The Joy of x

Moon Duchin on Fair Voting and Random Walks

Can geometry save democracy from gerrymandering? Mathematician Moon Duchin discusses the possibilities with host Steven Strogatz.

Moon Duchin, a professor of mathematics at Tufts University, uses metric geometry to help defend democracy against the threat posed by gerrymandering. But as she discusses with host Steven Strogatz, the problem of fair voting in a representative democracy can’t simply be reduced to an objective function. This episode was produced by Dana Bialek. Read more at Production and original music by Story Mechanics.

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Moon Duchin: You know there’s this great coffee table book of mathematicians’ portraits?

Steven Strogatz: Yes.

Duchin: Mathematical People, I think it’s called.

Strogatz: Yeah.

Duchin: So I just love those because they’re so alive, and they’re so diverse, and not just in kind of standard off-the-shelf diversity ways, but all kinds of diverse, human diversity.


And I reached out to the photographer, at one point, and asked if she’d consider making a series available for departments to buy, and she said no.

Steven Strogatz [narration]: From Quanta Magazine, this is “The Joy of x.” I’m Steve Strogatz. In this episode, Moon Duchin.

Duchin: So I have been talking to a couple of photographers about a portraiture project of just living math because it could do a few things at once. It could give people a sense that math has all this human diversity.

But it could do a second thing, at the same time, which is sort of dramatize or illustrate the fact that we know, which is that math is so alive and in progress, not settled at the time of Gauss. So I’m pretty excited about that as one of a trillion medium-term projects.

Strogatz: So let me tell you about Moon Duchin. She’s a mathematician who started her career working in a peculiar kind of geometry that concerns itself with shapes that live in more than three dimensions. They’re gnarly pretzel shapes. But somehow, she has found a way from that exotic kind of geometry into a very real-world concern with gerrymandering, with the practice in politics nowadays of twisting and manipulating congressional district boundaries to give one side an unfair partisan advantage.

She’s brought all kinds of powerful math, from the geometry she knew before to new tools in statistics and probability theory, to work on these interesting political questions. Her fame precedes her. I mean she really got a lot of notice for the work that she’s been doing on gerrymandering and math, so I was very eager to meet her and have a chance to talk to her.

We’ve had a couple of breakfasts together where she pulled an interesting jiujitsu move on me and made me feel like she wants to learn from me. And I found in those conversations that we have all kinds of overlapping interests in the history of math, in aspects of gender representation in our culture of math.

Strogatz: I think it’s something that’s pressing, that we need this all across the world, I would say, for math education, to show math as it really is.

Duchin: I think a lot about how to broaden participation in math. I think a lot about the pipelines that come into it, and the way that when you do graduate admissions, as I’m sure you know, it always seems like you’re … kind of arriving too late, somehow. That a lot of people have been diverted from the path before you get their files in your hands when you’re doing graduate admissions. And then if you go back to college, people feel the same way. And then for, boy, over a decade, I’ve been involved in an excellent summer program, Canada/USA Mathcamp, this really cool summer program for over 100 high school kids who are just intellectually alive to all things math-y. And then I know from doing the admissions there that now you’re admitting high school kids and it also feels too late.

So every time I get involved in a project that has an admissions component, it leads to thinking about what used to be called “the pipeline,” but now there are far better metaphors for this, but just about the problem of attrition, of identifying talent in ways that aren’t circular and elitist.

It’s always on my mind, these questions of setting up the culture in a way that’s inviting and taking outreach really seriously.

Strogatz: It seems that it’s been a longstanding interest of yours. I noticed that in college, you did a double major, math and women’s studies.

Duchin: Yeah, that’s right, the quaintly named “women’s studies” because it was the ’90s and we still had something called that. These days, it would be gender studies or something. I was really interested in questions of expertise, and community, and who gets to have access to truth, who gets to proclaim, who gets to certify. And so women’s studies was a place where I could think about those things, and so yeah. I did math and women’s studies, and ended up thinking it would be hard to unite those interest into one thesis, so I wrote two: one in spectral graph theory and one in feminist epistemology. And I leave you to guess which was for which major.

Strogatz: Okay, yeah, got that.

Duchin: But yeah, so the feminist epistemology question is just —. Epistemology is the philosophy of knowledge, how do we know, what do we know. And feminist epistemology grounds that in the social and thinks about who’s got the authority to make knowledge. And feminist epistemology of science has a literature that’s quite engaging, but I didn’t find that it was looking at math in a way that made a lot of sense to me as a budding mathematician. So I wanted to think about knowers and knowledge in math and what it takes to make a new thing in math.

