Condorcet methods
61-415: Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results The contingent vote is an electoral system used to elect a single representative in which
122-501: A 'cycle'. This situation emerges when, once all votes have been tallied, the preferences of voters with respect to some candidates form a circle in which every candidate is beaten by at least one other candidate ( Intransitivity ). 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
183-400: A 68% majority of 1st choices among the remaining candidates and won as the majority's 1st choice. 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 'Condorcet cycle', 'majority rule cycle', 'circular ambiguity', 'circular tie', 'Condorcet paradox', or simply
244-417: A candidate requires a majority of votes to win. It is a form of preferential voting . The voter ranks the candidates in order of preference , and when the votes are counted, the first preference votes only are counted. If no candidate has a majority (more than half) of the votes cast, then all but the two leading candidates are eliminated and the votes received by the eliminated candidates are distributed among
305-547: A contest between candidates A, B and C using the preferential-vote form of Condorcet method, a head-to-head race is conducted between each pair of candidates. A and B, B and C, and C and A. If one candidate is preferred over all others, they are the Condorcet Winner and winner of the election. Because of the possibility of the Condorcet paradox , it is possible, but unlikely, that a Condorcet winner may not exist in
366-532: 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, 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, or Condorcet consistent, because they will still elect
427-523: A specific election. This is sometimes called a Condorcet cycle or just cycle and can be thought of as Rock beating Scissors, Scissors beating Paper, and Paper beating Rock . Various Condorcet methods differ in how they resolve such a cycle. (Most elections do not have cycles. See Condorcet paradox#Likelihood of the paradox for estimates.) If there is no cycle, all Condorcet methods elect the same candidate and are operationally equivalent. For most Condorcet methods, those counts usually suffice to determine
488-427: A voter's choice within any given pair can be determined from the ranking. Some elections may not yield a Condorcet winner because voter preferences may be cyclic—that is, it is possible that every candidate has an opponent that defeats them in a two-candidate contest. The possibility of such cyclic preferences is known as the Condorcet paradox . However, a smallest group of candidates that beat all candidates not in
549-407: Is also a Condorcet method, even though the voters do not vote by expressing their orders of preference. 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 survives to be paired in a later round against another alternative. Eventually, only one alternative remains, and it
610-499: 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. Additionally, a voting system can be considered to have Condorcet consistency, or be Condorcet consistent, if it elects any Condorcet winner. In certain circumstances, an election has no Condorcet winner. This occurs as
671-411: Is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner , is formally called the Condorcet winner or Pairwise Majority Rule Winner (PMRW). The head-to-head elections need not be done separately;
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#1732783354664732-412: 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. Using a matrix like
793-468: Is declared elected. The supplementary vote and the Sri Lankan contingent vote are two implementation variations, in which voters cannot rank all of the candidates but rather are only permitted to express two or three preferences, respectively. This means that if a voter's marked preferences do not include either of the candidates who survive to the second round, then it will be impossible to transfer
854-401: Is holding an election on the location of its capital . The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are: The preferences of each region's voters are: 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
915-513: Is known as ambiguity resolution, cycle resolution method, 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, from top-choice to bottom-choice, and can only rank each candidate once, but
976-440: Is no preference between candidates that were left unranked. Some Condorcet elections permit write-in candidates . 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
1037-598: Is one of the two remaining candidates, their vote is transferred to the second-choice candidate. This means that the winning candidate has the support of a majority of voters who expressed a preference among the top two, although not necessarily a majority of votes cast in the first count. The supplementary vote was used in all elections for directly elected mayors in England , including the Mayor of London , and in elections for police and crime commissioners , until 2022, when it
1098-510: Is taken to be the one in the pair that the voter ranks (or rates) 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
1159-404: Is the winner. This is analogous 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
