Condorcet method

A **Condorcet method** (; ) is one of many election methods. It is part of a group of voting methods called ranked voting: people rank the candidates, by their preference. A condorcet method will elect the candidate that is preferred by more voters than their opponent in every head-to-head election against each of the other candidates, whenever there is such a candidate. This candidate, known as the *pairwise champion* or *beats-all winner*, is formally called the *Condorcet winner*.

When voters rank the people according to their choice, they can also assign the same rank to more than one candidate. To get the result, the people analyzing will look at how often a candidate is ranked higher than its competitors. Candidates that are always above their opponents are the Condorcet winners.

There are different Condorcet methods. When there is a Condorcet winner, they will agree on who the Condorcet winner is. There are situations, when there is no Condorcet winner. In these situations, the different methods use different ways to pick a candidate, who they then declare as successful.

Social choice theory looks at different methods of selecting a winner; each of the methods as benefits and drawbacks. 

There may be situations, where such a Condorcet winner might not exist in a particular election. This is  because the preference of a group of voters selecting from more than two options can possibly be cyclic — that is, it is possible (but very rare) that every candidate has an opponent that defeats them in a two-candidate contest. (This is similar to the game rock paper scissors, where each hand shape wins against only one opponent and loses to another). The possibility of such cyclic preferences in a group of voters is known as the Condorcet paradox. However, there always exists a smallest group of candidates that beats all candidates not in the group, known as the Smith set, which is guaranteed to only have the Condorcet winner in it when one exists; many Condorcet methods always elect a candidate who is in the Smith set when there is no Condorcet winner, and are thus said to be "Smith-efficient". The Condorcet winner is also usually but not necessarily the utilitarian winner (the one which maximizes social welfare).

Condorcet voting methods are named for the 18th-century French mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet, who championed such voting systems.  However, Ramon Llull devised the earliest known Condorcet method in 1299. It always elects the same candidate as Copeland's method in cases with no pairwise ties.

Condorcet methods usually use preferential (ranked) voting, though they can be done with separate rounds of runoff elections.

Most Condorcet methods have a single round of preferential voting, in which each voter ranks the candidates from most preferred (marked as number 1) to least preferred (marked with a higher number). A voter's ranking is often called their *order of preference.* The votes can be tallied in many ways to find a winner. Some - the Condorcet methods - will elect the Condorcet winner if there is one. They can also elect a winner when there is no Condorcet winner, and different Condorcet-compliant methods may elect different winners in the case of a cycle.

The procedure given in Robert's Rules of Order for voting on motions and amendments is also a Condorcet method, even though the voters do not vote by ranking the candidates. There are multiple rounds of voting, and in each round the vote is between two of the alternatives. The loser (by majority rule) of a pairing is eliminated, and the winner of a pairing is paired in a later round against another alternative. Eventually only one alternative remains, and it is the winner. This is similar to a single-winner or round-robin tournament; the total number of pairings is one less than the number of alternatives. Since a Condorcet winner will win by majority rule in each of its pairings, it will never be eliminated by Robert's Rules. But this method cannot reveal a voting paradox in which there is no Condorcet winner and a majority prefer an early loser over the eventual winner (though it will always elect someone in the Smith set). A lot of the literature on social choice theory is about this method since it is widely used and is used by important organizations (legislatures, councils, committees, etc.). It is not practical for use in public elections, however, since its multiple rounds of voting would be very expensive for voters, for candidates, and for governments.

## Summary 
In a competition between candidates A and B where voters have ranked the candidates, if more voters mark their ballots that they prefer candidate A over candidate B than the number of voters who mark their ballots to the contrary, then Candidate B is not (supposed to be) elected.

However, because of the possibility of the Condorcet paradox, it is possible, but unlikely, that this can't be done. This is sometimes called a *Condorcet cycle* or just *cycle* and can be thought of as Candidate Rock beating Candidate Scissors, Candidate Scissors beating Candidate Paper, and Candidate Paper beating Candidate Rock. If there is no cycle (and no ties), all Condorcet methods elect the same candidate.

