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Gerrymandering is a practice intended to establish a political advantage for a particular party or group by manipulating district boundaries, and can create large disproportions in voting results as we recently often hear in media.

I have recently heard very nice podcast about mathematican (Jonathan Mattingly) finally providing hard evidence for Gerrymandering: which is finally able to start convincing regulators, "recognizing what is too far".

Specifically, they define score of given division of state into districts, then (Monte Carlo) sample 24000 divisions, getting some distribution of elected democrats (e.g. 3 to 7 out of 13) - and it has turned out that elected 3 was below 1% of the sample - what can be seen as a strong evidence of Gerrymandering.

It is great that mathematics can now point hard evidence there, but from one side it is after the fact, from the other it can only point extreme Gerrymandering - like 3 seats for 3-7 distribution. In contrast, we would rather expect that the number of seats just agrees with vote proportions on the state level.

So the question is if mathematics could also offer a solution to prevent such problems in the first place. Mathematicians might be a better than politicians group for finding such solution, so I would like to ask for such possibilities, for example:

  1. Proportionality on state level - we calculate the number of democrats that should represent given state from vote proportion on state level, then choose them from districts with the largest proportions. However, some districts would have minority representative.
  2. Open competition for choosing the division: regulators define restrictions for "proper" map and concrete evaluation function (e.g. compactness + uniform population + some geographical score) using shared common data, and deadline for the competition - then e.g. everybody can send own proposal and the one having the highest score in the deadline moment wins - this map will be used. While finding the best one is a hard problem, this way many would be motivated to use computational power for this search, and finally there would be one objective winner. The main question here is existence of objective criteria for map which is "proper"? They might differ between states, but evaluation score could be chosen on federal level.
  3. Requirement of proportion agreement of chosen map - a map to be accepted requires that the number of its elected seats, according to the last election data, has to agree with state-level proportions.

The 3. option seems safe, but still might leave a place for some bad behavior, so the best would be just having a clear definition of a "proper" map and some evaluation score - which should be maximized, hence just choosing the best one accordingly to this score would leave no place for political fight.

This evaluation score should contain some compactness - e.g. if preferred shape is circle, we can find the smallest circle district fits in, largest circle which fits in district - and put their proportion into the score: 1 corresponds to perfect circle district (analogously for square as perfect shape). We can also put proportion of district circumference and circle there. Its another term should be uniform population distribution, what can be also easily evaluated. This kind of evaluation is party-neutral and could be chosen on federal level - ending the problem of Gerrymandering once for all: there is just chosen proposed "proper" map having the highest evaluation score.

What are other possible solutions?

Could one of them be implemented? Can the Gerrymandering problem be ended once for all?

Ps. Parallel Reddit discussion.

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    $\begingroup$ A few comments: 1. You should describe what Gerrymandering is (drawing geographical boundaries of voting districts to swing elections); 2. this question is very broad, as this is a potential problem in every place where there is voting based on boundaries, so; 3. can you limit the problem to some objective mathematical problem? For example, it seems quite plausible to me that there would be statistical methods to measure the likelihood that gerrymandering is happening, but this is far removed from actually changing how boundary lines are drawn. $\endgroup$ – Joppy Nov 23 '18 at 12:13
  • $\begingroup$ Gerrymandering problem seems to recently quickly grow, especially in USA, what is all over the media. Mathematicians seem better than politicians for finding its solution (if one exists) - the purpose of this (indeed very open) question is asking if there are known at least theoretical solutions, especially having a chance being realistically implemented? $\endgroup$ – Jarek Duda Nov 23 '18 at 13:13
  • $\begingroup$ Related: math.stackexchange.com/q/2311243 $\endgroup$ – Rodrigo de Azevedo Sep 5 '20 at 15:20
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If you want to keep the single-member districts, there's a known algorithm for guaranteed-unbiased districting. It has its problems (this monstrosity is what it does to Colorado), and it might accidentally gerrymander a state in maximally-bad circumstances, but it's a whole lot better than getting the politicians to do it.

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    $\begingroup$ Indeed we can find heuristic algorithms - which are still too complex to convince regulators. So instead of proposing division algorithm, maybe it would be better to choose only evaluation function - simple enough for regulators. The complexity is in optimizing this function - but it is only a problem for people who search, regulators don't need to know anything about it - only finally take the best proposed accordingly to objective criterion. $\endgroup$ – Jarek Duda Nov 23 '18 at 12:40
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    $\begingroup$ And while mathematically this can be said to be "optimal", it actually contains a ton of inherent bias, for example, always tending to split up urban areas and grouping the pieces with a large amount of rural area (which tends to swing a certain way), giving those urban denizens less of a voice. For the purposes of the math stackexchange this is a correct algorithm, but it demonstrates something that should be touched on in any practical talks about gerrymandering - sometimes "mathematically unbiased" does not equate to "unbiased". That's what makes this problem so hard in practice. $\endgroup$ – Nico A Nov 24 '18 at 1:09

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