# Condorcet method

Any election 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.

Condorcet is sometimes used to indicate the family of Condorcet methods as a whole.

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## Basic procedure

The basic procedure for casting ballots is identical to most preferential ballots, such as IRV and Borda ballots. However, the voter might prefer to order them differently. In most systems, first place is given additional consideration, as though first choice is a very strong preference. In ballots used in a Condorcet Method, order is the only consideration; first place is not a special rank with special consideration— it is simply preferred to second or third or fourth.

### Casting ballots

Each voter ranks the candidates in the order they prefer each candidate. The voter can include fewer than all candidates under consideration.

Usually when a candidate is not listed on the voter's ballot they are considered less preferred than listed candidates, and ranked accordingly. However, some variations allow a "no opinion" default option where no for- or against- preference is counted for that candidate.

Write-ins are possible, but are somewhat more difficult to implement for automatic counting than in other election methods. This is an implementation issue, but results in the frequent omission of the write-in option in ballot software.

### Counting ballots

Ballots are counted by considering all possible sets of two-candidate elections from all available candidates. That is, each candidate is considered against each and every other candidate. A candidate is considered to "win" against another on a single ballot if they are ranked higher than their opponent. All the votes for candidate Alice over candidate Bob are counted, as are all of the votes for Bob over Alice. Whoever has the most votes in each one-on-one election wins.

If a candidate is preferred over all other candidates, that candidate is the Condorcet winner. However, a Condorcet winner may not exist, due to a fundamental paradox: It is possible for the electorate to prefer A over B, B over C, and C over A simultaneously. This is called a majority rule cycle, and it must be resolved by some other mechanism.

#### Counting with matrices

A frequent implementation of this method will illustrate the basic counting method. Consider an election between A, B, C and D, and a ballot (B, C, A, D). That is, a ballot ranking B first, C second, A third, and D fourth. This can be represented as a matrix, where the row is the runner under consideration, and the column is the opponent. The cell at (runner,opponent) has a one if runner is preferred, and a zero if not.

A B C D
A 0 0 1
B 1 1 1
C 1 0 1
D 0 0 0

Cells marked "—" are logically zero, but are blank for clarity—they are not considered, as a candidate can not be defeated by himself. This binary matrix is inversely symmetric: (runner,opponent) is ¬(opponent,runner). The utility of this structure is that it may be easily added to other ballots represented the same way, to give us the number of ballots which prefer each candidate. The sum of all ballot matrices is called the sum matrix—it is not symmetric.

When the sum matrix is found, the contest between each candidate is considered. The number of votes for runner over opponent (runner,opponent) is compared the number of votes for opponent over runner (opponent,runner). The one-on-one winner has the most votes. If one candidate wins against all other candidates, that candidate wins the election.

The sum matrix is the primary piece of data used to resolve majority rule cycles.

## Resolving circular ambiguities

Just about any election system that treats every voter equally (anonymity) and every candidate equally (neutrality) has the possibility of ties. A Condorcet method isn't different in that regard. For example, it's possible for candidates to tie with each other and "pairwise defeat" everybody else.

However, "Condorcet" methods have an additional ambiguity: the problem of the Condorcet paradox. There may be cycles in the results.

For example, it would be possible for the totalled votes to record that A defeats B, B defeats C, and C defeats A. And while voters often vote so that there is a single Condorcet winner of a given election (see in that regard political spectrum), a Condorcet method is usually only considered for serious use if such cycles can be handled. Handling cases where there is not a single Condorcet winner is called ambiguity resolution in this article, though other phrases such as "cyclic ambiguity resolution" and "Condorcet completion" are used as well.

### Key terms in ambiguity resolution

The following are key terms when discussing ambiguity resolution methods:

• 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 a given electoral outcome.
• Schwartz set: 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
• Cloneproof: a method that is immune to the presence of clones (candidates which are essentially identical to each other). In some voting methods, a party can increase its odds of selection if it provides a large number of "identical" options. A cloneproof voting method prevents this attack. See strategic nomination.

### Different ambiguity resolution methods

There are a countless number of "Condorcet methods" possible that resolve such ambiguities. The fact that Marquis de Condorcet himself already spearheaded the debate of which particular Condorcet method to promote has made the term "Condorcet's method" ambiguous. Indeed, it can be argued that the large number of different competing Condorcet methods has made the adoption of any single method extremely difficult.

