[EM] Strong FBC, at last
fsimmons at pcc.edu
Thu Jan 30 09:43:48 PST 2003
On Wed, 29 Jan 2003, Alex Small wrote:
> I think the Partial Decisiveness condition removes the possibility of
> fractal boundaries, since I specified that the ties occur on a set of
> 4 dimensions (or N!-2 dimensions for N candidate races). I don't know
> much about fractal curves in a mathematical sense (although I know a
> tiny bit about experimental studies on fractals in physics and
> materials science), so I'm not certain that Partial Decisiveness
> removes fractal boundaries, but that's my best stab at it right now.
It depends on what kind of dimension you are talking about and how you
define "fractal." Mandelbroit deliberately left the definition of fractal
somewhat open. Some folks don't consider a fractal-like set (i.e. a set
with infinitely intricate detail) to be a genuine fractal unless its
Hausdorff dimension is strictly greater than its topological dimension.
Fractals can be generated randomly or deterministically. Some
deterministically generated fractals are hard to distinguish from the ones
generated randomly, just as deterministic chaos is hard to distinguish
from random chaos on the basis of superficial appearances.
Now suppose for the sake of argument that it takes randomness to get an
absolutely non-manipulable election method. Then pseudo-randomness could
be used to get a practically non-manipulable method.
How would this pseudo-randomness manifest itself in the geometry of the
victory regions? My guess is that the boundaries of these regions would
have a fractal appearance, similar to what Rob LeGrand observed in his
Note that pseudo-randomness is perfectly deterministic, but hard to
distinguish from genuine randomness (if there truly is such a thing).
One could use as the seed for a pseudo-random number generator the
fractional part of the square root of the common logarithm of the number
of voters, or some other ridiculous number.
[Imagine trying to explain this to the typical voter.]
However, in Declared Strategy Voting Cumulative Repeated Approval
Balloting (DSV-CRAB), the length of a cycle is a (natural) pseudo-random
number that varies chaotically (but deterministically) with the number of
voters in each faction, so no contrived seed is needed, and the typical
voter has no reason to worry about randomness.
Ordinary monotonicity goes out the window, but it is replaced with
monotone expectation, etc.
If we feel a need to approximate the Small Voting Machine much closer than
CSSD (winning votes/grade ballot version), then this "natural
pseudo-random" approach might be worth exploring in more depth.
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