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However, fast calculation methods based on integer programming allow a computation time in seconds for some cases with as many as 40 choices. The order of finish is constructed a piece at a time by considering the pairwise majorities one at a time, from largest majority to smallest majority.

For each majority, their higher-ranked candidate is placed ahead of their lower-ranked candidate in the partially constructed order of finish, except when their lower-ranked candidate has already been placed ahead of their higher-ranked candidate.

The three majorities are a rock paper scissors cycle. Ranked pairs begins with the largest majority, who rank B over C, and places B ahead of C in the order of finish.

Then it considers the second largest majority, who rank A over B, and places A ahead of B in the order of finish.

At this point, it has been established that A finishes ahead of B and B finishes ahead of C, which implies A also finishes ahead of C.

So when ranked pairs considers the third largest majority, who rank C over A, their lower-ranked candidate A has already been placed ahead of their higher-ranked candidate C, so C is not placed ahead of A.

The order of finish is "A, B, C" and A is the winner. An equivalent definition is to find the order of finish that minimizes the size of the largest reversed majority.

In the 'lexicographical order' sense. If the largest majority reversed in two orders of finish is the same, the two orders of finish are compared by their second largest reversed majorities, etc.

Any other order of finish would reverse a larger majority. This definition is useful for simplifying some of the proofs of Ranked Pairs' properties, but the "constructive" definition executes much faster small polynomial time.

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:. 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.

This is an example of "Unburying" or "Later does harm". The margin method would pick C anyway. This is an example of "Burying".

The winning votes method would pick B anyway. 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.

Though there won't always be a Condorcet winner or Condorcet loser, there is always a Smith set and "Smith loser set" smallest group of candidates who lose to all candidates not in the set in head-to-head elections.

Some voting methods produce rankings that sort all candidates in the Smith set above all others, and all candidates in the Smith loser set below all others, with this holding recursively for all candidates ranked between them; in essence, this guarantees that when the candidates can be split into two groups, such that every candidate in the first group beats every candidate in the second group head-to-head, then all candidates in the first group are ranked higher than all candidates in the second group.

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 and the later-no-help 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.

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:.

B is preferred by a — majority to A, and by a — majority to C. So, according to the Condorcet criterion, B should win, despite the fact that very few voters rank B in first place.

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.

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 all voting methods, [25] 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.

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

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.

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 [26] , later-no-harm , the participation criterion , and the consistency criterion.

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.

Organizations which currently use some variant of the Condorcet method are:. From Wikipedia, the free encyclopedia. This article needs additional citations for verification.

Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Proportional representation.

Mixed systems. Mixed-member proportional Additional member system Parallel voting mixed-member majoritarian Scorporo Majority bonus Alternative Vote Plus Dual-member proportional Rural—urban proportional.

Other systems and related theory. Cumulative voting Binomial voting Proxy voting Delegated voting Random selection sortition , random ballot Comparison of electoral systems Social choice theory Arrow's theorem Gibbard—Satterthwaite theorem Public choice theory.

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In , he published Vie de Voltaire , which agreed with Voltaire in his opposition to the Church. The journal's theme was that any sort of monarchy is a threat to freedom no matter who is leading and that liberty is freedom from domination.

It dealt with theoretical thought on perfecting the human mind and analyzing intellectual history based on social arithmetic.

Condorcet took a leading role when the French Revolution swept France in , hoping for a rationalist reconstruction of society, and championed many liberal causes.

As a result, in he was elected as a Paris representative in the Legislative Assembly , and then became the secretary of the Assembly. In April Condorcet presented a project for the reformation of the education system, aiming to create a hierarchical system, under the authority of experts, who would work as the guardians of the Enlightenment and who, independent of power, would be the guarantors of public liberties.

The project was judged to be contrary to the republican and egalitarian virtues, giving the education of the Nation over to an aristocracy of savants.

The institution adopted Condorcet's design for the state education system, and he drafted a proposed Bourbon Constitution for the new France.

Condorcet was not affiliated with any political party but counted many friends among the Girondins. He distanced himself from them during the National Convention , however, due to his distaste for their factionalism.

At the Trial of Louis XVI , Condorcet, who opposed the death penalty albeit supporting the trial itself, spoke out against the execution of the King during the public vote at the Convention — he proposed to send the king to the galleys.

Condorcet was on the Constitution Committee and was the main author of the Girondin constitutional project.

The constitution was not put to vote. When the Montagnards gained control of the Convention, they wrote their own, the French Constitution of Condorcet criticized the new work, and as a result, he was branded a traitor.

On 3 October , a warrant was issued for Condorcet's arrest. The warrant forced Condorcet into hiding. He hid for five or eight months in the house of Mme.

Vernet, on Rue Servandoni, in Paris. It narrates the history of civilization as one of progress in the sciences, shows the intimate connection between scientific progress and the development of human rights and justice, and outlines the features of a future rational society entirely shaped by scientific knowledge.

On 25 March Condorcet, convinced he was no longer safe, left his hideout and attempted to flee Paris. He went to seek refuge at the house of Jean-Baptiste Suard , a friend of his with whom he had resided in , [17] but he was refused on the basis that he would be betrayed by one of their residents.

The most widely accepted theory is that his friend, Pierre Jean George Cabanis , gave him a poison which he eventually used.

However, some historians believe that he may have been murdered perhaps because he was too loved and respected to be executed.

His coffin, however, was empty as his remains, originally interred in the common cemetery of Bourg-la-Reine , were lost during the nineteenth century.

In Condorcet married Sophie de Grouchy , who was more than twenty years his junior. Sophie, reckoned one of the most beautiful women of the day, became an accomplished salon hostess as Madame de Condorcet, and also an accomplished translator of Thomas Paine and Adam Smith.

