[EM] Condorcet cyclic drop rule

[Blake Cretney]

On Mon, 02 Apr 2001 07:52:19 -0000
"MIKE OSSIPOFF" <nkklrp at hotmail.com> wrote:
>> Of course Condorcet didn't really propose Ranked-Pairs
>> (defeat-support).  That method was invented recently by Mike, based
on
>> the method Tideman invented around 1985.  Condorcet wrote in the
>> 1700's.
>
> Wrong. Condorcet wrote around the 1780s or 1790s.

I'm not going to get into an endless debate about whether 1785 is in
the 1700's.

> Invented recently by Mike? Steve Eppley defined it before I did.
> But thanks for trying to give me credit for that better version.

Markus points out that he invented it.  My mistake.  In any case, it
is still impossible for someone who died in 1794 to have proposed a
method that was only introduced a few years ago.

But back to the plurality definition issue.  Once again, this is my
reference:

Iain Mclean, Fiona Hewitt, 1994
"Condorcet:  Foundations of Social Choice and Political Theory"
Edward Elgar Publishing Limited

All quotes are from "An Essay on the Application of Probability Theory
to Plurality Decision-Making", 1785.  Page references refer to the
translation.

p 136
> it would seem more reasonable to judge these propositions, not
> according to the preceding hypothesis, but according to the 
> degree of plurality they have obtained.

Point 1:  Condorcet says, use "degree of plurality."

Now, he never defines this term (at least in the text I have), but we
know his definition from context.

p134
> We must now look at the case in which a decision is made with the
> smallest possible plurality.  If we suppose each vote to have a 
> probability of 9/10 of being correct, then for 10 candidates, we 
> only require a plurality of four votes to have, even in the worst 
> possible case, a probability of 99/100 of having made a good 
> choice.

He's discussing the "smallest possible plurality".  Of course, he
doesn't literally mean the "smallest possible", which would be 1, but
a very small plurality.  He describes a "plurality of 4".

The question is, does he mean that there is a margin of 4, or
four votes on the winning side?  This is a clear reference to his jury
theorem, and that theorem uses margins.  If it didn't, it wouldn't be
true.  So, 4 must be the margin for his probability calculation to
make sense.  That means that he uses the phrase "plurality of four" to
mean a margin of 4, and that he describes this as a small plurality. 
If he measured plurality size by winning side, this could still be a
large plurality, if there were enough people on the winning side.  So,
he does not use the term as defined in Norm's French dictionary quote.

Point 2:  Condorcet measures plurality size by margin

p128
> Now, imagine that the three propositions with plurality 
> support form one of the two contradictory systems: if 
> there is no need to elect someone immediately, then we 
> should consider that no decision has been reached.  If, 
> however, an immediate decision is necessary, then we 
> must take the results of the two most probable 
> propositions.  For it is easy to see that, as we observed, 
> any two of the propositions give a decision which 
> contradicts the third, so that, for example in combination 
> 3, made up of the three propositions
>
>`A is better than B'
>`C is better than A' and
>`B is better than C',
>
> the first two give a decision in favour of C, the first 
> and third a decision in favour of A, and the second and the 
> third a decision in favour of C [printing error].  Now, if
> the proposition `B is better than C' has the smallest 
> plurality and `A is better than B' has the greatest, then 
> clearly the two propositions
>
> `B is better than C' and
> `B is better than A'
>
> each have a lower probability than the two propositions
>
> `A is better than B' and
> `A is better than C'
>
> B should therefore be excluded, and then C should be 
> preferred to A because the proposition `C is better than 
> A' has plurality support.

Point 3:  Condorcet says:  Use plurality size.

I think that Condorcet's reasoning is flawed here, but that's beside
the point.  This passage shows that Condorcet uses most probable and
greatest plurality interchangeably.  That only makes sense in the
context of his jury theorem and measuring plurality greatness by
margin.  Also, note that here Condorcet says that precedence should be
based on "greatest" plurality, and in the quote above (p136), he
talked about "degree of plurality" for precedence, so they must mean
the same thing.  Remember, this is all from the same essay.

Point 4:  To Condorcet, Plurality size = Degree of plurality = Margin
= Probability = Precedence

BTW, Markus has shown that Condorcet considered partial rankings, in
1793 if not in 1785.  If you believe that Condorcet was specifically
considering partial rankings when he recommended using margins, I
won't argue the point.

---
Blake Cretney   http://www.fortunecity.com/meltingpot/harrow/124/path

Ranked Pairs gives the ranking of the options that always reflects 
the majority preference between any two options, except in order to
reflect majority preferences with greater margins. 
(B. T. Zavist & T. Tideman, "Complete independence  of clones in the 
ranked pairs rule", Social choice and welfare, vol 6, 167-173, 1989)