[EM] Compactness

[Joe Weinstein]

COMPACTNESS  (Was:  Advantages of Dirichlet Region Districts)


It always struck me that conceptually the obvious and direct approach to 
compactness of a district is much as Forest has argued.  Namely, one should 
measure sprawl (inverse of compactness) by the average distance (or maybe 
average squared distance)  between residences, taken over all pairs of 
inhabitants.  (Variant: use the average, over all inhabitants, of the 
distance or squared distance from the residence to a suitable ‘centroid’: a 
‘central’ point chosen to minimize the resulting average.)

However, various writers on districting prefer instead cutely to use an 
extrinsic and sometimes more complex measure - boundary length.   Hitherto, 
out of a certain feeling of charity, I have tended to regard this 
substitution as innocuous and acceptable.

After all, many well-meaning people tend to believe that true credibility 
and worth require complexity, indirectness and even inscrutability.  [Aside. 
Such include some IRVites when they first hear about AV.  They instinctively 
reject AV precisely because AV is such a simple, easy, direct and obvious 
way to realize basic goals - in this case, of avoiding spoilage and getting 
an acceptable winner.]

I was wrong to be so casually accepting.  I want to thank Forest for making 
three important and persuasive arguments:

(1)  Substitution of boundary-lengths for inherent sprawl is NOT innocuous, 
because boundaries are inherently ambiguous: DIFFERENT boundaries can 
describe the SAME district.  Some boundary descriptions can be easier than 
others to comprehend, or lead to differing boundary lengths, but these 
variations do not alter what the district actually IS.  In effect, a 
district is really defined not by boundaries but by the set of its 
inhabitants’ ascribed residence positions - or information which is 
equivalent to that.

(2)  Compactness really IS or SHOULD BE intrinsic:  the sprawl of a district 
really does or should depend only on the district itself, not also on other 
districts.

(3)  For the sakes both of conceptual simplicity and of practical 
computation of a districting plan’s ‘goodness’ or ‘merit’, it’s preferable 
to define compactness or any other amenable concept intrinsically rather 
than extrinsically, whenever we can manage to do so.  In particular, it’s a 
lot simpler to compute just the average, over all SINGLE districts, of the 
district’s intrinsic sprawl, as versus having to compute the average, over 
all PAIRS of districts, of the weighted boundary between the pair.

Joe Weinstein

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