Condorcet method: Difference between revisions

{{Wikipedia}}

 

Any election method conforming to the [[Condorcet criterion]] - that is, one which always elects the [[beats-all winner]], a candidate who can beat any other candidate in a runoff, if one exists - is known as a '''Condorcet method'''. The name comes from the 18th century mathematician and philosopher [[Marquis de Condorcet]], although the method was previously described by [[Ramon Llull]] in the 13th century. Many Condorcet advocates agree that a further criterion that Condorcet methods should pass is the [[Smith criterion]], which means the Condorcet method will always elect someone from the [[Smith set]] when there is no beats-all winner (due to the [[Condorcet paradox]]).

 

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

 

If one candidate is preferred by more voters than all other candidates (when [[pairwise counting|compared one-on-one]]), that candidate is the [[Condorcet Criterion|Condorcet Winner]], abbreviated as CW. This can be determined through use of ranked or rated ballots (i.e. if a voter ranks or rates one candidate higher than another). On rare occasions, there is no Condorcet winner (because of either [[pairwise counting#Terminology|ties]] in the head-to-head matchups or the [[Condorcet paradox]]). In that case it is necessary to use some "tiebreaking" or cycle resolution/completion procedure; a very common standard for a Condorcet method's tiebreaking procedure is that it should be [[Smith-efficient]], that is, always elect someone from the [[Smith set]], the smallest group of candidates that win all their [[head-to-head matchup|head-to-head matchups]] against all candidates not in the group.

 

:''See also: [[Ballot]]''

 

 

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 and receive that ballot's vote in the matchup against their opponent if they are ranked or rated 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/matchup wins the matchup.

 

If a candidate is preferred over all other candidates i.e. wins all of their matchups, that candidate is the [[Condorcet winner|Condorcet candidate]] (Condorcet winner). However, a Condorcet candidate may not exist, due to a fundamental [[Voting paradox|paradox]]: It is possible for the electorate to prefer A over B, B over C, and C over A in each head-to-head matchup simultaneously. This is called a majority rule cycle or Condorcet cycle, and it must be resolved by some other mechanism (usually either by using another voting method, or [[:Category:Defeat-dropping Condorcet methods|"dropping" weak defeats]] in order to end the cycle).

:''Main article: [[Pairwise counting]]''

 

 

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[[:Category:Sequential comparison Condorcet methods|Sequential comparison]] is one such way: order all of the candidates in any manner desired, pairwise compare the first two, eliminate the loser of the matchup, and repeat until only one candidate remains. This requires ((number of candidates) - 1) pairwise comparisons, since for each comparison one candidate is eliminated, and all but one candidate must be eliminated. To check whether a Condorcet winner exists in a given election, do the previous procedure and then check whether the remaining candidate wins all of their pairwise matchups; this requires ((number of candidates) - 2) pairwise comparisons in the worst case, though if the ordering of the candidates in the procedure is done in such a way as to put candidates more likely to be Condorcet winners higher in the ordering, then in the best case 0 pairwise comparisons are required, since if the first candidate in the ordering turns out to be the Condorcet winner, all of their pairwise comparisons have already been done. Condorcet winners may often have a lot of 1st choice votes, especially in less contested elections, so it may be best to order the candidates descending by order of 1st choice votes, then 2nd choice votes, etc. These procedures can be used even for Condorcet PR methods by considering each winner set to be a candidate.

 

The following are key terms when discussing ambiguity resolution methods:

*[[Defeat strength|'''Defeat strength''']]: Different ways of measuring how "strong" a pairwise defeat is. Useful when deciding which defeats to "drop" in [[:Category:Defeat-dropping Condorcet methods|defeat-dropping Condorcet methods]].

 

 

There are a 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.

 

Examples of Condorcet methods include:

*'''[[Copeland's method|Copeland]]''' selects the candidate that wins the most pairwise matchups minus the number of matchups it loses (or simply, wins the most matchups). Note that if there is no Condorcet winner, Copeland will often still result in a tie.

*[[:Category:Condorcet-IRV hybrid methods|'''Condorcet-IRV hybrid methods''']]:

**'''[[Baldwin]]''' computes the [[Borda count]] for all candidates, then iteratively deletes (eliminates) the candidate with the lowest count.

 

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:

 

The text below describes (variants of) the defeat-dropping methods in more detail.

 

 

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.

 

The difference between 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).

 

The [[Schulze method]] resolves votes as follows:

 

 

In other words, this procedure repeatedly throws away the narrowest defeats, until finally the largest number of votes left over produce an unambiguous decision.

 

See also [[Pairwise counting#Terminology]]

*'''weak Condorcet loser''': a candidate who is defeated by or ties with every other candidate in a pairwise matchup. Similarly, there can be more than one weak Condorcet loser.

 

 

See also: [[Pairwise_counting#Example_with_numbers]]

[[Image:CondorcetTennesee.png]]</div>

 

 

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:

 

The results would be tabulated as follows:

 

<tr>

<th bgcolor="#c0c0ff">Nashville

<th bgcolor="#c0c0ff">Chattanooga

<th bgcolor="#c0c0ff">Knoxville

</tr>

 

 

<td bgcolor="#ffffff">4th

</table>

 

 

In this election, Nashville is the Condorcet winner (Nashville beats Memphis 58 to 42, and Chattanooga and Knoxville 68 to 32) and thus the winner under all possible Condorcet methods.

|+

!

!Nashville

!Chattanooga

|42 (-16 Loss)

|42 (-16 Loss)

|-

|Nashville

|Chattanooga

|58 (+16 Win)

| ---

|83 (+66 Win)

|58 (+16 Win)

|32 (-36 Loss)

|}

 

 

See also [[Score voting#Connection to Condorcet methods]].

See also [[Pairwise counting#Cardinal methods]] and [[Order theory#Strength of preference]].

 

Also see: [[Pairwise counting]]

 

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Any voting method can be made a Condorcet method by simply adding a condition that a Condorcet winner will win if one exists before running the voting method. It is possible to further make a voting method [[Smith-efficient]] by taking various approaches, such as eliminating candidates one by one until there is a Condorcet winner (like in [[Benham's method]]) or eliminating all candidates not in the [[Smith set]] before running the voting method's procedure. It is common terminology for Condorcet methods that start by electing the Condorcet winner if there is one, but otherwise run some other voting method, to be named as "Condorcet//voting method". For example, [[Condorcet//Score]] is [[Score voting]] modified to elect a CW. The Condorcet methods that start by eliminating all candidates not in a given set of candidates and then run some other voting method are named as "Given set//voting method" (sometimes with only one "/"). For example, [[Smith//IRV]] is [[IRV]] run on the [[Smith set]].

 

At one point, the synonymous phrase (to Condorcet voting) '''"[[Instant-Round-Robin Voting|Instant Round Robin Voting]]" (IRRV)''' was being coined to leverage the public's greater familiarity with [[IRV |Instant Runoff Voting]] (IRV). This phrase was being used in a [http://groups.yahoo.com/group/Condorcet legislative effort] to implement a Condorcet variant ([[CSSD]]) in the state of Washington.

 

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:

 

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[[Category:Condorcet methods]]