Elections: A Big Math Problem

Summary

Short Wave – "Elections: A Big Math Problem" (Nov 3, 2025)

Overview

On the eve of off-cycle elections across many U.S. states, hosts Emily Kwong and producer Hannah Chin dig into the “math of elections” — specifically, how different voting systems can significantly affect election outcomes. With a light and engaging tone, the episode uses expert interviews and real-world research to explain the hidden power of voting rules, introduces three major election systems, and discusses their tradeoffs. The ultimate takeaway: there’s no perfect way to vote, but understanding the math helps make sense of democracy’s complexities.

---

Key Discussion Points & Insights

1. Why Voting Systems Matter

• Emily introduces the concept: “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.” [00:50]
• Hannah defines a voting system as the rules that determine the winner of an election; these rules can dramatically affect the result.
• Math professor Dan Allman’s classroom experiment shows that simply changing election rules (not the voters or candidates) can yield different winners — highlighting the importance of the math behind voting.
  • Quote: “Any two different methods, if they're actually different, could conceivably lead to a different result.” – Dan Allman [02:56]

2. How Often Does the Voting System Change the Winner?

• Political scientist Roman Lachat (Sciences Po, Paris) analyzed data from around 200 elections in 50 countries using simulated different systems.
  • In about one-third of cases, the voting method changed the winner—especially where more candidates are present or where choices are very polarized.
  • Quote: “In approximately one third of the cases... Roman said there was a difference.” – Hannah Chin [06:55]

3. Three Voting Systems Explained

a) Plurality Voting ("First Past the Post")

• Most common in the U.S. and elsewhere.
  • Each voter picks one candidate.
  • Winner is the person with the most votes.
  • Pros: Simplicity and transparency—everyone understands how the winner is chosen.
  • Cons:
    • Often fails to reflect a majority choice and is prone to “vote splitting” and the “spoiler effect.”
      • Example: “There might not be an actual majority winner... more than half the voters didn’t like that person.” – Emily Kwong [08:29]
      • Quote: “Another problem… is that it's super vulnerable to something called vote split. So that's when candidates with some similarity… pull votes away from each other.” – Hannah Chin [08:47]

b) Ranked Choice Voting (RCV/Instant Runoff)

• Voters rank candidates by preference; lowest candidates are eliminated and votes redistributed until someone has a majority (>50%).
  • Pros: Tends to produce majority winners, encourages less negative campaigning, and can increase turnout.
    • Quote: “We found statistical evidence… that they perceive these elections as having less negativity.” – Carolyn Tolbert (Univ. of Iowa) [11:09]
    • Can change campaign strategies to seek second- and third-place votes.
  • Cons: More complex for voters, with risk of incorrectly filled ballots (“more choices, more problems”).
    • “Opponents… say this makes it harder to vote, or even that it causes more people to vote wrong.” – Hannah Chin [11:48]

c) Approval Voting

• Voters can select (“approve of”) as many candidates as they like.
  • The candidate with the most approvals wins.
  • Pros: Easy to understand and count; avoids vote splitting and the spoiler effect; encourages consensus.
    • Quote: “You can see that people take this opportunity to vote for several in order to both state what they want, really, and also what is reasonable to hope.” – Jean Francois Lallier [14:29]
  • Cons: Does not capture preference order or intensity; voters must decide where to draw the line between candidates they like and dislike.
    • Quote: “At some point, as a voter, you have to take responsibility… It's not so simple to vote.” – Roman Lachat [15:02]

4. Is There a 'Best' Voting System? (Arrow's Impossibility Theorem)

• Emily pushes: “I want to know which system is the best.” [15:29]
• Hannah explains Arrow’s Impossibility Theorem via Dan Allman: it’s mathematically impossible for any voting system to perfectly satisfy all fairness criteria all of the time.
  • Quote: “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.” – Hannah Chin [16:32]
• Approval voting partially skirts this theorem but cannot avoid all tradeoffs.
  • “No voting method… can address all the criterion of what makes a good voting method… you just need to choose which one of those matters most to you.” – Whitney Hua [16:57]
• Key insight: In democracy, we’re always choosing which flaws and tradeoffs we’re willing to accept.

---

Notable Quotes & Memorable Moments

• On Changing Rules Changing Outcomes:
  • “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.” – Dan Allman [03:13]
• On Majority Rule:
  • “Ranked choice voting has this huge advantage, which is that it provides a majority winner, like over half the votes.” – Carolyn Tolbert [09:58]
• On the Core Dilemma:
  • “Basically, Emily, you pick your problems… 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.” – Hannah Chin [17:07]
• On Democracy:
  • “Which really is just the recurring plight of democracy?” – Emily Kwong [17:21]

---

Timestamps for Key Segments

• [01:12] – What is a voting system? (Definition, Dan Allman’s classroom demo)
• [02:56] – The same votes, different counting rules, different winners
• [06:21] – Roman Lachat’s study: How often do different systems yield different winners?
• [07:35] – Plurality voting: strengths, weaknesses, and spoiler effects
• [09:25] – Ranked choice voting: how it works, pros & cons, expert analysis
• [12:18] – Approval voting: explanation, real-world examples, expert insight
• [15:37] – Arrow’s Impossibility Theorem and the impossibility of a perfect system
• [16:57] – Picking your priorities and tradeoffs in voting system design

---

Tone and Style

• Friendly, clear, and fast-paced, with relatable analogies (e.g., teddy bear elections, lunch orders) and playful banter.
• Experts provide clear, accessible explanations and highlight real-world consequences and mathematical realities.
• Final message: Understanding the math of elections demystifies how democracy works — and why no system is perfect.

