# [EM] Advantages of Dirichlet Region Districts

Forest Simmons fsimmons at pcc.edu
Sat Jan 17 11:55:04 PST 2004

```On Fri, 16 Jan 2004, Forest Simmons wrote:

>
> I'm not saying that you cannot specify a partition by drawing the
> boundaries.  I'm just saying that when comparing two proposals it whould
> be according to average within district voter distances according to some
> (perhaps further refined) metric.

In other words, let's use an intrinsic (as opposed to extrinsic) measure
of compactness.

Two distributions of voters with markedly different standard deviations
could have precisely congruent boundaries.  Since boundaries are extrinsic
they cannot detect the difference.

Here's another problem with extrinsic measures: two sets disjoint sets of
points can be separated by many different choices of boundaries. Consider
a planet with half the population above the arctic circle, and the other
half situated antipodally, i.e. the population is distributed
symmetrically with respect to the center of the planet, and nobody lives
anywhere near the equator.

Then any great circle will serve as a boundary to divide the population
precisely in half.  All great circles have the same length.  However, of
these the equator itself seems like the most natural boundary for
separating the population into two compact communities. Nevertheless, the
arctic circle (or the antarctic circle) would be better than any great
circle for measuring the compactness of one of the communities.

Thus we see with extrinsic measures, the compactness assessment depends on
choice of boundary.  A good partition could beat a better partition only
because the better partition wasn't specified by the optimal boundary.

Imagine three points A, B, and C, located at (0,0), (0,1), and (9,0),
respectively.

One proposed boundary is an ellipse enclosing points A and B, thus
excluding C.  The other proposed boundary is a small circle around A, thus
excluding B and C.

Of the two proposed partitions, the one with the largest boundary does a
better job at identifying the natural "communities."

One may object to these examples by saying that they are highly unlikely,
but they are offered in the spirit of the general principle which says
that the exceptional cases are the ones that prove the rule, i.e. all else
being equal, the rule with the better asymptotic behavior should be given
preference.

The third order Taylor polynomial for the arctan function is

x - (x^3)/3 ,

which has lousy asymptotic behavior, since it has no horizontal
asymptotes.

A better over-all approximation would be

x/(1 + 2*abs(x/Pi)),

which has the correct slope at the origin, as well as the correct
asymptotic behavior.

It is useful to keep this "exception proves the rule" principle in mind
when comparing election methods.  Often, IRV advocates will object to our
examples of its failures by saying that these contrived examples will
never happen in practice.  But the "contrived examples" are merely testing
the asymptotic behavior of the method, so to speak.  A method that fails
totally in an extreme case will do a lousy job near the extreme.

In the context of redistricting, boundary measures fail totally in the
case of random partitions, yet intrinsic measures are not phased by that
extreme.  Thus we see that intrinsic measures are inherently more
expedient in this context.

Forest

>
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