## 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

• "Mono-append": Appending X to ballots that left X unranked (and hence had treated X as ranked co-equal last) cannot decrease X's chances of winning.
• "Mono-add-plump": Adding ballots that "plump" for X cannot decrease X's chances of winning.
• Clone immunity.
• Woodall's "plurality criterion" and "symmetric completion" criterion.
• "Majority for solid coalitions": if a voter-majority prefers every candidate in some set S over every candidate not in S, then a member of S must win.
• "Smith set": If all the members of some candidate-subset S pairwise-beat all the non-members then the winner must come from S.
• "Condorcet criterion": If a "Condorcet winner" exists (who beats all opponents pairwise) then he must win. (Note: A Condorcet winner is a Smith set which has exactly one element.)
• "Dominant mutual third burial resistance": If there are three candidates X,Y,Z and X is top-ranked on more than a third of the ballots and wins, then if some Y>X>Z ballots are changed to "bury" X, i.e. becoming Y>Z>X, the winner can't change to Y.

### 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)

• Favorite Betrayal.
• "Participation": that is, unfortunately, casting an honest vote can be strategically worse for a voter, than not voting at all (e.g. it causes somebody that voter regards as worse, to win).
• Also fails numerous monotonicity criteria defined by Woodall including "mono-raise", "add-top", "remove-bottom", "raise-random", "sub-top", "raise-delete", "sub-plump". E.g, unfortunately, raising a candidate in your vote can cause him to lose.
• Subdistrict partitioning: that is, unfortunately, if X wins in district I and in district II, somebody else could win in the combined 2-district country. This prevents WoodSIRV from being "counted in precincts."
• "Later no harm": manipulations you make to the ordering of the candidates you rank below X on your ballot, should not harm X's winning chances.
• "Later no help": manipulations you make to the ordering of the candidates you rank below X on your ballot, should not help X's winning chances.