voting.wiki — Encyclopedia of Voting Systems

1Overview #

A voting system (or electoral method) is a set of rules that determine how votes are cast, counted, and translated into a collective outcome. Methods differ along three axes: the ballot format, the counting algorithm, and the representational target. Each axis carries trade-offs articulated by Arrow's impossibility theorem and the Gibbard–Satterthwaite theorem.

This article surveys the most widely deployed methods, contrasts their incentive structures, and provides illustrative ballot diagrams. For a deeper dive into specific algorithms, see §5 Counting Algorithms.

"The mathematics of choice is the mathematics of compromise — every method privileges one fairness criterion over another."
— Kenneth J. Arrow, Social Choice and Individual Values (1951)

2Voting Systems #

The four families below cover ~94% of national-level elections worldwide. Click any tab to switch the inline reference card.

2.1 First-Past-The-Post (FPTP)

Each voter selects a single candidate; the candidate with the most votes — even without a majority — wins the seat. Used in single-member constituencies in the United Kingdom, the United States, India, and Canada.

Ballot type: Single mark
Tally: Plurality
Spoiler risk: High
Year codified: 1429 (Knights of the Shire Act)

Spoiler dynamics produce Duverger's law: stable two-party systems emerge under FPTP because third parties siphon votes from their nearest ideological neighbor. See §3 Comparison.

2.2 Ranked-Choice Voting (RCV / IRV)

Voters rank candidates in order of preference. The lowest-ranked candidate is eliminated each round and their ballots redistribute to surviving choices until a candidate exceeds 50%. Australia (1918), Ireland (1922), and Maine (2018) use variants.

Ballot type: Ordinal ranking
Tally: Instant runoff
Spoiler risk: Low
Year codified: 1855 (Hare proposal)

RCV satisfies the later-no-harm criterion but fails monotonicity in pathological cases — a phenomenon explored in §5.

2.3 Approval Voting

Voters mark every candidate they "approve". Each approval counts as one vote; the candidate with the most approvals wins. Used by the United Nations Secretary-General selection process and several professional societies (IEEE, AMS).

Ballot type: Multi-mark
Tally: Sum of approvals
Spoiler risk: Very low
Year codified: 1977 (Brams & Fishburn)

Approval voting satisfies independence of irrelevant alternatives in the sincere-vote model and is mathematically equivalent to score voting on a binary scale.

2.4 Proportional Representation (PR)

Seats in a multi-member district are allocated in proportion to vote share. The two dominant families are party-list PR (Netherlands, Israel) and single transferable vote (Ireland, Malta). Allocation uses divisor methods (D'Hondt, Sainte-Laguë) or quota methods (Hare, Droop).

Ballot type: List or ordinal
Tally: Quota / divisor
Spoiler risk: None
Year codified: 1855 (C. Andræ, Denmark)

PR maximizes representativeness at the expense of single-member accountability — a trade-off central to comparative analysis.

3Comparison #

The matrix below summarizes how each method scores against five fairness criteria. Hover any cell to view its formal definition.

Criterion	FPTP	RCV	Approval	PR
Condorcet criterion	Fails	Partial	Partial	N/A
Monotonicity	Passes	Fails	Passes	Passes
IIA	Fails	Fails	Passes*	Partial
Proportionality	Low	Medium	Medium	High
Strategy resistance	Low	High	Medium	Medium

Pros — RCV adoption

Eliminates spoiler effect in single-winner races.
Encourages broad coalition-building.
Reduces negative campaigning (voters become #2 audience).
Compatible with existing single-member districts.

Cons — RCV adoption

Failure of monotonicity in pathological scenarios.
Complex centralized counting (re-tabulation per round).
Voter education requirements.
Ballot exhaustion if rankings are short.

4Ballot Diagrams #

The diagrams below illustrate ballot formats schematically. Selections are interactive — click a candidate to mark or rank.

First-Past-The-Post · single mark

Adelaide MercerCIV
Bram OkaforPRG
Camilla ReyesUNI
Devon WhitlockREF

Tap one candidate.

Ranked-Choice · ordinal preference

Adelaide MercerCIV
Bram OkaforPRG
Camilla ReyesUNI
Devon WhitlockREF

Tap to assign 1st, 2nd, 3rd…

Approval · multi-mark

Adelaide MercerCIV
Bram OkaforPRG
Camilla ReyesUNI
Devon WhitlockREF

Tap any number you approve.

5Counting Algorithms #

The pseudocode block illustrates an instant-runoff tabulation operating on the canonical ballot set B = {b₁, …, bₙ}.

// Instant-runoff voting (IRV) tabulation
function irv(ballots, candidates) {
  while (candidates.length > 1) {
    const tally = countFirstChoice(ballots, candidates);
    const top   = argmax(tally);
    if (tally[top] > ballots.length / 2) return top;
    const loser = argmin(tally);
    candidates = candidates.filter(c => c !== loser);
  }
  return candidates[0];
}

Live IRV round simulation

6Glossary #

Condorcet winner: A candidate who would defeat every other candidate in head-to-head pairwise contests.
Droop quota: The smallest number of votes guaranteeing election in a multi-member district: ⌊V/(S+1)⌋ + 1.
Duverger's law: The empirical observation that single-winner plurality elections tend toward two-party systems.
Monotonicity: The principle that increasing support for a candidate cannot cause them to lose.
Strategic voting: Casting a ballot that does not reflect sincere preferences in order to influence the outcome.

7References #

Arrow, K. J. (1951). Social Choice and Individual Values. New Haven: Yale University Press.
Brams, S. J. & Fishburn, P. C. (1978). "Approval voting." American Political Science Review, 72(3).
Gallagher, M. (1991). "Proportionality, disproportionality and electoral systems." Electoral Studies, 10(1).
Hare, T. (1859). A Treatise on the Election of Representatives. London: Longman.
Tideman, T. N. (2006). Collective Decisions and Voting. Aldershot: Ashgate.