1220-480: The Marquis de Condorcet , who championed such systems. However, Ramon Llull devised the earliest known Condorcet method in 1299. It was equivalent to Copeland's method in cases with no pairwise ties. Condorcet methods may use preferential ranked , rated vote ballots, or explicit votes between all pairs of candidates. Most Condorcet methods employ a single round of preferential voting, in which each voter ranks
1281-544: The Condorcet winner if there is one. Not all single winner, ranked voting systems are Condorcet methods. For example, instant-runoff voting and the Borda count are not Condorcet methods. In a Condorcet election the voter ranks the list of candidates in order of preference. If a ranked ballot is used, the voter gives a "1" to their first preference, a "2" to their second preference, and so on. Some Condorcet methods allow voters to rank more than one candidate equally so that
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#17327833546641342-464: 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 consistent method to another. 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 matchups, and eliminate all candidates not in
1403-555: The Copeland winner has the highest possible Copeland score. They can also be found by conducting a series of pairwise comparisons, using the procedure given in Robert's Rules of Order described above. For N candidates, this requires N − 1 pairwise hypothetical elections. For example, with 5 candidates there are 4 pairwise comparisons to be made, since after each comparison, a candidate is eliminated, and after 4 eliminations, only one of
1464-640: 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. [REDACTED] Suppose that Tennessee
1525-411: 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. An alternative way of thinking about this example if a Smith-efficient Condorcet method that passes ISDA is used to determine the winner is that 58% of
1586-605: The candidates from most (marked as number 1) to least preferred (marked with a higher number). A voter's ranking is often called their order of preference. Votes can be tallied in many ways to find a winner. All Condorcet methods will elect the Condorcet winner if there is one. If there is no Condorcet winner different Condorcet-compliant methods may elect different winners in the case of a cycle—Condorcet methods differ on which other criteria they satisfy. The procedure given in Robert's Rules of Order for voting on motions and amendments
1647-496: The complete order of finish (i.e. who won, who came in 2nd place, etc.). They always suffice to determine whether there is a Condorcet winner. Additional information may be needed in the event of ties. Ties can be pairings that have no majority, or they can be majorities that are the same size. Such ties will be rare when there are many voters. Some Condorcet methods may have other kinds of ties. For example, with Copeland's method , it would not be rare for two or more candidates to win
1708-424: The context in which elections are held, circular ambiguities may or may not be common, but there is no known case of a governmental election with ranked-choice voting in which a circular ambiguity is evident from the record of ranked ballots. Nonetheless a cycle is always possible, and so every Condorcet method should be capable of determining a winner when this contingency occurs. A mechanism for resolving an ambiguity
1769-739: The contingent vote and alternative vote can produce different results. Because, under the contingent vote, all but two candidates are eliminated in the first round, it is possible for a candidate to be eliminated who would have gone on to win had they been allowed to receive transfers in later rounds. Condorcet method Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results A Condorcet method ( English: / k ɒ n d ɔːr ˈ s eɪ / ; French: [kɔ̃dɔʁsɛ] )
1830-503: The contingent vote, systems like instant-runoff voting (IRV), Coombs' method , and Baldwin's method allow for many rounds of counting, eliminating only one weakest candidate each round. IRV allows a candidate other than the top two in the first count to win. A variant of the contingent vote has been used to elect the president of Sri Lanka since 1978. The supplementary vote was used to pick directly elected mayors and police and crime commissioners in England prior to 2022. In
1891-474: The eventual winner (though it will always elect someone in the Smith set ). A considerable portion of the literature on social choice theory is about the properties of 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 to administer. In
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1952-461: The first count then all but the two leading candidates are eliminated and their votes redistributed to help determine a winner in a second and final round. However, whereas under the ordinary form of the contingent vote voters can rank all of the candidates in order of preference, under Sri Lankan contingent voting the voter can only express their top three preferences (which can lead to exhausted ballots ). Each direct presidential election going back to
2013-410: The first in 1981 has seen a candidate from one of the two major parties or alliances at the time winning in the first count so never has the second round of vote counting ever been conducted. The supplementary vote (SV) is a variation of the contingent vote in which the voter ranks only two of the candidates in order of preference. If a voter's first-choice candidate is eliminated but their second choice
