Some of the most unassailable definitions result from criticism from
mathematicians. So I certainly dont object to Joes 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.
Im 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 couldnt be written. And, though there are obviously many
different ways the definition, via the approach that Im using, could be
worded, those are not mutually contradictory definitions, but are only
different wordings of the same definition.
Im new to the task of finding airtight or precise wording for a bias
definition, and I dont 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]
Thats what bias means to all of us, and maybe its enough as a definition.
My more elaborately-worded definition that Ive mentioned is more in the
nature of a test. Maybe theres a place for such a test, in addition to the
simple definition above.
But the test that I described in previous postings doesnt 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 cycles 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
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