---

Handy Summary Sentence

No election system is flawless; each has its own math and tradeoffs. But by understanding their differences — and the impossibility of the perfect system — we’re better equipped to make informed choices about the future of democracy.

Transcript

[Announcer]

This message comes from Memorial Sloan Kettering Cancer center, working to change the story of pancreatic cancer. And during Pancreatic Cancer Awareness Month, your support goes twice as far to accelerate breakthrough treatments. Donate [e-mail] give.

[Regina Barber]

Hey. Short 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 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 Han take it from here.

[Emily Kwong]

You're listening to Short Wave 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.

[Hannah Chin]

You, and that's the choice of which voting system to use.

[Emily Kwong]

Hey, Hannah Chin.

[Hannah Chin]

Hi, Emily.

[Emily Kwong]

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

[Hannah Chin]

Good question. So, for our purposes, a voting system is the set of rules that 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 and Politics. And on the very first day of class, he has all his students conduct a mock election.

[Dan Allman]

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.

[Hannah Chin]

The 99 voters are these made up people.

[Emily Kwong]

Oh, so it's not the students? No, no, no, no.

[Hannah Chin]

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.

[Emily Kwong]

A and C. Interesting. Okay, so then what happens?

[Hannah Chin]

So they have to decide which election rules they're going to use. So maybe they want to weight 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 gonna count each voter's first choice and just totally ignore their other preferences.

[Emily Kwong]

He's letting them, like, change the rules of the game.

[Announcer 2]

Right.

[Emily Kwong]

Okay.

[Hannah Chin]

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

[Dan Allman]

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

[Emily Kwong]

That's striking.

[Whitney Hua]

Yeah.

[Hannah Chin]

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.

[Dan Allman]

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.

[Emily Kwong]

So today on the show, the math of elections, we're going to cover three.

[Hannah Chin]

Different voting systems, three different ways that a ballot can look and that voter preferences can be taken into account.

[Emily Kwong]

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.

[Announcer 2]

This message comes from Memorial Sloan Kettering Cancer Center. At Memorial Sloan Kettering Cancer center, they know that a cancer diagnosis touches more than one life. It touches entire families. Make a gift to MSK in honor or in memory of someone you love. Your donation fuels groundbreaking research and compassionate care for patients and their families all over the world. Honor their legacy. Give hope to others. Your impact can be doubled for a limited time. Donate [e-mail] honor2x support for NPR and.

[Announcer]

The following message come from. Edward Jones. What does it mean to live a rich life? It means brave first leaps, tearful goodbyes, and everything in between. With over a hundred years of experience navigating the ups and downs of the market and of life, your Edward Jones financial advisor will be there to help you move ahead with confidence. Because with all you've done to find your rich, they'll do all they can to help you keep enjoying it. Edward Jones, Member, SIPC this message comes.

[Announcer 2]

From Synchrony bank, who can help you get your do nothing savings off the couch and working hard. No more sleeping late and needing all the chips. Time to Start earning your keep in a big way with their high yield savings account that has a great rate, easy access to your funds, and no monthly fees or minimum balance requirements. Put your lazy savings to work. Go to synchrony.com NPR Member, FDIC.

[Hannah Chin]

Okay.

[Emily Kwong]

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

[Whitney Hua]

Right?

[Hannah Chin]

Right, right.

[Emily Kwong]

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

[Hannah Chin]

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

[Roman Lachat]

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.

[Hannah Chin]

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.

[Roman Lachat]

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

[Hannah Chin]

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. 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.

[Emily Kwong]

Interesting. Okay, 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.

[Whitney Hua]

Exactly.

[Hannah Chin]

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 candidates wins the election.

[Emily Kwong]

This is how I had my teddy bears vote when I was in elementary.

[Hannah Chin]

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.

[Emily Kwong]

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

[Hannah Chin]

Right. It's super common. In single winner elections. And Roman says that's for a reason.

[Roman Lachat]

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 gets elected. So that's, for that reason at least. It's a very, very good system. But it has disadvantages.

[Emily 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%. And in that scenario, the second candidate would get elected even though more than half the voters didn't like that person.

[Hannah Chin]

No, totally. And another problem with plurality voting is that it's super vulnerable to something called vote splitt. 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.

[Emily Kwong]

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.

[Hannah Chin]

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.

[Emily Kwong]

Right?

[Hannah Chin]

Ranked choice voting. Some people also call this instant runoff or single transferable vote.

[Emily Kwong]

Yeah. We have this in the state where I used to live, Alaska, they use ranked choice for the general election. Okay.

