[EM] Tideman vs Beatpathwinner

[Blake Cretney]

It is often stated that the Tideman winner beats the Schulze winner
more often than the opposite.  I'm going to prove this.  Also, I'm
going to discuss Markus's point about the Schulze winner being less
defeated.

Let's say we have an example where the Schulze winner (which I will
call S) beats the Tideman winner (which I will call T).

For T to win, despite being beaten by S, it must be true that the S>T
majority was skipped.  This can only happen if there is already a
path locked from T to S that is greater than this majority.  The
existence of a locked path in Tideman implies the existence of a path
in Schulze.

Since S won in Schulze, and there is a path from T to S, there must
be an even higher path from S to T.

So, consider what would happen if the S>T victory was flipped, so now
T beats S with the same margin (or winning-votes depending on your
preference).  The answer is, that nothing would happen.  Schulze and
Tideman both still have higher paths that they would use instead.

So, my point is that every example where the Schulze winner beats the
Tideman winner is paired with an example where the Tideman winner
beats the Schulze.  This is constructed simply by flipping the
victory.

However, there are some examples where the Tideman winner beats the
Schulze winner, and which do not correspond to one of these pairs. 
For example, 

A>B 10
B>C 9
C>A 8
D>B 7
D>C 6
A>D 5

Tideman winner is A.  Schulze winner is D.  Clearly if the A>D
victory was flipped, D would win in both methods.

The basic point is this.  In Tideman, it is slightly harder to
construct a path leading between two candidates that can make the
direction of a victory irrelevant.  As a result, the direct
comparison is more often decisive between the two candidates. 
Although, I do not think we can say Tideman is superior just because
it more often beats Schulze than the other way around, I do think
that its lower tendency to over-rule direct comparisons is an
advantage.

It's worth pointing out that in the above example the Schulze
winner's greatest defeat is less than the Tideman winner's.  Markus
points out that this will more often be the case than the opposite,
and I think it is fairly easy to see that this is so, although I'm
not going to prove it in this email.  However, I don't see this as a
disadvantage for Tideman.

I think that the argument usually goes that this means we are
over-ruling a greater victory, and we should not do this.  But let's
think about what that really means.  It might mean, in the above
example that we elect D instead of A because we want to uphold C's
majority over A.  But let's think about the people who voted C over
A.  Are we doing them a favour?  Do they all prefer D to A?  If we
subtracted off the people who don't like the change, would we be left
with a majority at all?  Clearly not, since A in fact beats D on
direct comparison.

I guess my criticism is of the view that sees a majority as something
with a will of its own, that can go around demanding changes that
aren't supported by the people making up the majority.

---
Blake Cretney