This was performed in response to Mike's argument that Hill's apportionment method is more biased than Webster's. (As you will see, he's right.) In my simulation, I assumed:
* There are 50 states and 435 seats. * Each state is guaranteed one seat. That is, a state with population p is given max(1, r(p/q)) seats, where r is the appropriate rounding function for the apportionment method, and q is chosen so that the total number of seats comes out to 435. * State population is a uniformly-distributed random variable. * Bias is defined as Spearman's correlation coefficient between the states' populations and their seats/population ratios. A bias of +1 means that a high-population state will always be better-represented than a low-population state, and a bias of -1 means vice versa. * 10,000 simulations were performed for each method. ============ RESULTS ============ JEFFERSON'S METHOD: r(x) = floor(x) Sample mean: +0.35335543817527049 Standard deviation: 0.15941789295937267 95% C.I. for mean: (+0.3502, +0.3565) WEBSTER'S METHOD: r(x) = floor(x + 1/2) Sample mean: -0.11919100120047947 Standard deviation: 0.17043538286137358 95% C.I. for mean: (-0.1225, -0.1159) HILL'S METHOD: r(x) = x > gmean ? ceil(x) : floor(x) Sample mean: -0.17236094117647041 Standard deviation: 0.16535430662201497 95% C.I. for mean: (-0.1756, -0.1691) DEAN'S METHOD: r(x) = x > hmean ? ceil(x) : floor(x) Sample mean: -0.23259293157262878 Standard deviation: 0.16021643765947546 95% C.I. for mean: (-0.2357, -0.2295) ADAMS' METHOD: r(x) = ceil(x) Sample mean: -0.65397429051620526 Standard deviation: 0.10047298882085852 95% C.I. for mean: (-0.6559, -0.6520) Notice that the order of small-state favoritism is Adams>Dean>Hill>Webster>Jefferson, and that Webster's method is the least biased. These results are identical to Young's (http://www.brookings.edu/comm/policybriefs/pb88.htm) despite a different definition of bias. If the constitutional requirement for at least one seat per state is ignored (so that it's possible for a small state to get no seats), we get: JEFFERSON'S METHOD: r(x) = floor(x) Sample mean: +0.65593051455899887 Standard deviation: 0.099615581259577049 95% C.I. for mean: (+0.6540, +0.6579) WEBSTER'S METHOD: r(x) = floor(x + 1/2) Sample mean: +0.041516213579752617 Standard deviation: 0.17149636291830875 95% C.I. for mean: (+0.0382, +0.0449) HILL'S METHOD: r(x) = x > gmean ? ceil(x) : floor(x) Sample mean: -0.17763999039615813 Standard deviation: 0.16478759533582835 95% C.I. for mean: (-0.1809, -0.1744) DEAN'S METHOD: r(x) = x > hmean ? ceil(x) : floor(x) Sample mean: -0.22995669627851167 Standard deviation: 0.16273586509452664 95% C.I. for mean: (-0.2331, -0.2268) ADAMS' METHOD: r(x) = ceil(x) Sample mean: -0.65435152941176378 Standard deviation: 0.10090941303933143 95% C.I. for mean: (-0.6559, -0.6520) Webster is still the least biased of the five. (Note that the results for Hill, Dean, and Adams are not significantly different from the previous simulation; this is because these methods "naturally" enforce the one-seat minimum.) ---- election-methods mailing list - see http://electorama.com/em for list info
