Condorcet criterion

The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates. A Condorcet winner will not always exist in a given set of votes, which is known as Condorcet's voting paradox. When voters identify candidates on a left-to-right axis and always prefer candidates closer to themselves, a Condorcet winner always exists.^[1]

A voting system satisfies the Condorcet criterion if 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.

Relation to other criteria

The Condorcet criterion implies the majority criterion; that is, any system that satisfies the former will satisfy the latter. Because of this, Arrow's impossibility theorem shows that any method which satisfies the Condorcet criterion will not satisfy independence of irrelevant alternatives.

The Condorcet criterion is also incompatible with the later-no-harm criterion, the participation criterion, and the consistency criterion.

Compliance of methods

Complying methods

The following methods comply with the Condorcet criterion:

• Black
• Copeland
• Dodgson's method
• Kemeny-Young
• Minimax
• Nanson's method
• Ranked pairs
• Schulze
• Smith/minimax

Non-complying methods

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

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

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.^[2]

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

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

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}:

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.

In cases where there is a Condorcet Winner, and where IRV does not choose it, a majority would by definition prefer the Condorcet Winner to the IRV winner.

Majority Judgment

Majority Judgment is a system in which the voter gives all candidates a rating out of a predetermined set (e.g. {"excellent", "good", "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 give candidate A the rating "excellent", B "fair" and C "poor",

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

31 voters choose "excellent" for B, "good" for C and "fair" for A.

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

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 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%.

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.

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.

Unstrategic or "honest" range voting clearly does not comply with the condorcet criterion; for instance, if three voters vote for three candidates (10,9,0), (10,9,0), (0,10,0), then the first candidate is the Condorcet winner but the second candidate wins with 28 to 20 points. However, if all voters vote strategically, then Range is equivalent to approval voting. Thus with perfect information about all voters' motivations, any single Condorcet winner will be the Nash equilibrium, as cited above.

Further reading

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

See also

• Condorcet loser criterion
• Condorcet method
• Ramon Llull (1232–1315), who with the 2001 discovery of his lost manuscripts Ars notandi, Ars eleccionis, and Alia ars eleccionis, was given credit for discovering the Borda count and Condorcet criterion (Llull winner) in the 13th century

References