[EM] Kristofer Munsterhjelm suggests multiwinner voting method

[Kristofer Munsterhjelm]

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 (which is out of date and needs to be improved/revised - Forest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