[EM] Redistricting Paper w/ New Population Density Fairness (PDF) measure

[Kathy Dopp]

Thanks Warren for composing the counterexamples. You are right.  That
is the vestiges of a logic error I made earlier and thought I had
removed, but had not.

The paper still offers several valuable new insights including:

1. a new population density fairness measure that helps to judge the
proportional fairness of representation of a plan, and

2. debunks the validity of 11 proposed area compactness measures that
are unfortunately still in use today.

I'll read your page on the topic and revise my paper.

Kathy

> From: Warren Smith <warren.wds at gmail.com>
> Legislative Redistricting - Area and Population Compactness and
> Population Density Distribution Measures
> http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1945879
>
> Dopp (in internet post advertising above paper & in her abstract):
> "This article argues that area compactness is
> reliably measured using any of the area-to-square-of-perimeter
> measures (or their reciprocals or square roots) because
> ALL SUCH MEASURES RANK ANY TWO REDISTRICTING PLANS
> IN EXACTLY THE SAME ORDER."
> (emphasis mine.)
>
> --they do?
> Let X_k = A_k / P_k^2
> be the area / perimsquared
> measure for district k.
>
> If the measure for an entire multidistrict "plan" is
> sum_k  X_k
> then I claim that will rank plans in a different order than
> sum_k  squareroot(X_k)
> and in a different order than
> sum_k  1/X_k,
> in general.
>
> For example:
> say plan #1 has these X's for its three districts:
>   X1 = 10, X2 = 11, X3 = 12
> while plan #2  has these X's for its three districts:
>   X1 = 6, X2 = 11, X3 = 17
> then the goal of maximizing sum X_k says that plan #2 is better since 34>33
> contradicting the goal of maximizing sum squareroot(X_k)
> which says plan #1 is better since
> 9.943 > 9.889.
> (You also can scale all numbers in this 2-plan example by any constant factor.)
>
> If plan #3 has
>   X1 = 5, X2 = 11, X3 = 18
> then the goal of maximizing sum X_k says plan #3 is better than plan #1,
> contradicting the goal of minimizing sum 1/X_k
> which says plan #1 is better than plan #3.
> (Again you also can scale all numbers in this 2-plan example by any
> constant factor.)
>
> In view of these counterexamples, I suggest Dopp either rephrase the
> capitalized sentence, or
> perhaps much more alteration is needed than merely 1 sentence, like
> her whole paper is busted.
> I'm not saying the latter; I'm saying the true amount of repairing needed lies
> somewhere between those two extremes.   I think the truth is the the
> isoperimetric quotient indeed
> is a good idea, but it is not obvious to me what is the best way (from
> among the many
> inequivalent possibilities) to combine all the district values,
> to get a value for the entire multidistrict plan.  My web page on this
> topic is here:
>    http://rangevoting.org/TheorDistrict.html
>
>
> --
> Warren D. Smith
> http://RangeVoting.org? <-- add your endorsement (by clicking
> "endorse" as 1st step)
>

-- 

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