Mathematics research builds better ways to identify gerrymandering

By Meghan Chua


Gerrymandering, the practice of dividing the legislative districts in a state to give one political party a greater chance of winning seats, traces back to at least 1812 when Massachusetts governor Elbridge Gerry approved a district that looked like a salamander.

For a long time, strange district shapes like that were considered the hallmark of gerrymandering. But shape isn’t always an indicator of whether a district’s borders have been deliberately tampered with to achieve different results.

“These eyeball tests for whether something looks good or bad don’t work,” said PhD student Elle Najt. “What people have been trying to do is to establish a different kind of comparison.”

To that end, mathematicians analyze how to use algorithms to tell whether a political district is gerrymandered.

Najt studies algorithms that can build a large group of maps for a state, predict their outcomes, then compare those to the map that legislators selected. She works with UW–Madison mathematics professor Jordan Ellenberg as well as associate professor Justin Solomon and postdoctoral associate Daryl DeFord at MIT.

“What if we had a big bag of all the different maps that you could have?” Najt explained. “If we reached our hand in and took out maps randomly and then compared them to the one that you proposed, maybe your map would look extremely unusual in terms of its partisan advantage.”

Using a bag of maps to compare districts is better than an eyeball test, but even a computer-generated group of maps can be biased. Najt analyzes the processes that algorithms use to construct the bag of maps, called an ensemble, and what factors influence which maps end up in the ensemble. There are many methods people can use to build an ensemble.

Najt has looked closely at two methods that, when applied in different scenarios, may lead to different outcomes. One is based on a method for moving incrementally through the space of maps by randomly reassigning small census units to different districts, an example of a more general algorithmic technique called a Markov chain. The other method is based on a related combinatorial structure called a spanning tree.

Someone trying to gerrymander a district could tweak the algorithm, or the baseline information they feed the algorithm, to make it look like most of the maps in the ensemble are just like their gerrymandered map.

This is a concept Najt, Solomon, and De Ford call metamandering.

“The idea of metamandering is that you change something about the algorithm or the input to the algorithm so that the output maps become concentrated around a different portion of the space of maps, and in a way that preferentially changes the outcome of these ensemble methods,” she said.

The central question is how much flexibility a user has over the process of selecting a representative group of maps. Najt also works with undergraduates in the Department of Mathematics’ Directed Reading Program Matt Karman, Ian Gordon Mark, and Seanna Zhang (also in UW–Madison’s Undergraduate Research Scholars Program) to explore the question and identify specific reasons you might reject an ensemble as a bad sample.

Additionally, Najt is working with Annie Yun in MIT’s advanced undergraduate research opportunities program. Along with Solomon, Najt and Yun are developing sampling tools based on Schramm-Loewner evolution and conformal mappings of surface meshes. Najt said this approach avoids some metamandering tactics but also introduces different biases that the researchers do not yet understand well.

Ensembles are just one of the tools mathematicians and computer scientists are exploring to help determine whether political districts are gerrymandered. As researchers look further into other methods, Najt said they may be able to preemptively understand new techniques that politicians might use to gerrymander and use that knowledge to prevent it.

“I’m hoping people will eventually understand the signatures of when you’re deliberately manipulating sampling algorithms,” Najt said. “I’m optimistic about there being incremental ways to improve the performance of sampling methods for preventing really extreme excesses of gerrymandering.”