Jonathan Lundell wrote:
I haven't been following this thread in great detail, but I do have a
question: what is the distance function actually trying to measure and
minimize? What exactly are we trying to optimize when we minimize
"distance", by whatever measure? I might be close in a crow's-flight
sense to a neighbor on the other side of a river or freeway with a
driving distance of several miles.
OK, we could solve that in principle (though not too quickly) by using
Google Maps driving time, or the like. But what does driving time have
to do with grouping voters (unless we're drawing a precinct and
measuring travel time t the polling place)? Maybe we mean communication:
two people with telephones are closer in this sense than people without
(and people with unlimited long distance are close in this sense
independent of cartesian distance).
I live in several voting districts where geographical lines make
sense--my local fire district is a case in point, as would water and
sewerage (except that I provide my own in those cases, as it happens).
But legislative districts? I can't think what I'm trying to optimize for
which straight-line vs manhattan distance would be relevant. I've never
met my representatives on business; I communicate with them via email or
telephone, and I neither know nor care where there local offices are.
Proper redistricting has a few rules: the districts must be contiguous
(no empty spaces, exclaves or enclaves), and the districts must carry
similar population (otherwise "one person, one vote" suffers, and it
appears that United States court precedent have judged population
dissimilarity very strictly).
Minimizing over some distance is a side benefit. If you minimize over
Euclidean distance (or Haversine, if you want to be picky), the
districts are going to look compact - like squares or hexagons - rather
than the sort of contorted shape that "obviously gerrymandered"
districts have.
The underlying reason is that while there are very many ways of drawing
gerrymander-shaped districts, there are much fewer ways of drawing
districts that minimize with regard to some distance. Therefore,
gerrymandered districts (having no constraint on compactness) would
much more likely be gerrymander-shaped than compact. While there's no
direct reason to have districts that look nice on a map, it serves a
function similar to using expansions of pi or e in cryptographic
primitives: to show that no trickery is being done, because the
selection is so limited that finding trickery that's compatible with the
appearance (or the rules of the optimization) becomes nearly impossible.
By this reasoning, one could also turn your argument on its head. There
are various functions that leave too little to chance to be effectively
gerrymandered. If you pick Manhattan distance, fine enough; but if you
pick Euclidean or Haversine distance, then at least you have the added
improvement, however small, that if geographical lines *do* matter, then
the assignment optimizes for it, and that bonus is essentially free.
One could try to make districts that fit geographical lines even better
by weighting according to steepness (delta altitude; since it's hard to
cross mountains), or by using street distance (Google Maps driving
time). However, the methods discussed would need the distance metric to
be defined for all pairs of points, and I don't think there's street
distance data for that -- for instance, what's the driving time if you
start on a mountain peak?
(I wonder if candidates would try to make roads to nowhere to game the
metric :-)
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