So much later, I wrote my dissertation and I graduated, and then I discovered a mistake in my dissertation. I discovered a broken lemma. And the lack of surprise on my advisor’s part was alarming.

Strogatz: Oh.

Duchin: I was a postdoc at MSRI at the time, and I was really freaked out by this mistake, and I talked to lots of people, and I discovered that everybody I talked to had a mistake in their dissertation.

Strogatz: I see. That’s true.

Duchin: And since then, I’ve thought a lot about mistakes in math and kind of how they’re found, and how they’re fixed, and what they mean, and how deep they go. And some of the controversies about credit also has to do with what it means to be fully right in math. I think these questions are actually a lot harder than we admit to ourselves professionally, that things that are mostly true still are the foundation of huge important fields.

Strogatz: Yep, they absolutely are. Wow. I mean I’d like to hear more about your reaction to seeing your own mistake, how you found your mistake, and this question of being mostly right, and how we often shortchange that, as opposed to the person who dots the last i.

Duchin: Yeah, no, that’s right. I can’t quite remember how I figured out that the lemma was broken. It was probably when I was trying to extend it.

Strogatz: So you discovered it yourself. It wasn’t someone else pointed it out to you.

Duchin: I found it myself. I’m not sure anyone else has really read my thesis. But then my advisor’s reaction was, “Yeah, okay. I’m sure you can fix it because I’m really confident that the main theorem is right.” And that was fascinating to me because how can you be sure that the main theorem is right in a way that doesn’t rely on all the lemmas that you use to prove it?

Strogatz: So you keep using this word “lemma.” In case our audience is unfamiliar with that little bit of math jargon, what do you mean?

Duchin: Yeah, so in math, the main currency, the unit of truth, is a theorem, where that’s a statement of something, probably something kind of important, or at least you think so, and it comes with a proof. So the theorem is the main object, but then there are two smaller results that have their own names, lemmas and corollaries.

And so lemmas are smaller results, usually, that you use to prove a theorem, and corollaries are smaller results that follow or are entailed by the theorem.

Strogatz: Nice.

Duchin: Yeah, so often the sequence would be you establish a bunch of different lemmas. Those are the building blocks. You assemble those together to prove a major result or a theorem. And then you kind of mine that theorem for applications or corollaries.

Strogatz: Okay, cool, right. So you had this vision of what the big theorem was going to be, and along the way, you’re building up to it by assembling the building blocks into this edifice.

Duchin: And I would say that’s sort of typical. That’s how a theoretical math paper usually works, right? Wouldn’t you agree?

Strogatz: Yep. So you say there was one of the building stones that turned out to be a little bit wobbly, or broken, or something.

Duchin: Yeah, or wrong or something.

Strogatz: Wrong, yeah. Just dropped the metaphor for a second, yeah.

Duchin: No, but if you take really literally the standard math account of how we do what we do and how we know what’s right, if one of the building blocks or ingredients, if one of the results that’s used in the course of establishing the main one is compromised, then you have absolutely no reason to believe the main one.

Strogatz: On the standard account, yeah.

Duchin: On the standard account. But when you scratch just underneath the surface of that and try to understand what my advisor meant by, “I’m pretty sure your theorem’s right,” what you realize is that doing math is about building up lots of intuition in a very abstract area. So there are definitions. There are ways of thinking about them. There are standard facts about how they fit together. And for most high-level mathematicians, you have a feeling for the object that you’re working with. You also have a whole kit bag of examples to go to.

So when someone makes a claim to you, you think about whether it fits with your view of the world, and then you try all your examples and see if it’s upheld.

And that gives you strengthened or weakened belief, or maybe that gives you counterexamples to the claim. So I think people who have a lot of experience thinking about certain kinds of theoretical objects have ways of testing the likely truth against, “Is it predictive of these other facts that I know? Does it comport with these examples?” So there’s a whole testing ground that leads you to have raised or lowered confidence that something is right.

And my advisor was right about my thesis. The main theorem was fixable, but I didn’t know that at the time. For me, that was an important part of learning how math is actually done a little bit different from the standard narrative.

Strogatz: It’s very interesting. I think people who are following this discussion and have heard about math as this chain of logical reasoning building from the axioms — and the definitions up through the lemmas, the way it’s supposed to be in Euclid’s book, in The Elements. And it’s a very simplistic narrative because math is actually done by people trying to convince other people, and so it’s much more social than you might be led —. It’s not done by machines. I mean it is. It’s now starting to be done by machines.

Duchin: Also.

Strogatz: Right? So that’s a different game with them.

