[EM] Method definitions
Blake Cretney
bcretney at postmark.net
Mon Feb 28 23:43:19 PST 2000
This is in reply to Mike Ossipoff Re: Pairwise Matrices and Ballots
I am changing the subject heading because the topic changed.
MIKE OSSIPOFF wrote:
> > > 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.
You haven't addressed my point. If Simpson and Kramer describe a
method without considering partial rankings, and you take their words
out of context, and apply them to ballots with partial rankings, you
are not doing what they intended. For all we know, Simpson and Kramer
would be surprised and horrified to have their names attached to their
method as you describe it.
It isn't that they were vague. It is that by your own admission,
they never considered the issue of partial rankings. So, how can they
be taken as advocating a specific way of handling them?
> >
> >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.
Yes, but there is a difference between finding a new name for a new
concept, like "Schulze's method" and finding a new name for a well
known method.
> No MinMax other than MinMax(pairlos,wv) agrees with what
> Condorcet said, for instance.
I'll get to that a little further down.
> >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.
My understanding is that Condorcet only seriously considered the
situation of three candidates and complete rankings. If so, we have
the same problem as before. His words have been taken out of context
to appear to advocate Minmax(winning-votes) when in the context he was
using, margins was equal to winning-votes, and Tideman, Schulze and
many others are equivalent to Minmax.
He also mused briefly about extending his principles to more than
three candidates. He suggested that they could be solved the same way
as the 3 candidate example, by keeping the higher majorities and
losing the weaker ones. It is clear he assumed that the higher
majorities would be consistent. Since this is not the case, his
attempts to extend the method from beyond three candidate do not work.
> 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.
If that was the phrasing, I don't see it as obvious that he is
"literally" specifying margins. My dictionary gives as one definition
of a majority, "The number of votes cast for a particular candidate,
bill, etc., over and above the total number of remaining votes."
The English language (I don't know about French) is a little
ambiguous on this point.
As well, I note that in your recollection, Condorcet, like me,
assumes that all pairwise victories are in a sense a majority. What
do you read into that?
Of course, I am just playing the same game of taking Condorcet's
words out of context. In context, none of this mattered because
Condorcet was only considering full rankings.
> >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.
Why not call it Abraham Lincoln's method? He is even better known.
Of course, he didn't invent Minmax either. The fact that it is
frequently called Simpson-Kramer suggests to me that they are credited
with its invention.
MIKE OSSIPOFF wrote:
> I'm going to quote Brams & Fishburn's definition of Condorcet,
> because it shows the use of MaxMin to describe a procedure
> that looks at all of a candidate's pair-comparisons, rather
> than just at his pair-losses. If MaxMin is used in that way,
> it's reasonable that MinMax can mean that too. It seems to
> me that I've mostly only run across MaxMin definitions in
> academic writing.
>
> Brams & Fishburn's definition:
>
> "Condorcet's procedure [Condorcet (1785), Black (1958)] is
> a MaxMin procedure. [That would come as a surprise to Condorcet,
> judging by the translations that I've seen--Mike]
>
> Let v*(x) = min{v(x,y):y an element of X\{x}}. Then x an element
> of F maximizes v*(x) over X.
>
> v(x,y), I assume, is the number of people who voted x over y.
> I assume that F is the set of winners.
> I rendered the "an element of" symbol by words.
Once again, you have to consider context. Brams and Fishburn are,
like Condorcet, only considering complete rankings. Therefore, a
candidate gets the least number of votes against the candidate that
gives it its greatest defeat. The candidate with the most votes in
this defeat is least defeated. So, Maxmin and Minmax are actually
identical.
It is only when you take the definitions out of context, and apply
them to partial rankings, that this method becomes different than
Minmax. But that is clearly not what Brams and Fishburn intended.
They were just defining the method as efficiently as possible
considering that it wasn't intended for partial rankings.
I note that their definition of "Condorcet's method" when taken out
of context is not equivalent to the definition of Condorcet's method
that you get from taking his own definition out of context.
Of course, I would argue, based on the quote above, that Brams and
Fishburne have committed a similar error in describing "Condorcet's
procedure". Condorcet described a procedure for handling three
candidates that was equivalent to Minmax/Maxmin. He might have
extended this procedure to more than three candidates, but he did not.
In fact, his remarks on extending his procedure are very different
than this procedure. Either he considered the Minmax/Maxmin procedure
and rejected it, or he did not think in terms that allowed it as a
natural extension of his 3 candidate procedure. I suspect the latter
is the case.
I see it as appropriate, therefore, that most academics seem to call
this method Minmax or Simpson-Kramer instead of Condorcet's procedure.
There is no justification for deciding that Condorcet favoured
winning-votes, however.
> Their definition doesn't say anything about whether x beats
> y or vice-versa.
Yes, because in context, it doesn't matter.
---
Blake Cretney
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