Algebraic voting theory & representations of S_m≀S_n

Abstract

We consider the problem of selecting an n-member committee made up of one of m candidates from each of n distinct departments. Using an algebraic approach, we analyze positional voting procedures, including the Borda count, as $\mathbb{Q}{S}_m\wr S_n$-module homomorphisms. In particular, we decompose the spaces of voter preferences and election results into simple $\mathbb{Q}{S}_m\wr S_n$-submodules and apply Schur's Lemma to determine the structure of the information lost in the voting process. We conclude with a voting paradox result, showing that for sufficiently different weighting vectors, applying the associated positional voting procedures to the same set of votes can yield vastly different election outcomes.

Introduction

In this paper, we examine the process of electing an n-member committee comprised of one representative from each of n departments, with each departmental representative chosen from a field of m candidates. The committee selection process arises in many guises and is central to decision making procedures in a number of fields – for example, determining the best combination of features for a new product based on market research, or the best combination of treatments based on surveying medical professionals.

We examine voting procedures in which voters give complete or partial rankings of the set of possible committees from most to least desired. Using a positional voting procedure, points are awarded to committees based on their position in each voter's ranking, where a choice of weighting vector determines the amount of points awarded for each position. The committee with the most points is elected.

A more common and simpler approach to committee selection has voters rank the candidates within each department, and the wining committee is comprised of the most popular candidate from each department. The motivation for voters ranking all possible committees stems from Ratliff's work that shows voter preferences in the committee selection setting are often more complex and nuanced than their rankings of the candidates [12]. For example, voters can be influenced by a desire for diverse representation across the committee or consideration for the relationships among committee members. Examining data from a 2003 university election, Ratliff found strong evidence that voter committee preferences would not have been captured by having voters simply select their favorite candidate from each division. In particular, more than half the voters in the election had first and last choice committee preferences that were not disjoint. A subsequent paper by Ratliff and Saari [13] shows that undesirable election outcomes, stemming from asking voters to only rank individual candidates and ignoring the composition of the group, are not uncommon.

In what follows, we study committee voting procedures by examining the underlying algebraic structures of two spaces: the space of voter preferences and the space of possible election outcomes. This algebraic approach to voting theory was first introduced by Daugherty, Eustis, Minton, and Orrison [4]. Using representations of the symmetric group, Daugherty, et al., recover and extend several results of Saari, a pioneer in the field of mathematical voting theory who studied the geometry and subspaces of voting information (see, e.g., [14] & [15]). The purpose of such analysis is to uncover what aspects of voter preferences contribute to the outcome of an election when a positional voting method is used.

Following the lead of Daugherty, et al., Lee introduced the idea of using representations of wreath products of symmetric groups to study committee elections [10]. In this setting, the spaces of voter preferences and possible election results are viewed as wreath product modules. Our contribution is to find the module decomposition of these spaces, and to otherwise extend the results in [4] to committee selection voting.

In particular, we explore a paradox that arises from the fact that there is no voting system that perfectly translates the preferences of voters into a single election outcome and that results often say as much about the method of voting as they do about voter preference. To show how an election procedure can affect the election outcome when voting for individual candidates, Saari proved that given j sufficiently different weighted positional voting systems and any j orderings of n candidates, $A_1,\dots ,A_j$, there exist examples of voter preferences such that using voting system i results in the ranking of candidates $A_i$ [14, Theorem 1]. Essentially, this implies that if the weighting vectors are sufficiently different, the corresponding election results might not resemble each other at all, despite using the same set of voter preferences.

In [4, Theorem 1], Daugherty, et al. prove a stronger result using their algebraic framework. They show that given j sufficiently different weighting vectors and any j corresponding vectors of point totals for each candidate, there exist infinitely many voter preferences such that using weighting vector i results in the corresponding point total vector, for all $1\le i\le j$. Notice here that not only does the ranking of the candidates rely on the voting procedure, but so too does any (relative) difference in point totals.

We obtain an analog of [4, Theorem 1] for committee voting, showing again that if a set of weighting vectors are sufficiently different, the associated positional voting procedures can yield radically different outcomes. This troubling result shows that the choice of an inauspicious weighting vector could lead to the election of an undesirable committee.

In the committee selection procedure studied here, the algebraic structures of the space of voter preferences and space of results are much more complex than those examined by Saari and Daugherty, et al. In the single-candidate selection setting the result space, when viewed as a $\mathbb{Q}{S}_n$-module, decomposes into a direct sum of two irreducibles: $S^{(n)}\oplus S^{(n-1,1)}$, while in our committee-selection scheme, the result space decomposes into many irreducibles: ${\oplus }_{k=0}^n{\oplus }_{\left(\begin{array}{c}n\\ k\end{array}\right)}{S}^{((n-k),(k),\varnothing ,\dots ,\varnothing )}$. In the first case, weight vectors are “different enough” so long as they are linearly independent in $S^{(n-1,1)}$. In the second case, we must require that the weighting vectors be linearly independent in each component of the direct sum.

Connections between voting methods and decision making procedures have been used to extend results, like those proved here, to applications in non-parametric statistics and game theory (e.g. [3], [1], & [7]). It is the authors' hope that the results presented here might be applicable to other decision making settings.