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- [00:02]
Podcast Announcer
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Scott Bertram
Welcome to The Hillsdale College K12 classical education podcast, bringing you insight into classical education and its unique emphasis on human virtue and moral character, responsible citizenship, content, rich curricula and teacher led classrooms. Now your host, Scott Bertram.
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Thanks for listening. The Hillsdale College K12 Classical Education Podcast is part of the Hillsdale College Podcast Network. More episodes at podcast hillsdale.edu or wherever you get your audio. You also can find more information on topics and ideas discussed on this show at our website, k12 hillsdale.edu.
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Interviewer
we're joined by Dr. Thomas Trelor. He's professor of mathematics and dean of faculty here at Hillsdale College. Dr. Trelor, thanks for joining us.
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Dr. Thomas Trelor
Thanks for having me.
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Interviewer
Talking about a subject I know you've done a lot of work and research on the mathematics of voting theory. Now, when most people picture an election, they picture plurality voting. Everyone names one candidate, top vote getter wins. Is there something wrong with that? And if so, is there an example where it gives a result that most voters actually wouldn't want?
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Dr. Thomas Trelor
Yeah, so it's good to remember what the purpose of an election is for. Right? The purpose of an election is to take individual preferences and turn them into some sort of group action or group preference. And so it turns out if you only have one or two candidates, there is no problem whatsoever. Because at that point, if I say who my favorite candidate is, it's favorite with respect to what one other candidate. So if I like A, it means I like A better than B. The difficulty. And that is called the majority criterion, right? This idea that if I have a candidate that the majority of the voters prefer, maybe that candidate should win the election. Now, with three candidates, that majority criterion can come into play, right? Because what we have there is we could have a candidate who is has 40% of the votes and then let's just say candidate A has 40% of the votes, B has 30% of the votes, and then C has 30% of the votes. Well, in our plurality system that would mean candidate a wins, but 60% of the candidate of the voters may not like that particular candidate. In fact, they might like that candidate the least. So that would be the situation where you would find yourself in. You know, the one who inspires the most confidence out of a small group might end up winning the election, even though a lot of people don't particularly care for that one.
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Interviewer
You teach several different voting methods. Plurality runoffs, an instant runoff, Condorcet Bordeaux. Without getting too technical, what's the basic idea that separates these voting methods? What is each one trying to find out to capture?
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Dr. Thomas Trelor
So plurality, as we mentioned, is just what is the candidate that people are most enthusiastic about? And with two, we have a majority would end up ruling the problem with the other one. Or I'd say the other ones are trying to handle the case where you would have three or more candidates. And then how do you break it down? And so runoff. What runoff methods are usually trying to do is try to force a majority. So you have three candidates. That's a problem. Let's reduce it down to two candidates and get rid of one. So a runoff would force a majority. And then the other methods, Borda or Borda count, that is one where you start assigning points to candidates based off of how people like them. And so you may know this from Heisman Trophy, a lot of sports MVPs are decided by Borda counts. I've been on search committees here at the college where, you know, the chairman of the department in that will use Borda count as a method for picking that.
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Interviewer
I was on a committee to call a pastor and I was in charge, sadly. But the very first thing I did, we had all the resumes and we said, all right, write down your top. Well, I remember about top three. But then assign points. You have 10 points to assign. If you really love them, give them nine. So you can use this in other places too?
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Dr. Thomas Trelor
Yep, that's one of those big ones. And then the last one is that is oftentimes used or could be used is Condorcet. And the idea of Condorcet is again, you don't have a majority of the support. What it does is it breaks everything down to head to head matchups and say there's three candidates. How does A do against B? How does A do against C? How does B do against C? And if there's ever a candidate that would beat all the other candidates in head to head, we would call that the Condorcet win winner. And that would be another way to think about a voting method.
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Interviewer
This Condorcet winner that you just described, this candidate who would beat every other candidate One on one, head to head, it sounds like that's the person who should win. So why don't all our voting methods just get to that person?
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Dr. Thomas Trelor
Because that person doesn't always exist. Much like with the game Rock Paper Scissors, there's not always a strict ordering to this, these things. And you know, candidate A could be preferred to B and B to C, and then C back to A again. So sometimes that person just doesn't exist. And even if that person does exist, it can be difficult to get to that person. I'll just reference the recent California gubernatorial primaries. There were 63 candidates right in that. And if you wanted to get at every single voter's preference of every, every candidate in that, there's over 1800 head to head matchups that they would have had to choose between. And so that's an extreme example, but it's not something that's normal.
