 Thank you all so, so much for joining us here for Carnegie Mellon University's 2018 Constitution Day program, which commemorates the September 17th, 1787 signing of the United States Constitution. My name is Scott Levitt and I am here today representing the Office of Community Standards and Integrity, and I'm also here to introduce this year's speaker for the Constitution Day series. Before we move forward, I would like to recognize the Alumni Association for assisting with the live stream services that we have today, and I would also like to acknowledge the hard work that our University Library staff members have put into collaborating with us on this event. University Libraries has graciously provided us with their first edition copy of the Bill of Rights for viewing today up here at McLean Casing, which we will be able to speak on a little bit later in the program. It is with great pleasure that I introduce to you Dr. Wesley Hegden, who works on probability and discrete mathematics as an associate professor in the Department of Mathematical Sciences. Alongside key faculty here at Carnegie Mellon, Dr. Hegden developed rigorous ways of evaluating political maths for gerrymandering. Based on these methods, he testified as an expert witness in the recent lawsuit, which resulted in a new congressional district in Pennsylvania. In February 2018, the Pennsylvania Supreme Court found Pennsylvania's congressional district to be an unconstitutional partisan gerrymander, leading to the adoption of a new map for the 2018 midterm elections. In this talk, Dr. Hegden will discuss the evidence that the court used to reach this conclusion. Specifically, he will discuss a theorem that allows us to use randomness to detect gerrymandering of congressional districtings in a statistically rigorous way, and more generally, the ways in which mathematics can hope to productively interact with the law for years to come. At this point in time, I'd like us to give a warm welcome to Dr. Wesley Hegden. Okay, thanks. Thanks so much for the invitation. I'm really excited to talk about math and gerrymandering, two interesting subjects. Right, so I want to start off just by introducing as background this thing, the House of Representatives, which is somehow central to this whole concept. Like, what is this House of Representatives that all this gerrymandering has to do with? So the House of Representatives is one of the two major legislative houses in the country, and it has 435 members currently. It's not, it hasn't always been like that, but currently there's 435 members. And so the interesting point is how these members are chosen, right? So in the United States, what happens is there are 435 individual districts throughout the country, and individual elections are held in those districts to elect these members, right? So this is opposed to a more parliamentary system where you could hold a nationwide election and then assign representatives based on the proportions of the vote that each party got. So that's not what we do here. Instead, each state has a certain number of districts proportional to the population. So for example, Pennsylvania currently has 18 districts, and individual elections are held in those districts. So these districts are redrawn every 10 years in response to the U.S. Census. So like I said, the number of districts each state is apportioned is proportional to the population of the state. So this is why the census exists. And now also the districts are required to be roughly equal in population. So even when the number of districts that a state is apportioned doesn't change, they still redraw the districts so that the districts remain roughly equal in population. And so the important thing to notice is that there's nothing about this procedure of holding these individual 435 elections that guarantees any relationship between the votes cast for each party and the seats won by each party, right? There's no reason why if the two parties each get exactly 50% of the vote, that they have to get exactly 50% of the representatives, okay? So that's not how, there's nothing about this system which guarantees that. And in practice, all sorts of crazy exceptions to this property that you might hope holds do occur. So I'm just going to give a few examples. So one is in Pennsylvania here in 2012, Democrats won a strict majority of the vote. But out of those 18 seats that we have for Pennsylvania, they won only five, right? The Republicans got the other 13 seats with the minority of the vote. And actually so, I mean, this has been a remarkably stable result, right? So every election since this map after 2010 was implemented, there have been exactly five Democrat seats and 13 Republican seats until the recent Conor Lamb special election. In 2012 in North Carolina, the same thing happened. The Democrats won a strict majority of the votes, but again, won a small minority of the seats, only four out of the 13 seats with the rest going to the Republicans. So this is a slightly different example. If you look at Wisconsin 2012, but don't look at the U.S. congressional districts assigned to them. Instead, look at the state legislative equivalent of the House of Representatives, right? So there's some state legislative body equivalent to the House. That has 99 seats in it. There are 99 districts throughout Wisconsin. And Democrats won a quite strong majority of the vote in that election overall, 53 percent, but got only 39 out of 99 seats. It's a really big difference between, you know, the vote count and the seat outcome. 2012 was a crazy year overall because the Democrats actually won a plurality of the nationwide congressional vote. They won more votes than Republicans. But if you look at the final seat count, the Republicans got 234, Democrats 201. Notice that adds up to 435. There were no third parties here where I'm just counting Republicans and Democrats. So the Republicans had a strong majority of the seats, even though the Democrats got more votes. Now, when you see these results, it seems, I mean, at first glance, if you haven't thought much about gerrymandering, it should just seem weird that it's even possible, right? You wonder, like, how can this happen? And we said, you know, if you go back to how we said these representatives are chosen, they're chosen in these individual 435 elections. So if we think about Pennsylvania with this 5 out of 18 seats, there were 18 elections in Pennsylvania in individual districts, and Democrats only won 5 of them. So, okay, it seems natural that somehow we should go and we should look at these districts where these elections were held, okay? And so this is the congressional map of Pennsylvania that was implemented after the 2010 census, and it has been used since then for every election, congressional election that's happened so far. And, okay, so that's the district we're in right now. The most famous district here is this District 14. It's often called the Goofy Kicking Donald Duck District. So, like, this is Goofy Kicking Donald, right, Donald Duck's on the right, okay? Once you see it, it's pretty good. So, yeah, it's very strange. And when you see this map, right, you wonder, like, why would a district ever look like this, okay? And so let me, let's just talk about, like, why could a district ever look like this? Well, so this part of the state down here near Philadelphia, let's say, is generally Democrat-leaning, because it's close to a major city, and generally cities are more Democrat. Suppose, just for the sake of argument, that you wanted to carefully draw a district in which was favorable to the Republicans, okay? And you somehow wanted to get a district down here in the southeastern Pennsylvania, which would vote a Republican representative into office, even though generally this region is Democrat-leaning. What would you do? Well, you could go around to the different parts of the state, based on what you know about how people there have voted in the past, and try to cobble together different Republican-leaning areas from this region, and connect them into one coherent district, and then have that elect representative. And you imagine if you do that, if you're trying to paste together these different regions that are just, you know, just Republican-leaning enough, you might get some really weird shape, just like this, okay? And so this is exactly what gerrymandering is. It's where you go and carefully draw the boundary lines of districts to favor some political party or some political goal, okay? Like you set out with some goal in mind, and then you go and you try to make the districts work for you in that respect, okay? Now, okay, as was said in the introduction, actually we have a new map now for Pennsylvania, so the map no longer looks like this. So for the next election, we have this map. The map has changed, okay? And so how did this map