# [EM] Thought on Redistricting Algorithms

Alex Small alex_small2002 at yahoo.com
Thu Jun 23 00:46:53 PDT 2005

```>From time to time there have been discussions of automated algorithms for drawing electoral districts.  The basic motivation is to draw districts on some basis more objective and impartial than "let's protect the incumbents!"

I was thinking about one possible danger of impartial district boundaries, especially when drawing only a handful of districts (less than 10).  Say that a state is 60% one party and 40% the other.  (I realize that's an over-simplification, but simplifications can be useful for academic purposes.)  Ideally, one would want to see a delegation that's approximately 60% of one party and 40% of the other.  (And yes, I know, partisan balance isn't the only thing one might want, but the issue I'll get to here is academic rather than political.)

Now, for simplicity let's assume the state's population density is uniform.  This has the advantage of making every district have equal area, which will be useful for some of the ideas that I use below.

(I know, it's wildly unrealistic in most places, but you can always rescale the map, draw lines on that map, and then unscale it in the end to return to reality.  There are ways to rescale maps that weight areas according to their population.  I've seen it done for maps of the "red" and "blue" states.)

Anyway, if we look at a map that's colored in shades of red and blue to show partisan balance throughout the state, the map will not be a single shade.  The color will fluctuate from place to place, and we can associate with those fluctuations a correlation length (in the language of physics).  Conceptually, we can pick out regions in which the color varies "only slightly" (this concept can be made rigorous with equations).  The average size of such a region is called the "correlation length."

If the district size is significantly larger than the correlation length then it's clear that every district will have more or less the same composition, and the same party should win all or most seats.  If the district size is significantly smaller than the correlation length then it's clear that most districts will be homogeneous and be safe for one party or the other, but not every district will be safe for the same party.  The overall balance of seats will roughly reflect the partisan balance of the state.

Anyway, the first thought that I toss out is the concept of length scales:  In physics, length scales of fluctuations are crucial for understanding the properties of phyical systems and mathematical models.  In redistricting, the length scale of fluctuations may be crucial for understanding the results of a party-blind redistricting method.

All this assumes, of course, that the redistricting process is blind to politics, as well as natural geographical boundaries that may be proxies for political affiliation (e.g. regional differences, cities vs. suburbs, etc.).

So, I wonder if one could use some of the methods of physics to predict the effect of district size on party balance in the overall Congressional delegation.  I wonder if some of the redistrictin algorithms that we've discussed might exhibit phase transitions as the number of seats varies:  For small numbers of seats the result is single-party hegemony, but as the number of seats increases the district size becomes smaller and comparable to the correlation length.  When the district size is comparable to the correlation length, partisan balance sets in.

Here's another thought experiment that uses concepts similar to the renormalization group in condensed matter physics:  Say we start with a single state-wide race, and look at the results.  Then we start aggregating people into very small "districts" of equal size (say, 3 voters apiece) and each district gets a single electoral vote.  For very small districts the outcome should be the same (statistically) as you'd get with a straight popular vote.

As we make these districts larger and larger, the electoral vote should start to deviate more and more from the popular vote (but in a stochastic manner, so that the mean is still the popular result).  Then at some point, when the districts are quite large, they should have more or less the same composition and the electoral vote should be unanimous or nearly unanimous for one candidate (even though the popular vote is more closely divided).

Anyway, those are my thoughts.  I wonder if concepts of physics and phase transitions might yield insight on surprising properties of seemingly neutral redistricting algorithms.

Alex Small

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