[EM] concise apportionment formulas

Warren Smith wds at math.temple.edu
Sun Jan 28 12:28:41 PST 2007

WD Smith's apportionment methods concise formula as requested.

The new ones are   http://rangevoting.org/NewAppo.html
and an intro to apportionment generally is

All of the new ones are divisor methods.
The simplest one has "rounding point" between integers n and n+1 which is just
n+d  where d with 0<d<1 is a constant chosen to make it work best historically,
or if you want to ignore history and just rely on a theoretical model, then
employ the formula
   d = (1/K) * ln( K/[1-exp(-K)] ),     where    K = #states/#seats
which for the USA is K=50/435=10/87=0.11494252873... yielding d=0.495211255149063832...

That method does not give a 1-seat minimum; it will award small-enough states zero seats.
If you want to have a 1-seat minimum, then the first rounding point y (between 0 and 1)
needs to be replaced by  y=0.  The other rounding points y then are
still given by the usual formula   y=n+d,  for n=1,2,...  but now with a different value of d.
Again you can choose d to optimize historical performance, or in the same theoretical model
we have the exact formula
  d = ( ln(K) - ln(exp(K)-1) + ln(1/(1-K)) )/K
which when K=50/435 yields d=0.557504703.


There are also many other new methods described in  http://rangevoting.org/NewAppo.html
in the section titled "Alternative (less simple) theoretical models, goals, and results."
The rounding point formulas are given there in the form  y = (formula)
where y is the rounding point and the two consecutive integers are n and n+1 or
in some formulas A=n and B=n+1.
All of these formulas arise in the same way - you zero some "bias" notion in
some theoretical model.

In particular, the complicated formula which generalizes Mike Ossipoff's "bias free method"
is the following.  Choose the rounding point y, between integers A and B so that
   (A-B)*K*Ei1(K*y) + exp(-K*A) - exp(-K*B) - A*K*Ei1(A*K) + B*K*Ei1(B*K) = 0
where Ei1 is the exponential integral function and K is defined above.
The way in which you get Ossipoff's formula is, you take the limit K-->0+.
In this limit the rounding point formula simplifies to Ossipoff's
    y = (1/e) * B^B/A^A .

Warren D Smith

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