[EM] District continuity preserving re-districting

[atarr at purdue.edu]

Matt wrote:

> That may by OK if tweaking the model to get a preferred outcome is OK.  But
> if our goal is keep out gerrymandering then we are probably better off
> without placing tweak variables in the objective function.  Using the same
> units of measurement for all of the variables in the objective function
> avoids the need for such tweak variables.

Only if there is only one input variable in the objective function.  As soon as 
you have more than one input variable, there is some implicit weight being 
applied to each input.  All I'm doing is making it explicit.

If some scale is imposed on the initial metrics, then it is entirely possible 
to get away with a = b = 1.  For example, you could require that a single 
district covering the enitire area would produce a metric of zero, and one 
district for every census block would produce a metric of one.

> If we are going to introduce tweak variables then I think it is better to
> place them in the constraints.  Each constraint is for a single unit of
> measure so different units would have their own set of constraints. 
> Constraints can be iteratively tightened/loosened to find the boundary
> conditions that define which combination of constraints are infeasible for a
> given map input.  A well-defined formal procedure for doing this
> systematically can remove subjectivity.
> 
> For example, the maximum difference between district population size
> constraint can be reduced to the minimum feasible amount first.  Then another
> constraint, such as the district continuity constraint, can be added and
> iteratively tightened to find the feasibility boundary.  After the
> feasibility boundary is found the constraint can be loosened by some
> pre-determined method to provide slack for optimizing the objective.  This
> works best if the number of constraints is kept to a minimum.

There's still some built-in subjectivity here in the order you apply these 
constraints.

> It is better to avoid non-linearity in the model and to keep integers out of
> the objective function.  Non-linear integer optimization is more difficult
> than integer linear optimization which is more difficult than non-integer
> linear optimization.  

We can't really avoid discrete operations here.  People are not continuous.  
More to the point, I think it's a good idea to not allow census blocks to be 
split into more than one disctrict.  This means you must work in groups of 
several thousand people.

> All other things being equal the less complicated the
> optimization model the more opportunity for mathematically sophisticated
> citizens to be competitive re-districting participants and the more
> consistently the re-districting result will be close to reaching the
> objective and thus satisfy the public and the courts that the process is
> fair.

If the data containing the population, position, connections, and current 
district assignment of each census block is released to the public, writing a 
program that iterates solutions would not be too hard.  Graph theory is a 
pretty well-developed subject.

> Also, from this perspective it is better to provide proportional
> representation opportunity by implementing a PR election method within multi
> member districts than to add additional constraints in an effort to corral a
> majority of some minority into a single member district. 

I'm not sure exactly what you mean here, but these approaches can be used 
whether you're splitting California into 54 districts, or 7 super-districts.

> Optimizing for
> compact districts is not neutral, it can favor or disfavor minority
> representation depending on the geographic and demographic details. 

True, but I'm very comfortable with designing districts that place close 
communities into the same district.  If this results in some unintended bias 
one way or the other, that's fine.  EVERY apportionment method is going to 
benefit somebody.

> Of
> course, geographic and demographic details impact PR election methods also,
> but with multiple districts the overall impact may be more balanced.

Absolutely - this is yet another good argument for PR.

-Adam