# [EM] pairwise matrices and ballots

Blake Cretney bcretney at postmark.net
Sat Feb 26 20:06:19 PST 2000

```MIKE OSSIPOFF wrote:
> Blake--
>
> You have in your possession Bruce Anderson's proof that
> for every pairwise vote table there's a set of rankings that
> produces that pairwise vote table.

> It's in his paper entitled _How to Take Votes: New Ideas on
> Better Ways to Determine the Winners_.

Referring to page 184 (E-1) of his paper, he states:

> Theorem:  Given any round robin results array, there exists a set of
voter's preference
> over the alternatives involved whose corresponding pairwise matches produce
that array.

He then goes on to prove that result.  However, you have
misinterpreted what he means by a "round robin results array".  Check
his definition on p 80 (VI-32).  It is clear that a round robin array
only includes the information of wins, losses, and ties.

> r(X,Y)=1 if and only if X won its match against Y,

----------
> > > 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
> >were
> >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.

That's what I thought.  My point is, that if they aren't considering the
issue of incomplete rankings, they might say one of:

1.  Find the candidate who has the fewest votes against it in any pairwise
contest.
2.  Find the candidate who has the fewest votes against it in its greatest
loss.
3.  Find the candidate who has the smallest margin of defeat in its greatest
loss.

Knowing that all three are equivalent for their purposes.  If you take their
words out of that context, and instead apply them to incomplete rankings, you
have them arguing for a method that they likely never even considered, let

Note, the following is from a different posting by Mike Ossipoff, on the same
subject:
> Your own use of "MinMax" for Plain Condorcet shows you that
> that term is sometimes used for a method that considers only
> a candidate's defeats when determining his score.

True, but I'm still interested to know if it was used this way in any
published journal, where the issue of incomplete rankings was considered.  As
for my use, I find it very convenient to use the term Minmax for the basic
algorithm, and then specify separately, usually in brackets, which method I am
using to measure pairwise contests.  This seems simpler than having three
separate names for the three suggested ways of doing this.

As well, I dislike the term "Plain Condorcet" for the following reasons:

1.  It is unknown off this list.
2.  It implies that Condorcet invented this method.  This does not appear to
be the case, although his words may have been taken out of context, as I
described above.
3.  It is confusing because one would think that the Condorcet winner would
be identical to the winner of Plain Condorcet.

---
Blake Cretney

```