Episode Transcript
Representation Transcript
SH: So with apportionment, where did the U.S. start and where have we ended up?
IV: We started in a decent place and we're ending in a dumpster fire.
SW: Well that doesn't sound good.
SH: No, it certainly does not. But before we dig into that, let's introduce ourselves.
SW: Right-o!
SH: I am Sam Hansen
SW: And I’m Sadie Witkowski.
SH: And you are listening to Carry the Two, a podcast from the Institute for Mathematical and Statistical Innovation aka IMSI.
SW: The podcast where Sam and I talk about the real world applications of mathematical and statistical research.
SH: This is the second episode of our season all about the mathematics of democracy and politics. Last episode we talked all about voting and this time we will be digging into apportionment, or how states are assigned representatives
SW: And going by the voice we heard at the top, we have another similarity to last episode don't we?
SH: Good ears Sadie, you are right we are again joined by Ismar Volic, professor and chair of the mathematics department of Wellesley college, Director of the Institute for Mathematics and Democracy, and author of Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation
SW: He was a great guide to voting last time, so I am happy to hear from him about apportionment, although it does seem from the dumpster fire comment it might be fair to say he feels pretty similarly about both how the US does apportionment and how we handle voting?
SH: That would not be an unfair thing to say, but let's start with where apportionment was at the founding
SW: Ok, let's get into it then
IV: The apportionment was actually a big topic at the Constitutional Convention, right?
The founders, the framers, really talked about this a lot. They were trying to figure out a system by which a representative would represent a reasonable number of people. They would be someone who, you know, their constituents would be small enough in size so that a representative can have the sense of what they want, right? The representative can have the finger on the pulse of their constituents. So Madison in particular came up with some numbers and said, you know, we should have about 30,000 people per representative. He realized this may have to grow as the population grows. He said something like 50,000 would be, you know, as high as we should go.
SW: [laughs] Since we don't have thousands of members of the House of Representatives I am guessing we have gone a bit beyond where Madison thought we should land?
SH: Ummm, just by a little a bit
IV: So right now, there are 760,000 people per representative in the United States.
SW: Yeah, just off Madison’s recommendation by about an order or magnitude or so. Jeez
SH: To be fair to Madison, the population of the country has increased by more than an order of magnitude. Getting back to the current times, Ismar says asking one person to representent that many people is just not tenable
IV: It's kind of laughable. There's no way that a person can represent 760,000 people faithfully, right? Understand all of the concerns and desires and needs and political aspirations of 760,000 people. So there is a greater gap than ever between those who represent them. I'm talking about the House of Representatives with its 435 seats. There has never been such a huge gap between those who represent us and us who they're supposed to be representing. And, of course, any gap is very quickly filled by, you know, special interests and lobbyists, et cetera.
SW: Ok, I am starting to understand the dumpster fire comment
SH: Exactly, but remember apportionment in the states did not start out this badly
IV: As the population grew from the Constitutional Convention, so did the size of the House of Representatives. An attempt was made to keep this ratio reasonable.
SW: So what happened? Why is the ratio so large now?
IV: It was staying kind of reasonable until the 1920s where, because of the politics of the time that had to do with, you know, an influx of immigrants and, you know, rural versus urban divide and Republicans and Democrats. But whatever, it was the politics of the moment, and because of it, that the House size got frozen in time then. A legislation was passed, the Reapportionment Act, that froze the size of the House of Representatives where it was at that moment, and that happened to be 435.
SH: This was the Permanent Apportionment Act of 1929, and at that time this would have meant that a member of the house would be representing around 280,000 people
SW: [skeptical voice] That already seems like a lot of people
SH: Right? And the US population has grown 3 times as large since then.
SW: Yeah, that seems, not ideal. How does the US compare to other countries who do this kind of thing?
SH: Ummm, badly? Most other democratic countries change the number of representatives as their populations grow, leaving our ratio quite far afield
IV: There are these graphs of OECD countries, Organization for Economic Cooperation and Development. That's our sort of cohort that we're compared to in these statistics. And we are really an outlier when you look at where the representations are. You know, we're like off the chart somewhere.
