Short Wave - Elections: A Big Math Problem

Episode Date: November 3, 2025

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 [e-mail] 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+ at plus.npr.org/shortwave.See 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

Hey, Shaw Waivers, 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 U.S., 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 Hand take it from here. You're listening to Shortwave from NPR. 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 Shortwave right now. Regardless, before you even marked a ballot, there was a choice already made for you. And that's the choice of which voting system to use. Hey, Hannah Chinn. Hi, Emily. Hannah, our producer today, is going to tell us about some of those voting systems. Though, wait, hold on, what is a voting system? Good question. So for our purposes, a voting system is the set of rules that you use to determine the winner of an election.

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 D.C. His name is Dan Allman. And for the past 20 years on and off, he's been teaching this class called math. politics. And on the very first day of class, he has all his students conduct a mock election. 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. 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. Okay. Whether they'd like, say, candidate A more than candidate B or candidate B more than candidate C. Or maybe they like, can it be the most and they're indifferent to both A and C? Interesting. Okay, 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.

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

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 Shortwave, the science podcast from NPR. Okay, Anna, earlier you told me this kind of bonkers thing that changing the election rules could change the winner. Right, right, right. But how often does that happen in the real world, in real world elections?

So this is a really good question. And the issue is, in order to test these systems, you can't just run elections. because those have real-world consequences, right? Sure. But I found a researcher who did something really close. He and his colleagues ran a bunch of simulated scenarios using alternative voting systems. So we gather 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. This is Roman Le Chatt. He's an associate professor who studies voting behavior and electoral systems at Cience 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? In the 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 voting. But in approximately one third of the cases he simulated, Romance said there was a difference. These simulations led to different outcomes and different winners. And those cases tended to have two things in common. So the first factor is just the number of candidates. Like, the more people are parties that there are to vote for. And the second factor is polarization. like the more different that candidates are from one another.

Interesting, okay. So if in any given election there's either a lot of options or the options are particularly polarize, the voting system could matter a lot. Exactly. So we're going to look at those voting systems. 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. other candidates wins the election. This is how I had my teddy bear's vote when I was in elementary school and I held my mock election. I have questions. One vote per teddy bear. So in this scenario, yeah, you see the list of candidates. You choose one.

This is like most of the ballots I filled out in my life, though, to this day. Right. It's super common in single winner elections. And Wilman says that's for a reason. 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 we received and who are. who is elected. So that's for that reason, at least. It's a very, very good system. But it has disadvantages. 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%. 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. Okay. 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. 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. Okay, 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. 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. 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. 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 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. 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. And we found statistical evidence both among the candidates who were running and among the voters that they perceive these elections as. having less negativity. Right.

I mean, the way we do elections, 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 Carolyn 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 rank choice voting is by nature a more complicated system. Opponents of rank 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, 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 rank choice. what is the third system you want to cover?

So our last contender is approval voting. 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. It's simple to vote and it's simple to count. One of the big appeals for me as a scholar, but also as, you know, a second gen, right? is 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. This is Whitney Hua. She's the doctor of political science and the director of applied data in 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. And it generally discourages polarization. If anything, it encourages consensus, which I talked to Jean-Francée. Al-Alié 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. 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.

In order to both state what they want really and also, what it is reasonable to hope. That's a really sensible approach. Right. It makes a lot of sense. Yeah. But unlike in right 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.

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. 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. Okay. We started this episode with math. And it's only fitting that I answer this question with math.

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. 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. That sounds impossible or maybe irrational, but it absolutely can happen. 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. Correct. Error'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 Aeros 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. 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. 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 tradeoffs are you willing to make? 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. This episode was produced by Rachel Carlson. It was edited by our showrunner Rebecca Ramirez and fact-checked by Tyler Jones. Quasey Lee was the audio engineer. Special thanks to Sina LaFredo and Gilly Moon.

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