Condorcet ranking: Difference between revisions
→Smith set ranking
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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.)
The '''generalized Smith ranking''' is a Smith ranking where all candidates in
<blockquote>2 A
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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.
Another example: <blockquote>1 A>B>C
1 B>C>A
1 C>A>B
1 D>E
1 E>D
1 F=G </blockquote>
{| class="wikitable"
|+Pairwise comparison table (candidate on left is preferred by # of voters in their cell over candidate on top)
Pairwise victories are bolded, defeats are underlined, ties are struck through
Groups of candidates who pairwise beat all other lower groups are italicized; every odd group's numbers are made bigger and even group's numbers smaller
!
!A
!B
!C
!D
!E
!F
!G
|-
|A
|''<big>-</big>''
|'''''<big>2</big>'''''
|<u>''<big>1</big>''</u>
|'''3'''
|'''3'''
|'''3'''
|'''3'''
|-
|B
|<u>''<big>1</big>''</u>
|''<big>-</big>''
|'''''<big>2</big>'''''
|'''3'''
|'''3'''
|'''3'''
|'''3'''
|-
|C
|'''''<big>2</big>'''''
|<u>''<big>1</big>''</u>
|''<big>-</big>''
|'''3'''
|'''3'''
|'''3'''
|'''3'''
|-
|D
|<u>''<sup><small>2</small></sup>''</u>
|<u>''<sup><small>2</small></sup>''</u>
|<u>''<sup><small>2</small></sup>''</u>
|''<sup><small>-</small></sup>''
|''<sup><small><s>1</s></small></sup>''
|'''2'''
|'''2'''
|-
|E
|<u>''<sup><small>2</small></sup>''</u>
|<u>''<sup><small>2</small></sup>''</u>
|<u>''<sup><small>2</small></sup>''</u>
|''<sup><small><s>1</s></small></sup>''
|''<sup><small>-</small></sup>''
|'''2'''
|'''2'''
|-
|F
|<u>''<big>1</big>''</u>
|<u>''<big>1</big>''</u>
|<u>''<big>1</big>''</u>
|<u>''<big>1</big>''</u>
|<u>''<big>1</big>''</u>
|''<big>-</big>''
|''<big><s>0</s></big>''
|-
|G
|<u>''<big>1</big>''</u>
|<u>''<big>1</big>''</u>
|<u>''<big>1</big>''</u>
|<u>''<big>1</big>''</u>
|<u>''<big>1</big>''</u>
|''<big><s>0</s></big>''
|''<big>-</big>''
|}
The generalized Smith set ranking is A=B=C'''>'''D=E'''>'''F=G, or (A, B, C) tied for 1st, (D, E) tied for 2nd, and (F, G) tied for 3rd place. This is because 3 voters prefer (A, B, C) over all others (and at most 2 voters prefer any other candidates over (A, B, C)) but no smaller subset of (A, B, C) can say the same, 2 voters prefer (D, E) above all others when ignoring (A, B, C) (and at most 1 voter prefers other unignored candidates over (D, E)) but no smaller subset of (D, E) can say the same, and 1 voter prefers (F, G) above all others when ignoring (A, B, C, D, E) (and no voters prefer any other unignored candidates over (F, G)) but no smaller subset of (F, G) can say the same. Notice that when looking at the pairwise comparison table for the generalized Smith ranking that the Smith sets can be seen by looking at which groups of candidates have pairwise victories over all other candidates when ignoring the pairwise matchups between candidates in the group.
Another way of looking at the pairwise comparison table would be to put all candidates in each consecutive Smith set on their own line, like so:
{| class="wikitable"
|+
!
!A, B, C
!D, E
!F, G
|-
|A, B, C
|<big>-</big>
|'''Win'''
|'''Win'''
|-
|D, E
|<u><small>Loss</small></u>
|<small>-</small>
|'''Win'''
|-
|F, G
|<u><big>Loss</big></u>
|<u><big>Loss</big></u>
|<big>-</big>
|}
The simplest way to find the generalized Smith set ranking is to find the [[Copeland's method|Copeland]] ranking, put all candidates ranked 1st by Copeland in the Smith set and thus equally rank them 1st in the generalized Smith set ranking, and then check if any of the candidates ranked 2nd by Copeland can pairwise beat or tie any of the candidates ranked higher than them (1st) by Copeland who are in the Smith set; if any of them can, then all candidates ranked 2nd by Copeland are also part of the Smith set and thus are also equally ranked 1st in the generalized Smith set ranking, but if none of them can, then 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 equally ranked within their consecutive Smith set.
== Notes ==
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