Now suppose a new district with a population of 39,000 is added. In a
twenty-nine-seat house, its quota is 2.30, so it is entitled to two seats, and
a new house is constituted with thirty-one seats. Here is the table that
results.District Population (in thousands) Quota Number of Representatives
A 61 3.57 4
B 70 4.09 4
C 265 15.50 15
D 95 5.56 6
E 39 2.28 2District A has gained a seat at the expense of District C.
Possibly the Hamilton method could have survived two of the three
paradoxes discussed here, but the trifecta killed it. The method cur-
rently used, the Huntington-Hill method, adopted in 1941, is a round-
ing method that is arithmetically somewhat more complex than the
Hamilton method. However, as might be suspected, it, too, falls prey to
paradoxes. Two mathematical economists, Michel Balinski and H. Pey-
ton Young, were later to show that it couldn’t be helped.The Balinski-Young Theorem
As we have seen, representation is a consequence of the method chosen to
round fractions. A quota method is one that rounds the quota to one of the
two integers closest to it; for example, if the quota is 18.37, a quota method
will round it to either 18 or 19. The Balinski-Young theorem^6 states that it
is impossible to devise a quota method for representation that is impervi-
ous to both the Alabama paradox and the population paradox.
Although we have introduced this problem in what is probably its most
important and controversial context—the structure of the House of Repre-
sentatives and the Electoral College—the problem discussed here has other
important applications. Many situations require quantities to be divided into
discrete chunks. As an example, a city’s police department has obtained
forty new police cars; how should these be assigned to the city’s eleven pre-
cincts? A philanthropist has left $100,000 in his will to his alma mater for
twenty $5,000 scholarships in arts, engineering, and business; how should
the twenty scholarships be allocated among these areas? The Balinski-Young
theorem shows us that there is no fair way to do this, if we define fairness to
mean immunity to the Alabama and populations paradoxes.
The Smoke- Filled Rooms 233