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.