When I posted last night, I'd looked at Webster and noticed that each cycle
of the step function has overall seats per quota that's a little less than
one seat per quota. I saw that as bias, and found a different rounding
formula by which the sum, over a step-function cycle, of the function's
deviatian from 1 seat per quota, would be zero.
But now, looking at it again, it occurs to me that there's nothing wrong
with Webster having net deviation from 1 seat per quota in each cycle, as
long as it's the same in each cycle. Now it seems that all that is needed is
that the sum of the seats(quotas) function's summed displacement from the
1-seat-per-quota line be zero. And Webster achieves that. So, last night, I
was making it more complicated than it is. I shouldn't have so quick to
conclude that Balinski & Young were mistaken about Webster being unbiased.
Anyway, I retract my statement that SL/Webster has bias.
The other roundoff formula that I posted last night would be the unbiased
one if we wanted the sum of s(q)/q's summed displacement from 1 to be zero.
But if we instead want the sum of s(q)'s displacement from the
1-seat-per-quota line to be zero, we have an easier problem, a simpler
formula, and it's Webster.
By the way, my demonstration that LR/Hamilton is unbiased used the same
assumption that I've used with other methods. But it's a reasonable
assumption. For instance, if a state's quotas are between two and three,
I've assumed that it could equally well be anywhere between two and three.
So I also retract what I said about LR's unbias being less
distribution-dependant than that of Webster.
Mike Ossipoff
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