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Condorcet method

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The 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. Mainly because of Condorcet's paradox, a Condorcet Winner will not always exist in a given set of votes.

Any voting system which chooses the Condorcet Winner when it exists is known as a Condorcet method, after its deviser, the 18th century mathematician and philosopher Marquis de Condorcet, although the method was previously devised by Ramon Llull in the 13th century. [1]

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


Basic Procedure

Each voter ranks the candidates in order of preference. For each pair of candidates, it is determined how many voters prefer one candidate to the other by counting how many times it is higher ranked on the ballot (usually, candidates not placed on the ballot at all are considered to be less preferred than all those that are and equally preferred compared to each other). If one candidate is preferred to every other candidate, that candidate is declared the winner.

Resolving 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. In other words, there may be cycles in the results. 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.

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 who, when paired off in pairwise elections, can beat all other candidates outside the set.
  • 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 if it provides a large number of "identical" options. A cloneproof voting method prevents this attack. See strategic nomination.

Different Ambiguity Resolution Methods

There are countless number of "Condorcet methods" possible that resolve such ambiguities. The fact that Marquis de Condorcet himself already spearheaded the debate of which particular Condorcet method to promote, has made the term "Condorcet's method" ambiguous. Indeed, it can be argued that the large number of different competing Condorcet methods has hurt the adoption of any specific Condorcet method.

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) chooses the candidate whose worst pair wise defeat is less bad than that of all other candidates. [*]
  • Smith/Minimax restricts the Minimax algorithm to the Smith set. [*]
  • Ranked Pairs (RP) or Tideman (named after Nicolaus Tideman) with variations such as Maximize Affirmed Majorities (MAM) and Maximum Majority Voting (MMV) [*]
  • Schulze with several reformulations/variations, including Schwartz Sequential Dropping (SSD) and Cloneproof Schwartz Sequential Dropping (CSSD) [*]

[*] There are different ways to measure the strength of each defeat in some methods. Some use the margin of defeat (the difference between votes for and votes against), while others use winning votes (the votes favoring the defeat in question). Electionmethods.org argues that there are several disadvantages of systems that use margins instead of winning votes. The website argues that using margins, "destroys" some information about majorities, so that the method can no longer honor information about what majorities have determined and that consequently margin-based systems cannot support a number of desirable voting properties.

Ranked Pairs and Schulze are procedurally in some sense opposite approaches:

  • Ranked Pairs (and variants) starts with the strongest information available and uses as much information as it can without creating ambiguity
  • Schulze (and variants) repeatedly removes the weakest ambiguous information until ambiguity is removed.

The text below describes (variants of) these methods in more detail.

Ranked Pairs, Maximize Affirmed Majorities (MAM), and Maximum Majority Voting (MMV)

In the Ranked Pairs (RP) voting method, as well as the variations Maximize Affirmed Majorities (MAM) and Maximum Majority Voting (MMV), pairs of defeats are ranked (sorted) from largest majority to smallest majority. Then each pair is considered, starting with the defeat supported by the largest majority. Pairs are "affirmed" only if they do not create a cycle with the pairs already affirmed. Once completed, the affirmed pairs are followed to determine the winner.

In essence, RP and its variants (such as MAM and MMV) 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 difference betweeen RP and its variants is in the details of the ranking approach. Some definitions of RP use margins, while other definitions use winning votes (not margins). Both MAM and MMV are explicitly defined in terms of winning votes, not winning margins. In MAM and MMV, if two defeat pairs have the same number of votes for a victory, the defeat with the smaller defeat is ranked higher. If this still doesn't disambiguate between the two, MAM and MMV perform slightly differently. In MAM, information from a "tiebreaker" vote is used (which could be a distinguished vote such as the vote of a "speaker", but unless there is a distinguished vote, a randomly-chosen vote is used). In MMV all such conflicting matchups are ignored (though any non-conflicting matchups of that size are still included).

Cloneproof Schwartz Sequential Dropping (CSSD)

The "cloneproof Schwartz Sequential Dropping" (CSSD) method resolves votes as follows:

  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 narrowest defeats, until finally the largest number of votes left over produce an unambiguous decision.

The "Beatpath Winner" algorithm produces equivalent results.

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):

  • 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:

42% of voters (close to Memphis)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville

26% of voters (close to Nashville)
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis

15% of voters (close to Chattanooga)
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis

17% of voters (close to Knoxville)
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

The results would be tabulated as follows:

Pairwise Election Results
A
Memphis Nashville Chattanooga Knoxville
BMemphis[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 (by repeatedly removing Condorcet Winner): 4th 1st 2nd 3rd
  • [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.

Condorcet compared to Instant Runoff

There are reasonable arguments to regard the Condorcet criterion as a requirement of a voting system. If there is a Condorcet Winner (a candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate), many argue that that candidate should be selected by the voting system as the (sole) winner. From this point of view, instant-runoff is not as good as the Condorcet scheme, because there are circumstances in which instant-runoff will fail to pick the Condorcet Winner. Moreover, tactical voting can be in a voter's interest under instant-runoff, but never under Condorcet. Both systems, though, are susceptible to strategic nominations.

Assuming that preferences are sincerely expressed and remain consistent, in a head-to-head election, the Condorcet Winner of an election would always beat the Instant Runoff winner from the same set of ballots, provided the Condorcet Winner exists and differs from the Instant Runoff winner.

The Condorcet criterion is thus seen as being a natural extension of majority rule.

Technical advantages of Condorcet Voting

Since (using most of the proposed tiebreakers) Condorcet voting only uses the ballot totals in each pairwise race to determine the winner, the results can be tallied in a distributed fashion - ie, at the precinct level. This gives it added resistance to fraud, including the possibility of limited recounts. This advantage is shared by plurality voting (First_Past_the_Post_electoral_system) but not by instant-runoff.

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 Cloneproof Schwartz Sequential Dropping.
  2. The Free State Project for choosing its target state
  3. The voting procedure for the uk.* hierarchy of Usenet
  4. Five-Second Crossword Competition

External Resources

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