x j, if x i x j in the collective preference ranking determined in the k th iteration, then x i m x j in the collective preference ranking determined in the first iteration. Further, if for any voter profile q, q is ranking-invariant with respect to F, then F is ranking-invariant. Example Let m = 5. Suppose that the positional procedure for a voter profile q yields a collective preference ranking of x 1 5 x 2 5 x 3 5 x 4 5 x 5. 5 This notion of blotting out an alternative from each individual s preference order is consistent with economist Kenneth Arrow s property of Independence of Irrelevant Alternatives [2]. However, other methods could be used, such as having individuals revote or assigning a different positional procedure to each individual based on where she ranked the removed alternative(s).

45 29 Then for q to be ranking-invariant, the collective preference ranking at the iteration involving four alternatives must be x 1 4 x 2 4 x 3 4 x 4, the collective preference ranking at the iteration involving three alternatives must be x 1 3 x 2 3 x 3, and the pairwise ranking between x 1 and x 2 must be x 1 2 x 2. Definition Let F be an iterated positional procedure. Let 1 : s 2 : s 3 :... : s m 1 : 0 be the positional procedure used in the first iteration. Let q be a voter profile. Then q is winner-invariant with respect to F if the winner of the election under 1 : s 2 : s 3 :... : s m 1 : 0 is the same as the winner under F. 6 Further, if for any voter profile q, q is winner-invariant with respect to F, then F is winner-invariant. The following results are straight-forward consequences of Definitions and See Example later in this section for why the assumption on F and p is necessary for these results. Proposition Let F be a positional procedure. Let the unit vector p for the probability distribution of the individual preference orders be given. Assume that F and p are such that for any pair of alternatives x i and x j, there exists a voter profile which does not result in a tie between x i and x j. If a voter profile q is rankinginvariant, then q is winner-invariant. The converse fails. Proposition Let F be a positional procedure. Let the unit vector p for the 6 The winner of an election is also the alternative which is top-ranked by an election procedure. We only consider winner-invariance within the context of the IC condition, and we will later show in Theorem that the probability of obtaining any tie in the collective preference order is negligible. Thus, the notion of a winner is well-defined in our context.

46 30 probability distribution of the individual preference orders be given. Assume that F and p are such that for any pair of alternatives x i and x j, there exists a voter profile which does not result in a tie between x i and x j. On three alternatives, a voter profile q is ranking-invariant if and only if q is winner-invariant. Proof. The only iterated positional procedures on three alternatives are two-stage procedures whose second stage compares the top two alternatives from the first stage. Suppose the collective preference order on three candidates is x 1 3 x 2 3 x 3. If q is winner-invariant, then x 1 2 x 2. But by definition, this is the only ranking required to satisfy ranking-invariance on three alternatives. Definition Let F be an election procedure on m alternatives. Let q be a voter profile. Then q is pairwise-invariant with respect to F if, for any two alternatives x i and x j, x i m x j in the collective preference ranking determined by F, then x i 2 x j in the pairwise comparison of x i and x j under majority rule. Further, if for any voter profile q, q is pairwise-invariant with respect to F, then F is pairwise-invariant. By relaxing the assumption on IC, we will produce new results involving the consistency between a positional procedure s outcome and the pairwise rankings for a voter profile. Extensive research involving pairwise rankings and the particular conditions on the distribution of voter preferences (i.e. IAC, IC, DC, and homogeneity) has been completed, with a survey of the results discussed by Gehrlein [16]. For this research, discussed in Chapter 5, we will test for ranking-invariance and pairwiseinvariance.

47 31 Given the above-defined context of iterated positional procedures and invariance, we can now look forward to how we will analyze them. Democratic elections generally involve the creation of a voter profile by voters casting an anonymous ballot which expresses their preferences over the available candidates. Depending on the election procedure to be used, voters may or may not need to express a complete preference ranking over all of the alternatives. For example, in plurality rule elections, voters only express their top preference. Once the voter profile is created, it is interpreted by the given election procedure to determine a collective preference order on the alternatives. We wish to model this process via a statistical model under some basic assumptions. In Chapters 3 and 4, we will restrict our attention to iterated positional election procedures. In Chapter 5, our focus will be on the pairwise comparisons between the alternatives in relationship to positional procedures. In order for any positional procedure to have meaning, we must further assume that all individual preference orders are anonymous, total, and having only strict preferences between any distinct pair of alternatives. For simplicity, we will assume that voters cast sincere ballots which are independent of all other voters preferences. Further, we will assume that data about the population s distribution of individual preference orders over the alternatives is known. 7 With these assumptions, we can analogize the election process to the combi- 7 For practical elections, polling data can be used to approach this assumption up to some given margin of error.

48 32 natorics exercise of selecting particularly-colored balls from an urn. In particular, on m alternatives, there are m! possible strict preference orders. For each strict preference order, there is a fixed probability that a voter in the population will exhibit this preference order. Thus, to model an election, we could create an urn containing m! differently-colored balls such that the probability distribution of balls in the urn corresponds to the known population s distribution of voter preference orders. We can then create a voter profile containing n voters by sampling n balls from the urn with replacement. 8 The voter profile is then analyzed under the given election procedure to produce a collective preference order. This process is readily repeatable to produce as many voter profiles as desired, which will translate to exploring the probability of observing particular properties in iterated positional procedures. So we can later use the Central Limit Theorem to compute probabilities in the limiting case of arbitrarily large samples, we will consider a voter profile to be an arbitrarily large random sampling of individual preference orders from an infinite population whose probability distribution of preference orders is given beforehand. Reiterating that all individuals have independent preferences determined according to the given probability distribution of individual preference orders among the population, we can say that these samples are independent and identically distributed. Further, we can associate the probability that a random voter from the population has 8 Although this analogy might appear to suggest that some voters might vote multiple times, consider it instead as each of the n voters being assigned a strict preference order based on which ball is drawn from the urn. In this context, no voter casts more than one ballot.

49 33 each possible strict preference order by the unit m!-tuple p, where, for i = 1, 2,... m!, coordinate p i corresponds to the particular strict preference order which results from ordering the m! strict preference orders in lexicographic order. Then any voter profile q containing n voters can be considered as a random sampling with replacement of n voter preferences from the population such that the proportion of each strict preference order in the population is given by p. Example When m = 3, the preference order assigned to each coordinate of p is given in Table 2.1. Table 2.1: Preference Order Assignment for p Preferences p i A B C p 1 A C B p 2 B A C p 3 B C A p 4 C A B p 5 C B A p 6 We are ultimately interested in finding the limiting probability that iterated election procedures are ranking-invariant and winner-invariant. To do this, we require knowledge of some widely-used statistical constructs, namely the Central Limit Theorem (CLT) and the multivariate normal distribution. The Central Limit Theorem listed below uses generalized notation; we will define particular notation for our samples beginning in Section 2.2.

50 34 Theorem (Central Limit Theorem) [34] Let (x 1ξ, x 2ξ,..., x kξ ξ = 1, 2,..., n) be an independent and identically-distributed sample of size n from a k-variate distribution having finite mean vector µ = (µ i ), for i = 1, 2,..., k, and finite covariance matrix Σ = (σ ij ), for i, j = 1, 2,..., k. Let x be the vector of sample means. Then as n, n ( x µ) converges in distribution to the k-variate multinormal distribution with mean 0 and covariance Σ, denoted N k (0, Σ). Definition Let X 1, X 2,..., X k be k normally distributed random varibles with finite mean vector µ and covariance matrix Σ. Assume that Σ is positive definite. Then the probability density function (pdf) for the multivariate normal distribution is given by f (x) = ( ) 1 1 exp (2π) k 2 (x µ)t Σ 1 (x µ). Σ The multivariate normal distribution described in Definition is a wellstudied, continuous distribution in statistics. In general, to find the probability of an event given a continuous distribution, one integrates over the region corresponding to the event. Accordingly, the probability of observing any specific value under a continuous distribution is zero because the integral over a single point is zero. The CLT combined with this fact establishes the following theorem. 9 Theorem Let F be a positional procedure. Let p be given. Assume that F 9 To establish a rigorous proof, use a similar variates construction seen in Chapter 2.2. Then, instead of finding probabilities over the positive orthant of the multinormal integral, find the probability of obtaining the region defined by the desired preference order. For each tie present, the integral bounds will both be zero. Any definite integral from zero to zero is zero.

51 35 and p are such that for any pair of alternatives x i and x j, there exists a voter profile which does not result in a tie between x i and x j. 10 Then the probability that the collective preference ranking induced by F includes a tie between any two alternatives goes to zero for arbitrarily large electorates. Example We can now discuss why the assumption on F and p is necessary in Proposition Let m = 3, u = 1, and p = (p 1, 0, 1 p 1, 0, 0, 0). That is, we are considering an antiplurality election on three alternatives wherein all of the voters have either the preference order x 1 3 x 2 3 x 3 or x 2 3 x 1 3 x 3. Because every voter unanimously has x 3 ranked last while voting for two alternatives, we immediately have x 1 3 x 3 and x 2 3 x 3. However, x 1 and x 2 are indistinguishable under antiplurality for every ballot, implying that x 1 3 x 2 for every sample. However, for the comparison between x 1 and x 2, x 1 and x 2 are distinguishable for every voter. Thus, Theorem applies to conclude that the ranking x 1 2 x 2 occurs with zero probability after removing x 3 from consideration. Hence, any profile sampled from this distribution cannot be ranking-invariant. However, a voter profile sampled from this distribution can be winner-invariant. Let p 1 >.5. Then, by the Central Limit Theorem, we have x 1 2 x 2. Any fair procedure for breaking the tie between x 1 and x 2 has x 1 winning with probability.5. Thus, in half of the samples, the winner of the election between x 1 and x 2 is the same as the winner of the three-alternative antiplurality election. 10 This assumption is necessary due to a few extreme examples in which the probability of a tie can be one. See Example for a particular example.

