[EM] Exact spatial model probabilities?

[Kristofer Munsterhjelm]

My minimum-manipulability optimization system can find exact
probabilities that a voter will vote according to a particular ranking,
under impartial culture.

It's easy to approximately find such probabilities for spatial models as
well by just creating an instance and randomly placing voters on it to
determine what they vote, and hence in the limit of number of trials
going to infinity, get the probability.

But I was wondering, is it possible to do so exactly? Let's say that in
a spatial model, first c candidates are placed uniformly at random in an
unit d-cube. Then voters are also placed uniformly at random in this
cube and they prefer candidates closer to them to candidates further away.

Now I would suppose that the probability that a voter votes A first is
equal to the volume of the Voronoi cell that belongs to A. (And the
probability that a voter will vote A>B>C is the volume of the
intersection of the closest-neighbor Voronoi cell for A with the
second-closest-neighbor for B and the third-closest-neighbor for C.)

Well, that eliminates one source of randomness -- assuming I can exactly
calculate the vertices of these regions. But there's still the second
source in the randomness of the candidates. Do you know of any calculus
tricks to get the probabilities over every possible candidate position,
or is this simply too hairy?

One benefit of the spatial model over impartial culture is that it's
much more clear how one may assign utilities to each ranking, so the
optimization system could also be used to investigate tradeoffs between
VSE and manipulability. (Durand suggested sampling on a sphere for
associating utilities with impartial culture, but I haven't read the
suggestion in detail. https://arxiv.org/pdf/1511.01303.pdf)

-km