[EM] The Unique Winning Alliance method

[Rob Speer]

On Mon, Aug 04, 2003 at 04:12:08PM +0200, Markus Schulze wrote:
> Dear Rob,
> 
> I see the following problem with the Nash Set:
> 
>    Situation 1: A > B, A > D, B > C, C > A, D > B, D > C.
>    The Nash Set is ACD.
> 
>    Situation 2: Some voters rank candidate C higher so
>    that "D > C" is changed to "C > D". Now, the Nash Set
>    is ABC.

This is true, at least with some pairwise defeat values.

> In so far as I haven't made any presumptions about the
> strengths of the pairwise defeats, it is (at least for
> all those election methods X that are not identical to
> RandomCandidate in the circular 3-candidate case A>B>C>A)
> trivial to create an example where Nash//X chooses
> candidate C in situation 1 and doesn't choose candidate C
> in situation 2 resp. where Nash//X decreases the winning
> probability of candidate C from situation 1 to situation 2
> so that Nash//X violates monotonicity.
>
> Do you agree with my conclusions?

Yes. In fact, here's a specific example:

Situation 1:
   A  B  C  D
A  0  1 -2  3
B -1  0  2 -3
C  2 -2  0 -1
D -3  3  1  0

Situation 2:
Change one D>C to C>D.

For any reasonable X, under Nash//X, C wins situation 1, and B wins
situation 2.

Hmm. That's unfortunate.

What's interesting is that in both situations, SSD and Ranked Pairs
choose either A or C (down to a tiebreaker) - they avoid the candidates
who might not be in the Nash set. Although actually restricting the
election to the Nash set is not monotonic, do good Condorcet methods
tend to pick winners in the Nash set anyway? Are there cases where they
don't? Are they justified?
-- 
Rob Speer