# [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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