Definitions and important properties of Woodall's Method and two similar methods:
Woodall's method:[edit | edit source]
Do IRV till only one member of the initial Smiths set remains un-eliminated. Elect hir.
The Smith set is the smallest set of candidates such that every candidate in the set beats every candidate outside the set.
IRV definition (for the purpose of Woodall):
Repeatedly, cross-off or delete from the rankings the candidate who tops the fewest rankings.
Definition of "beats":
X beats Y if more ballots rank X over Y than rank Y over X.
A few properties of Woodall[edit | edit source]
Woodall meets the Mutual Majority Criterion (MMC), and has no chicken dilemma. Woodall meets the Condorcet Criterion (CC), and the Smith Criterion. Meeting Smith always implies meeting MMC, and Condorcet Loser as well.
Woodall doesn't meet FBC. FBC is necessary only under current conditions (dishonest, disinformational media, and an electorate who believe those media). Woodall isn't proposed for current conditions. Likewise for the similar methods proposed later at this page.
Consistency criteria: Woodall, like all Condorcet methods, fails Consistency, Participation, Mono-Add-Top, and Mono-Add-Unique-Top. Woodall fails Mono-Raise, but passes Mono-Add-Plump and Mono-Append.
Woodall's importance comes from its unmatched freedom from strategy-need, made possible by MMC, freedom from chicken dilemma, and CC. Advantages such as that come at a price. The abovementioned combination of properties appears to be incompatible with FBC and with Mono-Raise, Participation, Mono-Add-Top and Mono-Add-Unique top. Choice of a voting system always involves choice among properties.
The consistency criteria don't have strategic importance.
Definition of MMC:
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.
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 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 don't prefer X to Y.
Consequences of Woodall's properties[edit | edit source]
As with IRV, Woodall's MMC compliance and freedom from chicken dilemma mean that a MM have no need to not rank sincerely. They can, by merely ranking sincerely, ensure that the winner will come from their 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 chicken dilemma means that that MM have no need to not rank sincerely.
Therefore, IRV and Woodall guarantee automatic majority-rule enforcement for a mutual majority.
But Woodall additionally, as well as possible, guarantees automatic majority rule to all majorities, however constituted, by always electing the voted Condorcet winner (CW)
The voted CW is the candidate (when there is one) who beats each one of the other candidates (as "beat" was defined above).
Benham's method:[edit | edit source]
Benham is a method similar to Woodall. Benham can be defined a bit more briefly, because it doesn't mention the Smith set, though Benham, like Woodall, always chooses from the Smith set. But Woodall is more particular than Benham is, regarding which Smith set member it chooses.
Do IRV till there is an un-eliminated candidate who beats each one of the other un-eliminated candidates. Elect hir.
It should be pointed out that, of course, if there is a CW, then Woodall and Benham, by their above-stated definitions, will elect that CW without doing any IRV.
For current conditions (disinformational media and an electorate who believe those media), FBC is necessary.
FBC is important only for current conditions.
But, other than for current conditions, FBC would no longer be needed, and then the powerful above-described properties-combinations of IRV, Woodall, and Benham become important and decisive.
Schwartz Woodall[edit | edit source]
Schwartz Woodall is a variation of Woodall, and an improvement for small electorates, such as organizations, meetings or families.
Do IRV till only one member of the initial Schwartz set remains un-eliminated. Elect hir.
The Schwartz set has two equivalent definitions:
The beatpath definition of the Schwartz set:
There is a beatpath from X to Y if X beats Y, or if X beats something that has a beatpath 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 beatpath from Y to X, but not from X to Y.
Unbeaten set definition of the Schwartz set:
1. An unbeaten set is a set of candidates none of whom are beaten by 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.