Download all Excel spreadsheets for this chapter.

The arithmetic involved in applying the methods of Hamilton, Jefferson, Adams, and Webster is tedious and burdensome. Modern spreadsheets will ease much of our pain.

We start by applying Hamilton's method to the Congress of Parador, a new republic located in Central America. It is made up of six states: Azucar, Bahia, Cafe, Diamante, Esmeralda, and Felicidad (A, B, C, D, E, and F for short). The Congress of Parador is to have 250 seats, divided among the six states according to their respective populations. The population figures for each state are shown in Figure 4.1.

Figure 4.1: Data for the republic of Parador.

After entering our data into a spreadsheet, we find the standard divisor. To accomplish this, we compute the total population in cell B10 using the formula =SUM(B3:B8). We then divide by 250, the total number of seats, in cell B13 using the formula =$B$12/$C$1. The population of each state is then divided by the standard divisor to find each state's standard quota. To do this, we put the formula =B3/$B$12 into cell C3 and copy it down column C. The results are shown in Figure 4.2.

Figure 4.2: Computing the standard quotas for Parador's Congress.

The integer part of each standard quota is then found by using the ROUNDDOWN function of the spreadsheet. In cell D3 we place the formula =ROUNDDOWN(C3,0) and copy it down column D. To find the fractional part of each standard quota, we simply subtract the integer part from the entire standard quota for each state. Here, we place in cell E3 the formula =C3-D3 and copy it down column E. The resulting spreadsheet is shown in Figure 4.3.

Figure 4.3: More computation for Parador's Congress.

Finally, each state's apportionment is found by giving out the 4 extra seats available. After these seats are given out, the final Hamilton apportionment can be calculated in column G by adding each state's lower quota to any seats it gained in the apportionment of the extra seats. To do this, in cell G3 we use the formula =D3+F3 and copy it down column G. The final spreadsheet showing the Hamilton apportionment for Parador's congress is shown in Figure 4.4. The formulas used to generate the Hamilton apportionment are shown in Figure 4.5.

Figure 4.4: Hamilton apportionment for Parador's Congress.

Figure 4.5: Formulas for the Hamilton apportionment for Parador's Congress.

Finally, it should be noted that for a large number of states, giving out the extra seats is easy if one uses a SORT tool associated with the spreadsheet. In the Parador Congress example, we SORT A3:E8 according to column E. The result of the SORT is shown in Figure 4.6.

Figure 4.6: Using the SORT command to apportion Parador's Congress.

We will now apply Jefferson's method to apportion the Congress of Parador. The first step is to enter the data just as we had in applying Hamilton's method (See Figure 4.1). The next step is also the same as Hamilton's method -- we compute the standard divisor. Then, a modified divisor must be chosen. How do we choose such a divisor? This is somewhat tricky. The only thing that is clear is that such a modified divisor must be less than the standard divisor. We will simply start by trying a modified divisor of 49,900 in the case of the Parador congress.

The modified divisor located in cell B13 can now be used to compute modified quotas for each of the states of Parador. To compute each state's modified quota, we divide its population by the modified divisor. To accomplish this we put =B3/$B$13 into cell C3 and then copy this formula down column C. To compute the Jefferson apportionment, we round each state's modified divisor down in the next column as in Figure 4.7.

Figure 4.7: Finding the Jefferson apportionment for Parador's Congress.

As we can see from Figure 4.8, changes in the modified divisor cell B13 produce changes in the modified quotas in the next column automatically. This indeed illustrates the beauty of a spreadsheet and why it has been termed the killer application in the software business!

After making several educated guesses, the use of a modified divisor of 49,500 (among other possibilities) produces the Jefferson apportionment for Parador's congress (see Figure 4.8).

Figure 4.8: The Jefferson apportionment of Parador's Congress.

Walking

Exercise 4.1 [Excursions, Chapter 4, Exercises 1, 2, 7, 8] The Bandana Republic is a small country consisting of 4 states (Apure, Barinas, Carabobo, and Dolores). The populations of each state are given in Figure 4.9.

Figure 4.9: Population within the Bandana Republic.

(a) Use a dynamic spreadsheet to find each state's apportionment under Hamilton's method when the number of sets in the Bandana Republic legislature is M=160.
(b) Repeat your calculations using a legislature of size M=200. Are you convinced of the power of the modern spreadsheet?

