Winnie has convinced two friends, Louise and Tim, to buy lottery tickets. Winnie buys 18 tickets, Louise buys 4, and Tim buys 1. They are very lucky and win the top prize: a necklace with 100 diamonds! They decide to divide the diamonds in proportion to the number of tickets each bought.

Table 14.1 shows how the diamonds should be shared—if it made sense to cut one of the diamonds into three pieces! They round each share to the nearest whole number: When the fractional part is less than 0.500, they round down, and if any fractional part is greater than or equal to 0.500, they round up. Winnie’s share, 78.26 diamonds, is rounded down to 78; Louise’s share, 17.39 diamonds, is rounded down to 17; and Tim’s 4.35 diamonds are rounded down to 4. Because all three shares were rounded down, there is one diamond left. Winnie declares that since it was her idea to participate in the lottery, that diamond is hers.

Winnie, Louise, and Tim have an apportionment problem because their shares of the diamonds are not whole numbers but the sum of their shares is a whole number. Simple rounding leaves one diamond undistributed. Winnie imposed an arbitrary solution to this apportionment problem: That diamond is hers!

Self Check 1

Because we will be discussing the apportionment of seats in the U.S. House of Representatives based on the the census of 1790, it is useful to see the relevant text from Article I, Section 2, of the Constitution.

Representatives … shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons. The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct. The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative.

The italicized text includes the notorious “three-fifths rule” that, for the purpose of apportionment, a state’s population included three-fifths of the number of slaves residing in the state. In 1868, this wording was replaced by the part of the Fourteenth Amendment that was quoted at the beginning of this chapter. In this text, we use the term apportionment population to mean a state’s population for apportionment purposes. Until 1868, this would be the “Numbers” specified in the Constitution as quoted above. Currently, each state’s apportionment population includes those from the state who are stationed abroad for military or other United States government service.

The Constitution leaves to Congress the task of determining the apportionment method, subject to the following guidelines:

There was a vigorous discussion of apportionment methods in President Washington’s cabinet. Alexander Hamilton, the Secretary of the Treasury, proposed an apportionment method that will be the subject of Section 14.2. The Secretary of State, Thomas Jefferson—Hamilton’s rival—proposed another method that was eventually adopted. We will see Jefferson’s method in Section 14.3.

The Constitution does not specify the number of seats in the House of Representatives. It assigns to Congress the primary responsibility for all matters regarding apportionment, subject to the above guidelines. To accomplish the reapportionment of the House of Representatives, which is done following the decennial censuses, an apportionment bill must be passed by the House and the Senate, and signed into law by the president. Like any bill passed by Congress, the president has the option to veto it.

Congress interpreted the phrase, “one [seat] per thirty Thousand,” to mean that the number of seats in the House cannot be more than the total apportionment population of the United States, divided by 30,000. According to the census of 1790, the total apportionment population was 3,615,920, which, divided by 30,000, yields a quotient between 120 and 121. Thus, in the first apportionment bill, passed in 1792, Congress set the House size to be 120. The apportionment that resulted is shown in Table 14.2.

This bill was the subject of the first presidential veto in U.S. history. (President Washington only vetoed two bills during his time in office.) The reason for the veto can be explained in terms of district population.

President Washington’s interpretation of “one [seat] per thirty Thousand” was that no state was allowed to have a district population less than 30,000—and 8 of the 15 states had district populations that violated that limit. His veto message, probably written by Jefferson, also contains a statement about fractional parts being unrelated to the state’s populations. This indicates that the apportionment was determined by the Hamilton method, which we will see in Section 14.2.

Self Check 2

Now let’s see how to set up an apportionment problem. Although many apportionment problems do not involve the House of Representatives, our terminology refers to states, populations, and a house size. The first step in solving an apportionment problem is to identify the states, populations, and house size.

The standard divisor is the average district population. It is not the average of the district populations for each state, but the average district population for the nation as a whole.

The second step in addressing an apportionment problem is to determine the standard divisor. Continuing with our example of sharing the diamonds, the house size is 100 so the standard divisor is the total population divided by 100. The total population was the number of tickets bought, 23, and the standard divisor was $23÷100=0.23$.

Self Check 3

Because the standard divisor is equal to $\frac{\text{total}\text{population}}{\text{house}\text{size}}$, we can also say that

This formula emphasizes that a state’s quota is its fair share of the seats in the House, but it is less efficient for calculation.

The third step in solving an apportionment problem is to calculate the quota for each state. In the diamond-sharing example, the number of tickets is interpreted as the population of a state. The quota for each participant is calculated by dividing its population by the standard divisor. If there are many states, this may entail a bit of work. With a calculator, it is helpful to store the standard divisor in memory to avoid having to enter it repeatedly. If you would like to use a spreadsheet to compute an apportionment, there are pointers in the introduction to the Writing Projects (page 619).

Self Check 4

The quotas in the diamond-sharing scenario, $18÷0.23=78.26$ for Winnie, $4÷0.23=17.39$ for Louise, and $1÷0.23=4.35$ for Tim, are shown in the right column of Table 14.1.

All apportionment methods start with these three initial steps leading to the determination of the quotas. The next step, rounding the quotas to obtain whole numbers whose sum is the house size, is where the various methods differ.

Self Check 5

Ideally, each state’s apportionment should be close to its quota. It is unrealistic to expect that any state will be apportioned its exact quota because each apportionment is required to be a whole number and the quota is unlikely to be a whole number. In choosing an apportionment method, we must decide what we mean by the phrase “each state’s apportionment should be close to its quota.”

Apportionment always involves rounding, and there are many ways to round. “Rounding down” means discarding the fractional part of a number $q$ to obtain a whole number that we will denote as $\lfloor q\rfloor$. Thus, $\lfloor 7.0001\rfloor =7$, $\lfloor 7\rfloor =7$, and $\lfloor 6.9999\rfloor =6$. “Rounding up” gives the next whole number, denoted as $\lceil q\rceil$. Thus, $\lceil 7.0001\rceil =8$, but $\lceil 7\rceil =7$

No apportionment method is perfect for all occasions. Our goal is to understand how to choose a method that is appropriate for a particular apportionment problem.