[EM] More on Gerrymander prevention

[Anthony Simmons]

>> From: Adam Tarr <atarr at purdue.edu>
>> Subject: RE: [EM] More on Gerrymander prevention

Josh's proposal is indeed a very slick idea.  It groups
people according to social and economic criteria.  It would
work out well where I live -- on the west side of Puget
Sound. Seattle is not far away, in miles (kilometers,
lightseconds ...), but it may as well be another planet.

And the conductivity analogy is a good one, but it might have
to be modified a bit.  Certainly, conductivity of two
parallel roads is the sum of their conductivities, but two
roads in series shouldn't have any less conductivity (more
resistance) than the least conductive point in the circuit.
Or perhaps slightly less on average, since there's a greater
chance of a bottleneck from an accident.  So the rules would
be more like:

    C1, C2, C3, ..., Cn in parallel:  C = sum of C1, etc.
    C1, C2, C3, ..., Cn in series:    C = Smallest of C1, etc.

One thing I would try to add would be a measure to even out
the distribution of power (as measured by Banzhaf or
whatever).  It's step two, below.

As an alternative to maximizing conductivity of the entire
graph (if I might adopt the electrical metaphor, which I
think is inspired), how about this for breaking up the
country into districts of N people each:

    1.  Find the single collection of N people that has the
        smallest total conductivity.

    2.  Slide the boundaries in order to make the political
        makeup (as determined, perhaps, by party
        registration) as close as possible to the average in
        some larger area containing the district.  Do this by
        moving boundary sections with the greatest gradient -
        - that is, move the parts of the boundary that have
        the greatest effect on political makeup per unit area
        change.  Some parts get pushed out to enlarge the
        district; others get pulled in to reduce it by the
        same amount.  The idea is to slide the district in a
        fashion that gets it as close to the average makeup
        for the region with the least possible adjustment.
        The reason for this step is to eliminate, as much as
        possible, differences in individual power because of
        differing power indices for voters in different
        districts.

        I'm not sure if deliberately rendering the districts
        more typical of the surrounding political environment
        is the right correction.  It depends on what measure
        we choose in order to measure gerrymandering; the
        idea is to jiggle the borders to reduce that measure.
        I think this would be more reliable than just
        assuming that gerrymandering is minimized by
        selecting the right procedure.

    2.  Record the district thus created, and remove those
        people from the map.

    3.  If there are still people left, go to (1).

I'm not sure how it would work after most of the country has
been assigned to districts and only the sparsest areas are
left.  Might create some pretty weird rural districts.  But
even though they could be lanky and wander around a bit, I
expect they would pretty homogeneous socio-economically, so
perhaps it would work out right.

An alternative, instead of using conductivity, would be to
use population density instead.


>> Josh wrote:

>> >I think the map should be non-geographic, and, instead, road-based.
>> >
>> >Dense networks of roads should not be separated into separate districts.
>> >An urban area on two sides of a bridge could easily be divided.

>> Very slick idea Josh.  The question becomes, how do you
>> come up with a measure of road connectivity?  I would
>> propose basing it on the same principles as electrical
>> resistance.  The "resistance" of a road connecting two
>> points is proportional to its length, and inversely
>> proportional to the number of lanes.  Find the
>> "resistance" of every road connecting two adjacent census
>> blocks, and add them in parallel (1/total = 1/first +
>> 1/second + ...).  Invert this total resistance to get the
>> "conductance" or "road connectivity" of two adjacent
>> census blocks.

>> Build a graph (computer science-type graph, with nodes and
>> edges) out of the census blocks, with each edge (border)
>> weighted by the connectivity between those two nodes
>> (census blocks)  Now, we just tell the algorithm to build
>> equal-population districts that maximize total
>> connectivity.  It's a well-defined graph theory problem.

>> -Adam