[EM] Another look at Tideman

[Blake Cretney]

In my previous posting I suggested that Tideman was originally
rejected on faulty grounds.  Now, the arguments mentioned did not
influence me, because I disagreed with some premises behind them. 
However, I had suggested some other reasons for rejecting Tideman,
which have over time appeared less convincing.  In particular, my
justifications for favouring Schulze (using margins) over Tideman seem
in retrospect rather poor, and I am starting to lean towards Tideman
as the superior method.  The methods are so similar, that it is hard
to find useful points of differentiation.

Reverse-consistency

Some time ago, I suggested that Tideman was inferior because its
handling of ties did not appear to be internally consistent.

A > B 30
B > C 30
C > A 30
A > D 8
D > B 5
D > C 6

I reasoned that Tideman in effect made the winners
{ A, D }
suggesting that A and D are superior to B and C

But, if all ballots were reversed, the winners were
{ A, B, C}
which suggests that for the un-reversed ballots, D is superior to A,
B and C.

I note an equivalent situation in Schulze

5 A B C
5 D (A=B=C)

The potential winners are A and D.  The reverse gives C and D.
What the above ballot suggests is that A is preferred to B, but there
is no way to know which, or all, of A, B, and C are preferred to D.
The argument that D=A and A>B, therefore D>B, just does not follow.  A
tie should not be taken as a statement of equality, but of indecision.
 However, with this view, the complaint against Tideman evaporates.

---

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.

---
Blake Cretney