[EM] Bias definition and bias-test

[Michael Ossipoff]

Some of the most unassailable definitions result from criticism from 
mathematicians. So I certainly don’t object to Joe’s criticism about how I 
define bias. I welcome anyone to point out flaws, deficiencies, ambiguities, 
or possible problem situations, because only by finding and fixing those 
things can  a definition be perfected.

I’m not satisfied with my definition in its present form. If the complexity 
that Joe  mentioned refers to the job of finding airtight wording, then yes 
that can be a task. But it would be another thing to say that a good bias 
definition couldn’t be written. And, though there are obviously many 
different ways the definition, via the approach that I’m using, could be 
worded,  those are not mutually contradictory definitions, but are only 
different wordings of the same definition.

I’m new to the task of finding airtight or precise wording for a bias 
definition, and I don’t object to criticism of proposed definitions.

Maybe something simple:

A method is biased if (and to the extent that), when that method is used,  
larger states consistently have more (or consistently have fewer) seats per 
quota than do smaller states.

[end of simple bias definition]

That’s what bias means to all of us, and maybe it’s enough as a definition.

My more elaborately-worded definition that I’ve mentioned is more in the 
nature of a test. Maybe there’s a place for such a test, in addition to the 
simple definition above.

But the test that I described in previous postings doesn’t seem specific 
enough . Too many quantities are allowed to vary. For simplicity, as many 
quantities as possible should be fixed. And the manner in which a variable 
quantity can vary should be fully spelled-out. As for the fixed quantities, 
and initial values of variable quantities,, I chose numbers equal to those 
of the U.S.

Maybe something  like:

This specifies a bias-test, for use when one party (person or  group of 
people) wants to find bias, and another party wants to find no bias. Or when 
one party wants to show one method less biased than another, and the other 
party wants to show the opposite.

In this test there are initially 435 seats in the House. States can have 
anywhere from 0 to 52 Hare quotas of population. There are initially 50 
states.

The small states are those with from 0 to 26 Hare quotas. The large states 
are those with from 26 to 52 Hare quotas.

The  state-sizes, as measured by Hare quotas, are randomly chosen, and those 
random state-sizes could  have any probability distribution agreed upon by 
the two parties. For instance the distribution could be uniform, or could be 
of the form B*exp(-A*q), where A & B are positive constants., or it could be 
some other approximation of the distribution in the actual U.S.

Or, of course, it could be any distribution agreed-upon by the two parties 
as the distribution for which they want to do the bias test.

Given the above constraints, either party may increase the total number of 
states (which automatically increases the population), while proportionately 
increasing the house-size. And/or either party could increase the number of  
apportionments with randomly-chosen state-sizes. Either party may make these 
quantities as large as they want to, in an attempt to cause the test result 
to go the way they would like.

The method is unbiased if, when it is applied in this test, the ratio 
between the average s/q of  the large states and the average s/q of the 
small states can be made as close to 1 as desired, by sufficiently 
increasing the  number of states  and,  proportionately the house-size; &/or 
increasing the number of apportionments done with randomly-chosen 
state-sizes.   Method A is less biased than method B if sufficiently 
increasing those quantities will make the abovementioned ratio closer to 1 
for method A than for method B.

[end of suggested bias-test]

Cycle-Webster and Adjusted-Rounding make each cycle’s average s/q as close 
to 1 as possible, differing from 1 by a small random amount that would 
cancel and be less important as the variable quantities in the test are 
increased. I suggest that Cycle-Webster and Adjusted-Rounding are unbiased 
according to this test. Bias-Free, when tested with a uniform probability 
distribution, is likewise unbiased , by this test, for the same reason.


Mike Ossipoff