* Each voter ranks the candidates in order of preference (top-to-bottom, or best-to-worst, or 1st, 2nd, 3rd, etc.). The voter may be allowed to rank candidates as equals, to express indifference (no preference) between them. To save time, candidates omitted by a voter may be treated as if the voter ranked them at the bottom.
* For each pairing of candidates (as in a round-robin tournament) count how many votes rank each candidate over the other candidate. Thus each pairing will have two totals: the size of its majority and the size of its minority (or there will be a tie).

For most Condorcet methods, the pairwise counts are all that is necessary to figure out who won, who came in 2nd place, etc. and can be used to figure out if there is a Condorcet winner, and who it is.

## Definition 
A Condorcet method is a voting system that will always elect the Condorcet winner (if there is one); this is the candidate whom more voters prefer to each other candidate, when compared to them one at a time. This candidate can be found by checking if there is a candidate who beats all other candidates; this can be done by using Copeland's method and then checking if the Copeland winner has the highest possible Copeland score.

Note that there are ½*N*(*N*−1) pairwise comparisons for *N* candidates. For 10 candidates, this means there are 0.5*10*9=45 comparisons, which can make elections with many candidates hard to count the votes for.

In some situations, 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. Different Condorcet methods pick different winners in these situations. Some Condorcet methods involve the basic procedure described below, coupled with a Condorcet completion method, which is 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, ranked voting systems are Condorcet methods. For example, instant-runoff voting and the Borda count are not Condorcet methods.

## Basic procedure 
### Finding the winner 
In any Condorcet method that passes Independence of Smith-dominated alternatives, it can sometimes help to identify the Smith set from the head-to-head match-ups, and eliminate all candidates not in the set before doing the procedure for that Condorcet method.

Condorcet methods use pairwise counting. For each possible pair of candidates, one pairwise count indicates how many voters prefer one of the paired candidates over the other candidate, and another pairwise count indicates how many voters have the opposite preference. The counts for all possible pairs of candidates show all the pairwise preferences of all the voters.

Pairwise counts are often displayed in a *pairwise comparison matrix* or *outranking matrix* 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. Cells comparing a candidate to themselves are left blank.

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 on this voter's ballot over the 'opponent', while a '0' indicates that the runner is defeated.) When there is a Condorcet winner that IRV doesn't elect, a majority would by definition prefer the Condorcet Winner to the IRV winner. Supporters of the Condorcet criterion see it as an important criterion when choosing a voting method. They see the Condorcet criterion as a natural extension of majority rule. Condorcet methods tend to encourage the selection of broadly supported candidates who appeal to the median voter. Here is an example that is designed to support IRV while making Condorcet look bad:

499 A>B>C

3 B>C>A

498 C>B>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 goal of a ranked voting system is for voters to be able to vote sincerely and trust the system to protect their intent.

The importance of this scenario, of two parties with strong support, and the one with weak support being the Condorcet winner, may be less than one would expect, though, as it is common in plurality voting systems (see Duverger's law), but much less likely to occur in Condorcet or IRV elections, which unlike Plurality voting, punish candidates who are opposed by a significant block of voters.

Here is an example that is designed to support Condorcet while making IRV look bad:

33 A>B>C

16 B>A>C

16 B>C>A

35 C>B>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 strategy used and how often there is value in voting strategically differ in each method.

## Potential for tactical voting 
Like all voting methods, Condorcet methods are vulnerable to compromising. That is, voters can help avoid the election of a less-preferred candidate by insincerely (strategically) 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, or when one can be created.

All Condorcet methods are at least somewhat vulnerable to burying. That is, voters can sometimes help a more-preferred candidate by strategically lowering the ranking of a less-preferred candidate on their ballot.

## 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. However, the Condorcet criterion implies the majority criterion, and thus is incompatible with independence of irrelevant alternatives (though it implies a weaker analogous form of the criterion: when there is a Condorcet winner, losing candidates can drop out of the election without changing the result), later-no-harm, the participation criterion, and the consistency criterion.