[Hannah Chin]

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.

[Carolyn Tolbert]

We constantly rank things in our head.

[Regina Barber]

What shows we're going to watch, where.

[Carolyn Tolbert]

We want to travel, what we want to eat. It's not complex.

[Hannah Chin]

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.

[Whitney Hua]

Yeah.

[Emily Kwong]

The winner who gets that wins the election.

[Hannah Chin]

Yes. And that's what Caroline says most Americans want.

[Announcer]

So.

[Carolyn Tolbert]

So what it does is if a candidate doesn't come out with 50% or 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.

[Hannah Chin]

And so on and so on and so on.

[Emily Kwong]

Okay. This system has gotten really popular in the last few years, right?

[Hannah Chin]

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.

[Carolyn Tolbert]

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.

[Hannah Chin]

Right.

[Emily Kwong]

I mean, the way we do elections now, the plurality system is kind of a zero sum game where candidates are incentivized to take down the competition. 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.

[Announcer 2]

Yeah.

[Hannah Chin]

And Carolyn says it also makes the candidates more active in terms of voter outreach, which can increase voter turnout in the long term.

[Emily Kwong]

Participation in democracy is always good.

[Hannah Chin]

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.

[Emily Kwong]

Right. Because there's more room to express your preferences. There's also more room to fill out your ballot incorrectly.

[Hannah Chin]

Yeah. One expert I talked to called it more choices, more problems.

[Emily Kwong]

Okay, so we've talked about plurality, we've talked about ranked choice. What is the third system you want to cover?

[Hannah Chin]

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.

[Emily Kwong]

That's really different.

[Hannah Chin]

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.

[Emily Kwong]

Really? How so?

[Hannah Chin]

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.

[Emily Kwong]

Yes, yes, I get it. And you just go to the place with the biggest number of votes.

[Hannah Chin]

Exactly. Which makes the system pretty straightforward.

[Emily Kwong]

Right.

[Hannah Chin]

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.

[Whitney Hua]

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.

[Hannah Chin]

This is Whitney Hua. 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.

[Emily Kwong]

Oh, okay.

[Hannah Chin]

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.

[Emily Kwong]

To point to in St. Louis and Fargo. What have the results been?

[Hannah Chin]

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.

[Emily Kwong]

When candidates with overlapping voter bases pull votes away from each other.

[Hannah Chin]

Exactly. Which basically makes it spoiler proof. And it generally discourages polarization. If anything, it encourages consensus, which I talked to Jean Francois Lallier 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.

[Roman Lachat]

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.

[Emily Kwong]

In order to both state what they want, really, and also what it is reasonable to hope. That's a really sensible approach.

[Announcer 2]

Right?

[Hannah Chin]

It makes a lot of sense. 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.

[Roman Lachat]

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.

[Hannah Chin]

And Emily, that's our three. We're done.

[Emily Kwong]

No, we're not. No, we're not, because I want to know which system is the best.

[Hannah Chin]

Oh, we cannot have a whole episode.

[Emily Kwong]

About voting systems and then not pick a winner.

[Hannah Chin]

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, famous American mathematician and political theorist. And I'm actually going to go back to our math professor, Dan, to explain this one.

[Dan Allman]

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.

[Hannah Chin]

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.

[Emily Kwong]

Correct.

[Hannah Chin]

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.

[Emily Kwong]

So Arrow's theorem doesn't apply as much to say, approval voting, where you're not ranking all the candidates.

[Hannah Chin]

Exactly. But our approval voting expert Whitney says the main point still stands.

[Whitney Hua]

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.

[Hannah Chin]

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.

[Emily Kwong]

Which really is just the recurring plight of democracy?

[Hannah Chin]

Exactly.

[Emily Kwong]

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.

[Hannah Chin]

Thank you so much for having me.

[Emily Kwong]

This episode was produced by Rachel Carlson. It was edited by our showrunner Rebecca Ramirez, and fact checked by Tyler Jones. Kwesi Lee was the audio engineer. Special thanks to Sina Lofredo and Gilly Moon. I'm Emily Kwong. Thanks for listening to Shortwave, the Science podcast from NPR.

[Announcer]

This message comes from Vanta. What's your 2am Security worry? Is it do I have the right controls in place or are my vendors secure? Enter Vanta. Vanta automates manual work so you can stop sweating over spreadsheets, chasing audit evidence and filling out endless questionnaires. Their trust management platform continuously monitors your systems, centralizes your data and simplifies your security at scale. Get [e-mail] that's V A N T.

[Announcer 2]

A dot com this message comes from Mint Mobile starting at $15 a month. Make the switch at mintmobile.com Switch $45 upfront payment for 3 months 5 gigabyte plan equivalent to $15 a month Taxes and fees extra first 3 months only.

[Announcer]

See Terms support for NPR and the following message come from Warby Parker, the One Stop Shop for all your vision needs. They offer expertly crafted prescription eyewear plus contacts, eye exams and more. For everything you need to see. Visit your nearest Warby Parker store or head to warbyparker.com.