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
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.
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)
"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.