Condorcet ranking: Difference between revisions
→Smith set ranking
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Condorcet Criterion (XCC). In essence, it says that if the set of alternatives can be partitioned in such a
way that all members of a subset of this partition defeat all alternatives belonging to subsets with a higher
index, then the former should obtain a better rank than the latter.}}</ref> Each consecutive Smith set can be referred to as the 1st, 2nd, 3rd, etc. or the primary, secondary, tertiary, etc. Smith set.
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 simplest way to find a Smith ranking is to find the [[Copeland's method|Copeland]] ranking (the ranking of all candidates such that the candidates with the most ([[Pairwise counting#Terminology|pairwise victories]] minus pairwise defeats) are ranked first, the candidates with the second-most (pairwise victories minus pairwise defeats) are ranked second, etc.)
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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. There is an ambiguity to consider in that if, for example, there are five candidates, with three in the 1st Smith set and two in the 2nd Smith set, then the two 2nd Smith set candidates can be considered to either be in 2nd place (because they are worse than all the 1st place candidates but better than all other candidates) or 4th place (because in order for them to be the best candidates in the election i.e. 1st place, at least 3 candidates considered better than them must be removed from the election).
Another example: <blockquote>1 A>B>C
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Methods that produce Condorcet rankings (and also Smith rankings) include [[Copeland's method]], [[Pairwise Sorted Methods]], [[Schulze]], [[Ranked Pairs]], and [[Kemeny-Young]].
Smith set rankings can be given additional context by showing for each Smith set the weakest pairwise victory anyone in that Smith set has against anyone in a lower-ranked Smith set; one generalized way to show strength of victory is by % of votes earned in the matchup, which will always be over 50% for the pairwise winner. So, for example, if there are 4 candidates, with 3 of them in a cycle and each of them having a 65%, 60%, and 55% victory respectively against the fourth candidate, then 55% could be used to describe the minimum strength of all of the candidates in the set against the lower-ranked candidate.
The generalized Smith ranking can also 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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!Sp, Pa, Sa
| colspan="3" |SpongeBob, Patrick, Sandy (2-0)
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|-
!
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|3 Losses →
1 Win ↓
|Minimum of
51% margin
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!
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|10 (Sq) –
9 (Pl)
|Minimum of
62% margin
|}
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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. One way to do so is to show the minimum % of votes each candidate gets in the head-to-head matchups against all lower/worse Smith set candidates i.e. if A, B and C are in the Smith set, with D in the 2nd Smith set, and C has the smallest-size majority by %, say, 54%, then 54% can be used to indicate the minimum quality of all the 1st Smith set candidates against D.
Theoretical note:
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