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E Pluribus Confusion

There’s more than one way to turn census data into congressional seats

Barry Cipra

The Alabama Paradox

2010-07CompSciCipraFC.jpgClick to Enlarge ImageNeedless to say, Adams’s method was rejected in favor of Jefferson’s. But when the 1840 Census rolled around, Congress split the difference, so to speak, and used a method proposed by Daniel Webster. Webster’s method also uses a divisor D to produce quotients p/D, but instead of rounding everything up (Adams) or down (Jefferson), Webster rounds to the nearest integer—up if the fractional part is greater than 0.5 and down if it’s less. As a doling-out algorithm, Webster’s method amounts to keeping an ordered list of ratios p/(r+½).

Webster’s method was only used once. With the 1850 census, Hamilton’s method snuck back in, and it stayed on the books for the rest of the century—until it became too much of an embarrassment. Hamilton’s method, it turns out, does strange things.

This idiosyncrasy became apparent with the 1880 census. The chief clerk of the Census Office noted something odd when he computed how Hamilton’s method apportioned seats for various House sizes ranging from 275 to 350: In a House with 299 seats, Alabama, with a quota of 7.646, got rounded up, but in a House with 300 seats, Alabama’s quota of 7.671 got rounded down!

What accounts for the “Alabama paradox” is the fact that at 299, Alabama’s fraction, 0.646, barely beat out those of both Illinois (quota 18.640) and Texas (quota 9.640) for the “last” seat (meaning Alabama’s fraction was the smallest one to get rounded up). But at 300, Illinois’s quota went to 18.702, easily overtaking Alabama’s 0.671, whereas Texas’s quota squeezed by at 9.672. In effect, Illinois overtook Alabama for the 299th seat, and Texas grabbed the 300th.

Congress dealt with the Alabama paradox by settling on a House size 325, which just happened to be a number where the methods of Hamilton and Webster gave the same apportionment. Much the same took place with the 1890 census: Congress picked a size (356) for which Hamilton and Webster coincided. But in 1900, all hell broke loose.

When Hamilton’s method was applied to the 1900 census for a House size ranging from 350 to 400, Maine’s allotment bounced back and forth between three and four seats: three seats from 350 to 382, four from 383 to 385, back to three at 386, four again at 387 and 388, down again to three at 389 and 390, and back to four from 391 to 400. As Representative John E. Littlefield of Maine put it, “In Maine comes and out Maine goes.... God help the State of Maine when mathematics reach for her and undertake to strike her down.”

Politics being politics, Congress opted for a House of size 386—but jettisoned Hamilton’s method in favor of Webster’s, which allowed Maine to keep its fourth seat. In 1910, Congress again went with Webster, allocating 433 seats among the 46 states then comprising the union, with a specification that Arizona and New Mexico be given a seat each, if and when they became states. Thus Congress reached its current size of 435.

It would seem that Webster had won the day. Then the mathematicians got involved.

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