[EM] Dropping SD

[Markus Schulze]

Dear Blake,

you wrote (7 Jul 2000):
> Here's an example where SD violates monotonicity.  All unspecified
> victories are assumed to be lesser, and therefore dropped first.  The
> actual numbers aren't important, only the relative order of the
> victories.
>
> F:D 20
> D:E 22
> E:F 21
>
> B:C 13
>
> A:C 32
> C:B 31
> B:A 30
>
> G:H 42
> H:I 41
> I:G 40
>
> A:G 16
> G:D 10
> D:A 11
>
> 1st all unspecified victories are dropped, in arbitrary order.
> Drop G:D -- D:A:G
> Drop D:A -- A:C:B:F:D
> Skip B:D
> Skip A:G
> Drop F:D -- D:E:F
>
> D wins
>
> Now, set A:G to 9 instead of 16
> Drop A:G -- G:D:A
> Skip G:D
> Drop D:A -- A:C:B:F:D
> Drop F:D -- D:E:F
> Skip E:F
> Skip D:E
> Drop B:A -- A:C:B
>
> A wins
>
> So, A won by decreasing its pairwise victory over G.

I guess you mean "B:F 13" instead of "B:C 13".

******

you wrote (7 Jul 2000):
> Here's a description of Tideman's method.  I assume the reader has
> already been told what it means for one victory to be higher than
> another.  I will also not deal with the issue of ties.
>
> Tideman's method attempts to produce a complete ranking of the
> candidates, even if you only use the highest rank.  You start by
> choosing the highest victory, and resolving that no matter what, the
> winner in that victory will be ranked above the loser.
>
> You keep by selecting the victories one by one, and resolving that
> they will be reflected in the final ranking.  Eventually, however,
> you may reach a victory that contradicts the decisions you have
> previously made.  When this happens, you just ignore that victory and
> carry on the process with the others.
>
> -------------
>
> Anyone who claims that a method like Schulze, SSD, or SD is as simple
> or intuitive as Tideman should present a description of their method
> that is as simple or intuitive as the preceding.  In particular, it
> is no good to claim that a method is simple, and then describe it
> using terms like cycle and Smith set.  Since most people don't know
> what these are, you would have to define them as part of the
> description of the method.
>
> I particularly like the way Tideman's method deals with the idea of
> cycles.  Notice that I never had to use the term in describing the
> method, and the concept was introduced as a natural result of the
> need for consistency.  That is, I didn't have to describe them by
> talking about arrows on a directed graph, or some kind of mysterious
> circular reference of the candidates.

But the unique reason why your description of Tideman is so simple is
simply the fact that you avoid explaining how the Tideman winner is
actually calculated. As soon as the example is too large so that it
isn't obvious any more whether a given pairwise defeat is compatible
with already locked pairwise defeats, you have to use the concept of
cycles or beat paths to explain how to check whether this pairwise
defeat has to be locked or skipped.

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