Examples of Condorcet methods include:

• Black chooses the Condorcet winner when it exists and otherwise the Borda winner. It is named after Duncan Black.
• Smith/IRV is instant-runoff voting with the candidates restricted to the Smith set.
• Copeland selects the candidate that wins the most pairwise matchups. Note that if there is no Condorcet winner, Copeland will often still result in a tie.
• Minimax (also called Simpson) chooses the candidate whose worst pair wise defeat is less bad than that of all other candidates.1
• Smith/Minimax restricts the Minimax algorithm to the Smith set.1
• Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) with variations such as Maximize Affirmed Majorities (MAM) and Maximum Majority Voting (MMV)1
• Schulze (a.k.a. Schwartz sequential dropping (SSD), cloneproof Schwartz sequential dropping (CSSD), beatpath method, beatpath winner, path voting, path winner)1

1 There are different ways to measure the strength of each defeat in some methods. Some use the margin of defeat (the difference between votes for and votes against), while others use winning votes (the votes favoring the defeat in question). Electionmethods.org argues that there are several disadvantages of systems that use margins instead of winning votes. The website argues that using margins "destroys" some information about majorities, so that the method can no longer honor information about what majorities have determined and that consequently margin-based systems cannot support a number of desirable voting properties.

Ranked Pairs and Schulze are procedurally in some sense opposite approaches:

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

The text below describes (variants of) these methods in more detail.

### Ranked Pairs

In the Ranked Pairs (RP) voting method, pairs of defeats are ranked (sorted) from largest majority to smallest majority. Then each pair is considered, starting with the defeat supported by the largest majority. Pairs are "affirmed" only if they do not create a cycle with the pairs already affirmed. Once completed, the affirmed pairs are followed to determine the winner.

In essence, Ranked Pairs treat each majority preference as evidence that the majority's more preferred alternative should finish over the majority's less preferred alternative, the weight of the evidence depending on the size of the majority.

### Schulze method

The Schulze method resolves votes as follows:

1. First, determine the Schwartz set (the innermost unbeaten set). If no defeats exist among the Schwartz set, then its members are the winners (plural only in the case of a tie, which must be resolved by another method).
2. Otherwise, drop the weakest defeat information among the Schwartz set (i.e., where the number of votes favoring the defeat is the smallest). Determine the new Schwartz set, and repeat the procedure.

In other words, this procedure repeatedly throws away the weakest pairwise win, until finally the number of votes left over produce an unambiguous decision. Or, to put it another way, the fewest possible number of votes for a single preference are discarded to break the circular ambiguity.

## Related terms

Other terms related to the Condorcet method are:

• Condorcet loser: the candidate who is less preferred than every other candidate in a pair wise matchup.
• weak Condorcet winner: a candidate who beats or ties with every other candidate in a pair wise 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.

## An example

Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county):

• Memphis (Shelby County): 826,330
• Nashville (Davidson County): 510,784
• Chattanooga (Hamilton County): 285,536
• Knoxville (Knox County): 335,749

Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows:

 42% of voters (close to Memphis) 1. Memphis 2. Nashville 3. Chattanooga 4. Knoxville 26% of voters (close to Nashville) 1. Nashville 2. Chattanooga 3. Knoxville 4. Memphis 15% of voters (close to Chattanooga) 1. Chattanooga 2. Knoxville 3. Nashville 4. Memphis 17% of voters (close to Knoxville) 1. Knoxville 2. Chattanooga 3. Nashville 4. Memphis

The results would be tabulated as follows:

Pairwise Election Results
A
Memphis Nashville Chattanooga Knoxville
BMemphis[A] 58%
[B] 42%
[A] 58%
[B] 42%
[A] 58%
[B] 42%
Nashville[A] 42%
[B] 58%
[A] 32%
[B] 68%
[A] 32%
[B] 68%
Chattanooga[A] 42%
[B] 58%
[A] 68%
[B] 32%
[A] 17%
[B] 83%
Knoxville[A] 42%
[B] 58%
[A] 68%
[B] 32%
[A] 83%
[B] 17%
Ranking (by repeatedly removing Condorcet winner): 4th 1st 2nd 3rd
• [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption

In this election, Nashville is the Condorcet winner and thus the winner under all possible Condorcet methods. Notice how first-past-the-post and instant-runoff voting would have respectively selected Memphis and Knoxville here, while compared to either of them, most people would have preferred Nashville.

## Condorcet compared to Instant Runoff and First-past-the-post

There are circumstances, as in the example above, when both instant-runoff voting and plurality voting 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.

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 and strategic nominations.

## Use of Condorcet voting

Condorcet voting is not currently used in government elections. However, it is starting to receive support in some public organizations. Organizations which currently use some variant of the Condorcet method are:

1. The Debian project uses a modified version of the Schulze method for internal referendums and to elect its leader.
2. The Software in the Public Interest corporation uses the Schulze method to elect members of its board of directors.
3. The Gentoo Linux project uses the Schulze method.
4. The UserLinux project uses the Schulze method.
5. The Free State Project for choosing its target state
6. The voting procedure for the uk.* hierarchy of Usenet
7. Five-Second Crossword Competition (http://www.rsabey.pwp.blueyonder.co.uk/rpc/fscc/)

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