Duchin: I think another thing that’s really interesting is what happens when something is found to be wrong. So in theory, if you have a published paper and then it turns out to have a hole in it, in theory, you’d retract it and you’d try to fix it. And actually, if you look at math journals, there are just not that many errata and retractions.

Strogatz: True.

Duchin: And there’s actually some famous cases, or maybe cases that should be even more famous than they are. Here’s one that’s fun. The Busemann-Petty problem, which is a really cool question in convex geometry.

As you know, Steve, probably the most prestigious journal in math is the Annals of Math, and I think in the early-’90s, there’s a paper in the Annals where a mathematician proves the Busemann-Petty problem — has the answer “yes” in the last case that was open. And then behind the scenes, a mistake was found, and he spent some years trying to fix it up, but rather than retract the first paper, the most extraordinary thing happens, which is that several years later, a second paper appears in the Annals with the opposite theorem by the same author.

Strogatz: That covers the bases.

Duchin: Well, yes, but it sort of serves as the retraction to the first. I’m really interested in this question of how do we decide what’s right, what’s important, and what do we do when we’re wrong, and what are all the different flavors of wrong and all the different flavors of right. The lesson I draw isn’t, “Abandon ship. Math is unsound.”

It’s just that math is more like all the other human endeavors than the hype would have had it. But I actually take that to be a reason to do it, right?

So coming back to who chooses math, I think the idea that it’s timeless perfect proofs from the book of God, it draws one type of personality, but it turns away a whole lot of others. And instead, the idea that you gradually, through this mix of intuition and the activity we call proof, you gradually put together a better and better picture of an abstract world where you can find things you didn’t know to look for, to someone like me, that’s just way more appealing of a professional landscape to work in.


So I had the following kind of trajectory through research math.

As a graduate student, I was focused in the area that’s called Teichmüller theory, which is about the geometry of surfaces. So the kind of math that I started out doing as a graduate student was very much about taking a topological surface and thinking about all the different ways you can make it rigid, all the different rigid geometries that it admits. From there, it’s really natural to start thinking about groups and the geometries that they act on.

So ever since Poincaré, circa 1900, we take a group, an algebraic object associated to a surface, the fundamental group, which is all the loops that you can draw on the surface. And starting with Poincaré and Klein, there’s a whole kind of tradition of thought about the interplay between groups in geometry.

So as a postdoc, I kind of branched out in geometric topology and geometric group theory, and here’s an example.

So this is what my dissertation was about: What happens when you put those together? Now you could take a random walk on all the geometries on a surface. So here’s what I want you to picture. Imagine that you have a Genus 3 surface.

Strogatz: Better tell us what that is.

Duchin: I think I will. So what’s a Genus 3 surface? All right, well, a sphere is Genus 0, and then an inner tube, a torus, it’s like a hollow donut, that’s Genus 1, because it’s got one handle or one hole through it. And so Genus 2 would be just imagine surgery that attaches two inner tubes together so now you’ve got something that’s sort of like a figure eight that’s got two holes in it but it’s still a surface. And then you can keep going like that. You can just add more handles or more holes, however you want to think about it, and so you have Genus 3, 4, and so on.

And actually, there’s another beautiful theorem, classification of surfaces, that tells us that that’s basically everything for closed orientable surfaces, whatever that means. You have the sphere and then you have all of these Genus 1, 2, 3 and so on, and that’s pretty much it. You’ve seen everything. Okay.

Strogatz: So as far as handle — ’cause the word “handle,” I know what you mean, but if someone doesn’t have the right picture in their head… I’m thinking almost of a woman’s purse, the handle that you hold to hold the purse.

Duchin: Yeah. I think another way to visualize it is the handle on a coffee cup.

Strogatz: Or the handle on a coffee cup, sure.

Duchin: All right, so now I wanted you to imagine this Genus 3 surface and then here’s what I want you to think about doing to it. I want you to take a random walk on its geometries in the following way. Grab a handle and twist it all the way around. That’s an operation called a Dehn twist, and you can do Dehn twists lots and lots of different ways. And so basically what you’re doing is you’re twisting up the geometry of the surface.

And what that does to the surface is it takes some simple curves and when you twist them around, they become more complicated. So it’s a way of making curves more complicated. And so my dissertation was about the question what happens if you take a surface and you just do this for a super long time? You randomly grab handles and twist and you do lots and lots of these twists. So that’s a random walk on the space of geometry.

Strogatz: Nice.

