Condorcet method
pairwise-comparison electoral system / From Wikipedia, the free encyclopedia
A Condorcet method (English: /kɒndɔːrˈseɪ/; French: [kɔ̃dɔʁsɛ]) is one of many election methods that elects the candidate that is preferred by more voters than their opponent in every head-to-head election against each of the other candidates, whenever there is such a candidate. This candidate, known as the pairwise champion or beats-all winner, is formally called the Condorcet winner.
A Condorcet winner might not always exist in a particular election because the preference of a group of voters selecting from more than two options can possibly be cyclic — that is, it is possible (but very rare) that every candidate has an opponent that defeats them in a two-candidate contest. (This is similar to the game rock paper scissors, where each hand shape wins against only one opponent and loses to another). The possibility of such cyclic preferences in a group of voters is known as the Condorcet paradox. However, there always exists a smallest group of candidates that beats all candidates not in the group, known as the Smith set, which is guaranteed to only have the Condorcet winner in it when one exists; many Condorcet methods always elect a candidate who is in the Smith set when there is no Condorcet winner, and are thus said to be "Smith-efficient". The Condorcet winner is also usually but not necessarily the utilitarian winner (the one which maximizes social welfare).[1][2]
Condorcet voting methods are named for the 18th-century French mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet, who championed such voting systems. However, Ramon Llull devised the earliest known Condorcet method in 1299.[3] It always elects the same candidate as Copeland's method in cases with no pairwise ties.[4]
Condorcet methods usually use preferential (ranked) voting, though they can be done with separate rounds of runoff elections.
Most Condorcet methods have a single round of preferential voting, in which each voter ranks the candidates from most preferred (marked as number 1) to least preferred (marked with a higher number). A voter's ranking is often called their order of preference. The votes can be tallied in many ways to find a winner. Some - the Condorcet methods - will elect the Condorcet winner if there is one. They can also elect a winner when there is no Condorcet winner, and different Condorcet-compliant methods may elect different winners in the case of a cycle.
The procedure given in Robert's Rules of Order for voting on motions and amendments is also a Condorcet method, even though the voters do not vote by ranking the candidates. There are multiple rounds of voting, and in each round the vote is between two of the alternatives. The loser (by majority rule) of a pairing is eliminated, and the winner of a pairing is paired in a later round against another alternative. Eventually only one alternative remains, and it is the winner. This is similar to a single-winner or round-robin tournament; the total number of pairings is one less than the number of alternatives. Since a Condorcet winner will win by majority rule in each of its pairings, it will never be eliminated by Robert's Rules. But this method cannot reveal a voting paradox in which there is no Condorcet winner and a majority prefer an early loser over the eventual winner (though it will always elect someone in the Smith set). A lot of the literature on social choice theory is about this method since it is widely used and is used by important organizations (legislatures, councils, committees, etc.). It is not practical for use in public elections, however, since its multiple rounds of voting would be very expensive for voters, for candidates, and for governments.