Exploring a Generalized Partial Borda Count Voting System

The main purpose of an election is to generate a fair end result in which everyone\u27s opinion is gathered into a collective decision. This project focuses on Voting Theory, the mathematical study of voting systems. Because different voting systems yield different end results, the challenge begins with finding a voting system that will result in a fair election. Although there are many different voting systems, in this project we focus on the Partial Borda Count Voting System, which uses partially ordered sets (posets), instead of the linearly ordered ballots used in traditional elections, to rank its candidates. We introduce the Generalized Partial Borda Count Voting System, and explore which properties of Partial Borda are still satisfied in this general setting

An Exploration of Voting with Partial Orders

In this thesis, we discuss existing ideas and voting systems in social choice theory. Specifically, we focus on the Kemeny rule and the Borda count. Then, we begin trying to understand generalizations of these voting systems in a setting where voters can submit partial rankings on their ballot, instead of complete rankings

Win Big(k), Lose Big(k): A further exploration of the Win Big and Lose Big voting systems

This project focuses on the mathematical study of ranked voting systems, where voters rank their candidates in order of preference. There are many different voting systems used to determine a winner from a given set of ballots, for example Plurality, Borda Count, Coombs, Instant Runoff, and Anti-plurality. Looking further into a previous senior project where the author created a new voting system that selects an average alternative, we analyze the long term results of those systems. Taking the Win Big(k), and Lose Big(k) methods we compare them to various pre-existing ranked voting methods

A Strength Test for the Borda Count

When running an election with more than two candidates, there are many ways to choose the winner. A famous theorem of Arrow states that the only mathematically fair way to choose is to do so at random. Because this is not a desirable way to choose a winner of an election, many mathematicians have devised alternate ways of aggregating ballots. In my project I consider one of these ways -- the Borda Count, considered to be one of the most desirable from both the point of view of mathematics and economics -- and came up with a method to test the mathematical fairness of an arbitrary voting system against the known fairness of the Borda Count

Lose Big, Win Big, Sum Big: An Exploration of Ranked Voting Systems

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College

An algorithmic approach to detect non-injectivity of the Partial Borda Count

Voting is how we elect today’s voices, faces, and leaders in our country. It is argued to be a very essential right we have as a people. A voter votes, by listing their preferences. Their preferences are relating the candidates to one each other (i.e. whether they prefer candidate A to candidate B or if they are indifferent between the two). There are many different social choice functions that can be used to calculate the results of an election. This project glances over the theory of Condorcet, Borda, Arrow, and Young, all of whom had a great impact on voting theory and social choice theory. I experiment with partially-ordered preferences using the Partial Borda Count. The Partial Borda Count switches from being injective (one-to-one) to non-injective (multiple posets going to the same score vector) for all elections with 5-elements or higher. I created an algorithm that determines certain posets that go to the same score vectors for n-candidate elections (if n \u3e 5). My algorithm was able to detect all of the failures of injectivity for a 5-candidate election. I then use this algorithm to see if I can predict which posets go to the same score vector, for a 6-candidate election, without having to construct a 6-element database. It turns out my algorithm proved successful in locating some of the injectivity failures of 6-element elections

Neural networks would \u27vote\u27 according to Borda\u27s Rule

Can neural networks learn to select an alternative based on a systematic aggregation of conflicting individual preferences (i.e. a `voting rule\u27)? And if so, which voting rule best describes their behavior? We show that a prominent neural network can be trained to respect two fundamental principles of voting theory, the unanimity principle and the Pareto property. Building on this positive result, we train the neural network on profiles of ballots possessing a Condorcet winner, a unique Borda winner, and a unique plurality winner, respectively. We investigate which social outcome the trained neural network chooses, and find that among a number of popular voting rules its behavior mimics most closely the Borda rule. Indeed, the neural network chooses the Borda winner most often, no matter on which voting rule it was trained. Neural networks thus seem to give a surprisingly clear-cut answer to one of the most fundamental and controversial problems in voting theory: the determination of the most salient election method

Improving the Aggregation and Evaluation of NBA Mock Drafts

Many enthusiasts and experts publish forecasts of the order players are drafted into professional sports leagues, known as mock drafts. Using a novel dataset of mock drafts for the National Basketball Association (NBA), we analyze authors' mock draft accuracy over time and ask how we can reasonably use information from multiple authors. To measure how accurate mock drafts are, we assume that both mock drafts and the actual draft are ranked lists, and we propose that rank-biased distance (RBD) of Webber et al. (2010) is the appropriate error metric for mock draft accuracy. This is because RBD allows mock drafts to have a different length than the actual draft, accounts for players not appearing in both lists, and weights errors early in the draft more than errors later on. We validate that mock drafts, as expected, improve in accuracy over the course of a season, and that accuracy of the mock drafts produced right before their drafts is fairly stable across seasons. To be able to combine information from multiple mock drafts into a single consensus mock draft, we also propose a ranked-list combination method based on the ideas of ranked-choice voting. We show that our method provides improved forecasts over the standard Borda count combination method used for most similar analyses in sports, and that either combination method provides a more accurate forecast over time than any single author.Comment: 16 pages, 3 figure

Voting, the Symmetric Group, and Representation Theory

We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied QSn-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur\u27s Lemma), this allows us to recast and extend some well-known results in the field of voting theory