Algebraic Properties of Positional Voting Systems A Senior Comprehensive Paper presented to the Faculty of Carleton College Department of Mathematics and Statistics Peri Shereen, Advisor by Jordan Cahn John Eckert Margalo Mullaney Dana Neidinger 5/23/2016 Dedication Thanks to Peri Shereen for advising us, and to Zajj Daugherty, Alexander K. Eustis, Gregory Minton and Michael E. Orrison, and Donald Saari for their excellent sources. iii Abstract In this paper we explore the positional voting system through an algebraic lens, modeling voting procedures as linear transformations. We present results from Orrison et al. in the first chapter to set up the framework we use to discuss positional voting. In the second chapter, we present our results on preserving reversal - if everyone in the electorate reverses their votes, will the order of the results be reversed as well? The main theorem in this paper classifies reversal-preserving and reversal-ignoring positional voting procedures. v Contents Dedication iii Abstract v Contents vii 1 Positional Voting 1 1.1 Profiles . 1 1.1.1 The Voting Profile . 1 1.1.2 Group Action of Sn on Profiles . 3 1.2 Weighting Vectors and Voting Procedures . 3 1.2.1 The Positional Voting Procedure . 3 1.2.2 Partial Ranking Procedures to Full Ranking Procedures . 5 1.2.3 QSn Modules and Neutrality . 7 1.3 The Importance of the Weighting Vector . 8 1.3.1 Decomposition of Weighting Vectors and Equivalent Weighting Vectors . 8 1.3.2 Relationships Between Weighting Vectors and Results . 9 2 Reversal 13 2.1 Defining Reversal . 13 2.2 Preserving Reversal . 14 3 Conclusion 19 Bibliography 21 vii 1. Positional Voting This paper will explore positional voting systems through algebraic means. Saari has shown that many types of voting systems can be modeled geometrically; we aim to study voting systems algebraically. We studied an algebraic framework of positional voting systems explored previously by Michael Orrison et al. In this chapter, we will explore many results from their paper in order to set up our own theorem in the next chapter. A positional voting system has three main defining components: • Profiles: representations of the preferences of the voters in the electorate. We will view profiles as being in the vector space over combinatorial tabloids, which represent the voters’ rankings. • Weighting vectors: the vector that determines the voting procedure on a profile. We will view weighting vectors as w ∈ Qm where m is the number of rows in a given tabloid. Thus, the kth entry of the vector assigns a certain number of points to the kth ranked candidate(s). • Results: the outcome of an election when the weighting vector is applied to a profile. 1.1 Profiles In an election, the first step is to mathematically represent the votes of the electorate. Through the use of combinatorial tabloids, we will illuminate the system we used to represent our voter base. 1.1.1 The Voting Profile Let λ = (λ1, . , λm) be a sequence of positive integers such that the sum λ1 + ··· + λm = n. This λ determines the shape of a Ferrers diagram, which is a left-justified array of dots with λi dots in the ith row. A Young tableau of shape λ is such a shape where every dot is replaced with a box containing a single number from 1 to n with no repetition. Two tableau are said to be row-equivalent if they differ only by a permutation of the entries within the rows of each tableau. An equivalence class of tableaux under this relation is called a tabloid of shape λ. Tabloids are represented as diagrams with m rows, where each row i is an unordered set of λi numbers. From a voting-theoretic standpoint we can view tabloids as rankings, with the candidates listed in the top row being in first place and so on, with the candidates listed in the last row getting last place. Two candidates that are both placed in a voter’s kth ranking position will be treated with the same weight by positional voting, hence the reason we use row equivalences. 