 I'm going to bring up tonight's introducer, Dan Zaharapul. Let me just say that Dan has been a wonderful friend to me and to the museum since before the museum opened. We've worked together on lots of different projects. He's worked with our middle school students. He's worked with us at our galas. And he's just been a tremendous supporter of math outreach. I know you'll enjoy hearing from him. He is the executive director of the Art of Problem Solving Initiative, which runs BEAM, which is the bridge to enter advanced mathematics. So if children maybe don't come from a situation where they're being given enrichment in math and encouragement in math, Dan goes out and finds them and brings that to them. And we're delighted to partner with BEAM on many initiatives. And I'm delighted to give you Dan Zaharapul. Thank you, Cindy. It's an incredible pleasure to get to introduce Ben Blumsmith to you, because he is one of my favorite people. And I'm very excited for all of you to hear his talk. I am excited to hear his talk. I want to tell you a little bit of a story. And what I want to show you is how democracy infuses into Ben's soul. He taught for us many summers. But the first summer he did so, he was teaching a class called Group Theory, which is a fairly advanced mathematical class, especially for students the summer after seventh grade. We work with underserved students with talent in math. And you could see just from his being there that he cared very deeply about making mathematics accessible, including to those whose families maybe don't have the financial resources to get there themselves. He worked so late into the night, preparing this masterpiece of a class where they learned about symmetry by dancing. He would play the guitar, and his TA would call out dance moves. And they would see how their formation changed and use that to better understand group theory. And it was the sort of thing when Ben wears a suit, it's a sign of respect for his audience. And this was in the summer in a classroom that was air conditioned, but not necessarily that well. And every day he was there in a suit. And there was this. So one part of it is he cares deeply about this access. That is a side of math and democracy for Ben. But another part of it is that he infuses democracy into his classrooms, just into the environment of it. He's sort of like a Thomas Jefferson of the classroom, except without the bad parts. So what he's doing is that he doesn't move forward unless everyone in the class understands why and how things are working. His motto is honor your dissatisfaction. And so he's making sure that everyone is there. Everyone within it. It's a communal effort of the class to move forward. So Ben and democracy, in my mind, have been very closely linked, even if I didn't put that word to what he was doing back then. Now he's here to talk about how you can actually use math to study democracy. All of his work is about using math to advance society and to advance democratic principles. And it's basically remarkable. So let me tell you why he's a fancy person you should pay attention to. He got his undergrad degree from Yale. He went to Tufts to get a master of arts and teaching and taught for several years in both Boston and New York public schools. He then went to NYU to get a PhD in mathematics. He's now completing a TED residency. He'll be giving a TED talk, I believe, at the end of this month. And afterwards, we will see. But I'm sure that it's going to be a quest toward something great. So please welcome Ben. So thank you, Dan, for that absurdly lovely introduction. I was like, part of the talk, I was going to thank Dan for the introduction. I knew that that was going to happen. But I didn't know it was going to be like, I was going to plot. So thank you for that introduction. Cindy and the museum, thank you for hosting this talk. I'm going to be telling you things that I think that every citizen should know. So I really appreciate the museum supporting this conversation happening. And there's some other people I have to thank before I do anything else, which is the Tufts metric geometry and gerrymandering group. So this is run by Moon Duchen, the one in the middle there with the label. And she's a mathematician. Like me, her training in mathematics is in pure mathematics. She's a geometric group theorist, if that means anything to you. She did not start out doing something about how society works. But that's completely where she's landed. She started this group working on gerrymandering at Tufts. They've run this whole sequence of conferences in the last year. I went to three of them. And their work just really informed, like taught me about what I'm about to teach you I learned from them. So I had to thank them. And also explicitly Mira Bernstein, who's the other label one, I'm using several times in this talk, you'll see source MGGG somewhere on the slide. And that means that I got it from Mira. So thank you to those folks. They're amazing. Look them up. All right, gerrymandering. So just to make sure that we're all on the same page, what's gerrymandering? Every 10 years in the United States, the new census comes out. And when the new census comes out, it might affect how many representative states get in Congress. So they have to redraw the boundaries between their congressional districts every 10 years. And gerrymandering is redrawing those boundaries in such a way to benefit your political party or sometimes to try to cut out a racial minority from the political process. That's those two types of gerrymandering are the main types, partisan and racial gerrymandering. This talk is going to be just about partisan gerrymandering. Because they're both extremely rich problematic issues. And they intersect with each other in a lot of really rich problematic ways. So in the Q&A afterward, I invite questions about that relationship if you're interested. But I'm going to talk about partisan gerrymandering because the Supreme Court has found ways to set limits on racial gerrymandering. They're not ideal, but the Supreme Court has knocked down maps as a racial gerrymander. And it