[EM] Statistical analysis of Voter Models versus real life voting

[Kristofer Munsterhjelm]

Leon Smith wrote:
> On Fri, Jan 28, 2011 at 3:08 PM, Kristofer Munsterhjelm
> <km-elmet at broadpark.no> wrote:
>> One could generalize Yee diagrams to other distances than Euclidean, but
>> AFAIK, there's a theorem that says that with any centrosymmetric
>> distribution, the Yee diagram for a Condorcet method is the L2 Voronoi
>> diagram. Warren used this to argue that Range is better than Condorcet
>> because it would make more sense for voters with L1 (Manhattan distance)
>> utility functions to yield L1 win regions (which Range does) and not L2
>> (Euclidean) win regions, as Condorcet methods do.
> 
> Interesting,  but wouldn't you need slightly more stringent conditions
> than "merely" a centrosymmetric voter distribution?    For example,
> consider  four identical gaussian distributions added together,  with
> the peaks placed at four corners of a square.   Then place four
> candidates,  one at each peak,  and rotate the candidates around the
> center of the square by 20 degrees or so.   Now you have a
> centrosymmetric voter distribution and a condorcet paradox.    If your
> condorcet method resorts to IRV to resolve the ambiguity,  for
> example,  you certainly won't get a Voroni diagram.   (And I presume
> some of the other Condorcet methods would exibit the same behavior.)

I recalled it incorrectly. The actual version is that if you draw a Yee 
diagram with voters clustered centrosymmetrically around each pixel, 
whose utility is a function of Lp distance (for p = some L-norm, p = 1 
Manhattan, p = 2 Euclidean, etc) between the candidate and voter in 
question, then a Condorcet method renders the Lp Voronoi diagram, but 
when p is not equal to 2, the social optimal method may not be the Lp 
Voronoi diagram.

See the bottom of http://rangevoting.org/BlackSingle.html for an example.