Duchin: It’s extremely abstract, and it turns out that for a Genus 3 surface, that’s a 12-dimensional space that you’re wandering around in. So this was the research question, was what happens if you randomly twist for a really long time? Is the long-term behavior of the surface, is that well approximated by something else that you could do to it? And it turns out that the answer is yes. The answer is, believe it or not, if I take the original surface and I put it in appropriate flat coordinates, I flatten it out as much as possible, then long term, all that twisting is just like stretching the flat object, which is really surprising when you put it that way. I just wanted to situate the kind of math that I was trained to do, which first of all, has a taste for the random, and secondly, likes to think about shapes and working in high dimensions when you can’t necessarily get your hands on or construct everything but still being able to understand the big picture, the asymptotic, the long-term structure of these kinds of processes.


I have a taste for the very theoretical when it comes to math, and so I hadn’t been thinking about applications, and I’d been thinking that my political interests, my activist interests, would just be occupying a different part of my life than my research. So then I came to Tufts on a tenure-track job and what happened was that I got a chance to teach a class about elections during the 2016 primaries when my brain was very much in election space. And the class was about the math of elections, but I wanted to kind of draw that out and both engage with what was actually happening in then current events, but also take the math that I do and collide it with the political content.

So I started thinking about the shapes of districts, and so we think of gerrymandering as the abuse of redistricting that happens when you draw crazy-looking districts that are rigged to make a particular outcome likely or certain. So I figured that there must be a way to constrain the shape sensibly. How hard could it be to write down what it means to be a good shape and what it means to be a bad shape when it comes to political districts?

Well, here I am three years later to say not only is it hard to write down exactly what you mean but it turns out to be a little bit beside the point. The worst abuses when it comes to rigging outcomes aren’t really constrained by any notion of good shape in the districts. So yeah, it’s hard and useless.

Strogatz: Can you amplify on that? If it’s not about the shapes, what is the thing that’s the worst rigging?

Duchin: Yeah. Now let’s put on our math hats for a minute and think about what we’re doing when we do redistricting. So redistricting is a partition problem that says —. For instance, I’m sitting here in Massachusetts, and Massachusetts has something like 2% of the country’s population, and so by constitutional design, that means we should get something like 2% of the House of Representatives.

And so, well, you do the arcane rounding that you have to do, and you discover that at the moment, we have nine seats out of 435 in the House. Now we need to cut Massachusetts into nine parts with equal population, so that’s a partition problem. That’s taking an object and breaking it into pieces subject to some rules, and the kind of highest priority, the best-known rules, are population balance, connectedness. You want each of those pieces to be contiguous or connected, and some vague idea that they should be nicely shaped, but I’m about to say that that doesn’t do the work that you want it to do.

There are other rules because it’s about people and it’s about the way they live in cities and other formations. So some of the other rules say you should try not to split up other units like counties and cities unless you have to. That’s actually a rule. And then where there are communities that have a shared interest, you should try to keep those together.

Strogatz: Are these norms, or are these laws, or what?

Duchin: Well, interesting. We’re about to get to the only one that’s an actual law, a bill passed by Congress. So that’s the Voting Rights Act of 1965. So passed by Congress in ’65, but clarified, extended, strengthened over the years, until very recently, when it started to be weakened. So the Voting Rights Act is what tells us that we have to work to get minority opportunity.

Now let’s go back to being mathematicians. For a moment, we were being practical. Let’s go back to being mathematicians for a second and think about why would you need a special rule for minorities? Well, here’s why. Districts just on their own are really bad at representing minorities. Let’s think about that just from first principles. So suppose you have a state like Massachusetts, and we want there to be nine districts.

If we didn’t have any kind of geographical correlation, if we just assigned everyone a number randomly, uniformly randomly from 1 through 9 and that was their district, what would happen? If some candidate got 48% of the vote and some other candidate got 52, we would expect the 48% candidate or party to win no districts at all, and that’s just because we’ve got these great, classic probability theorems, law of large numbers, central limit theorem, that tell us that if we’re picking uniformly at random, then that 52-48 balance that we have statewide would be very nearly replicated in every district.

Strogatz: Meaning that the 48% candidates would be losing 48 to 52 in every district, and so they wouldn’t have a single seat, yeah.

Duchin: Yeah, so that’s no good. In representative democracy, you have a lot of different ideals in tension.

One of them is majority rule, so it should count for something to be in the majority, but the other is minority voice. You don’t want to completely exclude from representation the interests of a minority. So that kind of way of making districts doesn’t work because it absolutely, fundamentally locks out minorities, even very large ones.

And so part of the idea of districts is that, well, if you assume that there’s some geographical correlation between your views and your location, then maybe by making the districts be connected and not too distended, that they’ll sort of create zones where even a group that’s in the minority statewide could be in the majority in that district. That’s the concept of districts. And that sounds great, but if you actually — as I’ve been doing for the last few years — if you run a bunch of trials and look at how well that works, it turns out you get pretty significant under-, sub-proportional representation if you don’t do anything to work on that.

