In the example matrix above, B is the Condorcet winner, because they beat A and C in head-to-head elections. Proof of violation of fairness Note that the quantity of votes might not strongly favor a candidate, but that candidates only need to win the most number of contests in order to be a Condorcet winner. In the above example, B beats two other candidates (there are two green boxes indicating victory for B in the middle row) and A beats one. But the total margin by which a Condorcet winner is irrelevant: a Condorcet winner could win enough contests to be the Condorcet winner by just one vote each, while another candidate might win more votes but fewer contests. Condorcet-consistent voting systems can also, in rare cases, exhibit a preference cycle or paradox, although the circumstances that would cause this has not been known to occur yet in a governmental election using ranked ballots. Relation to other criteria The Condorcet criterion implies the majority criterion; that is, any system that satisfies the former will satisfy the latter. It further implies the mutual majority criterion whenever there is a Condorcet winner; the Smith criterion, which is a generalization of the Condorcet criterion, always implies the mutual majority criterion; not all Condorcet methods pass the Smith criterion. The Condorcet criterion is incompatible with the later-no-harm criterion, the favorite betrayal criterion, the participation criterion, and the consistency criterion. The Condorcet criterion satisfies the following criterion with some similarity to independence of irrelevant alternatives: removing losing candidates from the election can't change the result whenever there is a Condorcet winner. In addition, adding candidates who are pairwise beaten by the Condorcet winner can't change the winner when there is a Condorcet winner. (These two properties are related to, and implied by, the Independence of Smith-dominated alternatives criterion.) The Condorcet winner criterion is different from the Condorcet loser criterion. A system complying with the Condorcet loser criterion will never allow a Condorcet loser to win; that is a candidate who can be defeated in a head-to-head competition against each other candidate Compliance of methods Complying methods The following methods satisfy the Condorcet criterion: Black Copeland Dodgson's method Kemeny-Young Minimax Nanson's method Baldwin method Tideman Ranked pairs Schulze Smith/IRV Smith/minimax Tideman alternative method CPO-STV See Non-complying methods The following methods do not satisfy the Condorcet criterion. (This statement requires qualification in some cases: see the individual