Exercise 4.2 [Excursions, Chapter 4, Exercises 3, 9] The Scotia Metropolitan Area Rapid Transit Service (SMARTS) operates 6 bus routes (A, B, C, D, E, and F) and 130 buses. The buses are apportioned among the routes on the basis of average number of daily passengers per route, which is given in Figure 4.10.

Figure 4.10: Average number of daily passengers within SMARTS.

Use a dynamic spreadsheet to apportion the buses among the routes using Hamilton's method. How difficult is it to use your spreadsheet to reapportion the buses if another bus is added to SMARTS's fleet?

Exercise 4.3 [Excursions, Chapter 4, Exercise 47] Figure 4.11 shows the results of the 1790 census (the very first census of the United States taken after the Constitution was adopted).

Figure 4.11: Populations figures from the 1790 U.S. Census.

Based on the fact that the number of seats in the House of Representatives was set at 105, use a spreadsheet to find the apportionment that would have resulted under the original bill passed by Congress forcing the use of Hamilton's method.

Jogging

Exercise 4.4 [Excursions, Chapter 4, Exercises 15, 21, 27] The Bandana Republic is a small country consisting of 4 states (Apure, Barinas, Carabobo, and Dolores). The populations of each state are given in Figure 4.9. Use a spreadsheet to find each state's apportionment using a legislature of size M=160 under

(a) Jefferson's method.
(b) Adam's method.
(c) Webster's method.

Exercise 4.5 [Excursions, Chapter 4, Exercises 17, 23, 29] The Scotia Metropolitan Area Rapid Transit Service (SMARTS) operates 6 bus routes (A, B, C, D, E, and F) and 130 buses. The buses are apportioned among the routes on the basis of average number of daily passengers per route, which is given in Figure 4.10. Use a spreadsheet to apportion the buses among the routes using

(a) Jefferson's method.
(b) Adam's method.
(c) Webster's method.

Exercise 4.6 [Excursions, Chapter 4, Exercise 14] A mother wishes to distribute 10 pieces of candy among her 3 children based on the number of minutes each child spends studying, as shown in Figure 4.12.

Figure 4.12: Apportioning candy among three children.

(a) Find each child's apportionment using Hamilton's method and a dynamic spreadsheet.
(b) Suppose that, just prior to actually handing over the candy, mom finds another piece of candy and includes it in the distribution. Find each child's apportionment using Hamilton's method and 11 pieces of candy.
(c) Did anything paradoxical occur?
(d) Use Jefferson's method and a spreadsheet to apportion both 10 and 11 pieces of candy to the children. Does anything paradoxical occur?

Exercise 4.7 [Excursions, Chapter 4, Exercise 13] A mother wishes to distribute 11 pieces of candy among her 3 children based on the number of minutes each child spends studying, as shown Figure 4.12.

(a) Find each child's apportionment using Hamilton's method and a dynamic spreadsheet.
(b) Suppose that before mom has time to sit down and do the actual calculations, the children decide to do a little more studying. Say Bob studies an additional 2 minutes, Peter an additional 12 minutes, and Ron an additional 86 minutes. Find each child's apportionment using Hamilton's method based on the new total time studied.
(c) Did anything paradoxical occur?
(d) Use Jefferson's method and a spreadsheet to apportion the 11 pieces of candy to the children both before they do the extra studying and after. Does anything paradoxical occur?

Exercise 4.8 [Excursions, Chapter 4, Exercise 54] With the aide of a dynamic spreadsheet, make up an apportionment problem in which Webster's method violates the quota rule.

Exercise 4.9 [Excursions, Chapter 4, Exercise 45] The purpose of this example is to show that under rare circumstances, the use of a modified divisor method may not work. A small country consists of 4 states (A, B, C, and D) with populations of 500, 1000, 1500, and 2000. There are M=51 seats in the House of Representatives.

(a) Using a spreadsheet, find each state's apportionment under Jefferson's method. (b) With the aid of a dynamic spreadsheet, attempt to apportion the seats using Adams' method with the modified divisor of D=100. What happens if D < 100? What happens if D > 100?

Exercise 4.10 [Excursions, Chapter 4, Exercise 46] Design a spreadsheet that allow the user to input the population of a particular state, the standard quota for that state, and the size M of the House of Representatives in three cells and outputs to a fourth cell an estimate of the population of the entire country (all states).

