# Gerrymandering/Optimization of electoral districts for one particular party

I'm asking this on behalf of Zach Weiner (actually it's my own initiative in order to promote this site). Original text is here, and is as follows:

Hey-- This is Zach from SMBC, and I have a math question you may find of interest. I only mention who I am because it relates to the idea.

I had an idea for a comic about gerrymandering. As you may know, gerrymandering is a significant social problem, in that it stifles voters' opinions. So, my idea was this: Why not make a rule that perimeter/area always has to be under a certain value. I figured this would limit how salamander-like the districts could be made. Then, I tried to figure out the math of this on the assumption that it was a simple calculus min/max problem. It seems not to be...

The biggest problem I'm running into is how to formalize the idea of a shape being weird. My intuition tells me that the lower perimeter/area is, the less weird the shape. I.e. a wacky snakey shape designed to get several populations will have a higher perimeter/area than a more reasonable district shape, which should be vaguely rectangular or circular. But, I don't know how to mathematize that. If that could be proved, you could probably figure out a reasonable ratio.

Thoughts?

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If this question is not appropriate please let me know in this meta question – juan May 12 '11 at 21:10
The problem I see is that it is not trivial to define the perimeter of any electoral district, salamander or not. – Phira May 12 '11 at 21:14
Whatever you decide on the districts, if you make a comic where legislators use the Banach-Tarski paradox on their own constituents to get more votes I'll love you forever. – MartianInvader May 12 '11 at 21:19
For an actual law, I would compare the number of voters in the electoral disctrict to the number of voters in the convex hull of the electoral district, although one would want to make exceptions for natural non-convex shapes (e.g. curving mountain ridges). – Phira May 12 '11 at 21:21

You might look at Hodge, Marshall and Patterson, "Gerrymandering and Convexity", The College Mathematics Journal, Vol. 41, No. 4 (September 2010), pp. 312-324 http://www.jstor.org/stable/pdfplus/10.4169/collmathj.41.4.0312.pdf

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Yes, convexity seems like the main issue here. One would want to minimize something like the ratio of the area of the convex hull to the area of the shape... – Qiaochu Yuan May 13 '11 at 2:45
I don't believe convexity captures the issue well. The convexity test considers districts which are nearly one-dimensional non-gerrymandered, yet such district shapes could clearly be used to suppress or amplify voters' influence. – R.. May 21 '11 at 2:56

If $P$ and $A$ are the perimeter and area, then it's $P^2/A$ that you would want to limit. But it's not really workable. Firstly, there's the obvious problem of coastal constituencies, where the perimeter is not well-defined. Secondly, you could still get around the rule by making the boundary as nearly circular as possible, with a few fingers extending into (or out of) the territory of your supporters (or opponents) -- in a big city, these fingers would not have to be very long to make a difference in voter preference.

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I don't think coasts are really a big deal. You could probably even exclude the coast from the perimeter, or replace it with the max distance between any two coastal points. The fingers issue is what needs more attention... – R.. May 12 '11 at 21:40

Take a look at the paper: http://weblaw.haifa.ac.il/he/Events/eveFile/bizarreness090909.pdf and the many references there.

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Two years ago at the Joint AMS-MAA meetings there were special presentations dealing with some of the mathematical aspects of redistricting. Below is a list of the papers and presentations. Contacting the authors and/or finding the published versions of these items will provide useful information: eurekalert.org/pub_releases/2009-01/saef-jmm_1010509.php – Joseph Malkevitch May 20 '11 at 23:14

A google scholar search for "optimal gerrymandering" turns up the big results I'm aware of: Katerina Sherstryuk's "How to Gerrymander" http://www.hss.caltech.edu/SSPapers/sswp855.pdf and the American Economic Review's most recent post on the topic: http://web.mit.edu/rholden/www/papers/Friedman-Holden%20(AER,%202008).pdf.

I thought this, like fair division, was one of the areas where mathematicians and economists know the same literature.

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