[EM] [CES #3935] Dopp Population Density Fairness measure: I don't like it & here's why.

[Jameson Quinn]

2011/10/21 Warren Smith <warren.wds at gmail.com>

> 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.


I believe that the goal was to have it (including scaling factors which you
removed) within some predefined distance of the being 1, while also meeting
some compactness criterion.

>


> 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.
>

Yes, which is why she advocates using her measure AND a compactness measure.
She wasn't clear about how to combine the two, I'll admit; but she at least
thought of your issue.


>
> 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.
>


Again, the goal is 1, not 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.
>

See above.


>
> 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.
>

She does. Two measures, unclear how to combine them, but at least she's
clear that a good districting would be somewhere on the pareto front of
those two measures.


> 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.
>

You've misunderstood her in two significant ways.

I read her charitably and didn't actually look for worst cases of her
measure hitting 1 for a bad (nonproportional; compactness is irrelevant
here) districting. So I would be interested to hear your critical opinion.
But only once you've understood.

Jameson

>
> --
> Warren D. Smith
> http://RangeVoting.org  <-- add your endorsement (by clicking
> "endorse" as 1st step)
>
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