Short Wave

Elections: A Big Math Problem

November 3, 2025

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  • Different voting systems, even when applied legitimately to the same set of voter preferences, can produce different election winners, a concept illustrated by a mock election scenario. 
  • Plurality voting (first past the post) is simple but vulnerable to not electing a majority winner and suffering from vote splitting/spoiler effects, while Ranked Choice Voting aims to ensure a majority winner and may reduce polarization. 
  • Arrow's Impossibility Theorem suggests that no single voting system can satisfy all desirable criteria simultaneously, meaning election designers must choose which trade-offs (like consensus vs. majority rule) matter most. 

Segments

Introduction to Voting Systems
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(00:00:24)
  • Key Takeaway: The rules used to determine an election winner constitute the voting system, and different rules can yield different outcomes.
  • Summary: Regina Barber introduces the episode, which revisits a discussion on alternative voting systems ahead of off-cycle elections. A voting system is defined as the set of rules used to determine the winner of an election. A mock election scenario with 99 imaginary voters and three candidates demonstrates that varying the counting rules can lead to different winners.
Simulations on System Differences
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(00:06:03)
  • Key Takeaway: Simulations across nine voting systems showed that alternative methods led to different winners in approximately one-third of tested elections.
  • Summary: Researcher Romain Lachat simulated 200 elections using data from 50 countries to test system differences. The likelihood of a different outcome increases when there are more candidates or when the candidates are highly polarized. In cases without a clear majority winner, the voting system choice significantly impacts the result.
Plurality Voting Explained
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(00:07:49)
  • Key Takeaway: Plurality voting, or first past the post, elects the candidate with the most votes, but it often fails to produce a majority winner and is susceptible to vote splitting.
  • Summary: Plurality voting requires voters to choose only one candidate, and the one with the most votes wins. Its advantages are simplicity for voters and clear connection between votes and outcome. Disadvantages include electing candidates without majority support and vulnerability to the spoiler effect when similar candidates split the vote base.
Ranked Choice Voting Mechanics
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(00:09:32)
  • Key Takeaway: Ranked Choice Voting ensures a majority winner by iteratively eliminating the lowest-ranked candidate and redistributing their votes to the voters’ next choices.
  • Summary: Ranked Choice Voting allows voters to rank candidates by preference (first, second, third, etc.). This system provides a majority winner, as the process continues until one candidate secures over 50% of the votes. Implementation has been linked to less negative campaigning and increased voter outreach.
Approval Voting System
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(00:12:28)
  • Key Takeaway: Approval voting allows voters to select as many or as few candidates as they approve of, encouraging consensus and being spoiler-proof.
  • Summary: Approval voting is implemented by allowing voters to vote for every acceptable candidate on the ballot. This system is simple to count and has been implemented in cities like St. Louis and Fargo. Its main appeal is solving vote splitting and discouraging polarization by encouraging consensus-building.
Arrow’s Impossibility Theorem
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(00:15:48)
  • Key Takeaway: Kenneth Arrow’s theorem proves that no voting system can perfectly satisfy all desirable criteria when voters exhibit cyclical preferences (A>B, B>C, C>A).
  • Summary: Arrow’s theorem demonstrates that cyclical preferences can occur, meaning no system is universally perfect. This theorem applies strongly to systems requiring full ranking, like Ranked Choice Voting. Ultimately, voters must choose which criteria—such as consensus or majority rule—are most important to them.