Empowering the Voter: A Mathematical Analysis of Borda Count Elections with Non-Linear Preferences

Transcription

1 Empowering the Voter: A Mathematical Analysis of Borda Count Elections with Non-Linear Preferences A Senior Project submitted to The Division of Science, Mathematics, and Computing of Bard College by David Polett Annandale-on-Hudson, New York May, 010

2 Abstract This project explores the application of non linearly-ordered sets to the field of voting theory. In particular, a specific voting system known as Borda Count is studied. In order to analyze the mathematical properties associated with different voting systems the alternatives on a ballot are represented as elements of an ordered set. Traditional voting theory analysis has studied many properties of Borda Count that hold when voters ballots are limited to linearly ordered sets. This project expands the Borda Count election method to allow the use of bucket ordered and partially ordered sets as ballots instead of only linearly ordered sets. The different mathematical properties associated with traditional Borda Count are then analyzed in order to see if they still hold for these expanded Borda Count voting systems.

3 Contents Abstract 1 Dedication 5 Acknowledgments 6 1 Introduction 7 Preliminaries 1.1 Sets, Relations, and Orderings Voting Theory Linear Borda Count Bucket Borda Count Bucket Orders Bucket Borda Score Comparison of Bucket Borda to Linear Borda Satisfied Properties Unsatisfied Properties Random Thoughts on Resolvability Graded/Poset Borda Count Graded Posets Graded/Poset Borda Score Open Questions 110

4 Contents 3 References 113

5 List of Figures.1.1 Hasse Diagrams of a Set with Different Orderings Length and Height of a Partially Ordered Set Hasse Diagram of a Ballot Same Preference Order Represented in Different Ballot Formats Allocation of Borda Scores to a Linear Ballot Linear Borda Election Bucket Ordered Ballot Linear Extensions of a Bucket Ordered Ballot Monotonicity of a Bucket Borda Ballot Plurality Satisfied in Linear Borda Election Failure of Plurality in Bucket Borda Election Insertion of a Clone in a Linear Ballot Insertion of a Clone in a Bucket Ballot Insertion of Clone Changing Outcome of Linear Borda Insertion of Clone Changing Outcome of Bucket Borda Inverse of a Linear Ballot Saturated and Maximal Chains in a Partially Ordered Set Comparison of Graded Poset and Poset Graded Borda Score Computations

6 Dedication To ACORN, for supporting low- and moderate-income Americans in their fight for social justice throughout the last 40 years. In particular as it relates to this project, for greatly expanding the American electorate by registering an incredible 1.3 million new voters during the 008 election season.

7 Acknowledgments I would like to thank the following people for their support: John Cullinan, for spending the year teaching me about voting theory, learning with me about posets, and making my senior project experience more enjoyable than I could have ever imagined; Sam Hsiao, for providing me with excellent guidance and assistance throughout my four years at Bard; Matthew Deady, who was always willing to take a moment to discuss the game of basketball; Ellen Lagemann, who taught me more about education in one year than I d learned in my entire life; Greg Spanier, for introducing me to a style of teaching math which I can t wait to try out for myself; my friends, for making Bard a place I ll always love and never forget; and Mom and Dad, for opening up so many doors of opportunity.

8 1 Introduction An ordered set is a collection of elements that can be compared to one another in some manner. Mathematicians have come up with many different types of ordered sets. The two most commonly seen ordered sets are completely ordered sets, which we will refer to throughout this paper as linearly ordered sets, and partially ordered sets. As there names imply, a completely ordered set consists of elements which can all be compared to one another while a partially ordered set contains elements which have some order but are not necessarilly all comparable to one another. For example, the natural numbers are a completely ordered set since they can all be compared by value: 1 <, < 3, etc. In contrast, we can think of the list of my favorite foods as a partially ordered set. Let s suppose that I like chicken better than any other meat but have no preference between turkey and beef. Then chicken is comparable to both turkey and beef but neither of them is comparable to the other. In a more mathematical sense common partially ordered sets, which are also referred to as posets, include the natural numbers ordered by divisibility and the set of subsets of a given set ordered by inclusion. As the example above shows, ordered sets can be very useful in representing real life situations and preferences. In this

9 1. INTRODUCTION 8 paper, we take advantage of this aspect of ordered sets by utilizing them to model a voter s preferences in an election. This senior project considers the application of partially ordered sets in the field of voting theory. The motivation for this project initiated with my insistence that I did not want to spend a year working on a project which I could not relate to in some way. During my first three years at Bard I had several opportunities to speak with current seniors about their senior projects, and they all gave me the same piece of advice find a topic which I could be personally invested in and excited to devote many, many hours to. Toward the end of my junior year when I began to seriously think about what I d like to spend the next year studying I had two thoughts: the first was that the 300-level class I d enjoyed the most was Professor Hsiao s Combinatorics class and the second was that earlier that year Professor Cullinan taught a 100-level Voting Theory course that sounded very interesting to me. Studying the mathematics involved in voting systems and tying in combinatorics in some manner appealed to me as a topic which I could intuitively grasp the importance of and hopefully become passionately committed to working on. With these two thoughts in mind, I approached Professor Cullinan about potentially working with him on a project that incorporated voting theory and combinatorics in some manner. Professor Cullinan thought there was potential for a project and gave me the textbook from his voting theory course, Math and Politics: Strategy, Voting, Power, and Proof [5], to read over the summer in order to learn more about the field and begin formulating more specific project ideas. While reading the voting theory textbook over the summer was invaluable for introducing me to the field of voting theory and greatly increased my knowledge of the subject, it did not present me with any solid project ideas. I spent the first few weeks of senior year reading a large number of economic and mathematic papers in search of a project topic, and although I was intrigued by the topics in many of the papers I read about I wasn t able to find anything that seemed suitable for a project topic. Part of my dilemma