has yet to do that with a partisan gerrymander. So this talk is focused on the particular problematic of partisan gerrymandering. A little tiny bit of history. That's Elbridge Gary right there. It's spelled gerry, though, with a G. Elbridge Gary, in case you didn't know, is one of our founding fathers. He's a signer of the Declaration of Independence. He was the vice president to James Madison. And he was the governor of Massachusetts in 1812 when his legislature passed a districting of the congressional districts of Massachusetts that was designed to shore up support for his party. What was his party? The Democratic Republican Party. And it was very controversial at the time because there was this district. Anybody recognize geographically where we are based on some of these names, Ipswich and Andover? Yeah? Yep. This is just west of the North Shore. This is right north of Boston. And there was this district consisting of Salisbury, Amesbury, Haverhill, Mithuin, and so on. You could tell I'm from Massachusetts because I said Haverhill, right? And people were just like, what is that? Like the shape of it in connection with it, the obvious connection of that shape with the intention to shore up support for that party, was very controversial. What you see here is a political cartoon from the time mocking this district as some sort of like beast. Or maybe a salamander. And that's where the word gerrymander comes from. So that's your history lesson. Here's a contemporary such district. This is Pennsylvania 7. And this district was just struck down as unconstitutional under the Pennsylvania State Constitution by the Pennsylvania Supreme Court in January, very recently. So this talk is about gerrymandering. Before anything else happens, I need you guys to do some gerrymandering so we know what we're talking about. So I'm going to tell you what you're going to do. And you're going to do it on the you all got a piece of paper with eight copies of this image. If you need a pencil, the MoMath staff has some pencils for you. So raise your hand if you need a pencil and they'll get you a pencil. Here is what I want you to do. Think of this square as a model of a state. This image is from the MGGG, the Tufts Group that I mentioned before. And they call it Squaretopia. So I'm going to call it Squaretopia. Squaretopia is comprised of 90 voting precincts. The voting precincts are the squares. Each voting precinct, we should think of it as having the same number of voters as every other voting precinct just to keep things simple for ourselves. And each voting precinct, again, this is just for the sake of simplifying, we're going to think of each voting precinct as being entirely loyal to one party. And there are only two parties. The parties are the purple party and the white party. So here's what I want you to do. First, on one of those things, gerrymandered this map for purple. I want you to divide it into 10 congressional districts, each comprising nine of the precincts in such a way that you get as many of those 10 districts to be majority purple. After you gerrymandered for purple, take another of those Squaretopias and gerrymandered for white. So again, divide it into 10 districts of nine precincts each, nine little squares each. Except this time, try to have as many of them with a white majority instead of a purple majority. That's probably all you'll have time for in the time that I'll give you. But if you want a challenge and you're done with both those things, then see if you can create an even more partisan map for one or the other, or see if you can make a map whose outcome is as partisan as your white gerrymandered or whatever, except the districts look better, more blob-like and less tentatively. This is what I mean by a district. This is just an example. Every completely connected thing made of squares that are touching edge to edge, not corner to corner. Any questions about that? Last time somebody asked an additional question that I'll answer now just in case, which is you also have to have the districts be what a mathematician would call simply connected, which means don't have an O. A district is not allowed to completely surround some other space. The U shape one in Chicago, but it doesn't join up, though, right? It's just you're allowed to have a U. Actually, that raised a good clarification. A U and a C are OK. It just can't link up so that it completely contains something else and separates it from everything else. Does that make sense? OK, so I'm going to give you a few minutes to do this. Raise your hand if you got a gerrymander for red, for purple, with at least eight. Put your hands down if only eight, nine and up, nine and up. 10 and up. Well, a lot 10, right? So how about white? Let's talk about white. Anybody get five white districts? How about six? Six, six. How about seven? Anybody seven? Yes, some sevens. Yeah, so here's one. Here's a solution. Here's just one possible solution for purple, a gerrymander for purple, winning all 10. And here's a gerrymander for white, winning seven of the 10. They don't even look that crazy. They could, I mean, they're a little funny and long, but right, so the first take home lesson here is just the power of the pen, right? The votes were the same in the left and right scenarios. So that means that the person who controls the district thing is controlling the difference between 100% of SquareTopia's representatives in Congress being from the purple party and 30%. It's a lot of power. Take a second to reflect on how, what were the techniques that you used as a gerrymander? Because they come down to two basic techniques, right? One was to pack all of your, as much of your opposition as you could, into a small number of districts that they win in a landslide. And the other is to crack, to spread the rest of your opposition between a bunch of districts that they can't quite win. This is how you force your opposition to use their votes inefficiently. Either they're winning in a landslide and then half of them weren't needed to do that. They don't get anything more out of winning a landslide than they would out of winning