[EM] Voting space graphs, varying sigma

[Raph Frank]

On Thu, Dec 4, 2008 at 1:18 AM,  <fsimmons at pcc.edu> wrote:
> Imagine starting with a non-normal distribution with a sharp, infinite central peak at the origin of a coordinate
> system, say with a probability density function given by rho = (1/r)/e^r, where r is the distance from the
> origin.

I wonder would using logarithmic values for sigma give the best 'full' coverage.

Sigma lower than a critical value will just give condorcet (or nearest
candidate).

Sigma greater than a critical value will give the extremist that is
closest to the centre.  In effect all centerists will be centre
squeezed out.

For the pathology graph, it might be possible to only test a few
points.  For example, the star like test could be tested by drawing a
few test lines through each candidate position and only computing
elections on those lines.  If enough test lines were used, it is
likely that a pathology would be detected.

All the other pathologies will result in a failure of the star-like
test.  The only exception is the non-convex one and I am not sure if
that is actually a problem.  Also, if a candidate has no win-region,
then it would pass the non-star like test.

This means that the using the star-like test would be a reasonable way
to compute the pathology diagram.