[EM] apportionment, some new methods as food for thought; ossipoff&bias
Warren Smith
wds at math.temple.edu
Sat Jan 20 13:53:53 PST 2007
More on apportionment.
Ossipoff says the one crucial property is "unbiasedness" which tells us
that he really ought to formally DEFINE what he thinks unbiasedness is, so we can
have a clue what unbiasedness property(ies) his methods obey.
If unbiasedness is All, then I wonder why Ossipoff is not satisfied with the Hamilton method.
Does it obey his notions of unbiasedness?
Here are two more apportionment methods, which I basically do not understand at present,
but which would probably be good food for thought.
Notation:
P[k]=population of state k (sum over k is P).
S[k]=integer seatcount of state k (sum over k is S).
Method 1: Among all possible nonnegative-integer vectors S[] with the correct sum S,
choose the one which minimizes
sum(over k) P[k]^Q * |S[k]/P[k] - S/P| .
Here Q>=0 is a constant and we perhaps get different methods if Q differs.
Method 2 ["angle minimizing method"]:
Among all possible nonnegative-integer vectors S[] with the correct sum S,
choose the one which minimizes the angle between it and the population vector P
(both regarded as vectors in N-space, N=#states).
I wonder what properties these obey?
I also wonder if they are implementable via efficient algorithms?
(It's amazing what a tough subject apportionment is, one's first impression is it is a triviality.)
Warren D Smith
http://rangevoting.org
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