by 51%. And in the other cases, they're in districts where their votes are not electing anybody. So if you look at what you did, you'll see that's what you did. You can see that's what I did, especially. It's super clear in the white gerrymander on the right here, there are these three districts, this one, this one, and this one, where between the three, there's a single white precinct. So those are the packed landslide purple districts. And then in every single other district, white won by exactly five to four. So there were four purple votes that didn't elect anybody. Historically, both parties have gerrymandered every chance they get. The issue seems to me anyway to have a partisan tint at present. And the reason for this is because of a conscious Republican strategy aiming for down ballot races in the 2010 election so that they could control the redistricting process. So here's an op-ed by Karl Rove in the Wall Street Journal in 2010 discussing this all freely. For him, it's pure political strategy. If you, I encourage you to look this article up. And reading it, it's just like he's just like, he had the feeling it might be a Republican wave year. He was like, I think Americans don't like what Obama's doing. I think this could be a Republican wave. So we're just concentrating on riding that wave into control of state houses that are gonna allow us to control redistricting. And we think that's gonna make a big difference for the next decade or more. So it's not, there's no question about what the moral implications are, or anything, it's just pure tactics or the legality or anything, which is partly an illustration of the fact that the legality is murky, which I'll get to again later. Nonetheless, so in the present, there was a very successful Republican strategic move in 2010 that got control of a large number of state houses. So there are a large number of effective Republican gerrymanders at present, currently active in the house US congressional maps. But there are still Democratic controlled states that are suspected of gerrymandering. Illinois, I'm mentioning Illinois and Maryland. Maryland is one of the two partisan gerrymanders that's before the Supreme Court right now. So to me, it's clear that this should be seen as a non-partisan issue. And Americans, I mean, across the political spectrum, Americans hate the feeling that the system is rigged. And gerrymandering is the legislators choosing which voters are gonna be in the district that elects them. So gerrymandering is the legislators choosing the voters instead of the voters choosing the legislators. So Americans across the political spectrum don't like it. You might think given that, why isn't this somehow solved by now? But it's actually a hard problem to address. And I'm gonna mention two ways that it's hard. One is that there's no law against it. I was, you know, I pointed this out with the Carl Rove piece. Like he's just out in the open saying, let, yo guys, let's get control of redistricting. Let's put money into these down ballot races so that we can have those state houses and then we can control redistricting and then we'll just win. This is gonna be awesome. Like on the Wall Street Journal, like, you know, you might think it's illegal but it's clearly not. That's the point of that. Why is it not illegal because who would have to pass a law? I mean, okay, so how it could be changed, that's a big question and this talk is part of. But in the absence of a law, it falls to the courts to set limits. And the courts need to understand what is the legal theory of harm and who is being harmed in order to have a case that they can say something is unconstitutional. So is it, you know, is the issue at play the equal protection clause of the 14th Amendment? Is the issue at play the first amendments right to political free expression? It's not clear, but I am not a lawyer and this is not the Museum of Law. So the other problem is the one I'm gonna concentrate on. The other problem is how can you tell when a gerrymander has gone too far? And there are two methods of gerrymandering detection that loom large in the public mind. Funny shapes, right? There's Goofy again. And this goes back to the original, you know, the nominal, the name, the gerrymander that started it all. Not gerrymandering, but the one that started that word. And highly non-proportional outcomes. So this is just a quote from a report by the Brennan Center on the case, the Wisconsin case that's before the Supreme Court. Right now it basically just says, look at the wide gap between the percentage of votes statewide that Republicans got in this election and the number of Republicans that were elected. These two ideas dominate public discourse about why we think we're being gerrymandered at. This talk can be summarized in the following way. Neither of these is the right way to think about detecting gerrymandering. However, there is a much better way due to some fairly recent mathematical innovations and I'm gonna share it with you. That's the talk in a nutshell. So before I share you the way, I have to convince you that those aren't great ways. To do that, I'm gonna have you gerrymander a little bit more, just a little bit more. Here's what I want you to do this time. Not, you're not gonna gerrymander actually, you're gonna try to make a fair map, but I'm gonna ask you to try to make a fair map in two different, prioritizing two different things. The first time I want you to just prioritize making the districts compact and geometrically nice, like as close to a blob as possible. Avoid tentacles, avoid elongation. The second thing, once you do that, so do that, and then after you do that, I want you to try to draw another map where what you prioritize is proportionality of the outcome. By which I mean the vote is 60% purple. So the second time what I want you to do is try to draw a map where purple wins six of the 10 districts. If you complete both of those, which you won't, because I'm not gonna give you time, but if you do, then try to do a district thing that simultaneously meets both criteria. Six out of 10 and also geometrically nice and