So a quick example is it might be typical if you make a 10 x 10 grid and you put down a 40% minority. So there’s 100 nodes and you fill 40% of them with some color, and then you just sample a lot of districts. It turns out that in a toy example like that, you should probably expect somewhere between 20 and 25% of the representation for a 40% minority.

Strogatz: Wow, that’s interesting.

Duchin: Yeah, it is interesting.

Strogatz: I wouldn’t have guessed that.

Duchin: Yeah, I wouldn’t either before I got into it. Here’s the interesting thing about that particular experiment that I’m describing. You could put all kinds of different shape constraints down on the districts and it’s not going to matter too much for what you see. In particular, you can constrain away the most extreme examples of gerrymandering by dialing up your shape constraints, but you could still get quite extreme outcomes with nice shapes.

That’s true in the toy example and that’s even more true on real examples.


Strogatz: After the break, humble pie. Math really can’t cut the mustard when it comes to representative democracy. That’s ahead.


Strogatz: Back in 2018, the Pennsylvania Supreme Court struck down the state’s congressional district map. They found that it was an illegal partisan gerrymander that favored the Republicans. The governor of Pennsylvania then brought Moon in as a consultant to help analyze the proposed redistricting plan.

Duchin: So ordinarily when a legislature introduces a new districting plan, it gets introduced as a bill, and then it gets voted on, and passed like any other law. But in this case, everyone was under crazy time constraints because the court process was making everything move a lot faster than it ordinarily would, and whether because the time crunch or something else, the legislature did not put forward their new plan as a bill. They floated it on Twitter, true story.

Strogatz: Interesting.

Duchin: Well, so the Twitter plan, so that’s a do-over of the plan the legislature had enacted back in 2011, and they look really different. The 2011 plan has crazy shapes.

I liked to call it “tumors and fractals,” these kinds of characteristic windy boundaries, and skinny necks, and all the sins against kind of regular shapes that you can imagine. The new Twitter plan looks fine, but if you actually look at how much of an outlier it is in terms of partisan advantage, it behaves really nearly like the old plan it was replacing.

Strogatz: Earlier, you spoke about the space of all possible ways of twisting a surface.

Duchin: That’s right.

Strogatz: And now you’re going to speak to us about the space of all possible ways of making maps that satisfy certain constraints. So let’s talk about that and then how outliers fit into that idea.

Duchin: Yeah, this is my happy place these days. I really love thinking about this.

Strogatz: We love it. We want you in your happy place.

Duchin: And I hope what’s clear from this story is that I found myself thinking about redistricting from teaching it, and getting curious about it and trying to take the kinds of math that I like and bring them into the story. And then what I’ve discovered is that there’s a lot of work to do in redistricting. It could use a real math injection and I hope that kind of series of interventions is what my collaborators and I are providing, and lots of other people, too. But I take the kinds of math that I like and I bring them to bear on this problem, sometimes with more success and sometimes with less. But just like before I was taking a random walk in the space of surface geometries, now I want to take a random walk in the space of districting plans.

So you can think about that as you have a graph, which is all the chunks of the state and how they’re interconnected, and then you want to partition that graph. In Massachusetts, you want to cut it into nine pieces. Pennsylvania’s a little bigger, so you want to cut it into 18 pieces, and those are going to be the congressional districts.

I’ve got this giant graph of 9,000 precincts in Pennsylvania and I want to snip that 9,000-vertex graph into 18 pieces with roughly the same population. That’s the task. The number of ways to do that is — if you said it was astronomical, you’d be underselling the size. For a real districting problem, we’re probably looking at the google range, by which I mean 10 to the 100. Google before the search engine.

Strogatz: Right, so this is something young people will not know, that google, the word google, goes back a long time. It was a made-up big number.

Duchin: Yeah, 10 to the 100, or a 1 with 100 zeroes after it, which you just can’t think about how big that is, but that’s the universe of possibilities that you have for the districting problem, something to just tune your mind to the right scale.

Strogatz: In other words, these are all possible maps you could think of. There’s an unbelievably much-bigger-than-astronomical ways you could draw the map of the districts.

Duchin: Yeah. As the Pennsylvania example shows, I like to work on this problem not just theoretically, but practically, and so I followed very closely: all the legal developments that happened around gerrymandering in the last few years. This spring, the Supreme Court took up partisan gerrymandering again for the umpteenth time, and it was actually really interesting to hear how the justices were contending with the sheer size of the space of possibilities. Alito, at one point, wondered out loud if there might even be thousands of ways to do it.

