On Nov 19, 2009, at 5:35 PM, Kristofer Munsterhjelm wrote:
Juho wrote:
Well, this approach is also complex in the sense that the general
optimization algorithms may be as complex as you want, but the
optimization algorithms are totally independent of the politics and
the basic rules that determine what the final outcome should be
(the criterion) can be quite simple and intuitive.
(Additional criteria like favouring border lines that follow the
borders of states or rivers etc. can be easily included in the
agreed criterion. Maybe even higher cost of splitting cities etc.)
Splitline works because it's recursive. Any sort of divison
algorithm where you can smoothly control the relative sizes of the
two districts will work, also. Just subdivide into two, then freeze
one and subdivide the other. After you're done, unfreeze the first
(and so on). It may not produce the best result if the borders can
move on the unfrozen areas, but should work.
As for general optimization, if you're dealing with an election
method, then the optimization's approximation to the optimum (you
can't ensure it'll reach the true optimum if there are multiple
local optima and no additional structure) becomes a different rule
itself. For instance, Borda is a 5-approximation to the optimal
Kemeny ordering, but Borda is a completely different method from
Kemeny.
If you're dealing with redistricting, the competition solution that
you mentioned could work, but it might well be that, for
redistricting, capturing the exact tradeoff between looking like
"communities of interest" and being completely neutral is a task
best left to an independent commission. Of course, one can also
dissolve the problem rather than solve it, and employ some PR method
which would greatly diminish the incentive to do any gerrymandering
in the first place.
My thinking is that it might be easier to agree about the targets
rather than the whole procedure. The targets can be simpler to define.
Following Raph Franks model it would be thus enough to say that any N
points and the kn values and then derive the border lines and the
jointly agreed value of the solution from this data. That would not
leave much space for strategies and gerrymandering. The proposed
solutions would be evaluated and the one with best value would be
declared the winner.
The optimization procedures may not find the global optimum (but only
one of the local optima), but if there is an algorithm that can find
the global optimum then that solution will also be found. It is
possible that some party (that runs some optimization procedures)
would not publish the best solution it found (since the second best is
better to this party) but the field would be free for anyone else to
find that even better result. At some point one must freeze the
solution and ignore any better solution that someone might find later.
In most cases I guess it is quite improbable that better solutions
would be found later. And if they are found then they might not be
much better (probably true for most sensible criteria). No rules are
needed for the optimization algorithm (= just let the scientists and
politicians and private citizens do their best + maybe arrange some
official calculations too to make sure that at least someone makes a
serious attempt to find the best solution).
Juho
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