nice. All right, so you have a few minutes, go. All right, I'm gonna stop you. Okay, so here's my solution. There's lots of solutions for both of these, but here's one solution. On the left is my compact. I got as many three by three squares as I could, and then I made those as close to two by four rectangles, but they have to have nine. So it's a little funny, but like pretty good, don't you think? Nine out of 10 purple, 60% of the population elected 90% of the representation. Then I tried to make it proportional. Here's my attempted proportional. I got some squares in there. I got four little nice three by three squares instead of six, but four, but I had this funny guy, and this like, okay, this is not that bad, but it's clearly not as good as that. Take home lesson. Geographic compactness does not yield proportionality. This is like the first big message that I wanted to leave you with, because the public discourse about gerrymandering is like, look at that horrible district and look at these unfair non-proportional outcomes. We have to go after these horrible districts so that we can get proportional outcomes, and that's not gonna work. If what we want is proportional outcomes, which I'm not committing to what the objective should be, but people perceive wildly disproportionate outcomes as unfair, and if they attempt to address that unfairness by going after geometrically funky shapes, it's not gonna be successful. So that's the first big take home that I wanted to leave you with. These are two different competing objectives. Geometric compactness and proportional outcome. And these are issues in practice, not just theory. I'm gonna skip this slide for time, but I'm gonna go to this one. The Supreme Court has explicitly rejected proportionality as a standard by which to judge a partisan gerrymander as unfair. In the 1986 decision, Davis versus Bandemur, they wrote, the mere lack of proportional representation will not be sufficient to prove unconstitutional discrimination. If all or most they continue, if all or most of the districts are competitive, I'm gonna mention that Minnesota is such a state. This does happen, whereas state has mostly competitive districts. Even a narrow statewide preference for either party would produce an overwhelming majority for the winning party. So this is what in fact happens in Minnesota. When Minnesota is in a democratic mood, almost all the districts are slightly democratic and the delegation that goes to Washington is almost entirely democratic. When Minnesota is in an ever so slightly Republican mood, almost all the districts go Republican by small margins and almost the whole delegation is Republican. So this is a real thing. The Supreme Court was not just hypothesizing. Since 1986, several alternative measures attempting to quantify something that could be used to bring a case have been proposed. I'm gonna mention two of them very briefly because the thing that I really wanna get to is a third one of these. But these two I think are worth mentioning because partisan symmetry was the key to a case that was decided in 2006 and the other one I'll get to. So partisan symmetry is based on the idea of trying to predict a range of election outcomes that would occur as the mood in the state shifts between Democratic and Republican. And the question it asks is, does the way that the plan treats Democrats when the mood shifts Democrat equal the way that the plan treats Republicans when the mood in the state shifts Republican? So just to illustrate, I'm not gonna feel free to ask me in the Q and A at the end and I'll tell you where these graphs came from. I won't tell you now. But these graphs are attempting to capture how the representation changes as the vote share changes based on Minnesota's 2016 results extrapolating from that. If the Democratic, this is the graph is mapping Republican vote share along the x-axis and Republican seat share along the y-axis. Percentage of elected people who are Republicans along the y-axis. So it's saying that until the Republican vote share hits about 32%, all the elected officials are Democrats. And then once it hits 32%, they get one district electing a Republican. Once it hits 42%, they get a second district electing a Republican. Once it hits 40, like seven or 8%, they get a third district that goes Republican, et cetera. This is the same thing for Ohio based on again the 2016 congressional election. And what you can see is that Minnesota's, if you believe where the graph comes from, which I haven't even told you where it comes from so you can't evaluate that, but if the methodology underlying the graphs has some kind of soundness to it, the argument goes Minnesota is approximately symmetrical if you rotate it around the middle. Whereas Ohio is totally asymmetrical. So Minnesota is fair between the two parties while Ohio is not. That's the argument. The Supreme Court rejected this standard in 2006 in Lulac versus Perry. They said the assumptions that you built the graph on are not sound enough for us to trust this, to trust a constitutional violation based on this. The other idea I wanna mention briefly is the efficiency gap which has gotten a lot of press this past couple years because it was cited among a bunch of other types of evidence in the Wisconsin case at the district court level before it went to the Supreme Court. And because it was a new idea at the time and it was the court founded compelling, it just got totally up taken by the press. The idea of the efficiency gap is based on the idea of comparing the wasted votes between the two parties where wasted means either I'm a Republican in a district that went Democratic then my vote was wasted because it didn't elect anybody or I'm a Republican in a district that went Republican by a large margin and I'm in the half that was not beyond 50% that was not needed to elect a person. So comparing the wasted votes of the two parties, that's it in a nutshell, feel free to ask me more later. Who knows what the Supreme Court will think of it but to me efficiency gap shares a weakness with the proportionality standard and even with the partisan