Every invitation to Math Club ends with the same reminder: “As always, the pizza and pop are free.”
It’s not just the free food that draws between 30 and 60 undergrads to Math Club each week. Students crowd into the Nesbitt Common Room in East Hall to hear professors and postdoctoral researchers talk casually about math, with topics like “How to Make Money Gambling! Step 1: Cheat,” “Wallpaper Symmetry,” and “Modeling the Underlying Dynamics of the Spread of Crime.”
But there’s always that person who flips open the pizza boxes and says, “Where’s the pineapple?!” One of last year’s Math Club organizers, Sarah Koch, claimed that she could rectify the pineapple problem. Koch is an assistant professor in LSA’s Department of Mathematics; naturally, her solution involved running the numbers.
At one of the Math Club meetings, she asked the students to rank five pizza toppings—cheese, pepperoni, mushroom, pineapple, and ham—from 1 for their favorite to 5 for their least favorite. Ideally, the votes would yield the right mix of toppings for Math Club members.
The outcome was more complicated.
How (Un)Fair Are Elections?
Fifty-five pizza ballots were tallied. The results were announced at the club’s next meeting, whose topic was “How (Un)fair Are Elections?” A postdoctoral researcher in the math department, Angélica Benito, led the talk, and she began by evaluating the pizza ballots with the vote-counting method favored in U.S. elections—plurality.
In the plurality pizza vote, whichever topping had the highest number of No. 1 votes won: cheese, it turned out. But the problem with plurality is that a candidate can win even if the option isn’t well liked. In this case, cheese won with 32.7% of the vote, which meant that 67.3% of the students—the vast majority—did not prefer cheese pizza.
So Benito explored other vote-counting methods. The runoff method, for example, protects the populace from the candidate that most voters dislike. In the pizza vote, the runoff tally worked this way: Benito kept the two contenders with the most No. 1 votes—cheese and mushroom—and eliminated the rest. She then examined the discarded ballots. Wherever cheese ranked higher than mushroom, cheese received that vote, and vice-versa. Using the runoff method, mushroom won.
This was disconcerting. Should they order more cheese or more mushroom pizzas for Math Club? Benito thought maybe they could break the tie by counting a different way. So she tried sequential runoff, a variation of the runoff method that’s used in Australia, Canada, and some U.S. elections. Sequential runoff eliminates the least popular candidate, transfers the No. 2 votes from those eliminated ballots to their specified remaining candidates, then repeats the process until only one candidate is left. With sequential runoff, pepperoni won!
This was getting weird. It wasn’t clear at all how Math Club would order pizzas so that everyone would be happy. Plus, pineapple hadn’t even shown up yet. But Benito had a few more voting methods to try.
The Nesbitt Common Room of East Hall, named after former math professor Cecil J. Nesbitt, gives undergraduate math students the chance to sprawl in their own space during Math Club and between classes. Courtesy of LSA’s Department of Mathematics.
The Condorcet method, named after a French mathematician, works by staging two-way runoffs among every pair of candidates in the election. The winner is the candidate that outranks the others in each of the runoffs. Of all these counting methods, Condorcet best achieves consensus among voters—settling on ham, in Math Club’s case.
Benito introduced one more vote-counting method—the Borda Count, named after another French mathematician. U-M students use the Borda method to elect representatives in LSA’s Student Government and U-M’s Central Student Government. The Borda Count assigns a score to each of the ranks on each of the ballots—zero points for the topping ranked No. 5, one point for the No. 4 rank, and on up to four points for No. 1. All the scores on all the ballots were added together for each topping; the topping with the highest score won.
Because voters consider and rank all candidates, Borda “balances” voter preferences—the second- and third-ranked votes also have value, not just the No. 1 votes. But Borda can backfire. If voters try to game the system, middling candidates can accumulate points and win unexpectedly. Pineapple won the Math Club edition of the Borda Count.
By assessing the results of these five different elections, Math Club discovered that different vote-counting methods can produce wildly different results. Cheese won using plurality; mushroom won with runoff; pepperoni won the sequential runoff; ham won with Condorcet; and pineapple won the Borda Count. So, what was the answer to Benito’s original question, “How (un)fair are elections?”
Philosophically, it’s hard to say. Benito insists that before we find an answer, we first need to define “fairness.” Mathematically, it’s a little easier. Arrow’s Impossibility Theorem proves that it’s impossible for any single voting method to consistently satisfy everyone’s conception of a “fair” election, especially when more than two candidates are on the ballot.
A Vote for the Environment
Then came the big reveal. The Math Club pizza-topping election was a ruse. The whole point was not to choose pizza toppings. Rather, Koch actually wanted to improve environmental sustainability in the math department. And she’d enlisted Benito to help her.
As a relatively new professor, Koch had noticed that the math department—an otherwise refreshingly forward-thinking place—used Styrofoam cups for coffee and events. “I was just shocked that we weren’t at least using paper cups in the common room, so I wanted to do something about it,” she says, “and somehow involve the undergraduates and make it fun.” She thought that handmade, reusable ceramic mugs could replace the disposable Styrofoam cups, offering a much more sustainable solution.
Koch and Benito had invited students to submit design ideas for their personalized ceramic mugs. They planned for the students to elect their favorite design during Math Club.
“Now that we know no election method is ‘fair,’” Benito continued after running through the pizza vote results, “which voting method should we use to decide the winner of our Michigan math mug contest?” Because cries of “Borda!” rang more loudly than any other voting method, Benito used the Borda Count to tally the ballots on which the students had ranked the mug designs. Undergraduate Jinesh Shah’s design won. The Math Club had cast its vote in favor of the environment.
Koch and Benito recruited a local artist to create custom mugs at the non-profit Potters Guild in Ann Arbor. Read out loud as mathematical notation, the mug design says, “U of M mathematics.” Names on the opposite side personalize each mug. Photo by Angélica Benito.
Now that students and faculty have their reusable mugs, Styrofoam use has declined significantly around the math department. “It’s gratifying for us to go down to tea time,” Koch says, “and see people with their mugs. There’s solidarity here—cohesion with the people in the math department. They like their department, so we sold 125 mugs, which was awesome! And we’ve gotten requests for more.”
And as we head into another round of midterm elections, understanding that the very process of voting affects the outcome—especially when the outcome of the election can be as important as protecting the environment or our civil rights—definitely serves as food for thought.
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