52 36 We will eventually be interested in integrating the multinormal pdf over the positive orthant of R k. When k 3, closed-form expressions can be found. In special cases for k 4, such closed-form expressions can again be found. In other cases, the integral can be numerically estimated via a combination of Monte Carlo and quadrature methods. In particular, the MATLAB function mvncdf [20] performs this numerical estimation for k 25, and this function will be used in computing our probabilities. 2.2 Method In this section, we will describe the statistical model used to derive our results. This model is influenced by that defined by Gehrlein and Fishburn [16, 19], but we will expand it in two directions. One, by using the MATLAB function mvncdf for numerically estimating positive orthant probabilities via multivariate normal integrals for more than five variates, we can produce results on five alternatives with regard to iterated positional procedures. Two, by applying the Central Limit Theorem in cases where the variates do not have zero expected value, we can determine probabilities for three-alternative elections having particular properties given a probability distribution of the individuals preferences. Before we explore these applications, we will describe the general structure of this model. Let m be fixed. Let p be given as a unit m!-tuple representing the probability distribution of the individual strict preference orders organized by lexicographic order. When the IC condition is in effect, p i = 1 for all i = 1, 2,..., m!. Otherwise, m!

53 37 the entries of p may vary. As alluded to in the prior section, we will consider elections involving n voters as the random sampling of n strict preference orders with replacement from a population whose probability distribution of strict preference orders is given by p. This is analogous to the classic combinatorics exercise in selecting different colored balls from an urn with replacement and determining various probabilities based on the distribution of the balls in the urn. However, by comparison, we will let n become arbitrarily large in determining our desired probabilities. To illustrate our model for analyzing elections, we will first demonstrate the example for the basic case of determining the probability that a voter profile will produce the outcome A 3 B 3 C via the positional procedure 1 : u : 0 and the probability distribution p of individual strict preference orders across the population. After discussing this example, we will discuss how it will be generalized to the cases of interest in the coming chapters. Example Let m = 3 and let p be arbitrary. Fix the positional procedure as 1 : u : 0. Let us set up a model for determining the probability P ABC (u) that a sample election with an arbitrarily large number of voters produces the collective preference order A 3 B 3 C via the positional procedure 1 : u : 0. Step 1: Set up the variables. To satisfy A 3 B 3 C, we require A 3 B and B 3 C. If these two conditions are satisfied, A 3 C follows by transitivity. Thus, we have two variables to define. Let Y1 i denote the contribution of the i th voter toward A 3 B, and let Y2 i denote the contribution of the i th voter toward B 3 C. For each strict preference

54 38 order, Y i 1 and Y i 2 will be assigned a value based on the ranking of the two alternatives particular to each variable. For example, Y i 1 has the value of u 1 for the preference order B 3 A 3 C because A receives u points as the second-ranked alternative and B receives one point as the first-ranked alternative. Table 2.2 lists the values for the variables Y i 1 and Y i 2. Table 2.2: Values for Y i 1 and Y i 2. Preference Order Y1 i Y2 i A 3 B 3 C 1 u u A 3 C 3 B 1 u B 3 A 3 C u 1 1 B 3 C 3 A 1 1 u C 3 A 3 B u 1 C 3 B 3 A u u 1 Step 2: Compute the Expected Values We next compute the expected values for Y i 1 and Y i 2. This is found by multiplying the probability of each strict preference order by its value for the variable. We obtain the following expected values. E ( Y i 1 ) = (1 u) (p1 p 3 ) + (p 2 p 4 ) + u (p 5 p 6 ) E ( Y i 2 ) = (1 u) (p4 p 6 ) + (p 3 p 5 ) + u (p 1 p 2 ) Note that under the IC condition, both of the above expected values are zero.

55 39 In general, though, for arbitrary p, the expected values will almost always be nonzero. Regardless, by comparing the expected values to zero and utilizing the Central Limit Theorem, we can determine how to calculate P ABC (u). In particular, P ABC (u) corresponds to the probability that when an arbitrarily large sample with replacement is taken from the population, the sample means Ȳ1 and Ȳ2 are jointly positive. By computing the expected values of these variables (not the samples), we can infer some information about the samples which can be chosen. As seen in Step 3, there are precisely four cases which may occur. Step 3: Interpret the Expected Values If both of the expected values are zero, skip immediately to Step 4. If at least one of the expected values is nonzero, this step will be the last step in the process. Case 1: One or both of E (Y i 1 ) and E (Y i 2 ) are negative. Without loss of generality, assume that E (Y i 1 ) is negative. In this case, the Central Limit Theorem implies that for large n, the value of Ȳ1 will approach this negative value with variance approaching zero. Thus, for any ɛ > 0, we can find a large enough N so that the probability of finding a sample with positive mean is less than ɛ. Hence, we may conclude that P ABC (u) = 0. Note that if the probability of finding a sample with positive mean for one of the variables is zero, then the probability of finding a sample with positive mean for both variables is also zero. Case 2: Both of E (Y i 1 ) and E (Y i 2 ) are positive. The Central Limit Theorem implies that for large n, the values of Ȳj, for j = 1, 2, will approach their respective, positive expected values with variance approaching zero. Thus, for any ɛ > 0, we can

56 40 find a large enough N so that the probability of finding a sample with positive mean in both variables is greater than 1 ɛ. Hence, we may conclude that P ABC (u) = 1. Case 3: One expected value is positive and one expected value is zero. Without loss of generality, assume that E (Y i 1 ) is zero. Because E (Y i 2 ) > 0, the argument from Case 2 applies to conclude that the probability that Ȳ2 > 0 is one. Thus, the probability that Y i 1 and Y i 2 are jointly positive is the same as the conditional probability that Y i 1 is positive. Because E (Y i 1 ) = 0, we only need to check the variance σ 11 for Y i 1. If σ 11 0, finding P ABC (u) is equivalent to computing the probability of choosing a positive random number from a uniform distribution of real numbers along the real number line. Thus, P ABC (u) =.5. If σ 11 = 0, we then have a case where Y i 1 = 0 for all i. 11 Thus, A 3 B, implying that P ABC (u) = 0. Step 4: Set up the Central Limit Theorem in the Case that Both Expected Values are Zero. This is the base case for most of the prior research in this area using models of this type. Whenever the IC assumption is used, this case will necessarily apply. However, it is possible for both expected values to be zero without satisfying the IC assumption. For example, let u =.5 and choose any p satisfying the DC condition. To compute P ABC (u), first observe that E ( Ȳ j ) = 0 for any fixed n and for j = 1, 2. Then E ( Ȳ j n ) = 0 by the summation and scalar multiplication properties 11 One example where this can happen is when u = 1 and p = (.5, 0,.5, 0, 0, 0). Every voter casts an antiplurality ballot for A and B, and it can be verified from Table 2.2 that the entries for A 3 B 3 C and B 3 A 3 C are zero. Thus, the collective preference order is necessarily A 3 B 3 C. We assume that ties remain ties for these elections.

57 41 of expected values. The joint probability that Ȳj n > 0, for both j = 1, 2, is accordingly equivalent to the joint probability that Ȳj ) n > E (Ȳj n = 0, for both j = 1, 2. In the limit as n, the Central Limit Theorem implies that the joint distribution of these two variates is bivariate normal. Recalling that the probability of observing any specific value under a continuous distribution is zero because the integral over a single point is always zero, it suffices to consider the joint probability that Ȳj ) n E (Ȳj n = 0, for both j = 1, 2. We thus are interested in Φ 2 (Σ 1 ), which denotes the bivariate positive normal orthant probability of the binormal distribution with covariance matrix Σ 1 given by the covariances of the two variates. Step 5: Calculate Σ 1 by Computing Covariances. We next find the entries of Σ 1 in terms of u and p. This requires finding the covariance between each pair of variables (with repetition). In particular, the covariance between variables X j and X k is given by σ jk = E (X j X k ) E (X j ) E (X k ). However, we know that E ( ) Yj i = 0 for each j = 1, 2, which implies that for our two variates, σ jk = E ( ) ( ) Yj i Yk i. We list the computations for E Y i j Yk i below. E E ( (Y ) ) i 2 1 = (1 u) 2 (p 1 + p 3 ) + (p 2 + p 4 ) + u 2 (p 5 + p 6 ) ( (Y ) ) i 2 2 = (1 u) 2 (p 4 + p 6 ) + (p 3 + p 5 ) + u 2 (p 1 + p 2 ) E ( Y i 1 Y i 2 ) = u (1 u) (p1 + p 6 ) + (u 1) (p 3 + p 4 ) u (p 2 + p 5 )

58 42 Thus, we can write Σ 1 as follows. E Σ 1 = ( (Y i 1 ) 2) E (Y i 1 Y i E (Y i 1 Y i 2 ) E ( (Y i 2 ) 2 ) 2) Step 6: Calculate the Probability We can now set up the multinormal integral using Definition and Σ 1. We choose the bounds on the integral corresponding to the positive orthant. In the case of two variates, there exists an exact expression for the positive orthant probability. However, this exact expression requires the correlation matrix, which is a standardization of the covariance matrix. To compute the correlation ρ ij between two variables X i and X j, divide σ ij by the product σ ii σjj (i.e. the product of the standard deviations of the corresponding variables). Although we could compute the entire correlation matrix in this case, the correlation matrix derived from Σ 1 has ones on the diagonal entries and only one value of interest on the off-diagonal, namely ρ 12. In particular, ρ 12 = (Σ 1 ) 1,2 (Σ1 ) 1,1 (Σ1, and it can be computed in terms of u and p. To ) 2,2 ( evaluate P ABC (u), Sheppard s formula [32] tells us that P ABC (u) = cos 1 ρ 12 π The above model can be generalized to test various properties on three or ). more alternatives. 12 Properties which may be tested via this model are those which can be written as a series of conditions involving exactly two candidates wherein one candidate must defeat another via a positional procedure. In particular, such prop- 12 Arguably, m could be any value so long as k 25 in order to use mvncdf on MATLAB, but setting up the model becomes cumbersome and increasingly computationally expensive as m increases.

