After testing Hill and Bias-Free in the 10-state example, it occurred to me
to also test Hamilton. Hamilton's allocation was about 2.8 times less biased
than that of Hill. Bias-Free had tested more than 3 times less biased than
Hill.
I'd said that Bias-Free and Hamilton are the completely unbiased methods.
The probability that, by chance, Hill would finish last, just as I'd
predicted, is of course only 1/3.
Though both are unbiased, one would expect Hamilton to probably do not quite
as well as Bias-Free, due to Hamilton's randomness. The probability that, by
chance, those 3 methods would finish in the predicted order is only 1/6.
Surely Balinski & Young must have done apportionments for all the historical
censuses, by Hamilton, Webster and Hill, and compared those allocations for
bias. Has anyone done such comparisons?
For instance, someone recently posted that Webster and Hill gave the same
allocation for 2000. But, in the censuses where they differ, how does their
bias compare? And if anyone is going to do such comparisons, I'd suggest
testing Bias-Free along with Hamilton, Hill, and Webster.
I've been reporting measured bias as the ratio or difference of the average
seats per quota (or person) for the largest half of the states and the
smallest half of the states.
Mike Ossipoff
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