Voting Systems Compared
Kenneth Arrow proved in 1951 that no ranked voting system can simultaneously satisfy a short list of fairness conditions any reasonable person would demand. The proof is a theorem, not an opinion. Every voting method humans have ever used or proposed picks which fairness axiom to break. The interesting question is not "which system is fair" but "which failure mode is the society willing to live with."
What each system actually does
First-past-the-post (FPTP). Voters mark one candidate. Highest vote count wins. Used in the UK, US House, Canada, India for Lok Sabha seats. Maurice Duverger's 1951 observation: single-member plurality districts converge to two dominant parties over time, because votes for third parties "waste" themselves. The 2024 UK general election: Reform UK won 14.3% of the vote and 5 seats; the Liberal Democrats won 12.2% and 72 seats. Same country, same election, vote-to-seat ratio differs by an order of magnitude.
Proportional representation (PR). Seats allocated in proportion to vote share, usually with a minimum threshold (5% in Germany, 3.25% in Israel, 0.67% in the Netherlands). Result: parliaments with 5-12 parties and coalition governments as the default. The Netherlands' 2023 Tweede Kamer election seated 15 parties. Israel's Knesset has never had a single-party majority in 75 years.
Ranked-choice / instant-runoff (IRV). Voters rank candidates. The last-place candidate is eliminated and their votes redistribute to second choices, repeating until one candidate clears 50%. Used in Australian House elections since 1918, Ireland for the presidency, Maine and Alaska for federal races since 2020-21. Eliminates the spoiler effect FPTP creates but does not produce proportional outcomes — Australia's House has two dominant blocs much like Westminster's.
Single transferable vote (STV). Ranked ballots in multi-member districts. Achieves proportionality through preferences rather than party lists. Used in Ireland's Dáil and Malta. Mathematically elegant; ballot complexity scales with district size.
Approval and score voting. Voters approve any number of candidates (approval) or rate each on a scale (score). Used in some private elections and a few US municipal races. Avoids Arrow's framework by abandoning ranked ballots, but Gibbard's 1973 theorem still guarantees strategic voting is possible.
The arithmetic of representation
| System | Effective parties (Laakso-Taagepera, typical) | Coalition government | Disproportionality (Gallagher index) |
|---|---|---|---|
| FPTP (UK, Canada, India) | 2.0-3.0 | Rare | 10-25 |
| IRV (Australia House) | 2.5-3.0 | Rare | 10-15 |
| Mixed-member (Germany, NZ) | 4.0-6.0 | Default | 1-5 |
| Open-list PR (Netherlands, Brazil) | 5.0-10.0 | Default | 1-4 |
| STV (Ireland) | 3.5-5.0 | Common | 3-8 |
The Gallagher index measures the gap between vote shares and seat shares; 0 is perfect proportionality, 25+ means the legislature is shaped very differently from the electorate. The UK's 2024 figure was 23.4, among the worst recorded in any democratic election.
What Arrow actually proved
Arrow's impossibility theorem says: for three or more candidates, no ranked voting rule can satisfy all of these simultaneously:
- Unrestricted domain — the rule works for any set of voter preferences.
- Pareto efficiency — if everyone prefers A to B, the rule does too.
- Independence of irrelevant alternatives — A vs B should not depend on a third candidate C entering.
- Non-dictatorship — no single voter determines the outcome.
Real systems break (3). Adding a candidate can flip the winner even when no voter changes their A-vs-B preference. The 2000 US presidential election is the canonical FPTP example: Ralph Nader's 97,000 Florida votes plausibly flipped the outcome from Gore to Bush. IRV reduces but does not eliminate this — the 2009 Burlington, Vermont mayoral race produced a winner who would have lost head-to-head against another candidate on the ballot, a violation called non-monotonicity.
Gibbard-Satterthwaite (1973-75) added that any non-dictatorial system with three or more options is vulnerable to strategic voting. Honest preferences cannot always produce honest outcomes.
What's contested
Does PR produce better governance or just more parties? Lijphart's Patterns of Democracy (1999, updated 2012) argues consensus democracies (mostly PR) outperform majoritarian ones on quality-of-life measures. Critics including Iversen and Soskice note PR systems also correlate with longer coalition negotiations (Belgium took 541 days to form a government in 2010-11) and harder accountability — voters cannot easily "throw the bastards out" when ten parties share blame.
Does FPTP produce stable two-party systems, or does it just hide instability? Canada's Liberal-Conservative alternation looks like classic Duverger; India's FPTP system has produced 7+ effective parties because Duverger's law operates per-district, not nationally. The UK's 2024 result suggests FPTP can produce extreme disproportionality without producing two-party stability.
Is ranked-choice worth its complexity? Maine's IRV adoption survived multiple legal challenges. Alaska's 2022 special election produced a Democratic winner in a Republican-leaning district, sparking ongoing repeal efforts. The empirical record is too thin to settle the question yet.
Why this has to do with other realms
Voting-system design is a special case of concept aggregation problems — the same mathematical structure appears in juries, peer review, market-clearing mechanisms, and recommendation algorithms. Arrow began his career as an economist working on welfare functions; his theorem applies to any attempt to aggregate individual preferences into a collective ranking. The Condorcet paradox (cyclic majorities: A beats B, B beats C, C beats A) was discovered by a French marquis in 1785 trying to figure out how the Académie des Sciences should pick members. The math has not changed in 240 years. The institutions wrapping it have, and that's where every interesting argument lives.
An open question
If every voting system breaks at least one fairness axiom, what does it mean for a constitutional designer to "choose" one? Is the choice driven by which failure mode is most legible to voters, or by which one most benefits the designers' coalition at the moment of design? The historical record on the second question is not encouraging.
Key sources
- Social Choice and Individual Values, Kenneth Arrow (1951; 2nd ed. 1963) — the foundational text. The theorem is the first 50 pages; the rest is the framework.
- Political Parties, Maurice Duverger (1951, English 1954) — origin of Duverger's law and the sociological case for plurality systems producing two parties.
- Patterns of Democracy, Arend Lijphart (1999, 2nd ed. 2012) — 36-country comparison of majoritarian vs consensus democracies, the most influential empirical defense of PR.
- Gaming the Vote, William Poundstone (2008) — accessible tour of Arrow, Gibbard, and the case for approval/score voting.
- Gallagher, M. (1991), "Proportionality, disproportionality and electoral systems," Electoral Studies 10(1) — origin of the disproportionality index used above.
- to verify: ACE Electoral Knowledge Network database (aceproject.org) for current country-by-country system inventory.
Further reading
- Numbers Rule, George Szpiro (2010) — history of voting paradoxes from Pliny the Younger through Arrow, written for non-specialists.
- Electoral Systems and Party Systems, Arend Lijphart (1994) — denser than Patterns of Democracy, but the empirical workhorse.
- Donald Saari's lectures on geometric approaches to voting (UC Irvine, available on YouTube) — shows why Arrow feels less devastating once you see voting as a geometry problem.
- The Center for Election Science (electionscience.org) — advocacy-flavored but the technical writeups on approval voting are honest about tradeoffs.
- Australia's AEC site has clean public explanations of how IRV counts work in practice, with worked examples from real elections.
See Also
- concept arrow impossibility theorem (the proof itself, in more detail)
- concept duverger law (why FPTP gravitates toward two parties)
- concept aggregation problems (the cross-realm bridge — same math, different domains)
- event 2000 us election (the textbook FPTP spoiler case)
- concept coalition government (what PR produces in practice)
- person kenneth arrow (the economist who broke voting theory at 30)
- concept condorcet paradox (the 1785 puzzle that started it all)