SW: So what does Ismar think would be more reasonable?
IV: If we follow, sort of, the math that most countries follow, we should be at around 700. That should be the size of the House.
SH: The around 700, 693 to be exact, number comes from the cube root law devised in 1972 by political scientist Rein Taagepera, who observed that in many democracies the number of representatives was close to the cube root of the population.
SW: And that would be what, around half a million people per member of the house?
SH: That sounds right
SW: 500,000 still seems like a lot...
SH: For sure, but it is still a 1/3 less than they are covering now. Ismar knows that this would be a big ask of course
IV: I realize it's not a popular opinion. Like if I go out on the street and say, you know, we should increase. See how dysfunctional Congress is? We should have more of those people. That would not go over well, but ultimately it would be better, actually.
SW: [Laughs] Oh yeah, I can see how telling people that one of the best ways to fix our government being more politicians not going over too well
SH: Yeah I see that too too, though Ismar did tell me that a larger house is more likely to be encourage broader coalition and cross party cooperation
SW: I mean yeah sure, you might have me convinced but it still seems like a hard sell
SH: For sure, and there have been increasing attempts to make this change but none of them have really caught on as of yet
SW: Yeah, that’s is too bad. I do have another question about all of this though, as it seems to me that we are skipping past an important bit of mathematics when it comes to apportionment
SH: Oh yeah?
SW: Yeah. Sure it is important to figure out the number of representatives but how do we decide how many get assigned to each state?
SH: You’re right, that is an important bit of mathematics. One with a very specific issue
IV: As much as we'd like to, maybe sometimes we can't really chop up representatives into decimal fractional pieces so how do you round. Basically it's a rounding problem
SH: And at the founding there were two competing methods for solving this rounding problem, one proposed by Thomas Jefferson and the other by Alexander Hamilton, and, personally, I will never forgive Lin-Manuel Miranda for not making one of the cabinet battles about the greatest remainders vs greatest divisors fight.
SW: [laugh] It certainly would have been memorable, though it was probably already a hard enough fight to stage a rap musical about the founding fathers without bring math into it
SH: But just think, just think, about Daveed Diggs dropping bars like:
But Hamilton mis-calculates
His plan opens up multi-paradox gates
From New states to Alabama to population
Yet he moves forward with no hesitation
SW: [interrupts] Sam, Sam, stop. Please. Just stop.
SH: Ok, my Hamilton (Math's Version) fan fiction can wait I guess.
SW: Put it on AO3 or put it forever on ice. How about instead you tell me more about Hamilton and Jefferson's methods?
SH: Sure, I was going to rap about them. Both rely on calculating what is known as the standard divisor, which is the total population divided by the number of seats to be apportioned. Then you take a state's population and divide it by the standard divisor, which will give you a fractional number of seat a state would get
SW: So this is where Ismar's comment about chopping up representatives comes from?
SH: Exactly. Hamilton's method then rounds down to the nearest whole number to set the floor for the number of representatives a state gets. This of course assigns fewer seats than needed, so the method then assigns an extra seat to states depending on how large of a remainder there is left after the rounding
SW: So if the standard divisor is 4 and there are states A and B with populations of 10 and 15. B would get the extra seat since it would have a remainder of 3 instead A's remainder of 2?
SH: Exactly. Jefferson's method is a bit more confusing. Instead of rounding down and then assigning the extra seats, it keeps changing the standard divisor until you get the correct number of seats when you round down each state.
SW: Ok...
SH: What’s, what’s important to know is that both methods have their own issues. Jefferson's method violates what is known as the quota rule, which is that a state should receive a number of seats equal to rounding up or down the value you get when dividing it's population by the standard divisor
SW: That does sound reasonable
SH: Yeah, and in Jefferson's method it is possible for a state to go above that number of seats. And while Hamilton's method meets the quota rule, it can trigger the Alabama Paradox where a state loses a seat when the number of total seats is increased, the new states paradox, where the adding of a new state along with a proportional number of new seats still changes the overall apportionment, and the population paradox where state A grows more than state B but still loses a seat to state B
SW: Right....
