[EM] Dopp Population Density Fairness measure: I don't like it & here's why.

[Warren Smith]

Dopp's pdf file has vanished (?); the URL she gave
http://ssrn.com/abstract=1947297
apparently now gives me only the (revised) abstract, not the full paper
anymore.

Anyhow, let me concisely summarize her proposed
Population Density Fairness measure.
For a country to be subdivided into N equipopulous districts,
Dopp's measure (up to scaling factors which for any fixed country
at any fixed time do not matter so I removed them) is

DoppMeasure = [SUM(over k=1..N)OF  (1/Area_k - Q)^2 / Area_k]^(1/2)

where Q=N/SUM(Area_k)  does not depend on the subdivision
and  Area_k is the area of the kth district.
I got this from page 20 of her old draft dated 10/20/11. The goal is
to minimize it.

We can simplify by removing the final square-rooting without changing
the measure's
relative opinion about any two districting plans:

SimplifiedDoppMeasure = SUM(over k=1..N)OF  (1/Area_k - Q)^2 / Area_k

Now since
(1/Area_k - Q)^2 = (Area_k)^(-2) - 2*Q/Area_k + Q^2
we can rewrite this as

SimplifiedDoppMeasure = SUM(over k=1..N)OF  [ (Area_k)^(-3) -
2*Q*(Area_k)^(-2) + Q*Q*(Area_k)^(-1) ]

Anyhow, however you do it, I DON'T LIKE this measure.  Here's why.

Because, this measure depends ONLY on the district areas.
It does NOT depend on their perimeters, or their shapes, at all.

In other words: suppose Dopp constructs some nice districting.
Then ANY subdivision I construct having the same district areas as Dopp's
(and also equipopulous) -- no matter how many insane wiggles and evil tentacles
I add to all the districts to gerrymander them -- will have the same
DoppMeasure.   So this measure in no way discourages
gerrymandering, and it fails to have a unique optimum  (the "optimum"
districting according to it is extremely infinitely non-unique).

For example, say the country is a rectangle with uniform population
density, and N=2.
Then I'd say the best districting looks like this:

AAAAABBBBB
AAAAABBBBB
AAAAABBBBB
AAAAABBBBB

but if I gerrymandered it to be this:

ABBBBBBBBB
ABAAAAAAAB
AAAAABBBAB
AAAAABBBBB

then exact same DoppMeasure.

Also, even aside from this, I just do not agree with the
DoppMeasure-minimization goal
of causing all districts to have equal areas.
Note: if all districts have equal areas (and equal populations),
then DoppMeasure=0. Otherwise (not all areas equal) DoppMeasure>0.

I think urban districts really should
have smaller areas than rural districts.  DoppMeasure minimization would
abolish urban districts and cause every district to be a mix of urban and
rural in order to make all districts have the same area.

So, sorry.  I think this idea is a failure.   I had earlier got the
impression Dopp wanted
to use isoperimetric quotients as the basis for a districting-plan
quality measure.
I like that idea, though the best way to do it is not clear to me.
But the isoperimetric idea does not utterly abandon the use of perimeters.
DoppMeasure does abandon them.  That's a mistake.

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
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