Moon Duchin

“It just so happens, by what I feel is a truly lucky coincidence, that the kind of math that I do is ideally suited” to addressing gerrymandering, Duchin says. (Photography by Alonso Nichols/Tufts University)

Tackling gerrymandering with geometry

Moon Duchin, PhD’05, thinks she and her fellow mathematicians can help draw better electoral maps.

Gerrymandering—the hijacking of the redistricting process by redrawing a voting district’s lines to favor a specific person, group, or agenda—takes its name from Elbridge Gerry. As governor of Massachusetts in 1812, Gerry approved a party-serving district plan that critics mocked as looking like a salamander.

The opportunity for gerrymandering arises every 10 years, when district lines may need to be redrawn to reflect population shifts noted in each census. The next US Census takes place in 2020, so in 2021, we could see a fresh crop of districts resembling salamanders, lobsters, or, as one federal judge described a Maryland district, “a broken-winged pterodactyl, lying prostrate across the center of the state.” 

Although many people agree that partisan gerrymandering is, as Justice Samuel Alito has described it, “distasteful,” it’s also hard to prevent. Federal rules governing redistricting are minimal, and many state-level regulations are murky. For instance, 37 states require that districts be “compact,” but only a few bother to specify what that means. 

Tufts University professor Moon Duchin, PhD’05, thinks she can help—both in providing a better definition of compactness, and in bringing her fellow mathematicians into the legal conversation about redistricting. Duchin founded the Metric Geometry and Gerrymandering Group at Tufts. The Boston-based team is studying how geometry (Duchin’s specialty) and computing can be applied to redistricting, and training scholars in quantitative fields to serve as expert witnesses in court cases about gerrymandering. 

“Mathematicians have definitely thought about gerrymandering before,” says Duchin. “What’s new here is getting geometers together with computer scientists, and working with lawyers to understand how the math fits with practical measures we can take.” 

Gerrymandering has reemerged as a subject of public discussion and judicial scrutiny, thanks to a case currently before the US Supreme Court. Gill v. Whitford argues Wisconsin’s legislative districts favor Republicans so much that voters’ rights are violated. 

It’s one of several redistricting cases to appear before the court since the 1960s. In that time, the justices have intervened against racial gerrymandering but hesitated to take a decisive stance on partisan gerrymandering—in part because, as Justice Anthony Kennedy wrote in 2004, “there are yet no agreed upon substantive principles of fairness in districting” and no “clear, manageable, and politically neutral standards” with which to evaluate districts.

Duchin and her collaborators are trying to meet Kennedy’s challenge by proposing redistricting standards that are fair and rigorous enough for judges to adopt.

Their first line of attack is to develop a more sophisticated measure of compactness, because judges have made clear they care about it—even if they don’t know how to measure it. Existing definitions, rooted in 19th- century mathematics, have typically focused on how far the shape of a district is spread out from its center, or how relatively smooth or jagged its boundaries are. For example, a circle is more compact than a star. The more tortured a district’s shape, the theory goes, the more likely it is to have been gerrymandered. 

But there may be good reasons why a district looks strange. For instance, if a natural feature such as a mountain range cuts across a district, its edges could appear contorted. In some states, lawmakers are required to keep groups with shared interests together—a vague principle which has been applied to communities with shared socioeconomic status, education level, health concerns, religion, or ethnicity. So rather than just looking at a district’s shape to determine whether it has been gerrymandered, Duchin wants to focus on “where the people are in the shape and how the shape slices the population,” she explains. 

That idea is one basis of the refined definition of compactness she hopes to devise using contemporary geometrical techniques. By drawing a network of lines based on population density within a district and then digitally “folding” along those lines like origami, mathematicians can identify a district’s curvature—essentially, how “bendy” it is. And how districts look and bend when they are turned into origami, Duchin thinks, is a much better and more mathematically sound way to evaluate compactness.

Compactness isn’t the only way to evaluate whether a district is well or badly drawn, fair or unfair. Nicholas Stephanopoulos, assistant professor of law at UChicago, has proposed another measure called the “efficiency gap,” which uses the number of votes each political party “wastes” in an election to determine whether one has a systematic advantage over the other. A vote is considered “wasted” if it doesn’t contribute to a victory—such as a vote for a losing candidate or an unneeded, “extra” vote for a winning candidate.  

Duchin’s group is partnering with computer scientists who are developing programs that can factor in any combination of these parameters—compactness, shared interest groups, the efficiency gap, and others—and use them to produce a variety of reasonable redistricting plans. If the maps actually drawn by legislatures don’t resemble any of the computer-generated options, that’s a red flag that gerrymandering might be at work. The approach is relatively new, but Duchin believes it holds promise, if its theoretical underpinnings can be explained well in court—hence her group’s focus on expert witness training. 

Duchin got interested in gerrymandering while teaching a 2016 course on the mathematics of social choice, which examined the math underlying complex and shared decision making, such as voting. It took her just over a year to design a workshop she led this August, which brought together some 500 experts in various quantitative fields to think about the real-world subtleties of redistricting—including Shmuel Weinberger, the Andrew MacLeish Distinguished Service Professor of Mathematics at UChicago. 

Weinberger sees gerrymandering as “a mixture of a philosophical and political problem with mathematical overtones. The challenge is to figure out which kinds of maneuverings are genuinely illegitimate, and then to develop mathematical methods to [prove] it.”

In search of those quantitative methods, Duchin’s gerrymandering group is hosting workshops this fall and winter in four redistricting hot spots: Wisconsin, North Carolina, Texas, and California. And they’re gearing up for the redistricting to come in 2021. 

“I think we’re producing good map drawers—from teachers to coders, scholars, and professionals—but if that fails for reasons of political gridlock and the like, my hope is we’re producing good testifying witnesses for the court cases that will follow,” says Duchin. “It would be better to be on the front end of drawing good maps, but if we have to, we’ll be on the back end keeping the bad maps in check.”

This story was originally published in the Fall/17 print edition as “Line Items.”

Updated 10.25.2017 to indicate that a few districts specify what compactness means and to provide a more current explanation of shared interests.