change? So, right, so there was this lawsuit brought by the League of Women Voters against the Commonwealth of Pennsylvania, and there was a trial this past December. The Supreme Court ruled initially in January that the district was an unconstitutional partisan gerrymandering, and then the legislative respondents filed for an emergency state at the Supreme Court, which was denied on February 5th, and then the Pennsylvania Supreme Court adopted a new map on February 19th after taking suggestions from apps and not liking any of them. And then finally, the legislative respondents, which were the defense in the case, filed one more time for emergency state at the Supreme Court, which was finally denied on March 19th. So as of March 19th, this issue is somehow finalized, and now we have this map. I mean, I should say it's very recent, right? This all changed very quickly. And so I was an expert witness in this case, and I have to say I don't remember off the top of my head when they first contacted me about possibly being an expert, but my experience was generally like for like, you know, almost a year. I was like, is this case really happening? I don't know. There are these people that call me every, you know, every couple of months, and then suddenly, boom, it all happened in a week, and it was, you know, very fast moving. But from what I understand of the law, I think that's a little bit atypical. So this was heard on a really expedited schedule. Okay. Now the question is, in a trial like this, when you're, you know, suing the state, saying that there's a gerrymandering map, what kind of evidence can you hope to bring that the map is really gerrymandered? Okay? Now when I first, when I pose this question to you, at first I think a natural reaction is what kind of evidence do you really need? And when we saw this map, apart from looking ridiculous, with a 50-50 vote split, Democrats only get five seats. Isn't that evidence enough that it's gerrymandered? I mean, that seems kind of obvious. Case closed. That's a very natural, intuitive way to evaluate a map for gerrymandering, just look at the election results, and it seems to correspond to exactly what we care about. So there's a subtle thing that's wrong with this. So right now, the current legal framework under which people are suing states for having gerrymandered districtings is that we should throw out a districting if it was intentionally drawn to advantage one party. So certainly this is a districting which favors the Republicans. But if it does so just sort of by accident, because that just happens kind of naturally with districts in Pennsylvania, the current lawsuits are, you know, wouldn't necessarily argue that we should throw that out. And in particular, I want to consider, let's consider a quote, rend a map of Pennsylvania. So this is a map just drawn by a computer program that just, you know, draws 18 connected districts of roughly equal population. For this rend a map, if you simulate a hypothetical election using historical vote data and see how the Democrats and Republicans do in an election in this districting, with a 50-50 vote split, it's not 9-9 still, it's 7-11. So the Republicans still do better than the Democrats, even with this districting which was not at all drawn by some, you know, quote, evil bunch of politicians just trying to serve themselves. So what this shows is that it is quite possible for there to be some inherent bias in how the political geography of the state influences the, you know, the properties of districtings. So to see an example of this, let's look at that map of the, map of, this is the district of Pennsylvania that has been used for the election since 2010. And here what I've put in is the partisanship of each district measured from a race in 2010. So it's a race from 2010 because that was one of the last races before this map was drawn. So the red numbers indicate a Republican-leaning district and the blue numbers indicate a Democrat-leaning district. Notice that there's this district in Philadelphia here that's 90% Democrat. That's amazing, 90% Democrat. When you first see that, you might be like, wow, that's a great district for the Democrats. No, it's a terrible district for the Democrats, right? Because that's a district where the Democrats have all of these extra people voting in that district that they don't need. They could be in other districts that are close. Like here you have this district that's just barely Republican, 54%. Imagine if some of those 90% of Democrats were in that district. You could completely change how things are done. Now, if you look around, there's no Republican district that's 90% Republican. So just by this crude metric of where, how the voters are packed into districts, the Democrats are much more packed in this map. I mean, the second most Democrat district is also 79% Democrat, right? Okay, so when you look at this, it seems natural. Okay, it's obvious what happened. The Republicans went and they tried to pack the Democrats into a small number of districts and then create these weird-looking districts that would pull together some Republicans so that we could have some Republican seats. But then why is this random map still have a Republican bias? Well, you can imagine that this kind of packing could happen naturally more for one party than for another. So in particular, this district here, this 90% Democrat, I would challenge you to try to draw any district, just one district by itself in Pennsylvania that's 90% Republican. The point is there aren't actually 90% Republican regions of Pennsylvania. There's like 75% Republican regions, but 90% Republican. So there are more extremely Democratic regions than there are extremely Republican regions. And as a result, it's much easier to have a district pack Democrats and pack Republicans even by accident, even if you're not trying. Now, I'm not at all suggesting that what's going on in Pennsylvania is that this map happened by accident. But what I'm pointing out is that when you go and you argue this map is gerrymandered, what's going to happen, you know, in a trial, there are two sides. And the other side is going to come up and say, no, no, it's not. Look, just by accident, you can have an imbalance in the number of seats. And then you have to have some reason that you know that that's not what's going on. You have to have some way of rigorously arguing, no, in this case, yes, you're right. In principle, it can happen by accident. There's some bias. But for this map, we can tell. So that's somehow the goal. So this was about using election results to infer gerrymandering. What's another thing that we could try? Well, let's go back to Goofy Kick and Donald Duck. This seems like another obvious way of telling that something's going wrong. I mean, the shapes of these districts are just absurd. Isn't that evidence alone that something's going on? And to some extent, I think this is, you know, a powerful piece of evidence in the sense that it's very, I mean, you know, it feels very reasonable. If the district lines are crazy, it really suggests that somebody was trying hard to accomplish something. That said, it doesn't say what they were trying to accomplish. It doesn't tell you whether the Democrats were doing it or the Republicans. It's not quantitative. And you might worry that there could be other reasons for the districts looking really bad, although I'm not going to try to present any right now. Okay? But a really crucial problem with just worrying about shape to detect gerrymandering is that it can miss gerrymandering by a lot. Okay? So I'm going to show you three maps now. So remember I said that after the state supreme court ruled that the district was unconstitutional, they invited suggestions for maps. Okay? And so the two sides in the case both have been in maps. And I'm going to show you three maps. One was drawn by the legislative respondents who were defending the original gerrymandered plan. And they submitted a map which was again quite gerrymandered. It was again a 513 split. Okay? I can't believe the court didn't accept it. Right? And then there's two more maps in this list that I'll show you that were drawn by a computer, not by a computer program of mine, but by somebody else's that was just drawing districtings, just trying to minimize county splits and ignoring any partisan data. Okay? So these are the three maps. Map A, map B, and map C. Okay? And I guess the point here is that when I look at these three maps, it's not obvious to me that one of them looks really terrible. They both have kind of reasonable shapes. If you look at, and so, I mean, you could ask why these shapes are like weird at all. And a lot of it is just that they're following county lines, which is a reasonable thing to do. Okay? So just by eyeballing these maps, nothing stands out. But remember I told