10 1. INTRODUCTION 9 was that I really wanted a topic which combined combinatorics and voting theory and this was proving to be quite difficult. This all changed, however, when Professor Cullinan introduced me to a paper written by a number of mathematicians including Bard Master of Arts in Teaching professor Japheth Wood. This paper, titled Elections with Partially Ordered Preferences [1], explored the application of partially ordered sets as ballots in a number of different election methods. The motivation for this study was that traditional ballots in which voters must compare all the candidates to one another often force voters to overspecify their preferences in an election. Hence, it was the opinion of Wood and his co-authors that linearly ordered ballots should be replaced by partially ordered ballots in order to give voters the necessary freedom to express their true preferences in their ballots. Wood s paper studied the applicability of partially ordered sets (ballots) to a variety of voting systems including one election method known as Borda Count. In order to count the votes tallied in these partially ordered ballots Wood chose to look at the linear extensions of each voter s partially ordered preference set. I found this to be an interesting study in voting theory and appreciated that it involved some of the mathematical concepts I d been introduced to in combinatorics. In addition, there was a lot of room to further investigate the ideas presented in Wood s paper. I began studying the applicability of non-linear ballots to the Borda Count election system and quickly fell in love with the topic. In this paper we will discuss how replacing the linear sets used as ballots in a Borda Count election with non-linear sets effects the various mathematical properties associated with Borda Count. In particular, we hope to prove that one can allow partially ordered ballots in a Borda Count election and still retain all or nearly all of the desirable mathematical properties of Borda Count. In Chapter we begin by presenting some basic definitions and examples of ordered sets along with the definitions of some important aspects of these sets. We follow that with

11 1. INTRODUCTION 10 a brief introduction to voting theory in which we list some of the key definitions we will use to mathematically represent different aspects of an election. The chapter is concluded with the definition and an example of the Borda Count voting system which will serve as the basis for much of our analysis in the rest of the paper. In Chapter 3 we begin our application of non-linear sets to Borda Count elections by first more thoroughly defining and discussing the notion of a bucket ordered set. We then develop a new version of Borda Count that allows for the use of bucket ordered ballots, which we will refer to throughout the paper as Bucket Borda. Once Bucket Borda has been defined, we will compare the way Bucket Borda counts the votes in a bucket ordered ballot with the method of allocating points to a bucket ordered ballot using its linear extensions presented in [?]. Bucket Borda will serve as the focus for much of our paper, and we conclude this chapter by proving that it allocated points to the candidates in a ballot similar to the way Borda Count allocates points in a linear election. We begin Chapter 4 by presenting some mathematical properties of voting systems that hold in Borda Count elections and attempt to prove that these properties also hold in Bucket Borda elections. Through our analysis we develop three desirable mathematical properties that hold in Linear Borda elections and which we feel all Borda Count election methods should satisfy. Unforutnately, we then prove that it is impossible for any nonlinear Borda Count election method to satisfy these three properties simultaneously. This is one of the most important results shown in the paper. We then present mathematical properties which Borda Count fails to satisfy and attempt to prove that these properties are also unsatisfied in Bucket Borda elections. The chapter is concluded with a brief look into the resolvability of Borda Count and Bucket Borda voting systems. We were able to prove a few interesting theorems about the resolvability of Borda Count election methods, but there is still a lot of room here for further exploration.

12 1. INTRODUCTION 11 In Chapter 5 we finally are able to extend our findings to partially ordered sets. We begin by defining a new election method, referred to as Graded Borda, which extends the Bucket Borda election method to allow the input of graded poset ballots. This involves determining a function to count the votes in these less restricted ballots. The rest of the chapter is devoted to proving that Graded Borda satisfies the desirable base properties of Borda Count and Bucket Borda discussed in Chapters and 4. Throughout this chapter we note that the Graded Borda election method can be extended to allow partially ordered ballots without changing the way it is defined at all. This was a result which we had not anticipated but rather that we stumbled upon when trying to prove that certain properties of Borda Count held for Graded Borda. The final chapter is devoted to listing some open questions and ideas for further exploration into the topic. Some of these are questions I would ve liked to address in my project but did not have enough time for, and others are problems which occurred to me that I was unable to solve. It is my hope that these ideas could be useful for future Bard students when searching for a senior project topic of their own!