## Use of Condorcet voting 
Condorcet methods are not known to be 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 organizations. Organizations which currently use some variant of the Condorcet method are:

* The Wikimedia Foundation used the Schulze method to elect its Board of Trustees until 2013, when it switched to a ratings ballot with Support/Neutral/Oppose ballots.
* The Pirate Party of Sweden uses the Schulze method for its primaries
* The Debian project uses the Schulze method for internal referendums and to elect its leader
* The Software in the Public Interest corporation uses the Schulze method for internal referendums and to elect its Board of Directors
* The Gentoo Foundation uses the Schulze method for internal referendums and to elect its Board of Trustees and its Council
* The Free State Project used Minimax for choosing its target state
* The uk.* hierarchy of Usenet
* The Student Society of the University of British Columbia uses ranked pairs for its executive elections.
* Kingman Hall and Hillegass Parker House, two loosely affiliated student housing cooperatives, each use the Schulze method to elect their management teams.

## Other considerations 
*
* If every seat in a legislature is elected (in separate elections, by the same pool of voters – use of districts will avoid this) by the Condorcet method, the legislators would all be centrists and might all agree with each on what laws to pass. Some voters prefer to have opposites in the legislature so they can't pass laws easily. These voters might prefer the Condorcet method for electing executive offices.
* If 10 candidates run for governor in a Condorcet race, ballot counters may need to count 9+8+7+6+5+4+3+2+1 = 45 head to head runoffs to find the winner. While this is doable, it might be more practical to still use ballot access laws or primaries, defeating some of the original intent of the Condorcet method. Possible solutions:
** Computers can be used to speed up the counts, though some voters fear computers can be hacked and used for ballot counting fraud.
** Another option would be to allow several independent scanner owners count the ballots and compare results. Volunteer hand counters could then spot check various candidates and ranks to make sure they match the subtotals reported by the scanners.
** It is also possible to limit the number of ranks voters can use; for example, if every voter is only allowed to rank each candidate either 1st, 2nd, or 3rd, with equal rankings allowed, then only the runoffs between candidates ranked 1st and 2nd, 1st and 3rd, 1st and last, 2nd and 3rd, 2nd and last, and 3rd and last need be counted, as the runoffs between two candidates at the same rank will result in ties.
** The negative vote-counting approach to pairwise counting may reduce the amount of work the vote-counters have to do. For example, with 10 candidates, a voter who ranks candidate A as their 1st choice and doesn't rank any other candidate prefers A over 9 other candidates. In the regular approach, this means recording those 9 preferences; but with negative counting, it can simply be recorded that A is marked on 1 voter's ballot, with this itself indicating that A is preferred in every match-up. When a voter ranks a candidate 2nd, then a negative vote can be placed in the matchup between the 2nd choice and 1st choice to indicate that the 2nd choice is **not** preferred to the 1st choice, such that it will cancel out with the support the 2nd choice would receive against the 1st choice from being marked on the voter's ballot. Negative votes can likewise be applied to matchups where both candidates are ranked equally.
** If there are no more than 5 candidates ( or a larger number of candidates is short-listed to 5) then the amount of effort counting ballots could be reduced to normal acceptable levels by asking voters to select an order of preference from a predetermined list of the possibilities. This would mean that the ballots would just require to be counted once to determine the number of votes cast for each order of preference. The results would then be entered into a simple spreadsheet which would determine the Condorcet winner. For example where there are candidates A, B and C, there are six orders of preference, so voters could be asked to choose which of the six they wish to vote for. Counting would then be simply a matter of counting how votes were cast for each order of preference. The results could then be applied to a simple spreadsheet which revealed the Condorcet winner. If there were four candidates (options) then there would be 24 orders of preference; if there were five candidates then there would be 120 orders of preference and so on.
* Voters make an economic trade-off in the amount of time invested in researching and ranking candidates.  If voters rank too few candidates or rank such as to inaccurately represent their preferences, the Condorcet candidate cannot be correctly discovered. Nominating primaries reduce the number of candidates to avoid this, and the style of nominating primary can impact whether the Condorcet candidate—or at least a similar candidate—remains or if all such candidates are eliminated in favor of polarized options.

## Related pages
 

* Condorcet loser criterion
* Ramon Llull (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

## Notes and references 
## Further reading 
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## Other websites
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* . The Basic Motions page lists Rules of Order for discussions and the motions organized on ranked choice ballots used in Condorcet voting.
* . Multipage description of Condorcet method and Ranked Pairs from a Canadian perspective.
* . Demonstration and commentary on Condorcet method.
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