Condorcet winner criterion

An electoral system satisfies the Condorcet winner criterion (pronkɒndɔrˈseɪ) if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidates - that is, a candidate preferred by more voters than any others - is the Condorcet winner, although Condorcet winners do not exist in all cases. It is sometimes simply referred to as the "Condorcet criterion", though it is very different from the "Condorcet loser criterion". Any voting method conforming to the Condorcet winner criterion is known as a Condorcet method. The Condorcet winner is the person who would win a two-candidate election against each of the other candidates in a plurality vote. For a set of candidates, the Condorcet winner is always the same regardless of the voting system in question, and can be discovered by using pairwise counting on voters' ranked preferences. A Condorcet winner will not always exist in a given set of votes, which is known as Condorcet's voting paradox; however, there will always be a smallest group of candidates such that more voters prefer anyone in the group over anyone outside of the group in a head-to-head matchup, which is known as the Smith set. When voters identify candidates on a 1-dimensional, e.g., left-to-right axis and always prefer candidates closer to themselves, a Condorcet winner always exists. Real political positions are multi-dimensional, however, which can lead to circular societal preferences with no Condorcet winner. These terms are named after the 18th-century mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet. The concept had previously been proposed by Ramon Llull in the 13th century, though this was not known until the 2001 discovery of his lost manuscripts. Suppose the following matrix of pairwise preferences exists for an election: where the vertical axis labels of the above matrix indicate the runner and the horizontal axis labels indicate the opponent and votes in a pairwise contest can be found by comparing correspondences of runner/opponent.