Condorcet ranking: Difference between revisions
→Smith set ranking
No edit summary |
|||
Line 11:
* The secondary Condorcet loser (the new Condorcet loser if the Condorcet loser is eliminated), if one exists, is ranked next-to-last.
* The tertiary Condorcet loser (the new Condorcet loser if the Condorcet loser and secondary Condorcet loser are eliminated), if one exists, is ranked third-to-last.
* etc.
== 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. There will always be at least one Smith ranking, and it is possible to have more than one (each of which will differ only on the order in which Smith or Smith loser candidates are ranked).
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:
Line 26:
2 B
1 C▼
▲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. There will always be one and only one generalized Smith set ranking. ▼
▲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.
Line 41 ⟶ 40:
33 C>D>A>B</blockquote>
A pairwise beats B pairwise beats C pairwise beats D pairwise 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. ▼
▲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 ==
| |||