[EM] Dropping SD

[Blake Cretney]

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.

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Here's an example where SD violates GITC

E:A 8
A:D 30
D:E 9
E:F 11
F:D 12

1st, drop all lower victories

Drop E:A -- A:D:E

A wins.

Now, replace A by A B C, with
B:C 18
C:A 19
A:B 17

1st, drop all lower victories
Drop E:(A,B,C) in arbitrary order (A,B,C):D:E
Drop D:E -- E:F:D

E wins

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Isn't it time we dropped SD from discussion.  Tideman's method is
clearly superior.

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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.

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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.

---
Blake Cretney