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Condorcet Voting

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A Condorcet method is a single winner election method in which voters rank candidates in order of preference. There are multiple slightly differing methods, including Ranked Pairs and the Schulze method, that satisfy the definition to be considered a Condorcet method.

Condorcet methods are named for the eighteenth century mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet. Ramon Llull had devised a method that meets the Condorcet criterion in 1299, but this method is based on an iterative procedure rather than a ranked ballot.

Summary

  • Rank the candidates in order (1st, 2nd, 3rd, etc.) of preference. Tie rankings are allowed, which express no preference between the tied candidates.
  • Comparing each candidate on the ballot to every other, one at a time (pairwise), tally a "win" for the victor in each match.
  • Sum these wins for all ballots cast. The candidate with the greatest total wins is the one who is the most preferred, and hence the winner of the election.
  • In the event of a tie, use a resolution method described below.

A particular point of interest is that it is possible for a candidate to be the most preferred overall without being the first preference of any voter. In a sense, the Condorcet method yields the "best compromise" candidate, the one that the largest majority will find to be least disagreeable, even if not their favorite.

Definition

A Condorcet method is a voting system that will always elect the Condorcet winner; this is the candidate whom voters prefer to each other candidate, when compared to them one at a time. This candidate can easily be found by conducting a series of pairwise comparisons, using the basic procedure described in this article. The family of Condorcet methods is also referred to collectively as Condorcet's method. A voting system that always elects the Condorcet winner when there is one is described by electoral scientists as a system that satisfies the Condorcet criterion.

In certain circumstances an election has no Condorcet winner. This occurs as a result of a kind of tie known as a 'majority rule cycle', described by Condorcet's paradox. The manner in which a winner is then chosen varies from one Condorcet method to another. Some Condorcet methods involve the basic procedure described below, coupled with a Condorcet completion method– a special method used to find a winner when there is no Condorcet winner. Other Condorcet methods involve an entirely different system of counting, but are classified as Condorcet methods because they will still elect the Condorcet winner if there is one.

It is important to note that not all single winner, preferential voting systems are Condorcet methods. For example, neither instant-runoff voting nor the Borda count satisfies the Condorcet criterion.

Basic procedure

Voting

In a Condorcet election the voter ranks the list of candidates in order of preference. So, for example, the voter gives a '1' to their first preference, a '2' to their second preference, and so on. In this respect it is the same as an election held under non-Condorcet methods such as instant runoff voting or the single transferable vote. Some Condorcet methods allow voters to rank more than one candidate equally, so that, for example, the voter might express two first preferences rather than just one.

Usually, when a voter does not give a full list of preferences they are assumed, for the purpose of the count, to prefer the candidates they have ranked over all other candidates. Some Condorcet elections permit write-in candidates but, because this can be difficult to implement, software designed for conducting Condorcet elections often does not allow this option.

Finding the winner

The count is conducted by pitting every candidate against every other candidate in a series of imaginary one-on-one contests. The winner of each pairing is the candidate preferred by a majority of voters. The candidate preferred by each voter is taken to be the one in the pair that the voter ranks highest on their ballot paper. For example, if Alice is paired against Bob it is necessary to count both the number of voters who have ranked Alice higher than Bob, and the number who have ranked Bob higher than Alice. If Alice is preferred by more voters then she is the winner of that pairing. When all possible pairings of candidates have been considered, if one candidate beats every other candidate in these contests then they are declared the Condorcet winner. As noted above, if there is no Condorcet winner a further method must be used to find the winner of the election, and this mechanism varies from one Condorcet method to another.

Counting with matrices

In a Condorcet election votes are often counted, and results illustrated, in the form of matrices such as those below. In these matrices each row represents each candidate as a 'runner', while each column represents each candidate as an 'opponent'. The cells at the intersection of rows and columns each show the result of a particular pairwise comparison. Certain cells are left blank because it is impossible for a candidate to be compared with herself.

Imagine there is an election between four candidates: A, B, C and D. The first matrix below records the preferences expressed on a single ballot paper, in which the voter's preferences are (B, C, A, D); that is, the voter ranked B first, C second, A third, and D fourth. In the matrix a '1' indicates that the runner is preferred over the 'opponent', while a '0' indicates that the runner is defeated.

