Any election method conforming to the Condorcet criterion is known as a Condorcet method. It is named after the 18th century mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet. The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... A mathematician is a person whose area of study and research is mathematics. ... Jump to: navigation, search A philosopher is a person devoted to studying and producing results in philosophy. ... Marie Jean Antoine Nicolas Caritat, marquis de Condorcet (September 17, 1743 - March 28, 1794) was a French philosopher, mathematician, and early political scientist who devised the concept of a Condorcet method. ...

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

Basic procedure

The basic procedure for casting ballots is identical to most preferential ballots, such as IRV and Borda ballots. However, the voter might prefer to order them differently, due to differing strategic incentives.

Casting ballots

Each voter ranks the candidates in the order that he or she prefers each candidate. The voter may give some candidates equal rankings.

Usually when a candidate is not listed on the voter's ballot they are considered to be less preferred than all listed candidates for the purposes of the tally procedure. However, some variations allow a "no opinion" default option where no for- or against- preference is counted for that candidate.

Write-ins are possible, but are somewhat more difficult to implement for automatic counting than in other election methods. This is an implementation issue, but results in the frequent omission of the write-in option in ballot software. The voter is supposed to rank each candidate in numeric order of preference, 1st, 2nd, 3rd, 4th, and so on.

Counting ballots

Ballots are counted by considering all possible sets of two-candidate elections from all available candidates. That is, each candidate is considered against each and every other candidate. A candidate is considered to "win" against another on a single ballot if they are ranked higher than their opponent. All the votes for candidate Alice over candidate Bob are counted, as are all of the votes for Bob over Alice. Whoever has the most votes in each one-on-one election wins.

If a candidate is preferred over all other candidates, that candidate is the Condorcet winner. However, a Condorcet winner may not exist, due to a fundamental paradox: It is possible for the electorate to prefer A over B, B over C, and C over A simultaneously. This is called a majority rule cycle, and it must be resolved by some other mechanism. The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ... The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ... The voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i. ...

Counting with matrices

A frequent implementation of this method will illustrate the basic counting method. Consider an election between A, B, C and D, and a ballot (B, C, A, D). That is, a ballot ranking B first, C second, A third, and D fourth. This can be represented as a matrix, where the row is the runner under consideration, and the column is the opponent. The cell at (runner,opponent) has a one if runner is preferred, and a zero if not.

Cells marked "—" are logically zero, but are blank for clarity—they are not considered, as a candidate can not be defeated by himself. This binary matrix is inversely symmetric: (runner,opponent) is ¬(opponent,runner). The utility of this structure is that it may be easily added to other ballots represented the same way, to give us the number of ballots which prefer each candidate. The sum of all ballot matrices is called the sum matrix—it is not symmetric.

When the sum matrix is found, the contest between each candidate is considered. The number of votes for runner over opponent (runner,opponent) is compared the number of votes for opponent over runner (opponent,runner). The one-on-one winner has the most votes. If one candidate wins against all other candidates, that candidate wins the election.

The sum matrix is the primary piece of data used to resolve majority rule cycles.

Resolving circular ambiguities

Just about any election system that treats every voter equally (anonymity) and every candidate equally (neutrality) has the possibility of ties. A Condorcet method isn't different in that regard. For example, it's possible for candidates to tie with each other and "pairwise defeat" everybody else.

However, "Condorcet" methods have an additional ambiguity: the problem of the Condorcet paradox. There may be cycles in the results. The voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i. ...

For example, it would be possible for the totalled votes to record that A defeats B, B defeats C, and C defeats A. And while voters often vote so that there is a single Condorcet winner of a given election (see in that regard political spectrum), a Condorcet method is usually only considered for serious use if such cycles can be handled. Handling cases where there is not a single Condorcet winner is called ambiguity resolution in this article, though other phrases such as "cyclic ambiguity resolution" and "Condorcet completion" are used as well. A political spectrum is a way of comparing or visualizing different political positions, by placing them upon one or more geometric axes. ...