59 43 erties include the existence of Condorcet Winners and Condorcet Losers, agreement with pairwise outcomes, ranking-invariance, and winner-invariance. Gehrlein [16, 19] discusses the present research on the first two categories of properties in greater detail. In testing for a feasible property via the model, for each of the conditions the property requires, we can follow Step 1 to define a variable which takes on a particular value for each strict preference order. Once all of the variables are defined, the probability of satisfying the according property is equivalent to the joint probability that all of the variables are simultaneously positive. We next compute the expected values of these variables as in Step 2. In many cases, at most one of the variables will have an expected value equal to zero or at least one variable will have negative expected value. For these cases, Step 3 will apply to compute the probability. For the remaining cases, a modified form of Steps 4-6 will apply depending on how many of the variables have an expected value equal to zero. For the properties tested in the following chapters, we will describe Step 4 in further detail. As will be confirmed in Section 3.1, the probabilities determined by Fishburn and Gehrlein in Section 1.2 follow the prior example except for having more variates and utilizing the IC condition. In the cases involving two or three variates, exact values can be found [7]. For the four- and five-variate cases, exact values can be found in special cases as discussed by Johnson and Kotz [22]. Gehrlein [14] develops a numerical method for computing the probabilities for the four- and five-variate cases using the correlation matrix. More generally, the MATLAB program mvncdf, based on the method developed by A. Genz and F. Bretz [20], can be used to nu-

60 44 merically estimate positive orthant probabilities up to twenty-five variables using a combination of quadrature and Monte Carlo methods. This program allows for the use of covariance matrices (i.e. correlation matrices need not be calculated) and will be utilized to calculate the probabilies in Chapters 3 and 4. The following chapters will utilize similar models on more variates. In Chapters 3 and 4, which will discuss iterated election procedures on five alternatives under the IC assumption, our number of variates will range from five to ten. In particular, when computing probabilities under the IC condition, we will always be certain that the covariance/correlation matrix has full rank. However, when p is arbitrary, it is possible for the covariance matrix to have less than full rank possibly having a row of zeros as well. Thus, in these cases, the correlation matrix will either have less than full rank or be undefined. Further discussion on these cases will be found in Chapter 5, which considers iterated election procedures on three alternatives for any p.

61 45 CHAPTER 3 RANKING-INVARIANCE UNDER THE IMPARTIAL CULTURE ASSUMPTION In this chapter, we will discuss ranking-invariance for iterated positional procedures under the IC assumption for up to five alternatives. Section 3.1 will reproduce results from Fishburn and Gehrlein [18] in the three-alternative case. Sections will discuss new results in the four-alternative and five-alternative cases, respectively, for maximal iterated positional procedures. 1 Section 3.5 will explore the particular case of ranking-invariance for runoff elections for up to five alternatives. As a point of notation, for a general iterated positional procedure F, let P R m (F ) denote the probability that a voter profile q on m alternatives is ranking-invariant under the iterated positional procedure F. Let P P R m (F ) denote the probability that a voter profile q on m alternatives is ranking-invariant under the runoff iterated positional procedure F. The complement of P R m (F ) (respectively P P R m (F )) corresponds to the probability that iteration leads to a different ranking after some iteration (respectively the second iteration) compared to the ranking determined by the positional procedure during the first iteration. For this notation, when F is explicitly given by some set of positional procedures s, t, and/or u, then F will be replaced by the appropriate variables. 1 Recall that maximal iterated positional procedures remove the last-place alternative after each iteration until only one alternative remains.

62 Ranking-Invariance when m = 3 Let F be a maximal iterated positional procedure on three alternatives. Let the positional procedure on three alternatives be 1 : u : 0 and assume the IC condition for the individual preference orders. Then in the probability distribution vector p, for all i = 1, 2,... 6, p i = 1 6. We want to compute P R 3 (u). By following Example 2.2.1, we find that the probability for obtaining each strict collective preference order is Further, by neutrality, each alternative is considered equally under the positional procedure. Thus, we may restrict our attention to the collective preference ranking x 1 3 x 2 3 x 3 and multiply the resulting probability by six. When m = 3, for a voter profile q, ranking-invariance only requires the same pairwise outcome on the top two alternatives determined by the positional procedure 1 : u : 0 on q. For example, given a voter profile q on three alternatives whose collective preference ranking is x 1 3 x 2 3 x 3, for q to be ranking-invariant, q must satisfy the pairwise comparison x 1 2 x 2. We will set up the model to test these three variables: x 1 3 x 2, x 2 3 x 3, and x 1 2 x 2. After the model calculates a probability for voter profiles having the collective preference order x 1 3 x 2 3 x 3, we will multiply this probability by six to produce the probability over all voter profiles. To satisfy x 1 3 x 2 3 x 3, let Y i 1 denote the contribution of the i th voter toward x 1 3 x 2, and let Y i 2 denote the contribution of the i th voter toward x 2 3 x 3. 2 This provides an alternate proof to Theorem in the case of IC because the entire probability is assigned to the six strict preference orders, meaning that any preference order containing indifference between two alternatives must have zero probabilitiy. Such a result is necessary in order for us to focus our attention on strict collective preference orders.

63 47 To fulfill the pairwise comparison requirement, let Y i 3 denote the contribution of the i th voter toward x 1 2 x 2. For j = 1, 2, 3, Table 3.1 lists the values for Y i j for each preference order. Table 3.1: Values for the Three Variables in the Three-Alternative Case. Preference Order Y1 i Y2 i Y3 i x 1 3 x 2 3 x 3 1 u u 1 x 1 3 x 3 3 x 2 1 u 1 x 2 3 x 1 3 x 3 u x 2 3 x 3 3 x u 1 x 3 3 x 1 3 x 2 u 1 1 x 3 3 x 2 3 x 1 u u 1 1 Because we are using the IC condition, it can be readily verified that for all j = 1, 2, 3, E ( Y i j ) = 0. Thus, by following Step 4 from the prior section, the probability that a voter profile q is ranking-invariant will be equivalent to integrating a three-variate multinormal distribution over the positive orthant with covariance matrix Σ 2. To set up this integral, we must next compute the covariances. Recalling that σ jk = E ( ) ( ) Yj i Yk i, we list the computations for E Y i j Yk i below. E ( (Y ) ) i 2 1 = E E ( (Y ) ) i 2 2 = 2 ( ) 1 u + u 2 3 ( (Y ) ) i 2 3 = 1 E ( ) Y1 i Y2 i 1 ( ) = 1 u + u 2 3

64 48 E ( ) Y1 i Y3 i 2 = 3 E ( ) Y2 i Y3 i 1 = 3 Thus, we can write Σ 2 as follows. Σ 2 = 2 (1 u + 3 u2 1 ) (1 u + 3 u2 2 ) (1 u + u2 ) (1 u + 3 u2 1 ) We can now set up the multinormal integral using Definition and Σ 2. We choose the bounds on the integral corresponding to the positive orthant. In the case of three variates, using the correlation matrix, there exists an exact expression [7] for the positive orthant probability. This formula is generally attributed to Boole: P = 1 ( ) 2π cos 1 ρ 12 cos 1 ρ 13 cos 1 ρ 23. 4π To use this formula, we compute the correlation values from Σ 2. We find ρ 12 = 1 2 ρ 13 =, and ρ 1 3(1 u+u 2 ) 23 =. Thus, recalling that we must multiply the 6(1 u+u 2 ) resulting probability by six, we find that ( ( ) ( )) P3 R (u) = 3 4π 2π 3 2 cos 1 cos (1 u + u 2 ) 6 (1 u + u 2 ) 2, By letting u range over [0, 1], we can compare P R 3 over all positional procedures. A graph of these values is shown in Figure 3.1. We find that P R 3 ranges from a

65 49 Figure 3.1: P R 3 (u). minimum of.755 at both plurality and antiplurality rule to a maximum of.853 at the Borda count. P R 3 is also symmetric about u =.5. A proof for these absolute extrema values and the symmetry of P R 3 about u =.5 was given by Gehrlein and Fishburn [17]. Therefore, when m = 3, under the IC condition, the Borda count is least likely to produce a different ranking under iteration, whereas plurality rule and antiplurality rule are equally the most likely to produce a different ranking under iteration. 3.2 Ranking-Invariance when m = 4 Let F be a maximal iterated positional procedure. Let t and u be given as the positional procedure on four alternatives and three alternatives, respectively. Assume the IC condition for the individual preference orders. We want to compute

