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?

Solution 1:

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

Solution 2:

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.

Solution 3:

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