[EM] BeatpathWinner is SSD

[Markus.Schulze at alumni.TU-Berlin.DE]

Dear Mike,

I wrote (28 March 2005):
> Suppose V is the number of voters.
>
> Suppose d[X,Y] is the number of voters who
> strictly prefer candidate X to candidate Y.
>
> Suppose p(z)[X,Y] is the strength of the strongest
> path from candidate X to candidate Y when the strength
> of a pairwise defeat is measured by "z" (e.g. "z" =
> "margins", "z" = "winning votes", "z" = "votes against").
>
> Then I proposed the following criterion in 1997:
>
>    If p(wv)[A,B] > V/2 and p(wv)[B,A] < V/2,
>    then candidate B must be elected with zero
>    probability.
>
> Steve Eppley proposed the following criterion in 2000:
>
>    If d[A,B] > V/2 and p(wv)[B,A] < V/2,
>    then candidate B must be elected with zero
>    probability.

You wrote (29 March 2005):
> Yes, but to outdo a majority pairwise vote, it's necessary
> for that MPV to be in a cycle of MPVs _all of which are at
> least as strong as it is_.

I wrote (29 March 2005):
> In 1997, I proposed the following method (Schulze method,
> Schwartz sequential dropping, cloneproof Schwartz sequential
> dropping, beatpath method, beatpath winner, path voting,
> path winner):
>
>    If p(z)[A,B] > p(z)[B,A], then candidate B must be
>    elected with zero probability.

You wrote (30 March 2005):
> SSD and CSSD are two different methods, which can give two
> different outcomes with the same ballot-set. So BeatpathWinner
> can't be both SSD and CSSD.
>
> BeatpathWinner is equivalent to CSSD, but not to SSD.
>
> I mention that for your information, so that, if you want to
> be correct, you can leave SSD out of the list of names that
> refer to BeatpathWinner or methods equivalent to it. But it
> isn't important, and, as I said, I mention it only for your
> information.

BeatpathWinner _is_ SSD _is_ CSSD in so far as all of them
share this property:

   If p(z)[A,B] > p(z)[B,A], then candidate B must be
   elected with zero probability.

If you don't agree with this then please post an example
where this is not true.

Markus Schulze