Math and Politics: You Can’t Always Get What You Want

The students in Professor of Mathematics Michael Mossinghoff's "Math and Politics" class are asking and answering some intriguing questions–questions that fit squarely within the context of the 2016 election.

The question that is currently capturing the students' political passion is this: "How can a group elect a winner that a majority of its members dislike?"

"Plurality voting works by each voter only selecting their top choice, and the candidate with the most votes wins the election," explained Christian Fechter '17.

"... For example, in a three person election between Clinton, Trump and Johnson let's imagine that Johnson is taking more votes away from Clinton, and although those voting for Johnson or Clinton may have Trump as their least favorite candidate, Trump could easily win if Clinton and Johnson split the vote 15 percent Johnson, 40 percent Clinton–Trump could win with 45 percent of the vote despite being the least favorite candidate of 55 percent of the population; in that situation, you elect a president that the majority of the country dislikes, which seems to be a problem with plurality elections."

Rose Botaish '17 and Meranda Ma '17 explore the question from a less democratic angle.

"The course "Math & Politics" offers insight into various voting methods that produce such an outcome [election of disliked candidates]," write Botaish and Ma.

"For example, the social choice procedure referred to as a ‘dictatorship' simply means that one voter, the dictator, determines the outcome of the election. In other words, this voter's top choice candidate automatically becomes the winner, regardless of that candidate's position on every other voter's ballot. Therefore, if a majority of the voters dislike this candidate and place him/her in last place, but the dictator ranks him/her in first place, that candidate would still win the overall election."

Finally, Kathryn LeBey '20 describes a scenario in which luck and the lack of intense dislike (or strong enthusiasm) make for a winning combination.

"More convoluted is the anti-plurality method, which selects that candidate who the fewest voters dislike the most as its winner," LeBey explained. "That explanation itself warrants a second reading to be understood. If voters remain unenthused about a candidate enough to dislike them but without the extremity to rank them dead last out of all the choices, that candidate could very well win under the anti-plurality method. So, to win an election with this or a number of other methods in place, candidates don't need to concentrate on becoming well-liked; rather, they just have to get lucky!"

Power in Numbers

To further illustrate the ways in which mathematics and politics intersect, Prof. Mossinghoff weighs in with a few more broad-ranging questions and answers from his "Math & Politics" syllabus.

Q: How much power does the President of the United States really have?

We can look at how much legislative power the president has by asking how often the president provides critical support: so with the President's support, the bill in question becomes law, but without it, it fails. There are a number of mathematical models used in political science for measuring this, each with their own particular assumptions, and different models produce different values. The answer really depends on the kinds of coalitions that can form within the House and Senate.

Q: Is there an impartial way to test for gerrymandering?

There are mathematical tests one can apply to test for unusual [district] shapes, for example, perimeter-area comparisons. But a resolution is not that easy–after all, many states already have twisting and winding borders due to rivers and other natural features. Practical considerations like county and precinct boundaries matter, too–it might be useful for instance to avoid breaking up counties into multiple congressional districts, so that all voters in a county would use the same ballot. So this is an area where mathematicians, demographers, judges and lawmakers have all been quite active, and we should expect more activity in this area in the coming years.

Q: Why did Montana sue the U.S. Department of Commerce in 1991 based on how an average should be computed, in a case that was decided by the U.S. Supreme Court?

Montana has a single congressional district, encompassing the entire state–this is the largest district (by population) in the country, so Montanans are, in a sense, the least well represented in the House of Representatives. If you altered the method that is used to apportion seats to states, and used a method that was just slightly more beneficial to smaller states, Montana would get two representatives. This change boils down to a matter of how a particular average is computed. There are a number of ways in mathematics to compute averages: the average Montana preferred also comes up in the following SAT-type problem: if you drove 10 miles at 30 mph, and then 10 more miles at 40 mph, what is your average speed? (It's not 35 mph).

Q: Why did Rep. Charles E. Littlefield of Maine lament in Congressional committee in 1901: ``God help the state of Maine when mathematics reach for her and undertake to strike her down!''

After the 1900 census, government statisticians prepared a report showing how many seats would be allocated to each state for a range of possible house sizes under the current law. The method in use at the time (invented by Alexander Hamilton–take note, Lin Manuel!) assigned Maine either three or four seats over this range, but the value bounced between these over this range. So, Maine could lose a seat even when the size of the House increased. (Maybe even Littlefield's!). In the end, this issue, along with some other oddities that arose with Hamilton's method, caused Congress to change the method used to apportion seats.

Q: OK, but what can we learn from the Rolling Stones?

Throughout this course we look for the best way to perform various political procedures, from coming up with a procedure to pick a winner from a set of votes, to allocating seats to states according to population, to other problems involving the division of a fixed resource among multiple interested parties.

In many of these problems, we list some attributes that we would like to guarantee, but it turns out that in many cases one can prove that it is impossible to get all of the attributes that we would like. As the eminent English philosopher Sir Mick Jagger tells us, we can't always get what we want.