Condorcet winner criterion: Difference between revisions
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}</graph>Supposing this is the voter distribution, with 3 candidates Leftist, Centrist, Rightist who are each points on the left, center, and right of the distribution respectively, with all voters distributed from left to right, and voters on, for example, the left preferring Leftist>Centrist>Rightist, and the same holding vice versa. The Condorcet winner is the Centrist, because a majority prefer them over the "other side", for whichever side you look at.
The Condorcet winner/[[Smith set]] is a common [[equilibrium]] point in many voting methods. This is because a majority/plurality of voters have no incentive to deviate towards another candidate. Example for [[Approval voting]]:
35: A>B|>C
34: B>C|
31: C>B|>A
B is the CW. If voters approve everyone they ranked before the "|", then B is approved by all voters, and wins. If any of the three groups of voters here raises their approval threshold (only approves their 1st choice), then another group has an incentive to maintain their approval threshold where it is i.e. if C-top voters stop approving B, then the 69 voters who prefer B>C have an incentive to move their approval thresholds between B and C to ensure B is approved by a majority and C is not. Note that this requires both accurate polling and coordinated [[Strategic voting|strategic voting]].
Non-ranking methods such as [[plurality voting|plurality]] and [[approval voting|approval]] cannot comply with the Condorcet criterion because they do not allow each voter to fully specify their preferences. But instant-runoff voting allows each voter to rank the candidates, yet it still does not comply. A simple example will prove that IRV fails to comply with the Condorcet criterion.
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