Condorcet winner criterion

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In an election, a candidate is called a majority winner or majority-preferred candidate^[1][2][3] if more than half of all voters would support them in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the majority-rule principle^[4] or majority winner criterion, and are called majoritarian because they extend the principle of majority rule to elections with multiple candidates.

In situations where equal or tied ranks are allowed, a candidate who wins a simple or relative majority—more votes for than against, ignoring abstentions—is called a Condorcet (English: /kɒndɔːrˈseɪ/),^[2] beats-all, or tournament winner (by analogy with round-robin tournaments). However, precise terminology on the topic is inconsistent. Surprisingly, an election may not have a beats-all winner: it is possible to have a rock, paper, scissors-style cycle, when multiple candidates defeat each other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox,^[5] and is analogous to the counterintuitive intransitive dice phenomenon known in probability.

However, if voters are arranged on a left-right political spectrum and prefer candidates who are more similar to themselves, a majority-rule winner always exists and is the candidate whose ideology is most representative of the electorate, a result known as the median voter theorem.^[6] However, if political candidates differ substantially in ways unrelated to left-right ideology or overall competence, this can lead to voting paradoxes.^[7][8] Previous research has found cycles to be somewhat rare in real elections, with estimates of their prevalence ranging from 1-10% of races.^[9]

Systems that elect majority winners include Ranked Pairs, Schulze's method, and the Tideman alternative method. Methods that do not include instant-runoff voting (often called ranked-choice in the United States), first preference plurality, and the two-round system. Most rated systems, like score voting and highest median, fail the majority winner criterion intentionally (see tyranny of the majority).