[EM] Tideman and GMC

[Markus Schulze]

Dear Blake,

you wrote (11 Mar 2000):
> Markus Schulze wrote (29 Feb 2000):
> > Blake Cretney wrote (24 Feb 2000):
> > > Markus Schulze wrote (3 Feb 2000):
> > > > To my opinion, the parties and the voters are interested only in
> > > > which candidate wins and they are not interested in which candidate
> > > > gets which position in the ranking.
> > >
> > > For most, but not all uses, that will be the case.  For example, a
> > > party might want to use the method to sort a large number of their
> > > candidate for use in list PR.
> > 
> > For this situation, I would recommend neither the Tideman method nor
> > the Schulze method. I would recommend a method that guarantees some
> > degree of proportional representation.
>
> Using a majoritarian rather than a proportional method does have some
> advantages in this case.  It ensures that there will be some consistency
> between the candidates running for the same party, and it reduces the
> ability for parties and groups to "raid" the list.  I think that the
> parties themselves would prefer Schulze or Tideman to a PR system, for
> these reasons.  
>
> In particular, parties may find it embarassing to have certain ideologies
> represented in their slate.  If you have a party with a membership of
> which 10% are basically racist, then in a majoritarian method, the 90%
> can keep them out.  In a proportional system, this 10% will get on the
> list, and monopolize news coverage for the party.  Parties are likely to
> find they win more seats if they use a majoritarian system for choosing
> their list.

You have missed the point.

What I wanted to say is that the question which is the best single winner
election method and the question which method should be used to sort
candidates for use in list PR are two different questions and that it
is therefore not surprising if there are two different answers to these
questions.

******

You wrote (11 Mar 2000):
> I don't find your argument convincing, but after trying large numbers
> of examples, I have come to believe your conclusion, that Tideman
> usually relies on more contests than Schulze. That this offers greater
> opportunities for strategy seems doubtful to me.

Remember that you haven't yet presented any argument why the Tideman
method should be less vulnerable by burying or compromising than the
Schulze method.

******

You wrote (11 Mar 2000):
> Markus Schulze wrote (29 Feb 2000):
> > Blake Cretney wrote (24 Feb 2000):
> > > What I consider to be more important, however, is that Tideman
> > > is more likely than Schulze to honour the result of an individual
> > > pairwise victory. That is, if a majority of those with a preference
> > > rank A over B, Tideman is more likely to ensure that this is
> > > reflected in the final ranking, including the ranking for first
> > > place.
> >
> > Both methods (= Tideman and Schulze) minimize the maximum pairwise
> > defeat that has to be ignored to get a complete ranking.
>
> If I understand you correctly, that is true, but it isn't my point.
>
> What I am saying is, if all you know is that A is preferred to B,
> Which method is more likely to rank A over B?
>
> The answer is Tideman.  The reason is that Tideman only over-rules a
> majority (does not place it in the final ranking) when this is necessary
> to honour higher majorities (by placing them in the final ranking).
>
> That is, in Tideman, if a majority decision cannot be honoured in the
> final ranking, it isn't used at all.  This ensures that it does not
> over-rule a lower ranking that still can be honoured.  As a result, more
> majority decisions find their way into the final ranking.

You have missed the point.

Example:

   26 voters vote C > A > B > D.
   20 voters vote B > D > A > C.
   18 voters vote A > D > C > B.
   14 voters vote C > B > A > D.
   08 voters vote B > D > C > A.
   07 voters vote D > A > C > B.
   07 voters vote B > D > A = C.

   Then the matrix of pairwise defeats looks as follows:

   A:B=51:49
   A:C=45:48
   A:D=58:42
   B:C=35:65
   B:D=75:25
   C:D=40:60

The Tideman ranking is C > A > B > D and the Tideman
winner is candidate C. The Schulze ranking is A > C > B > D
and the Schulze winner is candidate A.

On the one side: The strongest pairwise defeat that the Tideman
method ignores to get a _ranking_ is C:D=40:60. But also the
strongest pairwise defeat that the Schulze method ignores to
get a _ranking_ is C:D=40:60. The Tideman method doesn't ignore
a second pairwise defeat to get a _ranking_. The Schulze method
ignores A:C=45:48 to get a _ranking_. Therefore you could
say that the Schulze method unnecessarily ignores a pairwise
defeat and that Tideman is more likely than Schulze to honour
the result of an individual pairwise victory.

But on the other side: The strongest pairwise defeat that the
Schulze method ignores to get a _winner_ is A:C=45:48 because
A:C=45:48 is the strongest defeat of the Schulze _winner_. The
strongest pairwise defeat that the Tideman method ignores to
get a _winner_ is C:D=40:60 because C:D=40:60 is the strongest
defeat of the Tideman _winner_. Therefore you can also say that
Schulze is more likely than Tideman to honour the result of an
individual pairwise victory.

In short:
1. Both methods (= Tideman and Schulze) minimize the maximum
   pairwise defeat that has to be ignored to get a complete
   ranking.
2. Both methods (= Tideman and Schulze) guarantee that if
   candidate X pairwise defeats candidate Y then this is
   reflected in the final ranking unless there is a beat path
   from candidate Y to candidate X which consists only of
   pairwise defeats that are stronger than the pairwise
   defeat X:Y. Therefore both election methods guarantee that
   if candidate X pairwise defeats candidate Y then this is
   reflected in the final ranking unless this would contradict
   a stronger pairwise defeat.
3. Whether Tideman or Schulze is more likely to honour the
   result of an individual pairwise victory depends on
   whether you are more interested in the _ranking_ or in
   the _winner_.

******

You wrote (11 Mar 2000):
> Here is another interesting point.  If one starts from a pairwise
> matrix with no cells equal, Tideman guarantees a complete ranking.
> Schulze does not.
>
> A>D b+2
> A>B b+7
> C>D b+6
> C>A b+9
> B>C b+5
> D>B b+8
>
> In this example, the best path from A to D is equal to the best path
> from D to A, because they share the B>C contest.

That isn't true.

Remember that I have sent you in a private correspondence (26 Nov 1999)
a proof that -when the pairwise matrix doesn't have equal cells- the
Schulze winner is always unique. Therefore your claim that the Schulze
method is less decisive than the Tideman method is simply wrong.

It is true that ties for the second place are possible. But there is a
trivial explanation for this observation: As I have proposed the Schulze
method only to calculate the _winner_, I have defined only the Schulze
_winner_ and not the Schulze _ranking_. Of course, the Schulze method
can also be defined in such a manner that also the _ranking_ is always
unique (e.g. whenever some candidates are tied then the Schulze method
is used as a tie breaker amongst these candidates).

Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de