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One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.
Number, Scientific American, 211, (Sept. 1964), 51 - 59.
Apportioning Representatives in the United States Congress
The Quota Rule
The fact that the affected states in the discrepancy just mentioned are Virginia and Delaware is no coincidence. In general, Jefferson’s method is biased in favor of larger states and against smaller ones. It violates what is called the “Quota Rule:
Here is a table that compares the populations and Representatives of Virginia and Delaware from 1790 through 1840:
By 1810, New York had overtaken Virginia as the most populous state in the Union. If we look at its numbers instead of Virginia’s, the discrepancy between that large state and Delaware is even more pronounced:
Using Jefferson’s method, New York always had its quota rounded up. On two occasions, its quota was rounded up more than one whole unit, in violation of the Quota Rule. In contrast, Delaware’s quota was rounded up only once. More striking are the cumulative results, showing New York well above and Delaware well below their expected totals.
There is a simple explanation for Jefferson’s bias toward the large states. The method works by lowering the divisor D by some d until the rounding fits the specified number of seats. But lowering the divisor causes the quotient to grow at a faster rate if the dividend is higher. For example,
Lowering the divisor by 7,000 in each case raises the quotient by more than 2.2 in the case of the large dividend but only by 0.75 in the case of the small one.
Caulfield, Michael J., "Apportioning Representatives in the United States Congress," Loci (November 2008), DOI: 10.4169/loci003163