Definition of "Pairwise Method"
landerso at ida.org
Mon Apr 15 05:41:06 PDT 1996
On Apr 14, 1:28am, Mike Ossipoff wrote:
> Subject: Re: Definition of "Pairwise Method"
> 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.
As I said, I'd like both, and "in-between's" also. I think that it is very
important to resolve basic issues before going on to more advanced topics. But
sooner or later we may be able to resolve basic issues and start in on advanced
topics. Then some formalism may be extremely important. Even then, however, it
would still be important (in fact, probably more important) to have English-only
interpretations of all formalistic statements.
> 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.
What about the Jean-Charles voting method that I defined in my last posting on
> 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.
First, as Mike points out, it doesn't make any difference for some calculations
of some types of voting methods, but it can in others. In particular, I think
it makes a major difference for various forms of Bucklin's voting method.
Second, it makes extending definitions of various voting methods much easier.
Third, it allows much easier application of criteria -- which is important for
my work, if not for others.
> 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.
Some day, I'd like to be able to post my Microsoft Word documents. If and when
this occurs, I could post my sample ballots. These ballots address the issue
Mike raises here.
> 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
Yes, which is my first point above. The reason I want to make it explicit is
for my second and third reasons. Of course, a fourth reason is that, for many
people, it is easier to understand careful and explicit explanations than
simplistic short ones. After all, look at the confusion even among people on
this list concerning how ties are treated in Condorcet's method. It seems to me
that Mike thought that his English was clear; but it also seems to me that Steve
benefited from the examples and, perhaps, the p's and q's. (I know I did when I
first learned about alternative methods for dealing with my in voter's
rankings.) So, in this case, Mike's plain English wasn't enough.
> 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
> 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.
Mike is correct in that I am defining Condorcet's method the way I want to in my
writings. Of course, I will continue to do so, precisely for the reasons I just
stated above. I believe that things will become much too confusing if different
versions of basically the same method are given very different names. I picture
calculating winners according to (the now defined) pairwise methods by using the
r(i,j) = p(i,j) + xq(i,j),
where either x = 0, or x = 1/2, or x = 1. When it is important to distinguish
among them, I would specifically refer to the Condorcet(0), Condorcet(1/2), or
Condorcet(1) method. I say that i is a Condorcet winner if it has the largest
minimum over j of r(i,j). I use x = 1/2; but Mike (and now, hopefully, everyone
else on this list who DOES NOT explicitly DISTINGUISH among the possibilities)
uses x = 1. I think there should be no problem here as long as I keep it clear
which is which whenever it makes a difference.
> 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".
Of course, I won't (intentionally) do this. In fact, I plan to do the opposite.
That is, unless I can prove otherwise, I'll assume that Copeland(1) passes all
of the criteria that Copeland(1/2) passes, but that (without proof otherwise)
Copeland(1/2) fails criteria that Copeland(1) passes. That's more than fair!
> 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.
I disagree with Mike's use of English here. For example, if I said that the
winners according to a certain voting method could be calculated using only a
pencil, paper, and slide rule, then that statement would be true if I really
only needed just pencil and paper; but it would be false if I needed a computer
(or an approval vote) also.
> 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
-- End of excerpt from Mike Ossipoff
I agree! I didn't start the use of "pairwise method." I just wanted to
understand it. And I still want to know how you would classify the Jean-Charles
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