Key terms in ambiguity resolution

The following are key terms when discussing ambiguity resolution methods:

• Smith set: the smallest set of candidates in a particular election such that every candidate in the set can beat all candidates outside the set. It is easily shown that there is only one possible Smith set for a given electoral outcome.
• Schwartz set: the union of all possible sets of candidates such that for every set:
  1. every candidate inside the set is pairwise unbeatable by any other candidate outside the set, i.e., ties are allowed
  2. no proper (smaller) subset of the set fulfills the first property
• Cloneproof: a method that is immune to the presence of clones (candidates which are essentially identical to each other). In some voting methods, a party can increase its odds of selection through strategic nomination, by providing a large number of "identical" options. A cloneproof voting method prevents this attack.

In voting systems, the Smith set is the smallest set of candidates in a particular election who, when paired off in pairwise elections, can beat all other candidates outside the set. ... The Schwartz set is a term used in regard to voting systems. ... Strategic nomination is the manipulation of an election through its candidate set (compare this to tactical voting, where the manipulation comes from the voters). ...

Different ambiguity resolution methods

There are a countless number of "Condorcet methods" possible that resolve such ambiguities. Marquis de Condorcet began this ambiguity by spearheading the debate. It can be argued that the large number of different competing Condorcet methods has made the adoption of any single method extremely difficult. Because the circular tie represents a real voter preference, and because all votes are treated equally, each method of breaking the cycle represents a different way of selecting votes to disenfranchise.

Examples of Condorcet methods include:

• Black chooses the Condorcet winner when it exists and otherwise the Borda winner. It is named after Duncan Black.
• Smith/IRV is instant-runoff voting with the candidates restricted to the Smith set.
• Copeland selects the candidate that wins the most pairwise matchups. Note that if there is no Condorcet winner, Copeland will often still result in a tie.
• Minimax (also called Simpson and Simpson-Kramer) chooses the candidate whose worst pairwise defeat is better than that of all other candidates.
• Smith/Minimax restricts the Minimax algorithm to the Smith set.
• Ranked Pairs (RP) or Tideman (first proposed by Nicolaus Tideman) with variations such as Maximize Affirmed Majorities (MAM) and Maximum Majority Voting (MMV)
• Schulze (a.k.a. Schwartz sequential dropping (SSD), cloneproof Schwartz sequential dropping (CSSD), beatpath method, beatpath winner, path voting, path winner)

Ranked Pairs and Schulze are procedurally in some sense opposite approaches (although they very frequently give the same results): The Borda count is a voting system used for single-winner elections in which each voter rank-orders the candidates. ... Duncan Black (May 23, 1908 - January 14, 1991) was responsible for unearthing the work of many early political scientists, including Charles Dodgson, and was responsible for the Black electoral system, a variant upon the Condorcet method whereby, in the absence of a Condorcet winner (e. ... Jump to: navigation, search When the Single Transferable Vote voting system is applied to a single-winner election it is sometimes called instant-runoff voting (IRV), as it is much like holding a series of runoff elections in which the lowest polling candidate is eliminated in each round until someone... Copelands method is a Condorcet method in which the winner is determined by finding the candidate with the most pairwise victories. ... Minimax is often considered to be the simplest of the Condorcet methods. ... Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) is a voting method that selects a single winner using votes that express preferences. ... T. Nicolaus Tideman (born August 11, 1943 in Chicago, Illinois) is a Professor of Economics at Virginia Polytechnic Institute and State University. ... Maximize Affirmed Majorities (MAM) is a voting method developed by Stephen Eppley that selects a single winner using votes that express preferences. ... Maximum majority voting (MMV) is a voting method that selects a single winner using votes that express preferences. ... Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ... AKA is an initialism for Also Known As. ...

• Ranked Pairs (and variants) starts with the strongest defeats and uses as much information as it can without creating ambiguity.
• Schulze repeatedly removes the weakest defeat until ambiguity is removed. (This approach is shared by minimax.)

The text below describes these methods in more detail. Minimax is often considered to be the simplest of the Condorcet methods. ...

