<br><br><div class="gmail_quote">2011/10/21 Warren Smith <span dir="ltr"><<a href="mailto:warren.wds@gmail.com">warren.wds@gmail.com</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
Dopp's pdf file has vanished (?); the URL she gave<br>
<a href="http://ssrn.com/abstract=1947297" target="_blank">http://ssrn.com/abstract=1947297</a><br>
apparently now gives me only the (revised) abstract, not the full paper<br>
anymore.<br>
<br>
Anyhow, let me concisely summarize her proposed<br>
Population Density Fairness measure.<br>
For a country to be subdivided into N equipopulous districts,<br>
Dopp's measure (up to scaling factors which for any fixed country<br>
at any fixed time do not matter so I removed them) is<br>
<br>
DoppMeasure = [SUM(over k=1..N)OF (1/Area_k - Q)^2 / Area_k]^(1/2)<br>
<br>
where Q=N/SUM(Area_k) does not depend on the subdivision<br>
and Area_k is the area of the kth district.<br>
I got this from page 20 of her old draft dated 10/20/11. The goal is<br>
to minimize it.</blockquote><div><br></div><div>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.</div>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"> </blockquote><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<br>
We can simplify by removing the final square-rooting without changing<br>
the measure's<br>
relative opinion about any two districting plans:<br>
<br>
SimplifiedDoppMeasure = SUM(over k=1..N)OF (1/Area_k - Q)^2 / Area_k<br>
<br>
Now since<br>
(1/Area_k - Q)^2 = (Area_k)^(-2) - 2*Q/Area_k + Q^2<br>
we can rewrite this as<br>
<br>
SimplifiedDoppMeasure = SUM(over k=1..N)OF [ (Area_k)^(-3) -<br>
2*Q*(Area_k)^(-2) + Q*Q*(Area_k)^(-1) ]<br>
<br>
Anyhow, however you do it, I DON'T LIKE this measure. Here's why.<br>
<br>
Because, this measure depends ONLY on the district areas.<br>
It does NOT depend on their perimeters, or their shapes, at all.<br>
<br>
In other words: suppose Dopp constructs some nice districting.<br>
Then ANY subdivision I construct having the same district areas as Dopp's<br>
(and also equipopulous) -- no matter how many insane wiggles and evil tentacles<br>
I add to all the districts to gerrymander them -- will have the same<br>
DoppMeasure. So this measure in no way discourages<br>
gerrymandering, and it fails to have a unique optimum (the "optimum"<br>
districting according to it is extremely infinitely non-unique).<br>
<br>
For example, say the country is a rectangle with uniform population<br>
density, and N=2.<br>
Then I'd say the best districting looks like this:<br>
<br>
AAAAABBBBB<br>
AAAAABBBBB<br>
AAAAABBBBB<br>
AAAAABBBBB<br>
<br>
but if I gerrymandered it to be this:<br>
<br>
ABBBBBBBBB<br>
ABAAAAAAAB<br>
AAAAABBBAB<br>
AAAAABBBBB<br>
<br>
then exact same DoppMeasure.<br></blockquote><div><br></div><div>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.</div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<br>
Also, even aside from this, I just do not agree with the<br>
DoppMeasure-minimization goal<br>
of causing all districts to have equal areas.<br>
Note: if all districts have equal areas (and equal populations),<br>
then DoppMeasure=0. Otherwise (not all areas equal) DoppMeasure>0.<br></blockquote><div><br></div><div><br></div><div>Again, the goal is 1, not 0.</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<br>
I think urban districts really should<br>
have smaller areas than rural districts. DoppMeasure minimization would<br>
abolish urban districts and cause every district to be a mix of urban and<br>
rural in order to make all districts have the same area.<br></blockquote><div><br></div><div>See above.</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<br>
So, sorry. I think this idea is a failure. I had earlier got the<br>
impression Dopp wanted<br>
to use isoperimetric quotients as the basis for a districting-plan<br>
quality measure.<br></blockquote><div><br></div><div>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.</div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
I like that idea, though the best way to do it is not clear to me.<br>
But the isoperimetric idea does not utterly abandon the use of perimeters.<br>
DoppMeasure does abandon them. That's a mistake.<br></blockquote><div><br></div><div>You've misunderstood her in two significant ways.</div><div><br></div><div>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.</div>
<div><br></div><div>Jameson </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
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Warren D. Smith<br>
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