Dual Orthogonal Redistricting

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Dual Orthogonal Redistricting

I assume most people know that, in the U.S., the States are divided up into "districts" for various political reasons. One major reason is that the laws, the Constitution, and the U.S. Census require each state to be able to appoint some certain number of Representatives to the U.S. House of Representatives, based on each state's population. And at least theoretically, each district is supposed to contain (very nearly) the same number of voters. However, the way in which each state is geographically divided up (the "shape" of the districts) can readily provide whatever regime performs the occasional redistricting scheme with tremendous advantages (this is called "gerrymandering"). And, unfortunately, whatever regime happens to be in power gets to dictate redistricting schemes (the courts occasionally attempt to remedy the very most flagrant abuses).

This has inspired comrade Warren D. Smith to come up with a "shortest splitline districting method". This method is algorithmic in nature, so it may be expected to remove any influence whatever regime happens to be in power may have in determining redistricting schemes. Overlooking some minuscule, readily remediable edge cases (such as off-by-one dilemmas), I will attempt to describe this method. I will ignore the complication that the earth is round and not flat for now. With the shortest splitline method, you take a map of the relevant region or state and find the shortest straight line that divides it into two geographic parts that contain an equal number of voters. Then you divide each of those two parcels into two more such parts until you have the requisite number of equally populated parcels corresponding to the required number of Representatives. And this is purely algorithmic and should be expected to eliminate the influence of whatever political regime happens to be in power. So far, so good.

But there are a few hairy edges. Actually, every redistricting plan developed by this method seems to somewhat resemble a broken pane of flat transparent glass. One has every expectation that each result is due to a totally chaotic process. In fact, it could reasonably expected that if a few voters should move in or out of any one district, the entire pattern of the plan would likely shift into a completely different pattern of division of districts. This could be quite disconcerting. Also, geography is notoriously (not to mention historically) fractal. What this leads to is that two algorithms that are sensitive to differing scales of measurement (one perhaps being sensitive to tiny incursions of territory that the other ignores, the potential number of these tiny features being virtually infinite) could very possibly generate two dramatically different redistricting plans. And the whole process really seems to provide a wonderful playground for chaos theorists. But this troubled me.

So after months of perplexity, I came across a few interesting things. Let's begin with a few thought experiments, which I will describe now. Let us obtain sixteen typical red building blocks, or to avoid heavy lifting, sixteen dominoes. These are easy to move around, since they are three dimensional rectangles (actually called "cuboids"). Now, let us arrange them to all have the same orientation, so if each one is, say, two inches by four inches, the total arrangement of all sixteen will be 4*2" by 4*4". OK this is nice. Suppose we have laid them on the floor in front of us and their east side is to our right, their west side is to our left, their north is farthest away, and their south side is closest to us. The north to south groups of blocks we will call "columns", and the east to west groups of blocks we will call "rows". Now suppose we move any one of the columns north by half an inch. This will lock all of the rows from moving either east or west. Or, if we were to move any row east by half an inch, we will lock all of the columns from moving north or south. Well this is a problem, as we can move all of the columns north or south as much as we like, or all of the rows east or west all we like, but we must choose one way of movement or the other, and cannot have both. Or... can we?

The earth is round, so we must deal with arc segments rather than straight line segments, but this makes no difference in the end. Latitude and longitude will work just the same way as Cartesian coordinates for our purposes. I will favor north-south distinctions over east-west distinctions simply because of minor climate distinctions. So let's say that some state, by virtue of population numbers, is entitled to ten Representatives. 10 = 2*5. So first dividing north to south, we could define five latitudes, each with an equal number of voters. Every voter in each of those latitudes gets to vote for potential Representatives who live in their latitude. Hopefully using strategic hedge simple score voting. But there are still two longitudes, and of course they are selected to have an equal number of voters. So each voter gets to vote in essentially two elections, one in which they get to choose a latitudinal Representative, and another in which they get to choose a longitudinal Representative. This is easy if each voter, who will automatically live in two kinds of districts, will have two ballots, one for each (overlapping) district.

This could be further rationalized. There could be three latitudinal, and three longitudinal elections, and one at-large election (there could be a rule requiring the number of latitudes and longitudes to be no greater than twice one or the other). This has great precedence since, for example, the state of Texas has often had at-large Representatives, mostly due to geographical impediments.

