Re: RFD: informal proposal

[Andrew Pimlott <andrew@pimlott.net>]

On Sun, Nov 17, 2002 at 12:01:28PM -0500, Raul Miller wrote:
> This is an informal request for discussion about how to handle quorum
> and supermajority requirements.

I just got a chance to catch up on the discussion.  I will give a
suggested solution, but first I would like to make a point that I
don't think has been stated directly (though it is at the heart of
Clinton's examples):

Any method in which an option can be eliminated "early"--ie, without
a fully head-to-head with all the other options--has the same
fundamental flaw as instant runoff, and should be rejected for the
same reason.

So for example, the clause, in most drafts, that first eliminated
options that were defeated by the default option, was a direct
invitation to insincere strategic voting.  It would encourage voters
to put the default option second, in an attempt to knock out the
other candidates early.  Exactly what we're trying to avoid with the
Condorcet method.

This clause is gone in Raul's most recent draft, though I think its
treatment of the default option is still flawed (as long as one
option defeats the default, any option with a supermajority
requirement is effectively relieved of that requirement).  I suspect
that any method with a special elimination rule involving the
default option is broken.

My understanding is that the spirit of the quorum and supermajority
requirements is that the winner should have the appropriate margin
over the default, not over the other candidates.  Given this, here
is my attempt:

1.  Create the matrix of pair-wise counts. P(A, B) is the count of
    voters prefering option A to option B.

2.  If there is a default option Z,

    a.  If there is a quorum Q, for all other options A add Q to
        P(Z, A).  So if the quorum is 10, a preference of 20 to 10
        for A over Z "acts like" 20 to 20.

    b.  For every option A that requires a supermajority x (a
        fraction between 1/2 and 1), multiply P(Z, A) by x/(1-x).
        So if A requires a 2/3 supermajority, a preference of 200 to
        100 for A over Z "acts like" 200 to 200.

    (Er, ignore the case of both a quorum and a supermajority
    requirement for the moment.)

    I increase P(Z, A) instead of decreasing P(A, Z) because a
    preference for the default ought to be strong in CSSD.

3.  Do a "textbook" Condorcet/CSSD algorithm on the modified
    counts.

Yes, you will find some odd outcomes, but I don't see anything too
outlandish.  For example, in a straight election with no
supermajority requirement, the winner might not be prefered to the
default.  I think you just have to accept this as a corner case
(which is definitely is--remember, this sort of thing doesn't often
happen in practice).

Most importantly, to the extent that Condorcet/CSSD is resistent to
strategy, this algorithm should be as well, since any strategy
against this would just be a scaled version of a strategy against
plain Condorcet/CSSD.

Well, I haven't thought this all the way through, but it has a nice
feel.

Andrew