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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 examples, voters rank candidates in order of preference. There are then multiple, slightly differing methods for calculating the winner, due to the need to resolve circular ambiguities—including the Kemeny-Young method, ranked pairs, and the Schulze method. Almost all of these methods give the same result if there are fewer than 4 candidates in the circularly-ambiguous Smith set and voters separately rank all of them.
Condorcet methods are named for the 18th-century French mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet. Ramon Llull had devised one of the first Condorcet methods in 1299,[1] but this method is based on an iterative procedure rather than a ranked ballot.
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.
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 be found by conducting a series of pairwise comparisons, using the basic procedure described above. For N candidates, this requires 1⁄2N(N−1) pairwise hypothetical elections. (For example, with 5 candidates there are 10 pairwise comparisons to be made.) 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 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, instant-runoff voting and the Borda count do not satisfy the Condorcet criterion.
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.
The count is conducted by pitting every candidate against every other candidate in a series of hypothetical one-on-one contests. The winner of each pairing is the candidate preferred by a majority of voters. Unless they tie, there is always a majority when there are only two choices. The candidate preferred by each voter is taken to be the one in the pair that the voter ranks higher 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.
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 summarize all the preferences of all the voters.
Pairwise counts are often displayed in 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. 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 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. | |||||
Using a matrix like the one above, one can find the overall results of an election. Each ballot can be transformed into this style of matrix, and then added to all other ballot matrices using matrix addition. 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) to find the Condorcet winner. In the sum matrix above, 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.
Cells marked '—' in the matrices above have a numerical value of '0', but a dash is used since candidates are never preferred to themselves. 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.
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near to the capital as possible.
The candidates for the capital are:
The preferences of the voters would be divided like this:
| 42% of voters (close to Memphis) |
26% of voters (close to Nashville) |
15% of voters (close to Chattanooga) |
17% of voters (close to Knoxville) |
|---|---|---|---|
|
|
|
|
To find the Condorcet winner every candidate must be matched against every other candidate in a series of imaginary one-on-one contests. In each pairing the winner is the candidate preferred by a majority of voters. When results for every possible pairing have been found they are as follows:
| Pair | Winner |
|---|---|
| Memphis (42%) vs. Nashville (58%) | Nashville |
| Memphis (42%) vs. Chattanooga (58%) | Chattanooga |
| Memphis (42%) vs. Knoxville (58%) | Knoxville |
| Nashville (68%) vs. Chattanooga (32%) | Nashville |
| Nashville (68%) vs. Knoxville (32%) | Nashville |
| Chattanooga (83%) vs. Knoxville (17%) | Chattanooga |
The results can also be shown in the form of a matrix:
| A | |||||
|---|---|---|---|---|---|
| Memphis | Nashville | Chattanooga | Knoxville | ||
| B | Memphis | [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: | 4th | 1st | 2nd | 3rd | |
As can be seen from both of the tables above, Nashville beats every other candidate. This means that Nashville is the Condorcet winner. Nashville will thus win an election held under any possible Condorcet method.
While any Condorcet method will elect Nashville as the winner, if instead an election based on the same votes were held using first-past-the-post or instant-runoff voting, these systems would select Memphis[2] and Knoxville[3] respectively. This would occur despite the fact that most people would have preferred Nashville to either of those "winners". Condorcet methods make these preferences obvious rather than ignoring or discarding them.
On the other hand, note that in this example Chattanooga also defeats Knoxville and Memphis when paired against those cities. If we changed the basis for defining preference and determined that Memphis voters preferred Chattanooga as a second choice rather than as a third choice, Chattanooga would be the Condorcet winner even though finishing in last place in a first-past-the-post election.
As noted above, sometimes an election has no Condorcet winner because there is no candidate who is preferred by voters to all other candidates. When this occurs the situation is known as a 'majority rule cycle', 'circular ambiguity', 'circular tie', 'Condorcet paradox', or simply 'cycle'. This situation emerges when, once all votes have been added up, the preferences of voters with respect to some candidates form a circle in which every candidate is beaten by at least one other candidate. For example, if there are three candidates, Candidate Rock, Candidate Scissors, and Candidate Paper, there will be no Condorcet winner if voters prefer Candidate Rock over Candidate Scissors and Scissors over Paper, but also Candidate Paper over Rock. Depending on the context in which elections are held, circular ambiguities may or may not be a common occurrence. Nonetheless there is always the possibility of an ambiguity, and so every Condorcet method must be capable of determining a winner when this occurs. A mechanism for resolving an ambiguity is known as ambiguity resolution or Condorcet completion method.
Circular ambiguities arise as a result of the voting paradox—the result of an election can be intransitive (forming a cycle) even though all individual voters expressed a transitive preference. In a Condorcet election it is impossible for the preferences of a single voter to be cyclical, because a voter must rank all candidates in order and can only rank each candidate once, but the paradox of voting means that it is still possible for a circular ambiguity to emerge.