2074-423: The first round has not been eliminated. It also guarantees that every voter has a chance to express a preference between the top two, unlike the limited forms of contingent vote . Voter turnout may also be higher in the second vote. The nonpartisan blanket primary is a variation of the two-round system except the first round does not pick a winner, but instead picks the two highest candidates who will compete in
2135-444: The first round, all but the top two are eliminated and there is a second round. However, in the two round system, voters are asked to return and vote a second time. Because of the similarities between them, the contingent vote and the two-round system can usually be expected to elect the same winner. However, in the two-round system, the voter is permitted to change one's mind from one round to another, even if their favourite candidate in
2196-470: The following sum matrix: 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
2257-445: The general election. Because the first round does not pick a winner, there will tend to be higher voter turnout in the second election. The contingent vote will generally pick the same winner as a blanket primary , except fewer voters in the primary round may lead to a different top-two candidates than if the whole electorate voted in both rounds. The top-four primary is a variant of the nonpartisan blanket primary which advances
2318-454: The group, known as the Smith set , always exists. The Smith set is guaranteed to have the Condorcet winner in it should one exist. Many Condorcet methods elect a candidate who is in the Smith set absent a Condorcet winner, and is thus said to be "Smith-efficient". Condorcet voting methods are named for the 18th-century French mathematician and philosopher Marie Jean Antoine Nicolas Caritat,
2379-483: The head of the commission, Raymond Plant , with the invention of SV, according to others, it was the brainchild of a Labour member of Parliament (MP) at the time, Dale Campbell-Savours and academic Patrick Dunleavy , who outlined and advocated for it in an article for the New Statesman magazine that was published September 29, 1989. In 2000, several districts in England introduced directly elected mayors. It
2440-644: The history of SV due to their similarities. The supplementary vote is said to encourage candidates to seek support beyond their core base of supporters in order to secure the second preferences of the supporters of other candidates, and so to create a more conciliatory campaigning style among candidates with similar policy platforms. SV is also likely to improve the chances of "third party" candidates by encouraging voters, who wish to do so, to vote sincerely for such candidates for whom, under systems such as first-past-the-post, they would be discouraged from doing so for tactical reasons. These positive effects are moderated by
2501-509: The incentives SV creates for voting, in some circumstances, for only candidates from among the leading three. Political scientists Colin Rallings and Michael Thrasher noted two flaws of SV: Under the two-round system (also known as runoff voting and the second ballot ) voters vote for only a single candidate, rather than ranking candidates in order of preference. As under the contingent vote, if no candidate has an absolute majority in
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2562-438: The most common ballot layout, they place a '1' beside their most preferred candidate, a '2' beside their second most preferred, and so on. In this respect the contingent vote is the same as other ranked ballot methods. There are then a maximum of two rounds of counting. In the first round only first preferences are counted. Candidates receiving an absolute majority of first preferences (i.e. more than half) are immediately declared
2623-425: 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
2684-419: The original 5 candidates will remain. To confirm that a Condorcet winner exists in a given election, first do the Robert's Rules of Order procedure, declare the final remaining candidate the procedure's winner, and then do at most an additional N − 2 pairwise comparisons between the procedure's winner and any candidates they have not been compared against yet (including all previously eliminated candidates). If
2745-417: The paradox of voting means that it is still possible for a circular ambiguity in voter tallies to emerge. Exhausted ballot In the alternative vote , ballot exhaustion occurs when a voter's ballot can no longer be counted, because all candidates on that ballot have been eliminated from an election. Contributors to ballot exhaustion include: This may occur because the voter chooses not to fill out
2806-563: The past, the ordinary form of the contingent vote was used to elect the Legislative Assembly of Queensland from 1892 to 1942. To date, this has been the longest continuous use of the system anywhere in the world. Contingent voting was used for Democratic party primaries in the US state of Alabama from 1915 to 1931. In an election held using the contingent vote, the voters rank the list of candidates in order of preference. Under
2867-426: The procedure's winner does not win all pairwise matchups, then no Condorcet winner exists in the election (and thus the Smith set has multiple candidates in it). Computing all pairwise comparisons requires ½ N ( N −1) pairwise comparisons for N candidates. For 10 candidates, this means 0.5*10*9=45 comparisons, which can make elections with many candidates hard to count the votes for. The family of Condorcet methods