Opponent
A B C D
R
u
n
n
e
r
A 0 0 1
B 1 1 1
C 1 0 1
D 0 0 0
A "1" indicates that the runner is preferred over the opponent; a '0' indicates that the runner is defeated.

Matrices of this kind are useful because they can be easily added together to give the overall results of an election. The sum of all ballots in an election is called the sum matrix. Suppose that in the imaginary election there are two other voters. Their preferences are (D, A, C, B) and (A, C, B, D). Added to the first voter, these ballots would give the following sum matrix:

Opponent
A B C D
R
u
n
n
e
r
A 2 2 2
B 1 1 2
C 1 2 2
D 1 1 1

When the sum matrix is found, the contest between each pair of candidates is considered. The number of votes for runner over opponent (runner, opponent) is compared with the number of votes for opponent over runner (opponent,runner). It is then possible to find the Condorcet winner. In the sum matrix above it can be seen that A is the Condorcet winner because A beats every other candidate. When there is no Condorcet winner Condorcet completion methods, such as Ranked Pairs and the Schulze method, use the information contained in the sum matrix to choose a winner.

In strict mathematical terms the cells marked '–' in the matrices above should be '0', but in these examples a dash is used for clarity. The first matrix, that represents a single ballot, is inversely symmetric: (runner,opponent) is ¬(opponent,runner). Or (runner,opponent) + (opponent,runner) = 1. The sum matrix has this property: (runner,opponent) + (opponent,runner) = N for N voters, if all runners were fully ranked by each voter.

Two method systems

One family of Condorcet methods consists of systems that first conduct a series of pairwise comparisons and then, if there is no Condorcet winner, fall back to an entirely different, non-Condorcet method to determine a winner. The simplest such methods involve entirely disregarding the results of pairwise comparisons. For example, Black is a method that involves finding the Condorcet winner if it exists, but using the Borda count instead if there is an ambiguity (it is named after Duncan Black).

A more sophisticated two-stage process is, in the event of an ambiguity, to use a separate voting system to find the winner but to restrict this second stage to a certain subset of candidates found by scrutinising the results of the pairwise comparisons. Sets used for this purpose are defined so that they will always contain only the Condorcet winner if there is one, and will always, in any case, contain at least one candidate. Such sets include the

  • Smith set: The smallest set of candidates in a particular election such that every candidate in the set can beat all candidates outside the set. It is easily shown that there is only one possible Smith set for each election.
  • Schwartz set: This is the innermost unbeaten set, and is usually the same as the Smith set. It is defined as the union of all possible sets of candidates such that for every set:
    1. Every candidate inside the set is pairwise unbeatable by any other candidate outside the set (i.e. ties are allowed).
    2. No proper (smaller) subset of the set fulfills the first property.
  • Landau set (or uncovered set or Fishburn set): the set of candidates, such that each member, for every other candidate (including those inside the set), either beats this candidate or beats a third candidate that itself beats the candidate that is unbeaten by the member.

One possible method is to apply instant-runoff voting to the candidates of the Smith set. This method has been described as 'Smith/IRV'.

Other systems

There are also various methods whose procedures do not fall back on an entirely different system. Most involve scrutinising the results of each pairwise contest and using this data to determine a winner of the overall election. These include:

  • Copeland's method: This simple method involves electing the candidate who wins the most pairwise matchings. However it often produces a tie.
  • Kemeny-Young method: Also known as 'VoteFair popularity ranking', this method ranks all the choices from most popular and second-most popular down to least popular.
  • Minimax: Also called 'Simpson' and 'Simpson-Kramer', this method chooses the candidate whose worst pairwise defeat is better than that of all other candidates. A refinement of this method involves restricting it to choosing a winner from among the Smith set; this has been called 'Smith/Minimax'.
  • Ranked Pairs: This method is also known as 'Tideman', after its inventor Nicolaus Tideman.
  • Schulze method: This method is also known as 'Schwartz sequential dropping' (SSD), 'cloneproof Schwartz sequential dropping' (CSSD), 'beatpath method', 'beatpath winner', 'path voting' and 'path winner'.

