[EM] Preferential voting system where a candidate may win multiple seats

[Kristofer Munsterhjelm]

On 07/18/2013 08:13 PM, Vidar Wahlberg wrote:

> Thoughts are welcome, and sorry for the amount of mails, I'm having a
> lot of spare time at the moment.

Could you try implementing Balinski's primal-dual method? It's somewhat 
explained in the Wikipedia article on biproportional apportionment, 
https://en.wikipedia.org/wiki/Biproportional_apportionment . Olli Salmi 
(who used to post to this list) wrote a document about it at 
http://www.uusikaupunki.fi/~olsalmi/vaalit/Biproportional_Elections.html .

If we're going to compare different methods, I also think we'd need some 
way of measuring the quality or proportionality of the result. The 
Sainte-Laguë index can be used in an ordinal manner on both axes, so you 
could say "according to the SLI, the nationwide result produced by 
algorithm X is more proportional than the one produced by algorithm Y". 
However, unlike say, the Gallagher index, it's much harder to say *how 
much* more proportional X is than Y, since the index isn't bounded.

In 2011, the department of Mathematics at KTH held a workshop on 
electoral methods. One of the talks dealt with biproportional 
representation, and the slides can be found here: 
http://www.math.kth.se/wem/Zachariasen.pdf . They used the Gallagher 
index (which is optimized by quota-obeying methods) and determined that 
the Balinski method produced the most proportional results. Though the 
Gallagher index is optimized by quota methods, not divisor methods, it 
may still be of interest.

(And now that I think about it: if it's desired, it should be possible 
to make n-proportional apportionment methods for n>2 -- e.g. a method 
that tries to balance regional representation, national representation, 
and representation of minorities according to their share of the 
population. The greater n is, though, the less intuitive the results 
will be.)