Moon Duchin, who leads the University of Chicago’s Data and Democracy Lab, explores how math can help solve gerrymandering and improve electoral systems.Credit...Meron Tekie Menghistab for The New York Times

A Conversation With

Moon Duchin on the ‘Mathematical Quagmire’ of Gerrymandering

Why the challenge of truly representative democracy is so complex.

• Nov. 3, 2025

Two decades ago, the mathematician Moon Duchin spent her summers teaching geometry at Mathcamp, a program for mathematically talented teens. Campers contemplated notions such as how to prove whether a given shape could be cut up into triangles. Another favorite was a famous “decomposability” paradox.

“You can take a solid ball, cut it into five pieces that don’t overlap, move those pieces around and reconstruct two solid balls that are the same size as the one you started with,” Dr. Duchin, a professor of computer and data science at the University of Chicago, said in a recent interview. “That’s really weird, right?”

These days, Dr. Duchin leads the university’s Data and Democracy Research Initiative and spends her summers in a similar yet different way. She runs internships centering around how math helps solve gerrymandering and designs alternative electoral systems.

On Oct. 20, Dr. Duchin delivered the annual Blackwell Seminar at the University of Washington, which honors one of her heroes, the mathematician and statistician David Blackwell. The topic: “How to Measure Political Representation.”

“Today I would say the whole world should be paying particular attention to this class of problem, which I’ll call the problem of democracy,” she noted in her preamble.

Dr. Duchin defines partisan gerrymandering as follows: The party in control draws district lines to get more representation and advance an agenda. It is sometimes described as politicians choosing voters, rather than voters choosing politicians; Dr. Duchin agreed with that characterization.

This summer she was preoccupied — overwhelmed, in fact — with what she called “the nuclear redistricting wars.” She serves as an expert in the Texas redistricting lawsuit, which challenges a congressional map aggressively redrawn in favor of Republicans. Since 2018, she has provided analysis in lawsuits over gerrymandered maps drawn by both parties; she was also an author of a Supreme Court amicus brief. Typically, she works for legal teams representing clients such as the N.A.A.C.P. and the League of Conservation Voters.

Map drawing and redistricting might seem like a basic geometry problem about avoiding weird shapes, but, as Dr. Duchin discovered, it is “sneaky deep.” “It turns out that just having nice shapes isn’t going to solve our problems,” she said.

She frames redistricting as a partition problem: Divide an object into pieces according to given rules. In this context, the object is a state and its voters; the pieces are the allotted number of districts. Although the rules vary by state, Dr. Duchin refers to “the big six”: population balance; compactness; contiguity; respect for existing civic boundaries (counties, cities, towns); respect for communities of interest and racial fairness (per the Voting Rights Act and the Constitution). Racial fairness is the crux of the Texas case, and a Louisiana case currently before the Supreme Court; it is an element of the complaint in the recently filed New York case, and many others.

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An illustration from a 2021 research paper by Dr. Duchin and collaborators. Given an initial partition, at left, the figure shows changes after one million flip moves, center, or just 100 more efficient “recombination” moves, right.Credit...DeFord et al, Harvard Data Science Review, 2021

As a math problem, assembling so many tiny pieces into large districts while balancing multiple parameters quickly explodes in complexity. Even for computers, it is intractably hard to enumerate all the possible maps.

In between deadlines for legal reports, Dr. Duchin discussed her research and what she sees as the gerrymandering endgame. The following conversation, conducted by videoconference and email, has been condensed and edited for clarity.

More than a half-century ago, the Supreme Court Justice Felix Frankfurter called gerrymandering a “political thicket” and “mathematical quagmire.” Why is it so tricky?

The big challenge, as courts have complained for decades, is that they lack a benchmark for what you’d expect to see in a map when nobody is gerrymandering. We need a canonical way to figure out, in a complicated situation, what we should expect — a notion of expected properties of districting plans. Scientists call this a “null model.”

Math can help with that.

In math, expectation is constructed through randomness. Given all the things that could happen, expectation is calculated by taking a weighted average based on how likely all those things are.

For example, if you want to know how many Democratic seats a typical nonpartisan Texas congressional plan might have, a mathematical sampling method called a Markov chain, a so-called random walk, shows you what is typical under the given rules. But this mathematical framework doesn’t get you all the way to fairness. Finding fairness requires debating the goals of representation.

How does it work?

We use carefully designed random walks to explore the vast universe of all possible redistricting plans, and to generate a large sampling, or ensemble, of maps that meet specified criteria. An ensemble provides a benchmark against which proposed maps can be measured.

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Dr. Duchin, presenting “How to Measure Political Representation” at the annual Blackwell Seminar at the University of Washington, discusses data about voting blocs from a recent Portland election.Credit...Meron Tekie Menghistab for The New York Times

You’ve spent a lot of time thinking about how to build random samples of reasonable districting plans. What’s your approach?

