At the end of last year, we had a discussion about which apportionment method was the most unbiased. Using a definition of "bias" as Spearman's correlation coefficient between population and seats/population ratio, and simulations including historical U.S. Census data, we concluded that Webster was the least-biased of the five classical divisor methods.
However, "least-biased" is not the same as "unbiased". So, I've been wondering if there was a divisor method with ZERO bias. For this, I considered "power mean" divisor methods. Recall that a divisor method gives a state S[k] = max(r(P[k] / Q), M) seats, where * M is the minimum number of seats per state * P[k] is the population of state k * Q is a "divisor" chosen to make sum(S) come out right * r is the rounding function r(x) = x > m(floor(x), ceil(x)) ? ceil(x) : floor(x), where m is a generalized mean function The power mean of degree p is m(x, y) = (x^p/2 + y^p/2)^(1/p). Special cases are: p -> -inf => m(x, y) = min(x, y) => Adams' method p = -1 => m(x, y) = 2*x*y/(x+y) = harmonic mean => Dean's method p -> 0 => m(x, y) = sqrt(x * y) = geometric mean => Huntington-Hill method p = 1 => m(x, y) = (x+y)/2 = arithmetic mean => Webster's method p -> +inf => m(x, y) = max(x, y) => Jefferson's method Spearman bias is a nondecreasing function of p. Based on the historical census data, a mean bias of zero occurs when p=2.68. Later, I shall post some simulation results with random populations. ---- Election-Methods mailing list - see http://electorama.com/em for list info
