[EM] Bias measures--There really are no two ways about it

[Michael Ossipoff]

I’ve posted the obvious bias definition that no one would disagree with, 
but, because it’s so short, let me repeat it:

A method or an apportionment is large-biased if the largest states have, on 
average, more seats per quota than the smallest states.

Then I added some definitions to make it clearer or easier to use:

1. We’re talking about a hypothetical country that has arbitrarily many 
states.
2. “The largest states” means an arbitrarily large number of states at the 
top end.
3. “The smallest states” are defined similarly.

As I said before, it doesn’t matter exactly how many the largest or smallest 
states are, because, if they are sufficiently many, the s/q disparities 
caused by 2 states being in different parts of their cycles will average out 
and cancel out. So the only s/q disparity that can remain is the kind 
between the  average s/q of the cycles.

That concludes my bias definition and its explanation.

Now, say someone wants to argue for a different bias definition, and let’s 
say that the two bias definitions give different results. If a method is 
unbiased by my definition, and we change the rounding points to make it 
biased by someone’s other definition, those new rounding points won’t let it 
be unbiased by my definition.
That’s because there’s only one rounding point that will give a particular 
cycle a s/q of 1. And as soon as you give one cycle an s/q different from 1, 
there must be  another cycle with s/q different from 1 in the opposite 
direction. So the changes required to make the method unbiased by that 
person’s other definition will bias the method by my definition. And,  with 
different cycles newly having different s/q, anyone would agree that unbias 
has been lost.

Or maybe the method or apportionment is just as unbiased as it can get by my 
definition. Then, when you change the rounding points to improve its unbias 
by someone’s other definition., it must be more biased by my definition, 
because it was initially as unbiased as it could get.

So, the best empirical test for bias is the correlation between the q and 
s/q of  the _cycles_, rather than the individual states.

Maybe, in a simulation with very many apportionments, the s/q disparities 
caused by states being in different parts of their cycles would cancel out, 
so that correlatoin by states would be the same as corelation
By cycles.  But with single apportionments by particular censuses, there 
could be a difference in results with the two kinds of correlation, cycle 
correlation and state correlation.

Mike Ossipoff