Condorcet ranking: Difference between revisions
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Methods that produce Condorcet rankings include [[Kemeny-Young]] and [[Ranked Pairs]], and [[Pairwise Sorted Methods]]. ▼
== Smith set ranking ==
A Smith ranking (Smith set ranking) is the same as a Condorcet ranking, except instead of being defined in terms of the Condorcet winner and Condorcet loser, it is defined in terms of the [[Smith set]] and [[Smith set#Smith loser set|Smith loser set]], and when there is a multi-member Smith set, the candidates in the Smith set may be ranked in any order so long as they are all ranked above non-Smith set candidates, with the same applying for the Smith loser set candidates with regards to being ranked lower than non-Smith loser set candidates. When there is a Condorcet ranking (one doesn't always exist), the Smith ranking will be the same as the Condorcet ranking.
The generalized Smith ranking is a Smith ranking where all candidates in the Smith set are equally ranked and the same holds for the Smith loser set; for example:
2 A
2 B
1 C
The generalized Smith ranking is A=B>C, meaning A and B are tied for 1st place, and C is in 2nd place. Note that any of A>B=C, B>A=C, or A=B>C would be valid Smith rankings, but only the third is a generalized Smith ranking; this is because the generalized Smith ranking assumes neutrality as to who is superior within the Smith and Smith loser sets. This may make it an appropriate tool to demonstrate how various groups of candidates would compare across all Smith-efficient methods.
Note that a variant of generalized Smith rankings can be created to address the unanimity/Pareto criterion. For example:
<br /><blockquote>34 A>B>C>D
33 B>C>D>A
33 C>D>A>B</blockquote>
A beats B beats C beats D beats A, so all candidates here are in the Smith set. However, C is unanimously preferred to D, so the Pareto-satisfying generalized Smith ranking would be A=B=C>D, whereas the regular generalized Smith ranking would be A=B=C=D.
== Notes ==
▲Methods that produce Condorcet rankings include [[Kemeny-Young]] and [[Ranked Pairs]]
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