This would make the algorithmic determination of districts vastly simpler. And would also make districting far more coherent to the voters.

Clarification For Dual Orthogonal Redistricting

The article above is my first attempt to describe dual orthogonal redistricting, so it's perhaps a bit clumsy and might lead to some misunderstandings. So I will try to clarify a few things. Again, I may overlook some minuscule, readily remediable edge cases.

I said that the complication that the world is spherical can be ignored, but some may still wonder why that is so. When my description is done this should be fairly obvious, but since most people are not used to thinking about this matter, I will point out a few basic facts. Typical local land surveys project geographic features onto flat plot plan paper for pragmatic reasons, although the earth is not actually flat. If it was perfectly flat you would not be able to sit up in bed. Of course if it was actually perfectly spherical you would still not be able to sit up in bed. But sailing ships, major roadways, microwave repeater systems, etc. must take account of the earth's sphericalness. The parallels and meridians (latitudes and longitudes) representation of the spherical geographic coordinate system indicated by circles that are inscribed on typical "globes" provides a quite useful means of indicating locations. It does this via "coordinates" representing intersections of specific latitudinal and longitudinal segments of circles.

Everything is based on two points in space: the northern end of the earth's axis of rotation (north pole), and the southern end (south pole). Any circle (which of course will have infinite pairs of opposing sides) with one side intersecting with the north pole and the corresponding opposite side intersecting with the south pole represents a longitude. An infinite number of these circles could define the surface of the sphere. Any circle on the surface of such a sphere for which three or more distinct straight lines from either the north or the south pole are of equal length represents a latitude. So begins your contemplation of spherical geography (and also of the reason why we can ignore all this for the purposes of dual orthogonal redistricting).

With dual orthogonal districting, each voter gets to vote in essentially two elections, one in which they get to choose a latitudinal Representative, and another in which they get to choose a longitudinal Representative. This is easy if each voter, who will automatically live in two kinds of districts, is provided with two ballots, one for each (overlapping) district. It seems practical to allow each voter to select one poling station in one of his or her two districts as the place to cast their votes, so each district must possess ballots for every orthogonal district that overlaps it. And also must have ballots for at-large candidates (sometimes required for practical purposes) as well. And all these ballots should all be able to accommodate write-in candidates.

Now, suppose we have a state or region that is entitled to have 33 Representatives. We should allow one Representative to be at-large, so we can have a neat 32 regional Representatives. Four longitudinal "slices" and four latitudinal "slices" will give us 16 districts. But we will split these in two, since we will require 16 longitudinal districts, and 16 latitudinal districts, and it will turn out that none of these will cover all of the same ground. Now, if all of these slices were of equal "width" (angular width actually) the districts they define are most unlikely to have an equal numbers of voters, and some could even have no voters at all. Remember that I said that we can deal with the longitudinal districts in exactly the same way we deal with the latitudinal ones. So let's deal with the longitudinal ones for now.

We should apportion each of the four longitudinal slices into four equally populated slices by adjusting "sliding bounds", which are simply (three) adjustable longitudinal segments. The key thing is (this is the cool part), we will now apportion each of these four longitudinal slices separately into four equally populated longitudinal districts by drawing three latitudinal (!) sliding bounds. We now have 16 equally populated longitudinal districts.

Now, we will apportion each of the four latitudinal slices via the exact same process, and we then will have 16 equally populated latitudinal districts. So we now have 16 longitudinal districts plus 16 latitudinal districts, and every one of them will have the exact same number of voters. Plus we will have the one big at-large district that elects one candidate, for a total of 33 Representatives.

(Merry Christmas!)

I think this is an

I think this is an interesting proposal. For me the main new idea here is allowing each voter to participate in more than one election.

What are some of the benefits and drawbacks of the present systems in the US for appointing a state's representatives? I can easily think of three benefits. One is that local issues and concerns can be, through that district's rep, spoken for within congress. The other is a way to handle direct communication with the public. We can all email our respective representatives, and they should be much more capable of handling that communication than, say, the POTUS because they have fewer people in their district. The third advantage is debatable but essentially gives the land itself a voice since states and areas with sparse population have a somewhat disproportionately-large number of representatives. (But only by about 25% I think.)

One obvious drawbacks is, as you mentioned, gerrymandering. Another one is the lack of proportional representation. There is some interplay between these two problems.

Are there any important benefits and drawbacks that I'm forgetting? I'm overlooking issues related to the performance of our reps and mechanisms within congress, like block voting, just to maintain scope.

As far as I can tell, Dual Orthogonal Redistricting reduces gerrymandering because it greatly restricts the amount of tweaking that can be done to the district boundaries.

But I think this proposal increases the size of each district by a factor of approximately sqrt(N) where N is the number of reps in the state. The average state has (435 / 50) = 8.7 reps, so the proposed redistricting would produce districts about 3 times larger for the "average" state. California would be the worst case here, where the new districts would be ... ugh 6x9 gives you 54 districts, one too many. Probably 5 x 10 with 3 at-large. So the new districts there would be 5 and 10 times larger than today in terms of population.

Because of larger districts, you weaken somewhat all three of my benefits above, and you probably also worsen the problem of minority interest under-representation (lack of proportional representation). Each representative would have, very roughly, triple the number of citizens to communicate with and represent, with a worst case of 10x. Since their districts would no longer follow any natural geographic or demographic boundaries, they don't really serve as a mouthpiece for raising local concerns. Consider, for example, the districting of the Eastern parts of California which will basically get combined with San Diego in the "vertical" districts, and the Northern "horizontal" district which would run all the way into Sacramento.

Take a look at some other proposals focusing on proportional representation, such as asset voting, reweighted range voting, party-list voting, and mixed-member proportional voting. With these systems, I think local interests find representation without any kind of districting at all. They do place a larger burden on voters (many more candidates to consider). And they don't AFAICT provide a natural way of assigning representatives to any particular population group within their state. So an additional mechanism would need to be invented to handle that. The representation of land/resources is also not handled within a given state, but it's not obvious that we really accomplish that today anyway, or that it's even important to do so.

Some Further Clarification

Each candidate must choose only one district, but each voter may vote in both districts in which he or she resides. The sum of vertical and horizontal districts must be an even number, because we must be able to divide the total number of districts into one set of vertical (longitudinal) and one set of horizontal (latitudinal) districts, and these must be of equal size.

To help prevent districts from becoming too lengthy, the largest of the closest pair of divisors of the number of vertical districts (and this closest pair will be the same for the horizontal districts) should not exceed 2 times the smallest of them. For example, if we need 20 representatives, we will need 10 vertical and 10 horizontal districts. 2 and 5 are the only divisors of 10, but 5 is more than twice as large as 2, so we should use a pair of 9 vertical and 9 horizontal districts for a total of 18. Then we will have 2 at-large representatives.

By this same logic, it is the case that California is currently entitled to 53 representatives in the House of Representatives but 53 is a prime number. However 4 and 13 are the closest pair of divisors of 52, and 13 is much more than twice 4, so we must try for 50 districts -- 25 vertical, and 25 horizontal districts. And we will have 3 at-large representatives.

This is a purely algorithmic process, so there can be no gerrymandering. Plus it guarantees that every voter will have a close regional affiliation with the districts in which he or she finds him or her self. And again, each district, whether vertical or horizontal, will have the same number of voters. Proportional representation and regional representation are quite distinct, but not mutually exclusive procedures. I have come across proportional representation methods that are vastly simpler and safer that those currently being practiced.


You're right, I completely botched my math with that sqrt(N) nonsense. Kind of embarrassing, really. Sounds like your proposal really just increases the number of citizens per representative by a factor of two, since each citizen belongs to two districts and all districts have approximately the same number of citizens. And any state with an odd number of reps will have one at-large representative. I guess California would be gridded 26x26, with one at-large rep.

I still wonder if anyone has come up with a proposal for electing representatives that implements proportional representation? Can this be designed in a way that does not specifically incorporate political parties into the system, and also maintains some kind of scheme to maintain a 1-to-1 or maybe just small-to-small relationship between citizens and their reps? It would also be nice for a politician to have a smallish target area they need to campaign within. Maybe dual orthogonal redistricting is a component of such a proposal.

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