[EM] Condorcet Flavored PR Methods

[Forest Simmons]

On Mon, 30 Sep 2002 matt at tidalwave.net wrote in part:

> I doubt that CPO-STV consistently gives the winning outcome "better PR
> representation" than the losing outcomes.  It only transfers votes when at least one
> candidate is common to both outcomes.  Since it is the transfer that makes it PR
> and since the proportion of outcome pairs that have common candidates decreases
> as the number of candidates increases relative to the number of winners, it seems
> to me that CPO=STV is semi- or quasi-PR.

I use Matt's comment as a natural launching point for another installment
on Condorcet Flavored PR Methods:

CPO-STV considers all head-to-head contests between potential winning
circles, so to speak.  In a given Head-to-head contest your vote goes to
the circle that includes your favorite F among those that are not in the
intersection of the two circles (i.e. your favorite in the symmetric
difference of the two circles).

Tideman asks why would anybody want to transfer such a vote?  If it
transfers within the circle, then it makes no difference.  If it transfers
to the other circle, the one that does not include F, then you are
betraying your favorite.

In other words, Tideman assumes that you prefer the circle that includes
F, and since transferring your vote cannot help the circle that you
prefer, there is no need to transfer.

There are two problems with this.

One problem can be seen on the following ballot:

A1>B1>B2>B3>B4>A2>A3>A4

The two circles are {A1,A2,A3,A4} and {B1,B2,B3,B4} with no overlap.

Candidate A1 is the favorite F, yet the voter of this ballot probably
prefers the B circle to the A circle.

This problem can be remedied in a way that I will explain later.

The other problem is, as Matt notes, the method seems to be only
semi-proportional in design, although in simulations it gives results
similar to ordinary STV most of the time.  Of course, even multi-winner
plurality (which is strategically equivalent to cumulative voting) gives
PR results on average for randomly chosen elections.

It seems to me that this latter problem could be remedied by the method of
discounted layers of redundant representation, which is the approach we
used in PAV instead of vote transfer.

Temporarily concentrating on the second problem while ignoring the first
we apply the method of discounted layers:

To find the (A,B) and (B,A) entries in the pairwise matrix, first
find the voter's favorite F of all of the candidates in the symmetric
difference (A-B)union(B-A) as in Tideman's method.  Then let a and b,
respectively, be the number of candidates in A and B, that are ranked
equal to or above F on the ballot.

The (A,B) and (B,A) entries are respectively 1+1/2+1/3+...+1/a and
1+1/2+...+1/b. [If either a or b is zero, then the corresponding sum is
taken to be zero.] If the ballot fully ranks all of the candidates, then
the larger of a and b will be only one greater than the smaller of the two
values, so that the margin will be 1/max(a,b), which is just the D'Hondt
value 1/(1+b) in the case that F is in A.

This makes sense because if F is in A, then the greater the number of
members of B preferred to F by the voter, the less important it is to the
voter for circle A to win over circle B.

Now, an attempt to take care of both problems at the same time:

Instead of using F, the favorite of the symmetric difference
(A-B)union(B-A), as the "approval cutoff" in the preceding paragraphs, use
M, the median candidate of the symmetric difference.

In the above sample ballot, where A and B are disjoint, the symmetric
difference is the entire set of candidates (A)union(B), and the median
M is between B3 and B4, so A1,B1,B2,and B3 are the "approved" candidates,
and the (A,B) and (B,A) contributions from this ballot are respectively
1, and 1+1/2+1/3.  On this ballot circle B is preferred over circle A by a
margin of 1/2+1/3=5/6.

Various other Condorcet Flavored PR methods similar to these can be
obtained by "approval cutoff" choices other than F or M used above, and/or
by using some other sequence of discounted layers of representation
besides the harmonic sequence 1,1/2,1/3, ...

If a dyadic approval ballot is used, then the natural approval cutoff for
the pairwise comparison is the strongest inequality straddled by the
symmetric difference of the two candidate circles being compared.

If a cardinal ratings or grade ballot is used, then (for example) the
approval cutoff could be the mean, median, or midrange value for
candidates in the symmetric difference of the two candidate circles.

Another source of variation: instead of using the entire symmetric
difference one could use the maximum of the medians of (A-B) and (B-A) as
the cutoff, thus giving more importance to the upper part of the ballot.

In our sample ballot above, that would move the approval cutoff to the
B2/B3 boundary, so that circle B would beat circle A by a margin of only
(1+1/2)-1=1/2 instead of the value of 5/6 that we got with the previous
choice of cutoff.

It makes sense that B would win by less of a margin if the first place
values are given more importance, since in this example F is a member of
A.

Now you have a glimpse of the vast variety of Condorcet Flavored PR
Methods that can be engineered along these lines.

Which ones have the best properties remains to be seen, but they all share
the "global advantage" over STV that Tideman and Richardson point out;
sequential methods sometimes inflict early elimination on a key member of
the beats-all winning circle, someone who doesn't do so hot alone, but
naturally complements the other members of the beats-all circle much
better than any other candidate.

Forest


----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em