Condorcet ranking: Difference between revisions
→Smith set ranking
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1 C</blockquote>
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 simplest way to find the generalized Smith set ranking is to find the [[Copeland's method|Copeland]] ranking, and then check if one of the candidates ranked 2nd by Copeland can pairwise beat or tie any of the candidates ranked higher than them by Copeland; if they can, then all candidates ranked 2nd by Copeland are part of the Smith set and thus equally ranked 1st in the generalized Smith set ranking, otherwise they are all part of the secondary Smith set and thus equally ranked 2nd in the generalized Smith set ranking. Repeat until all candidates are ranked into all of the various Smith sets.
Note that a variant of generalized Smith rankings can be created to address the unanimity/Pareto criterion. For example:▼
33 B>C>D>A▼
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 acknowledge that D must be at a lower rank than C, whereas the regular generalized Smith ranking would simply be A=B=C=D. ▼
== Notes ==
Methods that produce only Condorcet rankings may include [[Kemeny-Young]]; methods that produce Smith rankings and thus also Condorcet rankings when they exist include [[Copeland's method]], [[Pairwise Sorted Methods]], [[Schulze]], and [[Ranked Pairs]];
The generalized Smith ranking can be found by finding all candidates in the Smith set, equally ranking each of them, then eliminating all of them and finding the secondary Smith set (the Smith set now that the original Smith set has been eliminated) and equally ranking each of them lower than the candidates in the Smith set, and repeating until all candidates are ranked.
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{| class="wikitable"
| colspan="3" rowspan="2" |
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!Pl
!Sq
!Sp, Pa, Sa
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!
!Sp, Pa, Sa
| colspan="
|5 Wins ↓
|-
!
!Sq
| colspan="
|3 Losses
1 Win
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!
!Pl
|Plankton (0-4)
9 (Pl)
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▲|5 Losses - >
▲|4314 (BK) –
|}
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#
This makes it more ambiguous as to how to record the exact margins of some [[Pairwise counting#Terminology|pairwise matchups]] where groups of candidates are involved, as even if all candidates in a group of candidates pairwise beat all candidates not in the group, they may each do so with different margins.
Technical information:
▲Note that a variant of generalized Smith rankings can be created to address the unanimity/Pareto criterion. For example:<blockquote>34 A>B>C>D
▲33 B>C>D>A
▲33 C>D>A>B</blockquote>
▲A [[Pairwise counting#Terminology|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 acknowledge that D must be at a lower rank than C, whereas the regular generalized Smith ranking would simply be A=B=C=D.
[[Category:Voting theory]]
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