[EM] Adjusted Rounding--Webster by cycles

[MIKE OSSIPOFF]

Webster individually puts each state as close as possible to one seat per 
quota, but we've seen how optimization for states and pairs of states 
doesn't do much for overall unbias, uniform s/q expectation.

The important thing is that each _cycle_ be as close as possible to 1 seat 
per quota. So why not do Webster by cycle instead of by state?

For each cycle, each interval between integers, choose the round-up point so 
that the overall s/q of the states in that cycle will be as close as 
possible to 1.

It occurred to me that, in my 10-state example, population distribution was 
uniform, and that Hamilton's and Bias-Free's justification assumes that. 
But, if that isn't so in real apportionments, then the s/q of the biggest 
half of the states might systematically differ form that of the smallest 
half of the states. In particular, if the population-density distribution is 
single-peaked at the middle, maybe normally distributed, one would expect 
small states to be favored by methods optimized for uniform population 
distribution. Of course one could make a Bias-Free version for normal 
distribution, but Adjusted Rounding seems a more adaptable solution.

I haven't abandoned Bias-Free or Hamilton. I haven't done any apportionments 
based on censuses. Maybe Adjusted Rounding isn't needed, if Bias-Free and 
Hamilton do well in actual censuses. On the other hand, Adjusted-Rounding is 
easier to convince people of than Bias-Free, and won't put off people who 
don't like mathematical formulas.

Mike Ossipoff

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