Condorcet winner criterion

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In an election, a candidate is called a Condorcet (English: /kɒndɔːrˈseɪ/), beats-all, or majority-rule winner^[1][2][3] if a majority of voters would support them in a race against any other candidate. Such a candidate is also called an undefeated or tournament champion (by analogy with round-robin tournaments). Voting systems where a majority-rule winner will always win the election are said to satisfy the majority-rule principle, also known as the Condorcet criterion. Condorcet voting methods extend majority rule to elections with more than one candidate.

Surprisingly, an election may not have a beats-all winner, because there can be a rock, paper, scissors-style cycle, where multiple candidates all defeat each other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox.^[4]

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 also the candidate whose ideology is most representative of the electorate. This result is known as the median voter theorem.^[5] While political candidates differ in ways other than left-right ideology, which can lead to voting paradoxes,^[6][7] such situations tend to be rare in practice.^[8]