Schulze Method - Simpler Definition

Norman Petry npetry at
Sat Sep 5 22:50:38 PDT 1998

DEMOREP1 suggested that I should post an example of the Schulze method, as
I've defined it, applied to some problem.  My intent with the new definition
was just to simplify the definitions Markus and Mike had already provided,
by taking advantage of Markus' proof that Schulze wins cannot be cyclic.
Thus, since the method itself is exactly the same, any previous Schulze
example would do.

However, one thing that might be useful to show is an example in which there
is more than one beat-path undefeated candidate, to demonstrate the
equivalence of Markus' Schwartz set tiebreaker and my "undefeated"
tiebreaker.  As mentioned previously, this equivalence is guaranteed by
Markus' proof from August 31st ("Re: Maybe Schulze is decisive.")

Here goes...

Candidates: {A,B,C,D,E}


9: A>B>C>D>E
8: B>A>C>E>D
15: C>E>D>B>A
16: D>E>A>B>C

Pairwise Beats:

"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."

A>B 25:23
{A,B}>C 33:15
C>{D,E} 32:31
{D,E}>{A,B} 31:17
D>E 25:23

Beat-Path Defeats:

"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."

(A=B 31:31)
A>>C 33:31
A>>D 32:31
A>>E 32:31
B>>C 33:31
B>>D 32:31
B>>E 32:31
C>>D 32:31
C>>E 32:31
(D=E 31:31)

Winners: {A,B}

"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."

Reapplying Schulze's method, we get:

Candidates: {A,B}
Pairwise Beats:  A>B 25:23
Beat-Path Defeats:  A>>B 25:23
Winners: {A}


Note that the Schwartz set of the Beat-Path defeats was the same as the set
of unbeaten candidates.  Given:


Schwartz Set: {A,B}.  (Note that there are no cycles)


As we've seen in previous examples (August 9th -- "Re: Schulze: D>E 9
Bingo.", etc.), the Schulze method is as decisive as any of the other
methods we've been looking at.  Therefore, for large-scale public elections
it may be unnecessary to complicate the definition of the method by
mentioning either of the two tiebreakers I've included (Unbeaten(Schwartz) &
Random Ballot).  However, I expect this method has a greater chance of being
adopted by professional societies, committees, etc. than by the public (SD
is a better public proposal, in my view).  In these small-scale elections,
the tiebreaker(s) will occasionally be needed, so I think it's worthwhile
specifying them.

Norm Petry

-----Original Message-----
From: DEMOREP1 at <DEMOREP1 at>
To: npetry at <npetry at>
Date: September 5, 1998 2:16 PM
Subject: Re: Schulze Method - Simpler Definition

>You wrote-
>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?
>D- I suggest that you do at least one "complete" example (with ALL of the
>strange things that can happen- noting that there would be polls going on
>before an election).

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