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

[Warren Smith]

Kathy Anne Dopp:
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)