Math Department (Thesis/2007-08)

The Geometry of Multicandidate Voting (2-term thesis)

Under the direction of: William Zwicker

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Dear Students, I've found that it is very helpful for students to get a head start by doing some background reading before the official start of a thesis. If you will be working with me, please get in touch as soon as you know - in particular, at least 2 weeks before exams for Spring term 2007. Drop by my office (Bailey 200B) or send e-mail: zwickerw@union.edu.

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With only two candidates for office, there is no doubt about how to calculate who won an election; the winner is the candidate who gets a majority of the votes. How about elections with three or more candidates? In this case, voters can express their preferences by listing all the candidates, from their favorite on top of the list, down to their least favorite. One problem that can arise is a "Condorcet cycle": a majority of voters prefer candidate A to B, a majority prefer B to C, and yet a majority prefer C to A . . . strange! The existence of Condorcet cycles led Kenneth Arrow to his famous "Impossibility Theorem," which showed (speaking very loosely) that every possible election system for a multicandidate election (one with 3 or more candidates) suffers from at least one type of fairness problem. That is, no ideal voting system exists for multicandidate elections.

As a consequence, there are a number of different election systems: Borda count, Condorcet Rule, Approval Voting, etc. Experts disagree (strongly!) about their relative advantages and disadvantages. Recently, I've found that some of these systems are based on the mathematical idea of mean (average). That is, suppose we pretend that each voter casts her vote for a certain point p_i in space, that we then find the average position q of these votes (by averaging the vector coordinates), and that we finally use the location in space of this "average voter" q to determine the election outcome. For certain voting systems, if we position the original votes p_i in space in just the right way, we can duplicate the effect of the original system. Moving these positions changes the voting system.

But suppose our goal is not to duplicate old voting systems. Imagine we want to invent brand-new systems with different, attractive properties. Can we do this by using new ways to position votes in space?¹ I don't know - but, hey, if I did, it wouldn't be research!

¹"Space" here might have more than 3 dimensions.

Math Department web pages Created: 25 Apr 2007 Last modified: 25 Apr 2007 Comments to: math@union.edu

Thesis Topics - W. Zwicker Thesis Topics for 2007-2008 Thesis Topics - A. Taylor