[Election-Methods] [EM] Visualizations of Plurality, Hare, Approval, and Condorcet

[Gervase Lam]

> Date: Fri, 22 Dec 2006 19:06:41 +0100 (CET)
> From: Kevin Venzke
> Subject: [EM] RE :   Comments on the Yee/Bolson/et.al. pictures

> --- raphfrk at netscape.net a ?crit?:
> >  Btw, how hard is it to calculate the voronoi diagram ?
> 
> I think it is pretty hard. I also doubt I can afford to calculate
> a Voronoi diagram of voters for every election, since the point of it
> is supposed to be to save time.
> 
> >  It might be interesting to have the simulation automatically overlay
> >  the voronoi diagram lines on the results (maybe have it as a switch). 

Apparently, it can be quite quick to calculate the Voronoi diagram.
Have a look at the following interactive example:

<http://www.diku.dk/hjemmesider/studerende/duff/Fortune/>

At the bottom of the web page are some links to other very good
interactive Voronoi diagram generators.

More information can be obtained from the Wikipedia entry I
unintentionally found:

<http://en.wikipedia.org/wiki/Voronoi_diagram>

On Thu, 2006-04-20 at 17:47 -0500, Paul Kislanko wrote [in personal e-
mail]:
> No, the problem is that representation is already "3d" mathematically,
> projected into 2 dimensions. The next step would be to go to 8 dimensions
> and try and show it in a 3d projection onto a 2-d screen. Nobody can figure
> those out except for the maths folks who think in n-dimensions about
> particle physics, and they can't display those, either.

Looking at the Wikipedia entry, I am a bit sceptical about the validity
of using 2D versions of these diagrams.  One of the diagrams shown
"...is a slice of the Voronoi diagram of a random set of points in a 3D
box. In general a cross section of a 3D Voronoi tesselation is not a 2D
Voronoi tesselation itself."

So, I don't think a 3D diagram can be projected in to 2D space!?  I
suppose this is obvious considering that it is possible for two 3D
points to be miles apart in distance, but when projected into 2D space
(e.g. by taking a photograph), the distance between them would be close.
Given this, the only answer would be to use a many dimensional space.

However, the Wikipedia entry mentions: "The Voronoi diagram of n points
in d-dimensional space requires O(n^(d/2)) storage space. Therefore,
Voronoi diagrams are often not feasible for d>2. An alternative is to
use approximate Voronoi diagrams, where the Voronoi cells have a fuzzy
boundary, which can be approximated. [Diagram shown of Voronoi cells
with fuzzy boundaries]."

So, how was the 2D slice of the 3D Voronoi diagram generated?  Or is
this due to me not understanding at all what "storage space" means?

Actually, may be my question should be what does "feasible" mean in this
context?  Does it mean that humans can't clearly visualise these
diagrams beyond 2D?  Does having fuzzy boundaries truly help considering
that 3D Voronoi diagrams could be radically different from the 2D
version?

May be it is better to forget about these Issue space diagrams and
instead concentrate on Candidate space or Voter space.

Thanks,
Gervase.