[EM] Aha, think I understand Ossipoff now (& apologize for misunderstanding...)

[Warren Smith]

Here is an attempt to derive what Mike Ossipoff MAY have had in mind
for his "unbiased method."  I have no mindreading ability, so this is only a guess.
But I believe I now have the right guess.

What is the value y so that
  integral(from A to y)  1-A/x  dx  =    integral(from y to B)  B/x-1 dx  ?
This value magically causes the "disparity in seats per population"  (which Ossipoff had
once said was "bias") to average out to zero, if some multiple of population is assumed
uniform random in  the interval [A,B] and we round that multiple x of population up to B seats, 
or down to A seats, if x>y or x<y respectively.  Note that 1=x/x is the reason for the 1s in 
the integrands, and "disparity" here means 
"additive difference from the ideal noninteger apportionment."

Doing the integrals, we get
    y-A - A*ln(y/A) =  B*ln(B/y) + y-B;
with solution
    y = (B^B / A^A)/e
agreeing with Mike Ossipoff's "solution"!
AHA!

Hmm, oddly enough, this formula actually works despite my "proof" last post based on
number theory that it could not.  How can that be?  Because the two rational functions
in the two integrands are not the same, that is why (my sanity check had assumed
they were the same!).

OK, good - Ossipoff's method has at least now been promoted into the land of
understanding what it is.  However, it remains questionable in the sense that the uniform
distribution assumption is self-contradictory (as I pointed out last post re 1 and 1.4)
and unjustified (as discussed http://rangevoting.org/NewAppo.html).
If it really were uniform it'd cutoff at some point, and near that point Ossipoff's
formula would no longer be correct.  You could still make an apportionment
method out of Ossipoff's formula plus another formula for dealing with the high
end of the uniform distribution (Ossipoff did not) but even then you'd still have
a bogus method because what if a state population just happens to be fairly far ABOVE
the high end of the uniform distribution?  In that case the way of dealing with such a state 
would be unspecified.

So then we can (which is what Ossipoff presumably did) just ignore all of that
and proclaim we are going to use the  y = (B^B / A^A)/e  formula come hell or high water
(including in regimes where the mathematical justification above clearly is invalid).
That is fine if all that is asked is to define a method, not to present a justification
for how the method got there.

Now let us ask, what if we assume exponential not uniform distribution - which has
the advantage of being self-similar, and with no high-cutoff necessary, 
causing the formula we shall get to be valid everywhere - and ask for y so that
  integral(from A to y)  (1-A/x)*exp(-K*x)  dx  =    integral(from y to B)  (B/x-1)*exp(-K*x) dx  ?
then we get a nasty transcendental
equation involving "exponential integrals" (higher transcendental fns)
to solve numerically.  Specifically the equation is
   K*A*Ei(1,K*y)+exp(-K*A)-K*A*Ei(1,K*A)+B*K*Ei(1,B*K)-exp(-B*K)-B*K*Ei(1,K*y) = 0.
Here K=#states/#seats
if we want the exponential to have the right expectation value, namely #seats/#states
for the #seats that a state ought to get.

I had derived this equation in an earlier email to Ossipoff some days ago.
[In the LIMIT K-->0+ the y-formula you get this way, coincides with Ossipoff's formula.]

Because in the USA,  K=50/435=0.11494  is fairly small,  this limit K-->0+  is not
terribly far from the truth, and thus we expect Ossipoff's formula to have some
amount of real-world validity, thus presumably explaining why it did well in
numerical tests on US censuses that Ossipoff mysteriously alluded to once.

Warren D Smith
http://rangevoting.org