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, [EMAIL PROTECTED] 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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