[EM] pairwise matrices and ballots

MIKE OSSIPOFF nkklrp at hotmail.com
Thu Feb 24 21:39:17 PST 2000

> > Blake wrote:
> >
> > >Often people want to create examples involving pairwise methods,
> > >usually to show that the method behaves badly in some situation.
> > >Since not all pairwise matrices are possible, it is customary to
> > >provide a set of ballots instead of just providing a pairwise matrix.
> >
> > For any pairwise preference matrix, it's possible to devise
> > a set of rankings that will give that pairwise preference matrix.
> > So, for pairwise methods, it's unnecessary to furnish rankings
> > for an example--the pairwise preference table is sufficient.
>I don't know where you got that idea.  Try the following example:

I got that idea from Bruce Anderson. If I still have a copy of
his proof, I'll send it to you. I sent a copy to Markus. If
I no longer have a copy of the proof, maybe Markus does.

Markus: Would you post that proof if you still have it, or
send a copy to Blake?

>    A    B    C
>A  X    1    3
>B  2    X    1
>C  1    3    X

I'll send you rankings for that, in a subsequent message.
Of course no one said anything about a limit for the turnout.

> > By the way, MinMax is sometimes used to mean what we here call
> > Plain Condorcet, and is sometimes used to mean Simpson-Kramer--
> > two different methods.
>Could you please quote the sources you used for your definition of
>Simposon-Kramer, and the different uses of MinMax?  Make sure that they 
>actually considering the issue of incomplete rankings.

I didn't say they were considering the issue of incomplete
rankings. Or if I seemed to say that, I didn't mean to.

I don't know of a MinMax or Condorcet definition in an academic
article that says anything about incomplete rankings.

But unless they specifically say that they're using margins
as their measure, then it makes a difference whether they
consider all of a candidate's pairwise comparisons, or just
his defeats. Of course, if margins of defeat is what they
go by, then they're automatically looking at a candidate's
defeats when determining his score. But I'm not aware
of any academic definition of MinMax or Condorcet that specifies
that margins is the measure that it uses.

But some academic definitions of Simpson-Kramer & Condorcet
say to elect the candidate whose least votes-for in a pairwise
comparison is the greatest. Yes, I know that's MaxMin, not
MinMax. I've seen that type of definition called "MinMax" in
an article. I'll send you a copy.

Simpson-Kramer is defined in the Winter '95 issue of
_Journal of Economic Perspective_. I don't know the page number,
but it shouldn't be difficult to find. That issue has a number
of articles on voting systems. That definition, if I remember
correctly, says that the winner is the candidate whose
lowest number of votes for him in a pairwise comparison is
the greatest. I believe that Brams & Fishburn use that same
definition for Condorcet's method. I'll check again, but it'
seems to me that their definition of Condorcet is the same
as that _Journal of Economic Perspective_ article's definition
of Simpson-Kramer. But Brams & Fishburn's definition of Condorcet
doesn't recognizably resemble Condorcet's definition.

The term "MinMax" isn't so hard to find in academic articles.
I believe that Brams & Fishburn use it in a passage that I'm
going to send you.

You yourself used "MinMax", at your website, to refer to
the following circular tie solution: "The winner is the
alternative whose greatest defeat is the least." So you were
using it to refer to a method that looks only at a candidate's
defeats, not at all of his pairwise comparisons, to determine
his worst showing. So it only remains to show you a definition
of MinMax that doesn't specify that it only considers a
candidate's defeats. I'll find such an definition & send it
to you, naming its author, within a few days.

> > And they're still different methods even
> > if Simpson-Kramer counts votes-against in pairwise comparisons.
> > The difference is that Simpson-Kramer looks at all pairwise
> > comparisons, not just at pairwise defeats.
>Please illustrate this with an example.  I don't understand what you >mean.

Ok. I'll include the rankings, since we don't have agreement on
whether they're needed:

(The rankings are written in horizontal rows, as "ABC").

40: A
25: B
35: CB


B has 35 votes against him in a defeat, but he has 40 votes
against him in a pairwise comparison that isn't a defeat.
Simpson-Kramer would count 40 as B's greatest votes-against
, if Simpson-Kramer used votes-against.

So Simpson-Kramer, with votes-against, would score B differently
than Condorcet would.

If that can happen with that familiar votes-against example,
then I'd expect it to be possible with votes-for, which,
it seems to me, is the measure specified in the Brams & Fishburn
definition of Condorcet, and in the Simpson-Kramer definition.
I'll such an example along tomorrow.

Mike Ossipoff

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