13 Preliminaries.1 Sets, Relations, and Orderings Before we begin our discussion of voting theory it is important to discuss some basic properties of sets, relations and orderings. The definitions of relations, ordered sets, and their associated properties presented in this chapter all came from [3]. In this project we will be interested in restricting the ordering of the elements in a set by different properties. The relations we will use to define the differently ordered sets studied throughout this paper are listed in the following definition. Definition.1.1. Let A be an ordered set of elements. For all a, b, c A the binary relation is: irreflexive if (a a) asymmetric if a b, then (b a) transitive if a b and b c, then a c transitivity of equivalence if a b, then either a c or c b

14 . PRELIMINARIES 13 complete if either a b or b a. These properties can be used to define sets with different types of ordering. The three main ordered sets we will be analyzing in this paper are listed in the definition below along with their associated properties. We will define each of these sets more rigorously when they are first discussed in detail in the paper. Note that in the definition below we use the word strict before each of the sets we mention. The difference between a strictly ordered set and a non-strictly ordered set is simply that a strictly ordered set is irreflexive while a non-strictly ordered set is reflexive. Hence, in a non-strictly ordered set an element is related to itself while in a strictly ordered set an element is not related to itself. Since the sets we discuss in this paper are meant to represent voters preferences in an election we do not want an element in our sets to be related to itself. For that reason we will be working with strictly ordered sets throughout this paper. However, for ease of notation we will drop the strict from each of these sets when discussing them throughout the paper except in the definition that follows. Definition.1.. A binary relation on a set of alternatives A is a strict linear ordering if it satisfies asymmetry, transitivity, and completeness; a strict bucket ordering if it satisfies asymmetry, transitivity, and transitivity of equivalence; and a strict partial ordering if it satisfies asymmetry and transitivity. From Definition.1.1 it is clear that any set which is completely ordered must also satisfy transitivity of equivalence, so it follows from Definition.1. that any linearly ordered set is also a bucket ordered set. Additionally, since all bucket ordered sets satisfy asymmetry and transitivity it is clear that any bucket ordered set is also a partially ordered set. By the same argument, we see that any linearly ordered set is also a partially ordered set.

15 . PRELIMINARIES 14 Thus, we can see that the set of all linearly ordered sets is a subset of the set of all bucket ordered sets which is a subset of the set of all partially ordered sets. Throughout the rest of this section we will present definitions for many different aspects of the ordered sets which will be analyzed in the subsequent chapters of the paper. We begin by defining what it means for two elements to be comparable, incomparable, and equivalent in a given set. Definition.1.3. Let P be a partially ordered set such that a, b P. If a b or b a then a and b are said to be comparable. If (a b) and (b a), then a and b are incomparable. Definition.1.4. Let P be a partially ordered set such that a, b P. If for all c P a c if and only if b c, c a if and only if c b, and a and b are incomparable; then we say that a is equivalent to b, denoted a b. Of the three ordered sets we will study in this paper, linearly ordered sets are the most restrictive, followed by bucket ordered sets, and then partially ordered sets. This is because the elements in a linearly ordered set must be completely ranked which means each element in the set must be comparable to every other element in the set. Bucket ordered sets allow for a little more variation by requiring the elements in the set to only satisfy transitivity of equivalence instead of completeness. While completeness states that any two elements in a set must be comparable, transitivity of equivalence states that if two elements in a set are comparable then any other element in the set must be comparable to at least one of these alternatives. Therefore, we know that if two elements a and b are incomparable in a bucket order B then they must be located in the same level of B. Hence, they are

16 . PRELIMINARIES 15 preferred to and preferred by the same alternatives in B. It follows that if a and b are incomparable in a bucket order B, then a b. Intuitively this means that the elements in a bucket ordered set are partitioned into different levels of equivalent elements, and these levels are then linearly ordered so that elements located in different levels are all comparable to one another. Furthermore, we can view a linearly ordered set as simply a bucket ordered set in which the sie of each equivalent level, or bucket, is one. Partially ordered sets allow for the most variation in the ordering as there is no additional requirement that must be met as long as the elements in the set are asymmetric and transitive. This means the elements in a partially ordered set can be ordered in any manner as long as transitivity and asymmetry are satisfied. Since transitivity of equivalence does not always hold in a poset P, incomparable elements in P are not necessarily preferred to and preferred by the same alternatives, and we can not assume that they are equivalent. Hence, partially ordered sets are the only one of the three sets we will study that allow truly incomparable elements. Although they may appear to be somewhat similar, the fact that transitivity of equivalence must be satisfied in bucket ordered sets makes them significantly easier to analyze than partially ordered sets. An example of the differences between these three sets can be seen in Figure.1.1. In order to better comprehend and view these sets we will represent them through the use of Hasse diagrams. Intuitively, a Hasse diagram is a graphical representation of an ordered set in which the vertices of the graph represent the elements in the set, and a line between two vertices represents that their associated elements are related in the set. The relation we will be working with this in this paper is and we define it below along with a more rigorous definition of Hasse diagram. Definition.1.5. Let A be an unordered set of elements and P be a partially ordered set of A. For any a, b A, we say that a b in P if a is preferred to b.

