Aunc.1964
net.math
utcsrgv!utzoo!decvax!duke!unc!bts
Mon Mar 15 10:40:38 1982
Alabama paradox
     The U.S. Constitution says that seats in the  House  of
Representatives  should  be  apportioned  according  to  the
states' populations.  It does  not  give  an  algorithm  for
doing  this, however.  The Hamilton method works on the fol-
lowing plan: give each state the integer part of its  share,
then distribute the remaining seats to those states with the
largest fractional parts.  The paradox is this: if seats are
apportioned according to the Hamilton method, it is possible
that increasing the size of  the  House  will  decrease  the
number  of  seats  a  particular  state  receives. (Computer
scientists, accustomed to the peculiar things that can  hap-
pen with rounding-off, may be less dismayed than others.)

     This is best seen in an example.  Suppose we have three
states, A, B, and C, with 3%, 7%, and 90% of the population.
Compare the apportionments given by the Hamiton  method  for
totals  of  49  and 51 seats. (There is a tie at 50, so I'll
leave that out, to show that the problem doesn't  depend  on
it.)
	total	A	B	C

	49	1.47	3.43	44.1	actual share
		2	3	44	Hamilton method

	51	1.53	3.57	45.9	actual share
		1	4	46	Hamiton method

     The fact that it's called the "Alabama paradox" and the
fact that the Hamilton method is no longer used suggest that
this has happened.  For more on this  problem,  see  "Appor-
tionment Methods and the House of Representatives" by Donald
Saari, in the Dec. 1978 Math. Monthly.

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