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7. More Seats or Less?

The Alabama Paradox

The question:

Seats in a parliament are allocated according to the population size of the voting districts. Unfortunately, the proportions are nearly never integer numbers. So how many seats should districts get if their populations are, say, 10.2% and 16.8% of the nation’s population. Should the number of seats be rounded to 10 and 17? Unfortunately, rounding to the nearest integer won’t do, because that usually ends up with one seat short or long. In the US House of Representatives, rounding to the closest integer would almost certainly result in 434 or 436 seats, not exactly the desired 435.

In 1850, to avoid squabbles, Senator Samuel Vinton from Ohio, relying on a previous suggestions by Alexander Hamilton, the first Secretary of the Treasury, proposed a simple method: first, an appropriate divisor is sought, such that when the states’ populations are divided by this divisor and all results are rounded down only a few seats are left over. Next, allocate the leftover seats to the states with the highest leftover fractions. So far, so good.

From time to time, the size of the legislature needs to be increased to accommodate the nation’s ever-growing population. After the US Census of 1880, for example, Congress was considering an increase in the size of the House of Representatives from the then-current 293. Obviously, whenever the House is enlarged by a seat, one lucky state will gain an additional representative.



The paradox:

Let us look at the actual numbers of the 1880 census. The total population of the United States was 49,713,370. If the House had 299 seats, the appropriate divisor would be 165,120 and the number of representatives for the states Alabama, Texas and Illinois would be as follows:


                           Alabama          Texas           Illinois

Population                 1,262,505        1,591,749       3,077,871

“Raw” allocation            7.646             9.640           18.640

Seats in first round         7                 9               18

Fractional part             0.646             0.640           0.640

Additional seats             1                 0                0

Total seats                  8                 9                 18



Now see what happens if the size of the House is increased by one and 300 seats are to be allocated. The appropriate divisor becomes 164,580 and the calculations are as follows:


                           Alabama           Texas           Illinois

Population                 1,262,505         1,591,749       3,077,871

“Raw” allocation            7.671             9.672           18.701

Seats in first round         7                 9               18

Fractional part             0.671             0.672           0.701

Additional seats             0                 1                1

Total seats                  7                 10              19


Lo and behold! Alabama loses a seat, Texas and Illinois each gain one. A paradox!



For decades, after every census, there followed squabbling and wrangling about the apportionment of seats in the House. In order to give the congressmen the necessary ammunition for the infighting that would undoubtedly precede the 1880 apportionment of the House, C. W. Seaton, the chief clerk of the Census Office, did some computations. Using the census results, he did the long divisions to work out the apportionments according to the Hamilton-Vinton’s method for all House sizes between 275 and 350. Starting with 275 representatives everything worked out just fine all the way up to 299. Whenever he increased the House by one, the additional seat was picked up by some lucky state. But when he reached 300 seats, the above bombshell exploded. The delegation of the State of Alabama was decreased by one representative, from 8 to 7. In its stead, two states, Illinois and Texas, each gained one additional seat.

Congress went into a tizzy. The Hamilton-Vinton method of apportionment, which had by now become dear to most of them, was in danger. A short-lived method proposed by Senator Daniel Webster in 1842 – finding a divisor, such that the results, when rounded up or down to the nearest integer, gave the desired number of seats – was raised again. Tempers ran high. One congressman accused another of “committing a classic rape on a cloud of statistics, right in the face of the House.”



The reason for the Alabama Paradox becomes apparent when we delve a little deeper into the numbers. When the total number of seats in creases from 299 to 300 , the states’ “raw” numbers of seats grow on average by about one third of one percent . But Texas and Illinois start out with larger populations and therefore gain more in absolute numbers. Thus the number of “raw” seats grows by only 0.025 in Alabama (from 7.646 to 7.671), by 0.032 in Texas (from 9.640 to 9.672) and by 0.061 of Illinois (from 18.640 to 18.701). As a consequence, the larger states creep past Alabama.

Increasing the size of the House creates the problem. The obvious solution is to keep the size of the House unchanged and since 1911 the number of congressmen and -women is fixed at 435. That avoids the Alabama Paradox once and for all.

Ah, but now read the chapters on the Population Paradox.

Technical Supplement:

So what happened in 1880?

In order to avoid the contest between the proponents of the two methods from turning even uglier, Congress decided not to decide and resolved, instead, to enlarge the House to 325 seats. With this size the congressmen did not have to take sides beca use Webster’s and Hamilton-Vinton’s methods agreed and the problem could be postponed for at least another ten years . Maybe a wholly different apportionment method would be found in the meantime? Or the methods would again agree? Or the congressmen would no longer be in Congress and could let their successors worry about the Alabama paradox.

They were right. All it took in 1890 was an increase to 356 seats and the same compromise could be forged. With a House of that size, both methods agreed, and no state would lose a seat as compared to the previous apportionment. Ten years later, no such luck. When tables on the apportionment were prepared in 1901 for sizes of the House between 350 and 400, Maine’s apportionment oscillated between 3 and 4 seats and Colorado would receive 3 seats for every size of the House , except at 357 seats where it would be allocated just 2. Of course, the chairman of the Select Committee on the Twelfth Census, no friend of Colorado’s and Maine’s, suggested fixing the size of the House precisely at 357. Tempers rose and the atmosphe again became ugly. This time Congress did take a stand and the Hamilton-Vinton method was abandoned for the 1900 census data in favor of Webster’s. At least it did not suffer from the defect of the Alabama paradox. In addition, the House was enlarged to 386 seats which ensured that no State would lose a seat.

Comments, corrections, observations:          

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