[EM] Yee/B.Olson Diagrams (YBD's): the next step

[Brian Olson]

Disorganized thoughts:

Standard deviation could be considered to be interchangeable with the  
spacing between the choices.
Wider spacing is equivalent to tighter standard deviation.

I keep imagining a way to explain this as starting with a blank black  
space and colored dots representing the choices. Then visualize the  
population as a grey cloud of various intensity hovering over the  
choice space. The animation moves the population center to near a  
choice, fills in a pixel accordingly, jumps to near another choice,  
fills another pixel, and after a couple repetitions of jump and pixel  
fill the cloud jumps to the top left of the range and begins an  
accelerating scan over the space filling in a final image.

I like the idea of scanning over a quadrant to fully show what result  
images are possible and how they change. In the end a small-multiple  
layout would probably make sense. Printed it might be an 8x8 array of  
1" images.
Hmm, I could bash this out in perl this afternoon. Most of my spare  
compute cycles are running on my redistricting project right now, but  
I could probably make a rough draft with tiny 100x100 graphs and run  
that pretty quick.

I expect that showing the effect of standard deviation on IRV should  
be showable by a linear series varying standard deviation on an image  
that in some default case shows an interesting failure mode. I'm  
guessing that the size and shape of the defects will scale in a pretty  
regular way based on standard deviation.

On Dec 1, 2008, at 7:17 PM, fsimmons at pcc.edu wrote:

> One of the loose ends of Yee/B.Olson (YBD's)diagrams is the lack of  
> completeness
> of exposition
> in even the three candidate case as exemplified by two natural  
> questions that
> the skeptical observer might pose:
>
> 1.  How do we know that the candidate configurations in the extant  
> diagrams were
> not chosen by people biased against IRV and in favor of Condorcet  
> methods?
>
> 2.  What about the choice of standard deviation for the voter  
> distributions?
>
> A survey showing how the shape of the candidate triangle affects the  
> diagrams
> would answer the first question, and another survey that varies the  
> standard
> deviations would answer the second question. Appropriate graphical  
> summaries of
> the surveys and a good exposition of the whole would make an  
> outstanding article
> of Scientific American caliber.
>
> First, a few words about question 2:  The standard deviation of the  
> normal
> distributions of voters has no effect whatsoever on the YBD of a  
> Condorcet
> method.  An infinitesimal standard deviation would yield exactly the  
> same
> diagram as would a standard deviation with an infinitesimal  
> reciprocal.
>
> But YBD's for IRV can depend heavily on the standard deviation of  
> the voter
> distribution.  As the standard deviation is made to approach zero,  
> the IRV YBD
> will approach the Condorcet YBD, no matter the configuration of the  
> candidates.
>
> Now a few words about question 1:  We need a way of representing in  
> one graph
> all of the possible shapes of the candidate triangle.
>
> One way to do that is to fix the two endpoints of the side of  
> largest length,
> and let the third vertex vary enough to cover all of the  
> possibilities of shape.
> Suppose we put the endpoints of the largest side at (0, 1) and (0,  
> -1) on the
> x-axis of a rectangular coordinate system, so that the largest side  
> has length
> 2.  Then without loss in generality we can restrict the third vertex  
> to the
> first quadrant.  And since its distance to the vertex (0,-1) is no  
> more than
> two, we restrict further to the part of the first quadrant that is  
> inside a
> circle of radius 2 centered at (0,-1).
>
> For graphical results, each point of that region is colored  
> according to the
> type and extend of pathology associated with a candidate  
> configuration of that
> shape.
>
> Some of the pathologies are these, in order of increasing seriousness:
> (1) a non-convex win region.
> (2) a win region that is not star-like with respect to its candidate.
> (3) a win region that is not path connected or one with a hole in it.
> (4) a win region that doesn't contain its candidate.
> (5) an empty win region.
>
> Of course this shape based graph will have no regions of pathology  
> when the
> standard deviation is sufficiently small, but for large standard  
> deviation we
> expect more than half of the graph to represent one degree or  
> another of pathology.
>
> Yet another graph that relates the percentage of pathology in the  
> shape graph as
> a function of standard deviation would complete the picture.  In  
> fact, there
> could be several such graphs, one for each kind of pathology.
>
> There is potential for a great Scientific American article here, not  
> to mention
> a project for a master's degree.
>
> Forest
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