Tie-Breakers. Restarting w/ tie.

[Mike Ossipoff]

I want to say now that I agree that Bruce's suggestion to send
every reduced tie back to the 1st method, starting the whole 
count over with that reduced tie does seem the ideally best way
to do it--though I still don't consider it necessary, if its
added count labor or (possible) explanation difficulty is
a problem.

Or course, since Bruce & Steve have both expressed preference
for that procedure, and since it's ideally better, then if
we had to do a vote on EM, I certainly would agree with that
proposal. EM, as a group, is likely to prefer ideal merit
over count simplicity, especially since a computer program
could just as easily carry out that procedure as not.

***

About tie-breakers, I've only suggested 2, and suggested
that they be used in this order:

1. Beat-Something
2. Stepwise Plurality.

Of course there could be a need for more than 2, though that's
unlikely, since Stepwise Plurality, like Bruce's Plurality.ext,
is quite decisive. Still, though Beat-Something certainly seems to
me to deserve being 1st, and though Stepwise Plurality would be
unlikely to return a tie, I'd ideally want to specify more
tie-breakers.

At least, I'd like to slip another good one in after Beat-Something:

Fishburn's method:

If something beats B that doesn't beat A, but nothing beats A
without also beating B, then B is disqualified.

[Of course if A beats B, then something (A) beats B that
doesn't beat A]

The "Fishburn set" is the set of un-disqualified alternatives.
I got this definition from Bruce, and I trust that if I've
mis-stated it he'll say so. But I'm pretty sure that's how
it goes.

So, if it were permissible to add another tie-breaker to the
list, my list would then be:

1. Beat-Something
2. Fishburn
3. Stepwise-Plurality

Of course several Plurality-related methods have been defined
as tie-breakers, and if it were felt that the above 3 weren't
enough, then Plurality-Elimination & Bruce's Plurality.ext
could be added to the list after Stepwise-Plurality. Or just
in case.

I should say that the natural 1st method to use, before the
tie-breakers, would be Smith//Condorcet, since we've voted that
the best recommendation for public single-winner elections, and
since the Smith set confers all the benefits on small elections
that it confers on big ones.

On the other hand, since, as I said, the people on EM are people
who'd prefer ideal appeal to simplicity, I suspect that we'd
rather use Schwartz//Condorcet than Smith//Condorcet, if we
agree that it has special aesthetic appeal in a small election.

Of course a list of tie-breakers could be longer than what
I've implied so far in this letter, since one could list a
whole list of alternate methods, based on alternate standards,
and have them all ready to use if necessary. Not likely necessary,
so the time & effort to agree on a long list wouldn't seem 
justified. Of very likely there wouldn't be unanimous agreeement
even on a short list, like the one I've discussed in this
letter. 

Tom is correct to say that Smith has something in common with
Copeland, since it has to do with what something beats, and
since the number that something beats can be used to find the
Smith set. In fact, Smith, Schwartz & Fishburn all are about
what something beats. But only so as to define a dominant
group, or a dominated alternative. The problem is when
Copeland makes the number of victories & defeats its whole
basis for choice. Fishburn seems compelling enough to
use it early among the tie-breakers, though I don't know
whether its use _before_ Condorcet (as in
"Schwartz//Fishburn//Condorcet") would be too exclusive a
slection set and would interfere with Condorcet's method's
properties. That's why I'd only use Fishburn as a tie-breaker
, after Condorcet.

If EM had to do another multi-alternative single-winner
vote, it would be good to agree on a few tie-breakers. I'd
suggest the ones I've listed, during the discussion to choose
tie-breakers, in that eventuality.

Ideally it could be appealing to list a few methods with completely
different, but plausible, standards, ahead of the Plurality-related
methods. But any discussion about that would turn into a discussion
of all sorts of methods & standards, and that sounds like something
that well might be better avoided.


Mike





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