The Jefferson Method

Following President Washington’s veto of the first apportionment bill in 1792, Congress moved quickly to pass a new apportionment bill. Each state’s apportionment population was divided by 33,000, and the resulting quotient was rounded down to get a whole number, which was the state’s apportionment. Thus, Virginia, with apportionment population 630,560, received

Delaware’s apportionment was seat. Because the number of seats in the House was not specified in advance, the apportionment problem was I avoided. (It turned out that 105 seats were apportioned.)

If a state with population was apportioned seats, its district population would be Since

it follows that the district population

In words, the state’s district population would be at least 33,000.

The method just described was suggested by Thomas Jefferson, and it is called the Jefferson method.

With this method, a divisor is specified. Using , rather than the standard divisor , as the divisor, an apportionment quotient, rather than a quota, is obtained for each state. A state’s apportionment quotient is rounded down to obtain its apportionment.

586

The Jefferson method is one of a family of apportionment methods called divisor methods.

In an apportionment problem, there is a specified house size. When a divisor method is used in an apportionment problem, the challenge is to choose the divisor that achieves the intended house size. Select a divisor by trial and error, with the insight that to reduce the number of seats apportioned the divisor must be increased, and to increase the number of seats, a smaller divisor must be used.

The Jefferson Method is the divisor method associated with the rounding rule that rounds each apportionment quotient down. In this case, the divisor must be less than the standard divisor, because if the standard divisor were used, each state would be apportioned its lower quota—and we know that the lower quotas sum to less than the house size (except in the very unusual case where all the quotas turn out to be whole numbers). Let’s see how apportioning the diamonds (Example 5 on page 579) works with the Jefferson method.

We will implement this procedure for our high school mathematics teacher. The Jefferson divisor in this context can be interpreted as a minimum class size because the number of classes allocated for each subject is obtained by dividing the number of students enrolled by , and discarding any fractional section in the quotient. Thus, each section will have at least students.

Comparing the results of Example 11 and Example 4 (page 579), where the Hamilton and Jefferson apportionment methods were applied to the same apportionment problem, we see that the results differ. Examples 10 and 5 also exhibit different apportionments from another apportionment problem. Now let us apportion the TAs in the scenario we encountered in Example 8 on page 582. Will the Alabama paradox occur with the Jefferson method? Recall that with the Hamilton method, Calculus III had 3 TAs when 30 TAs were available, but only 2 TAs when another TA was hired.

Table 14.13 shows that when the house size increases from 30 to 31, Calculus II gets the new TA, and no course loses a TA. In fact, the Jefferson method prevents the Alabama paradox, as the following theorem confirms.

We will verify that the Alabama paradox is impossible if the Jefferson method is used.

Let be the divisor used to apportion a house with seats. For a house size of , the divisor must be less than . Consider a state with population . Its apportionment with house size will be , and its apportionment with house size will be .

Since , it follows that and hence

Therefore, an increase in the house size, with no change in any state’s population, cannot lead to a decrease in any state’s apportionment. This shows that the Jefferson method prevents the Alabama paradox.

Because divisor methods only differ by the rounding rule used, the same argument shows that any divisor method will prevent the Alabama paradox. The other two paradoxes also do not occur with divisor methods (see Exercises 26 and 27 on page 615).

Here is another distinction between the Hamilton and Jefferson methods— and this time it is in the Hamilton method’s favor. The Hamilton method always gives each state either its upper quota or its lower quota.

Although the house size was ignored when the Jefferson method was used to apportion seats in the House of Representatives, when the apportionment was complete, it was possible to compare each state’s apportionment with its quota. In 1822, there was a surprise.

590

The Jefferson method never gives a state fewer seats than its lower quota, but it takes only one example like the 1822 apportionment to show that a state’s Jefferson apportionment may be more than its upper quota. Therefore the Jefferson method does not satisfy the quota condition. In fact, if the House had continued to use the Jefferson method, it would have given some state more than its upper quota in every apportionment since 1850. For example, the Jefferson apportionment of the House according to the 2010 census gives California 55 seats, although its quota is 52.54; and Texas, with quota 35.55, would get 37 seats.