2928-469: The same number of pairings, when there is no Condorcet winner. A Condorcet method is a voting system that will always elect the Condorcet winner (if there is one); this is the candidate whom voters prefer to each other candidate, when compared to them one at a time. This candidate can be found (if they exist; see next paragraph) 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
2989-426: 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 summarize all the pairwise preferences of all
3050-546: The top four candidates from a single primary, regardless of party, and uses instant-runoff voting in the general election to pick a majority winner. As noted above, the instant-runoff voting (or alternative vote ) differs from the contingent vote in that it permits several rounds rather than just two. Under the alternative vote only candidate(s) for whom it is mathematically impossible to win are eliminated after each round, and as many rounds occur as are necessary to give one candidate an absolute majority. These differences mean that
3111-465: The two remaining candidates according to voters' preferences. The contingent vote can be considered a compressed or "instant" form of the two-round system (runoff system), in which both "rounds" occur without the need for voters to go to the polls twice. For this reason, the term instant-runoff voting has also been used for this method, though this conflicts with the more common meaning . It also has similarities to other ranked-choice systems. Unlike
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#17327833546643172-565: The use of the supplementary vote in 2022, citing voter confusion with a complex system. However, critics, including the Labour and Green parties, argued that the wasted votes were due to ballot layout and that the change was aimed at benefitting Conservative Party candidates. They also claimed that the supplementary vote was effective in increasing multi-party participation and was popular among voters. The histories of two-round voting and other forms of instant run-off voting may be seen as part of
3233-519: The vote, which is therefore declared "wasted" or "exhausted". In Sri Lanka, since the 1982 presidential election , a variant of the contingent vote electoral system is used to elect the country's president . As under the conventional contingent vote, in an election held using the Sri Lankan form of the contingent vote each voter ranks the candidates in order of preference, and if no candidate receives an overall majority of first preference votes on
3294-486: The voter might express two first preferences rather than just one. If a scored ballot is used, voters rate or score the candidates on a scale, for example as is used in Score voting , with a higher rating indicating a greater preference. When a voter does not give a full list of preferences, it is typically assumed that they prefer the candidates that they have ranked over all the candidates that were not ranked, and that there
3355-420: The voters, a mutual majority , ranked Memphis last (making Memphis the majority loser ) and Nashville, Chattanooga, and Knoxville above Memphis, ruling Memphis out. At that point, the voters who preferred Memphis as their 1st choice could only help to choose a winner among Nashville, Chattanooga, and Knoxville, and because they all preferred Nashville as their 1st choice among those three, Nashville would have had
3416-448: 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
3477-532: The winner is the candidate preferred by a majority of voters. When results for every possible pairing have been found they are as follows: The results can also be shown in the form of a matrix: ↓ 2 Wins ↓ 1 Win 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
3538-523: 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 and Knoxville 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, in this example Chattanooga also defeats Knoxville and Memphis when paired against those cities. If we changed
3599-422: The winner. However, if no candidate has an absolute majority, then all but the two candidates with the most first preferences are eliminated, and there is a second round. In the second round, the votes of the voters whose first preference had been eliminated are transferred to whichever of the two remaining candidates they ranked the highest. The votes are then counted, and whichever candidate has an absolute majority
3660-490: Was decided to use the supplementary vote for the election of these new mayors, including the Mayor of London, and for the election of police and crime commissioners across much of England and Wales. The supplementary vote was used for these offices from 2000 to 2022. In the 2021 London election, a record 5 percent of ballots were wholly rejected, and no candidate achieved a majority of the vote. The government responded by ending
3721-872: Was replaced by first-past-the-post voting (FPTP). In the early 1990s, the Plant Commission was established by the Labour Party to recommend a new voting system for the Parliament of the United Kingdom . When the Commission reported in 1993, instead of suggesting an already existing system, it recommended the supplementary vote system, which it said had never been used anywhere. In actuality, contingent voting had been in use in Australia as early as 1892. Although some commentators credit
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