SH: But to be fair, we mostly know about these issues in hindsight
SW: So then which one did the founders end up choosing, who won the rap battle?
SH: Well Hamilton convinced congress on his method, but
IV: Washington vetoed Hamilton's method. That was the first ever presidential veto in the history of the United States, one of only two that Washington issued, and he adopted Jefferson's method. Why? Because according to the Jefferson method, Virginia would have received one more seat, and Washington and Jefferson were from Virginia, over a Hamilton method. Hamilton amount would have given it one less. So, you know, right there, like the outset of our republic, you see this like political bickering and scheming, like who's going to get one more seat where, and that really has unfortunately followed the history of this problem over the last 250 years
SW: But that was then., I assume we don’t still use the Jefferson method?
SH: No, we do not. It has changed a number of times over the years, including using Hamilton's method at times. Then after the 1929 Reapportionment Act congress decided that it needed to officially set a method that would work well with the now capped 435 seats. There were 2 main contenders, the Webster and the Huntington Hill methods
SW: And are these methods more similar to the Hamilton or Jefferson methods?
SH: They are closer to the Jefferson method, as both rely on changing the standard divisor to get to the right number of seats. Where they differ is that they allow for rounding up as well as rounding down. Webster’s method relies on the typical .5 rounding cut off while Huntington Hill uses the Geometric Mean
SW: Which did they end up going with?
SH: Well it came down to a fight between Edward Huntington, a well known mathematician, and Walter Willcox, the founder of the US Census Bureau's statistical research office. On the positive side of things this is one of the few cases where congress consulted with members of the mathematical science research community
SW: This implies there is a negative side Sam
SH: And that negative side is that it still came down to politics
IV: A panel of mathematicians was assembled by the American Academy of Sciences. And Huntington was, and he wasn't on it, but he was such a large presence in the mathematical community at the time that these, this panel was sort of, you know, loathe to go against what he had preferred. So that might have influenced the final endorsement of the Huntington Hill method by this panel of mathematicians. And then there was another panel some 20 years later that reaffirmed this decision, but that panel had one of the same mathematicians on it. So, you know, they weren't going to decide differently this time around.
SW: Greeeaaattt
SH: I know it sounds bad, but according to Ismar it really isn't that big of a deal, even if he does slightly prefer Webster himself
IV: Webster and Huntington Hill are solid methods, and most of the time they give the same outcome, so it's not such a big deal that we're using one versus the other.
SW: That's good to hear, that at least both methods generally agree on the result. But I guess I can't help but wonder if there is something even better we could be using?
SH: Wellllllllll...
SW: Oh no, you are going to do it to me again. Aren't you?
SH: Do what to you again?
SW: Tell me that math has shown yet another thing is impossible. Like voting...
SH: Maybe
SH [from interview]: Can we at least do apportionment mathematically perfectly?
IV: No. We can't do anything perfectly in democracy, no matter how hard we try. There is yet another impossibility theorem in the apportionment arena called Balinski-Young, which basically says there's no perfect apportionment method either. Every apportionment method will fail. But again, the evidence that we have for Huntington Hill and Webster is that, the evidence is that they don't fail it in practice very often. I don't think they ever have in the last 70 years. And also we know probabilistically that theoretically they would fail some of these criteria very, very rarely.
SH: We also need to state that when we are talking about apportionment at the founding and the fight between the different methods of Hamilton and Jefferson, that they were assigning seats to states based on populations that were being calculated under the 3/5ths compromise where each enslaved peoples was counted as 3/5ths of a person for purposes of population and taxation without being afforded any of the rights of citizens
SW: Not to mention a lack of voting rights for all but the most privileged of white men. And even though the property owner restrictions went away over the first few decades, white women were not granted the universal right to vote until 1920, Indigenous Americans had to wait until 1947, and Black Americans until 1965
SH: And no amount of mathematics can make voting or apportionment work when the government does not allow large swaths of it population the right of suffrage
[music]
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[music end]
SH: There is one last thing I want to share with you Sadie, you could say it lies right at the middle of apportionment and voting
SW: Are we are finally getting to the Electoral College?
