COMPUTING SCIENCE

# E Pluribus Confusion

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

# Making Lists

Such issues notwithstanding, the notion that seats in Congress can be doled out one at a time until the House is full has a certain conceptual appeal. In fact, if you didn’t come up with Hamilton’s method earlier, it’s likely you came up with something along the following lines (and again, you may wish to see what comes to mind before reading further): Line the states up from biggest to smallest and give each one a seat to start with, thus taking care of the constitutional requirement right off the bat. At each step thereafter, keep a list, in decreasing order, of the ratios *p*/*r, r* being the current number of seats a state has been assigned. Now give the next seat to the state at the top of the list, decreasing its ratio to *p*/(*r*+1), and reinsert the state into the lineup to keep the list in decreasing order. This makes intuitive sense, since *p*/*r* is a measure of district size: It is the number of people represented by each of the seats.

If this is what you thought of, then you’re in league with John Quincy Adams. Jefferson’s method was used for the first five apportionments, but by the 1830s politicians from New England began grousing about being cheated out of seats. Jefferson’s method, it turns out, favors large states (of which his home state of Virginia was right at the top in 1790 and 1800) in a peculiar way: It has a penchant for assigning large states more seats than their quota suggests. In 1820, for example, New York and Pennsylvania had quotas of 32.503 and 24.917, respectively, yet wound up with 34 and 26 seats (out of 213 altogether). In 1830, New York was again granted an “extra” seat, receiving 40 despite a quota of 38.593.

Adams proposed a method that has the opposite effect. His approach is identical to Jefferson’s in that it divides a test number *D* into each state’s population *p,* but instead of rounding down, Adams’s method rounded *up* each quotient *p*/*D.* When all the math is said and done, it turns out this approach doles out seats in the fashion just described—in other words, according to the largest ratio *p*/*r.*

Indeed, the “pure” Jefferson method can also be described this way, except that the ordered list consists of ratios *p*/(*r*+1)—that is, states are prioritized according to how they would stand *after* another seat were added. A nice way to think of this algorithmically is to let the Adams approach run past 435, assigning all the way to 485 seats, and then ignore the first 50 assignments (thus taking away the first 50 seats, which went one to each state).

It’s instructive to see how the methods of Adams and Jefferson would have doled out the 105 seats of the first Congress, based on the 1790 census for the 15 states then comprising the United States. The populations are given in the second figure *(page 278), *along with their Hamiltonian quotas and the apportionment tallies for the two methods. The doling out of the seats appears in the third figure *(page 279). *The Adams apportionment is rows one through seven in the third figure; the Jefferson apportionment is rows two through eight. Note that Virginia and Pennsylvania appear twice in the last row, whereas Vermont and Delaware appear not at all.

Adams’s method tends to shortchange the large states. In 1830, it would have given New York (quota 38.593) only 37 seats and Pennsylvania (quota 27.117) only 26. In general, Jefferson’s method never gives less than the quota rounded down, and Adams’s method never gives more than the quota rounded up—and for the most part, it’s the large states that wander above or below what their quotas would suggest.

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