She was intelligent and well educated, fluent in both English and Italian. The marriage was a strong one, and Sophie visited her husband regularly while he remained in hiding.

Although she began proceedings for divorce in January , it was at the insistence of Condorcet and Cabanis, who wished to protect their property from expropriation and to provide financially for Sophie and their young daughter, Louise 'Eliza' Alexandrine.

Condorcet was survived by his widow and four-year-old Eliza. Sophie died in , never having remarried, and having published all her husband's works between and The Condorcet-O'Connors published a revised edition between and Condorcet's work was mainly focused on a quest for a more egalitarian society.

This path led him to think and write about gender equality in the Revolutionary context. One of the most famous Enlightenment thinkers at the time, he was one of the first to make such a radical proposal.

A visionary, he identified gender as a social construction based on perceived differences in sex and rejected biological determinism as being able to explain gender relations in society.

He denounced patriarchal norms of oppression, present at every institutional level, and continuously subjugating and marginalising women. Like fellow Enlightenment thinker Jean-Jacques Rousseau in his book Emile ou De l'Education , Condorcet identified education as crucial to the emancipation of individuals.

He stated: "I believe that all other differences between men and women are simply the result of education". In her book Essays in the Philosophy of Humanism , Hooks calls this new concept "feminine masculinity", "new models of self—assertion that do not require the construction of an enemy 'other,' be it a woman or the symbolic feminine, for them to define themselves against".

Condorcet's whole plea for gender equality is founded on the recognition that the attribution of rights and authority comes from the false assumption that men possess reason and women do not.

This is, according to Nall [ who? Scholars [ who? His detractors [ who? Condorcet's Sketch for a Historical Picture of the Progress of the Human Spirit was perhaps the most influential formulation of the idea of progress ever written.

It made the Idea of Progress a central concern of Enlightenment thought. He argued that expanding knowledge in the natural and social sciences would lead to an ever more just world of individual freedom, material affluence, and moral compassion.

He argued for three general propositions: that the past revealed an order that could be understood in terms of the progressive development of human capabilities, showing that humanity's "present state, and those through which it has passed, are a necessary constitution of the moral composition of humankind"; that the progress of the natural sciences must be followed by progress in the moral and political sciences "no less certain, no less secure from political revolutions"; that social evils are the result of ignorance and error rather than an inevitable consequence of human nature.

Condorcet's writings were a key contribution to the French Enlightenment , particularly his work on the Idea of Progress. Condorcet believed that through the use of our senses and communication with others, knowledge could be compared and contrasted as a way of analyzing our systems of belief and understanding.

None of Condorcet's writings refer to a belief in a religion or a god who intervenes in human affairs. Condorcet instead frequently had written of his faith in humanity itself and its ability to progress with the help of philosophers such as Aristotle.

Through this accumulation and sharing of knowledge he believed it was possible for any man to comprehend all the known facts of the natural world.

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. Added to the first voter, these ballots would give 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 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.

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.

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:.

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 [19] and Knoxville [20] 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 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.

Depending on 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 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 the paradox of voting means that it is still possible for a circular ambiguity in voter tallies to emerge.

The idealized notion of a political spectrum is often used to describe political candidates and policies. 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.

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 the various 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.

Some cycle resolution methods are Smith-efficient, meaning that they pass the Smith criterion. This guarantees that when there is a cycle and no pairwise ties , only the candidates in the cycle can win, and that if there is a mutual majority , one of their preferred candidates will win.

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 Minimax but done only after all candidates not in the Smith set are eliminated , Ranked Pairs, and Schulze. For example, with three candidates in the Smith set in a Condorcet cycle, because Schulze and Ranked Pairs pass ISDA , all candidates not in the Smith set can be eliminated first, and then for Schulze, dropping the weakest defeat of the three allows the candidate who had that weakest defeat to be the only candidate who can beat or tie all other candidates, while with Ranked Pairs, once the first two strongest defeats are locked in, the weakest can't, since it'd create a cycle, and so the candidate with the weakest defeat will have no defeats locked in against them.

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 fall-back methods involve entirely disregarding the results of the pairwise comparisons. For example, the Black method chooses the Condorcet winner if it exists, but uses the Borda count instead if there is a cycle the method is named for Duncan Black.

A more sophisticated two-stage process is, in the event of a cycle, 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. 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.

One way to think of it in terms of removing defeats is that Minimax removes each candidate's weakest defeats until some group of candidates with only pairwise ties between them have no defeats left, at which point the group ties to win.

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.

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 some cases with as many as 40 choices.

The order of finish is constructed a piece at a time by considering the pairwise majorities one at a time, from largest majority to smallest majority.

For each majority, their higher-ranked candidate is placed ahead of their lower-ranked candidate in the partially constructed order of finish, except when their lower-ranked candidate has already been placed ahead of their higher-ranked candidate.

The three majorities are a rock paper scissors cycle. Ranked pairs begins with the largest majority, who rank B over C, and places B ahead of C in the order of finish.

Then it considers the second largest majority, who rank A over B, and places A ahead of B in the order of finish. At this point, it has been established that A finishes ahead of B and B finishes ahead of C, which implies A also finishes ahead of C.

So when ranked pairs considers the third largest majority, who rank C over A, their lower-ranked candidate A has already been placed ahead of their higher-ranked candidate C, so C is not placed ahead of A.

The order of finish is "A, B, C" and A is the winner. An equivalent definition is to find the order of finish that minimizes the size of the largest reversed majority.

In the 'lexicographical order' sense. If the largest majority reversed in two orders of finish is the same, the two orders of finish are compared by their second largest reversed majorities, etc.

Any other order of finish would reverse a larger majority. This definition is useful for simplifying some of the proofs of Ranked Pairs' properties, but the "constructive" definition executes much faster small polynomial time.

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:. 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.

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