17 . PRELIMINARIES 16 Definition.1.6. The Hasse diagram of a poset P is an undirected graph representing the poset whose vertices are the elements of some base set A and where a b if there is a descending line connecting a to b in P. From the two definitions above we see that for any two elements a and b in a partially ordered set P, a is preferred to b if a is placed above b in the Hasse diagram of P. Hence, we can think of the Hasse diagram of a partially ordered set as listing the elements in the set in order from most preferred at the top to least preferred at the bottom. The base set A mentioned in the definitions above refers to an unordered set of elements while a partially ordered set P of A takes the elements in A and orders them in some manner. Since linearly ordered sets and bucket ordered sets are both suborders of partially ordered sets, the definition of Hasse diagram also applies to linearly ordered sets and bucket ordered sets. Similar logic applies to the rest of the definitions mentioned in this section. In Figure.1.1 we see examples of the Hasse diagrams of a linearly ordered set (L), a bucket ordered set (B), and a partially ordered set (P ). Note that each of these ordered sets consist of the same base set, A = {a, b, c, d}, but diferent orderings of the elements in this set. In the Hasse diagram of L each alternative is connected to every other alternative, and it follows that the elements in the set are all comparable to one another. Furthermore, we see from the Hasse Diagram of L that a b c d. From the Hasse diagram of B we see that b and c are equivalent since they are both preferred by a, preferred to d, and incomparable to one another. This also follows from our statement that any two elements in the same level of a bucket ordered set are equivalent. Thus, in B we have a b c d. In the Hasse diagram of P we see that b and c are incomparable since there is no line connecting b and c, and they are not equivalent since c d while b is incomparable to d. Thus, in P we have a b and a c d.

18 . PRELIMINARIES 17 Figure.1.1. Hasse Diagrams of a Set with Different Orderings We now provide the definitions for different terms associated with certain elements or suborders of an ordered set. For each of the following definitions, assume we are given a base set of alternatives A ordered in a poset P. Definition.1.7. An element a is called the maximum of P if a a for all a P. An element b is called maximal if there is no a P such that a b. Definition.1.8. An element a is called the minimum of P if a a for all a P. An element b is called minimal if there is no a P such that b a. It follows from the two definitions above that a poset can have many maximal or minimal elements but only one maximum or minimum. Furthermore, the maximum of a poset is by definition maximal as well, but a maximal element is not necessarily the maximum. If there is more than one maximal element in a poset then by definition neither element can be the maximum. The same holds true for minimal and minimum elements in a poset. We will now define what is meant by the height and length of an ordered set. These terms will be useful when determining how to allocate points to the elements in bucket and partially ordered sets in Chapters 3 and 5 of the paper.

19 . PRELIMINARIES 18 Definition.1.9. A chain in a partially ordered set is a subset C P such that for all a, b C, a and b are comparable. Hence, a chain is a linearly ordered subset of a partially ordered set. Definition The height, H(C), of a chain C is defined by H(C) = C 1 where C is defined as the number of elements in the chain. The height of an element a P is denoted H a, and is defined to be the height of the longest C P in which a is the maximum. The height of a poset P is defined as the height of the maximum element or maximal elements in P. Definition An antichain is a subset X of P such that for all a, b X, a and b are incomparable. Definition.1.1. The length, L(X), of an antichain X is defined by L(X) = X. The length of an element a B is denoted L a, and is defined as the number of elements located in the same antichain as a. Figure.1.. Length and Height of a Partially Ordered Set The partially ordered set P represented in Figure.1. has a maximum element a, no minimum element, and two minimal elements f and g. Let C = {d, e, f} be a subset of P. Then C is a chain since every element in the set is comparable to one another, and

20 . PRELIMINARIES 19 C has height two since it is composed of three elements. Note that this poset has height three since element a is the maximum of the poset and H a = 3. The poset contains two antichains, X 1 = {b, c, d} and X = {f, g}, where L(X 1 ) = 3 and L(X ) =. Note that P is also a bucket order since transitivity of equivalence holds for the elements in P. Because P is a bucket order we know that elements located in the same antichain are equivalent, so we have b c d and f g. Finally, it should be quite clear that P is not also a linear order since there are numerous elements in P which are not comparable to one another.. Voting Theory Voting theory is concerned with the mathematcal analysis of different social choice procedures, which are more commonly referred to as election methods or voting systems. The goal of all social choice procedures is to determine the will of the majority of a set of voters when trying to choose among several candidates. When there are only two candidates in an election this can be accomplished by simply letting each person vote for his or her preferred candidate, and declaring the candidate who receives the most votes the winner of the election, also referred to as the social choice. This style of voting is generally referred to as plurality voting. Plurality voting is one of the most well-known, widely used, and easily understood social choice procedures in the United States and many other countries around the world. In a plurality election each voter selects his or her single most preferred candidate, and the candidate who receives the greatest number of first-place votes is declared the winner. Clearly, in an election consisting of only two candidates plurality does an excellent job of determining the will of the majority. However, when voters are given three or more candidates to choose between this process becomes significantly more difficult to accomplish. The first dilemma encountered when extending elections from two to three or more candidates is that most traditional ballots only take into account a voter s most preferred

