[EM] Kristofer Munsterhjelm suggests multiwinner voting method

[Warren Smith]

>> 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(which is out of date and needs to be improved/revised - Forest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.

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.
Also in stupider combinatoric methods where voter needs ot specify
that many bits of info
in her vote, it would be even worse.

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html