Preferences in Referendum Elections

Suppose that, in a referendum election, your preferences on one proposal depend on the outcome of another. For example, you might like both Proposal A and Proposal B, but you might also feel that passage of both proposals would place an unreasonable tax burden on local residents. In your mind, the best outcome is for exactly one of the two proposals to pass. You don’t really care which one (since you like both of them), but you know that having both pass would be a real financial disaster. So how should you vote?

It turns out that, no matter what ballot you cast, you might wish that you had voted differently when the results of the election are announced on the 11:00 news. For instance, suppose that you vote NO on Proposal A and YES on Proposal B, only to find out that both proposals pass, Proposal A by a wide margin and Proposal B by one vote – your vote! I can’t imagine that you’d be too happy with this outcome. After all, had you voted NO/NO or YES/ NO instead, you would have gotten your wish of having exactly one proposal pass. As it is, you find yourself in precisely the situation you were trying to avoid.

The problem here stems from the fact that you had to vote on both issues simultaneously. If you had known that Proposal A was going to pass regardless of whether you voted for it or not, you may have still voted NO on it, but you definitely would have voted NO on Proposal B. The trouble is that your preferences are complicated in the sense that they can’t be boiled down to a YES or NO vote on each individual proposal. There is some interplay going on here that turns out to cause a whole mess of ugly election behavior (like an election where the winning outcome is the last choice of every single voter!).

Luckily for mathematicians such as myself, there also turns out to be a whole lot of interesting mathematics behind these sorts of preferences. That’s where my research has been focused. I use tools from set theory, combinatorics, abstract algebra, and even a little bit of computer simulation to analyze the properties and implications of interdependence within voter preferences. I’ve been quite surprised at the depth of mathematics that emerges from these sorts of investigations. At the same time, the problems that come up are easy to understand and accessible to undergraduates. I feel very fortunate to have found such a fun and exciting research area and am particularly excited about involving students in my work.

2008 REU Topic: Congressional Redistricting, Gerrymandering, and Measures of Compactness

Another area that I’ve recently become interested in is congressional redistricting and the gerrymandering problem. The basic idea is this: The Constitution of the United States requires that each state be allocated a number of congressional representatives proportional to its population. The Constitution does not, however, specify how states are to determine the geographic boundaries of each congressional district. This often leads to gerrymandering, which is the redrawing of congressional districts in order to gain or maintain seats for the dominant political party. During the 2008 REU, we will investigate and propose new ways to detect and measure gerrymandering. We will be particularly interested in developing new ways to measure the compactness of shapes. We will then investigate how these measures of compactness can be used to evaluate the desirability and fairness of a congressional redistricting.