1 2 2 1 tableaux: 3 6= 3 4 4 1 2 2 1 tabloids: 3 = 3 4 4 When λ = (1,..., 1) we call this a full ranking of the candidates. Otherwise we say this is a partial ranking. In our results we will be dealing primarily with the full ranking situation, because as we will see in section 1.3.2, all partially ranked positional systems can be modeled using fully ranked cases. Let Xλ denote the set of tabloids of shape λ. When λ = (1,..., 1), we can see that |X(1,...,1)| = n!, as there is a natural bijection between full rankings of n candidates and permutations of n. The bijection 1 2 CHAPTER 1. POSITIONAL VOTING involves assigning a number to each candidate. If candidate i is in place k we say that in the corresponding permutation k “goes to” i. In general, |Xλ| = n! , as we have all of the possible permutations of n λ1!...λm! candidates but divided by the row-equivalencies because every ordering of a row overcounts. In an election we will consider a fixed λ, and voters will choose from one of the rankings in Xλ based on which candidates they prefer. Thus, to determine the makeup of the entire electorate, we need to know how many people voted for each ranking x ∈ Xλ. Definition 1.1. A voting profile p is a function p : Xλ → Q such that p(x) is the number of voters that voted for the ranking x. Note that we do not require that p(x) be a positive integer; it may take any value in Q. We can consider the set M λ = {p : Xλ → Q}, the set of all profiles of shape λ. We see that M λ forms a vector space over the rationals: (p + q)(x) = p(x) + q(x) (αp)(x) = αp(x) We can easily find a basis of M λ, namely the indicator functions of M λ, where the indicator function of x ∈ Xλ is given by the Kronecker delta: ( 1 x = y δxy = 0 x 6= y Essentially, the indicator function of x is the function that gives one vote to a ranking x ∈ Xλ. Therefore, since the indicator functions correspond to the elements of Xλ, we can consider this basis as simply the elements of Xλ. We can view a profile as a formal linear combination of the tabloids in Xλ, where the coefficient of each tabloid x is the number of people that voted for the particular ranking, or p(x). For example, in the fully ranked 3-candidate case, we have 1 1 2 2 3 3 p = a 2 + b 3 + c 1 + d 3 + e 1 + f 2 3 2 3 1 2 1 1 1 for a, b, c, d, e, f ∈ Q, where a = p 2 , b = p 3 and so forth. Given a standard ordering, we can then 3 2 λ consider each profile p as a vector in Q|X |. For instance, in the previous example we would have a b c p = d e f where |Xλ| = 3! = 6, so p ∈ Q6. For our standard ordering we have chosen to use lexicographic ordering, as shown above, where when we order the rows of the tabloids in order of numerical ascendance. The tabloids fall in “dictionary order.” For instance, 1 2 1 3 2 3 3 4 3 > ··· > 2 > ··· > 4 > ··· > 2 4 4 1 1 We have already seen how there is a natural correspondence between full rankings in X(1,...,1)and elements (1,...,1) of the symmetric group Sn, so we can consider a bijection between fully ranked profiles in M and elements of the group ring S , where the group ring is simply the set of sums P a σ for coefficients a Q n σ∈Sn σ σ in the field Q. 1.2. WEIGHTING VECTORS AND VOTING PROCEDURES 3 1.1.2 Group Action of Sn on Profiles Instead of viewing tabloids as permutations, we can consider a different relation between permutations and tabloids (and therefore profiles), namely the action of a permutation on a profile. λ Let λ = (λ1, . , λm). We define the group action of Sn on X as a permutation of the entries of the tabloids. In other words, consider candidate k in tabloid x. In σx the position where k was will now have element σ(k). Example 1.2. Suppose we have n = 8, λ = (3, 2, 3), σ = (1 3 5)(4 2)(6 7), then 8 7 4 σ(8) σ(7) σ(4) 8 6 2 σ 5 6 = σ(5) σ(6) = 1 7 2 1 3 σ(2) σ(1) σ(3) 4 3 5 λ λ −1 This group action of Sn on X extends to an action of Sn on the profile space M by (σp)(x) = p(σ x) [1]. This seems strange, but we can in fact see that the inverse of σ is required in the definition of this action in order for it be a group action. Specifically it is needed to fill the associative requirement: (στ)p(x) = p((τ −1σ−1)x) = p(τ −1(σ−1x)) = (τp)(σ−1x) = σ(τp)(x). This action is key to viewing profiles as algebraic objects. 1.2 Weighting Vectors and Voting Procedures In positional voting, the voting procedure is determined by a vector of weights corresponding to the m rows of the tabloid of shape λ.