Condorcet winner
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A Condorcet winner (French: [kɔ̃dɔʁsɛ], English: /kɒndɔːrˈseɪ/) is a candidate who would receive more support than their opponent in a one-on-one race against any one of their opponents. Voting systems where a Condorcet winner will always win the election are said to satisfy the Condorcet winner criterion. The Condorcet winner criterion extends the principle of majority rule to elections with multiple candidates.
Named after Nicolas de Condorcet, it is also called a majority winner, a majority-preferred candidate, a beats-all winner, or tournament winner (by analogy with round-robin tournaments). A Condorcet winner may not necessarily always exist in a given electorate: for example, 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, and is analogous to the counterintuitive intransitive dice phenomenon known in probability. However, the Smith set, a generalization of the Condorcet criteria that is the smallest set of candidates that are pairwise unbeaten by every candidate outside of it, will always exist.
If voters are arranged on a single 1-dimensional axis, such as a left-right political spectrum, and always 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; this result is known as the median voter theorem. However, political electorates are inherently multidimensional in real-life, and the use of a one- or even two-dimensional model of such electorates would be inaccurate. For multi-dimensional spaces there might be no Condorcet winner according to the McKelvey–Schofield chaos theorem.
Previous research has found cycles to be somewhat rare in real elections, with estimates of their prevalence ranging from 1% to 10% of races.
Systems that guarantee the election of a Condorcet winners (when one exists) include Ranked pairs, Schulze's method, and the Tideman alternative method. Methods that do not guarantee that the Cordorcet winner will be elected, even when one does exist, include instant-runoff voting (often called ranked-choice in the United States), first-past-the-post voting, and the two-round system. Most rated systems (like score voting and the highest median rules) are not Condorcet methods, as they try to account for the strengths of different voters' preferences rather than always following the majority.