Elections: A Big Math Problem

Ahead of Election Day tomorrow, millions of ballots are being cast in statewide, local and special elections. So, today, we're revisiting an episode asking: What would happen if the rules of our electoral system were changed? Producer Hannah Chinn reported on that very question, and today, with host Emily Kwong, they dive into three voting methods that are representative of alternative voting systems. They look at where these systems have been implemented, how they work and what they may mean for future elections. Want to hear more about how math could change our lives? Email us at [email protected] mailto:[email protected] and we might cover your idea on a future episode! Listen to every episode of Short Wave sponsor-free and support our work at NPR by signing up for Short Wave+ atplus.npr.org/shortwave http://plus.npr.org/shortwave. See pcm.adswizz.com https://pcm.adswizz.com for information about our collection and use of personal data for sponsorship and to manage your podcast sponsorship preferences. NPR Privacy Policy https://www.npr.org/about-npr/179878450/privacy-policy

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• 0:00Speaker 00

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• 0:17Regina Barber?

  Hey, short waveers. Regina Barber here with a look back at a timely episode my co-host Emily Kwong did with one of our producers, Hannah Chin. Because tomorrow, in many states across the US, it's an off-cycle election day. So even though tomorrow's elections don't coincide with federal elections, many people will still be heading to the polls to vote in state, local, and special elections. And that got us thinking about all the possible ways for votes to be counted. I don't want to give too much away, so I'll let Emily and Hannah take it from here.

• 0:50Speaker 02

  You're listening to Short Wave from NPR.

• 0:55Regina Barber?

  If you can vote and you're listening to this, maybe you voted early. Maybe you mailed in your ballot. Maybe you're waiting at a polling place, listening to Short Wave right now. Regardless, before you even marked a ballot, there was a choice already made for you.

• 1:09Speaker 02

  And that's the choice of which voting system to use.

• 1:12Regina Barber?

  Hey, Hannah Chin.

• 1:13Speaker 02

  Hi, Emily.

• 1:14Regina Barber?

  Hannah, our producer today, is going to tell us about some of those voting systems. Though, wait, hold on. What is a voting system?

• 1:21Speaker 02

  Uh, good question. So for our purposes, a voting system is the set of rules that you use to determine the winner of an election.

• 1:29Regina Barber?

  Uh huh.

• 1:29Speaker 02

  And there are a lot of different rules that you can use. So to illustrate this, I talked to a math professor at George Washington University in DC. His name is Dan Ullman. And for the past 20 years, on and off, he's been teaching this class called Math and Politics. And on the very first day of class, he has all his students conduct a mock election.

• 1:51Speaker 03

  There are three candidates and there are 99 voters in this little scenario that I made up. But it's made to be a very close election.

• 1:59Speaker 02

  The 99 voters are these made up people. Oh, so it's not the students? No, no, no, no. But they know all the preferences of these 99 imaginary voters. Like whether they like, say, candidate A more than candidate B, or candidate B more than candidate C. Or maybe they like candidate B the most, and they're indifferent to both A and C. Interesting. OK, so then what happens? So they have to decide which election rules they're going to use. So maybe they want to wait the votes, right? Each voter's first choice gets two points, and each voter's second choice gets one, and their last choice gets zero. And then they tally up all those points to decide who wins. Yeah. Or maybe they have a first round of voting, and then the top two advance to like a runoff vote. Or maybe they're just going to count each voter's first choice and just totally ignore their other preferences. He's letting them like change the rules of the game. Right.

• 2:44Speaker 02

  OK. But again, Dan has created all of these imaginary voter profiles, right? He knows what's in them. Uh huh. So depending on the election rules that the class uses, they're going to end up with a different winner.

• 2:57Speaker 03

  Any two different methods, if they're actually different, could conceivably lead to a different result.

• 3:03Speaker 02

  That's striking. Yeah. And Emily, this isn't because of voter fraud or like double counting ballots or anything. These are all perfectly legitimate ways to count voter preferences.