Ranked Pairs and Schulze are procedurally in some sense opposite approaches (although they very frequently give the same results):

  • Ranked Pairs (and its variants) starts with the strongest defeats and uses as much information as it can without creating ambiguity.
  • Schulze repeatedly removes the weakest defeat until ambiguity is removed.

Related terms

Other terms related to the Condorcet method are:

  • Condorcet loser: the candidate who is less preferred than every other candidate in a pairwise matchup.
  • Weak Condorcet winner: a candidate who beats or ties with every other candidate in a pairwise matchup. There can be more than one weak Condorcet winner.
  • Weak Condorcet loser: a candidate who is defeated by or ties with every other candidate in a pair wise matchup. Similarly, there can be more than one weak Condorcet loser.

Comparison with instant runoff and first-past-the-post

There are circumstances, as in the example above, when both instant-runoff voting (IRV) and the 'first-past-the-post' plurality system will fail to pick the Condorcet winner. Proponents of the Condorcet criterion see it as a principal issue in selecting an electoral system. They see the Condorcet criterion as a natural extension of majority rule. Condorcet methods tend to encourage the selection of centrist candidates who appeal to the median voter. Here is an example that is designed to support IRV at the expense of Condorcet:

499 voters 3 voters 498 voters
1. A 1. B 1. C
2. B 2. C 2. B
3. C 3. A 3. A

B is preferred by a 501-499 majority to A, and by a 502-498 majority to C. So, according to the Condorcet criterion, B should win, despite the fact that very few voters rank B in first place. By contrast, IRV elects C and plurality elects A.

The significance of this scenario, of two parties with strong support and one with weak support which is the Condorcet winner may be misleading, though, as it is a common mode in plurality voting systems, but much less likely to occur in Condorcet or IRV elections, which unlike Plurality voting, punish candidates who alienate a significant block of voters.

Here is an example that is designed to support Condorcet at the expense of IRV:

33 voters 16 voters 16 voters 35 voters
1. A 1. B 1. B 1. C
2. B 2. A 2. C 2. B
3. C 3. C 3. A 3. A

B would win against either A or C by more than a 65-35 margin in a one-on-one election, but IRV eliminates B first, leaving a contest between the more "polar" candidates, A and C. Proponents of plurality voting state that their system is simpler than any other and more easily understood. All three systems are susceptible to tactical voting, but the types of tactics used and the frequency of strategic incentive differ in each method.

Potential for tactical voting

Like most voting methods, Condorcet methods are vulnerable to compromising. That is, voters can help avoid the election of a less-preferred candidate by insincerely raising the position of a more-preferred candidate on their ballot.

However, Condorcet methods are only vulnerable to compromising when there is a majority rule cycle. By contrast, instant runoff voting has compromising incentive when there is a majority rule cycle, but it may also have compromising incentive under other circumstances, i.e. when the Condorcet winner is eliminated before the final round.

Unlike IRV, Condorcet methods are vulnerable to burying. That is, voters can help a more-preferred candidate by insincerely lowering the position of a less-preferred candidate on their ballot. In general, this can be done by creating a false majority rule cycle that overrules a genuine pairwise defeat. Some Condorcet methods may be less vulnerable to the burying strategy than others.

Evaluation by criteria

Scholars of electoral systems often compare them using mathematically defined voting system criteria. The criteria which Condorcet methods satisfy vary from one Condorcet method to another. All Condorcet methods satisfy the Condorcet criterion and the majority criterion by definition; no Condorcet method satisfies participation or IIA.

Majority Monotonic Consistent Participation Condorcet loser IA independence Clone independence
Schulze Yes Yes No No Yes No
(see local IIA note)
Yes
Ranked Pairs Yes Yes No No Yes No
(see local IIA note)
Yes
Minimax Yes Yes No No No No No
Nanson Yes No No No Yes No No
Kemeny-Young Yes Yes No No Yes No No

Use of Condorcet voting

Condorcet methods are not currently in use in government elections anywhere in the world, but a Condorcet method known as Nanson's method was used in city elections in the U.S. town of Marquette, Michigan in the 1920s and today Condorcet methods are used by a number of private organisations.

References

Wikipedia.org


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