symmetry is more sophisticated idea but they all share this flaw. All of these methods set a standard for what counts as fair that doesn't look at the specific partisan structure of the state. It doesn't look at where the Democrats and Republicans live in the state. And political geography is biased against Democrats. What do I mean by this? Have a look at this totally amazing picture. Here's what this is of. For 20 different states, the state names are above the graph corresponding to that state. Each dot is a voting precinct. The x-axis is log population density so if you don't know what log is you can just think of it as population density. And the y-axis is the percent of vote in that precinct that went to George Bush in 2000 election against Al Gore. So I think this is fascinating for a lot of reason. First of all, look at that elbow macaroni shape that they all have. Why is this? Well, the left side of the graph is low population density so rural. The right side of the graph is high population density. So this elbow is saying well rural areas are more Republican and urban areas are more Democratic but also more specifically where the elbow is pronounced it's saying rural areas are Republican. Where's a good one here? Indiana, rural areas are Republican. Urban areas are both Republican and Democrat. That's why the elbow. I will just drop for you to notice and I'm not gonna say anything more about it that there are four states that have a kind of square pattern instead of that elbow pattern. Louisiana, Mississippi, Georgia and South Carolina. All those deep South states have this square instead of the elbow. That's for you to think about. It's deep but I'm not gonna say anything more about it. The thing I wanna point out to you now is this. Forget the x-axis entirely. Look only at the distribution along the y-axis. Look at Florida for example. The most Republican precinct in Florida, there is one precinct in Florida that's about 100% Republican but other than that the most Republican precincts in Florida are not more than 90% Republican. They have 10% Democrats in them. Whereas look at how many precincts there are in Florida that have no Republicans at all. Tons. You will notice that pattern in every one of these graphs except for the ones like Idaho which are just a really Republican state. The places that are Republican are not as Republican as the places that are Democrat. The Democratic places are Democrat. What does this mean? This means that Democrats are packing themselves as far as packing and cracking go. This is commonly believed to have something to do with population density. It has nothing to do with population density because if there were a ton of rural districts that were 100% Republican, they would be equally as packed from a gerrymandering point of view as urban districts that are 100% Democrat. It doesn't matter rural versus urban. What matters is what percent Republican versus Democrat and what we're seeing in these graphs is that there's a lot of every state has a big portion that's just all Democrats. And half those Democrats feel like they wasted their vote and the 25% Democrats who live in the places that are 75% Republican also feel like they wasted their vote. So that means that symmetry and the efficiency, like a perfect efficiency gap and perfect partisan symmetry and perfect as well as perfect proportionality are things that don't happen naturally if all you're doing is dividing up the state in a geometrically nice way. These are things that are not gonna happen. So all of these previous measures have been vulnerable to that criticism. Here is the solution. This is the other big takeaway that I wanna leave you with. There is a new game in town for how to measure gerrymandering called Outlier Analysis. These are the names of some people who have pioneered it. They're all awesome. Here's the idea. Using some method whose soundness needs to be thought about but this is its own problem but using some method start drawing random redistrictings of the state. Here I've got North Carolina. So these are from Wendy Cho's, Wendy Cho has an algorithm to do this. She's one of the people I listed before. Then draw some more. Then draw some more. Then draw a whole lot more. Tens of thousands at least. Wendy's algorithm actually does many, many more than that and she uses a supercomputer but at least tens of thousands. Now for each of these redistrictings we can ask the following question. Take real voter data from real elections to construct the partisan structure of the state. So here's a picture of North Carolina. The red dots are supposed to be Republican support. The blue dots are supposed to be Democratic support. This is based on actual election data. With that map any hypothetical district in any plan you can plop it down onto that map of partisan of the partisan structure of the state and then ask who wins that district. So if somebody wins that district you can look at the reds and blues and see. You see what I'm saying? This means now take all your little maps that you drew. You can ask if that had been the map how many Democrats and how many Republicans would have gone to Congress and you get a different answer for each one of your tens of thousands of maps. This gives you a distribution of partisan outcomes. So in this version of the, you know you can see here this means that almost every one of these maps as far as percentages go either six or seven Democrats won. This is how many Democrats across the bottom based on the 2012 election data. If these had been the maps and people voted exactly as they voted in 2012 in the vast majority of the cases six or seven Democrats would have won significant numbers of maps had five Democrats winning and significant numbers had eight Democrats winning almost no map had nine Democrats winning and practically nobody had four Democrats winning. These are the real maps. This was the one that was actually in play in 2012 and then it was ruled an unconstitutional racial gerrymander before the 2016 election. So this was the one that was in play in 2016. We can ask where their partisan outcome fits