you that one of these maps is a 513 map. It's basically as bad as the old map. And I don't think you can tell just by looking, but it's this map. So map A was this proposed replacement. It's just that Gerrimander is just as bad as the other one. So the point is that you can carefully draw a map and maintain nice shapes if you want. Before, they weren't trying to do that. So they were willing to use bad shapes to Gerrimander also. And then, yeah, they got crazy shapes. But if you want to maintain nice shapes, you can do them both at the same time. Maybe your Gerrimander will be a few percent less effective, but it will still be pretty crazy. So just to summarize, we shouldn't infer Gerrimandering just from election results. Right? And that's because the seat vote relationship could, in principle, favor one party just because of the political geography of the state. So if we have to show intent to Gerrimander, then that's not enough. Okay? And on the other hand, we don't want to infer Gerrimandering just from the shapes of districts. And the most important reason here is that if you just worry about the shape, like if we're going to fix Gerrimandering and we're going to do that just by passing a law, the shapes have to be really nice in this way. And we specify something really carefully. People just come back and Gerrimander with those shapes. Right? Like if you could make a rule, the shapes all have to be triangles. And you could still do it. Right? The point is the shape is not that much of a constraint because Gerrimandering works on basic principles, splitting up cities, packing cities, right? And you can do those with all sorts of shapes. So we're going to do something different. Okay? Instead, we're going to try to get at, so legally what is currently the heart of the question of whether something is Gerrimander, which is whether something was intentionally drawn to favor one political party. Okay? So our test will be based on the goal of trying to evaluate whether a district is somehow carefully crafted to give a political advantage to one party. Okay? And we're going to do this using randomness and math. Okay? So the first thing I'm going to do is tell you how randomness comes into this. Okay? We're going to have districtings. We're going to be doing something random to them. What could that be? Okay? Yeah? Was that a formal definition of Gerrimandering that the court would accept? No, so I am not a lawyer. So I cannot give you any, like, formal legal statement and claim the court will accept it. But so when I testified, this is, I was testifying that the districting was carefully crafted to ensure an advantage for the Republicans. And then that went into some complicated legal arguments that I shouldn't pretend to understand, but somehow this was a relevant point. Yeah. Okay. So this is a legislative district in Wisconsin. So this is that division of Wisconsin in 99 districts. Okay? So this is like their state's equivalent of the House of Representatives. Notice again, the shapes of these districts really aren't that bad. Actually, is it possible just to turn off the front lights? Does somebody know? I feel like it might be easier to see this. But if we don't know how, that's okay. Yeah. Okay. Yeah, maybe that's, I don't know. I don't want to do it because then it, I don't want to do it because that's my fault. Yeah. Okay. Okay. So anyways, so this is this legislative district of Wisconsin in 99 districts. And I'm going to make a small random change to it. Okay. So here's the change. What did I do? Okay. So if you look very carefully, or if I just give it away. So what I did is I switched something right there. So these little blocks in this, in this districting are precincts. They're voter precincts, which is the smallest unit, geographical unit of the state for which we know how people voted. Okay. So it's the smallest unit for which we can, you know, reliably conduct a hypothetical election. So what happened here is I randomly chose a precinct on the boundary of two districts and switched which district it was a member of. Okay. Now this one change doesn't change the districting a lot, but you can imagine doing this over and over again. Okay. So in practice, when we apply our method, we make trillions of changes to the districting over a very long period of time. So here there's something like hundreds of changes that you can't see because the precincts in the cities are actually very small. The precincts that you see are just a big row of precincts. And as these changes are made, the districting evolves over time. I mean, it takes a long time to change to be very different from how it started. But, you know, what you're doing is you're kind of, you're taking the districting and you're just massaging a little bit, you're kind of fuzzing it up. It ends up a little bit different than it started. And right. So when we make these changes, of course, we have to make sure that we don't somehow mess up the properties that we think the districtings should have to have to be about a comparison map for the original district. So in particular, we require that the districts remain contiguous. We require that the districts remain roughly equal in population. We require that the district states are somehow reasonable in some way that, you know, you can quantify this in different ways and try out different ways and make sure it doesn't affect your results. And there's other things you can worry about. So for example, I mentioned like following county lines. You can require that, you know, where possible you respect other political boundaries you can try to care about conforming to the voter rights act. You know, there's various complicated constraints you can impose on the process. But the point is once you have some list of constraints to impose in your districts, you can run this procedure just by choosing random precincts on the boundary of two districts and making the swap if they preserve the constraints that you've imposed. So I said that in this video, you're doing something like, you know, hundreds of changes a second. It's much slower than the algorithm that actually runs. So when we run the algorithm, it does something like, we look at a sequence of something like a trillion maps. So you'd have to watch this video for like a thousand years to see a full run of the algorithm, right? It's slowed down to be able to watch. So we see a ton of districtings. And what we do is for each of these districtings, we conduct a little hypothetical election looking at how people voted in the past to see how each party would do with that map, okay? Either in terms of the number of seats or in terms of some other metric that captures how, you know, which party that map favors, okay? And when we do that, the thing we pay attention to is how many of these maps that we see are as bad as the first map, okay? So the idea is we start with our map, we make the sequence of changes, and what we're looking to see is does making these small random changes to the map make it fairer, yeah? That's a very good question. So a natural thing to try to do is to just generate like random maps from scratch and then use those as comparisons. So the problem, or let's say a problem with this is that if you care about being completely mathematically rigorous, we don't have an algorithm to generate maps where we know in terms of actually having a mathematical proof that in a certain amount of time we have a random map from some distribution. Now that doesn't mean that you still shouldn't try to do this and there are people that evaluate gerrymandering this way. They use Markov chains to generate random maps and they use heuristics to test that they seem to be getting random maps and I think they're very careful and I think they're doing a good job and I absolutely think it's a reasonable thing to do. But the advantage of this method is you have some extra level of certainty where we're going to have a proof of a theorem that justifies our statistical claims and so we don't need to rely on saying that well we think we've run our process for long enough for this to work. But yeah, it's a good question and it's absolutely a reasonable thing to do. If I had a computer program that could generate a random map and I knew it's properties, that would be better. I would use that, that would be great. But somehow we don't perfectly have that. Okay, good, so let's go and look at results for Wisconsin. So I said when we do this, when we make the sequence, we see trillions of distributions of Wisconsin. And for each one we evaluate whether it's as Republican leaning as that first map. And so we can ask what fraction of these trillions of distrings are as Republican leaning as the first map and for Wisconsin it is, I always have to count the zeros. But I think it's 200 millionths, maybe somebody, okay if that's wrong maybe