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What Is Voting Theory?

Voting theory — also called social choice theory — is the mathematical and philosophical study of how collective decisions are made through voting. It examines different voting systems, analyzes their properties, identifies paradoxes and failures, and asks a deceptively simple question: when a group of people with different preferences need to make a collective choice, what’s the fairest way to do it?

The answer, it turns out, is that there is no perfect answer. But some methods are considerably better than others.

The Problem with Plurality Voting

Most Americans are familiar with plurality voting (also called first-past-the-post): everyone picks one candidate, and whoever gets the most votes wins. Simple. Intuitive. And deeply flawed.

The spoiler effect — A third candidate can split votes with a similar candidate, causing both to lose to someone a majority of voters oppose. Ralph Nader in the 2000 U.S. election is the textbook example — his 97,000 votes in Florida likely cost Al Gore the presidency, even though most Nader voters preferred Gore to Bush.

Wasted votes — In a plurality system, every vote for a losing candidate produces nothing. If your preferred candidate has no chance of winning, voting for them is strategically pointless, which discourages both third-party candidacies and honest voting.

Minority winners — A candidate can win with far less than majority support. In a five-way race, someone could win with 21% of the vote — meaning 79% of voters preferred someone else.

Two-party lock-in — Plurality voting mathematically tends toward two dominant parties (Duverger’s Law). Third parties can’t break through because voters fear “wasting” their votes, so they strategically vote for the lesser of two evils rather than their true preference.

Alternative Voting Systems

Ranked-Choice Voting (Instant Runoff)

Voters rank candidates in order of preference. If no candidate has a majority of first-choice votes, the last-place candidate is eliminated and those ballots transfer to voters’ next choices. This continues until someone achieves a majority.

Advantages: Eliminates the spoiler effect. Allows voters to express genuine preferences. Encourages more candidates and more civil campaigns (candidates want to be voters’ second choice, so they avoid attacking opponents personally).

Disadvantages: More complex for voters (though evidence from cities that use it suggests most voters adapt quickly). Can produce non-monotonic results in rare cases (where ranking a candidate higher actually hurts them).

Proportional Representation

Instead of single-member districts, voters elect a multi-member body where seats are distributed proportionally to vote share. If a party gets 30% of votes, they get roughly 30% of seats.

Most democracies outside the English-speaking world use some form of proportional representation. It produces legislatures that more accurately reflect voter preferences but can lead to fragmented multi-party systems requiring coalition governments.

Approval Voting

Voters can vote for (approve of) as many candidates as they like. The candidate with the most approvals wins. Simple to implement (same ballots, same machines), eliminates vote splitting, and lets voters support both their true preference and a “safe” choice simultaneously.

Condorcet Methods

These systems identify the candidate who would beat every other candidate in a head-to-head matchup. The “Condorcet winner” (when one exists) is arguably the most democratically legitimate outcome because a majority prefers them over each alternative individually.

The Mathematics: Why Perfection Is Impossible

Kenneth Arrow’s impossibility theorem (1951) is the central result of voting theory, and it’s genuinely surprising. Arrow proved that no ranked voting system can simultaneously satisfy all of these seemingly reasonable criteria:

1. Non-dictatorship — The result can’t be determined by a single voter’s preferences
2. Pareto efficiency — If every voter prefers A to B, the group should rank A above B
3. Independence of irrelevant alternatives — The group’s ranking of A vs. B shouldn’t change based on how they rank a third candidate C
4. Unrestricted domain — The system must work for any possible set of voter preferences
5. Social ordering — The output is a complete ranking, not just a winner

Each criterion seems obviously necessary. Yet Arrow proved they’re collectively impossible to achieve. Every voting system violates at least one of them.

This doesn’t mean all voting systems are equally bad — it means that choosing a system involves choosing which tradeoffs you’re willing to accept. Arrow’s theorem sets the boundaries of what’s mathematically possible.

The Gibbard-Satterthwaite Theorem

Related to Arrow’s result, this theorem proves that any non-dictatorial voting system with three or more candidates is susceptible to strategic voting — voters can sometimes get a better outcome by voting dishonestly (misrepresenting their true preferences) than by voting honestly.

Strategic voting is ubiquitous in plurality systems (voting for the “lesser evil” rather than your true preference). Different systems are susceptible to different strategic manipulations, but no system is immune.

Real-World Reform Efforts

Voting reform is gaining traction in the United States and elsewhere:

Alaska adopted a nonpartisan top-four primary with ranked-choice general election in 2020. The result: a more diverse field of candidates and the election of Mary Peltola, the first Alaska Native to serve in Congress.

Maine became the first U.S. state to use ranked-choice voting for federal elections in 2018.

New Zealand switched from plurality to Mixed Member Proportional representation in 1996, dramatically changing its political field from a two-party system to a multi-party democracy.

Over 50 U.S. jurisdictions now use ranked-choice voting for local or state elections.

The common argument against reform is “if it ain’t broke, don’t fix it.” But voting theory demonstrates quantitatively that plurality voting is broken — it systematically fails to elect candidates that majorities prefer, discourages honest preference expression, and locks voters into choosing between two options even when better alternatives exist.

Whether any specific alternative is politically achievable is a different question from whether it’s mathematically superior. Voting theory can answer the second question. The first is up to voters — and the system they use to decide.

Frequently Asked Questions

What is Arrow's impossibility theorem?

Arrow's theorem, proved by economist Kenneth Arrow in 1951, demonstrates that no ranked voting system with three or more candidates can simultaneously satisfy all of these reasonable criteria: non-dictatorship, Pareto efficiency, independence of irrelevant alternatives, unrestricted domain, and social ordering. In simpler terms, every voting system involves tradeoffs — there's no perfect method that always produces a 'fair' result by all reasonable definitions of fairness.

What is ranked-choice voting?

Ranked-choice voting (RCV), also called instant-runoff voting, lets voters rank candidates in order of preference. If no candidate wins a majority of first-choice votes, the candidate with the fewest votes is eliminated and their voters' ballots transfer to their second choice. This process repeats until someone has a majority. RCV is used in Australia nationally, in several U.S. cities and states (including Alaska and Maine), and in various other countries.

Why does the voting system matter?

The same voters with the same preferences can produce different winners depending on which voting system is used. In the 2000 U.S. presidential election, Ralph Nader received enough votes in Florida to likely have changed the outcome under different voting rules. Voting systems affect which candidates run, how campaigns behave, how many parties can be viable, and whether election results reflect voter preferences accurately.

Further Reading

• Stanford Encyclopedia of Philosophy — Voting Methods
• FairVote — Electoral Reform
• MIT Election Data + Science Lab