[EM] Yee/B.Olson Diagram Remarks

[fsimmons at pcc.edu]

Property (2) below is not always apparent in existing YBD's, because when sigma is small the notch 
that makes A's win region non-starlike will also be small and will be within sigma of the center of the 
circle that circumscribes the candidate triangle.  This center is (as often as not) outside the viewing 
window of the YBD when the triangle is obtuse.

To understand why this is so, contemplate the basic reason for the non-starlike effect:  

For each sigma, there is a point P in candidate C's win region for which there is a first place preference 
tie when the distribution of voters centered at P has standard deviation sigma.  If sigma is small P will be 
near the center of the circumscribed circle.   

When the center of the distribution is moved directly towards the long side of the triangle the first place 
preferences are in order of A>B>C, and A wins the runoff between A and B. 

But when the center moves directly from P towards any point A' sufficiently near A, the first place 
preferences change to the order order A>C>B, and C wins the runoff between A and C.  

This effect creates the notch that makes A's win region non-starlike.

> Here's the latest update on my investigation of "squeeze out" and "non-starlike" effects in Yee/B.Olson 
diagrams (YBD's) of IRV.

> I'm still concentrating on the three candidate case,

> If the triangle of candidates is scalene, then ...

> (1) for all sufficiently large values of sigma (the standard deviation of the voter distributions) candidate 
C (the one opposite the longest side of the triangle) will be excluded from her own win region.  The bigger 
sigma, the further outside her win region.  As sigma gets larger without bound the distance from C to the 
win region grows without bound.

> (2) for all sufficiently small values of sigma, the win region for candidate A (the candidate opposite the 
smallest side of the triangle) is not starlike relative to A.

> I am working on mapping the size of the sigma gap between these two kinds of pathologies.

Included in this investigation is a precise mapping of the set of triangles for which the two pathologies 
overlap, i.e. for which there exist values of sigma that are simultaneously large enough for (1) and small 
enough for (2).

Soon I will give details (pseudo code) of a procedure for determining the greatest lower bound for the 
values of sigma in (1) and the least upper bound for the values of sigma in (2).given any set of sides, 
vertices, and/or angles that determine the candidate triangle.

Forest