Condorcet winner criterion

The Condorcet candidate or Condorcet winner of an election is a candidate who, when compared with every other candidate, is preferred to him or her by more voters than prefer the other candidate. A Condorcet winner will not always exist in a given set of votes, which is known as Condorcet's voting paradox.

The Condorcet criterion for a voting system is that it chooses the Condorcet winner when one exists. Any method conforming to the Condorcet criterion is known as a Condorcet method.

It is named after the 18th century mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet.

Compliance of methods

Complying methods

Black, Copeland, Dodgson's method, Kemeny-Young, minimax, Nanson's method, Smith/minimax, ranked pairs and Schulze comply with the Condorcet criterion.

Non-complying methods

Approval voting, Range voting, Borda count, plurality voting, instant-runoff voting, and Bucklin voting do not comply with the Condorcet criterion.

Approval voting

Approval voting is a system in which the voter can approve (or vote for) of any number of candidates on a ballot.

Consider an election in which 70% of the voters prefer candidate A to candidate B to candidate C and vote to approve both A and B, while 30% of the voters prefer C to B to A and vote to approve both C and B. Candidate B would win (with 100% approval) even though A would be the Condorcet winner.

Note that this failure of Approval depends upon a particular generalization of the Condorcet criterion, which may not be accepted by all voting theorists. Other generalizations, such as a "votes-only" generalization that makes no reference to voter preferences, may result in a different analysis.

Range voting

Range voting is a system in which the voter gives all (or some maximum number of) candidates a score on a predetermined scale (e.g. from 1 to 5, or from 0 to 99). The winner of the election would be the candidate with the highest average score.

Range voting does not comply with the Condorcet criterion for the same reason as the Borda count. The Borda count can be viewed as a constrained type of Range voting where the scale is from 0 to the number of candidates minus 1 and the voter is not allowed to give two candidates the same score. See the case in the Borda count section for detail of why Range voting does not comply with the Condorcet criterion.

Borda count

Borda count is a voting system in which voters rank the candidates in an order of preference. Points are given for the position of a candidate in a voter's rank order. The candidate with the most points wins.

The Borda count does not comply with the Condorcet criterion in the following case. Consider an election consisting of five voters and three alternatives, in which three voters prefer A to B and B to C, while two of the voters prefer B to C and C to A. The fact that A is preferred by three of the five voters to all other alternatives makes it a Condorcet Winner. However the Borda count awards 2 points for 1st choice, 1 point for second and 0 points for third. Thus, from three voters who prefer A, A receives 6 points (3 x 2), and 0 points from the other two voters, for a total of 6 points. B receives 3 points (3 x 1) from the three voters who prefer A to B to C, and 4 points (2 x 2) from the other two voters who prefer B to C to A. With 7 points, B is the Borda winner.

Plurality voting

Consider an election in which 30% of the voters prefer candidate A to candidate B to candidate C and vote for A, 30% of the voters prefer C to A to B and vote for C, and 40% of the candidate prefer B to A to C and vote for B. Candidate B would win (with 40% of the vote) even though A would be the Condorcet winner, beating B 60% to 40%, and C 70% to 30%.

Instant-runoff voting

Instant-runoff voting (IRV) is a method (like Borda count) which allows each voter to rank all the candidates. Unlike the Borda count, IRV uses a process of elimination to assign each voter's ballot to their first choice among a dwindling list of remaining candidates until one candidate receives an outright majority of ballots. It does not comply with the Condorcet criterion. Consider, for example, the following vote count of preferences with three candidates {A,B,C}:

35: A>B>C

34: C>B>A

31: B>C>A

In this case, B is preferred to A by 65 votes to 35, and B is preferred to C by 66 to 34, hence B is strongly preferred to both A and C. B must then win according to the Condorcet criterion. Using the rules of IRV, B is ranked first by the fewest voters and is eliminated, and then C wins with the transferred votes from B.