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# Condorcet method

```Condorcet criterion
```

Any election method conforming to the Condorcet criterion is known as a Condorcet method. The name comes from a deviser, the 18th century mathematician and philosopher Marquis de Condorcet, although the method was previously devised by Ramon Llull in the 13th century.

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

 Contents

## 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 a counting 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 candidate. However, a Condorcet candidate 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 circular tie, 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 circular ties (also called circular ambiguities).

## 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 who, when paired off in pairwise elections, can beat all other candidates outside the set.
• 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:

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, Maximize Affirmed Majorities (MAM), and Maximum Majority Voting (MMV)

In the Ranked Pairs (RP) voting method, as well as the variations Maximize Affirmed Majorities (MAM) and Maximum Majority Voting (MMV), 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, RP and its variants (such as MAM and MMV) 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.

The difference between RP and its variants is in the details of the ranking approach. Some definitions of RP use margins, while other definitions use winning votes (not margins). Both MAM and MMV are explicitly defined in terms of winning votes, not winning margins. In MAM and MMV, if two defeat pairs have the same number of votes for a victory, the defeat with the smaller defeat is ranked higher. If this still doesn't disambiguate between the two, MAM and MMV perform slightly differently. In MAM, information from a "tiebreaker" vote is used (which could be a distinguished vote such as the vote of a "speaker", but unless there is a distinguished vote, a randomly-chosen vote is used). In MMV all such conflicting matchups are ignored (though any non-conflicting matchups of that size are still included).

### Cloneproof Schwartz Sequential Dropping (CSSD)

The "cloneproof Schwartz Sequential Dropping" (CSSD) 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 tie.

The "Beatpath Winner" algorithm produces equivalent results.

## 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

Only an explicit Condorcet based method will comply with the Condorcet criterion so that if there is a Condorcet winner (a candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate) then that individual is selected. So 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 that as the 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 may have a low level of "first choice" support, but a high level of "middle rank" support, especially if the voting system encourages all candidates to adjust their position to appeal to the median voter. To take another example, consider the following vote count of preferences with three candidates {A,B,C}:

 499: A,B,C 3: B,C,A 498: C,B,A

In this case, B is preferred to A by 501 votes to 499, and B is preferred to C by 502 to 498, hence B is preferred to both A and C. So according to the Condorcet criterion, B should win. By contrast, according to the rules of IRV, B is ranked first by the fewest voters and is eliminated, and C wins with the transferred votes from B; in plurality voting A wins with the most first choices.

Proponents of most Condorcet voting systems also claim a technical advantage in that since the ballot totals in each pairwise race are used to determine the winner, the results can be tallied in a distributed fashion - i.e., at the precinct level. Proponents of instant-runoff voting respond that fewer counts of votes are needed with their system, since only transferred votes need more than one observation; however, these counts must be done either with all ballots gathered centrally or with simultaneous counts and transfers at each precinct, which may be hard for large elections. 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 Cloneproof Schwartz Sequential Dropping for internal referendums and to elect its leader.
2. The Software in the Public Interest corporation uses Cloneproof Schwartz Sequential Dropping to elect members of its board of directors.
3. The UserLinux project uses Cloneproof Schwartz Sequential Dropping.
4. The Free State Project for choosing its target state
5. The voting procedure for the uk.* hierarchy of Usenet
6. Five-Second Crossword Competition