The Hamilton method satisfies the quota condition. This was obvious to Congress in 1850, and that is why it switched to the Hamilton method in that year.²

591

Congress has never used an apportionment method that satisfies the quota condition and avoids the paradoxes of the Hamilton method. In the 1970s, the mathematicians Michel L. Balinski and H. Peyton Young tried to devise such a method. Their research showed that the population paradox is possible for all apportionment methods except divisor methods. (The Hamilton method paradoxes are impossible with divisor methods.) However, no divisor method satisfies the quota condition! Thus, while Balinski and Young failed to meet their objective, they did prove that there is no apportionment method that satisfies the quota condition and prevents the population paradox. Their theorem is like Kenneth Arrow’s impossibility theorem, which tells us that there is no completely satisfactory way to decide multicandidate elections based on voter preference schedules (see Section 9.4, page 424).

The Jefferson method favors the larger states. It is not an accident that in every example that we have considered, the “state” with the largest population fared better with the Jefferson method than it did with the Hamilton method. Winnie, not Louise, received the extra diamond when the Jefferson method was used, as compared with the Hamilton method, and in the example of the high school math teacher, there were more sections of geometry and no sections of calculus when the Jefferson method was substituted for the Hamilton method.

Let’s see why the Jefferson method is biased in favor of larger states. Because the Jefferson method always rounds the apportionment quotients down, each state’s apportionment quotient must be greater than its quota (otherwise, each state would get its lower quota of seats or fewer). Therefore, the divisor used to obtain these apportionment quotients must be less than the standard divisor. If we compare the quotas to the apportionment quotients, we will find that their ratio is equal to the ratio of the standard divisor to the Jefferson divisor:

Denote this ratio by . Since the Jefferson divisor is less than the standard divisor, , so we will let . Then

and so the amount the state’s apportionment quotient exceeds the quota is

The larger the state’s quota is, the more it increases when the apportionment quotient is substituted. This is an advantage to the states with large populations.

Consider the congressional apportionment of 1822. The standard divisor is , and the Jefferson divisor is . The quotient is

thus . Now suppose that the quota for state is . The state’s apportionment quotient is obtained by increasing the state’s quota by . A state with a large quota will get a greater raise under these circumstances than a small state will.

To see how this works with numbers, consider a state with . The apportionment quotient is , or 1.003 greater than , which brings it to 20. The upper quota for is 19, but is awarded 20 seats! This violates the quota condition, and in fact, every state whose quota is 18.997 or more will be guaranteed to get at least its upper quota. If a state has a quota greater than , an identical calculation shows that it will receive at least its upper quota plus 1 seat. On the other hand, consider a small state whose lower quota is 1. To increase its apportionment to 2, its quota must be at least . Thus, a state with quota 18.997 gets more than its upper quota, and a state with quota 1.899 has to settle for its lower quota.

592

When parliamentary seats are being apportioned, the Jefferson method’s bias favors major parties, which typically attract many votes, over less popular parties. The Jefferson method is used by many nations that prefer voters to select party lists rather than to vote for individual candidates for parliament. It is believed that this practice favors more stable government because there will be less need for a major party to be forced into a coalition with a small party.

For historical reasons, when the Jefferson method is used to apportion seats in parliament, it is called the d’Hondt method, after Victor d’Hondt (1841–1901). Countries that use the d’Hondt method to apportion seats in their parliaments include Argentina, Denmark, and Israel.

The procedure for apportioning seats by the d’Hondt method is not the same as the one that we have used for the Jefferson method, but it leads to the same apportionment. With the d’Hondt method, seats are apportioned one at a time until the house size is reached.

