Woodall's method
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Woodall's method or Smith,IRV is a voting method that combines instant-runoff voting and Condorcet. It was invented by Douglas Woodall.
Woodall's method:
Do IRV till only one member of the initial Smith set remains un-eliminated. Elect hir.
Smith set:
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.
An alternative (but equivalent) definition of Woodall's used in James Green-Armytage's Condorcet-IRV paper[1]:
Score candidates according to their elimination scores, and choose the Smith set candidate with best score. That is, define each candidate’s elimination score as the round in which he is eliminated by AV [IRV].
A few properties of Woodall
Woodall meets the mutual majority criterion, and has no chicken dilemma. Woodall meets the Condorcet criterion, and the Smith criterion. Meeting Smith always implies meeting the mutual majority criterion, and Condorcet loser as well.
Woodall doesn't meet FBC. Like all Condorcet methods, Woodall 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 above-mentioned 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.
Consequences of Woodall's properties
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).
Schwartz Woodall
Schwartz Woodall is a variation of Woodall, and an improvement for small electorates, such as organizations, meetings or families.
Schwartz Woodall:
Do IRV till only one member of the initial Schwartz set remains un-eliminated. Elect hir.
Notes
Benham's method is similar, but always terminates in the same round as Woodall's or earlier. This is because the two methods are identical to IRV until their algorithms' completion, but Benham's method can potentially terminate in a round where there are still multiple members of the Smith set remaining i.e. a member of the Smith set whose only pairwise loss or tie is to one of the other candidates in the Smith set would become a CW if that other candidate is eliminated, and be the Benham winner.