[EM] Why Mike Ossipoff's "unbised apportionment method" cannot be taken seriously

[Warren Smith]

Regarding why Mike Ossipoff's "bias free apportionment method" is bogus.
I am sorry Mike has forced me to explain this.  I wished to inform him
re flaws in private, but Ossipoff has successfully forced me, quite against my
will, to make a fool of him in public.

Let me quote his original post which attempted to explain what his method was
[comments by me in these brackets]

--START QUOTE
Is there a quota and roundoff method that's free of bias? 
["bias" is not defined by Ossipoff here; he expects us to mindread]
For quota and roundoff methods, such as Webster, Hill, etc., freedom from bias is only 
possible with some particular probability density disrtribution for the 
states' populations or their numbers of population quotas.

So let's say that that distribution is uniform.  [which uniform distribution?
Ossipoff again expects us to mindread, but I shall presume he means uniform
on the interval from 0 to 2*CountryPopulation/#states, since this is the 
unique uniform distribution which avoids negative numbers, includes 0, and has the right
expectation value.  Now actually the whole idea of having a uniform distribution is simply
misguided - see my   http://rangevoting.org/NewAppo.html, but we shall follow Ossipoff anyhow]

The quota and roundoff method that is unbiased 
["unbiased" is not defined by Ossipoff, who expects us to mindread]
is the one that has, as its 
roundoff point (between the integers a & b):

(b^b/a^a)(1/e)

The first of its successive roundoff points (to the nearest hundredth) are:

1.47, 2.48, 3.49, 4.49, 5.49, 6.49, 7.49, 8.5, 9.5
[Ossipoff omits to mention a=0 b=1 which yields 1^1/0^0/e = 1/e = 0.37 assuming 0^0=1]

These roundoff points are much closer to those of Webster than to those of 
Hill, suggesting that Webster is the least biased of the 5 standard quota 
and roundoff methods.

Maybe the above-described method has already been described, but if not, or 
if it hasn't been named, I'll call it the Unbiased Method
--END QUOTE
[Ossipoff nowhere defines what his method actually is, expecting us again to mindread,
but I am able to do this mindreading and believe he means the following:
"choose a number q>0 so that the total #congressmen comes out right in the below;
award state k a number of congressmen   Round(StatePop[k]/q)
where Round(x), for noninteger x>=0, is rounding to either a=floor(x) or b=ceiling(x)
if x<(b^b/a^a)/e or x>that, respectively."]
[Ossipoff, although he did not define bias or even try to in this post, in another post
provided this "clarification":
"a state in the higher part of a cycle has a better s/q 
expectation than a state in the lower part of a cycle. So it only makes 
sense for bias to compare whole cycles to each other. That's really the 
unspoken intention when we speak of bias."   I personally find this clarification completely
uninformative, even if Ossipoff had bothered to define "s" and "q" or "better", 
which he did not, expecting us to mindread as usual.    Incidentally although Ossipoff
provided no clue in that post what a "cycle" was I know from other Ossipoff posts he
means an interval from one integer to the next, is a "cycle," which is
a new and original and poor use of the word "cycle."
In yet another post Ossipoff "clarified" by saying
"I defined bias as systematic disparity in seats per person. Or, more 
conveniently, seats per population quota."   Here he nowhere defines what "systematic disparity"
might mean quantitatively, again expecting us to mindread.]

So, it is hard to point out a flaw in something so pathetically undefined as Ossipoff's
"derivation" of his magic formula   (b^b/a^a)/e.   
As I remarked to Ossipoff it is NOT my responsibility to point
out a flaw, especially in a method he has nowhere ever defined and nowhere ever derived
and whose main usefulness is solely restricted to mindreaders.
It on the contrary is HIS reponsibility to define and derive it.  Nevertheless, because
I am kind and generous, I have here pointed a number of deficiencies out.  

Now let us consider the case where the states all have uniform distribution
on [0, 1] of pop/q.  Mike Ossipoff has stated his derivation works in such a case,
and that the "unbiased" method is then to give each state 0 congressmen
if it has <1/e=0.3678794412 quota, and 1 congressman if it has more.
Oddly enough, the exact same apportionment is also claimed by Ossipoff to be
"unbiased" if the interval of uniformity instead is [0, 1.4].  But we'll let that 
idiocy lie unchallenged.

Now, let us suppose that Ossipoff's mysterious bias measure which he did not define,
is some rational function of x, call it R(x), [because it seems ridiculous to suppose otherwise]
and he is choosing the "roundoff point"
y so that the expected value of R(x) for x in [0,1] is zero.
Then y must be determined by an equation of the form
    integral(from 0 to y) some rational function of x  dx
=   integral(from y to 1) same rational function of x  dx
(where by rational fn I mean ratio of two polynomials with rational-number coefficients).
What are the solutions y of such equations?
Well, they are always of the form
    ln(some rational fn of y) = some rational fn of y.

Which brings me to my point:  y=1/e  can never be a solution of
any such equation unless the right hand side is a rational CONSTANT, assuming
(basically) that e^e and 1 and e and e^2 and e^3 ... are algebraically independent.
But if so, then  y=(b^b/a^a)/e  can never be the solution of such an
equation where a and b are nontrivial integers (assuming ln(2), ln(3), ... are rationally
independent of the preceding)!

So, although it is hard for me to point out a flaw in something unstated that I have to mindread,
I nevertheless have overcome that hardness, and thus proved (under some commonly
believed number theoretic assumptions)  Ossipoff had to have a bogus formula -
even without actually knowing what his derivation was!

This kind of thing is called by mathematicians a "sanity check."  One can often see formulas
cannot be right even knowing next to nothing about how they supposedly got there.
I believe Mike's formula also fails some other sanity checks, and, purely because of my damaged 
psychology and angriness and weird spite, I will refrain from describing them.  You will just
have to be satisfied with what I have described, because, as I said, this far exceeds
my responsibilities on this matter.  Indeed, Mike expecting anybody to believe him and advcate
his method is really quite astounding, and indeed as far as I can tell, so far, nobody has 
claimed to have  understood  or supported Mike's "unbiased method" whatever, aside from himself.

I hope that Ossipoff will refrain from ranting about my "angry tone" in future and
indeed hope he will refrain from posting anything about apportionment whatever until such
time as he can actually fully define what he proposes and fully provide a PROOF his
proposal follows from some stated set of axioms.  Also, as a hint for the future Mike:
if I point out your math has to be bogus, then do not inform me I'm doing it out of "spite";
then launch a worldwide broadcast about same and my sad psychological hangups; 
instead either admit it, or try to produce a valid proof and actual definition.
E.g, you successfully defined "cycle Webster" in what I would call a "constructive response."

In the present case, your method suffers from so large a number of problems
that it probably is not even worth trying for any kind of salvage.  But of course,
even though we know your claims were bogus under a large class of mindreading
assumptions, the fact remains that I and everybody else are not perfect mindreaders,
and it is therefore possible that, with sufficiently generous mindreading assumptions
about what you could possibly have meant, you might be right.  
For example, under the axiom "Mike's formula is right", which quite possibly
constitutes the definitions that you omitted of bias, uniform, disparity, etc. we can derive
with perfect rigor the conclusion "Mike's formula is right."

Cheers, Warren D. Smith
http://rangevoting.org