the math of elections

Overview

This NPR Shortwave episode examines how the rules for counting votes determine election winners, sometimes even when voter preferences stay the same. It introduces three voting systems—plurality (first-past-the-post), ranked-choice voting, and approval voting—and uses a classroom-style mock election and real-world simulations to show how different tally methods can produce different winners. The discussion highlights that the likelihood of a system affecting outcomes rises with more candidates and higher polarization, and it touches on the trade-offs between majority rule, simplicity, and resistance to vote-splitting.

The piece also situates the conversation within broader political science concepts, including Arrow’s impossibility theorem, which suggests no voting method can satisfy all desirable criteria all the time. Viewers are left with the sense that choosing a voting system involves weighing competing values about legitimacy, turnout, and consensus.

Introduction and Core Question

The episode opens with a foundational question in political science: how does the method used to tally votes influence who wins? A math-focused professor guides students through mock elections to illustrate how different rules can yield different outcomes even when preferences are fixed. The central idea is that the winner depends not just on who is preferred, but on how those preferences are processed and counted.

Three Voting Systems in Focus

Plurality voting (first past the post) assigns victory to the candidate with the most votes, even if that total is not a majority. The system is simple and familiar, but it risks non-majority winners and vote-splitting that can distort outcomes when similar candidates divide the vote.

Ranked choice voting (instant runoff / single transferable vote) allows voters to rank candidates. If no one exceeds 50 percent, the lowest-ranked candidate is eliminated and their votes reassigned, continuing until a majority winner emerges. Proponents note that this encourages coalition-building and can reduce polarization, while critics point to ballot complexity and the potential for miscast ballots.

Approval voting lets voters approve any number of candidates, with the top-approved candidate(s) winning. It is simpler to vote and count than ranked choice and aims to reduce vote-splitting, but it cannot express the intensity of preference and may complicate decision thresholds for voters.

What Real-World Evidence Suggests

The podcast discusses simulations using data from dozens of elections across multiple countries, showing that the system can matter more when there are many candidates or when voters are highly polarized. The results imply that the choice of tally method is not neutral; it interacts with party systems, candidate profiles, and voter behavior to shape outcomes in meaningful ways.

The Theoretical Lens

Arrow’s impossibility theorem is introduced to frame the debate: no voting method can satisfy all fairness criteria simultaneously, so every system involves trade-offs. The conversation also addresses practical considerations for voters and policymakers, emphasizing that no method solves every problem, but different methods advance different democratic goals such as majority rule, inclusiveness, or reducing polarization.

Takeaways and Implications

Ultimately, the episode encourages readers to think critically about what they value in elections—the clarity of a majority winner, the inclusivity of voter expression, or the resilience of a system to strategic voting. The choice of voting system is a policy decision with societal consequences, not just a technical detail.