Coombs: Full count for each unranked

[Mike Ossipoff]

For Coombs' method to work when not everyone ranks every alternative,
whichs to say, for Coombs method to work in actual elections, it's
necessary that each of the alternatives that, among the un-eliminated
candidates, share lowest position in your ranking (anything
that you haven't ranked and isn't eliminated is of course in that
category) has to receive a full last-choice count from your ballot.

Don said that his reason for splitting the count between the
several unranked candidates on a ballot was mathematical, but
there's no mathematical reason why that should be done.

Don said that it's because he wants the number of "points" to
equal the number of voters, but there's no reason why the
concept of a point should enter into Coombs' method. In how
many rankings is candidate X at the bottom? What more need
be asked? Where do points have to come in?

Sure, if that's how you want to look at it, in terms of points,
and if it's really important to you that these points equal
the number of voters, then of course, if that's the most
important consideration to you, then it makes sense to
divide 1 by the number of unranked uneliminated candidates
on a ballot, and give each one of them a count equal to that
division-result. But you're then going to have a really hard
time selling the method, because it won't work. It'll fail
very badly whenever not everyone ranks all the candidates, which
will be virtually every election. But that depends on what you
want in a method, as I always say. If the ways in which 
Coombs(split) fails don't matter to you, but keeping the split
is what's important to you, as a standard in its own right,
then no one can criticize you for keeping the split. But,
as I said, you're not going to be able to sell a method
that fails every time, in terms of how most everyone, including
me & surely most on this list, judge what's important.
I doubt that anyone will like it when they find out what
happens if not everyone ranks every alternative.

I have no idea what Prof. Coombs said about unranked alternatives.
I'd guess that, most likely, he didn't say anything about it, since
the academic assumption has always been that everyone ranks all
of the alternatives. Just as Condorcet probably didn't say anything
about that question. In both methods, I define them in the way that
actually works. Similarly, that's why I define Condorcet's method
in terms of votes-against, though Condorcet, as I understand it,
didn't specify how defeats should be measured.

Don asked about more information about Coombs, but I don't
know what more there is to find out: Coombs successively
eliminates from the rankings the alternative lowest on
the most rankings. To deal with truncation, I change that
to: Coombs successively eliminates from the rankings the
alternative occupying or sharing lowest position in most
rankings. (With the understanding that any uneliminated
candidate(s) that you haven't ranked possess that honor).

But the 1st wording in the previous paragraph is the one that
you'll find in the academic literature, I expect, since
the authors probably don't consider truncation or having
several candidates at 1 rank position.

If you doubt that definition, of course then I'd urge you to
check for yourself. Go to any university library & look in
the subject catalog under "voting systems". I can even
name a book that will define Coombs, & some of the other
methods we've discussed (but it doesn't mention Condorcet's
method). I really hate to name a book here, because I can't
recommend it. I can't recommend any book, because I haven't
run across one that's any good. Anyway, the one that I'm
going to name, but not recommend, is a brief little book
that will give you the definitions, along with some academic
criteria, all which have been discussed here already:

Title: _Selected Topics in the Theory of Voting_

Author: Shiffrin

I hope I've got that title & author right. I might not have.

As I've said before, you can find a translation of Condorcet's
own words, his own definition of his method, in:

_The Theory of Committees & Elections_, by Duncan Black
 
Shiffrin seems to like Copeland. These guys live in a world
of their own, quite oblivious to the considerations that
are important to us, & to voters in general.


Mike








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