Extensions of Condorcet's Paradox of Voting
Under the direction of: William Zwicker
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 2012. Drop by my office (Bailey 200B) or send email: zwickerw@union.edu.
A student working on this topic should have strong programming skills.
Imagine that $a, b,$ and $c$ are three candidates for president, and three voters $v1, v2, v3$ have the following preferences: $\begin{array}{ccc} v_{1} & v_{2}& v_{3} \\ a& c &b \\ b& a &c \\ c& b &a \\ \end{array}$
This means that $v1$'s favorite candidate is $a$, her second favorite is $b$, and her least favorite is $c$; $v2$'s favorite is $c$, etc.
Notice that twothirds of the voters prefer $a$ to $b$, twothirds prefer $b$ to $c$ and twothirds prefer $c$ to $a$. This means that whichever candidate is elected, twothirds of the voters can name a single alternative candidate whom they prefer. This simple but puzzling situation is Condorcet's paradox of voting  it lies at the heart of much of the mathematical theory of voting.
Now suppose we were electing a committee $C$ of candidates, and we want the committee to be "stable" in the sense that for each candidate $y$ outside $C$ there is at least one candidate $x$ inside $C$ who is preferred to $y$ by a majority of voters. Condorcet's paradox tells us that to be stable, a committee may need to have more than one member. So, do twoperson stable committees always exist? Recently, a group of researchers from Paris and Singapore found a relatively simple example for which no stable twoperson committee exists; their example has 15 voters and 15 candidates. Even more recently, two of us at Union found smaller examples, which are more complicated. Do stable committees of size 3 always exist? How can we understand the structure of these examples? There is a wealth of related questions we can explore.
1 or 2term thesis