• 3:13Speaker 03

  The point is that different ways of thinking about it, different ideas you might have, different ways of eliminating or tallying the ballots can give different answers. You don't have to get people to change their mind. You just have to count their votes in a different way, and a different winner can happen.

• 3:33Speaker 02

  So today on the show, the math of elections. We're going to cover three different voting systems, three different ways that a ballot can look, and that voter preferences can be taken into account. And we'll look at the pros and cons of our current system, which is one of these voting systems, and what the alternatives could look like. You're listening to Short Wave, the science podcast from NPR.

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• 5:28Emily Kwong?

  Okay, Anna, earlier you told me this kind of bonkers thing that changing the election rules could change the winner.

• 5:35Speaker 02

  Right, right, right.

• 5:35Emily Kwong?

  But how often does that happen in the real world, in real world elections?

• 5:39Speaker 02

  So this is a really good question. And the issue is, in order to test these systems, you can't just run elections because like those have real world consequences, right?

• 5:48Emily Kwong?

  Sure.

• 5:48Speaker 02

  But I found a researcher who did something really close. He and his colleagues ran a bunch of simulated scenarios using alternative voting systems.

• 5:57Speaker 07

  So we gathered data on about 200 elections from about 50 countries where we have information about How the different parties are liked by voters, and not only the one they prefer, but their relative preferences over several parties.

• 6:15Speaker 02

  This is Roman Lachat. He's an associate professor who studies voting behavior and electoral systems at Sciences Po in Paris. And he ran simulations for nine different voting systems using this voter preference data. And one of his big questions in this study was exactly the same as yours, Emily. Basically, does it really make a difference?

• 6:35Speaker 07

  In a majority of cases, that wouldn't make a difference. And it makes sense, because there are really situations, cases, where clearly there is one party or candidate that is preferred by a majority of voters.

• 6:49Speaker 02

  But in approximately one-third of the cases he simulated, Roman said there was a difference. These simulations led to different outcomes and different winners. And those cases tended to have two things in common.

• 7:06Speaker 02

  So the first factor is just the number of candidates, like the more people or parties that there are to vote for.

• 7:11Emily Kwong?

  Hmm.

• 7:12Speaker 02

  And the second factor is polarization, like the more different that candidates are from one another.

• 7:19Emily Kwong?

  Interesting, OK. So if, in any given election, there's either a lot of options or the options are particularly polarized, the voting system could matter a lot.

• 7:28Speaker 02

  Exactly. So we're going to look at those voting systems.

• 7:33Speaker 02

  First up, plurality voting, often called first past the post. Emily, this one is probably going to sound the most familiar. Basically, everyone gets to vote for one and only one candidate. And any candidate who receives more votes than all the other candidates wins the election.

• 7:48Emily Kwong?

  This is how I had my teddy bears vote when I was in elementary school and I held my mock election.

• 7:52Speaker 02

  I have questions.

• 7:53Emily Kwong?

  One vote per teddy bear.

• 7:56Speaker 02

  So in this scenario, yeah, you see the list of candidates, you choose one.

• 8:00Emily Kwong?

  This is like most of the ballots I've filled out in my life, though, to this day.

• 8:03Speaker 02

  Right, it's super common in single winner elections. And Roman says that's for a reason.

• 8:08Speaker 07

  It's very simple to run, and it's very simple to understand for everyone. Everybody see a very clear connection between how many votes you received and who gets elected. So that's for that reason, at least. It's a very, very good system. But it has disadvantages.

• 8:24Emily Kwong?

  Right. I mean, one problem with having multiple candidates is that there might not be an actual majority winner. Like, the first candidate could get 40% of the vote, the second 45%, and the third 15%.

