Condorcet criterion
Encyclopedia
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.
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
.
; 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
.
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 others' motivations, and a single Condorcet winner exists, then that candidate will win under the Nash equilibrium
.
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.
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.
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.
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.
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.
Election
An election is a formal decision-making process by which a population chooses an individual to hold public office. Elections have been the usual mechanism by which modern representative democracy operates since the 17th century. Elections may fill offices in the legislature, sometimes in the...
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
Voting paradox
The voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic , even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other...
. When voters identify candidates on a left-to-right axis and always prefer candidates closer to themselves, a Condorcet winner always exists.
A voting system
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
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
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
.
It is named after the 18th century mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...
.
Relation to other criteria
The Condorcet criterion implies the majority criterionMajority criterion
The majority criterion is a single-winner voting system criterion, used to compare such systems. The criterion states that "if one candidate is preferred by a majority of voters, then that candidate must win"....
; that is, any system that satisfies the former will satisfy the latter. Because of this, Arrow's impossibility theorem
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
shows that any method which satisfies the Condorcet criterion will not satisfy independence of irrelevant alternatives
Independence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
.
The Condorcet criterion is also incompatible with the later-no-harm criterion
Later-no-harm criterion
The later-no-harm criterion is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a less preferred candidate cannot cause a more preferred candidate to lose.- Complying methods :Single...
, the participation criterion
Participation criterion
The participation criterion is a voting system criterion. It is also known as the "no show paradox". It has been defined as follows:* In a deterministic framework, the participation criterion says that the addition of a ballot, where candidate A is strictly preferred to candidate B, to an existing...
, and the consistency criterion
Consistency criterion
A voting system is consistent if, when the electorate is divided arbitrarily into two parts and separate elections in each part result in the same choice being selected, an election of the entire electorate also selects that alternative...
.
Complying methods
The following methods comply with the Condorcet criterion:- Black
- CopelandCopeland's methodCopeland's method or Copeland's pairwise aggregation method is a Condorcet method in which candidates are ordered by the number of pairwise victories, minus the number of pairwise defeats....
- Dodgson's methodDodgson's methodDodgson's Method is a voting system proposed by Charles Dodgson.-Description:In Dodgson's method, each voter submits an ordered list of all candidates according to their own preference . The winner is defined to be the candidate for whom we need to perform the minimum number of pairwise swaps ...
- Kemeny-YoungKemeny-Young methodThe Kemeny–Young method is a voting system that uses preferential ballots and pairwise comparison counts to identify the most popular choices in an election...
- MinimaxMinimax CondorcetIn voting systems, the Minimax method is one of several Condorcet methods used for tabulating votes and determining a winner when using preferential voting in a single-winner election...
- Nanson's methodNanson's methodThe Borda count can be combined with an Instant Runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method.- Nanson method :The Nanson method is based on the original work of the mathematician Edward J...
- Ranked pairsRanked PairsRanked pairs or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners....
- SchulzeSchulze methodThe Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners...
- Smith/minimaxMinimax CondorcetIn voting systems, the Minimax method is one of several Condorcet methods used for tabulating votes and determining a winner when using preferential voting in a single-winner election...
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 votingApproval votingApproval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...
- Borda countBorda countThe Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...
- Range votingRange votingRange voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...
- Plurality voting
- Instant-runoff votingInstant-runoff votingInstant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...
- Bucklin votingBucklin votingBucklin voting is a class of voting systems that can be used for single-member and multi-member districts. It is named after its original promoter, James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system...
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 others' motivations, and a single Condorcet winner exists, then that candidate will win under the Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
.
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.
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.
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 areal 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.
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.
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.
Bucklin voting
Bucklin is a ranked votingPreferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...
method that was used in some elections during the early 20th century in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
. The election proceeds in rounds, one rank at a time, until a majority
Majority
A majority is a subset of a group consisting of more than half of its members. This can be compared to a plurality, which is a subset larger than any other subset; i.e. a plurality is not necessarily a majority as the largest subset may consist of less than half the group's population...
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.
See also
- Condorcet loser criterionCondorcet loser criterionIn single-winner voting system theory, the Condorcet loser criterion is a measure for differentiating voting systems. It implies the majority loser criterion....
- Condorcet methodCondorcet methodA Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
- Ramon LlullRamon LlullRamon Llull was a Majorcan writer and philosopher, logician and tertiary Franciscan. He wrote the first major work of Catalan literature. Recently-surfaced manuscripts show him to have anticipated by several centuries prominent work on elections theory...
(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