21 . PRELIMINARIES 0 candidate. In an election with just two candidates this is not a problem because looking at a voter s most preferred candidate gives us sufficient information on the voter s preference toward both candidates. However, in an election with three or more candidates a ballot which only considers a voter s most preferred candidate ignores his preferences toward all the other candidates. The most common solution to this dilemma has been to extend voters ballots from a single most preferred candidate to a ranked linear preference order of the candidates in the election. Unfortunately, when we expand voters ballots in this manner, it becomes unclear how to most effectively determine the will of the majority. As a result of this ambiguity, many social choice procedures have been developed which when given a set of linearly ranked ballots are said to determine the candidate most preferred by the majority of the electorate. Unfortunately, as was shown in a famous 1950 voting theory proof by Kenneth Arrow there cannot possibly exist a fair social choice procedure which given an election consisting of three or more candidates is able to simultaneously satisfy three seemingly simple and desirable properties for all voting systems to have. The three properties mentioned in Arrow s proof are Pareto, monotonicity, and Independence of irrelevant alternatives. The Pareto condition states that if all of the ballots in an election are identical, then the winner on each of these ballots should be the winner of the election. The monotonicity condition states that if a candidate a is preferred to a candidate b in the overall election, and a voter who had b preferred to a in his ballot switches his preferences towards a and b (so that now a is preferred to b in his ballot), then a should still be preferred to b in the overall election. Independence of irrelevant alternatives states that given two candidates a and b in an election, the position of alternatives other than a and b in the ballots of the election should be irrelevant when deciding whether a is preferred to b in the overall election or b is preferred to a. We will discuss each of these conditions in more depth and define them more rigorously in Chapter 4. In his impossibilty theorem, Arrow proved that

22 . PRELIMINARIES 1 there can not exist a voting system which given an election consisting of at least three candidates can simultaneously satisfy the three conditions mentioned above except for a dictatorship (an election system in which a single voter s ballot is always taken to represent the electorate as a whole). For a more rigorous statement and proof of this theorem, see [5, 11-]. In order to mathematically represent an election we must first understand and define three important terms. The three components that come together to form an election are a set of candidates to be voted between, a set of ballots which represent the voters preferences towards the candidates in the election, and a voting system which analyzes these ballots in some pre-determined manner and outputs the winner of the election. The rest of this section is dedicated to rigorously defining these three terms, so that we have a good sense of how an election is represented and conducted mathematically. We will denote the candidates in a given election by the set A whose elements will be called alternatives most commonly denoted a 1, a,... a n where n is the number of alternatives being voted among in a given election. There will also be a set V whose elements will be called voters denoted v 1, v,..., v m where m is the number of voters in a given election. We will assume that in any election each voter v V arranges the alternatives in a list, or ballot, according to his preferences such that every alternative in A is present somewhere in his preference list. In general, a ballot is a function from the set of all voters to any possible partial ordering of the alternatives in a given set A. Given a specific v V, a ballot is a function from v to v s preference ordering of the alternatives in A. We state the definition of a ballot more rigorously below: Definition..1. Let A be a set of n alternatives such that A = {a 1, a,..., a n } and let V be a set of m voters such that V = {v 1, v,..., v m }. We define a ballot B(v) as a function B : V pos(a) such that the domain of our function is the set of all voters and

23 . PRELIMINARIES the range of our function is any possible partial ordering of A (unless specifically noted otherwise). Thus, B(v i ) is voter v i s preference ordering of the alternatives in A. From the definition above it is clear that a ballot is a partially ordered set of the alternatives in an election. These ballots, which we will sometimes refer to as preference lists, can be pictured as vertical orderings with the alternatives displayed from most preferred on top to least preferred on bottom. Recall from the previous section that the Hasse diagram of a partially ordered set P displays the elements of the set from most preferred at the top of the graph to least preferred at the bottom. Hence, a voter s ballot B(v) can be represented by a Hasse diagram of the alternatives in the base set A ordered by the relation such that for all alternatives a and b in B(v), if a is preferred to b in the ballot then a is located above b in the Hasse diagram of B(v). In Figure..1 we see the ballot for a voter v 1 consisting of the set of alternatives A = {a, b, c, d, e}. Since b is placed at the top of the Hasse diagram of B(v 1 ) we see that it is the most preferred alternative in the ballot. Similarly c is the least preferred alternative in the ballot since it is placed at the bottom of the graph. Furthermore, the vertices in the graph are all connected to one another which tells us that the alternatives in the ballot are all comparable to one another, and it follows that B(v 1 ) represents a linearly ordered preference for the alternatives in A. From the Hasse diagram of B(v 1 ) we see that v 1 s preference order for the alternatives in A can be expressed as b d a e c. Considering a voter s preferences toward a set of alternatives, it seems reasonable to place the following restrictions on a voter s ranked ballot. First, we will assume that a voter who prefers alternative a to alternative b and alternative b to alternative c will therefore prefer a to c. Hence, a voter s ballot should satisfy transitivity. Second, we will assume that a voter who prefers a to b will not prefer b to a. Hence, a voter s ballot should satisfy assymetry. At this point we feel there should be no further restrictions placed on

