The Seat Lost by Adding Seats

The United States Constitution says that seats in the House of Representatives are to be shared among the states according to their populations. It does not say how, and it cannot be followed literally, because exact shares are almost never whole numbers. A state entitled to 7.46 seats has to get 7 or 8, and the total has to come out right.

So somebody has to round, and rounding is where the politics went.

The first fight came in 1792. Congress passed a bill built on a method proposed by Alexander Hamilton, and George Washington vetoed it, the first veto any president used. The House adopted a method associated with Thomas Jefferson instead. That the very first presidential veto concerned rounding tells you how much was felt to be at stake.

Hamilton's method is the one most people would invent. Work out each state's exact share, its quota. Give every state the whole-number part. Then hand out the seats that are left over, one each, to the states with the largest fractional parts. It has an obvious virtue: every state ends up with its quota rounded either down or up, never further away.

In 1880 the chief clerk of the Census Office, C. W. Seaton, worked out the apportionment under this method for every House size from 275 to 350, so that Congress could choose. He found something nobody had expected. With 299 seats, Alabama received 8. With 300, it received 7.

Nothing had changed except the total. Adding a seat to the House took one away from Alabama, and the effect has been called the Alabama paradox ever since.

The mechanism is not mysterious once you look. Enlarging the House scales every quota up by the same proportion, but the fractional parts do not move in step: a big state's quota grows by more than a small state's, so its fraction climbs faster. The ranking of leftovers can reshuffle, and a state that was just inside the cut can find itself just outside it.

It got worse. When Oklahoma joined the Union in 1907, the House was enlarged by its fair share, five seats, so that nobody else need lose one. Recomputing by Hamilton's method would have moved a seat from New York to Maine anyway: admitting a new state changed the split between two old ones. And a state can grow faster than another and still lose a seat to it, because recomputing every quota shifts the remainders. That one is called the population paradox.

The alternative family is the divisor methods, and they work differently. Choose a number of people per seat, divide every state's population by it, and round each result by a fixed rule. If the seats do not add up to the size of the House, adjust the number and try again. Jefferson's method rounds down. Webster's rounds to the nearest whole number. The method the House has used since 1941, called Huntington-Hill, rounds at the geometric mean of the two whole numbers either side.

Divisor methods cannot produce the Alabama paradox. A bigger House means a smaller divisor, so every state's quotient goes up and no rounded figure can fall. They are immune to the population paradox too. What they give up is the guarantee Hamilton's method had: a divisor method can award a state more or fewer seats than its quota rounded either way. Jefferson's method, which rounds down and so favours large states, broke quota in practice while it was in use.

So each family buys one virtue with the other. Michel Balinski and Peyton Young proved, in their 1982 book Fair Representation, that this is not a failure of ingenuity. No method can both keep every state within its quota and guarantee that a state growing faster than another never loses a seat to it. You can have one or the other.

This is not abstract. After the 1990 census Montana dropped from two seats to one, and sued, arguing that a different divisor method would have treated it more fairly. The argument was serious: the methods measure fairness in different ways, and by some of those measures Montana had a case. The Supreme Court ruled against it unanimously in 1992, holding that the choice of method was for Congress to make.

That is the honest summary of the whole subject. There is no rounding that is simply correct. There is a choice between unfairnesses, and somebody has to make it.