• 8:35Speaker 02

  And in that scenario, the second candidate would get elected, even though more than half the voters didn't like that person. No, totally. And another problem with plurality voting is that it's super vulnerable to something called vote splitting. So that's when candidates with some similarity and maybe with overlapping voter bases pull votes away from each other. And that leads to something called the spoiler effect, which is when voting for a losing candidate ultimately affects which candidate wins. OK, in that case, voters are pressured to vote for one of the two candidates most likely to win, even if they don't actually prefer either. Right. Which means a lot of voters might be unhappy with the options that they're given. And this brings me to the second voting system: ranked choice voting.

• 9:22Speaker 02

  Some people also call this instant runoff or single transferable vote. Yeah, we had this in the state where I used to live, Alaska. They use ranked choice for the general election. OK, so you already know how this works. But for our listeners who don't, on the ballot, you fill in a bubble for your top candidate, but you also have the option to fill in one for your second choice candidate and your third choice candidate.

• 9:44Speaker 08

  We constantly rank things in our head, what shows we're going to watch, where we want to travel, what we want to eat. It's not complex.

• 9:53Speaker 02

  So this is Carolyn Tolbert. She's a professor of political science at the University of Iowa. And she says ranked choice voting has this huge advantage, which is that it provides a majority winner. Like, over half the votes. Yeah. The winner who gets that wins the election. Yes. And that's what Carolyn says most Americans want.

• 10:11Speaker 08

  So what it does is if a candidate doesn't come out with 50% plus one, then the candidate with the least votes is removed, and their second place votes are counted and added to the tallies. So at that point, if a candidate has 50% plus one, they're the winner. If not, then the next candidate with the lowest amount of votes is removed.

• 10:40Speaker 02

  And so on, and so on, and so on. Okay. The system has gotten really popular in the last few years, right? Yeah. I mean, Caroline's researched the impact of ranked choice voting after it's been implemented as well. Mm-hmm. And she says there are several kind of surprising additional outcomes, like candidates campaigning for second and third place, which creates a different and less polarized atmosphere compared to current elections.

• 11:03Speaker 08

  And we found statistical evidence, both among the candidates who are running and among the voters, that they perceive these elections as having less negativity.

• 11:15Speaker 02

  Right. I mean, the way we do elections now, the plurality system, it's kind of a zero-sum game where candidates are incentivized to take down the competition. Absolutely. You have to vote for me if I'm going to win at all. Ranked choice voting, though, sounds like different campaigning styles may result. Yeah. And Caroline says it also makes the candidates more active in terms of voter outreach, which can increase voter turnout in the long term. Participation in democracy is always good. Yeah. But other experts I talked to also pointed out that ranked choice voting is, by nature, a more complicated system. Opponents of ranked choice voting say this makes it harder to vote, or even that it causes more people to vote wrong and thus get their ballots tossed out. Right. Because there's more room to express your preferences,

• 12:00Speaker 02

  there's also more room to fill out your ballot incorrectly. Yeah. One expert I talked to called it "more choices, more problem." Oof. Okay. So we've talked about plurality, we've talked about ranked choice. What is the third system you want to cover? So our last contender is approval voting.

• 12:18Speaker 02

  This is when you have a ballot in front of you with all the candidates listed, and instead of just choosing one, you can vote for as many or as few as you want. That's really different. Yeah. It's one of the least common electoral systems used. But even though it's not used very much in elections, it's actually fairly common in everyday life. Really? How so? Yeah. So if you have a big group of people and you have to decide where you're going for lunch, and you might just be like, "Who wants tacos? Raise your hand." "Who wants ramen? Raise your hand." You can vote for as many as you want. Yes, yes. I get it. And you just go to the place with the biggest number of votes. Exactly. Which makes the system pretty straightforward, right? You can't fill in the bubble for too few or too many candidates the way you would in other systems.

• 13:02Speaker 02

  It's simple to vote, and it's simple to count.

• 13:07Speaker 09

  One of the big appeals for me as a scholar, but also as, you know, a second gen, right? Is its equity of access. We want to make sure that people who are from diverse backgrounds and from diverse education backgrounds can still understand how to vote.

