[EM] SD example--incomplete. Has error?

[MIKE OSSIPOFF]

EM list--

I'm not trying to defend SD out of any partiality or bias. If
SD can be nonmonotonic without pair-ties or equal defeats, I want
to know about that. But, though I could be mistaken, it seems to me
that Blake's example doesn't demonstrate that, for several reasons
any one of which would disqualify the example:

1. Blake said:

>All unspecified victories are assumed to be lesser, and therefore
>dropped first.

No, by SD's rules, lesserness isn't sufficient to allow a defeat
to be dropped. It has to be in a cycle. Not only must it have been
in a cycle at the beginning of the count, but, at the time when it
is dropped, it must still be in a cycle with other undropped defeats.

So all Blake has shown is that, if about 64% of the pair comparisons
are tied, then SD can be nonmonotonic. With 9 candidates, there are
36 pairs. Blake gives magnitudes for only 13 defeats, dropping the
other 23. Can Blake make his example more complete by actually
specifying those other 23 defeats? In such a way that they can all
be dropped before any of the other defeats?

If I may repeat SD's definition:

Drop the weakest defeat that's in a cycle. Repeat till there's
an unbeaten candidate.

So merely saying that 23 of the 36 defeats are weaker than the rest
is not enough to justify dropping them.

2. In Act II of the example, the number of ballots ranking A over G
is reduced from 16 to 9. Presumably by downranking A. But are you
sure that those voters can downrank A without disturbing any of the
other defeat magnitudes, or reversing any defeats? Blake hasn't
actually shown that his 2nd scenario of the example can happen via
rankings.

To summarize objections 1 & 2, Blake's example seems to me to be
incomplete. He's left out 23 defeats. He hasn't shown actual ballot
changes that can cause the change in the A>G defeat without making
any change in the other defeats.

***

As I said, if SD can be nonmonotonic without pair-ties or equal defeats,
I'd like to hear about it. But Blake, it seems to me, hasn't shown
that. Let's have a complete example.

***

Now, the above objections are enough to disqualify that example,
but doesn't it also contain an error? In Act II, after the A>G
defeat has been reduced, we first drop AG9. Then we skip GD10,
because it's not in a cycle (due to AG9 being dropped). But then
Blake drops DA11. But unless I mis-drew the arrow-diagram, the DA
defeat isn't in a cycle, and therefore can't be dropped.  What
cycle is DA11 in, after  AG9 has been dropped?

Now, if I've missed something, and misdrawn the graph, that still
doesn't change the fact that the example is incomplete.  But it
seems to me that its Act II was incorrectly carried out.

***

Mike Ossipoff


________________________________________________________________________
Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com