# Math and the Vote

A Geometer Examines Elections

Speaking at the MAA Carriage House on October 24, Moon Duchin (Tufts University) brought a mathematician's precision of thought to discussion of our often messy and contentious electoral system. In her Distinguished Lecture "Math and the Vote", Duchin tempered news of democracy's undeniable shortcomings with hopeful indications of how mathematical interventions might improve current methods of collective choice.

What is an election, Duchin asked, but an attempt to "take the preferences of a whole society's worth of people and aggregate them into an outcome"? And doesn't that sound like a math problem?

Perhaps, but it's a thorny one, as the Marquis de Condorcet recognized in the 18th century. For consider the preference schedule below. (In a preference schedule, the number atop each column specifies how many voters have the preferences in that column.) In the aggregate, these 150 voters prefer A to B, B to C, and C to A. Condorcet observed that, when collective preferences are cyclic like this (as they were in Burlington, Vermont's 2009 mayoral race), no voting system can make a principled choice of a winner.

But what do we mean by "system" exactly? Duchin defined a voting system as a "deterministic algorithm that takes a preference schedule and produces a winner set" and briefly described a selection of the infinitely many possibilities. Plurality, favored in the United States, declares the candidate with the most first place votes the winner. Most valuable players in baseball - and minority representatives in the Slovenian assembly - are selected via Borda count, a method in which more points are awarded for receiving higher rankings. There's elimination and pairwise comparison and beatpath and the Smith method. Smithifield Borda, even.

"And then there's always" - Duchin clicked "Dictatorship" onto her slide and the room erupted in nervous laughter - "in some ways the cleanest system of all, where only one voter counts."

Duchin also surveyed the countless criteria we might expect a voting system to satisfy. Maybe every voter should count the same, or voters should gain no advantage by voting dishonesty. Do you agree that "if a certain candidate is ranked number one by literally everyone then they should win"? If yes, you're in favor of a Pareto efficient system. Do you think that a candidate who wins every head-to-head should win overall? That's called Condorcet-fair. Now suppose you determine the outcome of an election according to your system and then change voter preferences such that each change is either neutral or favorable to Candidate X. If it's impossible for such a change to cost Candidate X the election, then your system is strongly monotonic.

"Does that seem like a reasonable thing to ask from a voting system?" Duchin asked. "Does it seem, maybe, essential?"

Well, too bad. A family of so-called impossibility theorems has established the disheartening fact that no conceivable voting system can meet even a mild list of criteria. The Muller-Satterthwaite theorem of 1977, for instance, states that, given more than two candidates, the only Pareto-efficient, strongly monotonic, single-winner system is...dictatorship. Yikes!

Duchin recommends reacting to this revelation flexibly and creatively - like a good mathematical modeler. "Whenever a model produces results that surprise or alarm," she told her Carriage House audience, we should examine the assumptions built into the model and re-evaluate their fit. (Does linearly ranking the candidates authoritatively reflect a voter's preferences in a stable way? Try ranking all of this year's primary candidates in a single list to see how difficult it is to capture your views.) Also, there is at least one voting system that satisfies enough criteria for a room full of mathematicians to feel okay employing it: When the Tufts University Math Department votes on a hire, they use beatpath.

Duchin devoted the rest of her talk to discussion of apportionment, the division of the electorate into districts that elect representatives. There too, she argued, mathematicians have insights to offer. The Constitution gives surprisingly little guidance on how to divide up states for voting purposes, and the traditional districting principles recognized by the courts can be hard to pin down. We could, however, consult a geometer about, say, how to assess the compactness of a voting district.

"Sometimes the power to draw boundaries is the power to determine outcomes," Duchin noted, also observing that districting has implications for fair representation for minority groups. Too often officials reverse-engineer or gerrymander districts to produce the election outcomes they desire, and we get "ghastly" districts like IL-4 - nicknamed the "earmuffs" and NC-12.

"If we're the enemy of tentacles, if we don't want things that are snaky and spread out," Duchin said, we can look for shapes that enclose a lot of area relative to their perimeters. We might award each district an "isoperimetric score" equal to the area of the district divided by the area of a circle of the same perimeter.

Though quick to acknowledge limitations of the proposed compactness score, Duchin stressed that now is no time for mathematicians to allow real-world messiness to discourage them from engaging with issues of electoral equity. She noted that "there's been an avalanche of changes to voting laws around the country," and that "we have to be very vigilant of the impact of those new regulations are going to have."

And then there's the 2020 census, the results of which will inform the next round of redistricting.

"I would like to think that geometers like me, that math modelers, that algorithmic auditors and a number of different people could get together before 2020 and have some really robust recommendations to make at that time," Duchin said. "That's a goal of mine."

Duchin's lecture, part of MAA's NSA-funded Distinguished Lecture series, was live-streamed on Facebook. Visit the Videos Section on MAA's Facebook page to watch the talk.

Katharine Merow is a freelance writer living in Washington, D.C.