2011/10/21 Warren Smith <[email protected]> > 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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