SH: We are finally getting to the Electoral College. So, during the Constitutional Convention when the election of the president came up it was quite clear that the South would not allow direct election of the president. As James Madison wrote "There was one difficulty however of a serious nature attending an immediate choice by the people. The right of suffrage was much more diffusive in the Northern than the Southern States; and the latter could have no influence in the election on the score of" enslaved peoples. It was also clear that the convention thought it had more important things to think about
IV: It was one of these things nobody wanted to talk about, nobody could agree on. I was relegated to this committee on unfinished business. Like what a nightmare, right? Can you imagine sitting on a committee of unfinished business?
SW: Monthly python style, the Committee of Unfinished Business? Yeah, that does not seem like top priority
SH: And after something like 30 votes the electoral college is approved as a compromise that allows the Slavery States to leverage the 3/5ths compromise in order to have more of a say in the presidential election. And specifics of the electoral college aside, Ismar thinks that the history of how it was developed holds an important lesson
IV: I mean, you know, we cling to these decisions that the framers made as like the end-all, be-all of the highest intellectual and moral thinking. And, you know, like, but no, they're humans. We just want to, like, get back home after how many weeks of sitting at this convention. And they just came up with this compromise that nobody, well, by definition of compromise, nobody was really happy with it.
SW: I am always in favor of remembering that people from history, especially the so-called great people from history, are just that: people
SH: Exactly, and like all people it is possible they made a mistake
IV: Very quickly it became clear that it didn't make any sense. And it certainly does not make sense today. I mean, you don't even need mathematical evidence to support these claims. The only thing you need to realize is kind of, you know, extract yourself from American exceptionalism for two seconds and look around and realize that no country in the world uses anything like the Electoral College. And all the new countries that were formed in the last—it's not like all the countries were there already, right? So it was too late. Lots of new countries came to be in the last 150 years. None of them have anything close to the electoral college.
SW: I’ve personally been arguing against the Electoral College for years, but we are a math and stats podcast so maybe we should take a look at the mathematical evidence
SH: That's a fair point. The first, and most obvious one, is that under the electoral college the winner of the popular vote may not be elected president.
SW: Yeah, if I remember correctly it has happened 5 times in the history of the US, and it even happened during my lifetime as a voter! I’m sure it has happened 2 times in the last 25 years
SH: And while that might be the most obvious mathematical evidence against the electoral college, it is by far from the only mathematical evidence
IV: The disproportionate power that voters have if they live in certain states. So if you look at, you know, the last elections, 2020, it would have been enough for a handful of voters, handful in relation to the, you know, millions and millions who voted, you know, something like, we're talking about a total of 40,000 voters or something. If they had lived in a different state with the same political reference, if they had just lived in a different state, the outcome would have been different.
SW: I think about that one a lot. We both live in Illinois, and when it comes to the presidential election our votes don't really matter
SH: I know, I moved rather recently I used to live in Michigan and Wisconsin. Two states where my vote had disproportionate power
SW: Ugh, which it never ceases to make me angry!. Why should where I live change the worth of my vote for a national office like the president.
SH: It shouldn't, and it can lead to massive disenfranchisement as Ismar notes:
IV: Or I think about the 30% of Republican voters in Massachusetts who also, they get no, they might as well stay home on election day, right? They get not only the presidential elections, they have no representation in Congress whatsoever because our entire delegation is Democrats, right? Millions of Republican voters in California, which their votes make no difference for presidential elections. There's a lot of disenfranchisement going on because of the Electoral College, because it ties, it anchors our elections to states, and that's not the right thing.
SW: But my understanding is we’re sort of stuck with it aren't we? After all it is in the constitution
SH: Yeah you’re right about that, the clearest way to move away from the electoral college would be with a constitutional amendment, though there are people who are trying to push for the National Popular Vote Interstate Compact where enough states get together so that they have 270 electors between them and they agree to assign their electors to the winner of the national popular vote
SW: Oh yeah, I have heard of that. Where does that stand?