correct me. But I think it's 200 millionths of the maps or as bad as the first one. It's pretty extreme, okay? For Pennsylvania it's similar, right? I mean it's four, 10 billions. So again it's a tiny fraction of the maps in our run or as bad as that first map. For one of the runs of Pennsylvania actually there's this amazing thing where like each step is you swap a precinct from one district to its neighboring precinct, right? And you do that then trillions of times. And for one of the runs we did, right, because for our analysis in the case we did several runs using different constraints. Remember I said you can have, you can list your constraints and then do runs with those constraints. We used several sets of constraints to show that the results were robust to the choice of constraints. And for one of the runs the first random swap made the districting a little bit fair and then it was never again in the trillions steps as unfair as the first district thing. So like one little move away and it was already fair forever. And so like I think that, yeah so what I said in the test one was that like it's almost like you can hear in the back of your mind you make that first change, you can hear the map maker saying no don't change that. That is exactly how we want it, right? So like somehow that's the way that this test is getting at this idea of being carefully crafted, right? Like do these small changes destroy the partisanship? And so here's the thing. Okay let me go back. So we have these crazy numbers, right? It's super gerrymandered because such a small fraction of these maps were as bad as the original map. But okay I'm proposing this, I'm saying look this is a great gerrymandering test. But we started out this talk with the great gerrymandering test which was look at the election results, okay? And if you see a 513 split with a 50-50 vote that seems like a good gerrymandering test. It's easier to understand arguably. So why is this better than that one? Remember the problem with that one was that it was possible for just a random map to fail the test in some states. Like for Pennsylvania it seems like random maps fail that test, which is a problem because then you can't really argue that there's an intent to gerrymandering just because you failed the test. So the question is, does this test have that same problem? Right, I've described the gerrymandering test which is you take your map, you do your random sequence of changes, you look at overwhelmingly the things you get are fairer than what you started with and if that's true you call it gerrymandered. Now is it possible that for some state with some really weird political geography, you know people live in weird places, actually typical maps fail the test? Is that possible? Okay, if that's somehow the question. Remember we had this random map. Like could this map, just a random map, tend to fail the test? This is the question. Could it be that random districts of the state have this pars and bias which goes away? And so then this is the map. So we prove the theorem, this is with, so this is the theorem, the theorem says no, this is not possible. In general the imprecise statement of the theorem is that if I have a typical object and I make a sequence of random changes to it, it shouldn't change in a consistent way. And this is true for districts but also in all sorts of settings. In a very general setting, you can make this statement that a typical object shouldn't change consistently under random changes. So this theorem was proved with Maria Chiquina at the University of Pittsburgh and Alan Fries also with me in the math department at CMU. And so this is what we apply then to this test, right? Because this test is, this is exactly what we did. We took an object, we made a sequence of random changes to it and we observed that it got consistently, it got different in a consistent way. It got consistently fair. And so we conclude that this thing we started with was very atypical. And of course, so this theorem gives you a formula to make statistical claims in a precise way based on what you observe. So for example, for Pennsylvania, we said that only a four, ten billionth fraction of the map's encountered by the test were as parsed as the original map. And so what that meant in terms of what we could say rigorously in our report was that the chance of this happening for a typical district of the state would be at most 300,000s, right? So if I just picked a district out of the bag which satisfied the constraints that we tried, then it would be extremely unlikely to look as bad as this, okay? And now this is a rigorous statistical statement that doesn't depend on having a way of drawing truly random maps. And, right, okay, so, and that's what you want, okay? So just to emphasize what is surprising about this, I'm claiming that this works without looking at every district in the state, right? Like, if you think, so maybe this is not obvious, but there's a huge number of possible districtings of a state, right? Like, depending on exactly how you discretize the state, it could easily be more than the number of elementary particles in the universe, okay? So there's some astronomical number of districtings of the state. You cannot hope to enumerate all of them, okay? This method also works, and this was the question before, without really looking at random districtings of the state. We don't put our hand in the bag and pull out a genuinely random map from scratch, okay? And the reason we don't do that is that we don't actually have a completely rigorous way of doing that. And, you know, we're very paranoid, so we only do things that we have completely rigorous ways of doing, okay? And in particular, the method claims to work, and this is maybe, should really be emphasized, without claiming to use anything about the political geography of Pennsylvania, right? Like, this method, I'm saying I just reply to any state. This point is really important, because when I was on the stand, the lawyers for the other side really tried to make a big point about the fact that I was not an expert in political geography, which I was very happy to admit that, right? Okay. But they seemed to think this was a big deal that I had no knowledge of political geography. But the point is that this method doesn't depend on that. You don't have to know anything about where the people live, right? Because this theorem that we proved is very general, right? So a typical object almost of any reasonable kind, obviously you have to set this up in this mathematical framework, shouldn't be changed in a consistent way by random changes. But then how is this possible? Like, how can you have this test with these properties? It seems kind of crazy. Like, how can I know that something really is unusual in this way with respect to everything? If I didn't look at everything, I'd even look at random samples of everything, and I don't really know that much about the whole set of possibilities. It seems kind of, it seems kind of crazy, okay? So at this point, I want to just kind of tell you a story about restaurants to kind of try to give you some intuition for why this is possible. Okay, but to nail it down, the question is like, how can we identify that our discerning is typical compared with everything else when we haven't looked at it, and we don't know that much about this space overall, okay? So here's a slide. There's not much on the slide, so I'm just going to describe this hypothetical scenario, okay? So the scenario I want you to think about is flying to a new city that you've never been to, that you know nothing about, and trying to get something to eat, okay? So the scenario is you land at the airport, you go to the taxi stand, you get in your taxi, and you tell the taxi driver, okay, I'm starving, I want to go somewhere to eat, but here's the important thing, I want you to take me to just this typical restaurant, because I've never been to the city before, I want to see what it's like, I want to get a feel for what the restaurant's here like. And your taxi driver says, absolutely, let's go, and he drops you off somewhere, he says, this is a totally random typical restaurant, you give him a really big tip, you go in, you eat your meal, and it's just disgusting, it's like the worst food you've ever had, okay? So before continuing, in this analogy, the taxi driver is like the map makers, it's whoever made the map, okay? This person in the taxi, this is like the people, okay? The restaurant is the district thing, right? The taxi driver chose this district thing, and he told us he was just going to give us a typical district thing, and instead, it's terrible, it's awful and we hate it, okay? And so we eat this terrible food, and then we're really mad at the taxi driver, we can't believe that he did this to us. But then we start thinking we're worried, well