[EM] multiwinner election methods

[Kristofer Munsterhjelm]

Warren Smith wrote:
> 1. the right way to compare election methods is "Bayesian Regret"
> (BR). http://rangevoting.org/BayRegDum.html
> 
> For a long time I thought this was only applicable for single-winner 
> voting methods. However, I eventually saw how to do it for
> multiwinner methods also: 
> http://groups.yahoo.com/group/RangeVoting/message/7706
> 
> it would be a substantial computer programming project to try to do 
> this, and so far, nobody has undertaken that project. But I recommend
> it!!  If Gregory Nesbit is looking for a project to undertake for,
> e.g. Intel Science Talent Search, he could do it :)
> 
> In the absence of BR, one is reduced to comparing voting methods
> using "properties." I also recommend that, but for multiwinner voting
> methods this too is in its infancy. A paper attempting to compare
> multiwinner voting methods (using properties) by me is here 
> http://www.math.temple.edu/~wds/homepage/works.html    paper #91. 
> However this paper is out of date and not fully satisfactory...
> 
> 2. About RRV (reweighted range voting) 
> http://rangevoting.org/RRV.html recent developments are these: Steven
> J. Brams found an example (in email to me) in which RRV violates
> "favorite betrayal." That is, there are elections in which foolishly
> voting your true favorite top, causes you to get a worse election
> result.
> 
> Warren Schudy found a beautiful theorem that EVERY multiwinner 
> election method in which the ballots are approval-style or
> range-style, must either 1. fail to be proportional 2. fail to be
> "invariant to reinforcement" (IR).
> 
> IR means that if a ballot is altered to increase score for X, that 
> should not stop X winning; similarly if decrease score for X, that
> should not stop X losing.

Does that include multiwinner methods that are based on ranked ballots?
I'm not sure, because on one hand, you only specify approval and range
style, but on the other, any range ballot can be reduced to a ranked
ballot with some additional information (how much better a certain
choice is to another), and so one could construct a rated pseudo-ballot
by running a ranked ballot through a weighted positional system.

I guess the theorem applies neither to asset (unless candidates pledge 
to transfer support in a certain way) nor to closed list PR, since both 
use single-vote type ballots.

> 4. systems based on every subset of the candidates is a 
> "pseudocandidate" are just nonstarters because there are far too many
> pseudocandidates.

Some of these might work if they have a single local optimum, so that a 
hill-climbing algorithm can find that optimum. But then it could be 
restated by including the hill-climbing algorithm into the method, and 
so might be exempt...