Dear Election List,
See some in-line comments.
Regards,
Joe
On Dec 7, 2006, at 5:40 PM, MIKE OSSIPOFF wrote:
I hadn't heard about the other methods being justified in terms of
transfers
between states after the allocation.
You can find the description in Balinski and Young's book Fair
Representation, page 102, of the revised edition which was relatively
recently released by Brookings Institution Press.
But, as for different standards for
judging the result of those transfers, by different standards of
proportional fairness, there doesn't seem to be much room for rival
standards.
Unfortunately, although it is not intuitive, seeming small changes in
the measure of "optimality" makes a big difference in which method
turns out to be optimal.
Webster optimizes when the comparison of seats divided by population
for pairs of states in "absolute" terms is made, while Dean's method
optimizes when the comparison of population divided by seats for
pairs of states in "absolute" terms is made. However, Huntington
showed that all of the different measures of fairness/proportionality/
optimization for the 5 "standard" methods when measured in terms of
relative differences leads to what today is usually called Huntington-
Hill.
The Constitution says that the seats should be allocated to the
states in
proportion to their populations. "Proportion" means that seats
should be
proportional to population. The (impossible) goal, therefore, is
for all the
states to have the same proportion, the same ratio, of votes to seats.
So, what relation between two states should be optimized with
respect to
seat transfers between them? Is there any room for disagreement?
Since the different methods have had supporters over the years there
does seem to be disagreement.
Starting
with the apportionment allocation, and then giving a seat from one
state to
another, should never cause their v/s to differ by a smaller factor
than it
did before the transfer. The words "proportional" and "proportion"
imply
that factor is what we're talking about.
Yes, Hill's procedure looks at factor where Webster's procedure
looks at
arithmetic rounding. But, as I said, no genuine justification can
be found
in the procedure definitions of Webster or Hill. If you can't make
the v/s
proportion the _same_ for all states, then who's to say which kind of
fudging is best? As I said, as soon as we round off to the nearest
integer,
like Webster or Hill, we're going from solid justification to
fudging and
word-games.
Perhaps it is word games, but it is also involves different views
about what should be the optimization criteria used, and what
fairness criteria are paramount.
So the transfer property is what can give solid justification. A
transfer
property doesn't work for Hill, because a fixed integral number of
seats
(one) is transferred. That's why Webster's arithmetic rounding,
placing the
state as close to its ideal number in terms of raw seat-count, is
what makes
it possible for Webster to have the transfer property. It means
that any
change in that party's seat total, such as receiving or giving a
seat, can
only put that party's seat total farther from the fractional seat
total
corresponding to its ideal v/s. And when a state is as close as it
can be to
that ideal fractional seat total, in terms of raw seat-count, then
it must
also as close as possible to its ideal v/s, as measured by the
factor by
which its v/s differs from the ideal. And that's true because
Webster rounds
arithmetically instead of geometrically.
As for Jefferson or the others, I've never heard of a transfer
property
claimed for them.
The work showing that this is so dates to the 1920's. It was done by
E. V. Huntington, who spent a large part of his career at Harvard
University. It turns out there are very different ways of thinking of
the 5 different "standard" apportionment methods (other than largest
remainders) and that these lead to different computational algorithms
that yield the same results. The three different points of view
involve "rounding rules," divisors, and rank functions. Thus, on the
web page (Census Bureau) that describes the current apportionment
method (Huntington-Hill) used by the US the method is described in
terms of producing a table of numbers and assigning the seats after
the first 50 of the 435 seats are assigned, one to each state, as
required by the Constitution, in order of the size of the numbers in
this table.
http://www.census.gov/population/www/censusdata/apportionment.html
Jefferson, for instance differs from Webster in rounding
down instead of rounding to the nearest whole seat.
Since there's no solid justification in the procedures, we can justify
according to how transfer affects the factor by which the 2 states'
v/s
proportions differ. If the transfer of a seat between two states
makes their
v/s differ by a smaller factor than it did before, then something
is wrong
with the initial allocation.
Mike Ossipoff
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