actually, wait, maybe we can't be mad at the taxi driver. For all we know, the restaurants in the city are just terrible, right? We don't know anything about the city, okay? That's somehow the problem. How can I rigorously say, yes, the taxi driver was not being nice to me, okay? And so at first, it maybe seems like you can't do anything, because you don't know everything, the city's huge, you can't enumerate all the restaurants, we don't know anything about typical restaurants in the city, we're not a city planner, right? We don't know what restaurants typically are like, okay? But we do something simple, which is we just start wandering around this little part of the city that we're in. We start looking at the restaurants near the restaurant that he dropped us off at. And what we observe is that not only is this restaurant that he dropped us off terrible, but the restaurants surrounding it are all better. And now we're really, really upset at the taxi driver, right? Because, well, it's possible that there are cities where all the restaurants are bad, so he takes you to a typical restaurant and it's a bad restaurant. There are no cities like even hypothetical made up cities where all the restaurants are bad restaurants surrounded by good restaurants, right? That's not possible even in principle, right? You could try to draw one, like try to, like, put down the restaurants and pick which ones are good or bad. And no matter what you do, you can't make a city where typical restaurant is a bad restaurant surrounded by a sea of better restaurants. Okay, and that's exactly what we do, basically, right? We start with this district ring, this terrible district ring that we were given by the taxi driver, by the whoever drew the district ring, and we start making these little random changes to it, and we observe that these districts that you get in this neighborhood of this original district ring are fair than the one we started with. And so this means exactly that, okay, this was not just typical district ring. Some weird process chose this thing, okay? Okay, and so that's how this works. Okay, how much time do I have? Okay, I have, like, 10 minutes or something? 15. 15 minutes, okay, great. So then there's the next, so, okay, this figure, I don't know how useful this is, this is supposed to be, like, the bad restaurant in here, all the food is green or something, and then, like, you wander around and you notice that, like, all the food is better around it, which is for some reason represented by red, and, right, and it's not possible for this to happen for a typical restaurant in the city, okay? Now, next, okay, so that's sort of, like, evaluating gerrymandering and, you know, how you bring this courtroom and how you talk about evidence that a district ring is gerrymandering. So the next question is, okay, so what do you do about this, like, more broadly? Like, how do you fix gerrymandering more generally? And, okay, so here I know a lot less, but I'm going to talk about it in any ways, okay? So the first thing that you can do is, okay, well, there are these lawsuits. There was this lawsuit in Pennsylvania that was very successful. There have been lawsuits in other states which are, okay, so no other maps have changed in this cycle, but there's some potential for that to happen. In particular, a key difference with the Pennsylvania lawsuit versus the other lawsuits is the Pennsylvania lawsuit over the state constitution. The point is that the argument was being made that the district ring was unconstitutional under the Pennsylvania state constitution. And the important point about that is that the U.S. Supreme Court doesn't review that decision. So I said, like, the Supreme Court was denying those emergency stays, and that's not necessarily because they agreed that the district ring was gerrymandered or that was bad, but just because they were saying, we don't have jurisdiction over this question. And so there are other states where you might imagine that similar challenges to gerrymandering might work, but there are problems with this lawsuit approach. So in particular, lawsuits probably will only be able to address the absolute worst cases of gerrymandering, where it's really clear that the district ring is extremely gerrymandered. I think it's probably unlikely, just in my personal opinion, that any judges will want to weigh in on the question of whether some close case should be thrown out. In all the cases so far where maps have been thrown out, you have experts testifying that the district ring is gerrymandered, and on the other side, you don't have any experts testifying that it's not. So for example, in this case that I was in, there were experts on the other side, but they were just there to kind of try to, like, you know, poke us. They were just kind of trying to call us names and stuff, okay, or like say, like, they were there to cast doubt on the general idea of deciding that something is gerrymandered. Nobody ever does an analysis in these cases and say, look, I did my analysis and it's not gerrymandered, because these are such extreme cases that's not possible. But once you get to close cases where you could really kind of try to have arguments on both sides, I think it's very hard to imagine judges wanting to be the ones making the call, okay? So what else can you hope to do? So there's another idea, which is independent commissions. So this is an idea which is used in several states now. So for example, California has independent commission, Arizona has independent commission. There's a movement to get independent commissions in Pennsylvania. The idea here is that you take the role of making maps out of the hands of politicians, and you give it to some sort of non-Parisian body. And typically this body is somehow assembled, actually to some extent randomly, through some algorithm, which chooses people based on their partisanship. And yeah, so it's an interesting idea and it's being used in some places and it seems to be working better than just having one party do everything. But there are interesting criticisms of independent commission. So one that I heard only recently that really made me think was that in some sense you can view them as undemocratic. So when I first heard this, I thought it was a little bit funny, but remember I said that you're going to have some non-Parisian commission. I mean, what this means is we don't elect the commission. The commission is somehow chosen by some procedure, which involves randomly selecting people. And so there are interesting questions about how accountable the commissions are. Like if they started, so far it's working great. If they started doing a bad job, who are they accountable to? It's kind of an interesting question. But in any case, mostly what I want to talk about is, is there anything else you can do? And so for this, I'm going to talk about some work that I did with Ariel Procaccia here at the Computer Science Department at CMU and Dingling Yu, who was visiting here when we did this work. So the idea here is, so there's a theoretical question we were interested in, which is, is it possible to somehow leverage the competition between the two political parties in a state to generate a fairer district thing? Is there something we can do other than just having one party do everything, or having an independent commission that we generate somehow? And so one thing that this name should remind you of is I cut you choose. If you think back to children dividing up a cookie, if you have two children and they have to split a cookie or a piece of cake or something, they know that while one of them should cut it and the other one should pick. And a simple analysis shows that if you do this, they both get at least half of what they want. And so notice that that protocol doesn't require the children to be honest or to cooperate or play nice or anything. They can both be very selfish and just want as much of the cookie as possible, but still they both get some fairish outcome. And so the question is, can you do something similar for district thing? So here you're not trying to divide up the state into my part and your part. You're trying to divide it up into a district thing, which we'll both use. But can you somehow have some protocol through which we'll agree on a district thing through competition and still have it somehow have some reasonable properties? So this is the protocol that we came up with. This is a protocol to divide the state into K districts. So in Pennsylvania, K would be 18. So first, the first player divides the state into 18 districts. Just boom. That's if he does it. And you would have to respect any legal requirements in the state. If you have to follow county boundaries to the extent possible, whatever the rules are, you would follow those rules. And now, right now in Pennsylvania, that would be it. That's