[EM] Fw: IBCM, Tideman, Schulze

[Markus Schulze]

Dear Steve,

you wrote (2 Jun 2000):
> It's not really an open question.  My software generates voters' 
> rankings (randomly), and turns up scenarios where IBCM & MTM 
> (majoritarian Tideman) are decisive and disagree.  Here's an 
> example which closes the question affirmatively:
>
>    32 voters' rankings:    
>    3: ABCD
>    3: ABDC
>    1: ADBC
>    2: BADC
>    1: BCAD
>    2: BCDA
>    2: CABD
>    2: CBAD
>    2: CBDA
>    2: CDAB
>    1: CDBA
>    2: DABC
>    2: DACB
>    1: DBAC
>    3: DBCA
>    3: DCAB
>
>    Majorities: AB18,CA18,DA18,BC18,BD17,DC17
>
>    IBCM elects:    B
>    MTM elects:     D
>    Schulze elects: D

The unique IBCM winner is candidate B. But when the ADBC
voter switches to ABDC then the majorities look as follows:
AB18,CA18,DA18,BC18,BD18,DC17. Now (if RandomBallot is used
as a tie breaker) candidate B is elected only with a
probability of 5/32.

Therefore, my main arguments against the IBCM method and
in favour of the Schulze method are:

(1) The IBCM method violates monotonicity.
(2) The IBCM method violates beat path GMC.
(3) The IBCM winner usually depends on more elements of the
    pairwise matrix than the Schulze winner.

You wrote (2 Jun 2000):
> Each reader must judge the relative importance of criteria.

But then you have to accept that I consider these criteria
important.

******

You wrote (2 Jun 2000):
> Norman wrote (27 May 2000):
> > I have confirmed the result of Steve Eppley's simulation
> > comparing the pairwise winners of Schulze, Tideman, and IBCM,
> > which he announced to the list on February 23, 2000: 
> > 
> > >  The same software which shows that Tideman's winner tends to
> > >  beat the Schulze winner when the two methods disagree also shows
> > >  that the IBCM winner tends to beat the Tideman winner pairwise
> > >  when IBCM and Tideman disagree, and the IBCM winner tends to
> > >  beat the Schulze winner pairwise when IBCM and Schulze disagree.
>
> Furthermore, IBCM and MTM both have what appears to be a 
> significant edge over Schulze in the head-to-head comparison, 
> much larger than IBCM's edge over MTM.  
>
> In a message being posted separately ("Head-to-head: Schulze vs. 
> MTM") I have posted some raw data, calculated by my simulation 
> software, which supports the contention that MTM dominates 
> Schulze in the head-to-head comparison.

Your argumentation is problematic.

In your simulations, you implicitely presume that the voters vote
independently. But we all agree that the voters usually don't vote
independently. Otherwise there was absolutely no justification for
the Independence from Clones Criterion. And Norman concludes
correctly that then we have to prefer the Copeland method.

******

You wrote (2 Jun 2000):
> Here's another criterion which might not be considered very 
> important.  But it seems desirable and it distinguishes between 
> MTM and Schulze:
>
>    Define the "1st place finisher" as the given method's winner.
>    Define the "Nth place finisher" as the alternative which
>    would be chosen by the given method if preferences for the
>    1st thru (N-1)th place finishers were neglected.
>
>    Immunity From Squawking criterion:  The 2nd place finisher
>    must never beat the winner pairwise.
>
> Another criterion which distinguishes between MTM and Schulze is 
> the Subsequence Invariance criterion I posted in February (which 
> Markus kindly informed us was not original, indicating there are 
> other people who place some value on it).  
>
> There is a weaker form of Subsequence Invariance which Schulze 
> also fails: "The outcome must not change if preferences for the 
> last place finisher are neglected."  Assuming that a candidate 
> who expects to finish last has the least incentive to compete, 
> it would be nice if the outcome is unaffected by his/her 
> decision whether or not to compete.  Thus, this appears to be a 
> desirable criterion (though it might not be considered very 
> important).

I want you to remember that you haven't yet explained how the
IBCM method is used to calculate a ranking of the candidates.
If your answer is "I don't use the IBCM method to calculate a
ranking." then your argumentation becomes quite meaningless
because this answer would demonstrate that you agree with me
that the task of an election method is to find a winner and not
a ranking.

******

You wrote (2 Jun 2000):
> If RandomVoterHierarchy is the tie-breaker, aren't IBCM and
> MTM completely independent from clones, in whatever "strong"
> formulation Norm referred to?

Of course, you can use RandomBallot. But then your claim that
"IBCM and MTM are more decisive than Schulze or Path Voting"
(26 Feb 2000) becomes false.

******

You wrote (2 Jun 2000):
> Markus tweaked Tideman's definition of clones and demonstrated
> independence of "tweaked" clones of the Schulze method.  But
> that tweak was not a strengthening of the independence
> criterion.  It was a weakening.

That isn't true. Tideman only discusses situations where every
voter makes a complete ranking and where the winner is decisive.
I generalized the Independence from Clones Criterion to
situations where voters don't necessarily make complete
rankings and where the election method could contain random
tie breakers. Therefore, my definition of the Independence from
Clones Criterion is stronger than Tideman's original definition.

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