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Interviewer
It's a real example.
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Dr. Thomas Trelor
It's a real example. And so it's more complicated. You're asking a lot of the voters to know the subtleties between each of those potentially many candidates.
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Interviewer
Yeah. What are the axioms of fairness in voting theory?
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Dr. Thomas Trelor
Okay, so we talked about what's called the majority criterion, which is the idea that if, if somebody gets the majority of the votes, if they're preferred by the majority of the voters, they should win the election. And when we think about voting methods, usually we should start with some, some underlying assumptions like that one that we'd say, okay, whatever voting method we use, we would like to build it up out of these things. We'd like these things to be true. They're like, they're axioms. Just like Euclid has his five axioms in his elements. And so everything else is built off of those underlying assumptions. So majority criterion is one of those. Oftentimes the Condorcet criterion is one where if there's a candidate that would win all the head to head matchups, maybe we want that person to win. There's one called the monotonicity criterion, which is basically if a candidate gets more support, it shouldn't hurt them. Okay, but there are instances where a candidate could win an election if it were held today, but they could go out and work really hard to get more support. And by getting more support, they could actually end up losing this, the, the election. So that would be violating the monotonicity criterion, which we generally would not want to happen if we could. And then the fourth one that is oftentimes listed is independence of irrelative alternatives. So that's the Idea. If you introduce or if you eliminate a candidate who really is not viable, they have no chance of winning the election. You would like, if you were to add or subtract one of those, it wouldn't change the result, right? Well, a lot of voting methods will violate that independence of irrelevant alternative. So those would be the, typically the four fundamental acts. If we would like, whatever voting method we decided on, it would satisfy those four things.
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Interviewer
Dr. Thomas Trelor with us talking about the mathematics of voting theory. And that brings us to Arrow's theorem, which is sort of the famous bombshell result here. In plain terms, what did Arrow prove and why did it earn him a Nobel Prize?
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Dr. Thomas Trelor
So what arrow showed, and sometimes it's called the arrow impossibility theorem. What Arrow said was if there are three or more candidates, that it is impossible to build a voting method that satisfies all four of those what we just called axioms of fairness, that it's not possible to do so and outside a dictatorship. So if you just have one voter who's deciding on this, then you're okay, but three or more candidates, it's impossible to build a voting method, which is good to know because it's good to know that it's not that we just haven't found the right voting method because we're not clever enough. It's impossible to find that voting method. And then you start thinking about it in terms of, okay, when we design a voting method, we know we're going to have to have some trade offs. There are going to be some good, there's going to be some bad to it. What is fundamentally important to us? What do I need every voting method to do? And is it okay if my voting method violates this other one?
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Interviewer
A natural reaction to all of this conversation we've had is voters should just be honest. You, though, treat strategic voting as something that's kind of baked into the system and not a flaw in people. Why is insincere voting almost unavoidable?
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Dr. Thomas Trelor
We just have to recognize that it is a product of the system that we put in place. And so I could say voting theory falls under the category of another area called game theory. And the idea of game theory is you have these players and they can take certain actions and then they have payoffs. And the goal in game theory is to maximize your payoff. Well, an election is like that. I can cast a vote and I could cast a vote for my favorite person who has no chance of winning. So I could cast instead of vote for my second favorite person person, because maybe that person would have a chance of winning. And so that would be considered an insincere vote. But that's the sort of thing that could allow me to maximize my payoff. So when we, when we put a, you know, when we decide on a voting method, we probably should acknowledge that people are going to follow certain strategies to try to maximize their payoffs. Now, personally, I prefer to people or the, the method that we have in place doesn't cause people to play so many games, because people like to play games and they don't know the real rules or the best strategies in that. So we want to be a little bit careful with that. But we should always assume that people are going to try to cast a vote and it's going to maximize their total payoff or what they prefer to happen.
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Interviewer
If we shift gears a little to power in something like the electoral College, a bigger state has more votes, more electoral votes. But you argue that raw vote count isn't the same as actual power. So how can a voter or a state have more votes, but not proportionally more influence?
