Equal defeats in RP(m)

[MIKE OSSIPOFF]

I'd asked:

>Someone has designated Ranked Pairs(margins), and so it's necessary
>to ask for a complete definition of it. In particular, how exactly
>does it deal with equal defeats, in all the kinds of situations
>in which they can occur and in all the procedural questions they can
>raise in an RP(m) count?
>
>Obvioiusly it would be inappropriate for that definition to be
>supplied after the ballots are posted.

Blake replied:

I've been assuming that no specific tie-breaker is proposed for the
purposes of this election.  Unless I'm mistaken, that is the case for
most of the methods suggested.  For example, IRV, plurality, Approval,
etc.

I reply:

If someone designated IRV, we'd have to ask how to deal with
equally lowest candidates. Ties in Approval? You apparently missed it,
but we discussed what to do when a designated method returns a tie.

It's the internal ties during a method's procedure for which solutions
should be specified before the ballots are posted.

Rob LG: Maybe we shouldn't post the ballots till Blake decides to tell
us how his method deals with equally lowest defeats during the count.
Alternatively, we could just determine all the winners that are possible,
from all the conceivable ways of dealing with such ties, and call the
outcome a tie between those alternatives. Or, thirdly, we could write
our own equally-lowest-defeats rules for RP(m).

Of course a 4th alternative would be for Blake to tell us how he
wants those equally lowest defeats to be dealt with. Blake says he
described it "recently". Would he be so good as to tell us approximately
on what day he described it on EM, so that we can find it in the
archives?

Here's my suggestion for equal defeats in RP(m):

If 2 or more defeats whose margin must be compared have the same margin,
then we consider weaker the one with less defeat-support. If 2 or more
have the same margin & defeat support, then we treat them as all equally
weak.


But suppose there are 2 defeats in one cycle, and another defeat in
another cycle that all have the same margin. For the 2 defeats in the
same cycle, that margin is the weakest in that cycle. For the other
defeat, that margin is also the weakest in that defeat's cycle.

Let's call the 2 defeats A & B, and the 1 defeat C.

And say that that margin is the strongest that's the weakest in a cycle.

Now, do we drop ("skip") C
, because it's the strongest defeat that's (uniquely)
the weakest in a cycle? Or do we find out which of the A & B is weaker,
by comparing their defeat support, and then find out whether that weaker one 
is stronger than C, by comparing its defeat support with that of C?

For that matter, say A & B are stronger than C. Should we still drop
C instead of A or B, because it's uniquely weakest in its cycle, as
long as it's stronger than any other defeat that's uniquely weakest in
its cycle?

Rob LG: It may be up to you & me to rule on this, unless Blake
has a suggestion. But the trouble with that is that we've seen the
ballots. That's why it would be better if Blake could tell us how
he wants these situations to be dealt with. But I admit that
equal defeats make a real mess of Ranked-Pairs.

Blake continued:

Personally, I recommend the tiebreaker Zavist and Tideman give, and
which I described recently.

I reply: Would you mind re-posting it, or telling us the approximate
day on which you previously posted it?

Blake continued:

However, most of the people who will vote
have already voted, so it's a little late to add to the proposal.

I reply:

If you've already posted it, then that's what we'll go by. I trust that
your posting answers the questions that I asked in this e-mail.

If not, then, even though 7 people have voted, either you tell how
you want those situations dealt with, or else any situation that
hasn't been covered will have to be considered to result in a tie between
all the alternatives that could win under the various possible
interpretations. Because it wouldn't really be appropriate for Rob LG
or me to rule on that, since we've seen the ballots.

The fact that people have voted doesn't seem such a problem. The
problem would be if you told us how you want those situations dealt
with after we posted the ballots. Shall we wait till you tell us how
you want the situations dealt with, before we post the ballots, or
shall we just post them as soon as possible after the balloting deadline,
whether or not solutions have been specified, and then just declare
a big tie if there's a situation that hasn't been covered?

Blake continues:

I should point out that most cases of equal defeats won't affect the
result, and that it isn't hard to write a program which detects these
cases (or at least the common ones).  In fact, I've written such a
program.

I reply:

Blake, I don't think the problem is the detection of equal defeats.
The problem is about whether or not you tell us how you want the
situations that I described in this e-mail to be dealt with. And, for
that matter, are there other ambiguous situations that you should
specify solutions for in Ranked Pairs?

Mike Ossipoff

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