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COMPUTING SCIENCE

# E Pluribus Confusion

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

# Whole Numbers

Most people, when asked for a reasonable way to turn fractions into integers (and you may wish to stop reading for a moment to see what comes to mind), propose something along the following lines. First, note that each fraction Sp/P—called the state’s quota—lies between two consecutive integers, so start by rounding to the nearest integer, with whatever rule you like for handling the unlikely possibility that the fractional part of Sp/P is exactly one-half. If the sum of all these rounded numbers turns out to be exactly S, you’re done. If it’s less than S, find the states with the largest quotas that got rounded down, and give them an extra seat until the total gets up to S; similarly, if it’s more than S, find the states with the smallest quotas that got rounded up, and take seats away from them (subject to the constitutional guarantee that you don’t reduce a 1 to a 0).

If you don’t like the notion of taking seats away after initially allotting them, an equivalent method is to round everyone down at first, then give an additional seat each to states with the largest fractional parts until you get to S. Whichever rounding route you take, this approach is named after Alexander Hamilton, who offered it as the rationale behind the first apportionment bill passed by Congress in 1792 and sent to President Washington for his signature.

It was the first bill ever vetoed.

The veto came at the urging of Thomas Jefferson, who had his own idea for apportioning seats. Jefferson’s approach was to pick an ideal number D for the size of a district—he liked the number 30,000—then divide it into each state’s population p, and round the result p/D down to the nearest integer to get s. If the total number of seats misses the target, Jefferson noted, you can go back, modify D either up or down, and try again.

You might worry that Jefferson’s method might never find a value of D that produces a House of size S—or, worse, that two different values of D would apportion the S seats differently. But neither worry is necessary. There is always a range of values for D that produces an apportionment with S seats, and you get the exact same apportionment for anything in the range.

One way to see this is to imagine starting with a value of D so unrealistically large that every state’s quotient, p/D, is less than 1, so nobody gets a seat. Then start reducing D. As you do so, each state’s quotient increases, and every so often some state’s quotient will reach and then exceed an integer value. If you ignore the unlikely possibility that this happens to two or more states simultaneously (or if you invent a tie-breaking rule to handle such a scenario), you can think of the House as growing in size from 0 all the way up to P, with seats being doled out one at a time.

If, once all the seats are doled out, some states are unrepresented, it may be necessary to take the last few doled-out seats back and reassign them to the unrepresented states, in order to satisfy the constitutional requirement. Were Jefferson’s method currently in use, something like this would have happened in the last apportionment: Based on the 2000 census, neither Wyoming nor Vermont would have received a seat under a “pure” Jeffersonian method; the last two seats would have gone to Minnesota and Pennsylvania.