24 . PRELIMINARIES 3 Figure..1. Hasse Diagram of a Ballot a voter s ballot. However, in nearly every ranked voting system utilized in today s society voters are forced to provide a complete preference ordering for the alternatives in the election. These three criteria lead us to the following definition of a traditional linearly ranked ballot. Definition... Let B(v) be a traditional linear ballot in a ranked voting system. Then for all a, b, c B(v) the following three properties must hold: 1. If a b and b c, then a c. (Transitivity). If a b then (b a). (Asymmetry) 3. For all a b, either a b or b a. (Complete) Clearly, the three restrictions placed on a traditional linear ballot in a ranked voting system are equivalent to the mathematical properties of a linearly ordered set. Hence, we can utilize linearly ordered sets to express a voter s preferences in a traditional ballot. While voters around the world are nearly universally required to vote using a ballot that restricts their preferences to a linear ordering, we agree with the argument in [1] that these

25 . PRELIMINARIES 4 ballots force voters to overspecify their preferences and often fail to accurately reflect a voter s true preferences in his ballot. Hence, we wish to eliminate the completeness restriction currently placed on voters ballots and allow them to vote in elections utilizing partially ordered ballots. As we ve pointed out before, since linearly ordered sets are a subset of partially ordered sets voters could still provide a complete ranking of the alternatives, but they would now have much greater freedom in ranking the alternatives in their ballots if they desired. By removing the ordering restrictions currently placed on voters ballots we are improving their ability to express their true preferences for the alternatives in an election by greatly increasing the number of possible orderings they can utilize to rank the alternatives. The two ways we will expand our voters ballots in this paper are through the use of bucket ordered ballots and partially ordered ballots. The majority of this project involves gradually removing the restrictions placed on voters ballots and analyzing how these changes effect the mathematical properties of the Borda Count voting system defined in the following section. Ideally, we would like to give voters the freedom to vote on partially ordered ballots but these sets are significantly more difficult to analyze, so we will begin our research in Chapter 3 by focusing on the extension from linearly ordered ballots to bucket ordered ballots and then extend our findings to partially ordered ballots in Chapter 5. At times throughout this paper we will want to limit the output of the ballot function to only linear or bucket orderings of a given set of alternatives A. As we showed in Section.1, any linear order is also a bucket order and any bucket order is also a partial order, so it follows that the set of all partial orders contains the set of all linear orders and the set of all bucket orders. Hence, our general functon B : V pos(a) can be adjusted to restrict the voters ballots to only linear orderings by limiting the output of B(V ) to the set of all

26 . PRELIMINARIES 5 linear orderings of A. Similarly, we can also restrict voters ballots to bucket ordered sets by limiting the output of B(V ) to the set of all bucket orderings of A. We will now provide a simple example to motivate why it seems necessary to expand the ballot types we allow voters to use in an election. Let A = {a, b, c, d} and v be a voter whose actual preference ordering of the alternatives in A is a b, a c, and c d (so by transitivity a d). The three ballots in Figure.. represent voter v s preference ordering expressed through a linearly ordered ballot B(v 1 ), a bucket ordered ballot B(v ), and a partially ordered ballot B(v 3 ). In the linear ballot, v is forced to select a preference between b and c even though he views these alternatives as incomparable. The linear orderings a c b d and a c d b could also be possible linear ballots for voter v based on his actual preferences towards the alternatives in A. We have arbitrarily picked one of the possible linear orderings compatible with his preferences to express his ballot. It should be clear that in the linearly ordered ballot v is forced to overspecify his preferences towards the alternatives in A, and thus his ballot does not accurately reflect how he truly feels about the alternatives. In the bucket ordered ballot, v is forced to compare b and d even though he views these alternatives as incomparable. This increases c s weight in the ballot and decreases the weights of b and d relative to voter v s actual preference ordering of the alternatives. Similar to the linear ballot, the bucket ordered ballot forces v to compare alternatives where he actually has no preference. Only in the partially ordered ballot is voter v s ballot an accurate representation of his preference ordering. Since linear ordered ballots and bucket ordered ballots would both still be allowed in a partially ordered election we feel there is no reason to not give voters the option to use ballots which allow for greater freedom in ranking the alternatives. We conclude this section by mathematically defining the other two important political terms associated with an election: a profile and a voting system. The set of all ballots in a given election will be called a profile. This leads us to the following definition:

27 . PRELIMINARIES 6 Figure... Same Preference Order Represented in Different Ballot Formats Definition..3. Let A be a set of alternatives and B(v j ) be a ballot of A where j {1,,..., m} and m is the number of voters in an election. The set of all ballots in an election is called a profile, denoted B, where B = {B(v 1 ), B(v ),..., B(v m )} and ballot B(v j ) represents the individual preference ordering of voter v j. Voting theory is concerned with analyzing the mathematical properties associated with different social choice procedures, or voting systems. These voting systems can be intuitively thought of as functions which input a profile of ballots associated with a given set of alternatives and output the winning alternative, or social choice, of the given election. From Alan Taylor [5] we are given the following definition of a social choice procedure: Definition..4. A social choice procedure is a function for which a typical input is a profile (set of preference lists) B of some set A (the set of alternatives) and the corresponding output is either an element of A, a subset of A, or NW. While Taylor s definition of a social choice procedure only returns the single alternative in A which is the social choice according to a given profile and social choice procedure, the definition of voting system we will use in this paper returns the complete social preference ordering of the alternatives in A according to a given voting system. We denote the social