The idealized notion of a political spectrum is often used to describe political candidates and policies. This means that each candidate can be defined by her position along a straight line, such as a line that goes from the most right wing candidates to the most left wing, with centrist candidates occupying the middle. Where this kind of spectrum exists and voters prefer candidates who are closest to their own position on the spectrum there is a Condorcet winner (Black's Single-Peakedness Theorem). Real political spectra, however, are at least two dimensional, with some political scientists advocating three dimensional models.
In Condorcet methods, as in most electoral systems, there is also the possibility of an ordinary tie. This occurs when two or more candidates tie with each other but defeat every other candidate. As in other systems this can be resolved by a random method such as the drawing of lots. Ties can also be settled through other methods like seeing which of the tied winners had the most first choice votes, but this and some other non-random methods may re-introduce a degree of tactical voting, especially if voters know the race will be close.
The method used to resolve circular ambiguities is the main difference between Condorcet methods. There are countless ways in which this can be done, but every Condorcet method involves ignoring the majorities expressed by voters in at least some pairwise matchings.
Condorcet methods fit within two categories:
Many one-method systems and some two-method systems will give the same result as each other if there are fewer than 4 candidates in the circular tie, and all voters separately rank at least two of those candidates. These include Smith-Minimax, Ranked Pairs, and Schulze.
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, the Black method chooses the Condorcet winner if it exists, but uses the Borda count instead if there is an ambiguity (the method is named for 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 scrutinizing 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
One possible method is to apply instant-runoff voting to the candidates of the Smith set. This method has been described as 'Smith/IRV'.
Some Condorcet methods use a single procedure that inherently meets the Condorcet criteria and, without any extra procedure, also resolves circular ambiguities when they arise. In other words, these methods do not involve separate procedures for different situations. Typically these methods base their calculations on pairwise counts. These methods include:
Ranked Pairs and Schulze are procedurally in some sense opposite approaches (although they very frequently give the same results):
Minimax could be considered as more "blunt" than either of these approaches, as instead of removing defeats it can be seen as immediately removing candidates by looking at the strongest defeats (although their victories are still considered for subsequent candidate eliminations).
The Kemeny-Young method considers every possible sequence of choices in terms of which choice might be most popular, which choice might be second-most popular, and so on down to which choice might be least popular. Each such sequence is associated with a Kemeny score that is equal to the sum of the pairwise counts that apply to the specified sequence. The sequence with the highest score is identified as the overall ranking, from most popular to least popular.
When the pairwise counts are arranged in a matrix in which the choices appear in sequence from most popular (top and left) to least popular (bottom and right), the winning Kemeny score equals the sum of the counts in the upper-right, triangular half of the matrix (shown here in bold on a green background).
| … over Nashville | … over Chattanooga | … over Knoxville | … over Memphis | |
|---|---|---|---|---|
| Prefer Nashville … | — | 68 | 68 | 58 |
| Prefer Chattanooga … | 32 | — | 83 | 58 |
| Prefer Knoxville … | 32 | 17 | — | 58 |
| Prefer Memphis … | 42 | 42 | 42 | — |
In this example, the Kemeny Score of the sequence Nashville > Chattanooga > Knoxville > Memphis would be 393.
Calculating every Kemeny score requires considerable computation time in cases that involve more than a few choices. However, fast calculation methods based on integer programming allow a computation time in seconds for cases with as many as 40 choices.
In the Ranked Pairs method, pairwise defeats are ranked (sorted) from strongest to weakest. Then each pairwise defeat is considered, starting with the strongest defeat. Defeats are "affirmed" (or "locked in") only if they do not create a cycle with the defeats already affirmed. Once completed, the affirmed defeats are followed to determine the winner of the overall election. 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.
The Schulze method resolves votes as follows:
In other words, this procedure repeatedly throws away the weakest pairwise defeat within the top set, until finally the number of votes left over produce an unambiguous decision.
Some pairwise methods—including minimax, Ranked Pairs, and the Schulze method—resolve circular ambiguities based on the relative strength of the defeats. There are different ways to measure the strength of each defeat, and these include considering "winning votes" and "margins":
If voters do not rank their preferences for all of the candidates, these two approaches can yield different results. Consider, for example, the following election:
| 45 voters | 11 voters | 15 voters | 29 voters |
|---|---|---|---|
| 1. A | 1. B | 1. B | 1. C |
| 2. C | 2. B |
The pairwise defeats are as follows:
Using the winning votes definition of defeat strength, the defeat of B by C is the weakest, and the defeat of A by B is the strongest. Using the margins definition of defeat strength, the defeat of C by A is the weakest, and the defeat of A by B is the strongest.
Using winning votes as the definition of defeat strength, candidate B would win under minimax, Ranked Pairs and the Schulze method, but, using margins as the definition of defeat strength, candidate C would win in the same methods.
If all voters give complete rankings of the candidates, then winning votes and margins will always produce the same result. The difference between them can only come into play when some voters declare equal preferences amongst candidates, as occurs implicitly if they do not rank all candidates, as in the example above.
