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 CRAB simulations. 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. Forest ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
