[EM] Fw: IBCM, Tideman, Schulze

[Markus Schulze]

Dear Steve,

you wrote (2 June 2000):
> Norman wrote (27 May 2000):
> > That said, I am not persuaded that this is a particularly important
> > consideration -- if we thought this was an important factor in
> > selecting a method, we would probably choose Smith//Copeland, since
> > its results are even superior to those of IBCM.  It is likely that
> > a method has to sacrifice some of the desireable properties of
> > Schulze's method in order to do better here. Smith//Copeland, for
> > example almost certainly violates clone criteria, whereas Schulze's
> > method is the only method (that I'm aware of) that has been proven
> > to satisfy one strong formulation of Clone Independence criteria.
> > Tideman satisfies a weaker definition of clone independence, SD
> > is 'mostly' clone independent, but fails in rare cases; IBCM and
> > Tideman both violate Beatpath GMC, etc.
>
> I'm not aware that IBCM or Tideman fail to satisfy the stronger
> form of clone independence.  I do recall that a couple of years
> ago there was a discussion in EM about Tideman, which didn't use
> the RandomVoterHierarchy tie-breaker discussed later for
> Schulze's method.  Also, that discussion may have been about
> Tideman's 1987 version, not the better version defined in Zavist
> & Tideman's paper, "Complete Independence of Clones in the
> Ranked Pairs Rule." (Social Choice and Welfare, 1989).  If
> RandomVoterHierarchy is the tie-breaker, aren't IBCM and MTM
> completely independent from clones, in whatever "strong"
> formulation Norm referred to?

Unfortunately, the definitions in your 26 Feb 2000 mail were very
vague. Therefore it was not possible to check whether the claimed
criteria were really met by the mentioned methods.

The definitions in your 3 June 2000 mail are still very vague. But
they are precise enough to demontrate that the Tideman method and
the MTM method violate the strong definition of Clone Independence
criteria.

You wrote (3 June 2000):
> Tideman's minileximax used margins, whereas MTM's minileximax
> uses majorities.
>
>    "Thwarted majorities":  If more voters ranked x over y than
>    vice versa, but the social ranking does not rank x over y,
>    then the social ranking "thwarts" the majority who ranked
>    x over y.

Example:

   30 voters vote A > B > C.
   30 voters vote B > C > A.
   30 voters vote C > A > B.

   A:B=60:30
   B:C=60:30
   C:A=60:30

   Tideman and MTM (and every other anonymous and neutral election
   method) choose candidate A with a probability of 1/3,
   candidate B with a probability of 1/3 and candidate C with a
   probability of 1/3.

   Suppose that candidate A is substituted with a set of clones
   A[1],...,A[n]. Suppose that the pairwise defeat between two
   clones A[i] and A[j] is always strictly weaker than 60:30.

   Due to your 3 June 2000 definition of the Tideman method and
   the MTM method, the methods choose the winner of that ranking
   for which the vector in which the strengths of the thwarted
   majorities are sorted in an decreasing order is
   lexicographically minimal. Correct?

   The vector of the thwarted majorities of the ranking
   C > (all A[i]) > B contains only one pairwise defeat of
   strength 60.

   On the other side, the vector of the thwarted majorities of
   a ranking that starts with candidate B contains at least
   n pairwise defeats of strength 60.
   
   Therefore the vector of the the thwarted majorities
   of a ranking that starts with candidate B is always
   lexicographically strictly larger than the vector of
   the thwarted majorities of the ranking C > (all A[i]) > B.
   Therefore candidate B cannot be elected any more.

   But this is a clear violation of the strong formulation of
   Clone Independence criteria. Therefore the Tideman method
   and the MTM method (due to your 3 June 2000 definition)
   violate the strong formulation of Clone Independence
   criteria.

******

You wrote (2 June 2000):
> Markus wrote that I claimed IBCM is monotonic, but I believe
> I've never claimed that. (I reread the messages I've posted
> which mention BCM, and didn't find such a claim.  Perhaps he
> will provide a quote of the alleged claim?)

I asked you whether the IBCM method meets monotonicity
(13 May 2000). In your reply (17 May 2000), you called my
question a "false claim," an "unwarranted attack" and an
"antisocial behavior." You wrote that my question demonstrates
the "antisocial quality" of my "tactics." If you didn't pretend
that my question whether the IBCM method meets monotonicity
was unjustified then why did you call this question a "false
claim" and an "unwarranted attack"? And why did you call me
"nasty" and "antisocial" for asking that question? Or is
this your usual tone? In other words: When you use words like
that when somebody asks an obviously justified question then
what kind of words do you use when somebody really asks an
unjustifiable question?

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