[EM] Duncan Black on Condorcet

[Blake Cretney]

Dear Markus,

Before I comment on Duncan Black, I'll give an interesting quote from
Condorcet that is relevant to the current subject.  He's advocating
the Condorcet criterion (though not in name).  He gives an example of
three candidates, where C pairwise beats the other two, and B pairwise
beats A.  In justifying the selection of A, he makes the following
point:

p 126 (same reference as my prior postings)
> 1.  Candidate A clearly does not have the preference, 
> because there is a plurality of votes against him whether 
> he is compared to B or to C (and this is always the case 
> in such situations).  The choice is therefore between B 
> and C.  As the proposition `B is better than C' has only 
> minority support, we must conclude that it is C who has 
> plurality support.

Condorcet is implying the Condorcet loser criterion.  As I understand
it, the reason A shouldn't win, is that it is defeated by every other
candidate.  Furthermore, he implies a Condorcet loser independence
criterion.  That is, the Condorcet loser shouldn't even affect the
result.  He seems to think that each of these criteria is actually
more basic than the Condorcet winner criterion.  I disagree, and
therefore don't find this argument persuasive.  However, it's
important because it shows that Condorcet was aware of the possibility
of a Condorcet loser, and felt that it was clear that such a candidate
should not win.

This is a problem for anyone who argues that MinMax, or Mike's "Plain
Condorcet", or "Condorcet's method" is a natural extension of
Condorcet's principles.  Since these methods can elect a Condorcet
loser, it seems safe to assume that Condorcet would have rejected
these methods, if he had been aware of them.

On Wed, 11 Apr 2001 16:10:03 +0100
Markus Schulze <schulze at sol.physik.tu-berlin.de> wrote:
> Dear Blake,
> 
> Black wrote (Duncan Black, "The Theory of Committees
> and Elections," Cambridge University Press, 1958):
--snip--
> >    As Nanson says "The general rules for the case of any
> > number of candidates as given by Condorcet are stated
> > so briefly as to be hardly intelligible ... and as no
> > examples are given it is quite hopeless to find out what
> > Condorcet meant." The following three interpretations,
> > however, might be suggested.

You and I seem to agree on what Condorcet actually meant, based on the
complete context of his essays.  So, we both disagree with Nanson.

> >    (i) We might write down the list of propositions for
> > A in relation to each of the other candidates, those for
> > B, etc., as shown; and then deem the largest size of
> > minority to be a majority, then the next largest minority,
> > and so on, until all of the propositions relating to
> > one of the candidates become majorities (either true or
> > deemed) and elect him.

Interesting that Black is measure majority strength as inverse to the
size of the conflicting minority.  This is the opposite of Mike's
method of using the number of voters on the winning side.  I suspect
that Black wasn't really considering the issue of incomplete rankings
in this definition.  This method is equivalent to MinMax (ignoring the
majority strength issue).

> >    (ii) It would be more consistent with Condorcet's
> > words, though not with his lists of symbols, to take into
> > account all of the propositions derived from the voting
> > and disregard the proposition G > H, say, which has the
> > smallest majority in its favour, and the corresponding
> > H > G. Then if propositions of the type H > I, I > G
> > existed, we would deem H > G. Proceeding to delete the
> > smallest, next-smallest, ... majorities, we would select
> > that candidate who was the first to secure a majority or
> > deemed majority over each of the others.

I find this definition rather baffling.

> >    (iii) It would be most in accordance with the spirit
> > of Condorcet's previous analysis, I think, to discard
> > all candidates except those with the minimum number of
> > majorities against them and then to deem the largest
> > size of minority to be a majority, and so on, until one
> > candidate had only actual or deemed majorities against
> > each of the others.

This looks like Copeland//MinMax.  Black may be referring to the fact
that this method meets Condorcet loser, and Condorcet's analysis
suggests that he considers that necessary.

> >    Contrary to what Condorcet says, however, each of these
> > three interpretations would give rise to the choice of a
> > definite candidate (ties in the voting apart), so that
> > none of them may be what he had in mind. It is a pity
> > that on this crucial question his argument should be so
> > fragmentary.

As far as I can tell, Condorcet never gives the impression that his
procedure won't lead to a definite candidate.  Of course, it doesn't. 
But Condorcet never suggests that he is aware of this fact.

My criticism of Black's interpretations are as follows:

1.  He comes with Delphic expectations.  That is, he seems to expect
that Condorcet will be right, but cryptic.  But Condorcet describes
his procedure in different ways at least three times.  So, he had
plenty of opportunities to express himself.  Writing persuasively was
his life's work, and he is usually very good at clearly expressing his
ideas.  So, it would not be merely a "pity", but quite surprising if
Condorcet was indeed "fragmentary" on such a crucial point.  It is
much more likely that Condorcet and Black are using a different set of
assumptions.

2.  Black words each procedure in terms of finding a winning
candidate.  Condorcet always views finding a winner to be linked to
finding a system of propositions free of contradiction.  Of course,
Condorcet acknowledges that you can take a Condorcet winner off the
top, but this is just a short-cut.  Black suggests procedures that
find a winner, and give no indication how the propositions would be
made consistent.

3.  Black says, "The following three interpretations, however, might
be suggested."  It sounds to me like we shouldn't take these methods
too seriously as meticulous reconstructions of Condorcet's actual
thoughts.  Otherwise, Black wouldn't have three contradictory
interpretations.  Black is more concerned about suggesting plausible
methods.

It think it would be a big stretch to take one of Black's
interpretations and call this "Condorcet's method", or claim that
Condorcet actually proposed it, or even that Black claimed Condorcet
did.  The most one can say is that Black believed that Condorcet might
have meant one of these interpretations.

---
Blake Cretney