> From: 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.
Yes. I uploaded changes and I can't download the paper either. Will check it again in the a.m. I can send it to anyone who wants it. > > 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) False. This is not remotely my PDF measure. You somehow neglected all density in the algebraic expression because there are no population numbers in the formula and you can *Not* eliminate the weights, or the measure gives completely different answers. FYI, density means the number of people living there divided by the area. d_i is the density in district i. Have no idea what measure you are evaluating but it has zero to do with what I proposed. > > 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. Even if you had the measure correct, which you do not, the goal is *Not* to minimize it. You didn't read my paper or the examples where I calculated the measure and showed the exact data and calculations for each example. > > We can simplify by removing the final square-rooting without changing > the measure's > relative opinion about any two districting plans: Only if you claim (a_squared + b_squared = a + b)! Comeon Warren, I know you can do algebra better than that! > > SimplifiedDoppMeasure = SUM(over k=1..N)OF (1/Area_k - Q)^2 / Area_k Nope. Not even remotely true. > > Now since > (1/Area_k - Q)^2 = (Area_k)^(-2) - 2*Q/Area_k + Q^2 > we can rewrite this as Yes true. But has ZERO to do with anything I wrote in my paper. > > 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. Neither do I like this nonsensical measure you've invented having nothing to do with anything in my paper! > > Because, this measure depends ONLY on the district areas. FALSE. my measure is called POPULATION DENSITY FAIRNESS (PDF). Notice the words *POPULATION DENSITY*. Population density depends on the numbers of persons living in each district proposed within a plan and the number of persons living in the state. The number of persons does *Not* cancel from the expression. > It does NOT depend on their perimeters, or their shapes, at all. TRUE. My PDF measure is NOT a measure of compactness, as is clearly stated in my paper, it is a measure of POPULATION DENSITY FAIRNESS. Think about it awhile and I'm sure you'll be able to figure it out Warren. I clearly listed all the small number of variables and the formula is very simple, especially for a person with your mathematical background if you spend a few minutes reading my paper rather than fabricating silly things. > > In other words: suppose Dopp constructs some nice districting. Huh? Well I did construct some simple examples of a redistricting plan. > 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. CONDITIONALLY TRUE. As long as you do not change the population density of the district and its area (or the density or area of its contiguous neighboring districts), you can wiggle it all you want without changing the measure. > So this measure in no way discourages > gerrymandering, Your definition of gerrymandering apparently involves making sure political parties that tend to be distributed according to population density have representation proportional to their population numbers. Wierd definition for gerrymandering - sort of the opposite definition than the common meaning. Sounds like gerrymandering or proportional representation is a good thing by your definition. I'm confused, are you against proportional representation then or does gerrymandering have a good connotation to you? > and it fails to have a unique optimum (the "optimum" > districting according to it is extremely infinitely non-unique). If your claim is true, that is good because that gives options to redistricting boards to try to meet the plethora of other concerns that are important to meet when redistricting, by choosing the most proportionately fair plan for representing people living in regions with varying density that meets the other important considerations. > > For example, say the country is a rectangle with uniform population > density, and N=2. Do you know of ANY state or county or municipality with uniform population density? Wow. What planet do you live on? My PDF measure does not even consider that case because that case is NONEXISTENT in real life. > Then I'd say the best districting looks like this: > > AAAAABBBBB > AAAAABBBBB > AAAAABBBBB > AAAAABBBBB If by A and B you mean people of different political parties, it seems unlikely to have exactly 50% of each in every district. Ties in every district! How do people elect ANY legislative representatives in that case? It hardly seems ideal to me to be unable to elect any leaders at all. > > but if I gerrymandered it to be this: > > ABBBBBBBBB = goes to party B > ABAAAAAAAB = goes to party A > AAAAABBBAB = goes to party A > AAAAABBBBB = goes to no one Yes. That hardly seems ideal either if the parties are equal in numbers. > > then exact same DoppMeasure. Warren, you dropped the population densities, and the weights, and obliterated the algebra by assuming b^2 + a^2 = a + b, etc and therefore fabricated your own measure that is as nonsensical as your uniform population density example. This imaginary example of situation which NEVER occurs in real life in any US state, county, or town doesn't seem very useful even if