Yes, that would work. The paper has a note about excluding districts like this:
> "Valid districts are contiguous and have equal population. Strict constraints on compactness, geographic splits, or other restrictions are not necessary, but such limitations could be included. However, valid districts may not include “donuts,” where one district entirely encircles another."
Yes! A lot of modern political science research uses computational methods, big data, etc. Here are some interesting papers on redistricting, by the research group that wrote the package we use in this paper. https://alarm-redist.org/applications.html
We have an analytic solution in the supplementary materials [1]. One of the biggest challenges to optimality proofs is how to capture the importance of geography as a constraint. I don't believe there are any papers on redistricting that include geographic constraints in analytic/formal solutions. That's why we prefer simulations for our main results.
Requiring geometric contiguity allows almost-unchanged gerrymandering through the backdoor.
Here's the Stonewall algorithm to gerrymandering with contiguous Define-Combine:
1. Start by gerrymandering a map with N contiguous districts the usual way.
2. Pick one of the districts to be your "mortar", the others are your "bricks".
3. Shrink all bricks minimally to open gaps between them without changing the population or election outcome.
4. Fill in the gaps with your mortar, so that all bricks are completely enveloped.
6. Split all bricks and the mortar in half, ensuring that one of the mortar halves does not touch any bricks. This gives you your final map of 2N contiguous districts for the second mover to combine.
Then the mortar half that doesn't touch any bricks can only be combined with the other mortar half. And since all bricks are completely surrounded by mortar (which they can no longer be combined with) the second mover can only combine a brick half with the other half of the same brick.
This leads to a map with the same outcome as the original gerrymandered map.
Of course it would be blindingly obvious if anyone actually attempted to do that, but there might be subtler ways to use contiguity to create a forced-choice situation.
That's a clever argument, though I wonder if it would be defeated by some restrictions on the chromatic polynomial. In your example, the number of 3-colorings up to isomorphism is extremely large, because the colors of any brick and its cobrick can be switched, and the mortar condition implies the districts are 3-colorable.
Of course, the chromatic polynomial is #P-complete, so this may pose some difficulty.
Concentric circles, or pie chart design would be problematic also.
The real problem with this strategy is that it accepts self-interested gerrymandering, rather than starting by rejecting it as wrong; immoral; anti-productive.
The whole gist of democracy is to start with the assumption that people are immoral and anti-productive. But if we pit them against each other, half of the immoral people will cancel out the other half. It takes only a tiny fraction of informed, intelligent people all arriving at the same solution (because it's correct) to nudge the result in the right way, most of the time.
It would be nice for that to work, because it means that you don't have to overcome the automatic argument of "I'm not immoral and anti-productive, you're immoral and anti-productive". You never have to tell anybody they're wrong or bad. You only have to tell them that they're not in the majority, which is an objective statement.
It would similarly be nice if we could have an objective but blame-free way to resolve the meta-question of electing representatives. Or at least, to have people say to their own partisans, "Hey, this is obviously unfair, could we tone it down a bit?" But the reply is always "If we don't do this unfair thing, the other people will do MORE unfair things, and then it will get even worse".
This is a great example of why the paper's methodology is so utterly bogus. A Markov chain algorithm isn't going to come up with strategies like this, but humans certainly will.
Hm. This doesn't seem like it does much if there's a sufficiently high concentration of cracked districts, relative to packed districts.
I gerrymandered during the define phase using classic packing / cracking strategies, such that I had 8 majority-B districts (2:3) and 2 majority-A districts (4:1 and 5:0), and unsurprisingly, the only districts I was able to combine that were majority-A were those that included the packed districts.
If the overall split was, say, 27:23 instead of 25:25 such that we could define 9 majority-B districts in the define phase, then I would only have been able to define a single majority-A district in the combine phase.
(And yes, all of these gerrymandered districts would be considered safe B seats, as one would expect with a 20% margin)
There are also potentially issues if the packed districts are geographically clustered -- we see this a lot in states with a single predominant urban center (e.g. Kansas, Minnesota, Kentucky). In those cases, you might be forced to combine multiple packed districts due to pathological maps. For instance, consider a map where a Democratic bastion is districted into concentric rings -- that satisfies the contiguity requirement, yet only the outermost district abuts any Democratic-minority districts.
Suppose you're defining as Party B, and you draw 8 majority-B districts (2:3) and 2 majority-A districts. Then, when Party A is combining, they would pair each of the majority-A districts with a majority-B district with a smaller margin, resulting in 2 A districts and 3 B districts. This is an improvement compared to if B drew 5 districts unilaterally, where it could draw 4 majority-B districts.
Yes, it is an improvement in this specific scenario, since there's limited granularity in how we can draw the districts.
If instead we had 9 population nodes we could assign to each Define district, then we might be able to draw nine 4:5 districts in favor of B, and a single 9:0 district in favor of A. In that case, A cannot recover any of the unfairness that B introduced.
Biggest potential weakness seems like the ability for the merging party to strategically collapse districts such that two legislators from the opposing party are made to reside in the same district, creating a new district with no incumbent. Not all legislatures require that representatives live in the district, and there would be real trade-offs to doing so, but it would be a pretty powerful hammer to wield against up-and-coming opposition candidates.
It doesn't matter at the abstraction of proportional representation, but it potentially matters quite a bit when you get into the nitty-gritty of actual elections.
You're right that this method doesn't protect incumbents. However, protecting incumbency and avoiding open-seat elections isn't necessarily a bad thing, and could increase electoral competition in some places. Some states don't allow incumbency to be taken into account when redistricting already.
Potentially in some form, but we haven't investigated it. The utility functions for each party would be very different. Instead of trying to maximize the seats that they win, parties would also need to think about the coalitions that could form if no party won a majority of the seats.
> "Valid districts are contiguous and have equal population. Strict constraints on compactness, geographic splits, or other restrictions are not necessary, but such limitations could be included. However, valid districts may not include “donuts,” where one district entirely encircles another."