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The Census Office reported this problem to Con gress, but Congress was not persuaded to abandon the method. In fact, they chose N = 325 precisely because the methods of Hamilton and Webster re turned the same apportionment at this value. This process repeated itself in 1890. The Census Office said the method of Hamilton is flawed, but enough of the representation in Congress still liked it, and they compromised with N = 356, because the two methods agreed. The breaking point came in 1900. The House Se lect Committee on the Twelfth Census asked the Census Office to use Hamilton ’ s method from N = 350 to 400 seats. Table 3 demonstrate the problems that were evident to the committee. The committee, wary of the populist leanings of Colorado, chose N = 357. However, this caused Maine to lose a seat, prompting Maine Rep. John E. Littlefield to say, “ God help the State of Maine when mathematics reach for her and undertake to strike her down. ” The full House rejected the com mittees ’ proposal. Instead, they chose the Webster Method with N = 386, which was large enough so no state lost a seat. Two other interesting paradoxes may appear with Hamilton ’ s method. In the “ New States Paradox, ” adding a new state and sufficient seats for its ap portionment, may shift a seat from one existing state to another. In the “ Population Paradox, ” two states may both grow in population, but the faster growing state may lose a seat to the slower growing state. These are both signs of a method that seems suboptimal. Divisor Methods like Jef ferson and Webster do not have these paradoxes. However, Hamilton Method does have one ad vantage over the Divisor Methods. It “ stays within the quota. ” That is, Hamilton ’ s method, a state will always gets an integer above or below its quota. We saw earlier with New York in 1820 that, Jef ferson rounded its quota of 32.50 up beyond the next integer to 34 seats. Balinski & Young (2001) demonstrate that it is not possible for a method to both be free from these three paradoxes and stay within the quota.

Hill Method

In 1902, the Census Bureau became a standing part of the federal government instead of a temporary office for each census. They began to collaborate with university mathematicians, statisticians, and economists to lay a stronger theoretical foundation for their work. Cornell professor and future presi dent of the American Statistical Association Wal ter F. Willcox, worked with them to convince Con gress that Webster ’ s Method was the soundest, and it was used in 1910 with N = 433 so that no state lost a seat. Provisions were made so that Arizona and New Mexico would each get one seat if they became states, which happened in 1912, pushing the House of Representatives to 435 members. In 1910, Joseph A. Hill, the chief statistician in the Division of Revision and Results at the Census Bu reau, started to push back against Webster ’ s Meth od. Back in 1830, University of Vermont professor James Dean submitted another divisor method to Congress. Congress promptly ignored it, but it is worth noting here. Dean recommended rounding the quota by comparing each state ’ s divisor ________ to the overall D and choosing to minimize ______. Equivalently, if _____ , this can be accomplished by rounding not with respect to n +0.5, the arithmetic mean of n and n +1 but with respect to the harmonic mean of n and n +1. The reciprocal of the harmonic mean is the arithmetic mean of and ____, or This number is between n and n + 1, and so it can also be used as a rounding threshold, but it is slightly smaller than n + 0.5. For example, to round 4.48, we would look at the harmonic mean of 4 and 5, which is ___________, and so 4.48 would round up to 5 using this method. This may not seem like an obvious way to round, but it has a justification—maximizing equity of representation. Ideally, the people per representa tive for each state would be same as the nation as a whole, but due to rounding, some states have greater representation than others. .

1920 - present: Deadlock and the Huntington

Virginia Mathematics Teacher vol. 48, no. 1

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