you were applying my actual PDF measure Warren. The reason for my deriving a population density measure is because population densities *vary* and political party affiliation in the US tends to vary with population density. > > Also, even aside from this, I just do not agree with the > DoppMeasure-minimization goal I never had any such goal of minimizing your fictional account of my measure. > of causing all districts to have equal areas. In your imagination. In real life, Impossible. Why on earth would I suggest weights for the weighted variance measure be the area of the district divided by the total area of the state (I.e. the proportion of the state's area that the district is) if "all districts have equal areas"?! Wow. You must imagine me to be a hopeless idiot to see that on the page rather than what I actually wrote. If all the weights are equal, of course we don't need any weights do we Warren, so of course you can just cancel the weights since you magically assume all districts have the same area. Warren, we're down here on earth. Please come join us down here in real life. Thank you. :-) > Note: if all districts have equal areas (and equal populations), > then DoppMeasure=0. Otherwise (not all areas equal) DoppMeasure>0. Perhaps true with your version of Doppmeasure. > > I think urban districts really should > have smaller areas than rural districts. Warren, that is inevitable because urban districts have higher density, THUS urban-only districts MUST HAVE SMALLER AREAS. I think everyone agrees with you there. > 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. Such a plan would give all seats to the majority political party in most states. However, your claim is necessarily true in all cases though. It depends on the population distribution in a state. It sounds like you would rather have small strictly urban districts so that Democrats could be concentrated in these districts in high proportion, so that Republicans could have a disproportionate share of the rest of the seats of state legislatures in relation to their numbers in the state. This is a common form of gerrymandering that can be done along with compact districts. Are you really calling plans with proportional representation gerrymandering? > > So, sorry. I think this idea is a failure. I had earlier got the I Totally agree with your opinion of your your fictional bastardization of all semblance of my PDF measure. Yep. What is your purpose in sending this to this list Warren? I hope not and am hoping you are just making an unbelievable mistake given how simple my formulas are to understand. > impression Dopp wanted > to use isoperimetric quotients as the basis for a districting-plan > quality measure. Yes. That was *before* I fully understood how compactness tends to disadvantage Democrats and give a disproportionate number of seats to Republicans due to Dems usually being concentrated in urban areas. I thought a proportionately fair measure for redistricting plans would be a positive thing (whereas you call it gerrymandering assuming you think of gerrymandering and proportionally fair representation as a negative thing.) > I like that idea, though the best way to do it is not clear to me. What idea do you like? The isoperimetric quotient so that we can have proportionately more Republicans than Dems in comparison to the number of Dems and Repubs in the population? > But the isoperimetric idea does not utterly abandon the use of perimeters. True. You obviously did not read my paper at all or are deliberately fabricating straw men. Try skimming it Warren. My paper is actually quite good and I think I'm the first person, at least to my knowledge, to devise a way to measure the proportional fairness of redistricting plans for various measures, but remember to look carefully at the simple expression of PDF which involves densities, which involves population numbers, and which cannot be simplifed any further than it is. I already spent a week simplifying the expression as compared to its form the way I derived it. Believe me, you can *Not* simplify it further. You have to input the numbers AS IS, and watch where the parenthesis are in the expression, and do not cancel out all the numbers of people in districts and in the state because they do *Not* cancel. > DoppMeasure does abandon them. That's a mistake. Yes. PDF does not involve any measures of perimeters, but if you actually read my paper, you would find I suggested first finding a plan having population density fairness close to one so that it encourages proportional representation of political parties, and then evaluating plans for area or population compactness, although those measures are less important than PDF. I am really surprised that you seem to define proportionately fair redistricting plans as gerrymandering -- or did I misunderstand what you've said here? Kathy Dopp http://electionmathematics.org Town of Colonie, NY 12304 "One of the best ways to keep any conversation civil is to support the discussion with true facts." "Renewable energy is homeland security." Fundamentals of Verifiable Elections http://kathydopp.com/wordpress/?p=174 View some of my research on my SSRN Author page: http://ssrn.com/author=1451051 ---- Election-Methods mailing list - see http://electorama.com/em for list info