28 . PRELIMINARIES 7 preference ordering of a given voting system as A, where A is a ranked ordering of the alternatives in A based on their final preference in the overall election. If the social choice of a voting system is a single alternative in A, we often will denote it as a where a A such that a is the most preferred alternative in A. It is also important to recognize that given the same set of alternatives A and the same profile of voters B, two different voting systems can output different social preference orderings. With these thoughts in mind, we present the definition of a voting system, or election method, as a function from a profile to the social preference ordering. Definition..5. A voting system E is a function which inputs a profile B (set of ballots) of some set A (the set of alternatives) and outputs the bucket ordered set A where each alternative in A appears in A and the alternatives in A are ordered by E. We will call A the social preference ordering according to E (note that we allow A to be a bucket ordered set in order to allow for the possibility of ties in the social preference ordering). More rigorously, we re-state the definition as follows: Definition..6. Let A = {a 1, a,..., a n } be a set of alternatives and B = {B(v 1 ), B(v ),..., B(v m )} be a profile of A. A voting system E is defined as a function E : B A where A is the ranked preference ordering of the alternatives in A according to the ballots in B and voting system E. It is important to note from the two definition given above that a voting system is a mathematical function which inputs a set of ballots and outputs the social preference ordering of the alternatives being voted between in the election. This is the definition of a voting system we will utilize throughout the rest of this paper to explain and analyze the Borda Count voting system introduced in the following section.

29 . PRELIMINARIES 8.3 Linear Borda Count One of the more mathematically interesting and socially effective voting systems is the ranked voting system devised by Jean-Charles de Borda in 1781 known as Borda Count. Intuitively, Borda Count determines the social choice of an election by allocating points to the alternatives based on how intensely they are preferred in each ballot of the election. In a given ballot, an alternative is allocated points equal to the number of alternatives ranked lower than it in the ballot. Hence, in a ballot with n alternatives the most preferred alternative in the ballot receives (n 1) points, the second most preferred alternative receives (n ) points, and so on down to the least preferred alternative which receives 0 points. We can rigorously define the allocation of Borda Scores in a linear ballot in the following manner: Definition.3.1. Let A = {a 1, a,..., a n } be a set of alternatives and B(v) be a linearly ordered ballot of A. Let a i be the i th most preferred alternative among the n total alternatives in B(v). The Linear Borda Score of a i in ballot B(v) is defined as BS ai (v) = n i. Note that for ease of notation the Linear Borda Score of an alternative a in a ballot B(v) will be written as BS a (v) rather than the more notationally complex BS a [B(v)]. Utilizing the equation defined above, we now rigorously define a Linear Borda voting system. Definition.3.. Let A be a set of alternatives and B be a profile of A. The Linear Borda election method, BS : B A, is defined as the voting system which allocates points to the alternatives in each ballot of B according to the Linear Borda Score equation in Definition.3.1.

30 . PRELIMINARIES 9 In Figure.3.1 we have a linearly ordered ballot B(v) of the set A = {a, b, c, d} in which voter v s preference ordering is b c a d. The Linear Borda Scores allocated to the alternatives in B(v) in a Linear Borda election are as follows below: Figure.3.1. Allocation of Borda Scores to a Linear Ballot BS b (v) = n 1 = 4 1 = 3 BS c (v) = n = 4 = BS a (v) = n 3 = 4 3 = 1 BS d (v) = n 4 = 4 4 = 0 Note from this example that the most preferred alternative in the ballot is allocated a Borda score of (n 1) and the least preferred alternative in the ballot is allocated a Borda score of 0. These are two properties of the Linear Borda voting system which we will want to retain when extending Borda Count to non-linear ballots in the following chapters. The third basic mathematical property of Linear Borda which we would like for any non-linear

31 . PRELIMINARIES 30 Borda election to also satisfy involves the total amount of Borda scores allocated to the alternatives in a single ballot and is presented in the following theorem. Theorem.3.3. Let A be a set of n alternatives and B(v) be a linear ballot of A. Then n n(n 1) BS ai (v) =. i=1 Proof. Let A = {a 1, a,..., a n } be a set of alternatives and B(v) be a linear ballot of A. Without loss of generality we can assume the alternatives in A are ordered such that a 1 a... a n. By Definition.3.1 we know that BS a1 (v) = n 1 BS a (v) = n. BS a( n 1)(v) = n (n 1) = 1 BS an (v) = n n = 0. It follows from the calculations above that n i=1 BS a i (v) = (n 1) + (n ) (n (n 1)) + (n n). Hence, we have n BS ai (v) = (n 1) + (n ) (n (n 1)) + (n n) i=1 = (n + n... + n) ( (n 1) + n) = n n(n + 1) = n n + n = n n n(n 1) =.

