[EM] IRV's Starlike Noncompliance

fsimmons at pcc.edu fsimmons at pcc.edu
Sun Dec 7 16:21:54 PST 2008

Recently we have seen how easy it is to construct examples of IRV's "Squeeze Effect" which results in 
candidates being isolated from their own win regions in Yee/B.Olson Diagrams.

In the case of three candidates, the squeezed out candidate is always the one opposite the longest side 
of the candidate triangle.

It also turns out to be fairly easy to construct Yee/B.Olson Diagrams for IRV in which one or more of the 
end points of the longest side has a win region that is not starlike relative to that candidate.  This means 
that moving towards the candidate on some line you will move out of the win region for that candidate 
and then back into it.

For simplicity I will show how to construct an example in the case of an obtuse isosceles triangle:

Orient the triangle so that the horizontal base is the long side, i.e. the side opposite the obtuse angle 
vertex V.

Find the center C of the circle determined by the three candidates.

Let P be a point strictly between this center C and the midpoint of the base.

Adjust the standard deviation of the voter distribution so that when it is centered at P, each of the three 
candidates gets one third of the first place votes.  Two facts make this possible: (1) the fact that P is 
interior to the Dirichlet Region for the top vertex V, and (2) the fact that the other two Dirichlet regions 
have central angles greater than 120 degrees.

Then with this standard deviation the Yee/B.Olson Diagram (for IRV) will give non-starlike win regions to 
both of the end points of the base.

Sorry to be so terse, but whoever will carry out this construction will see why it works.  The region below 
P will be split between the two endpoint win regions (right down the middle), but a horizontal line through 
P will follow the win region for the top vertex V to the radii of the circle connecting the endpoints to the 
center C.

Bye for Now

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