[EM] Replies to Warren's apportionment webpag, in which I am featured.

[Michael Ossipoff]

Warren's apportionment webpage was recently (accurately) quoted to say
that Webster's method minimizes, for every pair of states, the
difference between the s/q of those two states.

Warren actually posted that statement at his website. When that quote
was posted, I immediately posted a demonstration of why Warren's
statement can't be true.

Where Warren's webpage lists many muddled, poorly-justified, versions
of what he calls "improved Webster", he said that he prefers the
version that judges the importance of an s/q deviation by considering
the population of the state for which it occurs. Divide the s/q
deviation by the state's population, to find the s/q wrongness per
person. ...because, the more people in a state, the more people are
treated unfairly by the s/q deviation (if it's in the negative
direction), the more people whose utility is thereby lowered. But what
about a state with _too much_ s/q? Is it good that the utility of all
those people in a large state is raised thereby? Or is it worse,
because a larger number of people have, for their state, an unfairly
large s/q?  :-)

It's most peculiar for that to be the rationale for what was billed as
a way to achieve unbias. Somehow attempting to maximize utility in an
allocation, in that way, can be Warren's goal, but it has nothing to
do with unbias. Divisor methods achieve unbias by giving to everyone
and equal expected s/q.

But if that's the principle of Warren's best method for unbias, then
that might help to explain the following:

Someone recently quoted Warren's favorite recommendation for an
unbiased apportionment method:

In each interval between two integers, the rounding point is the lower
of those two integers + .4954.

After that was quoted, I said that it implies that Webster is only
biased by 1/3 of one percent. But then I found out that, even with a
uniform distribution, Webster is biased by about 1.9%.

For comparing the s/q of a large state to that of a small state, it's
the small state's s/q that varies a lot, and whose rounding point, in
these methods, differs most from .5

So consider the interval between 1 and 2.

Warren's method's rounding point is at 1.4954

Webster's rounding point is at 1.5

BF's rounding point is at about 1.47  (BF stands for "Bias-Free)

(I've showed told why BF is the unbiased divisor method when the
distribution is uniform)

Warren's rounding point is about 5 times closer to that of Webster
than to that of BF.

For uniform distribution, BF is unbiased, and Webster is biased by 1.9%.

But Warren's method is intended for a certain non-uniform
distribution. One that is a decreasing function, decreasing at a
decreasing rate. That causes more unbias.
Therefore, with his own distribution function, his method is even more biased.

We needn't wade through the (probably gibberish) jargon at Warren's
website, to work on the evaluation of his method proposals. The above
brief discussion is enough to show that Warren's favorite
recommendation for an unbiased method is biased nearly as much as
Webster, nearly 1.9%. large-biased.

And even with Warren's distribution function, his method is much more
biased than BF.

Why am I posting about Warren's apportionment webpage? Because I'm
mentioned in it.

Let me quote Warren:

"Ossipoff came up with the idea of trying to seek an unbiased
apportionment method"

[endquote]

Actually, long before that webpage was written, I had proposed,
posted, defined, described, BF, and described its derivation. As I've
said, BF is the unbiased divisor method, for a uniform probability
density for states, by population.

I'd also proposed Weighted-BF, defined it, and described its
derivation, given instructions for its derivation. Both definitions
and descriptions were brief, clear and concise. They were like the
definitions and descriptions that I've posted for those methods during
the past few days.

Weighted BF takes into account a non-uniform probability density
distribution, approximated by some function. I mentioned a few
possible approximating functions that could be used.

I'd posted a ready-to-use formula for BF's rounding point.  R = (b^b/a^a)(1/e).

I had posted those things before Warren entered the discussion, well
before Warren posted about them.

Warren then said:

"...although his execution of the ideas was poor."

[endquote]

I'd asked Warren for specifics about that, about what he meant, about
what, in particular, was poorly executed. He was never able to give
any specifics.

BF was completely and correctly "executed", right to the point of
posting the formula for its rounding-point.

I'd defined and described Weighted-BF (WBF), and outlined its
derivation, giving instructions for its derivation.

I don't know if Warren would call that "execution". If so, I executed
it. But if "execution" means posting a formula for the rounding point,
then I didn't "execute" it, but neither did I "poorly execute" it.

Again, Warren didn't tell what he meant.

If Warren wanted to "execute" it, by following my instructions and
writing the formula for its rounding point, that would be fine,
instead of raging claims that I'd poorly executed it.

"...due to incorrect mathematical manipulations."

[endquote]

Warren had been making that claim on EM, and I'd repeatedly asked him
what, in particular, was incorrect. Warren never was able to specify
anything incorrect.

When I introduced BF, Warren began, with hysterical anger, repeating
that I must have made some error in its derivation, because he felt
that a formula for that purpose couldn't take that form.

After a long time in which he raged that the formula and its
derivation must be incorrect, he eventually admitted that it was
correct.

My definition, description, and derivation outline and instructions
for WBF were correct, brief, and concise. I hadn't posted any
"mathematical manipulations" to write a formula for its
rounding-point. Warren never explained what he meant or was talking
about.

"...and/or confusing expositions"

[endquote]

My definitions and descriptions were concise and clear. They were like
the definitions, descriptions and derivation outlines and instructions
that I've posted in recent days, for BF and WBF.

"Ossipoff also did not appreciate the desirability of the exponential
density as an underlying model."

[endquote]

On the contrary, I'd suggested it. When Warren advocated it, I agreed
with its desirability. But I'd considered other, easier, approximating
functions.

Warren was apparently unaware that it is not unusual to use as an
approximating function, a function that, though it is not the one most
closely related to the "model" or the principle by which the variable
varies, is nevertheless convenient because it's easier, &/or because
it leads to a more tractable problem.

"The present work can be viewed as simply carrying out Ossipoff's
idea, but now with correct execution..."

[endquote]


As I said, Warren never answered the question of what "execution" was incorrect.

I'd have no objection if Warren had merely carried out the actual
writing of the rounding-point formula for WBF. What he began writing,
however, bore little resemblance to the WBF that I'd outlined, and it
is very unlikely that his muddled and vague 'work" carries out WBF. It
was something else, something that he didn't justify or really even
define.  ...always accompanied by much acrimonious ranting that I had
done it incorrectly.

I've proposed, defined and outlined the derivation of BF and WBF. I've
supplied a rounding-point formula for BF. I don't know what it is that
Warren has written at his webpage, but it is unlikely to "carry out"
WBF.

...especially given the bias that I demonstrated for it, above in this posting.

Mike Ossipoff