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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