[EM] pairwise matrices and ballots
MIKE OSSIPOFF
nkklrp at hotmail.com
Sat Feb 26 22:51:20 PST 2000
[Mike]:
> > It's in his paper entitled _How to Take Votes: New Ideas on
> > Better Ways to Determine the Winners_.
>
[Blake]:
>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.
Ok, I'd thought that round robin results array meant the same
as pairwise vote table. If not then I was mistaken about what
he said he proved, and that mistake was the basis for my
statement about every pairwise vote table having a set of
rankings that produce that table.
>
> > r(X,Y)=1 if and only if X won its match against Y,
>
>----------
> > 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
>alone advocated.
Brams & Fishburn, whom I'll quote in this letter, speak of
MaxMin votes-for. It seems to me that the Simpson-Kramer definition
in the _Journal of Economic Perspective_ is written that way too.
So it isn't usually necessary to guess what they meant. And if
someone _were_ completely vague about it, then they could mean
votes-against or votes-for, and that would mean it would make a
difference whether we look at all of a candidate's pairwise
comparisons or just at his defeats.
>
>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
I don't know if it's used that way in a published journal. I've
probably only seen MinMax or MaxMin used to refer to methods
that look at all of a candidate's pairwise comparisons to determine
his score, in journal articles. But your use of it with
the other meaning here means that I was right to say that the
terms are used with both meanings.
>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.
Sure, in fact MinMax is fine on this list, because Plain Condorcet
is pretty much the only MinMax method that's been proposed
here & much mentioned here.
It's only elsewhere, as at the website, that the information
in brackets would be necessary, to indicate that the
method counts votes-against (winning-votes) or votes-for
or margins, and whether it looks only at defeats. So
Plain Condorcet could be called MinMax(pairloss,wv).
>
>As well, I dislike the term "Plain Condorcet" for the following reasons:
>
>1. It is unknown off this list.
True, but so seem the interpretations of Condorcet's method
that are discussed on this list, except for Tideman.
No MinMax other than MinMax(pairlos,wv) agrees with what
Condorcet said, for instance.
>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.
The translations of Condorcet's own words for his bottom-up
iteration proposal have Plain Condorcet as their literal
interpretation. Yes some of us, including me, believe that
Mr. Condorcet meant more than Plain Condorcet, but what I
call Plain Condorcet is the literal, simplest interpretation
of that proposal, the name seems reasonable, to distinguish
it from the more refined interpretations, that I call
Cycle Condorcet interpretations, because, though they solve
circular ties by dropping defeats, they won't drop a defeat
unless it's the weakest defeat in some cycle. I fit Schulze
into that category since Schulze, SSD, & SD are equivalent when
there are no pairwise ties or equal defeats.
As nearly as I can remember, this is what Condorcet is translated
as saying:
If those propositions [pairwise defeats] cannot all exist
together [because of a cycle], then ignore the proposition having
the smallest majority. Proceed in that way till there's
an unbeaten candidate.
That could be said: Drop the weakest defeat. Repeat till somone
is unbeaten. That's equivalent to saying that the winner is
the candidate whose greatest defeat is the least.
***
Of course if we're dropping defeats because they cannot all
exist together, then wouldn't it make more sense to drop only
from among those that cannot all exist together, when it comes
to picking the winner? That's why it was suggested to drop
defeats only from among the members of the current Schwartz set.
Hence, SSD.
Or, "defeats that cannot all exist together" could be more
broadly interpreted as the defeats that are in cycles. It's
that broader interpretation that leads to the Cycle Condorcet
methods.
>3. It is confusing because one would think that the Condorcet winner would
>be identical to the winner of Plain Condorcet.
That's true, but I hate to give up the name "Condorcet's method",
because it carries the prestige of the founder of voting theory.
Mike Ossipoff
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