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

[Kathy Dopp]

> From: 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.

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