Goldfish (single-winner method)

[Blake Cretney]

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On Sun, 23 Aug 1998 18:08:29   Norman Petry wrote:

>As far as I know, the only decisive, pairwise methods which fully satisfy
>GITC are: (1) Schulze's method, and (2) Tideman's _improved_ method, which
>uses a tiebreaker ballot, probably in the manner you describe (see "Complete
>Independence of Clones in the Ranked Pairs Rule", Soc Choice Welfare (1989)
>6:167-173).  Schulze's method seems to be better in this respect, since a
>special tiebreaker isn't required under any circumstances we've yet
>discovered.  This seems to me to be a definite advantage of Schulze, since
>adding special tiebreaker-ballot rules to other methods increases
>complexity, and may be unintuitive to voters.  At first glance, it might
>seem unfair to allow one particular voter to affect the election outcome.
>If you've discovered any GITC violations for Schulze that we don't know
>about, I'd be interested in seeing them!

Here is my argument for why any pair-wise method, even if it uses a random selection procedure between some alternatives will fail GITC unless augmented somehow.

Balloting 1:  No clones
1 A B
2 A B
3 A B
4 B A
5 B A
6 B A
  A  B
A X  3
B 3  X
So, based only on the matrix, a random tie-breaker would give
P(A)=.5 P(B)=.5

Balloting 2:  B has a clone named C

1 A C B
2 A C B
3 A B C - C cannot be a clone of A
4 C B A
5 B C A
6 B C A

  A B C
A X 3 3
B 3 X 3
C 3 3 X

Judging by this matrix we would have to give
A B and C equal chance P(A) = 1/3 P(B)=1/3 P(C)=1/3
but to avoid rich party P(A) = P(B)+P(C)

That is, the pair-wise matrix does not provide enough information to avoid GITC in this case.

So, any pair-wise system will have to be augmented with something, like a tie-breaker ballot, in order to avoid GITC, although perhaps only in very rare circumstances.


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