Condorcet winner criterion: Difference between revisions
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==Commentary==
On a one-dimensional [[political spectrum]], the beats-all winner will be at the position of the median voter. Example (also see https://i.stack.imgur.com/z4bjy.png):<graph>{
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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.
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In this case, B is preferred to A by 501 votes to 499, and B is
preferred to C by 502 to 498, hence B is preferred to both A and C. So
according to the Condorcet criteria, B should win. By contrast, according to the rules of IRV, B is ranked first by the fewest voters and is eliminated, and C wins with the transferred voted from B; in plurality voting A wins with the most first choices. Note that B and C are a [[Mutual majority criterion|mutual majority]], so most majority rule-based methods would rule A out of winning. If A drops out, then B becomes the majority's 1st choice; so this is an example of IRV failing [[Independence of irrelevant alternatives|independence of irrelevant alternatives]].
Range voting does not comply because it allows for the difference between 'rankings' to matter. E.g. 51 people might rate A at 100, and B at 90, while 49 people rate A at 0, and B at 100. Condorcet would consider this 51 people voting A>B, and 49 voting B>A, and A would win. Range voting would see this as A having support of 5100/100 = 51%, and B support of (51*90+49*100)/100 = 94.9%; range voting advocates would probably say that in this case the Condorcet winner is not the socially ideal winner. In general however, it is expected that the Condorcet winner (and Smith Set candidates in general) will almost always be very high-utility when compared to the utilitarian winner.
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