Schulze Method - Simpler Definition

[Norman Petry]

Dear Markus,

On August 31, 1998 you wrote ("Re: Maybe Schulze is decisive."):

>I have demonstrated, that if candidate A defeats candidate B
>via beat-paths and candidate B defeats candidate C via
>beat-paths, then candidate A defeats candidate C via
>beat-paths. Thus: Schulze wins cannot be cyclic.

This is excellent.  I had hoped this would be the case, as it allows us to
define the Schulze method much more simply, without reference to the
Schwartz set.  Earlier (October 3, 1997 -- "Re: Condorect sub-cycle rule")
you defined your method as:

>Step1: Calculate the Smith set and eliminate all those
>candidates, who are not in the Smith set.
>
>Step2: Calculate the beat-path-defeats of the remaining
>candidates. The winner is that candidate, who defeats every
>other candidate via a beat path.
>If there is no unique candidate, who defeats every other
>candidate via a beat path, then the Schwartz set of
>the beat-path-defeats is calculated, those candidates,
>who are not in this Schwartz set, are eliminated and
>the algorithm is re-started with the remaining candidates.

Now, since you've proven that the beat-path matrix cannot contain cycles,
the beat-path Schwartz set simply consists of all the unbeaten candidates
(i.e.: those who beat or tie all other candidates).  Unless I'm mistaken,
Step 1 of the process is merely an optimisation, designed to reduce the set
of winners as much as possible before computing beat-paths.  Therefore, it's
not strictly necessary to use the Smith-set reduction prior to calculating
beat-path defeats, so we can simplify the definition by leaving out this
step.  Thus, one simple way to define your method would be:

Schulze's Method (brief definition):

"Calculate the beat-path-defeats of the candidates.  The winners are those
candidates who are unbeaten via a beat-path.  If there is more than one
winner, repeat this step with only the winning candidates until a single
winner is found."

To me, this description has two main advantages: (1) the method now serves
as it's own tiebreaker -- we no longer need to resort to a different method
(Schwartz) to resolve beat-path ties, and (2) much shorter and simpler
definition.

***

For completeness, we could also incorporate your proposed random-ballot
tiebreaker to deal with cases where the set of winning candidates cannot be
reduced to a single winner by the beat-path tiebreaker:

Schulze's Method (complete definition):

"Step 1: Calculate the beat-path-defeats of the candidates.  The winners are
those candidates who are unbeaten via a beat-path.  If there is more than
one winner, repeat this step with only the winning candidates until no
further reduction in the set of winners is possible.

Step 2: If there is still more than one winner, pick a random ballot.  The
winners are those previous winners who received the highest ranking on this
ballot.  If there is more than one winner, repeat Step 1 with only the
winning candidates."

***

Please let me know if the above definitions contain any errors.  I'm sure
the wording I've used could be improved upon, so if you or anyone else have
alternate definitions of the method, I'd be interested in seeing them.


Norm Petry