Adjusted Rounding, which I'm going to call "Cycle Webster", is nothing other
than Webster applied to cycles instead of to individual states. A cycle is
an interval from one whole number of quotas to the next. Like from 4 quotas
to 5 quotas.
The states in a particular cycle have, together, a certain number of quotas,
and so the cycle wins, in Webster, a certain number of seats. The
rounding-up goes to the more populous states in the cycle, till all the
cycle's seats are given to its states.
Of course, in the event that, with a particular quota, some states get zero
seats, it's necessary to give them one anyway, as in ordinary Webster.
Cycle Webster is the completely and unconditionally unbiased method.
Everyone in the country would have exactly the same expectation for
representation. No other method can offer that. With even the best
fixed-rounding-point method, Bias-Free, vagaries of the distribution of
population frequency density could cause some bias.
That distribution function is a decreasing function, decreasing at a
decreasing rate. That means that Bias Free will actually very slightly favor
larger states, but not as much as Webster, which adds a little large-bias of
its own. And not as much as Hill, whose bias isn't subtle at all.
In the event that only a fixed-rounding-point method were acceptable, one
could use weighting to try to account for the distribution, to make a
Weighted-Bias-Free. Then, having made it thereby as unbiased as possible,
either adjust the weighting function or lower the rounding points a little,
to make it slightly but reliably small-biased, so that no one can say that
you're trying to replace Hill with something that favors large states.
But Cycle-Webster is my best proposal.
One could likewise define a Cycle-Hamilton. It too would be completely
unbiased, but wouldn't have the immediate optimality of Cycle-Webster.
Mike Ossipoff
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