I defined and described BF in my most recent posting in the thread called Sainte-Lague and bias, or something like that.
I mentioned that, in December 2006, I posted here that BF is the unbiased divisor-method for allocation, if one is equally likely to find a state anywhere on the population-scale. Then, a few months later, I, for some reason, posted that that was mistaken, and that Webster is the unbiased one after-all. Looking at the problem now, I have no idea why I said that. Looking at the problem now, it's obvious that it's BF. ...for the reason that I describe in my most recent post to the Sainte-Lague & Bias thread. For a quotient, q, (result of dividing a state's population by some common divisor) between the consecutive integers a and b, BF's rounding point, R, is: R = (b^b/a^a)(1/e). For the assumption of uniform probability distribution, BF is the completely unbiased divisor method. With uniform distribution, Webster is about 1.9% large-biased. By that I mean; Say there's a small state whose quotient is between 1 and 2, and is equally likely to be anywhere in that interval. Say there's a large state whose quotient is between 53 and 54, and is equally likely to be anywhere in that interval. The large state's expected s/q is 1.9% greater, that's 1.019 times greater, than that of the small state. Those integers (1,2,53 & 54) were chosen to cover the range between the largest and smallest states for which s/q should be equal, so as to give the greatest factor by which s/q can differ with whatever method is being discussed. But Webster's round-off point is closer to that of BF than Hill's round-off point is. Between 1 and 2, Webster's round-off point is about twice as close to that of BF as Hill's is. For a non-uniform distribution, Weighted-BF is the unbiased divisor method. In the other posting, my most recent one at the Sainte-Lague & bias thread,I described how one could find the round-off point for Weighted-BF (for a non-uniform distribution). Mike Ossipoff ---- Election-Methods mailing list - see http://electorama.com/em for list info
