[EM] Yee/B.Olson Diagrams (YBD's): the next step

[fsimmons at pcc.edu]

One of the loose ends of Yee/B.Olson (YBD's)diagrams is the lack of completeness
of exposition
in even the three candidate case as exemplified by two natural questions that
the skeptical observer might pose:

1.  How do we know that the candidate configurations in the extant diagrams were
not chosen by people biased against IRV and in favor of Condorcet methods?

2.  What about the choice of standard deviation for the voter distributions?

A survey showing how the shape of the candidate triangle affects the diagrams
would answer the first question, and another survey that varies the standard
deviations would answer the second question. Appropriate graphical summaries of
the surveys and a good exposition of the whole would make an outstanding article
of Scientific American caliber.

First, a few words about question 2:  The standard deviation of the normal
distributions of voters has no effect whatsoever on the YBD of a Condorcet
method.  An infinitesimal standard deviation would yield exactly the same
diagram as would a standard deviation with an infinitesimal reciprocal.

But YBD's for IRV can depend heavily on the standard deviation of the voter
distribution.  As the standard deviation is made to approach zero, the IRV YBD
will approach the Condorcet YBD, no matter the configuration of the candidates.

Now a few words about question 1:  We need a way of representing in one graph
all of the possible shapes of the candidate triangle.

One way to do that is to fix the two endpoints of the side of largest length,
and let the third vertex vary enough to cover all of the possibilities of shape.
 Suppose we put the endpoints of the largest side at (0, 1) and (0, -1) on the
x-axis of a rectangular coordinate system, so that the largest side has length
2.  Then without loss in generality we can restrict the third vertex to the
first quadrant.  And since its distance to the vertex (0,-1) is no more than
two, we restrict further to the part of the first quadrant that is inside a
circle of radius 2 centered at (0,-1).

For graphical results, each point of that region is colored according to the
type and extend of pathology associated with a candidate configuration of that
shape.

Some of the pathologies are these, in order of increasing seriousness:
(1) a non-convex win region.
(2) a win region that is not star-like with respect to its candidate.
(3) a win region that is not path connected or one with a hole in it.
(4) a win region that doesn't contain its candidate.
(5) an empty win region.

Of course this shape based graph will have no regions of pathology when the
standard deviation is sufficiently small, but for large standard deviation we
expect more than half of the graph to represent one degree or another of pathology.

Yet another graph that relates the percentage of pathology in the shape graph as
a function of standard deviation would complete the picture.  In fact, there
could be several such graphs, one for each kind of pathology.

There is potential for a great Scientific American article here, not to mention
a project for a master's degree.

Forest