[EM] Bias-Free in the historical census apportionments

[Michael Ossipoff]

In the recently-posted historical census apportionments, BF did worse than 
Webster in every census apportionment where those 2 methods apportioned 
differently.

Dan, did you definitely do the Bias-Free that I defined in the posting just 
before this one?

If so, then the only possible explanation is that the free seats required by 
the seats-for-every-state rule caused so much small-bias that Webster's 
slight large-bias is needed to cancel it out, resulting in less measured 
bias with Webster.

Bias-Free is genuinely bias-free if the frequency distribution is flat. With 
a flat distribution, Webster is sliglhtly large-biased. Add a slight 
large-bias due to the distribution, and both methods' apportionments are 
large-biased, but Webster moreso, because Webster has some large-bias of its 
own.

So it must be that the free-seat small-bias is making Webster the best one, 
because it comes closest to exactly canceling out the free-seat small-bias.

Of course Bias-Free and Weighted Bias-Free could have their rounding points 
carefully moved upward in order to very nearly cancel out the free-seat 
small-bias.

The best solution is to make sure that the House is large enough that all 
the states _qualify_ for at least one seat, in every census. That could be 
done separately, or a method could carry it out as part of the method's 
rules.

In any case, since that's how the House should be, I claim that it's 
important to also report correlations based on not having a free-seat rule.

Mike Ossipoff