My collaborators and I started thinking about alternative Markov chain methods. The spark came from a particularly brilliant student, Elle Najt. She observed that a graph theory construct called a spanning tree — sort of like a skeleton of a graph — gives a much more efficient way of mutating the districting plans than trying to change the picture by flipping one pixel at a time. Over the last few years, we’ve developed this into a class of redistricting algorithms with promising properties. We call it “recombination,” or the ReCom algorithm.

How does that work?

There’s a powerful analogy to card shuffling. The traditional randomization method applied to redistricting was like repeatedly taking a single card off the top of the deck and putting it into the middle at a random place. Eventually, that process will shuffle your deck, but it will be really slow.

Instead, with spanning-tree methods, we do the equivalent of a riffle shuffle: You divide the deck in two parts and then interleave the two sides randomly. So rather than a local move where you change one thing, say the district assignment of one census block, you do a global move where you make a big change, changing a lot of block assignments, at every step. Specifically, we fuse two adjacent districts, draw a random spanning tree, and split that to create two whole new districts with nearly the same population.

Because it’s not only fast and efficient, but also reaches a steady state that is nicely suited to redistricting, recombination has made a big impact in the field.

Why haven’t courts adopted it?

Well, you’d be surprised: All nine Supreme Court justices have signed on to one opinion or another calling algorithmic sampling useful in their voting rights cases. Most of the conservatives called recombination ensembles “surely probative” (i.e., giving relevant evidence) in the recent race-based case in Alabama. The liberals embraced the ensemble method for partisan gerrymandering in North Carolina. It’s probably not too cynical to say it depends on what conclusion they want to reach.

But they have lots of quibbles, such as worrying that computers don’t take all human factors into account. I think this is partly based on a mistaken assumption that the algorithms are trying to simulate human judgment in mapmaking — they’re not. They’re just trying to show you a random sample of alternatives under the rules.

Video

Recombination algorithms generate a sample of districting plans for the state of Texas.CreditCredit...Moon Duchin

With the redistricting wars, what’s the endgame?

One possibility is that in a year or two pretty much every state will be a one-party state. You’ll have your red states and blue states with fully red and blue delegations. That’s probably not too good for American democracy. In October, during the Texas trial, the state put on the mapmaker who said “it’s a purely partisan draw,” with the expectation that today’s courts will license that.

Where do you go from here?

For now, the courts are not the place to work out big ideas about representation and fairness. I’ve decided to focus on doing “democracy science” with a timeline of, say, 20 years. Because pendulums do swing, we hope, and I want the best possible science in place for when the courts are receptive again.

I’m doing a lot of work on alternative systems of election that can robustly give you proportional outcomes without having to do careful line drawings. I think taking a hard look at the design of electoral systems could get us out of this nuclear moment.

What are the options?

One system I’m studying is proportional ranked choice voting, also known as the single transferable vote method, or S.T.V. In this scenario, you have multi-member districts using ranked choice.

For example, rather than one representative per each of nine districts in Massachusetts, reduce the number of districts to three, and have each elect three representatives by a ranking method. That way Massachusetts would probably get two or three, rather than zero, Republicans in Washington.

The biggest American city that uses S.T.V. is Portland, Ore., and it was a great success there in last November’s City Council election. All kinds of groups were able to elect their preferred candidates in proportion to their voting numbers.

But this system is only one possibility, and there are many other ways to ask people about their preferences and then aggregate them into good representation. I think of nurturing alternative systems on the local level as a way to leverage the laboratories of democracy and figure out what works.

Where else might math take us?

A bunch of math and computer science people have taken the notion of alternative systems and run with it far beyond what would be practical, just as an engine of ideas. In the field of computational social choice theory, one simple idea that’s very useful to “think with” is called random dictatorship.

Do tell.

Suppose you have a lot of people casting their votes, expressing their preferences. Then, after the votes have been cast, you pick one random person to be the temporary dictator, and adopt all their preferences and ignore everybody else. This system has a couple of nice properties. One feature is that rationally you should always vote honestly because sometimes you’re going to be the dictator, so you better say what you really want. And the rest of the time, you don’t matter at all, so you have no incentive to vote dishonestly or strategically.

If you run a random dictatorship over and over and over again, then on average it’s going to get you something with great expected properties. Any particular time that you run it, one person has all the say, but if you play it over and over, then again randomization is your friend. A little bit of randomization can give systems more robust fairness properties.

Nobody’s proposing we actually vote by random dictatorship in real life. But it’s an example of mathematicians innovating. Some of those creative ideas are going to produce reasonable voting systems that get us closer to mutually agreed standards of fairness.

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