Strogatz: Oh, you’re kidding.

Duchin: No, I’m not.

Strogatz: Really? So there’s a lot more than thousands.

Duchin: Yeah, he said, “Maybe dozens, maybe hundreds, maybe even thousands of ways.”

Strogatz: Oh, come on. You’re killing me.

Duchin: But to be fair, he didn’t just mean to partition a graph. He meant to make a plan that meets all those rules that I was outlining before.

There’s a lot of those rules and they’re not all kind of easy for mathematicians to fold into their thinking about the space of possibilities.

Strogatz: Good. All right, I feel better now. Okay. So there’s 1,000 that meet all these other criteria, and what is the real number? There are probably a lot more than 1,000.

Duchin: Aside to Alito, there are trillions and trillions, in fact.

Strogatz: Okay, but still.

Duchin: Actually, an interesting thing about thinking through districting is that the possibility space is so big, my motto is anything you can do one way, you can do a million ways. Most mathematicians, when they start thinking about the redistricting problem, they desperately want it to be an optimization problem. They want there to be what, in math and in engineering, we call an objective function, some score that measures how good a plan is and then your job is just to optimize it, find the best plan. But as on-the-nose optimization problems, we’re trying to literally find the best.

These are definitely computationally intractable, — NP-hard with a bullet, let’s just say. You could never do it on the nose. And that means you’re trying to find something good or near optimal, not just something that’s best. And now you’re in the range where there are just billions of fundamentally different ways to do it. So I try to convince technical people when they start working on redistricting to break the mindset that it’s about optimization. It’s really not about that. It’s about understanding how your priorities trade off rather than trying to find the best of something, in my view.


Strogatz: To optimize something is to find the best way of doing it. Now, of course, “best” might be ambiguous. Best might mean fastest, cheapest, most efficient. We have to decide what would be best in a given context, but the optimum solution is the best solution and it’s very natural for someone with our kind of training, Moon’s or mine, to think there is an optimal way of drawing the districts that would be most fair, in this case. We should draw the districts in a way that best represents the will of the people. Well, how do you do that? Trying to solve that is trying to view the districting problem as an optimization problem, and if I understand what Moon is saying, she says it’s a little more — it’s actually a lot more complicated than that.

Duchin: Okay, so I’ve got this giant number of ways to cut up Pennsylvania and what I can do then is, I can ask —. For instance, suppose I want to think about Democrats and Republicans, which I have to say is not the only or even maybe the most interesting viewpoint on the redistricting problem, but a lot of people care about it, so let’s talk about that one. Well, and I can look at all these different ways to redistrict Pennsylvania and I can ask what kinds of partisan properties do they have?

If I look at recent patterns of how people in Pennsylvania vote and I lay a districting plan on top of the voting pattern, how many districts have more red than blue? And I can just count up the number of Republican districts, the number of Democratic districts, and I get some sort of temperature of the partisan composition. In principle, I can do that for every one of these shmillion. That’s my new technical term for how many. So I can do this for all the shmillion possibilities, and I can just get some kind of bell curve of how many, say, Republican seats I would expect out of 18.

Keep in mind, Pennsylvania recently has been about really close to a 50-50 state. Actually, if you look in 2016 at both the presidential race and the Senate race that happened that year, it’s closer, I think, than 51-49 for both of those, if you just look at the two-way share. So it’s razor close, bright purple state, not red, not blue, but in the middle. And still, you can wonder if the fact that Democrats live in cities and where those cities are.

The fact that Philadelphia’s on the edge of the state and not in the middle. Something about the layout, something about that distribution of voters, something about the rural, the various kind of parameters of the problem — they might cause there to be some structural advantage for one part or the other even though there’s 50-50 voting.

That’s a big message that I think mathematicians have to give here. It’s that there’s no reason to be confident that in the absence of partisan intent, 50-50 voting would lead to 50-50 representation. It’s just not necessarily the case. It depends on all this rich combination of these distributional, geometric and combinatorial aspects.

Strogatz: So you’re saying even if in good faith the maps were drawn, lots of them would just naturally tend to favor, I suppose, red.

Duchin: And it varies by state.

Strogatz: Given the way it is in that particular state, as you said, with Philadelphia on the extreme east and stuff.

Duchin: Yeah, my group did an analysis of our home state of Massachusetts, and what we found is that particularly in the 2000 to 2010 census cycle, there was actually no way. So even though Republicans get a third of the votes and more here in Massachusetts, even though there are nine seats, your heart may desire a third of the votes to produce a third of the seats, what we found is it’s not only difficult to get a third of the seats to go Republican. It is impossible to get even one.

