Woodall's "Smith,IRV" Condorcet voting method

By Chris Benham & Warren D. Smith

T.N.Tideman, in his book, considered a particular voting method that combined the "Smith set" notion with "Instant Runoff Voting," to be tentatively the "best" of the voting methods examined in his book. However, Tideman unfortunately was unaware of the following [Douglas R. Woodall: Monotonicity of single seat preferential election rules, Discrete Applied Maths. 77,1 (1997) 81-98]. As Chris Benham pointed out, and so did Woodall in that paper, there is reason to believe Tideman's method is clearly inferior to the following related

Woodall's Smith+IRV-type voting method (WoodSIRV):
Proceed by successively eliminating the candidates with the fewest top-rank votes (just as in IRV) except that before each IRV elimination, check to see if there is a single candidate X with no (among remaining candidates) pairwise losses. As soon as such an X appears, elect X.

On what basis can we claim that WoodSIRV appears superior? Well, WoodSIRV appears to be easier to describe and appears to obey the same set of Tideman's properties. And it also obeys these two properties that Tideman's Smith+IRV method fails: "mono-append" and "mono-add-plump."

Properties obeyed by WoodSIRV

The mono-append and mono-add-plump properties that Tideman fails but WoodSIRV satisfies

We demonstrate both property failures in one example:

#voters their vote
10 A>B>C>D
6 B>C>D>A
2 C
5 D>C>A>B

In this election, all the candidates are in the Smith set (which Woodall calls the "top tier"), and the IRV winner – and hence winner with Tideman's Smith+IRV method – is A. But if you add two extra ballots that "plump" for A (i.e. vote for A and leave the rest unranked; unranked candidates being regarded as ranked coequal bottom) or which append A to the two C ballots, then the top tier becomes {A,B,C}, and (when you delete D from all the ballots before applying IRV) then, according to Tideman and paradoxically, C wins.

Meanwhile WoodSIRV behaves reasonably: A still wins.

So, at least based on Tideman's own properties (and provided the WoodSIRV modification does not injure the method's "strategy resistance") the WoodSIRV modification seems clearly superior to Tideman's "best" method.

Properties failed by WoodSIRV (which Tideman's method also fails)

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