Warren Smith wrote:
It's an interesting question whether there can be a proportional
multiwinner voting method
without needing to use "reweighting"... but this is not it. "Asset
voting" works
http://rangevoting.org/Asset.html
but it is "unconventional."
--Actually, now that Kris.Munk. points out "combinatoric" methods, I recall that
"penalty function" methods with cleverly chosen functions can indeed work.
You consider all binomial(C,W) possible winner-sets, where C=#candidates,
W=#winners, 0<W<C, and for each you evaluate a function. The set with the
greatest (or least, depending on your defn) value of the function wins.
The function has to be defined very carefully, but it is possible to
do so in such a way that you can prove a proportionality theorem.
(If you just try any old definition, it will almost certainly fail to
yield proportionality.) If you look at my paper #91 here
http://www.math.temple.edu/~wds/homepage/works.html
(which is out of date and needs to be improved/revised - Forest
Simmons and I were planning on doing that... maybe there should be
other authors too like KM himself perhaps...) then you will see the
LPV method (logarithmic penalty voting) basically invented
by Forest Simmons, does the job. See section 7.9.
More generally, one might consider a method to be "combinatoric"
(combinatorial?) if it treats councils like pseudo-candidates in a
single-winner election, where each council is given a score according to
some function.
Then LPV is "Range" (simple optimization from cardinal ballots). CPO-STV
and Schulze STV are pseudo-candidate methods based on Condorcet. If the
council function is simple identity for a council of size one, the
method naturally reduces to the base method in the single-winner case
(like CPO-STV reduces to the base Condorcet method and Schulze STV
reduces to Schulze).
LPV is a beautiful idea, but its "representativity" property actually
is probably not a good thing (we have some results indicating it
conflicts with other properties) and
also problem with this and every other "combinatoric" method (at least
if implemented by brute force) is binomial(C,W) can be huge, causing
enormous work by the election
authority. It may be that "branch and bound" methods can cut this
work down to acceptable levels, but that is not yet confirmed
experimentally.
If there are either no local optima but the global optimum, or the
method obeys house monotonicity, one can get around the brute force
problems mentioned. For the former case, a simple hill climbing
algorithm would work; for the latter, one could start by finding the
single-winner candidate, then generate n-1 sets of that candidate + some
other, determine the winning set/council, then generate n-2 sets of that
candidate + some other, etc.
Also in stupider combinatoric methods where voter needs ot specify
that many bits of info
in her vote, it would be even worse.
That would be out of the question, IMHO. Voters wouldn't know how to
rank the councils, and even if they did, asking them to rank all the
possible councils would be insane.
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