[EM] Automatic districting. Juho: Largest-Remainder.

[Michael Ossipoff]

By the way, isn't it true that gerrymandering is really only a problem when
the elections are between two roughly equal parties. I wonder how much
gerrymandering would matter in our elections when we no longer consider the
Republicans and Democrats to be "the two choices". I suppose that, even
then, if there are regions where Democrats and Republicans can still win,
then the districts could be carefully drawn so that they'd "lose big and win
small", as Gilmour described it. But I doubt that that would help the
Republocrats much, under those new conditions.


Automatic Districting:

When I refer to a district's "diameter", I mean the maximum straight-line
distance across it.

I'd like to revise the optimization criterion by which the line, for the
bottom of a band, is positioned:

Position the line so as to minimize either the maximum or the arithmetic
mean of the diameters of the districts in that band.

Diameter seems the simplest and most useful measure of maximum
campaign-travel distance in a district.

Shortest-Split-Line isn't too much more complicated than Band-Rectangle, and
Shortest-Split-Line's computational requirements are a lot less than some
other automatic districting proposals. Shortest-Split-Line would be a good
alternative choice, though nothing matches the simplicity of Band-Rectangle.

I'll abbreviate Band-Rectangle Districting as "BR". I'll abbreviate
Shortest-Split-Line as "SSL".

Though BR is neater, resembling the way that many states are divided into
rectangular counties, SSL, in an irregularly shaped state like California,
will look more artistically interesting and creative.

Maybe the full name for SSL should be Successive Halving for Shortest
Split-Line. Though, if the number of district population units in the area
to be "halved" is odd, of course it won't really be halving, but it's a
division that's as close to halving as possible. When I say "halving", I
mean the kind of halving or near halving that Warren spoke of.

SSL makes sense, because, if, when halving a region, you divide along that
region's shortest suitable crossing line, you're also dividing the region's
longer dimension, tending to keep the diameters of the resulting two regions
low. A good goal.

But then, why not make that goal explicit? Instead of dividing along the
shortest crossing line that halves the district, why not choose the halving
division so that it minimizes either the maximum or the sum of the two
resulting subregions? 

As for the other automatic districting method whose URL Ted posted, it's
obviously a _lot_ more computation-intensive than SSL or BR. I  don't think
that it's completely or well specified either. For example, it refers to a
point's distance from the "center". Is he suggesting that the centroid of
each putative district be calculated for that purpose?

But, if computation-time were no object, one could just look for a division
of the state, by straight lines (great circles) into districts, so as to
minimize either the maximum or the sum of the diameters of the districts.

It's simple enough to suggest, but finding that division would be very
computation-intensive.

I'd expect hexagons. If the district's population were uniformly-distributed
(and at least if flat surface were (contrafactually) assumed), wouldn't it
be mostly regular hexagons?

Juho: Largest Remainder:

I mentioned someone I used to know who fiercely advocated Largest-Remainder
against Sainte-Lague, saying that LR would be easier to enact because people
would be more likely to accept it, due to its simplicity.

Of course that sounds like my argument for Approval. But there's a big
difference: Approval's simplicity is _elegance_.  LR's simplicity involves a
parting-of-the-ways with proportionality, and I didn't find that to be
elegant. I found it to be unaesthetic.

As I was saying, I can understand the justification for using d'Hondt to
allocate seats to parties. And I know why you don't use d'Hondt to allocate
seats to districts: Proportionality is much more important when you're
allocating seats to districts. Each district should have the same voting
power per person. At least on the average, as it does, an average unaffected
by Largest Remainder's random fluctuation.

And so, you don't want a biased method for allocating seats to districts.
Largest-Remainder was chosen because it's unbiased. Why wasn't Sainte-Lague
chosen? Because it's often mistakenly said to be biased in favor of large
parties. But it isn't. As I've been saying, Sainte-Lague is unbiased, and is
the optimally proportional allocation.

Balinsky & Young verify that in their book "Fair Proportionality" or "Fair
Representation", a book about apportioning the U.S. House of
Representatives.  I'm not sure of the title. Probably "Fair Representation".
As you know, in U.S. House of Representatives apportionment, Sainte-Lague is
called "Webster's method", having been proposed by Daniel Webster. We used
Webster's method for a while for HR apportionment. We've used a number of
different methods. Now we're using a large-biased method that could be
called "Hall's Folly". Thomas Jefferson proposed the method known as
d'Hondt, and, for HR apportionment, it's called Jefferson's method. We used
it for a while too.  We used Largest Remainder too, and it was abandoned due
to its paradoxes. Of course, given the other things that work against equal
voting power per person here, it makes not the slightest difference which
apportionment method we use. In fact, our only real problem is Plurality.

I've also demonstrated the optimal proportionality of Sainte-Lague at the
Barnsdale (barnsdle) electoral reform website.

So, seats should be allocated to districts by Sainte-Lague.

Mike Ossipoff