[EM] Advantages of Dirichlet Region Districts

[Forest Simmons]

On Fri, 16 Jan 2004, Forest Simmons wrote:

> The only vote would be to decide on which objective measure of over-all
> compactness to adopt.
>
> If anybody is thinking about a boundary based measure, please tell me how
> to find the geographical boundary of the partition suggested by Stephane
> (based on birthdays) or the one that divides the state into a hundred
> districts according to the last two digits of your social security number.
>
> The average-within-district-distance measure that I suggest could handle
> these two cases as easily as any other partition of the voters, without
> ever having to worry about boundaries.


To shed a little more light on this:

The planar shape with minimum boundary length (for a given area) is the
circle.

The planar shape with the minimum average distance between points (for a
given area) is the circle.

Both measures agree in this benchmark example, so why not use the more
generally applicable measure, i.e. the second one, which will work for any
partition in any (totally bounded) metric space of any dimension.

Furthermore, in our application we are seeking regions that represent
communities.  The average distance between voters in the same district is
a very direct way of measuring that objective, while boundary length is
only an indirect way of getting at it.

[Imagine a planet with all of the population on its equator in two small
antipodally related communities. Any pair of antipodal points would divide
the equator into two equal population districts. So boundary size doesn't
really address the issue. Consideration of boundary size alone would as
likely as not split both of the communities in half.]

Furthermore, it makes direct sense to average the distances between voters
along roads, but requiring roads for boundaries doesn't make as much
sense.  Better to have rivers, mountain ridges, and other sparsely
populated natural barriers as boundaries.  This is what my measure will
tend to do, but not what the boundaries-as-roads approach would do.

I hope that nobody takes offense at these remarks, but it seems to me that
the within district average voter distance measure is uniformly superior
to any boundary based measure, and that if we want to make further
progress we should concentrate on refining the average distance approach
and forget about the boundary based approaches.

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.


Forest





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