[EM] Summable opinion space discovery.

[Dan Bishop]

Kristofer Munsterhjelm wrote:
> While trying to find a solution to another problem, I discovered 
> something that might be used to opinion space from ballots (and the 
> candidates' position in that space) in a summable manner.
>
> Consider a rating- or approval matrix m, where m{voter_1, A} is 
> voter_1's rating or approval (0 or 1) of candidate A. Say there are v 
> voters and c candidates. Each candidate can now be assigned a 
> v-dimensional point according to the ratings by every voter, and we 
> can calculate ballot similarity between two candidates by determining 
> the distance (according to some metric) between the two candidates' 
> assigned points.
>
> "m" itself is obviously not summable, because it depends on the number 
> of voters v. However, we might build a distance matrix q where q{A, B} 
> is the distance between A and B according to the metric in question. 
> As long as the metric itself is summable, q will be.
http://wiki.electorama.com/wiki/Candidate_correlation
>
> ...
>
> However, if these problems can be solved, this summable opinion space 
> concept could possibly be used to make a summable method that is PR. 
> It would probably not be Droop proportional, but if it directly 
> detects opinion space and then allocates candidates to reproduce it, 
> it would at least be proportional.
>
> Perhaps there is a more direct way of building opinion space (by using 
> SVD or PCA), but I never got that to work. In any case, this is a 
> rough idea, so don't pick too much on the details :-)
>
I've considered an approach like that before. The problem is that 
knowing a voter chose C>>B>A doesn't tell you whether the spectrum is 
A------------C-v----------B or A-B----------C------------v, so it's 
rather difficult to build the opinion space.