[EM] RE : Voting geometry, with interactive examples

[mrouse1 at mrouse.com]

I did some checking, and I found out that plotting candidates on the
triangle ABC *did* violate IIA. Worse yet, it violates the majority
criterion. For example, if you had 50.1% in the triangle ACB and 49.9% in
the triangle CBA, the center of mass would be somewhere in CAB (the center
of mass moves through that triangle as the votes shift from ACB to CBA).

I then decided it might be more consistent (and violate fewer criteria) to
take the surface of a cube rather than a triangle. If you define the
following cube faces for a three candidate election:

Top: C>A>B
Bottom: B>A>C
Left: A>C>B
Right: B>C>A
Front: A>B>C
Back: C>B>A

Weight each face by how many votes matches it. For things like
bullet-voting A, you'd put half of a vote in A>B>C and half in A>C>B, and
if you had an A=B>C vote, you'd put a half of a vote apiece in A>B>C and
B>A>C. Each option covers a pyramidal volume defined by the square face as
the base and the center of the cube as the apex. Once all of the weights
have been assigned, find the center of mass. The pyramid it lies in will
give the rank order.

The nice thing about this compared to the triangle version is that it
doesn't appear to violate the majority criterion. Now I just need to find
out what previously-defined voting method this corresponds to, and what
*other* voting properties it violates. :D

Michael Rouse
mrouse1 at mrouse.com