voting.wiki
Folio I

The Architecture of Choice

Voting is humanity's oldest technology for collective decision-making. From Athenian ostraka to modern digital ballots, every system encodes assumptions about fairness, representation, and the nature of preference itself.

This encyclopedia catalogues every known voting method — their mathematical properties, historical contexts, and practical trade-offs. Each entry is a complete reference, with interactive demonstrations that let you experience how different systems transform identical preferences into divergent outcomes.

" The method of voting is not a neutral container — it is itself a political choice.

The study of voting systems, or social choice theory, was formalized by the Marquis de Condorcet in 1785, though its roots extend to Ramon Llull's 13th-century manuscripts.

Folio II

Plurality Voting

The simplest and most widespread system: each voter selects one candidate, and the candidate with the most votes wins. Also known as First-Past-the-Post (FPTP), this method is used in the United States, United Kingdom, Canada, and India.

A
42%
B
35%
C
23%
Candidate A wins with a plurality, despite 58% of voters preferring someone else.

Plurality voting satisfies the majority criterion — if one candidate has more than 50% of the vote, they always win. However, it fails the Condorcet criterion, as the winner may lose head-to-head matchups against every other candidate.

See also Duverger's Law — the tendency of FPTP systems to converge on two-party competition
Folio III

Ranked Choice Voting

Voters rank candidates in order of preference. If no candidate receives a majority of first-choice votes, the candidate with the fewest votes is eliminated and their voters' second choices are redistributed. This process repeats until one candidate reaches a majority.

Round 1
A
40%
B
35%
C
25%
Round 2
A
45%
B
55%
Candidate B wins after redistribution, despite trailing in first-choice votes.
" RCV allows voters to express nuance — not just who they want, but who they would accept.

Also known as Instant Runoff Voting (IRV) or the Alternative Vote. Used in Australia (since 1918), Ireland, Maine, Alaska, and numerous municipal elections worldwide.

Folio IV

Approval Voting

Each voter may approve of (vote for) as many candidates as they wish. The candidate with the most approvals wins. This eliminates the spoiler effect entirely, as supporting one candidate never hurts another.

ABC
Voter 1 ✓ ✓
Voter 2 ✓ ✓
Voter 3 ✓ ✓
Total 2 2 2
A three-way tie — approval voting can produce ties more readily than plurality systems.

Approval voting was first formally studied by Brams and Fishburn in 1978. It is used by the Mathematical Association of America, the American Statistical Association, and several UN agencies.

See also Score Voting — a generalization where voters assign numerical scores rather than binary approve/disapprove
Folio V

Arrow's Impossibility

In 1951, Kenneth Arrow proved that no ranked voting system can simultaneously satisfy all of a small set of fairness criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.

This theorem fundamentally constrains the design of democratic systems — every method requires trade-offs. The question is not which system is perfect, but which imperfections are acceptable for a given context.

Theorem (Arrow, 1951)

For three or more candidates, no ordinal voting system can convert individual rankings into a community-wide ranking while satisfying unanimity, non-dictatorship, and IIA simultaneously.

" The search for the perfect voting system is, in a precise mathematical sense, futile.

Arrow's theorem earned him the Nobel Prize in Economics in 1972. It remains one of the foundational results in social choice theory and has influenced fields from economics to philosophy to computer science.