66 50 P4 R (t, u). Similar to the prior section, for all i = 1, 2,... 24, p i = 1 for the probability 24 distribution vector p, and the probability for obtaining each strict collective preference order is 1. Further, by neutrality, we may similarly restrict our attention to the 24 collective preference ranking x 1 4 x 2 4 x 3 x 4. When m = 4, for a voter profile q with the collective preference ranking x 1 4 x 2 4 x 3 x 4, ranking-invariance requires the outcomes x 1 3 x 2 3 x 3 and x 1 2 x 2. We will set up the model to test these six variables: x 1 4 x 2, x 2 4 x 3, x 3 4 x 4, x 1 3 x 2, x 2 3 x 3, and x 1 2 x 2. After the model calculates a probability for voter profiles having the collective preference order x 1 4 x 2 4 x 3 4 x 4, we will multiply this probability by 24 to produce the probability over all voter profiles. To satisfy x 1 4 x 2 4 x 3 x 4, let Y i 1 denote the contribution of the i th voter toward x 1 4 x 2, let Y i 2 denote the contribution of the i th voter toward x 2 4 x 3, and let Y i 3 denote the contribution of the i th voter toward x 3 4 x 4. In order to satisfy x 1 3 x 2 3 x 3, let Y i 4 denote the contribution of the i th voter toward x 1 3 x 2, and let Y i 5 denote the contribution of the i th voter toward x 2 3 x 3. To complete the pairwise comparison requirement, let Y i 6 denote the contribution of the i th voter toward x 1 2 x 2. For j = 1, 2,..., 6, Table 3.2 lists the values for Y i j for each preference order. Because we are using the IC condition, it can be readily verified that for all j = 1, 2,..., 6, E ( Y i j ) = 0. Thus, by following Step 4 from the prior section, the probability that a voter profile q with collective preference ranking x 1 4 x 2 4 x 3 4 x 4 is ranking-invariant will be equivalent to integrating a six-variate multinormal

67 51 Table 3.2: Values for the Six Variables in the Four-Alternative Case. Preference Order Y1 i Y2 i Y3 i Y4 i Y5 i Y6 i x 1 3 x 2 3 x 3 x 4 1 t 2 t 2 t 3 t 3 1 u u 1 x 1 3 x 2 3 x 4 x 3 1 t 2 t 2 t 3 1 u u 1 x 1 3 x 3 3 x 2 x 4 1 t 3 t 3 t 2 t 2 1 u 1 x 1 3 x 3 3 x 4 x 2 1 t 2 t 2 t 3 1 u 1 x 1 3 x 4 3 x 2 x 3 1 t 3 t 3 t 2 1 u u 1 x 1 3 x 4 3 x 3 x 2 1 t 3 t 3 t 2 1 u 1 x 2 3 x 1 3 x 3 x 4 t t 3 t 3 u x 2 3 x 1 3 x 4 x 3 t t 3 u x 2 3 x 3 3 x 1 x 4 t t 2 t u 1 x 2 3 x 3 3 x 4 x t 2 t 2 t u 1 x 2 3 x 4 3 x 1 x 3 t t 2 u x 2 3 x 4 3 x 3 x t 3 t 3 t u 1 x 3 3 x 1 3 x 2 x 4 t 2 t 3 t u 1 1 x 3 3 x 1 3 x 4 x 2 t t 3 u 1 1 x 3 3 x 2 3 x 1 x 4 t 3 t 2 t u u 1 1 x 3 3 x 2 3 x 4 x 1 t 2 t t 3 u u 1 1 x 3 3 x 4 3 x 1 x 2 t t 2 u 1 1 x 3 3 x 4 3 x 2 x 1 t 3 t t 2 u u 1 1 x 4 3 x 1 3 x 2 x 3 t 2 t 3 t u u 1 x 4 3 x 1 3 x 3 x 2 t 2 t 3 t u 1 x 4 3 x 2 3 x 1 x 3 t 3 t 2 t 2 1 u x 4 3 x 2 3 x 3 x 1 t 2 t 2 t 3 t u 1 x 4 3 x 3 3 x 1 x 2 t 3 t 2 t 2 1 u 1 1 x 4 3 x 3 3 x 2 x 1 t 3 t 3 t 2 t 2 1 u u 1 1

68 52 distribution over the positive orthant with covariance matrix Σ 3. To set up this integral, we compute the covariances. We list the computations for E ( Y i j Y i k ) below. ( E Y i2 j ) = 1 (3 (1 + 6 t2 2 + t 2 3) 2 (t 2 + t 3 + t 2 t 3 )) := η, for j = 1, 2, 3. 6 E (Y i 1 Y i 2 ) = E (Y i 2 Y i 3 ) = η 12. E (Y i 1 Y i 3 ) = 0. E (Y i 1 Y i 4 ) = E (Y i 2 Y i 5 ) = 1 4 (2 t 2 t 3 + t 2 u + t 3 u) := λ 4. E (Y1 i Y5 i ) = E (Y2 i Y4 i ) = E (Y7 i Y9 i ) = λ. E (Y i 8 1 Y6 i ) = 3+t 2 t 3 := ω. E (Y i Y6 i ) = ω ( ) ( ) E (Y i 3 Y i 4 ) = E (Y i 7 Y i 10) = 0. E Y i2 4 = E Y i2 5 E (Y i 4 Y i 5 ) = ν 3. E (Y i 4 Y i 6 ) = 2 3. E (Y i 5 Y i 6 ) = 1 3. E ( = 2 (1 u + 3 u2 ) := 2ν. 3 ) Y i2 = Thus, Σ 3 is given by the following matrix: η 6 η λ λ 8 η η λ λ ω η λ Σ 3 = 2ν ν ν ω 6 We can now set up the multinormal integral using Definition and Σ 3. We choose the bounds on the integral corresponding to the positive orthant. In the case of six variates, there is no known method for obtaining an exact answer for our covariance matrix. However, we can use the MATLAB program mvncdf to compute the probability numerically. The result is then multiplied by 24 to obtain P4 R (t, u)

69 53 with accuracy to at least four decimal places. 3 In presenting the data for P R 4 (t, u), we note that as there are three parameters among t and u, we must isolate particular slices of the data. We will consider three restrictions of the parameters. One, we will choose t to correspond to plurality rule, Borda count, or antiplurality rule. Two, we will choose u to be plurality rule, Borda count, or antiplurality rule. Three, we will choose u based upon t via a process we will call natural iteration. It is defined for any number of candidates in Definition Definition Let F be a maximal iterated positional procedure on m alternatives. For 2 l m, let the positional procedure on l alternatives be given by 1 : v 2 : v 3 :... : v l 1 : 0. Then F satisfies natural iteration if and only if for every integer l such that 2 l < m, the positional procedure on l alternatives used by F satisfies one of the following two conditions: 1. If the positional procedure used by F on l + 1 alternatives is not antiplurality rule, then the positional procedure on l alternatives is given by 1 : v 2 v l v 1 v l : v 3 v l v 1 v l :... : v l 1 v l v 1 v l : If the positional procedure used by F on l + 1 alternatives is antiplurality rule, then the positional procedure on l alternatives is again antiplurality rule. 3 At the cost of increased run time, greater precision can be obtained.

70 54 Example Let 1 :.7 :.2 : 0 be the positional procedure on four alternatives. The natural maximal iterated positional procedure has 1 :.625 : 0 and 1 : 0 for its positional procedures on three and two alternatives, respectively. Example Baldwin s Method, which is IRV with the Borda count used at each iteration instead of plurality rule, is a maximal natural iterated positional procedure. Our first graphs consider the popular positional procedures of plurality rule, Borda count, and antiplurality rule. In Figures , the graphs of P R 4 are shown for these procedures. Observe the symmetric relationship between plurality rule and antiplurality rule, wherein the plots are reflected over the line u =.5. For plurality rule, the numerical minimum of.2367 occurs when u = 1 (i.e. alternating plurality and antiplurality rule); the numerical maximum of.4036 occurs at approximately u =.148. For antiplurality rule, the same results occur at the complementary values for u. For the Borda count, the numerical minimum of.4159 occurs twice: when u = 0 and u = 1. The numerical maximum of.6194 occurs at u =.5, which corresponds to Baldwin s method. In general, when choosing the positional procedure on four alternatives, the Borda count is least likely to produce a change in the collective preference order after iteration. Among these three positional procedures, while the location of the maximum varies, the minimum is produced by choosing u to be the farthest away from the initial method (e.g. choose u to be plurality rule when t is antiplurality rule). We pause to note two observed reflections from Figures One, Figure 3.2 and Figure 3.3 are reflections of each other. Two, the graph of the Borda count

71 55 Figure 3.2: P R 4 ((0, 0), u) for plurality rule on the first iteration. Figure 3.3: P R 4 ((1, 1), u) for antiplurality rule on the first iteration.

72 56 Figure 3.4: P R 4 (( 2, ) ) 1 3 3, u for the Borda count on the first iteration. in Figure 3.4 reflects over the line u =.5. These reflections are not by accident. By considering the complement of an iterated positional procedure and the reversal of individual preference orders, we can prove that the according probabilities are equal. Definition Let v = (v 1 = 1, v 2,..., v n 1, v n = 0) correspond to a positional procedure on n alternatives. Define the complement ( v) of the positional procedure by 1 : 1 v n 1 :... : 1 v 2 : 0. Example The complement of the positional procedure 1 :.7 :.4 : 0 is 1 :.6 :.3 : 0. Definition Let i be an individual preference order. The reversal i is defined as follows: For any two alternatives x j and x k, if x j i x k, then x k i x j. Example The reversal of the preference order x 1 x 2 x 3 x 4 is x 4 x 3 x 2 x 1.

