By Christoph Börgers
The following is a short and timely reflection from the author of Mathematics of Social Choice: Voting, Compensation, and Division, which was published by SIAM in 2010.
American democracy has widely recognized flaws. We vote on a workday, when many people—especially the underprivileged—find it hard to take time off to make it to the polling stations. Lines in front of polling stations are longer in poor neighborhoods than in wealthy ones. Gerrymandering distorts election outcomes. In many elections, voters who support small parties choose one of the two major parties instead, for fear of “throwing their vote away.”
Although these are political issues, they do have mathematical aspects. Among them is the question of how to structure a fair election that selects a single winner from a field of candidates. This is the subject of the first half of my book, Mathematics of Social Choice. I restrict myself to election methods that are based on voters’ rankings of candidates. We call a collection of candidate rankings by all voters a preference schedule. Denoting the set of all possible preference schedules by \(\mathcal{R}\) and the set of all candidates by \(\mathcal{C}\), an election method is a map
\[\mathcal{R} \rightarrow \mathcal{C}.\]
One can easily think of criteria that this map should satisfy — criteria on which most people would agree. We then ask whether and how a map that satisfies these criteria can be designed.
Nicolas de Condorcet (1743-1794) proposed the most famous such criterion. I use the notation
\[X \succ Y\]
to indicate that more than half the voters rank candidate \(X\) over \(Y\). Condorcet’s criterion then states that \(X \in \mathcal C\) should win if
\[\forall Y \in \mathcal C ~~~~~ X \neq Y \Rightarrow X \succ Y.\]
One calls \(X\) the Condorcet candidate. A method that satisfies Condorcet’s criterion is called Condorcet-fair.
The most common election method in the U.S. is the plurality method. The winner is the candidate whom the largest number of voters name as their top choice. This is not Condorcet-fair. For example, in the 2000 presidential election in Florida, Al Gore lost to George W. Bush by 537 votes, assuming the votes were counted correctly. However, Green Party candidate Ralph Nader received over 97,000 votes. It seems likely that most Nader voters preferred Gore to Bush, so Gore was probably the Condorcet candidate. Had Gore won Florida, he would have won the presidency.
The economists Eric Maskin and Amartya Sen have advocated for adherence to Condorcet’s criterion, which they call “majority rule” [1]. If we agree, we still have to decide what to do when there is no Condorcet candidate. This is not likely, so it is a secondary issue, but it is by no means impossible and therefore must be addressed. There are many possible answers and many fairness criteria to examine, some of which I discuss in my book.
Most voting reform activists in the U.S. favor a different method, called instant runoff or ranked choice voting (RCV). It is not Condorcet-fair, but it violates Condorcet’s criterion less frequently than the plurality method. The strength of RCV is the so-called later-no-harm property: voters’ second, third, etc. choices cannot affect whether their first choice wins. No Condorcet-fair method has this property.
The second half of Mathematics of Social Choice is about fair resource allocation, particularly “cake cutting.” The mathematical formalization of this problem is due to Hugo Steinhaus [2], the mentor and collaborator of Stefan Banach. It is based on measure theory, that is, additivity (the language in my book is less formal). This strikes me as a weakness, as the idea that 100 apple pies should please me 100 times as much as one pie seems far-fetched. However, the subject does have real applications to sharing computing resources, assigning seats in sought-after college courses, etc.
I wrote my book for a course at Tufts University that is intended for students who are majoring in non-mathematical subjects; they take the course to satisfy a breadth requirement. Nonetheless, the book contains many theorems and their proofs, along with examples and discussions of the significance of these theorems.
In all my classes, I have always asked myself, “How can I make what I am talking about interesting to my students?” An even better question is, “How can I talk about what my students are interested in?” Many college students care about societal fairness, perhaps now more than ever, which is why fair voting and fair resource allocation are good topics for college mathematics courses, even for mathematics majors.
References
[1] Maskin, E., & Sen, A. (2016, May 1). How to let the majority rule. The New York Times, p. SR-7.
[2] Steinhaus, H. (1948). The problem of fair division. Econometrica, 16, 101-104.
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Christoph Börgers is a professor of mathematics at Tufts University.