Ranked Pairs

In the Ranked Pairs (RP) voting method, pairwise defeats are ranked (sorted) from strongest to weakest. Then each pairwise defeat is considered, starting with the strongest defeat. Defeats are "affirmed" only if they do not create a cycle with the defeats already affirmed. Once completed, the affirmed defeats are followed to determine the winner. Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) is a voting method that selects a single winner using votes that express preferences. ...

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.

Schulze method

The Schulze method resolves votes as follows: Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ...

1. First, determine the Schwartz set (the innermost unbeaten set). If no defeats exist among the Schwartz set, then its members are the winners (plural only in the case of a tie, which must be resolved by another method).
2. Otherwise, drop the weakest defeat information among the Schwartz set (i.e., where the number of votes favoring the defeat is the smallest). Determine the new Schwartz set, and repeat the procedure.

In other words, this procedure repeatedly throws away the weakest pairwise defeat within the top set, until finally the number of votes left over produce an unambiguous decision.

Defeat strength

Many pairwise methods (including minimax, ranked pairs, and Schulze) resolve majority rule cycles based on the relative strength of the defeats. There are different ways to measure the strength of each defeat, which include "winning votes" and "margins": Minimax is often considered to be the simplest of the Condorcet methods. ... Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) is a voting method that selects a single winner using votes that express preferences. ... Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ...

• Winning votes: The number of votes on the winning side of the defeat
• Margins: The number of votes on the winning side of the defeat, minus the number of votes on the losing side of the defeat.

Sometimes these two approaches can yield different results. For example:

Pairwise defeats:

• B beats A, 55 to 45 (55 winning votes, a margin of 10 votes)
• A beats C, 45 to 44 (45 winning votes, a margin of 1 vote)
• C beats B, 29 to 26 (29 winning votes, a margin of 3 votes)

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 in minimax, ranked pairs, and Schulze; But, using margins as the definition of defeat strength, Candidate C would win in the same methods. Minimax is often considered to be the simplest of the Condorcet methods. ... Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) is a voting method that selects a single winner using votes that express preferences. ... Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ...

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 truncate their ballots, as in the example above.

The choice between margins and winning votes is open for debate. Because all Condercet methods always choose the strong 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. Despite this, margins appear to be a more intuitive measure of defeat strength to many people. Plurality criterion is a voting system criterion devised by Douglas Woodall. ...

Related terms

Other terms related to the Condorcet method are:

• Condorcet loser: the candidate who is less preferred than every other candidate in a pair wise matchup.
• weak Condorcet winner: a candidate who beats or ties with every other candidate in a pair wise matchup. There can be more than one weak Condorcet winner.
• weak Condorcet loser: a candidate who is defeated by or ties with every other candidate in a pair wise matchup. Similarly, there can be more than one weak Condorcet loser.

An example

Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county): Jump to: navigation, search State nickname: Volunteer State Other U.S. States Capital Nashville Largest city Memphis Governor Phil Bredesen (D) Senators Bill Frist (R) Lamar Alexander (R) Official languages English Area 109,247 km² (36th)  - Land 106,846 km²  - Water 2,400 km² (2. ...

• Memphis (Shelby County): 826,330
• Nashville (Davidson County): 510,784
• Chattanooga (Hamilton County): 285,536
• Knoxville (Knox County): 335,749

Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows: Census. ...

The results would be tabulated as follows:

• [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption

In this election, Nashville is the Condorcet winner and thus the winner under all possible Condorcet methods. Notice how first-past-the-post and instant-runoff voting would have respectively selected Memphis and Knoxville here, while compared to either of them, most people would have preferred Nashville. The first-past-the-post electoral system is a voting system for single-member districts, variously called first-past-the-post (FPTP or FPP), winner-take-all, plurality voting, or relative majority. ... Jump to: navigation, search When the Single Transferable Vote voting system is applied to a single-winner election it is sometimes called instant-runoff voting (IRV), as it is much like holding a series of runoff elections in which the lowest polling candidate is eliminated in each round until someone...