32 . PRELIMINARIES 31 Thus, we see that the sum of the Linear Borda scores allocated to the alternatives in a single linear ballot always equals n(n 1). The result seen in Theorem.3.3 is very nice because it tells us that every ballot in a Linear Borda election contributes the same weight toward determining the winner of the overall election. This property along with the already mentioned facts that the most preferred alternative in a linear ballot is allocated a Borda Score of (n 1) and the least preferred alternative in a linear ballot is allocated a Borda Score of 0, are the three desirable properties of the Linear Borda voting system which we will hope to retain in our non-linear Borda methods. These three properties will be more rigorously defined in Section 4.1 and provide the motivation for one of our most interesting results concerning Borda elections with non-linear preferences. The following definition clarifies the distinction between the individual Borda Score an alternative receives from a given ballot and the sum of the Borda Scores an alternative receives in a given election. Each of these quantities will be mentioned numerous times in the definitions, theorems and proofs presented throughout the rest of this paper, and defining them now makes it much easier to discern which of the two is being discussed at a given moment in later sections. Definition.3.4. Let A be a set of alternatives and B = {B(v 1 ), B(v ),..., B(v m )} be a profile of A. Let a A and B(v j ) B. The ballot Borda Score of alternative a in ballot B(v j ) is defined as BS a (v j ). The total Borda Score of alternative a in an election BS : B A is defined as BS a (v j ). Now that we have defined a Linear Borda voting system we must also define how to determine the winner of a given Linear Borda election. As we stated earlier, the alternative in a Linear Borda election which receives the largest total Borda Score is declared the winner of the election. Hence, we have the following definition.

33 . PRELIMINARIES 3 Definition.3.5. Let A be a set of alternatives, B be a profile of A such that B = {B(v 1 ), B(v ),..., B(v m )}, and BS : B A be a Linear Borda election. The winner of a Linear Borda election, a A, is defined as the alternative whose total Borda Score is greater than the total Borda Score of each other alternative in the election. Hence BS a (v j ) > BS a (v j ) for all other a A. We call A the Borda Count social preference ordering and a the Borda Count social choice. In Figure.3. we present a profile of ballots for a hypothetical Linear Borda election. The election, BS : B A, consists of four alternatives (A = {a, b, c, d}) and a profile of five voters whose preference rankings of the alternatives in A are expressed through their linearly ordered ballots (B = {B(v 1 ), B(v ), B(v 3 ), B(v 4 ), B(v 5 )}). From the ballots seen in Figure.3. we compute the total Borda score for each alternative in the election: Figure.3.. Linear Borda Election

34 . PRELIMINARIES 33 5 BS a (v j ) = 11 5 BS b (v j ) = 8 5 BS c (v j ) = 7 5 BS d (v j ) = 10 Hence, we see that 5 BS a(v j ) > 5 BS d(v j ) > 5 BS b(v j ) > 5 BS c(v j ). Thus, the social preference ordering for this election is a d b c, and it follows that a is the social choice of the Linear Borda election. In subsequent sections of this project we will spend much time exploring the properties of Borda Count that hold when we extend the domain of our elections to include bucket and partially ordered ballots. We conclude this section by introducing three essential properties which we feel are absolutely necessary for any non-linear voting system to satisfy in order to be considered a valid extension of Linear Borda Count. These properties, which we will refer to as the Axioms of Borda Count, are extremely simple and should not be confused with the desirable properties of Borda Count mentioned earlier in this section. Definition.3.6. Let A be a set of alternatives and B(v) be a ballot in a Borda Count election. The Axioms of Borda Count which must be satisfied in any valid Borda Count voting system are: 1. For all a A, BS a (v) 0.. If a b in B(v), then BS a (v) > BS b (v). 3. If a b in B(v), then BS a (v) = BS b (v).

35 . PRELIMINARIES 34 It should be quite clear that these three axioms are not only desirable but necessary for any extension of Linear Borda to be a valid Borda Count voting system. Axiom (1) states that the minimum Borda Score an alternative can be allocated in a ballot is zero. If (1) did not hold, then it would be possible for voter s to severely penalize candidates by allocating them a negative ballot Borda Score and actually lowering their total Borda Score in the overall election. This not only seems rather unfair, but it would also make it much more difficult to compare ballots among several voters. Axiom () states that if one alternative is preferred to another in a given ballot, then the alternative that is more preferred must be allocated a greater ballot Borda Score than the alternative that is less preferred. If () did not hold, then voter s would have no incentive to rank the alternatives in order of preference because alternatives that were less preferred could receive larger Borda Scores than alternatives that were more preferred. Axioms (1) and () can be inferred directly by retaining desirable properties from Linear Borda. In contrast, Axiom (3) is inapplicable when considering a linearly ordered ballot since two alternatives can never be equivalent in a linear ordering. It is, however, extremely important to establish this axiom now because it formalizes the notion of two alternatives being tied in a bucket ordered or partially ordered ballot. As was mentioned in Section.1, if two alternatives are incomparable in a bucket ordered ballot then they are preferred by the same set of alternatives, preferred to the same set of alternatives, and incomparable to the same set of alternatives in the ballot. Thus the two alternatives are equivalent in the ballot, and it seems rather obvious that they should be allocated the same Borda Score. The Axioms of Borda Count provide the basis for determining if a voting system which extends Linear Borda to allow non-linear ballots is a valid Borda Count voting system. It is important to keep these axioms in mind throughout our discussion of non-linear Borda