Condorcet winner criterion

The Condorcet candidate (a.k.a. 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 voting system satisfies the Condorcet criterion (English: /kɒndɔːrˈseɪ/) if it always chooses the Condorcet winner when one exists. Any voting method conforming to the Condorcet criterion is known as a Condorcet method.

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 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.

Example

Suppose the following pairwise preferences exist for an election:

Wins are bolded

	A	B	C
A	---	186	405
B	305	---	272
C	78	105	---

B is the Condorcet winner, because they beat A and C in head-to-head elections.

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.)

Compliance of methods

Complying methods

Main article: Condorcet method

The following methods satisfy the Condorcet criterion:

• Black
• Copeland
• Dodgson's method
• Kemeny-Young
• Minimax
• Nanson's method
• Ranked pairs
• Schulze
• Smith/IRV
• Smith/minimax
• Fully strategic Approval voting

Non-complying methods

The following methods do not satisfy the Condorcet criterion. (This statement requires qualification in some cases: see the individual subsections.)

• Borda count
• Bucklin voting
• Instant-runoff voting
• Majority Judgment
• Plurality voting
• Honest Approval voting
• Range voting

Borda count

Main article: 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.

Bucklin voting

Main article: Bucklin voting

Bucklin is a ranked voting method that was used in some elections during the early 20th century in the United States. The election proceeds in rounds, one rank at a time, until a majority is reached. Initially, votes are counted for all candidates ranked in first place; if no candidate has a majority, votes are recounted with candidates in both first and second place. This continues until one candidate has a total number of votes that is more than half the number of voters. Because multiple candidates per vote may be considered at one time, it is possible for more than one candidate to achieve a majority.

Instant-runoff voting

Main article: Instant-runoff voting

Instant-runoff voting (IRV) is a method (like Borda count) which requires each voter to rank 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.

Note that 65 voters, a majority, prefer either candidate B or C over A; since IRV passes the mutual majority criterion, it guarantees one of B and C must win. If candidate A, an irrelevant alternative under IRV, was not running, a majority of voters would consider B their 1st choice, and IRV's mutual majority compliance would thus ensure B wins; in this way, IRV's failure of the Condorcet criterion here also implies a spoiler effect. In cases where there is a Condorcet Winner, and where IRV does not choose it, a plurality would by definition prefer the Condorcet Winner to the IRV winner.

Majority judgment

Main article: Majority judgment

Majority judgment is a system in which the voter gives all candidates a rating out of a predetermined set (e.g. {"excellent", "fair", "poor"}). The winner of the election would be the candidate with the best median rating.

Consider an election with three candidates A, B, C.

35 voters rate candidate A "excellent", B "fair", and C "poor",

34 voters rate candidate C "excellent", B "fair", and A "poor", and

31 voters rate candidate B "excellent", C "fair", and A "poor".

B is preferred to A by 65 votes to 35, and B is preferred to C by 66 to 34. Hence, B is the Condorcet winner. But B only gets the median rating "fair", while C has the median rating "good" and hereby C is chosen winner by Majority Judgment.

Plurality voting

Main article: Plurality voting system

With plurality voting, the full set of voter preferences is not recorded on the ballot and so cannot be deduced therefrom (e.g. following a real election). Under the assumption that no tactical voting takes place, i.e. that all voters vote for their first preference, it is easy to construct an example which fails the Condorcet criterion.

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 voters 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%.

The assumption of no tactical voting is also used to evaluate other systems; however, the assumption may be far less plausible with plurality precisely because plurality accommodates no other way for subsidiary preferences to be taken into account.

Approval voting

Main article: Approval voting

Approval voting is a system in which the voter can approve of (or vote for) any number of candidates on a ballot. Depending on which strategies voters use, the Condorcet criterion may be violated.

Consider an election in which 70% of the voters prefer candidate A to candidate B to candidate C, while 30% of the voters prefer C to B to A. If every voter votes for their top two favorites, 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. Also, if all voters have perfect information about each other's motivations, and a single Condorcet winner exists, then that candidate will win under the Nash equilibrium.

Range voting

Main article: Range voting

Range voting is a system in which the voter gives all candidates a score on a predetermined scale (e.g. from 0 to 9). The winner of the election is the candidate with the highest total score.

Range voting doesn't satisfy the Condorcet criterion. Consider an election with three voters and three candidates with the following range votes:

Candidate	Voter 1	Voter 2	Voter 3
A	5	5	1
B	4	4	4
C	0	0	0

In pluralistic head-to-head elections, two voters prefer A to B, and all three prefer both A and B to C, making A the Condorcet winner. However, candidate B is the range winner with 12 points compared to 11 points for A.

Range voting satisfies the Condorcet criterion as long as voters score candidates in the head-to-head elections as they do in the full election. For example, let's say three voters vote for three candidates (A,B,C) as follows:

The second candidate is the Condorcet winner and the winner of the normal election with 12 to 10 and 0 points. In the case where all voters are voting strategically, range voting is equivalent to approval voting, and any Condorcet winner will win because of the Nash equilibrium as mentioned above.

However, if voters change their voting strategy from honest to strategic only for the head-to-head elections, then range voting does not satisfy Condorcet. For the same example shown above, the head to head elections involving A would look like this:

Voter Candidate (A or B) Voter 1 A Voter 2 A Voter 3 B

Voter Candidate (A or C) Voter 1 A Voter 2 A Voter 3 A

Since in both cases, A would be the winner, the Condorcet winner is A, but B still wins the full election. Some, like the authors of rangevoting.org, say that defining the Condorcet criterion in this way makes the criterion not always desirable. If the winners of the head-to-head contests were determined by range voting rules rather than pluralistic voting, range voting would satisfy Condorcet.

Further reading

• Black, Duncan (1958). The Theory of Committees and Elections. Cambridge University Press.
• Farquharson, Robin (1969). Theory of Voting. Oxford: Blackwell. ISBN 0-631-12460-8.
• Sen, Amartya Kumar (1970). Collective Choice and Social Welfare. Holden-Day. ISBN 978-0-8162-7765-0.

See also

• Condorcet loser criterion
• Condorcet method