What is the Condorcet paradox?

The Condorcet paradox occurs when collective preferences are cyclic: a majority prefers A over B, a majority prefers B over C, and a majority prefers C over A. There is no single 'best' choice — every candidate loses to some other in a head-to-head matchup. This is one reason Arrow's Impossibility Theorem proves no voting system can satisfy all reasonable fairness criteria simultaneously.

What is Arrow's Impossibility Theorem?

Kenneth Arrow's 1951 theorem proves that no ranked voting system with three or more options can satisfy a small set of fairness criteria simultaneously (unanimity, independence of irrelevant alternatives, non-dictatorship, transitivity). Every system makes trade-offs. The choice of voting system is therefore a value choice, not a technical one.

Why does plurality create two-party systems?

Duverger's law states that single-winner plurality systems (First-Past-the-Post) tend toward two-party competition. Voters fear 'wasting' votes on small parties, so they consolidate behind the two largest. This is why the US, UK, and Philippines (presidential plurality) all see effective two-party or two-coalition dynamics, while proportional or ranked-choice systems sustain more parties.

What is ranked-choice voting?

Ranked-choice voting (also called Instant-Runoff Voting / IRV) lets voters rank candidates by preference. If no candidate has a majority of first-choice votes, the candidate with the fewest is eliminated and their voters' next preferences are redistributed. The process repeats until someone has a majority. IRV is used in Australian House elections, Irish presidential elections, and US cities including New York and San Francisco.

Voting system comparison

Same voters. Same candidates. Six different winners.

Drag the candidate dots along the political spectrum. Six voting systems recompute the result in parallel — when they crown different people, the counter ticks.

5 candidates · 200 voters · — different winners

analysing…

Political spectrum ← Left · Centre · Right →

Drag any candidate dot. The voter density curve below shows where the electorate sits.

Voter distribution

Pre-set elections

Number of candidates5

Approval threshold40%

Each voter approves all candidates within this share of the spectrum's width.

How to use

Drag the candidate dots along the political spectrum canvas. Pick a voter distribution (bell, polarised, left-leaning…). Click a pre-set election (PH 2022, spoiler effect, Condorcet paradox) to load that scenario. The six outcome cards below the spectrum recompute every system in parallel — when they disagree, the "different winners" counter ticks up.

The six systems

Plurality (FPTP)

Most first-place votes wins, even without a majority. Used in PH (president), US, UK House. Vulnerable to spoilers — a third candidate splitting the vote.

Two-round runoff

If no one gets 50% in round 1, the top two face off in round 2. Used in French presidential elections.

Instant-runoff voting (IRV)

Rank candidates. If no majority on first preferences, lowest is eliminated, their votes redistribute to next preferences. Repeat until someone has 50%. Used in Australia, Ireland.

Borda count

Rank candidates. For N candidates, 1st = N−1 points, 2nd = N−2... Sum across all voters. Highest total wins. Used by sports rankings and some academic awards.

Approval voting

Vote for as many candidates as you "approve of." Highest approval total wins. Used by IEEE, UN Secretary-General selection.

Condorcet

The candidate who would beat every other candidate in head-to-head matchups wins. Sometimes no such candidate exists — the Condorcet paradox — exposing fundamental incompleteness of majority rule.

Arrow's Impossibility Theorem

In 1951, Kenneth Arrow proved that no ranked voting system with three or more candidates can simultaneously satisfy these reasonable fairness criteria:

Unanimity — if everyone prefers A to B, the system must rank A above B.
Independence of irrelevant alternatives — adding or removing a losing candidate shouldn't change the relative ranking of others.
Non-dictatorship — no single voter's preference dictates the outcome.
Transitivity — preferences must be consistent.

Every system breaks at least one. The choice of voting system is therefore a value choice, not a technical one. Arrow won the Nobel Prize in 1972 for this.

Real-world cases

2000 US Presidential election. Bush won Florida by 537 votes. Ralph Nader (Green) received 97,488 votes, mostly from voters who'd have preferred Gore to Bush. Under IRV, Gore likely wins. This is the canonical spoiler effect.

1992 Philippine presidential election. Fidel Ramos won with just 23.6% of the vote across seven candidates. A clear majority preferred someone else, but the system delivers the largest minority.

2016 PH elections. Rodrigo Duterte won with 39%. Mar Roxas and Grace Poe together had 45%. Under runoff or IRV, the outcome could have been different.

2022 Alaska special election. First major US race decided by ranked-choice voting. Mary Peltola (D) won over Sarah Palin (R) after IRV redistribution, despite Palin leading first preferences.

Strategic voting

The Gibbard-Satterthwaite theorem (1973) extends Arrow's result: every reasonable voting system is vulnerable to strategic manipulation — voters can sometimes get better outcomes by misrepresenting their true preferences.

Examples:

Plurality: "Don't waste your vote on a third party." Voters consolidate on viable candidates → two-party systems (Duverger's law).
Borda: rank your favourite first and least-favourite last to maximise their spread.
IRV: the "favourite betrayal" problem — sometimes ranking your true favourite first hurts your second favourite.

No system is fully strategy-proof. Different systems just have different attack surfaces.

Citations

Arrow, K. (1951) — Social Choice and Individual Values
Gibbard, A. (1973), Econometrica — strategy-proofness
Poundstone, W. (2008) — Gaming the Vote
FairVote / electionscience.org — voting reform datasets
Duverger, M. (1954) — Political Parties

This sim uses 1D ideological proximity to derive voter preferences — a useful but oversimplified model. Real voters weigh policy, identity, charisma, partisan loyalty, and tactical considerations beyond ideological distance. The approval-voting threshold is also a heuristic: in reality, voters set their own approval cutoffs.

Built by Joshua Chua · Fable rebuild · vanilla JS + Canvas