SH: They are close, but still short of the 270 electors needed. Plus it is not clear that the compact would stand up to legal review even if it did reach the threshold. Though not everyone thinks that getting rid of the electoral college entirely is the right thing to do
TJ: I do think that a little bit of flux where the college doesn't follow the popular vote in close elections, like 300,000 across the country, I think that's a feature of the system, and encourages both campaigns to fight a little bit harder next time out for both parties.
SW: Wow, that's not an opinion I have heard much. Who was that sharing this?
SH: I will let him introduce himself
TJ: So I am Theodore R. Johnson, or Ted, and I am a columnist at the Washington Post and a senior advisor at a think tank in D.C. called New America.
SH: And Ted comes at this question from a rather different perspective than our last guest
TJ: Look, I'm a retired military guy who reinvented himself in his 40s to be a scholar on race and democracy.
SH: In particular one of the big reasons Ted was drawn these areas of scholarship was because of
TJ: just how lopsided the Black vote has been really since the 15th Amendment in 1870, but in particular since 1960s in the Civil Rights Act, Voting Rights Act of 65. And I've always wondered whether voting so uniformly helps Black Americans achieve their policy desires and goals, or whether it creates a vacuum of competition wherein one party assumes it has it, takes the vote for granted, the other party doesn't even attempt it. Does that create a quandary wherein the political demands of a group are ignored because it's sort of captured by one side?
SH: Which leads directly to questions about the Electoral College since, as Ted told me
TJ: Because I think competition for every voter matters in a democracy that pretends to be responsive to its public.
SW: And as we know there is little competition for voters in most states due to the electoral college. But then, why does Ted not want to abolish it entirely as well?
SH: For those same competition reasons, just instead of only competing in swing states he’s worried about competition becoming all about the most populous states
TJ: I'm less a fan of the national popular vote because it allows for wide margins of popular states to basically become insurmountable pretty quickly. If you're winning by a million and a half votes in California, then it's hard to pick that up in Rhode Island and Wyoming and Alabama, it's just, it becomes a bit too much.
SW: So then what does he think we should do about the electoral college?
SH: He shared two potential paths toward reforming the electoral college. In the first one
TJ: One is to do what Maine and Nebraska do, which is basically partition out electors by congressional districts.
SW: And what about the two electoral votes that aren't tied to congressional districts?
SH: Those would go to the state popular vote winner
SW: Ok, I can definitely see how this is better than what we have now. But I feel like this is still hiding some downsides?
SH: There are, and one of them is pretty big. So big in fact Carry the Two has covered it before
TJ: So a congressional model relies on how the districts are drawn in a state with a lot of partisan gerrymandering is going to bake in inherent advantages for one party over the other, even when you go to that partitioning of electors within a state.
SW: Of course, the G word. I was wondering when we were first going to mention gerrymandering.
SH: Well we do not have time to dig into it right now, so if you want to learn more go check out the Moon Duchin episode of Carry the Two for a deep dive, it was a great one. Sadie was the producer back then. Instead let's talk about the other approach for reforming the electoral college
TJ: So what I like is the proportional allocation approach. And what this does is allows a state to partition its electors based on the statewide popular vote. So, you win 60% of the popular vote, you get 60% of that state's electors rounded up to the next whole number.
SH: And you keep going down past the second top vote getter too
TJ: Ross Perot would have gotten seven, eight, nine electoral votes in California and in Texas based on how many popular votes he received.
SH: Ted see's this method of proportional allocation as providing a good balance between the popular vote and the positive parts of the electoral college
TJ: The results of a state in the electoral college more closely mirror the popular vote. And that is a good thing. And it does so without eradicating some of the protections the electoral college built in for low population states, for example, to ensure that the tyranny of the majority doesn't, you know, 50% plus one doesn't get to ram its agenda down the other side's throat.
SW: He says it more closely mirrors the popular vote. Does that mean we wouldn't get the popular and electoral vote split anymore?
