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

The same result can be proved as in the 2d Yee diagram ... as long as the distribution of voters is centrally symmetric, of any two candidates, rhe one closer to the center of symmetry will defeat the other one pairwise [or possibly tie if there are big gaps in the support of the voter distribution]. On Tue, Nov 29, 2022, 12:27 PM Kristofer Munsterhjelm <km_elmet@t-online.de> wrote: > 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 > ---- > Election-Methods mailing list - see https://electorama.com/em for list > info >