73 57 Theorem Let t and u be given. Then P R 4 (t, u) = P R 4 ( t, ū). Proof. We can pair each individual preference order with its reversal. Then the contribution by to each Y i j, for j = 1, 2,..., 6, for P R 4 (t, u) is the same as the contribution by to each Y i j, for j = 1, 2,..., 6, for P R 4 ( t, ū). Because reversal pairs of individual preference orders partition the space of individual preference orders, the ensuing calculations for P R 4 (t, u) and P R 4 ( t, ū) are the same. We can also explore the impact of u on the probabilities for P R 4. We plot the graphs for u corresponding to plurality rule, the Borda count, and antiplurality rule. For each plot, we show three angles of the graph. The first is the threedimensional graph plotting P R 4 across all t. The other two angles are slices of the three-dimensional graph, one with t 2 =.7 and the other with t 3 = 0.3. These slices are representative of the behavior of the three-dimensional graph over all t. For plurality rule, the numerical estimate of the maximum is.4716, occurring at approximately 1 :.392 :.059 : 0. The numerical estimate of the minimum is.2367 when antiplurality rule is used on four alternatives. Figures show the graph for plurality rule on four alternatives. Antiplurality rule is the mirror of plurality rule, and by Theorem 3.2.8, we obtain the same probabilities on complement election procedures. Its plots are shown in Figures The Borda count attains some of the higher probabilities for P4 R. The numerical minimum of.3593 occurs at plurality and antiplurality rule. The numerical maximum of.6194 occurs at the Baldwin method. Because u =.5, complementary procedures on t will yield the same values. Thus, the plot for the Borda count on three alternatives is symmetric over the plane

74 58 Figure 3.5: P R 4 (t, 0) for plurality rule in the second iteration. t 3 = 1 t 2. The graphs for the Borda count are shown in Figures Overall, when the Borda count is used in the second iteration, the probabilities are generally higher by ten to twenty percentage points. Accordingly, using the Borda count in the second iteration implies a decreased probability of changing the collective preference order via iteration. Also, the probabilities are smallest when the positional procedure used in the first iteration is furthest away from the positional procedure used in the second iteration (e.g. choose t to be plurality rule when u is antiplurality rule). We now consider how P R 4 changes over t when u is derived from t via natural iteration. In Figure 3.14, P R 4 is plotted for t at =.005. The numerical maximum is approximately.6194 at 1 :.665 :.335 : 0, the closest point in the mesh to Baldwin s method. The numerical minimum is approximately.2377 at 1 :.995 :.995 : P R 4 is not continuous for natural iteration whenever antiplurality rule is used in an iteration involving three or four alternatives.

75 59 Figure 3.6: Slice of the graph of P R 4 (t, 0) with t 2 =.7. Figure 3.7: Slice of the graph of P R 4 (t, 0) with t 3 =.3.

76 60 Figure 3.8: P R 4 (t, 1) for antiplurality rule in the second iteration. Figure 3.9: Slice of the graph of P R 4 (t, 1) with t 2 =.7.

77 61 Figure 3.10: Slice of the graph of P R 4 (t, 1) with t 3 =.3. Figure 3.11: P R 4 (t,.5) for the Borda count in the second iteration.

78 62 Figure 3.12: Slice of the graph of P R 4 (t,.5) with t 2 =.7. Figure 3.13: Slice of the graph of P R 4 (t,.5) with t 3 =.3.

79 63 Figure 3.14: P R 4 (t, u) for natural iteration. Figures plot slices of the three-dimensional graph for t 2 = 0.7 and t 3 = 0.3, respectively. Slices containing these general curves can be seen across all fixed t 2 and t 3. Compared to other iterated positional procedures with the same positional procedure t on four alternatives, the probabilities for ranking-invariance for natural iteration tend to be among the higher values (but not necessarily the maximum). Accordingly, if one wanted a simple rule for choosing an iterated positional procedure on four alternatives, by considering the numerical evidence for ranking-invariance, natural iteration would be an effective option. Overall, based on our discussion, our numerical evidence suggests that on four alternatives, iteration is more likely to alter the original collective preference order in two regards. One of them is when the initial positional procedure differs greatly from the Borda count. The other is when the positional procedure used in the second

80 64 Figure 3.15: Slice of the graph of P R 4 (t, u) for natural iteration with t 2 = 0.7. Figure 3.16: Slice of the graph of P R 4 (t, u) for natural iteration with t 3 = 0.3.

81 65 iteration is the opposite of the positional procedure used in the first iteration. 3.3 Ranking-Invariance when m = Method In this section, we set up the method for establishing P R 5 (s, t, u). Similar to the setup for m = 3, 4, we will focus on x 1 5 x 2 5 x 3 5 x 4 5 x 5 as the collective preference order. Under the IC condition, the probability of obtaining each strict collective preference order for five alternatives is uniformly 1. Accordingly, we 120 will find the probability that a profile is ranking-invariant while having the collective preference order x 1 5 x 2 5 x 3 5 x 4 5 x 5 and multiply the result by 120 to obtain P R 5. For such a profile to be ranking-invariant, we must find that the second-stage (t) outcome is x 1 4 x 2 4 x 3 4 x 4, the third-stage (u) outcome is x 1 3 x 2 3 x 3, and the final-stage (majority rule) outcome is x 1 2 x 2. This model requires ten variates. We ll first define the four variates for the assumed collective preference order x 1 5 x 2 5 x 3 5 x 4 5 x 5. Let Y i 1 denote the contribution of the i th voter toward x 1 5 x 2, let Y i 2 denote the contribution of the i th voter toward x 2 5 x 3, let Y i 3 denote the contribution of the i th voter toward x 3 5 x 4, and let Y i 4 denote the contribution of the i th voter toward x 4 5 x 5. The next six variates are similar to those defined for m = 4 except that they range over 120 strict preference orders instead of 24. To satisfy x 1 4 x 2 4 x 3 x 4, let Y i 5 denote the contribution of the i th voter toward x 1 4 x 2, let Y i 6 denote the contribution of the i th voter toward x 2 4 x 3, and let Y i 7 denote the contribution of the

82 66 i th voter toward x 3 4 x 4. To satisfy x 1 3 x 2 3 x 3, let Y i 8 denote the contribution of the i th voter toward x 1 3 x 2, and let Y i 9 denote the contribution of the i th voter toward x 2 3 x 3. To fulfill the pairwise comparison requirement, let Y i 10 denote the contribution of the i th voter toward x 1 2 x 2. Appendix A contains the values for all ten variables over the 120 strict preference orders. Because we are using the IC condition, it can be readily verified that for all j = 1, 2,..., 10, E ( Y i j ) = 0. Thus, by following Step 4 from Section 2.2, the probability that a voter profile q with x 1 5 x 2 5 x 3 5 x 4 5 x 5 as the collective preference ranking is ranking-invariant will be equivalent to integrating a ten-variate multinormal distribution over the positive orthant with covariance matrix Σ 4. To set up this integral, we will compute the covariances. We list the computations for E ( Y i j Y i k ) below. ( E Y i2 j ) = 1 (2 (1 + 5 s2 2 + s s 2 4) s 2 s 3 s 2 s 4 s 3 s 4 s 2 s 3 s 4 ) := α, for 5 j = 1, 2, 3, 4. E (Y i 1 Y i 2 ) = E (Y i 2 Y i 3 ) = E (Y i 3 Y i 4 ) = α 10. E (Y i 1 Y i 3 ) = E (Y i 1 Y i 4 ) = E (Y i 2 Y i 4 ) = 0. E (Y i 1 Y i 5 ) = 2 15 (3 t 2 t 3 + 2s 2 t 2 s 2 t 3 + s 3 (t 2 + t 3 1) s 4 (1 + t 2 2t 3 )) := 2β 15. E (Y i 2 Y i 6 ) = E (Y i 3 Y i 7 ) = 2β 15. E (Y i 1 Y i 6 ) = E (Y i 2 Y i 5 ) = E (Y i 2 Y i 7 ) = E (Y i 3 Y i 6 ) = E (Y i 4 Y i 7 ) = β 15. E (Y i 1 Y i 7 ) = 0. E (Y i 1 Y i 8 ) = E (Y i 2 Y i 9 ) = 1 10 (4 + s 2 s 3 2s 4 + u (s 2 + 2s 3 + s 4 2)) := γ 10. E (Y i 1 Y i 9 ) = E (Y i 2 Y i 8 ) = E (Y i 3 Y i 9 ) = γ 20. E (Y i 1 Y i 10) = 2+s 2 s 4 5 := θ 5. E (Y i 2 Y i 10) = θ 10. E (Y3 i Y5 i ) = E (Y3 i Y8 i ) = E (Y3 i Y10) i = 0. E ( Y i ( ) E Y i2 j 4 Y i j ) = 0, for j = 5, 6, 8, 9, 10. = 1 (3 (1 + 6 t2 2 + t 2 3) 2 (t 2 + t 3 + t 2 t 3 )) := η, for j = 5, 6, 7. 6

