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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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