With the d’Hondt method, it is unnecessary to compute the standard divisor or the quota for any party. The d’Hondt method provides the order in which members of parliament receive their seats. The first member to be seated is the first on the list of the party with the plurality of votes, and so on. However, it achieves exactly the same result as the Jefferson method. Think of the parties’ priority numbers as divisors. When a party receives the last seat (because its priority is highest), its priority number (before that seat is awarded to it) is the divisor that will produce the Jefferson apportionment. Here is why this is so: Suppose the party that is about to receive the last seat got votes and already has seats. Its priority number is . Thus, (exactly—no rounding needed), so with divisor , party will get seats. The seats awarded to the other parties were awarded with priority numbers larger than , so the divisor will award those seats to them as well. They will not get any additional seats, though, because their current priority numbers are less than .

593

594

The Webster Method

A second divisor method was introduced in 1832 by Senator Daniel Webster of Massachusetts.

The Webster and Jefferson methods are free of the paradoxes that we have seen with the Hamilton method, but neither satisfies the quota condition. However, the Jefferson method favors the large states, while the Webster method is neutral, favoring neither the large nor the small states. Furthermore, the Webster method rarely violates the quota condition by giving a state more than its upper quota, or fewer seats than its lower quota, and would not have done so in any of the 23 congressional apportionments that have occurred so far.

It is possible that first use of the Webster method was before Webster invented it! The apportionment bill that was vetoed by President Washington was consistent with the Webster apportionment, with 30,000 as the divisor. The resulting apportionment was unconstitutional, but perhaps it would have not been vetoed if a large enough divisor had been used.

Let’s revisit the three friends who won 100 diamonds in a lottery (see Example 10 on page 586).

To see why the Webster method is not biased in favor of large states or small states, we will use the same analysis that we applied to Jefferson’s method. Let be the standard divisor, let be the divisor that is used in the Webster method, and let . Each state’s apportionment quotient is equal to

596

as with the Jefferson method. When (this happens when the number of seats tentatively apportioned is less than the house size), the states all receive an across-the-board increase in their quotas; and just as in a company where the workers receive the same percentage raise, the larger states are favored. When , the reverse is true because a large number multiplied by will decrease more than a small number would. Thus, the Webster method favors neither large nor small states when the tentative apportionment exactly fills the house; it favors small states when the tentative apportionment must be reduced, and it favors large states when the tentative apportionment must be expanded. On balance, the Webster method is neutral because it is equally likely that the tentative apportionment will be less than the house size or that it will be greater than the house size.

When seats are apportioned to parties after an election, a method that is equivalent to the Webster method can be used. This method, called the Sainte-Laguë method (after André Sainte-Laguë, 1878–1950), involves a process that is analogous to the d’Hondt method. A Sainte-Laguë table is constructed with a column for each party. The number of votes received is at the top of each party’s column, and below that the number of votes is divided by successive odd numbers, 3, 5, 7, …. Seats are apportioned to the parties by treating the entries in the table as a priority list. The party with the highest priority gets a seat, and then that priority number is crossed out. The next seat goes to the party with the largest remaining priority in the Sainte-Laguë table. Germany and New Zealand use the Sainte-Laguë method to apportion seats to their parliaments.

If you are curious to know why the Sainte-Laguë method produces the same apportionment as the Webster method, refer again to the party that is about to get the last seat. Let be the number of votes that it received, and suppose that it has seats already, before the last seat is awarded. Its priority, , is the highest remaining in the table. [The odd number is .] Multiply this priority by 2 to get a number . Using this as the divisor, we see that

(The s cancel each other out, as we divide the quotients by inverting the divisor and multiplying.) This quotient can be rounded up to get seats. The other parties have lower priority, and so they get no new seats with as the divisor.

597

So far, three of the four methods that have been used to apportion seats in the U.S. House of Representatives have been introduced. The fourth is the Hill-Huntington method, to be described in Section 14.4. This divisor method was adopted by statute following the 1940 census and is still used. In Table 14.16, the properties of the four apportionment methods are described.