[EM] The Sainte-Lague index and proportionality

[Michael Ossipoff]

Kristofer:

This posting could have been entitled: "In which I admit that I don't
know what I'm doing"  :-)

In my most recent posting to this thread, I said that from December
2006 to January 2007, I'd been judging unfairness as s/q deviation per
person. That discussion was a long time ago, and I got that impression
of it from looking at Warren's website. But, looking at my postings
from that time, I noticed that that wasn't what I was saying.

Here's what led me, in December 2006, to day that Bias-Free, and not
Webster, is the method that is unbiased if the probability
distribution is assumed to be uniform:

Let's speak of intervals between to successive integers, N and N+1. In
those days I was calling that a "cycle". We could instead call it an
"interval", but maybe I should stick with "cycle".

Let's name the two integers bounding that cycle "a" and "b". Where b = a+1.

My understanding is that, at first, I calculated the expected number
of seats for a state that is between a and b. And also the expected
amount of population (n terms q, the quotient when that state's
population is divided by a particular divisor). I said that the
expected s/q, in that cycle, is the expected seats divided by the
expected q.

(That's how I've been calculating it lately too)

But then I said that that is a little rough or crude. I said that what
we should really do is calculate it directly, by integrating s/q, with
respect to q, from a to b. Since the probability density is assumed
constant throughout (a,b), that gives the expectation for s/q in
(a,b).

I was wording it a little differently then. but it seems to me that
what I said meant the same thing as what I've just described.

At the time, that direct calculation sounded more reliable than
assuming that expected s divided by expected q gives expected s/q. To
tell the truth, lately I've not felt entirely comfortable with that
assumption either.

So, maybe I was right in December 2006, and have been wrong afterwards.

When I later decided that expected s divided by expected q was the
right way, that meant that Webster would be the unbiased method, with
a uniform distribution. In a way that's a relief, that such a simple
and precedented method is the unbiased method. But I have to admit
that I was a little disappointed too, because I really liked the
formula that I got for the roiundoff point R, with the more direct
solution.

R is the roundoff point in (a,b). Between a and R, s/q = a/q.  Between
R and b, s/q = b/q. So the integrals of those functions over those
ranges are added, to get the integral of s/q from a to b.

Then the result is set equal to 1, and the resulting equation solved for R.

The formula that I got for R was:

R = (b^b/a^a)*(1/e)

That's b to the b power, divided by a to the a power, the result divided by e.

Now isn't that an interesting and appealing formula?

How nice if it turns out to be right.

By the way, Warren agreed at his website that that formula is the
right expression for R, to give an unbiased divisor method if the
distribution is uniform.

When I proposed that method in December 2007, I called it "Bias-Free" (BF).

Of course, since the distribution isn't really uniform, I also
suggested a Weighted BF. That's what Warren and I were discussing, and
using different distribution approximations for.

Then, at some point in the 1st two months of 2007, decided that
expected s/q should be calculated by dividing expected s by expected
q. But now it seems to me that I might very well have been right when
I preferred that more direct calculation, the one that led to
Bias-Free.

For Weighted BF, the function to be integrated would be s/q multiplied
by the probability density function (a function of q, for which an
approximation would be used).

When I decided that Webster and Weighted-Webster were better, Warren
still preferred BF and Weighted-BF. (but he didn't use the same names
that I used).

Now it seems to me that Warren might have been right all this time.

Mike Ossipoff