[EM] Apportionment bias simulation

[Dan Bishop]

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.)