Schulze Method - Simpler Definition

Norman Petry npetry at
Sat Sep 5 09:01:04 PDT 1998

Yesterday, I wrote:

>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."

I realised after posting this new definition that my use of the term
"unbeaten" is ambiguous, and would cause confusion without definition.  By
"beaten", I mean of course _beat-path_ beaten, not _pairwise_ beaten.
Adding these definitions will, of course, increase the length of the
method's description, but considering that the previous definition makes
reference to both the Smith and Schwartz sets (each of which requires
probably a paragraph to define), this isn't that bad.

I'll try again, this time starting from Mike's declarative definition from
July 27th ("Re: Party List P.S.").  To keep the above two meanings both
distinct and brief, I'll use the word "beat" to refer to a pairwise victory,
and "defeat" to refer to a beat-path victory.  Mike's definition has the
additional advantage of being more thorough, explaining how to determine the
pairwise wins as well.  I've used the word "victory" instead of "defeat" to
avoid ambiguity where necessary, since I don't think doing so causes any
problems in this context (one candidate's victory is another candidate's


Schulze's Method, revised (brief definition):

Candidate A "beats" candidate B if more voters rank A over B than
vice-versa.  The strength of that victory is the number of voters who ranked
A over B.

There's a "beat-path" from A to B if either A beats B, or if A beats
something that has a beat-path to B.  The strength of a beat-path is
measured by its weakest victory.

Candidate A "defeats" candidate B if A has a stronger beat-path to B than B
has to A.

The winners are those candidates who are undefeated.  If there is more than
one winner, exclude all the defeated candidates from the election and
reapply the method until a single winner is found.


To include the random-ballot tiebreaker, add the following:

"If there is still more than one winner, pick a random ballot.  The winners
are the undefeated candidates who received the highest ranking on this
ballot.  If there is more than one winner, exclude all non-winners from the
election and reapply the method until a single winner is found."


Note that the above definition is somewhat subtle, in that it doesn't
explicitly say to use the _strongest_ beat-paths when determining defeats.
I think this is clear enough though if the definition is read carefully.
Also, when I'm referring to the "method", I mean the _complete_ Schulze
method, not just a particular step.

Clear as mud, eh?

Norm Petry

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