Markus Schulze schulze at sol.physik.tu-berlin.de
Tue Feb 1 06:20:33 PST 2000

Dear Blake,

you wrote (31 Jan 2000):
> What about
> 25 A D B C
> 35 B A D C -> B
> 30 D C A B
> 4  A D C B
> 6  C A D B
>             winning-votes
> A->D 35-30   35
> A->B 65-35   65
> C->A 36-29   36
> B->C 60-40   60
> D->B 65-35   65
> D->C 59-6    59
> Here, A is in the SASW; D is not.  A wins in Tideman (winning-votes),
> but D wins in Schulze.

I don't understand your example. What do you want to demonstrate with
your example? Do you want to criticize the mathematical formulation of
beat path GMC? Or do you want to criticize the Schulze Method?

First: The intention of beat path GMC is that a voter should rather be
punished than rewarded for truncating his votes. In your example, beat
path GMC does exactly what it was designed to do. Those 35 voters whose
sincere opinion is B > A > D > C truncate their votes and change the
winner from candidate A to candidate D. Thus the truncators are
punished. In so far as beat path GMC does exactly what is was designed to
do, your example cannot be interpreted as a criticism of beat path GMC.

Second: In your example (25 voters vote A > D > B > C; 35 voters vote B;
30 voters vote D > C > A > B; 4 voters vote A > D > C > B; 6 voters vote
C > A > D > B.), not enough information is available to calculate the
SASWs. In your example, Tideman accidently elects a SASW while Schulze
accidently elects a non-SASW because you presumed that the sincere
opinion of the 35 voters who vote for B only is B > A > D > C. But in so
far as we usually don't have any information about the sincere preferences
but only about the reported preferences, it is also possible that Tideman
accidently elects a non-SASW while Schulze accidently elects a SASW in
your example. The difference between Tideman and Schulze is that Schulze
can elect a non-SASW only if not enough information is available to
calculate the SASWs while Tideman can unnecessarily elect a non-SASW.

You wrote (31 Jan 2000):
> Markus Schulze wrote (29 Jan 2000):
> > Blake Cretney wrote (24 Jan 2000):
> > > I have made some attempts to show that Schulze (path voting) is in
> > > some way intuitive.  That is, it seems to rely on "arguments"
> > > composed of majority views, where the strength of the argument is
> > > equal to its weakest link.
> > >
> > > So, if we have
> > >
> > > A>B 30
> > > B>C 20
> > > C>D 15
> > >
> > > If we view pairwise decisions as more probably correct than
> > > incorrect, we have to view this as evidence that A is better
> > > than D.  But what if we also knew that
> > >
> > > C>E 30
> > > E>B 25
> > >
> > > Clearly, this gives evidence contrary to one of the links in our chain
> > > of argument (B>C).  It suggest that it is more likely that C is better
> > > than B, than the contrary.  The whole chain of argument falls apart.
> > > In fact, it is Tideman that makes use of this additional information.
> >
> > It is not correct to say that "Tideman makes use of additional
> > information." For every pair of election methods it is possible to
> > create a situation such that both election methods lead to the same
> > result and such that -if this situation is slightly modified- election
> > method 1 still leads to the same result as before and method 2 leads
> > to a different result. But it is not correct to conclude that election
> > method 2 uses more information. All election methods use the same
> > information; they only interpret this information differently.
> My point was not that Tideman makes use of more information, in
> general.  My point was only that additional information is used in a
> particular situation, and that in that situation the additional
> information seems relevant.  Of course, this assumes an attempt to
> base the method on probability, evidence, or some similar grounds.
> That is, if we justify Schulze on the basis that a path of victories
> A>B, B>C, C>D
> should be granted as evidence that A>D, as an obvious result of
> believing that each pairwise victory is evidence of the superiority of
> the winner over the loser, then we must also acknowledge that a path
> contradicting one of these pairwise victories (a side-path) must
> contradict the entire chain of argument.  Tideman ensures that if a
> path is to be "locked", each victory must not contradict a higher
> "locked" path.  So, it is using the information provided by these
> side-paths.

Whether candidate A wins should -to my opinion- not unnecessarily
depend on whether candidate C is better than candidate B.

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

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