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Dr. Thomas Trelor
I like to give a simple example for this. A simple example. Imagine you have a group of 100 people, 51 women and 49 men. Okay? So men and women make up approximately half and half. A little more than half, a little less than half. Now, what happens if all the women decide to form a coalition and cast their votes together? So you have 26 of the. They just decide if 26 or more of us agree to this, this is what we're going to do as a group. Okay, so they have about half the weight in the electorate. They have all the power. Because at that point, 26 women would decide, or more right. Would decide on 51 votes, which would be enough to pass anything. So even though we have, in that particular example, men and women with about the same amount of weight in the electorate, if the women form a coalition like that, there is nothing the men can do. And it's not that the men did anything. It was just part of the structural ability. So that is true. This is true with our political parties. This is why our political parties have majority and minority whips, right? To keep people in line with their party. So that if you have the majority of the seats in, say, the House or the Senate, you want to be able to pass whatever legislation that's going to occur. And again, you can have something very close. Most of the time, we have something very close to 50, 50 split. And yet we will clearly acknowledge that one party over the other has the power.
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Interviewer
Let's talk apportionment. This is the process of dividing seats, like say, seats in the House of Representatives among the states based off of population. Sounds like pure arithmetic. Is there an underlying idea with the processes here?
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Dr. Thomas Trelor
Yeah. So with, with apportionment, there's the attempt to give each se state the number of representative seats according to its state population. And when you do this, and when you perform the calculations, which start out being arithmetic, you get numbers like that state should get 7.6 seats, but you can't give us, you know, six tenths of a seat. And so then you end up rounding and you need to round down to seven or you round up to eight, or you round it somewhere else. And the question would become, how do you fairly round? And it ends up being a question like a lot of these questions here, a question of fairness. And so there are, through the history of our country, there's a Hamilton's method, Jefferson, so Alexander Hamilton, Thomas Jefferson, there's an Adams method, there's a Webster method, and now the one we use is Huntington Hill, which has been in place since about the 1940s, I believe. And all of them just have different ways of breaking up the total number of seats that we have into some sense of fairness between the states.
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Interviewer
What is the Alabama paradox where adding a seat can cost a seat 1. How does something like that happen?
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Dr. Thomas Trelor
So that is actually something that happened with Hamilton's method. And the idea here is it was seen in the 1880s where they were determining how many seats should we have in the next Congress, in the House of Representatives, and then what should be our apportionment method at that point it was Hamilton's method. And I forget who probably who was performing the calculations at the time. But what we had is a situation where they went from 275 to 350. They said, well, let's just figure out what the apportionment would be in all of those cases. And what they found is at 299, if we had 299 seats in the house, Alabama would get eight of them, and at 300, Alabama would get seven of them. So you add another representative to Congress and it causes Alabama to lose a seat, meaning it went from Alabama to someone else. And I think it went to New York at that point. And so that's a flaw in the system because you wouldn't want to say, oh, you know what, we should really have another candidate, you know, another elector in there, and then have it negatively affect someone. You would Just expect that extra one would go to someone else.
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Interviewer
Yeah.
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Dr. Thomas Trelor
So when it violates that, that's called the Alabama paradox. And Hamilton rule does violate that on occasion.
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Interviewer
Am I correct that the first presidential veto in the history of our country was concerning apportionment?
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Dr. Thomas Trelor
Yes. So that was back in 1792, when President Washington issued the first ever presidential veto when it was on apportionment and the Congress had passed. Had said we should have. Hamilton's rule, the one we were just talking about, should be the way to set apportionment. And Washington's cabinet was deeply divided on this. Jefferson was on one side, Hamilton certainly was on the other side. And Washington agreed with the Jefferson side of things and ended up vetoing that, saying it was unconstitutional, it was unfair. And fundamentally, the idea is, with the Hamilton method, you divide through, you figure out, you get that 7.6, and then you always round down, and then some states get a plus one. The argument was, well, you're treating states differently. Some of them you're giving a plus one, some of them you're not giving a plus one. And so what happened is 10, 10 days later, Congress passed Jefferson's method, and then it was approved. This wasn't given as part of the argument, but both Washington and Jefferson were from Virginia. And Jefferson's method did benefit large states, and so Virginia did. Now, was that part of their thinking along the way? Maybe. We don't know.