Strogatz: What, really?

Duchin: Yeah, true fact.

Strogatz: So the Republicans have a big structural disadvantage in Massachusetts.

Duchin: Huge structural disadvantage which just has to do with the fact that they don’t have enclaves in the state, and they’re distributed really uniformly around the state.

Strogatz: I see.

Duchin: So you might see an election where Republicans have a third of the statewide vote, but they also have a third of every town and a third of every precinct.

Strogatz: So it’s just like that 52-48 example you gave us earlier.

Duchin: It is.

Strogatz: It’s an instantiation of that effect.

Duchin: It’s just like that.

Strogatz: So that must drive Republican voters crazy for decades forever in Massachusetts.

Duchin: Even though we have congressional elections every two years, Massachusetts has not sent a Republican to Washington in its House delegation since 1994.

Strogatz: Really? I didn’t appreciate that. That’s interesting, wow.

Duchin: That is a long run of futility. And so at a first level of analysis, your gut feeling that that sure sounds like a Democratic-favoring gerrymander. And when we went to look at that, we were able to prove that it isn’t, that not only is the neutral tendency to get no Republican seats. It’s actually literally impossible. Even, it turns out, if you drop contiguity and you just grab precincts from all over the state greedily for Republicans, you will not find a single Republican district for those voting patterns that I was talking about.

Strogatz: So I think it’s a very important example to keep in mind because it always sounds to me that the discussion of the math of gerrymandering is code for let’s figure out a way to help the Democrats.

And your point is that isn’t really necessarily the case. The Republicans are getting underrepresented in Massachusetts, you might say. This does not have to be a partisan topic, per se.

Duchin: No, no.

Strogatz: It shouldn’t be, we hope.

Duchin: Truly, the math here is counterintuitive. It’s rich. It’s hard. It’s deep. And through luck or through sheer force of will, the kinds of math that I’m trained to do actually give you insight into the problem, so that’s a treat for me.


Okay, so then coming back to Pennsylvania, so what does it mean to be an outlier? Okay, well, that means that if I had access to the full space of possibilities, I could just say whether a particular plan like the Twitter plan, is it extreme in its partisan outcome relative to all of the other ways to do it?

The problem is, and here’s where random walks come in, I can’t access that full world of possibilities. It’s too big. It’s too complicated. But good news. I can random walk around it. I can take a random sample and I, and especially my colleagues — Justin Solomon; Daryl DeFord is a postdoc that works with our group; and we collaborate with lots of other people on this work. We’ve really tried to understand what you’d have to do to sample effectively and representatively in a short time.

We have a new paper that’s about to come out that kind of proposes what’s called a Markov chain approach to that, but essentially a random walk just like the one in my dissertation. But this time, a political, practical random walk and not a 12-dimensional space of three-holed surfaces.

Strogatz: So the punchline being then that the Twitter map just visually looks like, oh, that’s not really gerrymandering.

Duchin: It looks beautiful.

Strogatz: That looks reasonable, and yet sort of under the hood, it ended up being much more extremely partisan than you might have thought.

Duchin: Yeah. You do a forensic analysis in the way that I’m describing and you find that it’s just as much of a partisan outlier as the 2011 plan it was replacing.


Strogatz: Partisan outliers. This is a really interesting statistical idea that came up in the conversation. That you could imagine drawing districting maps many possible ways. In fact, there are trillions upon trillions of ways you could do it, and they might all meet reasonable criteria of the shapes are not too freaky looking. You keep districts contiguous, that’s all in one piece. And so you have a bunch of criteria, and even with those criteria, there’s still so many ways you could do it. Instead of picking one map, you look at this population of many possible maps and see, well, what happens if we try them all.

But if a lot of them give the same result, maybe that’s really the result that you should be getting, whereas if you get a result that only occurs one out of a million times, that’s an outlier and that seems wrong. If we’re trying to represent the will of the people, we want the result to fall right in the middle of the bell curve of all possible results.

Duchin: If you want to exercise your humility muscles, redistricting is a really good gym for you, because you can’t really get anywhere in this problem without listening to practitioners and understanding the needs of communities for representations.

I mentioned before that on-the-nose optimization is probably out of reach because the computational complexity makes it… If you could find the best districting plan for just about any of these criteria that we talked about, then you could also crack internet crypto systems and win a million dollars because you proved P equals MP, and so on.