SH: Not exactly
TJ: In this proportional allocation method, Bush still wins. Bush still wins. But the difference is, there's a number of them. One of them is the vote doesn't come down to Florida. The election doesn't come down to Florida.
SW: But why does the election not coming down to Florida matter?
SH: It all comes back to the idea of competition. Not only does it not come down to Florida, the election would not have come down to any single state. Which Ted thinks would make a huge difference
TJ: Republicans now campaign more in California, so they don't lose by 15 percent. They only lose by seven and they win electors for the effort. Same for Democrats in Texas.
SW: And our votes in Illinois would mean something
SH: Exactly, and they still wouldn't over power our Iowan neighbors' votes even though we have 4 times the population. Which is core to Ted's vision
TJ: The protection of political minorities across the country is a feature of liberal democracies. It's actually part of what the college was created to do. It employed mathematics from an electoral calendar or electoral map from the 18th century. And so it's just time for us to update the math. But the idea that political minorities should be protected in a democracy is a constitutional one. It's an American one and one most people support.
SH: That is not say Ted doesn't see limits to this need
TJ: I am all for allowing for political minorities to have an opportunity to win office, but not when they're in the minority by 8 million votes nationwide in an election where about 150, 140 something million people voted.
SW: Seems like a reasonable caveat
SH: It certainly does. But do you know what I like the most about these reforms versus the popular vote?
SW: No, I am not sure I do
SH: They are much easier to implement
TJ: So if every state independently decided to partition its electors like Maine or Nebraska or to do a proportional allocation like we've discussed they absolutely could. And the Constitution allows that. Both the Elections Clause and, I even think maybe, the 10th Amendment would permit it.
SW: Oh, hands down that would be my favorite part too
SH: Right? Now, before we sign off I want to share one last thing that Ismar told me about why it is important for us to use mathematics and statistics to look at our democratic systems
IV: Why? You know, why are we talking about math and democracy? Well, there's this simple realization that once you make it, it opens lots of doors very widely. And of course, everything is simple retroactively once you figure it out, right? But this is really simple. When you think about many of our basic democratic processes, they are really mathematical in nature, right? There are some algorithms, some formulas, some equations that are determining how you go about the mechanics of our democracy. And we've talked about voting. That's how do you tally the votes? How do you decide the size of the House of Representatives? How do you actually apportion these seats? How do you divide 435 into 50 states? So that's some kind of a mathematical process. Once you make the realization and you start digging into what math governs these processes, you quickly also realize that a lot of them are completely broken. They're dysfunctional. They are causing, you know, well, they are just outright discriminatory often. They do not represent us. They are not capturing our will and all these things. At the same time, math can suggest better processes like using Instant Run off instead of plurality, right? Increase the size of the House of Representatives to a certain number that can be mathematically determined, optimized mathematically. And what is the formula by which you allocate seats of the House of Representatives, right? Are we using the right one? What is the right one? What does that even mean? Should we be using a different one? And so many of our processes function mathematically like this, and this is what I'm trying to kind of impart, right? We should be studying democracy from this impartial, apolitical, mathematical point of view, and we should be asking, do we have the best math governing our processes? And if we implement better ones, it'll increase representation, it'll increase diversity, it'll really help our democracy.
[music]
SH: Don’t forget to check out our show notes in the podcast description for more Ismar and Ted, including links to their work we discussed on this episode
SW: And if you like the show, give us a review on apple podcast or spotify or wherever you listen. By rating and reviewing the show, you really help us spread the word about Carry the Two so that other listeners can discover us.
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SW: And do you have a burning math question? Maybe you have an idea for a story on how mathematics and statistics connect with the world around us. Send us an email with your idea!
SH: You can send your feedback, ideas, and more to Sam AT IMSI dot institute. That’s S A M at I M S I dot institute.
IM: We’d also like to thank Blue Dot Sessions for the music we use in Carry the Two.
SW: Lastly, Carry the Two is made possible by the Institute for Mathematical and Statistical Innovation, located on the gorgeous campus of the University of Chicago. We are supported by the National Science Foundation and the University of Chicago.