Exercise 4.11 Election to Cambodia's National Assembly with 122 members is by parties on a proportional basis, with the number of seats each party wins determined by the percentage of votes the party received. The results of the July 1998 elections are given in Figure 4.13.

Figure 4.13: Results of the July 1998 Cambodian elections.

Apportion Cambodia's National Assembly using a spreadsheet and

(a) Hamilton's method.
(b) Jefferson's method.
(c) Adam's method.
(d) Webster's method.

The actual apportionment of the National Assembly was: Cambodian People's Party - 64, United National Front - 43, Sam Rainsy Party - 15.

Exercise 4.12 Election to Austria's Nationalrat (National Council) with 183 members is by parties on a proportional basis, with the number of seats each party wins determined by the percentage of votes the party received. The results of the 1998 elections are given in Figure 4.14.

Figure 4.14: Results of the 1998 Austrian elections.

Apportion the Nationalrat using a spreadsheet and

(a) Hamilton's method.
(b) Jefferson's method.
(c) Adam's method.
(d) Webster's method.

The actual apportionment of the Nationalrat was: Social-Democratic - 65, Nationalist - 52, Conservative - 52, and Green - 14.

Exercise 4.13 Election to Israel's Knesset (Parliment) with 120 members is by parties on a proportional basis, with the number of seats each party wins determined by the percentage of votes the party received. The results of the May 1999 elections are given in Figure 4.15.

Figure 4.15: Results of the May 1999 Israeli Knesset elections.

Apportion the Israeli Knesset using a spreadsheet and

(a) Hamilton's method.
(b) Jefferson's method.
(c) Adam's method.
(d) Webster's method.

The actual apportionment of the Knesset was: Yisrael Akhat - 26, Likud - 19, Shas - 17, Meretz - 10, YBA - 6, Shinui - 6, ha-Merkaz - 6, Mafdal - 5, Yahadut HaTorah - 5, Ra'am - 5, ha-Ikhud ha-Leumi - 4, Hadash - 3, YB - 4, Balad - 2, AE - 2.

Exercise 4.14 Election to South Africa's National Assembly with 400 members is by parties on a proportional basis, with the number of seats each party wins determined by the percentage of votes the party received. The results of the June 1998 elections are given in Figure 4.16.

Figure 4.16: Results of the June 1998 South African elections.

(a) Hamilton's method.
(b) Jefferson's method.
(c) Adam's method.
(d) Webster's method.

The actual apportionment of the National Assembly was: African National Congress - 266, Democratic Party - 38, Inkatha Freedom Party - 34, New National Party - 28, United Democratic Movement - 14, African Christian Democratic Party - 6, Freedom Front - 3, United Christian-Democratic Party - 3, Pan African Congress of Azania - 3, Federal Alliance - 2, Minority Front - 1, Afrikaner Eenheidsbeweging - 1, Azanian People's Organisation - 1.

Exercise 4.15 Suppose Canada were to use a congress which included a 435 seat House of Representatives as the United States. Look up current population information on the Canadian provinces on the internet and use a spreadsheet to apportion the new Canadian congress using

(a) Hamilton's method.
(b) Jefferson's method.
(c) Adam's method.
(d) Webster's method.

Running

Exercise 4.16 [Excursions, Chapter 4, Exercise 42] The apportionment method called Lowndes' method is described in the paragraph preceeding Exercise 42 in Chapter 4 of Excursions.

(a) Design a dynamic spreadsheet that will find the apportionment of Parador's Congress under Lowndes' method.
(b) Which states do better under Lowndes' method than under Hamilton's method.
(c) Does Lowndes' method seem to favor larger or smaller states?

(a) Use a spreadsheet and the Huntington-Hill method to apportion Parador's Congress. Hint: You will likely use the ROUNDDOWN command. For example, if we wanted to round the value in cell A1 using a cutoff of 0.4 (instead of the usual 0.5) and put the result in cell A2, we would use the formula =ROUNDDOWN(A1+0.6,0) in cell A2.
(b) Why is it important to use the ROUNDDOWN command to implement the Huntington-Hill method instead of, say =ROUNDUP(A1-0.4,0) in cell A2?
(c) How does the Huntington-Hill apportionment compare with that of other methods?

Exercise 4.18 The method currently used to apportion the United States House of Representatives is the Huntington-Hill method. Look up current population data for the 50 states on the internet and use the Huntington-Hill method to apportion the U.S. House (435 seats). Compare your apportionment to the actual apportionment.

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