how you would have the district thing. But in this protocol, what would happen next is you would give that district thing to the other player who would have to freeze one of those 18 districts. So they would look at all those districts and pick their favorite one, maybe, and freeze it. And then they would get to redraw the other 17, however they want. And then they give the map back to the first player who from those 17 freezes one and redraws the remaining 16. So now we have two frozen districts, one from each player. And that continues. So just to see an example. So in this case, we can divide up the state, which is just a rectangle. And normally, like the two players have some goals related to, you know, wanting to win a lot of seats. But to make something which sort of geometrically you can interpret, let's assume that the blue player wants the state divided into horizontal districts for some reason. And the red player wants the state divided into vertical districts for some reason. Okay. I don't know why this would ever happen, but let's just assume this for the sake of making a figure. Okay. So the way this would work, so suppose we're dividing this into five districts and the red player is going first. So first, the red player would divide the district thing into five districts. And if they're hoping for vertical districts, they might divide it like this. Okay. Now blue can freeze any of these districts. So if they, apart from wanting horizontal districts, have some secondary considerations, they can pick which one of these they like the most. Okay. So maybe they really don't want a vertical district over here or something. So, okay. So maybe they'll freeze this one. Okay. And after they freeze this district, they get to divide up the rest into now four districts. Right. Because one district is already there. And red gets to pick which one of these to keep. And then divides up the rest. And blue chooses one of these. Okay. And this keeps going until the, we should notice at the last stage there's really no choice. Okay. The last stage just gets chosen. And at the end you have some district thing where each district was drawn by one player and sort of accepted. Right. Because they froze it by the other player. And notice that each player got something, kind of some of what they wanted. Right. There's some horizontal districts and some vertical districts. Okay. So what can we say about this protocol? So we have this paper where we analyze this in, you know, various simple models of how a state works. And we show that there's a sensible relationship between the seats that you win and the votes cast for each party. Unlike for a district thing in general. Right. So for example, if I have, if there's a state where one party has 51% of the vote, it's possible for them to win every seat. Right. If the district thing is such that in every district, they have 51% of the vote, they can get every seat, which is pretty crazy. So we show that in an after model of the state, if you use this protocol, then you won't have that. There's some nice relation that you can determine between the seats and the votes. And the other thing that we showed is that either player can prevent packing of voters. Right. So suppose, for example, that, you know, the blue player really is trying to pack a group of voters right here into one district. We show that the red player can prevent them from doing that. They have a strategy to make sure that this population won't be packed into one district or even into a few districts. Okay. Now, okay, this is all sort of abstract though. I'm not necessarily proposing this as a concrete solution that we should implement tomorrow. For that, independent commissions are a much better developed option. But it's interesting just to think about abstractly what the advantages and disadvantages of this are. Right. This is another example of sort of math interacting with policy. So what are some problems with this method? So one problem that really stands out is that it really is based on a two-party system. Right. So at least the analysis that we've done. So we haven't analyzed the case of more parties. Right. So this is based on there being two parties, they're going to fight with each other. A lot of existing independent commission systems are also based on this idea in the sense that, you know, the commissions are assembled, some from the majority party, some from the minority party, and then some independence, which are Democrats, Republicans, and independence. But still, this is a real criticism. Of course, so an interesting thing about this is, so currently there are no states, I think where there's a third party which is so powerful that it would seem to have a reasonable claim to participate in the districting process, but you can imagine that changing. However, one interesting question is like, should both parties even always have a say? Right. So this always gives both parties somehow an equal seat at the table. If I take, you know, a very red state like Utah, where the voters don't think the Democrats represent their interests, should the Democrats have equal say in districting or vice versa in a Democrat state? It's kind of, I mean, that's a philosophical question that, I mean, I'm not prepared to answer, but, you know, this imposes an equal role for both parties, and it's not really, I mean, maybe there should be some threshold for that to be the case. Finally, I mean, a very important thing about this is it doesn't address what people call bipartisan gerrymandering, which is where people are carefully drawn districts for a goal, which is not trying to be better than the other party. It's like maybe to protect incumbents, right, or to somehow enshrine the old guard, okay? And this doesn't address that at all. And, I mean, basically the problem here is that it doesn't take district drawing out of the hands of politicians, okay? Now, what are some advantages? So, I want to start with districts are still drawn by politicians. I'm going to call this an advantage also. So, the reason I call this an advantage is that in principle, this means that you still are leveraging the Democratic system, right? There is, I mean, yeah, I mean, sometimes we forget this, but politicians are in principle supposed to be accountable, right? In principle, I say. So, this has at least theoretical advantage that it's still tethered to the Democratic system. And depending on how your independent commissions are set up, that might not be the case. It does address the balance of power between two parties, right? So, like, if you're going to have politicians draw the districts, at least fixes this issue of the seat balance. And something else that's nice about it is that it would be relatively simple to imagine implementing it. So, what I mean by this is there are other proposals to replace dishing with, say, a computer algorithm to draw the districts, right? Like, I'm going to write a complicated computer algorithm to draw the districts and then let's make that the law. So, in that situation, basically this whole program has to become the law, like this whole algorithm. And most people won't understand it. This is a protocol where you can actually understand it and it's relatively simple. And the reason is that it's still using the politicians, right? It's still using people. And in the same way, it also interacts with the existing legal framework, right? So, currently, there's a lot of law about what properties districts have to have. And since the politicians are going to still be drawn the districts in this hypothetical system, that can still all be respected. Now, that said, I still consider this, I mean, I don't know about Ariel, but I still consider this mostly a theoretical idea, which is interesting. I mean, the point of this, I think, is just to show that math can let you leverage competition to produce a fair outcome. That is, you don't have to sort of leave the big world completely. Like, I can just take these people that maybe don't like each other and are just trying to be selfish and have them carry out some protocol. And we can still somehow prove something about what can be the outcome of that procedure, okay? So, just to tie it back to the beginning, I said Pennsylvania has a new map, maybe the most important part of this talk. So, this map only matters if we use it, okay? So, midterm elections are November 6th. I think the absentee ballot is sometime in October, the deadline for that. So, you know, if you're out of town, you should look into that. In general, all this matters because this is democracy where we vote, okay? So, I think we should all make the map matter, okay? Yeah. So, that's it. And I'll take questions. Yeah, and then I'll come back. So, did you use your algorithm on this map? We did, actually, yeah. So, I don't remember the exact numbers, but