83 67 E (Y i 5 Y i 6 ) = E (Y i 6 Y i 7 ) = η 12. E (Y i 5 Y i 7 ) = 0. E (Y i 5 Y i 8 ) = E (Y i 6 Y i 9 ) = 1 4 (2 t 2 t 3 + t 2 u + t 3 u) := λ 4. E (Y i 5 Y i 9 ) = E (Y i 6 Y i 8 ) = E (Y i 7 Y i 9 ) = λ 8. E (Y i 5 Y i E (Y i 7 Y i 8 ) = E (Y i 7 Y i 10) = 0. E ( Y i2 8 ) ( = E 10) = 3+t 2 t 3 Y i2 9 E (Y i 8 Y i 9 ) = ν 3. E (Y i 8 Y i 10) = 2 3. E (Y i 9 Y i 10) = 1 3. E ( 6 Y10) i = ω. 12 := ω. E (Y i 6 6 ) = 2 (1 u + 3 u2 ) := 2ν. 3 ) = 1. Y10 i2 Thus, Σ 4 is given by the following matrix: 2β 15 β 15 0 α α θ α α β 2β β γ γ θ α α β 2β γ α β η η λ λ ω Σ 4 = η η λ λ ω η λ ν ν ν γ 10 γ 20 We can now set up the multinormal integral using Definition and Σ 4. We choose the bounds on the integral corresponding to the positive orthant. In the case of ten variates, we use the MATLAB program mvncdf to compute the probability numerically. The result is then multiplied by 120 to obtain P R 5 (s, t, u) with accuracy to at least four decimal places. In general, this method will compute P R 5 for any

84 68 iterated positional procedure. However, as there are six parameters for the positional procedures, compared to m = 4, we have a greater challenge for m = 5 to represent the results. We will again focus on the popular iterated positional procedures involving plurality rule, Borda count, and antiplurality rule Results We will first focus on particular choices for s and explore how varying t and u influence P5 R. As we still have four variables with t 2, t 3, u, and P5 R, in order to produce graphs for P5 R, we will isolate the particular values of u as 0,.5, and 1. For other values of u, the graphs are similar to those depicted. The graphs for plurality rule can be seen in Figures Although it is not graphed, the numerical maximum of.150 occurs approximately at 1 :.21 :.07 : 0 and 1 :.21 : 0. The numerical minimum of.0324 occurs when t is antiplurality rule and u is plurality rule, or when plurality rule and antiplurality rule are alternated with each iteration. Figures show the graphs for the Borda count. The numerical maximum of.3862 occurs when the Borda count is used at each iteration. The numerical minimum of.0776 occurs when t is antiplurality rule and u is plurality rule (or vice versa). Last, for antiplurality rule, Figures depict the according probabilities. Similar to plurality rule (by taking the complement positional procedures), the numerical maximum of.150 occurs approximately at 1 :.93 :.79 : 0 and 1 :.79 : 0, and the numerical minimum of.0324 occurs when t is plurality rule and u is antiplurality rule. Overall, Baldwin s method is the numerical maximum across all iterated posi-

85 69 tional procedures (at P R 5 =.3862), and the iterated positional procedures comprising plurality rule and antiplurality rule in alternating order both produce the numerical minimum of In general, for each s, when the positional procedures for t and u are similar to those for s, P R 5 will tend to be higher. However, no matter the iterated positional procedure used, in the vast majority of cases, iteration will cause some change in the collective preference order. The result about complement iterated positional procedures is proven for five alternatives in Theorem The proof follows similarly to that for four alternatives, and it can be generalized to any number of alternatives. Theorem Let s, t, and u be given. Then P R 5 (s, t, u) = P R 5 ( s, t, ū). Proof. We can pair each individual preference order with its reversal. Then the contribution by to each Y i j, for j = 1, 2,..., 10, for P R 5 (s, t, u) is the same as the contribution by to each Y i j, for j = 1, 2,..., 10, for P R 5 ( s, t, ū). Because reversal pairs of individual preference orders partition the space of individual preference orders, the ensuing calculations for P R 5 (s, t, u) and P R 5 ( s, t, ū) are the same. We next consider some two-dimensional slices of R 6 which provide some insight into the behavior of P R 5. Our first consideration involves natural iteration. Even with t and u determined by s, there are still four variables under consideration: s 2, s 3, s 4, and P R 5. To provide meaningful data amidst these four variables, we assume that s 2 is given and compute the average value for P R 5 across a numerical mesh of values for s 3 and s 4 such that s 2 s 3 s 4. In particular, in Figure 3.26, we plot the average

86 70 Figure 3.17: Probability that plurality rule is ranking-invariant for u = 0. Figure 3.18: Probability that plurality rule is ranking-invariant for u =.5.

87 71 Figure 3.19: Probability that plurality rule is ranking-invariant for u = 1. Figure 3.20: Probability that the Borda count is ranking-invariant for u = 0.

88 72 Figure 3.21: Probability that the Borda count is ranking-invariant for u =.5. Figure 3.22: Probability that the Borda count is ranking-invariant for u = 1.

89 73 Figure 3.23: Probability that antiplurality rule is ranking-invariant for u = 0. Figure 3.24: Probability that antiplurality rule is ranking-invariant for u =.5.

90 74 Figure 3.25: Probability that antiplurality rule is ranking-invariant for u = 1. probabilities for P5 R for s 2 with =.05 and P5 R as the average of all data points with s 3 and s 4 chosen at intervals of =.05 such that s 2 s 3 s 4. Although these are not graphed, for natural iteration, the values range from.0403 (at 1 :.95 :.95 :.95 : 0) to.3862 (at 1 :.75 :.5 :.25 : 0). Similar to what was found for four alternatives, when s is fixed, P R 5 for natural iteration is among the higher probabilities which we can obtain across all choices for t and u. Accordingly, we have further evidence that natural iteration is a straight-forward rule for iterated positional procedures which compares favorably to other iterated positional procedures. However, we should reiterate that natural iteration still leads to a vast majority of samples which are not ranking-invariant. We next explore how the choice of t and u affects P R 5. We let s 2 and s 4 vary on multiples of.01 while setting s 3 = s 2+s 4 2. In Figures , we let t and u be both plurality rule, both the Borda count, both antiplurality rule, and antiplurality

91 75 Figure 3.26: Average Values for P R 5 under Natural Iteration. rule for t with plurality rule for u. Observe that the choice of t and u makes a major difference in P R 5. With plurality rule, the numerical estimate of the maximum of.1687 occurs at 1 :.3 :.15 : 0 : 0 while the minimum of.0372 occurs at 1 : 1 : 1 : 1 : 0. For antiplurality rule, the numerical estimate of the maximum of.1687 occurs at 1 : 1 :.85 :.7 : 0 while the minimum of.0372 occurs at 1 : 0 : 0 : 0 : 0. This is expected considering Theorem For the Borda count, the numerical estimate of the maximum of.3862 occurs at the Borda count while the minimum of.1102 occurs at 1 : 0 : 0 : 0 : 0. If antiplurality and plurality rule are alternated for t and u, the numerical estimate of the maximum of.1006 occurs at 1 :.96 :.81 :.66 : 0 while the minimum of.0325 occurs at 1 : 0 : 0 : 0 : 0. Of note, the minimum value for using the Borda count for t and u is greater than the maximum value possible when t and u alternate between plurality rule and antiplurality rule. Overall, using the Borda count for t and u suggests a better chance to obtain ranking-invariance than

92 76 Figure 3.27: The graph of P R 5 when t and u correspond to plurality rule. the use of other positional procedures for t and u, but in all cases, the majority of samples will not be ranking-invariant and possibly almost 97% of them could fail to be ranking-invariant. We can also explore how the choice of s 2 and t 2 influences P R 5 when we assume that s 3 =.5, s 4 = 1 s 2, and t 3 = 1 t 2, noting that s 2 and t 2 are always at least.5 so that the resulting positional procedures are valid. For this exploration, we assume that u =.5. The positional procedures are then set up to have symmetry between the second and second-to-last values. With =.01 for s 2 and t 2 on the interval [.5, 1], we find the numerical minimum of.1014 occurs at s 2 =.5 and t 2 = 1. The numerical maximum is approximately.3861 at s 2 =.75 and t 2 =.67, which is our closest point in the mesh to the Borda count for each iteration. A plot for P R 5 based on s 2 and t 2 is shown in Figure Consistent with our prior results, the greater probabilities occur when s 2 and t 2 are chosen closer to their respective Borda count values of s 2 = 3 4

93 77 Figure 3.28: The graph of P R 5 when t and u correspond to antiplurality rule. Figure 3.29: The graph of P R 5 when t and u correspond to the Borda count.

94 78 Figure 3.30: The graph of P R 5 when t is antiplurality rule and u is plurality rule. and t 2 = 2. Also, these procedures tend to produce higher probabilities for ranking- 3 invariance compared to some of the values we have previously seen for combinations of plurality rule and antiplurality rule. That said, much of this observation might be due to our choice of u = Summary of Results In observing the probabilities for ranking-invariance for up to five alternatives under the Impartial Culture assumption, we notice by m = 5 that in most cases, voter profiles are not ranking-invariant. That is, iteration will have some influence on the collective preference order. Considering the dramatic decrease in values from three alternatives to five alternatives, including some probabilities around 3.2% but never higher than about 38.6%, it is reasonable to conjecture that for any iterated positional procedure, Pm R will tend to zero as m increases.

95 79 Figure 3.31: P R 5 when s and t are defined with symmetry. Among all iterated positional procedures, Baldwin s method appears to consistently obtain higher probabilities of ranking-invariance compared to all other iterated positional procedures. This is not surprising considering the positive properties which are associated with the Borda count. The values for Pm R for Baldwin s method for m 5 are shown in Table 3.3. The value for m = 3 was proven to be a maximum by Gehrlein and Fishburn; the values for four and five alternatives are numerically supported as the maximum values. However, it is worth noting that Pm R for Baldwin s method decreases from 85.3% to 38.6% in going from m = 3 to m = 5, suggesting that iteration (and the removal of least-desired alternatives) has a significant impact on the collective preference ranking.