inside the distribution. Yeah, that's the punchline and we're pretty much out of time so I wanna give us time to ask questions so I'm not gonna go further. So let's do questions. You mentioned for this that they used the real election data from the previous election what if you pick like four of the past elections and the histogram like totally changes from one to the other. Beautiful question, beautiful question. So they do that. So they look at this across multiple past elections. Joey Chen, one of the people who did it does a whole thing where he kind of averages the last seven elections together to try to get what the partisan baseline is but that's a beautiful question because one challenge is okay. I mean, fundamentally this method assumes that the partisan behavior of the voters doesn't change that much over the 10 years of the redistricting. And that assumption needs to be justified. I mean it's empirically verifiable to a certain extent but sometimes you have a really awful candidate and that candidate changes the, you know. And then sometimes there's other political effects like I think in Massachusetts if you compare gubernatorial races to presidential races you'll get very different partisan behavior. So that's a beautiful question and the methods attempt to deal with that by either looking at multiple elections or by somehow averaging the elections. Did that answer? I have a question. So how do you sample from the set of all legal maps? It seems like there should be exponentially many. Yes, how do you generate the random maps? That's the question? Yeah, so that's a beautiful question and that's the mathematically most challenging and interesting part of this method. Where do these maps come from? Where do the random maps come from and why is it reasonable to say that they're random? I think that the story with this is basically that this method was pioneered by political scientists. I mentioned Joey Chen and Wendy Tamcho and then mathematicians got involved. So Mattingly, John and Mattingly and his group at Duke is mathematicians. I was, Joey Chen, Wendy Tamcho, these are political scientists. Mattingly is a mathematician. Maria Chiquina is a mathematical biologist and she has a team of mathematicians. So these guys are really more on the mathematical side and that question evolved when this method went from the hands of political scientists to mathematicians. Political scientists didn't really worry very much about it. They just wrote a computer program that would create some redistricting of the state. They used some kind of random process to generate that redistricting. They did not put partisan information into that random process and they said it's neutral. But they didn't, there's not a theorem underlying their method that says that they are sampling from the true space of possible maps according to some well-defined probability distribution. The mathematicians did what, how they, I mean, they can't prove it either, but here's what they did. They're using, they're using something called Markov Chain Monte Carlo, which is a method that permeates the study of probability distributions on spaces that are hard to understand. So this, what I mean space that are hard to understand. In our situation, the space of possible maps. If you think of every possible map as a point in a big space, where like if you tweak the boundary just a little bit, then you just moved to a close by point, that space is some crazy high dimensional, we have no idea what it's shaped like, type of a space of possibilities of what all the maps could be. And MCMC, Markov Chain Monte Carlo, is the only tool we have for probing a probability distribution on a space like that. So it basically, how it works is you pick a starting point, which is a map that for Mattingly's group, they actually used the 2012 and 2016 real maps. Then you change it in a random way and then you evaluate whether the change makes the map better or worse with respect to traditional redistricting criteria like geometric compactness, but also don't split counties, try to keep communities of interest together. They programmed a measure for like how good a map is in relation to those nonpartisan traditional redistricting criteria. And then, so they start with a map, then they randomly change it in some way. They ask, is it better or worse with respect to that measure? Then if it's better, they move to that new map. If it's worse, then they move to the old map with some probability based on how much worse. And in that way, they kind of like create an algorithm that walks around the space of possible maps and creates different possibilities. And if that algorithm runs for long enough, then there's a theorem that says that the maps that you get are sampled from a certain random distribution that you can explicitly write down on the original space. We don't know how to tell what long enough means. And that's the gap. But Chiquina's group has a way of, has written a paper that proves that they have a way of saying a map is highly atypical with respect to that distribution, even if I don't know that I've sampled the distribution yet. So there's really, I mean, that was the most technical, that little last five minutes was the fanciest stuff I said today. I'm sorry if that was over. You know, I just, trying to answer that question, it's a real, this is really where a lot of the juicy math is. So did that answer? So once you have like done the analysis and done the random maps and the distribution, like how exactly do you find like the map and does the thing you were just talking about take into account like the ideal distribution? Can you ask that question one more time? Okay, so once you've done the outlier analysis and with all the random maps to find like the ideal distribution, how does, like how do you find like the perfect map? Beautiful question. Okay, I understand the question now I think. Okay, so the goal of this whole program is not to find the right map. The goal of the program is only to have evidence that an actual map is highly gerrymandered. So all the whole thing is just building a case against the North