Condorcet winner criterion

In an election, a candidate is called a Condorcet (English: /kɒndɔːrˈseɪ/), beats-all, or majority-rule winner if more than half of voters would support them in any one-on-one matchup with another 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 Condorcet criterion. Condorcet voting methods extend majority rule to elections with more than one candidate.

Surprisingly, an election might not have a beats-all winner, because there can be a rock, paper, scissors cycle with multiple candidates each defeating the other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox. When there is no single best candidate, tournament solutions (like ranked pairs) choose the candidate closest to being an majority winner.

If voters are arranged on a left-right political spectrum and prefer candidates who are more similar to themselves, a beats-all 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. While political candidates differ in ways other than left-right ideology, which can lead to voting paradoxes, such cases tend to be rare in practice.