The choice between margins and winning votes is the subject of scholarly debate. Because all Condorcet methods always choose the Condorcet winner when one exists, the difference between methods only appears when cyclic ambiguity resolution is required. The argument for using winning votes follows from this: Because cycle resolution involves disenfranchising a selection of votes, then the selection should disenfranchise the fewest possible number of votes. When margins are used, the difference between the number of two candidates' votes may be small, but the number of votes may be very large—or not. Only methods employing winning votes satisfy Woodall's plurality criterion.
An argument in favour of using margins is the fact that the result of a pairwise comparison is decided by the presence of more votes for one side than the other and thus that it follows naturally to assess the strength of a comparison by this "surplus" for the winning side. Otherwise, changing only a few votes from the winner to the loser could cause a sudden large change from a large score for one side to a large score for the other. In other words, one could consider losing votes being in fact disenfranchised when it comes to ambiguity resolution with winning votes. Also, using winning votes, a vote containing ties (possibly implicitly in the case of an incompletely ranked ballot) doesn't have the same effect as a number of equally weighted votes with total weight equaling one vote, such that the ties are broken in every possible way (a violation of Woodall's symmetric-completion criterion), as opposed to margins.
Under winning votes, if two more of the "B" voters decided to vote "BC", the A->C arm of the cycle would be overturned and Condorcet would pick C instead of B. This is an example of "Unburying" or "Later does harm". The margin method would pick C anyway.
Under the margin method, if three more "BC" voters decided to "bury" C by just voting "B", the A->C arm of the cycle would be strengthened and the resolution strategies would end up breaking the C->B arm and giving the win to B. This is an example of "Burying". The winning votes method would pick B anyway.
Other terms related to the Condorcet method are:
Some Condorcet methods produce not just a single winner, but a ranking of all candidates from first to last place. A Condorcet ranking is a list of candidates with the property that the Condorcet winner (if one exists) comes first and the Condorcet loser (if one exists) comes last, and this holds recursively for the candidates ranked between them.
Methods that satisfy this property include:
Many proponents of instant runoff voting (IRV) are attracted by the belief that if their first choice does not win, their vote will be given to their second choice; if their second choice does not win, their vote will be given to their third choice, etc. This sounds perfect, but it is not true for every voter with IRV. If someone voted for a strong candidate, and their 2nd and 3rd choices are eliminated before their first choice is eliminated, IRV gives their vote to their 4th choice candidate, not their 2nd choice. Condorcet voting takes all rankings into account simultaneously, but at the expense of violating the later-no-harm criterion. With IRV, indicating a second choice will never affect your first choice. With Condorcet voting, it is possible that indicating a second choice will cause your first choice to lose.
Plurality voting is simple, and theoretically provides incentives for voters to compromise for centrist candidates rather than throw away their votes on candidates who can't win. Opponents to plurality voting point out that voters often vote for the lesser of evils because they heard on the news that those two are the only two with a chance of winning, not necessarily because those two are the two natural compromises. This gives the media significant election powers. And if voters do compromise according to the media, the post election counts will prove the media right for next time. Condorcet runs each candidate against the other head to head, so that voters elect the candidate who would win the most sincere runoffs, instead of the one they thought they had to vote for.
There are circumstances, as in the examples above, when both instant-runoff voting and the 'first-past-the-post' plurality system will fail to pick the Condorcet winner. 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. 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 goal of a ranked voting system is for voters to be able to vote sincerely and trust the system to protect their intent. Plurality voting forces voters to do all their tactics before they vote, so that the system does not need to figure out their intent.
The significance of this scenario, of two parties with strong support, and the one with weak support being the Condorcet winner, may be misleading, though, as it is a common mode 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 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.
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, or when one can be created.
Many 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.
Example with the Schulze method:
| 46 voters | 44 voters | 10 voters |
|---|---|---|
| 1. A | 1. B | 1. C |
| 2. B | 2. A | 2. B |
| 3. C | 3. C | 3. A |
| 46 voters | 44 voters | 10 voters |
|---|---|---|
| 1. A | 1. B | 1. C |
| 2. C* | 2. A | 2. B |
| 3. B* | 3. C | 3. A |
Supporters of Condorcet methods which exhibit this potential problem could rebut this concern by pointing out that pre-election polls are most necessary with plurality voting, and that voters, armed with ranked choice voting, could lie to pre-election pollsters, making it impossible for Candidate A to know whether or how to bury. It is also nearly impossible to predict ahead of time how many supporters of A would actually follow the instructions, and how many would be alienated by such an obvious attempt to manipulate the system.
| 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 |
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; the Condorcet criterion is incompatible with independence of irrelevant alternatives, later-no-harm, the participation criterion, and the consistency criterion.
| Monotonic | Condorcet loser | Clone independence | Reversal symmetry | Polynomial time | |
|---|---|---|---|---|---|
| Schulze | Yes | Yes | Yes | Yes | Yes |
| Ranked Pairs | Yes | Yes | Yes | Yes | Yes |
| Minimax | Yes | No | No | No | Yes |
| Nanson | No | Yes | No | Yes | Yes |
| Kemeny-Young | Yes | Yes | No | Yes | No |
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,[4] and today Condorcet methods are used by a number of private organizations. Organizations which currently use some variant of the Condorcet method are: