It's been nearly four years since we discussed gerrymandering. I've been thinking about it lately, since it's been back in the news, and since I've come up with what I think is an original way to measure it. Last time we discussed it, the discussion was all about the borders, e.g. minimizing the variation in their curvature, or minimizing the ratio between the square of the perimeter and the area enclosed. But that's obviously not the right measure, since making a border frizzy on a centimeter scale would greatly change such a measure without changing the real amount of gerrymandering at all. My proposal is to start with a database listing, to high precision (e.g. 10 meters) the home location of everyone in the state. The gerrymandering in a district would be measured by summing the squares of the distances between each two people in the district. The gerrymandering in a state would be measured by summing the squares of the gerrymandering numbers for each district in the state. (Of course the number of districts is kept constant, and their populations are kept as equal as possible, i.e. the most populous district can have at most one more person than the least populous district.) There's probably no practical algorithm for finding the unique lowest score. But everyone would be free to draw boundaries however they liked, so long the populations were equal, and submit their scores. Whoever came up with the lowest score before the deadline would have their submission accepted. Simulated annealing would probably be a good way to find low-score redistricting plans. This is fairly CPU-intensive, but I think today even laptop computers could easily handle it. With N people in a district, there would be N*(N-1)/2 comparisons to make. Is census data available that gives everyone's home location to within ten meters? Of course all other information (name, age, income, citizenship, etc.) would be stripped for privacy reasons.
I don't think bad boundaries is the problem. In many cases geometric proximity isn't a measure of common interests. Sure it is for some local issues like getting pot holes fixed and managing elementary schools. But for Congress critters? Engineers, teachers, investors, doctors,...all living in the same area may have quite different interests at the national level. The objection to gerrymandering has been how it has been used to "pack" and "crack" the electorate to give a large majority in representatives to one party out of proportion to its popular vote. My solution would be to enlarge districts so, instead of electing one representative, each district would elect two or three. A further step in this direction would be to drop the requirement that representatives reside in the district that elects them. Brent On 4/5/2018 7:52 PM, Keith F. Lynch wrote:
It's been nearly four years since we discussed gerrymandering.
I've been thinking about it lately, since it's been back in the news, and since I've come up with what I think is an original way to measure it.
Last time we discussed it, the discussion was all about the borders, e.g. minimizing the variation in their curvature, or minimizing the ratio between the square of the perimeter and the area enclosed.
But that's obviously not the right measure, since making a border frizzy on a centimeter scale would greatly change such a measure without changing the real amount of gerrymandering at all.
My proposal is to start with a database listing, to high precision (e.g. 10 meters) the home location of everyone in the state. The gerrymandering in a district would be measured by summing the squares of the distances between each two people in the district. The gerrymandering in a state would be measured by summing the squares of the gerrymandering numbers for each district in the state. (Of course the number of districts is kept constant, and their populations are kept as equal as possible, i.e. the most populous district can have at most one more person than the least populous district.)
There's probably no practical algorithm for finding the unique lowest score. But everyone would be free to draw boundaries however they liked, so long the populations were equal, and submit their scores. Whoever came up with the lowest score before the deadline would have their submission accepted.
Simulated annealing would probably be a good way to find low-score redistricting plans.
This is fairly CPU-intensive, but I think today even laptop computers could easily handle it. With N people in a district, there would be N*(N-1)/2 comparisons to make.
Is census data available that gives everyone's home location to within ten meters? Of course all other information (name, age, income, citizenship, etc.) would be stripped for privacy reasons.
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I already live/vote in the worst gerrymandered district in California, and perhaps the nation (CA#24). I agree with Brent that metrical compactness isn't the main problem -- especially in this age of the Internet. But even if some sort of metric could be defined, I don't believe that the solutions are unique, and would depend substantially on the overall shape of the state. Suppose that one had a very long thin state which forced all of the districts into *line segments* -- i.e., each person lives "closer" (under the chosen metric) to a state border than to any other person. If the total state population is less than the number needed for 2 Representatives, then we have only 1 Representative for the entire state and we're done and unique. Suppose we have enough total state population for 2 Representatives. We need to find the dividing line that separates the population with a difference of at most 1. If the state has an odd number of people, then we have two solutions, so no uniqueness. If we have enough population for 3 Representatives, then we need to find 2 dividing lines, which provides for 2 sources of non-uniqueness. Suppose now that our state consists of 2 disconnected pieces -- e.g., Michigan. How does the metric work between the disconnected pieces? We could even have a dumb-bell-shaped state, or a panhandle state. Do we use a different metric in the panhandle? If we have a more-or-less compact 2D state, we could start with a Voronoi diagram about each person. But what if the population density was maximal along a lazy river which approximated a space-filling curve through the whole of the state? At 09:56 PM 4/5/2018, Brent Meeker wrote:
I don't think bad boundaries is the problem.
In many cases geometric proximity isn't a measure of common interests. <snip> On 4/5/2018 7:52 PM, Keith F. Lynch wrote:
It's been nearly four years since we discussed gerrymandering.
I think it's important to consider the metric of the "distance" to the polling place(s), as well as the "distance" to others in the district. Distance in quotes because getting from A to B is dependent on many things besides physical distance. For example, A and B could be 100 meters apart but on opposite sides of a highway or river with no nearby crossings. Time between A and B might be better, but that still might depend on modes of conveyance and that might vary between people of different means. So measuring the "goodness" of a districting probably requires various kinds of map data in addition to population data. Of course, if we had a way of secure remote voting, some of these considerations could go away. But I won't argue that the ability of people to physically get together isn't important. On 06-Apr-18 13:44, Henry Baker wrote:
I already live/vote in the worst gerrymandered district in California, and perhaps the nation (CA#24).
I agree with Brent that metrical compactness isn't the main problem -- especially in this age of the Internet.
But even if some sort of metric could be defined, I don't believe that the solutions are unique, and would depend substantially on the overall shape of the state.
Suppose that one had a very long thin state which forced all of the districts into *line segments* -- i.e., each person lives "closer" (under the chosen metric) to a state border than to any other person.
If the total state population is less than the number needed for 2 Representatives, then we have only 1 Representative for the entire state and we're done and unique.
Suppose we have enough total state population for 2 Representatives. We need to find the dividing line that separates the population with a difference of at most 1. If the state has an odd number of people, then we have two solutions, so no uniqueness.
If we have enough population for 3 Representatives, then we need to find 2 dividing lines, which provides for 2 sources of non-uniqueness.
Suppose now that our state consists of 2 disconnected pieces -- e.g., Michigan. How does the metric work between the disconnected pieces?
We could even have a dumb-bell-shaped state, or a panhandle state. Do we use a different metric in the panhandle?
If we have a more-or-less compact 2D state, we could start with a Voronoi diagram about each person. But what if the population density was maximal along a lazy river which approximated a space-filling curve through the whole of the state?
At 09:56 PM 4/5/2018, Brent Meeker wrote:
I don't think bad boundaries is the problem.
In many cases geometric proximity isn't a measure of common interests. <snip> On 4/5/2018 7:52 PM, Keith F. Lynch wrote:
It's been nearly four years since we discussed gerrymandering.
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I live in CA#26 and the only sign of gerrymandering I see is along our mutual boundary, which appears tweaked to put Ventura into #24. Just by eyeball it doesn't look gerrymandered at all and no less compact that many other CA districts. Brent
On 06-Apr-18 13:44, Henry Baker wrote:
I already live/vote in the worst gerrymandered district in California, and perhaps the nation (CA#24).
this is a good resource created (i believe) by moon duchin, a mathematician at tufts who has been energetic organizing the mathematical community to discuss gerrymandering and even prepare mathematicians for expert testimony in legal proceedings. various well-attended math and gerrymandering meetings have already been held in the last 18 months with more upcoming. https://sites.tufts.edu/gerrymandr/resources/ On Fri, Apr 6, 2018 at 10:44 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I already live/vote in the worst gerrymandered district in California, and perhaps the nation (CA#24).
I agree with Brent that metrical compactness isn't the main problem -- especially in this age of the Internet.
But even if some sort of metric could be defined, I don't believe that the solutions are unique, and would depend substantially on the overall shape of the state.
Suppose that one had a very long thin state which forced all of the districts into *line segments* -- i.e., each person lives "closer" (under the chosen metric) to a state border than to any other person.
If the total state population is less than the number needed for 2 Representatives, then we have only 1 Representative for the entire state and we're done and unique.
Suppose we have enough total state population for 2 Representatives. We need to find the dividing line that separates the population with a difference of at most 1. If the state has an odd number of people, then we have two solutions, so no uniqueness.
If we have enough population for 3 Representatives, then we need to find 2 dividing lines, which provides for 2 sources of non-uniqueness.
Suppose now that our state consists of 2 disconnected pieces -- e.g., Michigan. How does the metric work between the disconnected pieces?
We could even have a dumb-bell-shaped state, or a panhandle state. Do we use a different metric in the panhandle?
If we have a more-or-less compact 2D state, we could start with a Voronoi diagram about each person. But what if the population density was maximal along a lazy river which approximated a space-filling curve through the whole of the state?
At 09:56 PM 4/5/2018, Brent Meeker wrote:
I don't think bad boundaries is the problem.
In many cases geometric proximity isn't a measure of common interests. <snip> On 4/5/2018 7:52 PM, Keith F. Lynch wrote:
It's been nearly four years since we discussed gerrymandering.
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
It's worth remembering that congressional districts *must* be a union of census tracts (since otherwise the decennial "population" is undefined). However, the recent mathematical gerrymandering work has, quite rightly, focused on things that courts might plausibly adopt as standard tests. Since Roberts looked at the proposed and quite straightforward "efficiency gap" metric and said in oral arguments that "It may be simply my educational background, but I can only describe it as sociological gobbledygook", I don't think throwing better math at finding boundaries is going to help much. Many wise folks today (Moon Duchin included) are putting their eggs into the approach of sampling from the space of *all possible* districting plans that meet the relevant legal criteria. This space is too large to exhaustively enumerate, but they're trying various techniques to reasonably uniformly sample from it. The goal here is not to say which districting plan is "The Best", but rather to show what the whole distribution of possible plans looks like, so that if one that's proposed is obviously a massive outlier according to some metric (like how partisan it is), it's easy to draw a picture which shows that. This turns the gerrymandering fight from a mathematical one into a *data visualization* exercise, which I think is brilliant. Courts have gone for an "I know it when I see it" standard before, and given all the gamesmanship that any mathematical definition would lead to, this seems to me like a great way to side-step the problem. --Michael On Fri, Apr 6, 2018 at 3:01 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
this is a good resource created (i believe) by moon duchin, a mathematician at tufts who has been energetic organizing the mathematical community to discuss gerrymandering and even prepare mathematicians for expert testimony in legal proceedings. various well-attended math and gerrymandering meetings have already been held in the last 18 months with more upcoming.
https://sites.tufts.edu/gerrymandr/resources/
On Fri, Apr 6, 2018 at 10:44 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I already live/vote in the worst gerrymandered district in California, and perhaps the nation (CA#24).
I agree with Brent that metrical compactness isn't the main problem -- especially in this age of the Internet.
But even if some sort of metric could be defined, I don't believe that the solutions are unique, and would depend substantially on the overall shape of the state.
Suppose that one had a very long thin state which forced all of the districts into *line segments* -- i.e., each person lives "closer" (under the chosen metric) to a state border than to any other person.
If the total state population is less than the number needed for 2 Representatives, then we have only 1 Representative for the entire state and we're done and unique.
Suppose we have enough total state population for 2 Representatives. We need to find the dividing line that separates the population with a difference of at most 1. If the state has an odd number of people, then we have two solutions, so no uniqueness.
If we have enough population for 3 Representatives, then we need to find 2 dividing lines, which provides for 2 sources of non-uniqueness.
Suppose now that our state consists of 2 disconnected pieces -- e.g., Michigan. How does the metric work between the disconnected pieces?
We could even have a dumb-bell-shaped state, or a panhandle state. Do we use a different metric in the panhandle?
If we have a more-or-less compact 2D state, we could start with a Voronoi diagram about each person. But what if the population density was maximal along a lazy river which approximated a space-filling curve through the whole of the state?
At 09:56 PM 4/5/2018, Brent Meeker wrote:
I don't think bad boundaries is the problem.
In many cases geometric proximity isn't a measure of common interests. <snip> On 4/5/2018 7:52 PM, Keith F. Lynch wrote:
It's been nearly four years since we discussed gerrymandering.
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
participants (6)
-
Brent Meeker -
Henry Baker -
Keith F. Lynch -
Michael Kleber -
Mike Speciner -
Thane Plambeck