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Condorcet winner criterion: Difference between revisions
Popping the example out of the Template:Image frame and into a #Detailed example section.
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{{Wikipedia}}<!-- {{cleanup|date=February 2020}} -->
{{Wikipedia}}<!-- {{cleanup|date=February 2020}} -->{{Image frame|width=450|content=Suppose the voters had been polled on their preferences among the candidates, and the following preferences in head-to-head matchups are produced between French Fries (FF), Hamburger (H), and Cookies (C) (FF>C shows the number of voters who prefer FF over C, for example):▼
The '''Condorcet criterion''' for a [[voting system]] is that it chooses the beats-all winner when one exists. Any method conforming to the Condorcet criterion is known as a [[Condorcet method]].▼
Mainly because of [[Condorcet paradox|Condorcet's voting paradox]], a beats-all winner will not always exist in a given set of votes. However, there will always be a smallest group of candidates such that more voters prefer anyone in the group over anyone outside of the group. If the beats-all winner exists, they will be the only candidate in this group, which is called the [[Smith set]]. Voting methods that always elect from the Smith set are known as "[[Smith-efficient]]". ▼
== Example ==▼
As an example, if there are 3 candidates, with head-to-head matchups indicating a 51% majority prefers the second candidate over the first, and a 43% plurality prefer the second over 37% preferring the third (with 20% of voters having no preference), then the second candidate gets more votes than their competitors in all matchups and so they are the Condorcet winner.
=== Detailed example ===
▲
FF>H:51, FF>C:60
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FF (French Fries) is the CW here.
▲|caption=Example of finding the Condorcet winner|border=|max-width=}}The '''Condorcet candidate''', '''Pairwise Champion''' (PC), '''beats-all winner''', or '''Condorcet winner''' (CW) of an [[election]] is the candidate who is preferred by more voters than any other candidate in [[Pairwise counting|pairwise matchups]]. This is determined by observing whether more voters rank or score the Condorcet winner higher than each of the other candidates than the other way around.
▲The '''Condorcet criterion''' for a [[voting system]] is that it chooses the beats-all winner when one exists. Any method conforming to the Condorcet criterion is known as a [[Condorcet method]].
▲Mainly because of [[Condorcet paradox|Condorcet's voting paradox]], a beats-all winner will not always exist in a given set of votes. However, there will always be a smallest group of candidates such that more voters prefer anyone in the group over anyone outside of the group. If the beats-all winner exists, they will be the only candidate in this group, which is called the [[Smith set]]. Voting methods that always elect from the Smith set are known as "[[Smith-efficient]]".
▲== Example ==
▲As an example, if there are 3 candidates, with head-to-head matchups indicating a 51% majority prefers the second candidate over the first, and a 43% plurality prefer the second over 37% preferring the third (with 20% of voters having no preference), then the second candidate gets more votes than their competitors in all matchups and so they are the Condorcet winner.
== A more general wording of Condorcet criterion definition ==
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