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Interviewer
Talking with Dr. Thomas Trelor from Hillsdale's mathematics department on the mathematics of voting theory. What is the Balinsky Young theorem?
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Dr. Thomas Trelor
So the Balinsky Young theorem is the equivalent of arrows impossibility theorem. And so it says that in apportionment, if we start with four basic assumptions and one is called the quota criterion, and then we ask for it to not violate the Alabama paradox, the new state paradox, the population paradox. The example of the new state paradox is. I forget the year, but when Oklahoma joined the union, it was decided that given their population of the territory, that they would get five seats. The number of seats were recalculated with five more representatives, and then everyone else, you know, plugged back into the system, and in that process, one of the states lost a seat. Right. So again, the whole idea was that maybe just Oklahoma should gain a seat or gain the five seats and everyone else be the same. No, that's not what happened. So that would be called the new state paradox.
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Interviewer
Okay.
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Dr. Thomas Trelor
I don't know why they didn't call it the Oklahoma paradox, but there you go. And then population, Right? So if two states are changing. If their population of one is growing faster than the other, they could actually end up losing. So if we ask for the same thing that we did out of the arrows theorem is it's fair if it doesn't satisfy one of these paradoxes. And if you have this other quota criterion and what the Balinsky Young theorem says, it's impossible to build an apportionment system that will satisfy all of that. So again, there's going to be a judgment call out of all of this and what's the most important thing to avoid and what's okay to have?
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Interviewer
So between arrow on voting methods and the impossibility there, and Balinsky Jung on apportionment and the impossibility there, the math tells us at least here twice, that there's no perfect system to be built. Is that a discouraging conclusion for you? Is there something even useful in knowing what the limits are?
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Dr. Thomas Trelor
Yeah, I would say it's definitely useful to know what you can do and what you can't do. And I would say it's also hopeful because I think a lot of people look around and say, well, if we're just more clever, we can do this better. And the answer is nice to know, no, we can be as clever as we want to, probably too clever for our own good, but sometimes it's impossible to do things. And so it reframes the question rather than we're just not doing it right into the question of, okay, what is most important to us and how should we lay this out? How should we decide what's the most valuable things and what are the things that we will secure and what are those things that we have to let set aside. And so I would say in both of those theorems, it's nice to know where you're at, and it's very useful. But it's also encouraging to know. Let's think about what's most important to us and realize, you know, we need to be a little careful in terms of trying to find all the answers all the time, at least knowing when we can and we can't.
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Interviewer
So most of us are still going to the ballot box and choosing a name. There are some who have a different system now called ranked choice voting. And there's debate, discussion. There's a lot of. There's a lot of inputs that go into that, of course. Do you want to wade into this, what we know about ranked choice voting thus far?
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Dr. Thomas Trelor
Yeah. So ranked choice, like an instant.
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Interviewer
Essentially. An instant. Right.
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Dr. Thomas Trelor
Essentially instant runoff. Runoff, yes. Yeah. I would say you just have to be a little careful with these. So the instant runoff is one of those situations that you can violate the monotonicity. You can have people going out and getting extra, working hard and trying to increase that. The vote totally could actually end up hurting them. As long as your system isn't encouraging people to kind of game the system, then things like that can be okay. And it's not just on the voter side of things that you have to be careful on. It's on the elector and the candidate side of things. Because in certain voting systems, you can start adding extra candidates in there only to punish your opponents. And Borda Count's a good example of that, where you could say, I'm gonna introduce a couple of candidates that people can put above my biggest rival. So you have to be careful about those sorts of things also.
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Interviewer
Dr. Thomas Trelor is professor of mathematics, Dean of faculty here at Hillsdale College as we talk about the mathematics of voting theory. Dr. Trelor, thanks so much for joining us here on the Hillsdale College K12 Classical Education Podcast.
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Dr. Thomas Trelor
Thank you for having me.
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Scott Bertram
I'm Scott Bertram. We invite you to like us on Facebook search for Hillsdale College K12 classical education. You also can follow us on Instagram hillsdalek12. That's Hillsdale K12 on Instagram. Thank you for listening to The Hillsdale College K12 classical education podcast, part of the Hillsdale College Podcast Network. More at Podcast Hillsdale. Edu or wherever you get your audio.