[Election-Methods] Spearman-unbiased apportionment

[Dan Bishop]

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.