Woodall's method: Difference between revisions

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== Woodall's method: ==
== Woodall's method: ==
Do IRV till only one member of the initial Smiths set remains
{{definition|Do IRV till only one member of the initial Smiths set remains
un-eliminated. Elect hir.
un-eliminated. Elect hir.}}
[end of Woodall definition]
Smith set:
Smith set:
The Smith set is the smallest set of candidates such that every
{{definition|The Smith set is the smallest set of candidates such that every
candidate in the set beats every candidate outside the set.
candidate in the set beats every candidate outside the set.}}

[end of Smith set definition]
IRV definition (for the purpose of Woodall):
IRV definition (for the purpose of Woodall):
Repeatedly, cross-off or delete from the rankings the candidate who
{{definition|Repeatedly, cross-off or delete from the rankings the candidate who
tops the fewest rankings.
tops the fewest rankings.}}
[end of IRV definition for the purpose of Woodall]


Definition of "beats":
Definition of "beats":
X beats Y if more ballots rank X over Y than rank Y over X.
{{definition|X beats Y if more ballots rank X over Y than rank Y over X.}}
[end of "beats" definition]

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=== A few properties of Woodall ===
=== A few properties of Woodall ===
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Definition of MMC:
Definition of MMC:
A mutual majority (MM) is a set of voters comprising a majority of the
{{definition|A mutual majority (MM) is a set of voters comprising a majority of the voters, who all prefer some same set of candidates to all of the other candidates. That set of candidates is their MM-preferred set.
voters, who all prefer some same set of candidates to all of the other
candidates. That set of candidates is their MM-preferred set.
If a MM vote sincerely, then the winner should come from their MM-preferred set.
If a MM vote sincerely, then the winner should come from their MM-preferred set.
A voter votes sincerely if s/he doesn't vote an unfelt preference, or
A voter votes sincerely if s/he doesn't vote an unfelt preference, or fail to vote a felt preference that the balloting system in use would have allowed hir to vote in addition to the preferences that she actually does vote.
fail to vote a felt preference that the balloting system in use would
have allowed hir to vote in addition to the preferences that she
actually does vote.
To vote an unfelt preference is to vote X over Y if you prefer X to Y.
To vote an unfelt preference is to vote X over Y if you prefer X to Y.
To vote an unfelt preference is to vote X over Y if you don't prefer X to Y.
To vote an unfelt preference is to vote X over Y if you don't prefer X to Y.}}
===Consequences of Woodall's properties===
[end of MMC definition]
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Consequences of Woodall's properties:
As with IRV, Woodall's MMC compliance and freedom from chicken dilemma
As with IRV, Woodall's MMC compliance and freedom from chicken dilemma
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ranking sincerely, ensure that the winner will come from their
ranking sincerely, ensure that the winner will come from their
MM-preferred set. They can assure that, even while fully, freely and sincerely choosing
MM-preferred set. They can assure that, even while fully, freely and sincerely choosing
_among_ that MM preferred set by sincere ranking. And freedom from
''among'' that MM preferred set by sincere ranking. And freedom from
chicken dilemma means that that MM have no need to not rank sincerely.
chicken dilemma means that that MM have no need to not rank sincerely.
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But Woodall additionally, as well as possible, guarantees automatic
But Woodall additionally, as well as possible, guarantees automatic
majority rule to _all_ majorities, however constituted, by always
majority rule to ''all'' majorities, however constituted, by always
electing the voted Condorcet winner (CW)
electing the voted Condorcet winner (CW)
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of the other candidates (as "beat" was defined above).
of the other candidates (as "beat" was defined above).

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== Benham's method: ==
== Benham's method: ==
Benham is a method similar to Woodall. Benham can be defined a bit
Benham is a method similar to Woodall. Benham can be defined a bit
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Benham:
Benham:
Do IRV till there is an un-eliminated candidate who beats each one of
{{definition|Do IRV till there is an un-eliminated candidate who beats each one of
the other un-eliminated candidates. Elect hir.
the other un-eliminated candidates. Elect hir.}}
[end of Benham definition]


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Benham become important and decisive.
Benham become important and decisive.

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== Schwartz Woodall ==
== Schwartz Woodall ==
{{Merge to|Schwartz Woodall|date=August 2019}}
{{Merge to|Schwartz Woodall|date=August 2019}}
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Schwartz Woodall:
Schwartz Woodall:
Do IRV till only one member of the initial Schwartz set remains
{{definition|Do IRV till only one member of the initial Schwartz set remains un-eliminated. Elect hir.}}
un-eliminated. Elect hir.


[end of Schwartz Woodall definition]
The Schwartz set has two equivalent definitions:
The Schwartz set has two equivalent definitions:
The beatpath definition of the Schwartz set:
The beatpath definition of the Schwartz set:
There is a beatpath from X to Y if X beats Y, or if X beats something
{{definition|There is a beatpath from X to Y if X beats Y, or if X beats something that has a beatpath to Y.
that has a beatpath to Y.
X has a beatpath to Y if there is a beatpath from X to Y.
X has a beatpath to Y if there is a beatpath from X to Y.
X is in the Schwartz set if there is no Y such that there is a
X is in the Schwartz set if there is no Y such that there is a beatpath from Y to X, but not from X to Y.}}
beatpath from Y to X, but not from X to Y.
[end of beatpath definition of the Schwartz set]
Unbeaten set definition of the Schwartz set:
Unbeaten set definition of the Schwartz set:
1. An unbeaten set is a set of candidates none of whom are beaten by
{{definition|1. An unbeaten set is a set of candidates none of whom are beaten by anyone outside that set.
anyone outside that set.
2. An innermost unbeaten set is an unbeaten set that doesn't contain a
smaller unbeaten set.
3. The Schwartz set is the set of candidates who are in innermost unbeaten sets.
2. An innermost unbeaten set is an unbeaten set that doesn't contain a smaller unbeaten set.
3. The Schwartz set is the set of candidates who are in innermost unbeaten sets.}}
[end of unbeaten set definition of the Schwartz set]


End of page
[[Category:Condorcet methods]]
[[Category:Condorcet methods]]