• 13:26Speaker 02

  This is Whitney Huah. She's a doctor of political science and the director of applied data and research at the Center for Election Science, which advocates for voting reform and specifically focuses on approval voting. Oh, okay. And approval voting has been implemented in two cities, St. Louis, Missouri, and Fargo, North Dakota. But the reality is, we don't have a lot of other real-world examples to point to. In St. Louis and Fargo, what have the results been? I mean, so far, it seems like it's really worked. And supporters say that that's because it basically solves the problem of vote splitting when candidates with overlapping voter bases pull votes away from each other. Exactly. Which basically makes it spoiler-proof. Mm-hmm. And it generally discourages polarization.

• 14:06Speaker 02

  If anything, it encourages consensus, which I talked to Jean-François Lalié about. He's a director at the French National Center for Scientific Research. He specializes in social choice, game theory, and economics. And he's done a ton of research on approval voting in the past 20-some years.

• 14:23Speaker 10

  You can see that people take this opportunity to vote for several in order to both stay what they want, really, and also what is reasonable to hope.

• 14:36Speaker 02

  In order to both state what they want, really, and also what it is reasonable to hope. Yeah. That's a really sensible approach. Right. It makes a lot of sense. Yeah. But unlike in ranked choice voting, you can't express intensity of preference. Like, all of your approvals are equally weighted. So it might make it harder for you to make a decision.

• 14:57Speaker 10

  So the problem with approval voting for the voters is that they wonder, where should I put the bar between the candidates that I like and the candidates that I do not like? At some point, as a voter, you have to take responsibility. You have to take decisions. It's not so simple to vote.

• 15:21Speaker 02

  And Emily, that's our three. We're done. No, we're not. No, we're not. Because I want to know which system is the best. Oh. We cannot have a whole episode about voting systems and then not pick a winner. OK. We started this episode with math. And it's only fitting that I answer this question with math.

• 15:40Speaker 02

  In social choice theory, there's this famous thing called Arrow's impossibility theorem. It was proved in the 50s by Kenneth Arrow, this famous American mathematician and political theorist. And I'm actually going to go back to our math professor, Dan, to explain this one.

• 15:54Speaker 03

  The thing that underlies Arrow's theorem is the prospect that an electorate might like A more than B, B more than C, and C more than A.

• 16:09Speaker 03

  That sounds impossible, or maybe irrational, but it absolutely can happen.

• 16:17Speaker 02

  And as any group project leader or oldest sibling can tell you, sometimes it feels like there are no decisions that you could make that would leave a majority of the group happy. Correct. Arrow's theorem tells you, you're right. In the words of Mr. Kenneth Arrow himself, "Most systems are not going to work badly all of the time, but all systems can work badly at times, assuming voters have candidate preferences." So Arrow's theorem doesn't apply as much to, say, approval voting, where you're not ranking all the candidates? Exactly. But our approval voting expert, Whitney, says the main point still stands.

• 16:51Speaker 09

  No voting method, right, can address all the criterion of what makes a good voting method, right? Like, you just need to choose which one of those matters most to you.

• 17:02Speaker 02

  Basically, Emily, you pick your problems, right? Is it more important to avoid vote splitting? Or is it more important to find consensus? Is it more important to have majority rule? And what trade-offs are you willing to make?

• 17:16Speaker 02

  Which really is just the recurring play of democracy. Exactly. Hannah Chin, thank you for this math-driven story for Election Day. My hat's off to you. Thanks for bringing this to the show. Thank you so much for having me.

• 17:31Speaker 02

  This episode was produced by Rachel Carlson, it was edited by our showrunner Rebecca Ramirez, and fact-checked by Tyler Jones. Kwasi Lee was the audio engineer. Special thanks to Sina LaFredo and Gilly Moon.

• 17:44Speaker 02

  I'm Emily Kwong. Thanks for listening to Short Wave, the science podcast from NPR.

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