But there’s another thing that I think is easy to forget which is that it’s not just that it’s hard to optimize. It’s that we fundamentally don’t know how to and should not try to turn the whole complicated picture of representative democracy and its ideals into an objective function. So there’s two steps there that make it not an optimization problem. You can’t find the best thing once you have an objective function, but also, no objective function really captures the complexity of what we’re trying to do when we vote.

Strogatz: Also, I mean to put it in plain English that my mother would have understood if she were still alive, would be what do you value? What do you care about? I mean the objective function is an expression of what matters, and it’s fine to say if I had the objective function, then, well, even optimizing it would be a hard math problem.

But of course, people have a lot of trouble agreeing on what the objective function should be. What are our values? That’s not something math can solve.

Duchin: No, that’s right. When you think about kind of how democracies work best, they usually work best with engagement and deliberation, and so I think one of the things that I find exciting about all of the attention that’s being paid to redistricting at this moment is that it forces you to have those conversations.


These days… So I run a group at Tufts and with close collaboration with MIT called the Metric Geometry and Gerrymandering Group, MGGG. We’re growing pretty fast and we do a mix of research, outreach, and consulting. So we try to prove theorems about how all this works and how it fits together.

We do a lot of outreach. We run a big summer program called the Voting Rights Data Institute that just ended a few weeks ago. And the last two summers, we had 85 students, grad and undergrad, come through this program where they come for six weeks and they do real-world data analysis problems to do with voting rights and redistricting. We also work with teachers trying to help people think about getting math and civics fused in their courses at different levels, and we do consulting.

So like that job for the Pennsylvania governor, we love to help people think about how they might want to change the rules and whether they’re living up to the rules. One of the things we’ve developed is a toolkit for analyzing plans and trying to understand their properties, like you said, under the hood. So that’s pretty ambitious, and of course, I have a day job, too, as a math professor.

So there’s a lot going on around here, but it’s really rewarding. I was talking before about thinking that research and activism would be different parts of my life, and I don’t know that I think of this as activism, but it’s very engaged with practicalities. Our group runs this summer research program and we’ve had 85 students come through in the last two years, and they’re bursting with talent. They’re interdisciplinary. They’re energetic. They want to make the world a little bit better in one way or another.

And then as the program ends, a lot of them, who are sort of wrapping up undergrad, and some who are wrapping up grad school, they’re asking, “What should I do next? What are maybe some graduate programs or some kinds of jobs that tap this crossover interest in science, technology, and the world?”

And I do think that data science programs are cropping up everywhere, but I would love to see more of them think about meaningful interdisciplinarity between math, computing, statistics and politics, the social world, and taking advantage of social science with more than a gesture. So I’d love to see more data programs having people read the anthropology canon, engage and talk with sociologists, think about urban planning.

A beautiful thing about the American education system compared to say Europe and a lot of the rest of the world is that it’s less siloed. In principle, it’s easier to be interdisciplinary, but not enough of that interdisciplinary crosses over between science and the humanities and social sciences.

And this is a moment when we’re starting out and conceiving of data science when we should really be trying to make that a more porous division.


Strogatz: How does Moon engage others in this gerrymandering problem?

Duchin: I mentioned our summer research programs for students, so that’s separate. We also do a lot of different kinds of training workshops. We do educator training. We’ve done five expert training sessions, and some of those folks who have come to our expert sessions have gone on to do important work. So for instance, one person, through us, got on to the census citizenship case as a consulting expert. It’s a long-term process plugging people in as — especially from more theoretical backgrounds — to get to the place where you can be on call as an expert for these kinds of cases.

There’s a whole learning curve and we also have to build up trust with litigators. It’s a long process. I think we’re definitely now reaching the point where we can take people who’ve trained with us and try to plug them in to important cases, and that feels good.

Strogatz: Well, I think you’ve given us an answer to the question that so many parents ask their kids if the kid likes math. I know my parents used to ask me this, “What are you ever going to do with that?” And so I think we both feel, and a lot of mathematicians feel, wow, there is so much to do with math. The whole world feels increasingly mathematical, so I think you’ve given us just a magnificent case study of the power of math to shed light on really important, in this case, issues of democracy.

Duchin: Well, that means a lot coming from you.


Strogatz: “The Joy of x” is a podcast project of Quanta Magazine. We’re produced by Story Mechanics.

Our producers are Dana Bialek and Camille Petersen. Our music is composed by Yuri Weber and Charles Michelet. Ellen Horne is our executive producer. From Quanta, our editorial advisors are Thomas Lin and John Rennie. Our sound engineers are Charles Michelet, and at the Cornell University broadcast studio, Glen Palmer and Bertrand Odom-Reed, who I like to call Bert. I’m Steve Strogatz. Thanks for listening.


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