Author: Brent Bradberry Title: Geometric View of Some Apportionment Problems Source: Mathematics Magazine, February 1992 issue Summarized by: Kathie Barnes
This article analyzes various apportionment laws for the House of Representatives that have been suggested. Particular attention is given to Hamilton's method, and the paradoxes it created. Apportionment laws are the rules by which states are alloted seats in the House of Representatives. The author, Brent Bradberry first defines a "fair and monotone" apportionment with 5 axioms. These axioms ensure that when populations change, or new states are added, the representation of each state with respect to the other states is "fair," which Bradberry defines mathematically through these axioms. For a method to be monotone, no state can lose a Representative when new states are added to the Union without a change in population (this would create a larger House of Representatives, though). Thus, a fair and monotone apportionment is not biased toward any state or group of states.
Bradberry then describes various methods of apportionment. The basic form is always the same: find the ratio of the population of state i to the entire population, call this number the quota, q, of state i, and then divide the number of seats accordingly. The question that always arises is what to do with the decimals, as no state can have .67 of a Representative. Thus the question becomes one of how and when to round off numbers.
Before embarking on an actual proof, Bradberry defines a geometric representation of all possible apportionments of h house seats to s states, called a (s-1)-dimensional simplex. Thus, for 1 state, there is only 1 possibility: state 1 gets all h seats, which corresponds to the 0-simplex, which is a point. With two states, there are more possibilities, represented by the one simplex that is a line between the points which represent apportionments of "all or nothing"; the point where state 1 has all h seats, and state 2 has 0 seats, and the point where state 1 has 0 seats, and state 2 has all h seats. Three states get a bit more complicated, but can be represented by a triangle and its interior. Again, the vertices correspond to the all or nothing apportionments. Similarly, the apportionments for four states are represented by a solid tetrahedron.
While one cannot easily visualize the simplex for 5 states (after all, it is 4-dimensional), it is an easy generalization from the tetrahedron. For instance, this simplex would have 5 vertices instead of 4. Keep in mind that while all possible apportionments of h house seats to s states are represented in the (s-1)-simplex, not all points in the simplex represent valid apportionments. In fact, most of them don't. Since states can't have .67 of a Representative, only the discrete points that have integer apportionments are allowed.
The need for integer apportionments is why this problem is not a simple matter of arithmetic. Bradberry proves that it is impossible to have a fair and monotone method for apportionment. Using geometric simplices, Bradberry illustrates where each of the five axioms would fail (if they do) by shading these areas on the simplices. It turns out that the only types of methods that are monotone are so called divisors methods; methods where a common divisor is found such that the integer portion of the quota divided by this common divisor is used as the number of Representatives for each state. Thus, the integer portions for each state must add to the total number of seats.
For instance, Hamilton's method is not a divisor method. His method can be described as follows: Take the quota for each state, and multiply it by the number of seats h, and take the integer component of this multiplication. Then parcel out the remaining seats in order of largest fraction. Obviously, if there are remaining seats, the integer component does not add to the total number of seats h. A divisor method would be described as follows: take the quota of each state, and find a common divisor, e, such that, if all the q/e are rounded up (or down, using a different e), (summation of q/e) = h (when the summation is taken over all states). Remember that q depends on i.
However, the need for a divisor method creates a problem, as Bradberry also proves that divisor methods cannot be fair. Thus, since divisor methods are the only possible monotone methods, it is impossible to satisfy all 5 axioms for a fair and monotone apportionment of seats. Right now, the US uses some combination of the two, in order to create as just a system as possible.
See Brent Bradberry's "Geometric View of Some Apportionment Problems" in Mathematics Magazine, February 1992 for further information.