[EM] BeatpathWinner is SSD

[Markus Schulze]

Dear Mike,

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.

I wrote (30 March 2005):
> 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.

You wrote (30 March 2005):
> If sharing a property makes methods the same, then
> BeatpathWinner is Approval, because both methods meet WDSC.

Of course, the question is what a "criterion" is and what a
"method" is.

Example: The Borda method chooses that candidate who has the
largest Borda score. Of course, it is possible that there is
more than one candidate with the same largest Borda score.
There are many different proposals how to solve this problem;
but nevertheless I would summarize all these proposals under
the name "Borda method". I would say that these proposals are
only "different tie-breaking strategies for the Borda method".

In my opinion, the statement "If p(z)[A,B] > p(z)[B,A], then
candidate B must be elected with zero probability" defines a
_method_ and not a _criterion_ because:

1. When the number of voters goes to infinity then the
   probability that there is more than one candidate C who
   can be elected without contradicting this statement goes
   to zero. This cannot be said about WDSC.

2. In my opinion, a "criterion" is a "desirable property".
   But this statement itself is not a desirable property,
   it only describes a possible way to get compliance with
   some other desirable properties (e.g. monotonicity,
   Smith-IIA, Schwartz, resolvability, reversal symmetry).

Therefore, what you call "BeatpathWinner" or "SSD" or "CSSD"
are only different tie-breaking strategies for the Schulze
method that I proposed in 1997.

Markus Schulze