96 80 Table 3.3: P R m for Baldwin s Method and m 5. m Probability Reducing Some of the Variates: Primary Elections In many democratic societies, there exists practical value in holding runoff elections wherein an initial election is held over numerous candidates in order to establish a later election over just two candidates. If we assume that all voters preferences over the two finalists remain the same as their preferences during the initial election, we can explore the probability that the winner of the initial election also wins the runoff election against the second-place candidate. Gehrlein and Fishburn [18] discussed the three-alternative case P P R 3, which is the same as that discussed in Section 3.1 because any iterated positional procedure on three alternatives is necessarily a runoff iterated positional procedure. We will reproduce the results for P P R 4 which are stated in Gehrlein [19]. 5 Finally, we will establish new results for P P R 5. For both m = 4 and m = 5, all of the hard work was completed in computing ranking-invariance for maximal iterated positional procedures. We only need to use a particular subset of the variables to compute the probabilities for runoff elections. In the four-alternative case, we require x 1 4 x 2 4 x 3 4 x 4 and x 1 2 x 2. This 5 The method for obtaining these results is discussed in Gehrlein and Fishburn [18], but they do not discuss these results in that paper.

97 81 Figure 3.32: P P R 4 (t) for various four-alternative positional procedures. corresponds to the set of variates Y i j from Section 3.2 when j {1, 2, 3, 6}. Thus, to compute P4 P R, we follow the process defined in Section 3.2 for just the variates Yj i with j {1, 2, 3, 6}. P P R 4 has a numerical minimum of.6805 at both plurality rule and antiplurality rule. P P R 4 has a numerical maximum of.7998 at the Borda count. Its graph for varying t can be seen in Figure In the five-alternative case, we require x 1 5 x 2 5 x 3 5 x 4 5 x 5 and x 1 2 x 2. This corresponds to the set of variates Y i j from Section when j {1, 2, 3, 4, 10}. Thus, to compute P P R 5, we follow the process defined in Section for just the variates Y i j with j {1, 2, 3, 4, 10}. Although P P R 5 can be computed for any positional procedure s on five alternatives, we focus our attention on s 2 and s 3 so we can provide a graph of P P R 5. Values for P P R 5 were numerically computed for =.02 in each parameter, and their averages are plotted for each fixed (s 2, s 3 ) in Figure In exploring the particular data points for P P R 5, we find that

98 82 Figure 3.33: P P R 5 (s) for various five-alternative positional procedures. the positional procedures which least often agree with the primary result are plurality rule and antiplurality rule, which are ranking-invariant with probability As for the maximum, the Borda count is ranking-invariant with probability It is worth noting that even though plurality rule is most commonly used in each stage of runoff elections, plurality rule always has the lowest probability of rankinginvariance for m 5. That said, across the board, runoff elections have a significantly greater probability of being ranking-invariant than their maximal-iteration counterparts. This is intuitively expected because of the decreased number of iterations and thus fewer opportunities for an iterated election procedure to change the collective preference ranking.

99 83 CHAPTER 4 WINNER-INVARIANCE UNDER THE IMPARTIAL CULTURE ASSUMPTION We now turn our attention to the impact of iteration on the winner chosen by a positional procedure. In many elections, the voting population seeks to choose only one candidate. Thus, in these cases, the voting population might consider which alternatives finish in second, third, fourth, etc., to be of lesser consequence. Accordingly, we can relax our earlier condition of ranking-invariance to winner-invariance and explore how the probabilities differ. In the case of three alternatives, it was previously established that ranking-invariance and winner-invariance are equivalent. Thus, the data presented from Gehrlein in Fishburn in Section 3.1 are the same for this exploration. Let Pm W (F ) denote the probability that a voter profile q on m alternatives is winner-invariant under the iterated positional procedure F. The complement of Pm W (F ) corresponds to the probability that iteration determines a different winner for the election compared to the winner determined by the positional procedure during the first iteration. In this notation, note that by the discussion above, for any iterated positional procedure F on three alternatives, P3 R (F ) = P3 W (F ). 4.1 Method In the cases of four and five alternatives, we will modify the methods for ranking-invariance to produce probabilities for winner-invariance. When we are just

100 84 concerned with obtaining the same winner as that determined by the initial positional procedure, we must consider all cases where the desired winning alternative is not eliminated at each step of the process. For maximal iterated positional procedures, this means that if x 1 is the desired winning alternative, then winner-invariance is equivalent to ensuring that x 1 never finishes last in each iteration. Thus, we expect numerous cases which result in x 1 ultimately winning under the iterated positional procedure. Proposition Let F be a maximal iterated positional procedure on m 4 alternatives. Suppose that the positional procedure for a winner-invariant voter profile q produces the collective preference ranking x 1 m x 2 m... m x m. Then there are m 2 k=2 k k! possible distinct combinations of rankings over all of the iterations such that x 1 wins under F. 1 Proof. The iterations under F which involve m alternatives and two alternatives have fixed outcomes. For m alternatives, the outcome has to be x 1 m x 2 m... m x m. For two alternatives, x 1 must win against whichever candidate is also still under consideration by F. Thus, there is only one acceptable outcome for these stages of the iteration. Consider the iteration under F which involves 3 l m 1 alternatives. x 1 must be among these alternatives because q is winner-invariant. For x 1 to advance to the ensuing iteration, we only require that x 1 does not finish in last place among 1 Not all of these combinations will occur for all F. For example, the Borda count does not produce all possible combinations of collective preference rankings on subsets of alternatives.

101 85 the l alternatives. For l alternatives, there are l! possible collective preference orders. x 1 will finish in last place in (l 1)! of these collective preference orders. Then l! (l 1)! = (l 1) (l 1)! by factoring (l 1)! from each term in the difference. As each stage of iteration is independent of all other stages, we multiply (l 1) (l 1)! over all 3 l m 1 to determine the total number of possible outcomes. Finally, by substituing k = l 1, we obtain the desired quantity. Because positional procedures must produce a unique collective preference order, each voter profile under F must produce a unique set of collective preference rankings over all of the iterations. Thus, we can compute the probability that F will produce each possible winner-invariant set of collective preference rankings over all of the iterations. The sum of these probabilities will correspond to P W m (F ), the probability that F is winner-invariant. Each of the cases can be handled as they were in the ranking-invariant case. In particular, the ranking-invariant case is just one way in which a voter profile can be winner-invariant. If we desire a different collective preference ranking at some stage of the iteration, we need only to change those variables. For example, under ranking-invariance with m = 4, we set up variables Y i 4 and Y i 5 to test that x 1 3 x 2 3 x 3 is the collective preference order for the iteration involving three alternatives. For winner-invariance, we could instead have the collective preference order x 3 3 x 1 3 x 2. Thus, for this case, we will redefine Y i 4 to test for x 3 3 x 1 and Y i 5 to test for x 1 3 x 2. Then, as the iteration on two alternatives involves x 1 and x 3, we will redefine Y i 6 to test for x 1 2 x 3. The values for each variable for each individual preference order are computed similarly as before,

102 86 by comparing the points each alternative in the individual preference order would receive under the according positional procedure. Noting that the IC condition is still in effect, the entire process which followed for ranking-invariance is exactly the same for computing the probability for a particular set of collective preference rankings over all of the iterations. Essentially, to compute winner-invariance for F, we follow the method for ranking-invariance m 2 k=2 k k! times, once for each possible set of collective preference rankings. We then sum the results and multiply by m! to produce P W m (F ) (as we assumed that the collective preference ranking on m alternatives was x 1 m x 2 m... m x m ). In the following subsections, we will analyze the cases for m = 4, 5. Following that, we will summarize our results, including some particular discussion on IRV considering that IRV has practical importance in modern political elections. 4.2 Winner-Invariance for m = 4 Let m = 4. The results considered in this section are similar to those computed in the ranking-invariance exploration for m = 4. We will explore the probabilities for when t and u individually correspond to plurality rule, the Borda count, and antiplurality rule, with the other parameter arbitrary. We will then explore the probabilities under natural iteration. Our first graphs fix t to be the popular positional procedures of plurality rule, antiplurality rule, and the Borda count. In Figures , the graphs of P W 4 are shown for these procedures. As in the ranking-invariant exploration, we maintain a

103 87 symmetric relationship over complement iterated positional procedures. For plurality rule, the numerical minimum of.5756 occurs when u = 1 (i.e. alternating plurality and antiplurality rule); the numerical maximum of.6308 occurs at u = 0. Further, the graph suggests that P W 4 ((0, 0), u) constantly decreases in u. For antiplurality rule, the same results occur at the complement values for u, also suggesting that P W 4 ((1, 1), u) increases in u. For the Borda count, the numerical minimum of.7650 occurs twice: when u = 0 and u = 1. The numerical maximum of.7838 occurs approximately at u =.405 and u =.595. The probability at Baldwin s method is As our error tolerance for m = 4 is approximately , numerical error cannot be the reason why Baldwin s method did not produce the maximal value. Overall, we see that we have a small range of values across u for each fixed t, with this range around.02 when t is the Borda count. Further, comparing to Figures , these values for winner-invariance are about twenty to thirty percentage points higher compared to their ranking-invariance counterparts. We next explore how u influences the probabilities for P W 4. We plot the graphs for u corresponding to plurality rule, antiplurality rule, and the Borda count. For each plot, I show three angles of the graph, each with t 2 and t 3 varying with =.005. The first figure is the three-dimensional graph plotting P W 4 across all t. The other two angles are slices of the three-dimensional graph, one with t 2 =.7 and the other with t 3 =.3. These slices are representative of the behavior of the three-dimensional graph over all t. For plurality rule, the numerical estimate of the maximum is.7705 at approximately 1 :.585 :.255 : 0. The numerical estimate of the minimum is

104 88 Figure 4.1: P W 4 ((0, 0), u) for plurality rule on the first iteration. Figure 4.2: P W 4 ((1, 1), u) for antiplurality rule on the first iteration.

