[EM] Methods with more nearly perfect unbias
Michael Ossipoff
email9648742 at gmail.com
Mon Jul 2 18:26:35 PDT 2012
Warren Smith and I have discussed methods with less bias than
Sainte-Lague/Webster. We discussed in for apportionment of congressional
seats to states, so I'll discuss in in those terms.
Warren suggested "Random-Rounding", in which, when a state's number of Hare
quotas is rounded up with a probability equal to the fractional part of its
number of Hare quotas.
Do that for each seat. If the total number of seats doesn't equal the
desired number, then do the randomizations again...until the desired number
of seats is allocated.
Of course randomness is sometimes not popular, but it's a method to
consider.
Someone suggested to Warren (and I've heard or read this suggestion
elsewhere too) that what should be kept as proportional as possible, is the
_time average_ of the states' s/p.
Those are good simple solutions.
A more conventional solution would be deterministic, and would consider
each decade's apportionment separately.
Warren and I have both suggested various less biased methods for that. Some
of mine were unworkable, because of (at that time) unexamined assumptions
that I'd made about the relation between the number of seats and the number
of states.
But one proposal of mine, which I called "Weighted Webster" is workable.
Warren and I both proposed implementations of it and variations of it.
Say we number the states, from smallest to largest.
Each state, then, has a "state-number". I let "S" stand for state number,
and "x" stand for population.
Though all of the states' state numbers are integers, find an interpolating
function to find the state number at population _between_ states, in some
region of interest. Interpolate
S(x) based on the S and the x for several states.
Warren suggested an exponential function, and that sounds fine.
Express population in terms of the quotient of dividing the state's
population by a common divisor, as the divisor methods do.
Consider the interval between x = a and x = b, where a and be are
consecutive integers. I'll call that interval "that interval".
What is the probability of some state in that range rounding up?
Say R is the roundoff point in that interval.
(S(b)-S(R))/(S(b)-S(a)) is the probability of rounding up. I'll abbreviate
that as "p".
What is the expected number of seats? It's :
a + p.
What is the expected population? It's:
a + 1/2
Set a + p equal to a + 1/2. (Of course write p out as defined above).
Solve for R.
That's the roundoff point that will make the expected s/v in that interval
equal to 1.
Solving for R, you first must solve for S(R). Then, knowing what S(R) is,
solve for R.
the x value for R will be a logarithmic function of S(R), if we've used an
exponential interpolating function.
There would be some rule for which states' S and x values to use for the
interpolation, for use in a particular a to b interval.
I've written that out hurriedly, because I have things to do, and can't
stay on the computer long. But I hope that well describes Weighted Webster
as I define it.
Of course, because this is hurried, I could have overlooked something, and
be posting nonsense. I'll take my chances. It's probably valid.
Now, to get about my household tasks.
Mike Ossipoff
Mike Ossipoff
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