Carolina map. You know, basically it's like, look, this map is way out there. So I don't know what the map that most ideally represents the will of the people is, but I know it ain't this one. Like that's what, all this gives you is the range of partisan outcomes that it would be reasonable to expect from a fair map. It doesn't actually give you a fair map. Did that answer? Okay. Okay, we live in a country that has an appalling, appalling rate of people not voting. Now here in New York City when they gerrymandered, they don't gerrymander on the number of people. They gerrymander on the number of people who they know will vote. And there is a big, big difference. And I wonder, does that go into your computation? I mean, we're talking to Clinton, when he came in, he came in with 48% of the people who could vote voted. So he won 24%, Bill Clinton, 24% of the people who could have voted, that's appalling. Right, so the answer is yes, and it's because this, I mean that the method accounts for that gap because the partisan structure of the state that you're getting here is based on actual election data. So it's based on who is voting. So, you know, that, but the method, the only way that this addresses that issue as a net, you know, because you're talking about something being appalling and I agree, it's like, where's our democracy if half the people are not even engaging in, you know. The only way that I feel that this addresses it is that I believe that a lot of voters are alienated by the feeling that their vote doesn't matter. And if this can help combat that by allowing the courts to rule out maps that make people really feel like their vote doesn't matter, then that'll help people feel like their vote matters. But it's not, you know, everything, I'll say this because it came up in the last question session too. This thing that I've shown you here is a very, it's participating in a very concrete reform effort. It's not like a general, this is the way things should be. It's like there's a very specific thing going on, which is there are a lot of extremely gerrymandered maps and as technology increases, gerrymanders are gonna continue to make their gerrymanders more effective using the new technological tools. So it feels very urgent to stem the flow of those gerrymanders so that we can feel like our votes matter at all rather than not mattering at all. But there's a thousand other issues to do with the political process being successful that this is only, like this is only doing one thing. The one thing it's doing is how can we build a case that courts will find compelling so that courts can throw out gerrymanders now? It's not an activist program to reform the whole voting system. So it's really inserting itself at a very specific point in a very specific reform trajectory. That's like litigation is a really important part of it. The Supreme Court is a really important part of it. I think that it's broader than that because if people see this picture, that that's gonna matter and it's gonna help settle, to me I believe it's gonna help break some of these conversations out of a partisan deadlock. I hope anyway because it's like let's get science involved and find out if this map is typical of the represents the will of the people or not. But it was designed to participate in one point which is to build a legal case. So that only makes sense, that only matters if the path for reform that you're looking at is the legal path for reform. And the issue of inadequate voter turnout might be addressed by this problem being solved partly through the indirect way that I said but I think it needs to be addressed 100 other ways too that involve totally different reform pathways. Does that speak to that? Yes, except that we've stored pictures of people standing online for five and six hours and we were told that their districts had been changed so people are planning gerrymandering based on the fact that they know that some people will vote and some people may not. Right, right. So I guess so it comes down to as far as this goes what it comes down to is the evaluation of gerrymandering is using the same, is ideally using the same data set that the gerrymanders are using. Like this is the map that the gerrymanders are using ideally to figure out where to gerrymanders so that their gerrymander works. And to the extent that we can get hold of the same data they have, then we can show that they gerrymandered. That's what this is trying to do. So it's just, it doesn't step to the side of that. You know what I'm saying? It's like that's part of the landscape that they're working with and that this is working with. So just to summarize, I think what you're really saying is that this last method, it doesn't tell you how to design an ideal division of districts. It gives you a measure of how much they cook the books. I mean after the fact, they measure how much books were cooked. That's right. And that's what makes it of legal interest that it shows that somebody must have cooked the books without you knowing even exactly how they did it. Right, that's right. But could I ask you, you ended up, this was a fascinating talk, but then you ended abruptly because the hour was up. Yes. Could you give us a quick summary of what you would tell us with more time? Yeah, I'll try to do that. I'll try to do that extremely fast because otherwise I feel like I'm just giving the rest of the talk. So, all right. So the first thing I was gonna show you was this, which is a more detailed version of the thing that I just showed you, because this breaks it down by district. Each little box plot is the distribution of the percent of Democrats in the like say sixth most Democratic, sixth least Democratic district. So this right here, I just picked one. So the sixth least Democratic district, the box plot is the distribution of Democrats in the random plans in that district. The purple dot, green dot and orange dot are three actual maps. The purple dot and the orange dot are the 2012 and 16 plans that were actually used. The green dot is a plan that was created by a non-partisan panel of judges commissioned by a