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Condorcet method
A Condorcet method (; ) 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).
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. It always elects the same candidate as Copeland's method in cases with no pairwise ties.
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.
In a competition between candidates A and B where voters have ranked the candidates, if more voters mark their ballots that they prefer candidate A over candidate B than the number of voters who mark their ballots to the contrary, then Candidate B is not (supposed to be) elected.
However, because of the possibility of the Condorcet paradox, it is possible, but unlikely, that this can't be done. This is sometimes called a "Condorcet cycle" or just "cycle" and can be thought of as Candidate Rock beating Candidate Scissors, Candidate Scissors beating Candidate Paper, and Candidate Paper beating Candidate Rock. If there is no cycle (and no ties), all Condorcet methods elect the same candidate.
For most Condorcet methods, the pairwise counts are all that is necessary to figure out who won, who came in 2nd place, etc. and can be used to figure out if there is a Condorcet winner, and who it is.
A Condorcet method is a voting system that will always elect the Condorcet winner (if there is one); this is the candidate whom more voters prefer to each other candidate, when compared to them one at a time. This candidate can be found by checking if there is a candidate who beats all other candidates; this can be done by using Copeland's method and then checking if the Copeland winner has the highest possible Copeland score.
Note that there are ½"N"("N"−1) pairwise comparisons for "N" candidates. For 10 candidates, this means there are 0.5*10*9=45 comparisons, which can make elections with many candidates hard to count the votes for.
In some situations, an election has no Condorcet winner. This occurs as a result of a kind of tie known as a "majority rule cycle", described by Condorcet's paradox. Different Condorcet methods pick different winners in these situations. Some Condorcet methods involve the basic procedure described below, coupled with a Condorcet completion method, which is used to find a winner when there is no Condorcet winner. Other Condorcet methods involve an entirely different system of counting, but are classified as Condorcet methods because they will still elect the Condorcet winner if there is one.
It is important to note that not all single winner, ranked voting systems are Condorcet methods. For example, instant-runoff voting and the Borda count are not Condorcet methods.
In any Condorcet method that passes Independence of Smith-dominated alternatives, it can sometimes help to identify the Smith set from the head-to-head matchups, and eliminate all candidates not in the set before doing the procedure for that Condorcet method.
Condorcet methods use pairwise counting. For each possible pair of candidates, one pairwise count indicates how many voters prefer one of the paired candidates over the other candidate, and another pairwise count indicates how many voters have the opposite preference. The counts for all possible pairs of candidates show all the pairwise preferences of all the voters.
Pairwise counts are often displayed in a "pairwise comparison matrix" or "outranking matrix" such as those below. In these matrices, each row represents each candidate as a 'runner', while each column represents each candidate as an 'opponent'. The cells at the intersection of rows and columns each show the result of a particular pairwise comparison. Cells comparing a candidate to themselves are left blank.