[EM] Voting space graphs, varying sigma

[fsimmons at pcc.edu]

I like these ideas for varying sigma and keeping down the computational cost of detecting failure of starlike 
win regions.

Let sigma1 and sigma2, respectively, be the distance between the two closest and the distance between 
the two most widely separated candidates.  Then a range of values between half sigma1 and 50 times 
sigma2, done logrithmically as per Raph's suggestion, should give the full picture.


----- Original Message -----
From: Raph Frank 
> 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.
>