[EM] Primaries? (a PR Condorcet proposal)

[Andrew Myers]

> Dave Ketchum wrote:
> Not necessarily, sure, but I don't think that Condorcet is clearly the best 
> method to elect two candidates.  It seems likely that it would end up 
> picking two candidates from the center of a party, and nobody from a wing 
> (think Kerry and Edwards, in stead of Kerry and Dean).  But there have been 
> some stabs taken at Condorcet-flavored proportional representation.  The 
> best attempt is probably this one:
> 
> http://groups.yahoo.com/group/election-methods-list/message/10308
> 
> It's pretty complicated, but worth the read.  Try to sell that to the 
> public, though...
> 
> -Adam

Here is another idea for Condorcet-style proportional representation that I
have been considering. Feedback is welcome.

The rule used is a generalization of ordinary Condorcet voting. The basic idea
is to 'lift' preferences expressed on individual candidates to preferences on
whole committees of k members, then use the Condorcet criterion to pick the
'best' committee. The trick is to lift the preferences in a way that doesn't
simply let a majority dictate the composition of the committee.

We say that a given voter has an *f-preference* for possible committee A over
another, B, if voter prefers A to B when considering only the f most preferred
candidates on each committee. We can compare any two possible committees for
all values of f from 1 to k by determining for each voter and each value of f
whether they have an f-preference for one of the two committees over the other,
and summing up the number of voters with a f-preference for one and the number
with a f-preference for the other. Suppose that the number of voters with a
preference for committee 1 is v and there are n voters overall. Then an
f-preference for committee 1 is *valid* if

    ceiling(vk/n) >= f.

The key idea is to ignore invalid preferences; to decide whether committee 1 is
preferred over committee 2, we find the largest f such that v voters have a
valid f-preference for one of the two committees. This v becomes the strength
of the preference on committees in the sense used by Condorcet methods.

For k=1, this voting scheme is simply the ordinary Condorcet scheme because of
the ceiling operation. For larger k, it yields proportional results. Here is an
example: Suppose that 60% of the voters want candidates A>B>C>D>E,
and 40% want X>Y>Z>U>V. Ordinary Condorcet criteria will rank
the candidates A, B, C, D, and E higher than the other five candidates, because
a majority likes them better.  For some tasks, ABCDE is the right result.
However, if the election is selecting a five-person committee to represent the
whole population, 40% would have no representation at all. The proportional
representation algorithm above would elect the candidates ABCXY, achieving
perfect proportional representation.  The 60% have no valid preference for
ABCDE over ABCXY because their maximum valid preference is a 3-preference
(60% = 3/5). The 40%, on the other hand, have a valid 2-preference for the second
result, so ABCXY is the winning committee, as desired.

-- Andrew Myers