Condorcet compared to Instant Runoff and First-past-the-post

There are circumstances, as in the example above, when both instant-runoff voting and plurality voting will fail to pick the Condorcet winner. Proponents of the Condorcet criterion see it as a principal issue in selecting an electoral system. They see the Condorcet criterion as a natural extension of majority rule. Majoritarianism is a political philosophy or agenda which asserts that a majority (sometimes categorized by religion, language or some other identifying factor) of the population is entitled to a certain degree of primacy in society, and has the right to make decisions that affect the society. ...

Condorcet methods tend to encourage the selection of centrist candidates who appeal to the median voter. Jump to: navigation, search In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ...

Here is an example that is designed to support IRV at the expense of Condorcet:

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.

Here is an example that is designed to support Condorcet at the expense of IRV:

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. In voting systems, tactical voting (or strategic voting) occurs when a voter misrepresents his or her sincere preferences in order to gain a more favorable outcome. ...

Potential for tactical voting

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. In voting systems, tactical voting (or strategic voting) occurs when a voter misrepresents his or her sincere preferences in order to gain a more favorable outcome. ...

However, Condorcet methods are only vulnerable to compromising when there is a majority rule cycle. By contrast, instant runoff voting has compromising incentive when there is a majority rule cycle, but it may also have compromising incentive under other circumstances, i.e. when the Condorcet winner is eliminated before the final round. When the single transferable vote voting system is applied to a single-winner election it is sometimes called instant-runoff voting (IRV), as it is much like holding a series of runoff elections in which the lowest polling candidate is eliminated in each round until someone receives majority vote. ...

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. In general, this can be done by creating a false majority rule cycle that overrules a genuine pairwise defeat. Some Condorcet methods may be less vulnerable to the burying strategy than others. In voting systems, tactical voting (or strategic voting) occurs when a voter misrepresents his or her sincere preferences in order to gain a more favorable outcome. ...

Use of Condorcet voting

Condorcet voting is not currently used in government elections. However, it is starting to receive support in some public organizations. Organizations which currently use some variant of the Condorcet method are:

1. The Debian project uses the Schulze method for internal referendums and to elect its leader.
2. The Software in the Public Interest corporation uses the Schulze method to elect members of its board of directors.
3. The Gentoo Linux project uses the Schulze method.
4. The UserLinux project uses the Schulze method.
5. The Free State Project for choosing its target state.
6. The voting procedure for the uk.* hierarchy of Usenet
7. Five-Second Crossword Competition

See also: Use of the Schulze method Jump to: navigation, search Debian, organized by the Debian Project, is a widely used distribution of free software developed through the collaboration of volunteers from around the world. ... Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ... Software in the Public Interest, Inc. ... Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ... Jump to: navigation, search This article is about the Linux distribution. ... Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ... UserLinux is a Linux distribution that aims to provide businesses with a freely available, high quality operating system accompanied by certifications, service, and support options designed to encourage productivity and security while reducing overall costs. ... Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ... The Free State Project (FSP) is a plan to have 20,000 or more liberty-oriented people move to a single state of the United States, with the intent of influencing local political policy. ... Jump to: navigation, search Usenet is a distributed Internet discussion system that evolved from a general purpose UUCP network of the same name. ... Jump to: navigation, search The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ...

See also

• List of democracy and elections-related topics

Politics is the process and method of gaining or maintaining support for public or common action: the conduct of decision-making for groups. ...

External resources

• Ranked Pairs by Blake Cretney
• Condorcet Voting Calculator by Eric Gorr
• Voting methods: tutorial and essays by James Green-Armytage
• Voting Systems by Paul E. Johnson
• Condorcet's Method By Rob Lanphier
• Accurate Democracy by Rob Loring
• Voting and Social Choice (PDF) by Herve Moulin. Demonstration and commentary on Condorcet method.
• CIVS, a free web poll service using the Condorcet method by Andrew Myers
• Software for computing Condorcet methods and STV by Jeffrey O'Neill
• Maximum Majority Voting by Ernest Prabhakar
• A New Monotonic and Clone-Independent Single-Winner Election Method (1, 2) (PDF) by Markus Schulze