Condorcet criterion

Voting systems
Part of the Politics series
Plurality/majoritarian Simple majoritarianism Limited voting Single non-transferable vote Plurality First-past-the-post Two-round Exhaustive ballot Preferential/ranked/ordinal systems Borda count Bucklin voting Condorcet methods (Copeland's, Dodgson's, Kemeny-Young, Minimax, Nanson's, Ranked pairs, Schulze) Coombs' method Instant-runoff (alternative vote) Contingent vote Oklahoma primary electoral system Cardinal/graded/rated systems Approval voting Proportional approval voting Sequential proportional approval voting Satisfaction approval voting Majority judgment Range voting
Proportional representation Party-list (Open lists, Closed lists, Local lists) Highest averages (D'Hondt, Sainte-Laguë) Largest remainder (Hare, Droop, Imperiali, Hagenbach-Bischoff) Single transferable vote (CPO-STV, Gregory, Schulze STV, Wright)
Mixed-member Mixed-member proportional Majority bonus system Additional member system Scorporo, Av+ Parallel voting Majoritarian representation Plurality-at-large Preferential block voting General ticket
Other Proxy voting Delegable voting Delegated voting Random selection Sortition Random ballot Social choice theory Arrow's theorem Gibbard–Satterthwaite theorem Voting system criteria Public choice theory (separate but related) Multiple Ballot Voting Cumulative voting
Politics portal

The Condorcet candidate or Condorcet winner (English pronunciation: /kɒndɔːrˈseɪ/) 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. Sometimes this winner is called the Pairwise Champion. 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 single-dimension political axis (e.g. left to right) and always rank higher candidates who are closer to themselves on this axis, 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

Main article: Condorcet method

The following methods comply with the Condorcet criterion:

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

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.^[2]

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.

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

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", "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

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 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

Main article: 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.

References

↑ Black, Duncan (1948). "On the Rationale of Group Decision-making". The Journal of Political Economy 56 (1): 23–34. doi:10.1086/256633. JSTOR 1825026.
↑ Laslier, Jean-Francois (2006). "Strategic Approval Voting in a Large Electorate" (PDF). IDEP Working Papers (Marseille, France: Institut D'Economie Publique) 405.

Voting systems

Part of the Politics and Election series

Single-winner voting system	Approval voting Borda count Bucklin voting Contingent vote Coombs' method Copeland's method Exhaustive ballot First-past-the-post voting Instant-runoff voting Kemeny–Young method Simple majoritarianism Minimax Condorcet Nanson's method Plurality Positional voting system Range voting Ranked pairs Schulze method Two-round system

Proportional representation	Mixed-member Party-list Single transferable vote Schulze STV CPO-STV Highest averages method Sainte-Laguë D'Hondt Largest remainder method Alternative vote Plus Closed list Open list Overhang seat Underhang seat

Semi-proportional representation	Parallel voting Single non-transferable vote Cumulative voting Limited voting Proportional approval voting Sequential proportional approval voting Satisfaction approval voting

Usage	Table of voting systems by country

Voting system criterion	Condorcet criterion Condorcet loser criterion Consistency criterion Independence of clones Independence of irrelevant alternatives Independence of Smith-dominated alternatives Later-no-harm criterion Majority criterion Majority loser criterion Monotonicity criterion Mutual majority criterion Pareto efficiency Participation criterion Plurality criterion Resolvability criterion Reversal symmetry Smith criterion

Voting system quotas	Droop quota Hagenbach-Bischoff quota Hare quota Imperiali quota

Other	Ballot Vote counting system Election threshold First-preference votes Spoilt vote Sortition

Portal — Project

This article is issued from Wikipedia - version of the Wednesday, April 13, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.