this map did not come out as germanted when we did an identical analysis using the same data, the same metrics. It didn't come out as germanted. So, and we also, so those computer simulated maps I showed, maps B and C, also didn't show up as germanted in our analysis when we analyzed them. Yeah. Question. Your analysis in the first half of the talk relies on, I think, the assumption that both those cast independent of the maps. Absolutely, yeah. And that's a real, right. So, yeah. So, the question is like, when I changed the map, couldn't that change how people are voting? Because I'm using this historical voting data. And this gets at, I mean, a whole host of issues. I'm also just assuming that this past voting data is somehow reliable and like really capped. I mean, that was the data that we used as a 2010 Senate race. So, there's various reasons why that's a good proxy for the Parsonship guess in 2010. But absolutely. I mean, you could criticize the choice of data, how well will this correspond, especially when I changed the map. However, all of this should only make it harder to detect gerrymandering in the state. Right? So, the thought experiment is suppose that I do my analysis over and over again. And each time I use worse and worse data. Like, first I use 2010 data, then 2000 data, then 1990 data, 1980 data. Once I'm back to like 1900, there will be almost no correspondence between the voting data then and how people are distributed in Pennsylvania now or very little. And in particular, the analysis won't find that this districting is carefully drawn with respect to that old data. Okay? And so, the worse your data corresponds to the truth, the harder it is for there to be statistical significance in your outcome. Like, in particular, what this analysis finds at the end is that, to say it very carefully, is that there's this strong statistical significance with respect to how this map looks with respect to this particular data. So, there has to be some explanation for that. And it seems like the most logical one is, well, this data is somehow associated with the true citizenship for how people guess, people would vote. And so, you know, it's as a result of this careful drawing. So, I can't imagine another explanation for why there's a correlation between the SES-TAC-TUMI race and this map, other than this map was carefully drawn. Right? I mean, I'm the wrong person to ask this question of, but I never heard the judge talk about how likely it had to be for, yeah. But certainly, at the numbers that we were presenting, we're beyond any threshold he would have imagined. I think people are put on death row with lower probabilities of confidence, right? So, but yeah, I don't know whether people, yeah, think about this issue of what the correct probability threshold should be. Yeah? Yeah, excellent question. So, this is why, this is an important, right. So, an important property of the algorithm is that at every round, you have a complete districting. So, you always know it can be completed. Right? The first step is I draw it into 18 districts, okay? And then I give that to the other person who freezes one and gets to redraw the rest. Worst case scenario, he can always use my division. So, there always is a division of the rest, which respects all the legal constraints. There might not be many others, but there is always a completion. So, this is a, so if instead I use an algorithm where I draw one district and then you draw one district and then we have this problem. But that's not how this works. If I draw a complete district thing, you freeze one and then redraw. And since you are starting from a drawing of the district, you know there's at least one completion. That's a good question. So, I never really show those numbers because they depend to a large, they depend to a surprising degree on exactly what constraints you choose. Like how compact you require the districts to be. Are you freezing counties or are you not? You get different numbers of seats. So, the thing that's consistent is that you always get a district which seems fair between the Democrats and Republicans. But changing the geometric properties of the districts and how you constrain that changes this number. So, I can't really answer the question of how many seats you should get, especially because remember I'm not claiming that my Markov chain is mixing. So, I'm not claiming that I'm really sampling the distribution. So, theoretically I don't want to claim that, but even practically. So, I actually don't think you do. So, I think that there's a good chance. Yeah, so. So, here's evidence that. Yeah, so here's evidence that you're not getting random district thing. So, I don't have the pictures ready right now, but if you look at like one of the, at the end of the trillion steps, you look at one of the maps. What you'll see is that, okay, so let me at least go back to the Pennsylvania map, the old Pennsylvania map. Okay, so what you'll see is for example, so like this gray district here, after your trillion steps, there might still be a gray district down here. It will look, be a different shape, but in particular it's in the roughly the same space. If I had really mixed the Markov chain, I would expect sort of the colors to be in random places. Like I would expect really everything to have moved around. And you don't see that. So, I think that's actually good evidence that it's not mixing. That doesn't, it's not to say that you could, couldn't run the chain better. Oh, okay, I'm saying even if you don't freeze the counties. Let's say you don't freeze the counties. Let's say you just preserved population and connectivity. Yeah. Yeah, so the, I mean, the seats always change, but like we're talking about integers. Like is it two or three or four? Is there a change of two or three or four seats? And it's like all those answers occur, depending on how I constrain it, right? So, okay, so here's an important point, which is that there's not actually, so like the legal constraints on discharging aren't precise enough to lay out how you have to constrain the district, right? Because there's this rule that districts have to be compact, which somehow means that the, you know, they're not geometrically too crazy. They can't be a fractal. What does that mean? So it could mean a lot of different things. So there's different ways of constraining that. And so like one natural way of constraining that is by, let's say, the ratio of the area to the square of the perimeter. Or I could constrain it in other ways. And all those choices matter. And, yeah, so I think it's kind of, but anyways, we should talk about this more. Yeah. Yeah. The West, as you know, I have reservations about the eye-cut you choose. Absolutely, yes. Yeah. And for everybody here, let me just comment. That besides fairness, one of the other things that most reformers like to have is responsiveness or sometimes called competitiveness. You want to have districts that are likely to change as the mood of the country changes. You don't want to have them locked in so that you already get nine and nine. You want to be able to go from seven to 11 on if the Republicans have a wave and 11 vice versa. Right. So that's an important consideration, I think. Now, maybe you've changed it since the last heard of your paper that you guys wrote, but my analysis of that said that that cut and choose thing would essentially lead to districts that were not responsive. Yeah. So I would definitely say that our analysis is not, we don't analyze how competitiveness responds to this because I'm not even sure we have a good theoretical model for how to do that because it depends on your model for what the parties care about. It's not more about preserving their incumbents or more about having more seats in the other party. It's kind of, you know, it's a little bit subtle. I will say that, so our main comparison point for the eye-cut you choose, so like what is the current system in Pennsylvania? So what is the algorithm? The first player, the same, the first step is the same. The first player draws the districts and then the algorithm is over. That's it. That's the end of it. Right. And so compared to that, I think that there's no reason to expect, it's hard to even imagine how it's possible for what we propose to be actually worse on competitiveness than the existing system of just having one party do everything. So we're fixing one problem, which is seats. And I think it's true that, for example, in California... I haven't done any analysis since the paper that you gave me. No, we haven't analyzed... Like I said, I'm not sure... I did analyze that using theorems, and I came out and said you'll have districts that are either more red and more blue and you won't have very much competitiveness. Yeah, I mean... But that just, it's just an issue of, you know, we've come up with nice