# [EM] FW: Re: Transfers of seats between states

Joseph Malkevitch malkevitch at york.cuny.edu
Thu Dec 7 15:51:58 PST 2006

```Dear Election List,

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
>
> 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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------------------------------------------------
Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

Phone: 718-262-2551 (Voicemail available)

My new email is:

malkevitch at york.cuny.edu

web page:

http://www.york.cuny.edu/~malk

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