Condorcet method: Difference between revisions
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{{Wikipedia}}
Any election method conforming to the [[Condorcet criterion]] - that is, one which always elects the
At present, the synonymous phrase '''"[[Instant-Round-Robin Voting|Instant Round Robin Voting]]" (IRRV)''' is being coined to leverage the public's greater familiarity with [[IRV |Instant Runoff Voting]] (IRV). This phrase is currently being used in a [http://groups.yahoo.com/group/Condorcet legislative effort] to implement a Condorcet variant ([[CSSD]]) in the state of Washington.
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== Simple explanation ==
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 procedure; the most common minimum standard for a Condorcet method's tiebreaking procedure should be [[Smith-efficient]], that is, always elect someone from the smallest group of candidates that win all their [[head-to-head matchup|head-to-head matchups]] against all candidates not in the group.
== Casting ballots ==
:''See also: [[Ballot]]''
Each voter fills out a [[preferential voting|ranked ballot]] or [[Cardinal voting|rated ballot]]. The voter can include less than all candidates under consideration. Usually when a candidate ''is not listed'' on the voter's ballot they are considered less preferred than listed candidates, and ranked accordingly. 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 a counting issue, but results in the frequent omission of the write-in option in ballot software.
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If a candidate is preferred over all other candidates, 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 simultaneously. This is called a majority rule cycle or Condorcet cycle, and it must be resolved by some other mechanism.
=== Counting with matrices ===
:''Main article: [[Pairwise counting]]''
A frequent implementation of this method will illustrate the basic counting method. Consider an election between A, B, and C, and a ballot (B, C, A, D). That is, a ballot ranking B first, C second, A third, and D forth. 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.
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Other terms related to the Condorcet method are:
* '''Condorcet loser''': the candidate who is less preferred than every other candidate in a
* '''weak Condorcet winner''': a candidate who beats or ties with every other candidate in a
* '''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.
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== Connection to cardinal methods ==
Score Voting can be thought of as a Condorcet method where a voter is allowed to give a fraction of a vote to a candidate in a pairwise matchup against other candidates, rather than a full vote or nothing. Further, the amount of a vote the voter gives in one runoff directly alters the amount they give in another; if they arrange their scores such that they give 0.4 of a vote to help one candidate beat another, this automatically means they can at best arrange their scores such that they give up to 0.6 of their vote to help the second candidate beat someone else. Assuming a voter would vote the exact same way in a Score Voting runoff between all possible pairs of candidates as they did in the original Score election, Score elects the Condorcet winner using this modified definition.<ref>https://rangevoting.org/CondDQ.html</ref>
It's possible to modify Score to be more like a traditional Condorcet method by allowing voters to write the scores they would give to every possible pair of candidates in a Score runoff, and then using a Condorcet method to process this, treating a score of, say, A5 B3 (where the max score is 5) as 0.4 votes for A>B. As this would be utterly infeasible with just a few candidates running however, one way to accomplish most of the same objective is to allow voters to mark on their ballots that they want their vote strategically optimized, meaning that if their cardinally expressed preferences are A5 B3 Z2, instead of having their vote considered as B3 Z2 in an B vs. Z runoff, it would be considered as B5 Z0 (if the max score is 5), which is functionally equivalent to the Plurality voting runoffs that are used for the traditional Condorcet winner definition. This strategic optimization can be done fractionally to allow a voter to customize how much optimization they want to be done with their scores in each runoff. It is also possible for voters to indicate an approval threshold, meaning that for all approved candidates, no strategic optimization is applied to pairwise matchups between them, but all pairwise matchups between approved and disapproved candidates are strategically optimized. With this modification, if all voters use strategic optimization, Score becomes a traditional Condorcet method (which will need a cycle resolution method to be applied at times), but if no voters strategically optimize, it remains Score (which never needs cycle resolution methods to be applied).
Note that the above schemes can make Score fail the logical property that a voter's strength of preference between any pair of candidates must equal the sum of the strengths of preference between all sequential pairs of candidates in a beatpath from the first candidate of the pair to the second. The failure of this property seems to be the major reason traditional Condorcet methods can have Condorcet cycles and one major reason for why they fail certain properties such as Favorite Betrayal and Independence of Irrelevant Alternatives.
See also [[Pairwise counting#Cardinal methods]] and [[Order theory#Strength of preference]].
== Demonstrating pairwise counting ==
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