nonprofit. So what these graphs show is that the green dots track the medians of the random distribution pretty well. So the non-partisan map is mirroring the random distribution. But the purple and orange dots are extreme outliers, especially not so much on the left side of the graph, but in the middle third and the right. Because these are the packed Democratic districts. This was a Republican gerrymander. These are the places where there are extra Democrats on the plans created by the legislature. And this middle third is the districts that would have been competitive in the random scenario or a non-partisan scenario, but are Republican in the gerrymandered scenario. So that's that. All right. So yes, yay, that was just a more nuanced image of how outlier analysis works. I was gonna point out a subtlety, which is that the Wisconsin gerrymander and the North Carolina gerrymander had fundamentally different goals. Because the North Carolina gerrymander, US House of Representatives, it's like what I had you do. Get as many candidates as you can over there. But the Wisconsin gerrymander was a state house. So all they need is a supermajority. They don't need everybody. So once they have 60 of the 99, they're good. And therefore the goals are different. You don't wanna get as many districts as you can. You wanna get a supermajority worth of safe as possible districts. So I was, this I knew I wasn't gonna have time to do, but just I had the idea beforehand of having you try to do that again with this map. And here's a solution that this is gerrymandered for red, seven districts out of 10. So I figure that's a veto proof majority. But they're all seven to two wins. So what's interesting about this is that the partisan outcome, just the number is not that extreme. You know that when you did the compact districts, most of you got eight. So this is like less red part, less purple party than just a geometrically regular thing. However, the purple majorities are extremely safe. And that means that even if the state swings quite a bit toward the white party, you could lose two, you could have two purple seats flip white in every district and they would still be winning seven seats. So right now they have 60% of the vote. If that happened, they would have like 40% of the vote and they'd still be winning seven seats. So this is a highly gerrymandered map and the reason that's relevant is that this is how the Wisconsin gerrymander works. So you only see the gerrymander when the vote, when the Republican vote share is below 50%. So you saw it in 2012. In 2012 it was less than 50% and the outcome is an extreme outlier. But when it's above 50%, it actually looks like a totally typical outcome. So in order to see it this way, you have to do this, that image that I showed you for North Carolina, you can see the gerrymander here because the red dots, which are what they did are above the distribution on the Democratic districts and below the distribution on the competitive districts. So this last part is not just cooking the books, but adding spices to the mixture. Right, exactly. And then I was just gonna mention some frontiers for further work. That apropos of Solidar's question in the back, how do we create good sampling techniques to build the random maps? How do we understand the geometry of the space of possible maps? Here's some totally different ideas. How about instead of having the legislatures and charge redistrict, we come up with some good game theoretic protocols that two parties could use without needing an independent commission to create a fair map. So I'm involved with a project in the pretty early stages, I don't know if this is gonna go anywhere, but we're trying to develop protocols like that with Stephen Brams. Thank you for that introduction, Sylvan. Another is because single member districts lead to non-proportional outcomes, if you care about proportionality, you might wanna consider reforming the system of single member districts. So people are thinking about how to create multi-member district systems that don't shut out minorities, which is the traditional problem with multi-member districts. So that's another thing going on. And last but not least, what's the relationship of shape to gerrymandering? Cause I've been at pains to show you that shape is not determinative, right? Here is the Wisconsin map, I didn't mention this before, it's not that bad looking. Like there is no Snoopy kicking Donald Duck in this map. And on the left, that's the 2016 North Carolina map. The 2012 looks ridiculous, but the 2016 is not that crazy looking. Nonetheless, the partisan outcomes you could see in that distribution were equally extreme. So if shape doesn't just tell you, but we all wanna believe shape, ever since the beginning, it's part of our intuition about it. So to me, there's a really interesting scientific question, and I've initiated a project about this too, I don't know if it's gonna go anywhere either, but to try to understand what the real role of shape is in limiting gerrymandering. To what extent does it limit gerrymandering? Oh, and this, I should credit MGGG because they have some really great ideas about this. So Moon Duchin, the person who runs MGGG, she comes from a background that deals with graph theory, if you know anything about graph theory. And her idea is to re-envision the geometric structure of the state so that it sees more of the political structure of the state and it's not just a map. So these individual points are like voting precincts and then there are some kind of voter unit and then they're linked by a line if they're next to each other. And maybe we can put some kind of weights or numbers on these links that represent something like how economically tied in these two areas are or how demographically similar they are or other kinds of politically relevant information so that the state's geometric structure sees what's going on politically and not just the map. Maybe if we do that, then we'll be able to see gerrymanders in this shape that we can't see in this shape. So that's the whole thing, now you got the whole talk. A hand for Ben.