Definition of "Pairwise Method"

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Definition of "Pairwise Method"

Craig Carey-2
Yes, it would be roughly true to say that I'd consider a method to be
a Pairwise method if it meets Bruce's definition, and the Condorcet
criterion. But I'd be perfectly willing to say that a method can be
Pairwise without meeting the Condorcet criterioin, for the same of
completeness, in case, as Bruce says, someone proposes such a method
tomorrow. But no one, except maybe a mathematician, would propose such
a method. For all the proponents of Pairwise methods, the whole purpose
of doing the pairwise comparisons is to find if there's something that
beats everything else. That's the 1 thing that every Pairwise method,
& every Pairwise proponent have in common. So, maybe for mathematical
completeness, but not for any conceivable practical reason, I'd agree
that a method can be Pairwise if it doesn't meet Condorcet's Criterion.

But I still prefer my definition to Bruce's. My main protest against
Bruce's definition is that it's starting to turn this list into
one that uses mathematical symbolic language. Everything that can be
said about voting systems can be said in plain English. For instance,
the definition that I gave differed from Bruce's in a few material
ways, but, most important, it differed by being said in quite plain
English. My concern is that EM will lose members if we let it be
turned into a mathematicians-only list, a list where things are
said in the language of mathematicians when they could instead
be said in plain English. As I've said, there could, of course, be
some times when formulese could be very brief & clear. I don't want
people to unsubscribe from EM, or start deleting its messages, or to
quit the Single-Winner-Committe because the proceedings of these
groups are beginning to be done in a foreign language! I want an
"English-only" law for EM. No Formulese, except in a few special
cases where it really is necessary for brevity &/or clarity. Those
situations are rare or nonexistent in the voting systems topic, where
pretty much everything is more easily, briefly & clearly said in English.

Anyway, I don't object to Steve's & Bruce's proposal to count
as Pairwise methods methods that don't meet the Condorcet Criterion,
but it's for mathematicial completeness purposes only, since we're
not going to find any such animal.

There are a few minor material differences between Bruce's definition
& mine, by which I mean differences other than differences in language:

Bruce includes something about counting how many people rate x & y
equal, either by ranking them equal or by ranking neither. I
know of no reason to count that. All that matters, it seems to me,
is how many rank x over y, & vice-versa. I realize that Bruce
believes that, if you don't rank x or y, you should be counted
as voting half of a preference for x over y, and half of a preference
for y over x. I don't understand how Bruce could say that, since
mathematicians usually take things literally, and if you don't
rank x or y, than you're not expressing _any_ preference between
them, full, half, or otherwise. I don't know why Bruce would make
up a preference that you didn't vote.

Anyway, though, just as we can include methods that don't meet
the Smith Criterion, for completeness, I also have no objection
to including methods that count the people who equally rate
x & y, for the same reason. But wait--isn't that information
already available from the number of candidates, the number of
voters, & the number of people ranking x over y, & the number of
people ranking y over x? So it seems redundant to include the q
array.

To briefly return to how unranked or equally-ranked alternatives
in your ranking should be counted, I've noticed that Bruce has
used his way of counting them in a definition of Condorcet's method
that he has sent to a few members of this list. I insist that the
proponents of a method, not an opponent of it, be the ones to define
it, for the purpose of our discussions & comparisons. Besides, I
don't believe that Mr. Condorcet said to count preferences that weren't
voted.

But of course if Bruce were to abandon Copeland & be a Condorcet
proponent, then he'd have justification to define whatever version
of Condorcet's method he likes & wants to advocate.

Of course he could claim that his version of Condorcet is better than
ours, if he wanted to, but what he mustn't do is define it his way for
the purposes of comparing it to Copeland, explicitly or implicitly.
"Your method is as I define it, and therefore doesn't do what you claim
& isn't as good as my method".

***

Other, more minor material differences exist between Bruce's & my
definitions of Pairwise methods. Bruce's definition seems to imply
that all of arrays p & q will be used, where I emphasize that it
might not be necessary to do all of the pairwise comparisons before
finding something that beats everything else.
Maybe Bruce's wording doesn't imply what I've just said it does. That's
open to interpretation.

The other minor difference is that Bruce says that a Pairwise method
mustn't use any information other than arrays p & q, the information
about how many people have ranked, say, x over y, y over x, & the
number who have not ranked them or ranked them equal. I've proposed
several Pairwise methods that use other information. I've proposed
methods that sometimes use a 2nd balloting by Approval. I've proposed
refinements (which I haven't discussed here yet), for dealing
with really devious electorates, which consider such things as
whether certain rankings are consistent with eachother, in the
sense of agreeing on what alternative is "between" what other
alternatives, and how unanimous a various sets of voters are
about pairs of alternatives other than their favorite.

All of those things I've mentioned count things other than the
voters ranking x over y & vice-versa, & the number ranking them
equal or not ranking either.

So though we're now only talking about 1-balloting methods, &
though Condorcet, as I now propose it, only uses array p, and though
it would be only for very devious electorates that I'd propose
those 1-balloting refinements of Condorcet that use other informtion--
I still don't want the definition limited to methods that use only
arrays p & q.

***

But I re-emphasize that the different Pairwise methods differ from
eachother so outrageously, in terms of merit, that it isn't that
helpful to classify them together as a group. If I say "Pairwise",
you have no way of knowing if I'm speaking of a good method, or
a really lousy one. Better to just use the names of the particular
methods.

***

Mike






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