[EM] Condorcet cyclic drop rule

[MIKE OSSIPOFF]

> > the MinMax interpretation of Condorcet's wordings has been
> > proposed by Black (Duncan Black, "The Theory of Committees
> > and Elections," Cambridge University Press, 1958).
>
>Is that really an interpretation, or a new method based on Condorcet's
>principles?

If you're talking about what you call "Minmax", it's an interpretation
of Condorcet's drop-weakest definition. The way I currently define
PC is the literal interpretation of Condorcet's drop-weakest proposal.
Yes, if he was interested in dropping the proposition least likely]
to be "correct", then that suggests margins, but the literal wording
of his proposal suggest defeat-support. But--aside from the issue
of defeat-support vs margins--PC, as I define it now, is the literal
interpretation of Condorcet's drop-weakest proposal.

I assume that by "Minmax", you mean PC.

>
>Here's what Mike has to say, at
>http://www.barnsdle.demon.co.uk/vote/condor2.html
>
> > We call that method "Plain Condorcet" ("PC").
>
>Note that originally Mike called this "Condorcet's Method", and some
>people, like Rob Lanphier, still do.

Others call it Condorcet's method. That name is found for it in
journal articles, though we don't know how those articles would
measure defeats, with incomplete rankings. It especially deserves
the name Condorcet's method, since it's the literal interpretation.

>
> > It turns out that this method was proposed in the late 18th
> > century, by the founder of voting theory.
> >
> > The Marquis de Condorcet, in the period just after the
> > French Revolution, participated in the discussion of how
> > the new government should be set up. This included voting
> > system proposals.
> >
> > He proposed that if there's a candidate such as I've called
> > BeatsAll winner, that candidate should win. And he proposed
> > a natural & obvious solution for when there's no BeatsAll
> > winner.
> >
> > Here's what he suggested. What I call pairwise defeats,
> > he called "propositions":
> >
> > If the propositions can't all exist together [because
> > there's a cycle instead of a BeatsAll winner], then, one
> > at a time, drop the proposition with the smallest majority,
> > until there's an unbeaten candidate.
>
>The square brackets, and context seem to imply this is a quote, but it
>isn't offset or in quotation marks.  Mike, is this intended as a
>quote?  Do you have a reference?

Yes there was a reference for that, and I'll find it and post it
again, just as I posted it here before.

It isn't a dircect quote. It's close to it. It says it largely in the words
used by a translator of Condorcet, but I wasn't copying the translation
when I wrote tha--I didn't have it with me at that time.


>Indeed this statement is equivalent to MinMax,

No, that depends on what Minmax is. I've heard conflicting definitions
of Minmax. Minmax is a hopelessly vague term. I don't use it.

>although an odd way of
>describing it.

An odd way of describing it? Yes, if what I mean to say was your
definition of Minmax, it would be odd to say something else that's
different but equivalent. I said it as I did for 2 reasons: 1)That
was how Condorcet said it; 2) It's more compellingly plausible when
worded as I worded it--as opposed to saying "Elect the candidate whose
greatest defeat is the least".

>
> > In other words, drop the weakest defeat. Repeat till there's
> > an unbeaten candidate.
> >
> > (The strength of B's defeat by A is measured by how many
> > people ranked A over B).
>
>Here's another quote from Condorcet as a comparison.
>
>Iain Mclean, Fiona Hewitt, 1994
>"Condorcet:  Foundations of Social Choice and Political Theory"
>Edward Elgar Publishing Limited
>
>p 238 (of the translation) from "On Elections" 1793
> > A table of majority judgements between the candidates taken
> > two by two would then be formed and the result -- the order
> > of merit in which they are placed by the majority --
> > extracted from it.  If these judgements could not all exist
> > together, then those with the smallest majority would be
> > rejected.

There, you've saved me the trouble of finding a reference; you've posted it 
for me. "If those judgements could not all exist together"
is the wording that I was closely copying in the wording of mine that
you quoted. "...those with the smallest majority would be rejected"
has been taken to mean that we keep dropping the weakest defeat till
someone's unbeaten.


>
>That sounds a lot like Ranked Pairs.

It obviously isn't Ranked Pairs. Condorcet, in that quote, says
to drop the weakest defeat. He didn't say, there, to drop the
strongest defeat that's the weakest defeat in a cycle. But Condorcet's
top-down proposal is Ranked-Pairs. Condorcet proposed 2 well-known
circular tie solutions, one top-down, and the other bottom-up. Ranked
Pairs is his top-down proposal, which was reasonably interpreted by
Tideman as meaning that we drop the strongest defeat that's the weakest
defeat in a cycle, and repeat that till there are no cycles. I realize
that Tideman worded that differently but equivalently.

Your own brief Ranked Pairs definition is incomplete.

>Reject smaller majorities, but
>only in favour of larger ones in the complete order.

I don't know what that means, and, in any case, I can't find it in
the Condorcet quote above.

On the other
>hand it seems oddly vague.  I suspect that Condorcet thought that if
>you kept successively dropping majorities (from smallest up), you
>would have a valid ranking as soon as you eliminated all the cycles.

If you have no cycles, then you have a transitive ordering (but if there are 
pairties, then it might not be what you'd call a ranking).

You seem to think that Condorcet wanted to eliminate all cycles.
But he said to elect the voted CW. If he wanted to get rid of all
cycles, then he wouldn't have just said to elect the voted CW.

What caused him to say that the propositions cannot all exist together
is if there's no voted CW, eveyone has a defeat. Since that's the
only reason why he's dropping defeats, then we stop dropping them
as soon as someone's unbeaten.

Again, his top-down proposal does imply getting rid of all cycles,
and was reasonably interpreted by Tideman as what
Tideman called Ranked Pairs.

>If this were true, his description would not be ambiguous.
>Unfortunately, this isn't the case.  So, you can take Condorcet's
>statement as vague, but it's probably better seen as incorrect.

What are you talking about? If the propositios cannot all exist
together, then drop the one with smallest majority. If you repeat
that, then eventually you'll only have propositions that can exist
together. As I said, since Condorcet simply said to elect the candidate
who beats each of the others pairwise, if there is one, regardless of
whether there are cycles among the other candidates, his goal was
obviously not so get rid of all cycles. We're dropping defeats only
because there's no one unbeaten. So we stop when someone is unbeaten.

>
>It certainly doesn't look like MinMax.

Since Minmax has various definitions, I wouldn't argue on what does
or doesn't look like Minmax. But when we drop the weakest defeat,
and repeat till no one's unbeaten, that's PC. And PC is equivalent
to electing the candidate whose greatest pairwise defeat is the least.
And I suspect (but am not sure) that that's what you mean by Minmax.

There's no "until there's an
>unbeaten candidate."  Condorcet implies that you will drop majorities
>until all the judgements can exist together.

Implies :-) There you go again with your own personal interpretations.

Condorcet says to elect a candidate who beats everyone pairwise, if
one exists. If, all the judgements cannot exist together merely
because they're inconsistent with anyone beating everyone, or being
unbeaten, then when someone is undefeated, then all the judgements
can now exist together, because they're consistent with a certain
candidate being unbeaten, like the voted CW that Condorcet initially
wanted.

In any case, I think
>it's clear now that if this could be "interpreted" as specifying
>MinMax, it would be the marginal form of MinMax, since Condorcet
>specified margins.

Not by his literal wording in his proposal. His interest in
the probability of a defeat being "correct" suggests that he'd
use margins, however.

Mike Ossipoff

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