[EM] Antisocial behavior

[Markus Schulze]

Dear Mike,

Definitions:

(1) Candidate A is "strongly Pareto inferior" to candidate B
means: Every voter strictly prefer candidate B to candidate A.

(2) Candidate A is "strongly Pareto superior" to candidate B
means: Every voter strictly prefer candidate A to candidate B.

(3) Candidate A is "weakly Pareto inferior" to candidate B
means: No voter strictly prefers candidate A to candidate B and
at least one voter strictly prefers candidate B to candidate A.

(4) Candidate A is "weakly Pareto superior" to candidate B
means: No voter strictly prefers candidate B to candidate A and
at least one voter strictly prefers candidate A to candidate B.

******

With the definitions above Steve's false claim can be
formulated as follows: "The Schulze method can choose a weakly
Pareto inferior candidate."

******

Mike wrote (18 May 2000):
> Markus wrote (18 May 2000):
> > If there is no voter who strictly prefers candidate A to
> > candidate B and at leat one voter who strictly prefers
> > candidate B to candidate A, then candidate A cannot defeat
> > candidate B via beat paths. Therefore candidate A is
> > eliminated either in a decisive step of the Schulze method
> > or in a random step of the Schulze method where a voter
> > is chosen randomly.
>
> Is that situation impossible because it isn't possible to supply
> a set of rankings for it? Of course, aside from that, of itself,
> it the fact that no one ranks A over B doesn't meant that A can't
> have a strong beatpath to B. And the fact that someone ranks B over
> A doesn't mean that B has a strong beatpath to A. But maybe it's
> that that situation can't be created by a set of rankings. I don't
> know.

Suppose that no voter strictly prefers candidate A to candidate B
and at least one voter strictly prefers candidate B to candidate A.
Suppose that A > C[1] > ... > C[n] > B is the strongest beat
path from candidate A to candidate B. Then the the beat path
B > C[1] > ... > C[n] > A must be at least as strong as the beat
path A > C[1] > ... > C[n] > B because of the following two reasons:

(1) The pairwise defeat B > C[1] must be at least as strong as the
pairwise defeat A > C[1] because every voter who strictly prefers
candidate A to candidate C[1] must also strictly prefer candidate B
to candidate C[1] and every voter who strictly prefers candidate C[1]
to candidate B must also strictly prefer candidate C[1] to candidate A.

(2) The pairwise defeat C[n] > A must be at least as strong as the
pairwise defeat C[n] > B because every voter who strictly prefers
candidate A to candidate C[n] must also strictly prefer candidate B
to candidate C[n] and every voter who strictly prefers candidate C[n]
to candidate B must also strictly prefer candidate C[n] to candidate A.

Therefore candidate A cannot defeat candidate B via beat paths. In
so far as beat path defeats are transitive, candidate A cannot be a
unique Schulze winner. Therefore candidate A cannot win the elections
decisively.

Question: Can candidate A win the elections in a random step? Answer:
No! Reason: In every random step of the Schulze method, a voter is
chosen randomly and all those potential winners are eliminated to whom
this randomly chosen voter strictly prefers at least one other potential
winner. Therefore in any case, candidate A must be eliminated before
candidate B can be eliminated.

******

Mike wrote (18 May 2000):
> Markus wrote (18 May 2000):
> > My 28 Feb 2000 mail was a reply to your 23 Feb 2000 mail and
> > your 26 Feb 2000 mail in which you wrote complete nonsense
> > about the different methods.
> 
> Maybe that exemplifies the kind of hostility that was being
> referred to.

Another problem of Steve's 23 Feb 2000 mail and his 26 Feb 2000
mail is the fact that he doesn't define his IBCM method properly
and nevertheless expects from the participants of this maillist
to discuss his method in great detail. Before Steve can expect
that his method is discussed in this maillist, he has to post an
exact and authoritative definition of his method.

Example: It is not even clear whether Steve's IBCM method differs
from the Tideman method. Therefore Steve should post some
non-critical examples where the IBCM winner differs from the
Tideman winner and explain why the IBCM winner differs from the
Tideman winner. With "non-critical" I mean examples where the
IBCM winner differs from the Tideman winner really because these
are two different methods and not because they interpret equal
rankings differently or because they handle ties differently.

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