sounding and by the way, it sounded really good to me when you first saw it. But the problem is there's things, procedures which don't seem to have anything to do with fairness or responsiveness. Like, for example, the Pennsylvania Constitution says compactness and following district boundaries, not splitting counties and things like that. But the problem with those innocuous sounding procedures is that due to the political geography of Pennsylvania, you're going to always favor the Republicans. And so the new map, unfortunately, is going to give you seven Democrats and 11 Republicans with 50% of the vote. So what I hear you saying is that, yes, maybe this thing is better than having one party do it in terms of the seat performance, but there's some other way in which it's not better. And I would say that's almost certainly true. Probably competitiveness is not the only example of a way in which it's not better. There's probably other ways, too. But I think it's better in some ways. And I don't think it's worse in any ways in having one party do it. And the question is really what's the comparison? So I don't know of a silver bullet. Like, I don't know of a solution to gerrymandering that I can't attack. Like, I can think of problems with just about everything. And so the question is... Yeah, that's a good idea. Yes? In your algorithm or system, tell us about a district other than its voting tendencies. Does it tell us anything about the socioeconomic status of demographics? And one wonders whether or not certain districts ought to be defined more by those characteristics than by their voting path. One could see an area with a very large number of retirees with different sections of the retirement community actually having sort of different voting patterns. 70, 30 Republican in some areas and 64 Republican in other areas. While the borders of the 70, 30 say match up with the very rural area that also tends to vote Republican. But the socioeconomic status, the demographics of that other area that simply happens to have the same voting pattern as 70, 30 voting pattern as another adjacent area, those two get matched up. Whereas the retirement community gets split up simply because it's voting in different ways. I'd rather keep the retirement community together because that makes more sense as a district than caring about the subsections disambiguating the retirement community into its political affiliations. Right, so I guess you're talking about whether you should constrain the districtings that you look at by sort of how they split up these communities or something like that. Yeah, so that's something you could certainly try to do. One thing that makes me cautious about going in that road is that it involves just a whole bunch of judgment calls. And when you do this kind of analysis and you go in court and you say, this is what I did, I wanted to be clear that I didn't have a lot of choice in how to apply the method. I wanted to be clear that I couldn't cherry pick what I did to get the outcome that I wanted. Right, and if I can go and make the list, well I think this community should be preserved and this one should be preserved but that one's not a real community and this one should be preserved. That gives me a lot of freedom to influence what districtings will be a set of comparisons. And there's actually an interesting story so I don't remember where this article was published but if you look for this, it's not so hard to find. There was an interesting piece about one of the early redistrictings with the California Independent Commission the Democrats were able to hack the commission so in the following way. So the independent commission was made up of these citizens who I think nobody thinks that they were intentionally trying to rig the process but the Democrats did this ingenious thing which is that they sort of manufactured a bunch of these communities of interest and then had them go and testify before the committee. They said, so like we have this community of dog walkers that goes every Sunday and we're this very close knit group and so please keep us in one district and they went around and they just created all these communities and had them testify before the committee and as a result, the Democrats were able to achieve a lot of what they wanted in that district and they were preserving incumbents and all sorts of stuff. I mean, again, this is not my expertise but the point is that there's just a lot of selection and judgment that goes into these kind of calls that make me reluctant to implement it especially when it's not my area of expertise. If I do this, if I go and say, I'm going to preserve these communities of interest on the stand the first question I'm going to get is, are you an expert in the geography of Pennsylvania and its communities of interest? And I'll have to say no, I'm not, right? So that's a problem for me. No, so that's a good question. I haven't tried to do that. I have so, you know, one just sanity check that we did is so 538 at some point, the website 538 has this gerrymandering site where I think they actually by hand did this. They by hand created a bunch of Republican gerrymanders and Democrat gerrymanders and other districtings of states just as an exercise. And we did at some point, you know, run our algorithm on some of those to, you know, just as a sanity check. So, you know, the Democrat districtings look like they're gerrymandering for the Democrats and the Republican districtings look like they're gerrymandering for the Republicans. But no, we haven't tried to run an opt, like an optimization problem to find an extreme case. I mean, that would involve slightly different techniques but yeah, it's a reasonable thing to try. Yeah? On the topic of optimization problems, is there a way to like measure or redistribute? Yeah, so there's all sorts of ways that people do this. I don't know all the details. Like one thing that people talk about is like the likelihood that your vote will be the pivotal vote in a district. You can measure that. I mean, that's getting at the question of how competitive the district is, right? Like if you are a very powerful voter in the sense that your vote is very likely to matter, your district is very close. And yeah, and so I mean, I guess one answer to this question is that if you have something like that that you care about, you could analyze the districtings using our method with that thing that you care about. Like if what you care about is competitiveness and you want to show that your districting is anti-competitive, you could have some metric of that. And you could apply, so remember what we did, we looked at this long sequence of districtings and we evaluated how favorable it was to one party. Instead of doing that, we could ask how competitive is it and check, you know, if this thing that we started with was the least competitive or, you know, the bottom one billionth of the competitive districtings, then that would be evidence that it was, you know, drawn in some way which carefully reduced its competitiveness. Thank you so much to everyone. And thank you so much, Dr. Pegg-Luckhold. Thank you so much. Mary Kay Johnson, who's hanging out on the corner over here, who served as Carnegie Mellon's federal election like that far away. And you can do that. And so Mary Kay will be elaborating briefly on the first edition, Bill of Rights, that we have at the front of the room. So I'll turn it over to Mary Kay. Welcome and... Welcome? Is that okay? Back? So that night is for the live stream, so Mary Kay, if you wanted to come over here and use this mic. We have to bring that one in here. Welcome and thank you for making today such a wonderful success. Thank you, thank you, thank you. You guys are doing a lot for our statistics. As guests. Anyway, so in the Posner Center, this is a multi-purpose building, but part of it in the corner is a vault that contains the collection of the Posner Fine Arts Foundation. And one of the items in it is a very intriguing Bill of Rights. One of four copies known of this particular edition, which is a real dull government document where Thomas Jefferson as Secretary of State is communicating with the other state legislatures, saying this is how all of the other guys voted on those 12 amendments. Remember those 12? And all of the states are writing back, we don't like number one and two. That's just administrative stuff. Yes, we want the other 10. And that's what this document does. It's available to you online at the Posner Collection online. What you're seeing today is just page one of the real document. Again, thank you. So as we conclude our Constitution Day program, you are more than welcome to come on up to the front of the room and check out the first edition of the Bill of Rights. And we also have some refreshments out in the hallway. We have cookies and brown...