105 89 Figure 4.3: P W 4 (( 2, ) ) 1 3 3, u for the Borda count on the first iteration when antiplurality rule is used on three alternatives. Figures show the graph for P W 4 (t, 0). Antiplurality rule is the mirror of plurality rule, and like the ranking-invariant case, we obtain the same probabilities on complement election procedures. Its plots are shown in Figures The Borda count attains some of the higher probabilities for P W 4. The numerical minimum of.6071 occurs twice: at plurality and at antiplurality rule. The numerical maximum of.7836 occurs at 1 :.665 :.335 : 0, which is the closest point in the mesh to Baldwin s method. Because u =.5, complementary procedures on t will yield the same values. Thus, the plot for the Borda count on three alternatives is symmetric over the plane t 3 = 1 t 2. The graphs for the Borda count are shown in Figures For each t, the choices for u for the second iteration generally differ by only a few percentage points. Thus, winner-invariance for four alternatives is not sensitive to the choice for u. By comparison to ranking-invariance, when t is closer to plurality rule or antiplurality

106 90 Figure 4.4: Graph of P W 4 (t, 0) for plurality rule in the second iteration. rule, the values for winner-invariance can increase by up to forty percentage points. For t closer to the Borda count, the values for winner-invariance only increase by fifteen to twenty percentage points. This discrepancy suggests that when t is further away from the Borda count, it is more likely for iteration to change the collective preference between non-winning alternatives. We finally consider how P W 4 changes over t when u is derived from t via natural iteration. In Figure 4.13, P W 4 is plotted for t at =.005. The numerical maximum is approximately.7836 at 1 :.665 :.335 : 0, the closest point in the mesh to Baldwin s method. The numerical minimum is approximately.5771 at 1 :.995 :.995 : 0. 2 In Figures 4.14 and 4.15, slices of the three-dimensional graph are plotted for t 2 = 0.7 and t 3 = 0.3, respectively. The curves in these two figures are representative of those 2 P W 4 is not continuous for natural iteration whenever antiplurality rule is used in an iteration involving three or four alternatives.

107 91 Figure 4.5: Slice of the graph of P W 4 (t, 0) with t 2 =.7. Figure 4.6: Slice of the graph of P W 4 (t, 0) with t 3 =.3.

108 92 Figure 4.7: Graph of P W 4 (t, 1) for antiplurality rule in the second iteration.. Figure 4.8: Slice of the graph of P W 4 (t, 1) with t 2 =.7.

109 93 Figure 4.9: Slice of the graph of P W 4 (t, 1) with t 3 =.3. Figure 4.10: Graph of P W 4 (t,.5) for the Borda count in the second iteration..

110 94 Figure 4.11: Slice of the graph of P W 4 (t,.5) with t 2 =.7. Figure 4.12: Slice of the graph of P W 4 (t,.5) with t 3 =.3.

111 95 Figure 4.13: Graph of P W 4 (t, u) for natural iteration. seen for any fixed value of t 2 and t 3. Overall, the values for P W 4 under natural iteration are among the higher values which can be attained for each fixed t. Overall, based on our discussion, our numerical evidence suggests that on four alternatives, iteration is significantly less likely to alter the winner in comparison to the original collective preference order. This is expected because winner-invariance is a weaker condition than ranking-invariance. Overall, winner-invariance tends to be more likely for iterated positional procedures which are closer to Baldwin s method, but, for some still unknown reason, methods near Baldwin s method produce slightly higher probabilities for winner-invariance. 4.3 Winner-Invariance for m = 5 Let m = 5. The results considered in this section are similar to those computed in the ranking-invariance exploration for m = 5. However, due to the increased run-

112 96 Figure 4.14: Slice of the graph of P W 4 (t, u) for natural iteration with t 2 = 0.7. Figure 4.15: Slice of the graph of P W 4 (t, u) for natural iteration with t 3 = 0.3.

113 97 time for computing P W 5, we have generally limited our precision to a maximum error of.003. Increased precision could be obtained at the expense of increased run-time. We will first focus on particular choices for s and explore how varying t and u influence P W 5. As we still have four variables with t 2, t 3, u, and P W 5, in order to produce graphs for P W 5, we will isolate the particular values of u as 0,.5, and 1. For other values of u, the graphs are similar to those depicted. The graphs for plurality rule can be seen in Figures The numerical maximum of.545 occurs when t and u correspond to plurality rule (i.e. IRV), and the numerical minimum of.462 occurs when t and u correspond to antiplurality rule. As Theorem involving complement positional procedures follows similarly for winner-invariance, the complement results hold when s is antiplurality rule. The graphs for antiplurality rule can be seen in Figures Figures show the graphs for the Borda count. For the Borda count, the range of values is much smaller. The minimum value of.706 occurs at plurality and antiplurality rule. The maximum value is around.737, but it does not occur at the Borda count. Increasing our precision to.0003, we find that the Borda count is winner-invariant with probability.7360 whereas numerous values (e.g. t 2 = 17, t 24 3 = 7, and u = 3 ) produce a probability around outside of the margin of error for the Borda count. Nonetheless, as can be seen from the graphs for the Borda count, the choice of t and u has little effect on the probability that the Borda count is winner-invariant. Overall, in comparison to the four-alternative case, the values for P W 5 are significantly higher than their rankinginvariant counterparts. The smallest value for P W 5 is larger than the largest value for

114 98 Figure 4.16: Probability that plurality rule on five alternatives is winner-invariant for u = 0. P R 5. Still, we find that the winner will change under iteration at least 25% of the time regardless of the positional procedures used and potentially over 50%. We next consider some two-dimensional slices of R 6 which provide some insight into the behavior of P W 5. Our first consideration involves natural iteration. In Figure 4.25, we plot the probabilities for P W 5 for s 2 with = 1 36 and P W 5 as the average of all data points with s 3 and s 4 at intervals of = 1 36 such that s 2 s 3 s 4. Although these are not graphed, for natural iteration, the values range from.470 (at 1 : 35 : 35 : : 0) to.736 (at 1 :.75 :.5 :.25 : 0). We next explore how the choice of t and u affects P W 5. We let s 2 and s 4 vary on multiples of =.01 while setting s 3 = s 2+s 4 2. In Figures , we let t and u be both plurality rule, both the Borda count, both antiplurality rule, and antiplurality rule for t with plurality rule for u. Observe that the choice of t and u

115 99 Figure 4.17: Probability that plurality rule on five alternatives is winner-invariant for u =.5. Figure 4.18: Probability that plurality rule on five alternatives is winner-invariant for u = 1.

116 100 Figure 4.19: Probability that antiplurality rule on five alternatives is winner-invariant for u = 0. Figure 4.20: Probability that antiplurality rule on five alternatives is winner-invariant for u =.5.

117 101 Figure 4.21: Probability that antiplurality rule on five alternatives is winner-invariant for u = 1. Figure 4.22: Probability that the Borda count on five alternatives is winner-invariant for u = 0.

118 102 Figure 4.23: Probability that the Borda count on five alternatives is winner-invariant for u =.5. Figure 4.24: Probability that the Borda count on five alternatives is winner-invariant for u = 1.

119 103 Figure 4.25: Average values for P W 5 under natural iteration. makes a major difference in P W 5. With plurality rule, the numerical estimate of the maximum of.717 occurs at 1 :.64 :.4 :.16 : 0 while the minimum of.462 occurs at 1 : 1 : 1 : 1 : 0. For antiplurality rule, the numerical estimate of the maximum of.717 occurs at 1 :.84 :.6 :.36 : 0 while the minimum of.462 occurs at 1 : 0 : 0 : 0 : 0. This is expected considering Theorem For the Borda count, the numerical estimate of the maximum of.736 occurs at the Borda count while the minimum of.508 occurs at both plurality and antiplurality rule. If antiplurality and plurality rule are alternated for t and u, the numerical estimate of the maximum of.715 occurs at the Borda count while the minimum of.504 occurs at plurality rule. By comparison to fixing s, we find a much larger disparity in the possible values when t and u are fixed. Accordingly, we can conjecture that the choice of s has more influence on P W 5 than the choices of t and u. We can also explore how the choice of s 2 and t 2 influences P W 5 when we assume

120 104 Figure 4.26: The graph of P W 5 when t and u are plurality rule. Figure 4.27: The graph of P W 5 when t and u are antiplurality rule.

121 105 Figure 4.28: The graph of P W 5 when t and u are the Borda count. Figure 4.29: The graph of P W 5 when t is antiplurality rule and u is plurality rule.

122 106 Figure 4.30: P W 5 when s and t are defined with symmetry. that s 3 =.5, s 4 = 1 s 2, and t 3 = 1 t 2, noting that s 2 and t 2 are always at least.5 so that the resulting positional procedures are valid. For this exploration, we assume that u =.5. The positional procedures are then set up to have symmetry between the second and second-to-last values. With = 1 90 for s 2 and t 2 on the interval [.5, 1], we find the numerical minimum of.621 occurs at s 2 =.5 and t 2 = 1. The numerical maximum is approximately.737 at numerous values but not including the Borda count. A plot for P W 5 based on s 2 and t 2 is shown in Figure Application to Instant-Runoff Voting Our primary practical motivation for considering winner-invariance is to analyze the probability that IRV is winner-invariant, or that the use of IRV will produce a different winner than the one produced by plurality rule alone. Our model, as defined, produces this probability. True, there is one slight difference in that IRV stops