[EM] High precision Yee diagrams

[Kristofer Munsterhjelm]

Raph Frank wrote:
> I implemented one that can handle PR methods, but that didn't use randomness.
> 
> It also uses java to let the user move the voter centre around.  Two
> circles are use.  The inner one contains 50% of the voters (if I
> remember correctly).
> 
> http://raphfrk.com/ping_yee/results.html
> 
> Under the system, if asked for 100 voters, it would create a 10x10
> rectangular grid of voters and then transform them using a rectangular
> grid to guassian mapping function.

That's interesting, though it would by necessity be a truncated Gaussian.

Another way to make the mean exactly the center would be to mirror every 
voter. Draw a random voter from a Gaussian centered on (0,0), and if you 
get the position (x,y), also admit (-x, -y). For that matter, since the 
Gaussian is separable, you could add (x, y), (x, -y), (-x, y), and (-x, -y).

If the Gaussian isn't centered on (0,0), do the required translation 
before mirroring.

-

As for PR (although it's a bit off topic), you could show the entire PR 
result in a single image by overlapping the win regions for the 
candidates. Consider basing color on HSV. Hue wraps around, so you 
simply assign a certain hue value to every candidate, then the pixel 
value is the sum of the hues for the candidates that won. This leaves 
saturation and value to be either fixed or used for some other 
parameter, such as the degree to which the method is indecisive 
(requires a tiebreak) at the point in question.

Assigning random hues could lead to the problem where two adjacent 
regions have nearly the same compound hue. if A and B win in one region, 
and C and D win in another, it could be the case that the hue of (A + B) 
is very close to that of (C + D). To counter this, you could try to find 
an optimal set of hues.
Say the PR election is for two seats. Then set the hue values in a 
minmax fashion so that the minimal distance between one sum of two hues 
and another sum of two hues is maximized. Since the number of candidates 
n, and number of seats, k, have to be small in order for the maximal 
minimal distance to be large enough, it should be feasible to do this by 
linear programming. Unless I'm mistaken, the number of constraints would 
be proportional to n choose k. Another option would be to use brute 
force (keep a record of the best hue assignments), but that would be slower.