Kristofer: This posting could have been entitled: "In which I admit that I don't know what I'm doing" :-)
In my most recent posting to this thread, I said that from December 2006 to January 2007, I'd been judging unfairness as s/q deviation per person. That discussion was a long time ago, and I got that impression of it from looking at Warren's website. But, looking at my postings from that time, I noticed that that wasn't what I was saying. Here's what led me, in December 2006, to day that Bias-Free, and not Webster, is the method that is unbiased if the probability distribution is assumed to be uniform: Let's speak of intervals between to successive integers, N and N+1. In those days I was calling that a "cycle". We could instead call it an "interval", but maybe I should stick with "cycle". Let's name the two integers bounding that cycle "a" and "b". Where b = a+1. My understanding is that, at first, I calculated the expected number of seats for a state that is between a and b. And also the expected amount of population (n terms q, the quotient when that state's population is divided by a particular divisor). I said that the expected s/q, in that cycle, is the expected seats divided by the expected q. (That's how I've been calculating it lately too) But then I said that that is a little rough or crude. I said that what we should really do is calculate it directly, by integrating s/q, with respect to q, from a to b. Since the probability density is assumed constant throughout (a,b), that gives the expectation for s/q in (a,b). I was wording it a little differently then. but it seems to me that what I said meant the same thing as what I've just described. At the time, that direct calculation sounded more reliable than assuming that expected s divided by expected q gives expected s/q. To tell the truth, lately I've not felt entirely comfortable with that assumption either. So, maybe I was right in December 2006, and have been wrong afterwards. When I later decided that expected s divided by expected q was the right way, that meant that Webster would be the unbiased method, with a uniform distribution. In a way that's a relief, that such a simple and precedented method is the unbiased method. But I have to admit that I was a little disappointed too, because I really liked the formula that I got for the roiundoff point R, with the more direct solution. R is the roundoff point in (a,b). Between a and R, s/q = a/q. Between R and b, s/q = b/q. So the integrals of those functions over those ranges are added, to get the integral of s/q from a to b. Then the result is set equal to 1, and the resulting equation solved for R. The formula that I got for R was: R = (b^b/a^a)*(1/e) That's b to the b power, divided by a to the a power, the result divided by e. Now isn't that an interesting and appealing formula? How nice if it turns out to be right. By the way, Warren agreed at his website that that formula is the right expression for R, to give an unbiased divisor method if the distribution is uniform. When I proposed that method in December 2007, I called it "Bias-Free" (BF). Of course, since the distribution isn't really uniform, I also suggested a Weighted BF. That's what Warren and I were discussing, and using different distribution approximations for. Then, at some point in the 1st two months of 2007, decided that expected s/q should be calculated by dividing expected s by expected q. But now it seems to me that I might very well have been right when I preferred that more direct calculation, the one that led to Bias-Free. For Weighted BF, the function to be integrated would be s/q multiplied by the probability density function (a function of q, for which an approximation would be used). When I decided that Webster and Weighted-Webster were better, Warren still preferred BF and Weighted-BF. (but he didn't use the same names that I used). Now it seems to me that Warren might have been right all this time. Mike Ossipoff ---- Election-Methods mailing list - see http://electorama.com/em for list info
