[EM] Half-exact spatial models

[Kristofer Munsterhjelm]

I was reading about JGA's spatial model when it occurred to me that the 
Voronoi trick we talked about a while ago to get near-exact ballots for 
Yee diagrams could also be used for spatial model calculations (e.g. 
strategic susceptibility).

In JGA's model, both candidates and voters are drawn iid from a 
d-dimensional symmetric unit normal distribution. So the process would 
go like this:

First, choose a number of candidates' positions from a d-dimensional 
symmetric unit normal distribution. This is the inexact part.

Then, as with the Yee map, the candidates' positions divides the space 
R^d into a number of sectors (convex polytopes) enclosing the volume 
where, if a voter is located there, that voter would rank the candidates 
in a particular way.

Then in theory, to get the exact ranked ballots for this particular 
assignment of candidates, take the integral of the d-dimensional unit 
normal over each polytope. The integral then determines what fraction of 
the infinite number of voters who would've voted according to the 
ranking that polytope represents.

In practice, it's not that easy because the value would be irrational 
and (if I recall correctly) there's no general closed form expression 
for d>2. For small d, numerical integration could work, but if d gets 
large, perhaps you have to resort to Monte-Carlo anyway, in which case 
there's no need to go through the whole Voronoi business.

Also, the process wouldn't reduce Monte-Carlo to a fully exact process; 
it would reduce MC over candidates and voters into just MC over 
candidates (since the candidate positions still have to be chosen 
randomly). I don't expect there's anything remotely close to a neat 
expression for the integral over all candidate positions of the exact 
result below - in particular, I don't see any way to integrate over the 
space of possible Voronoi polytopes.

-km