Chapter Four: The Mathematics of Apportionment

Transcription

1 Chapter Four: The Mathematics of Apportion: to divide and assign in due and proper proportion or according to some plan. 6 New states are being created and Congress is allowing a total of 250 seats to be divided amongst the 6 new states according to population How do we decide how many seats each state gets? A B C D E F Standard Divisor (SD) = (total pop.)/(# seats (M)) (SQ) = (state s pop.)/sd State Population SQ Rounded A B C D E F With standard rounding practices, how many seats did we hand out?

2 1. The Bandana Republic is a country consisting of four states: Apure, Barinas, Carabobo and Dolores. There are 200 seats in the Congress. The population of each state is given in the table. State Population (p i) Upper Quota Apure 3,310,000 Barinas 2,670,000 Carabobo 1,330,000 Dolores 690,000 a. Calculate the standard divisor. SD = b. Describe what the standard divisor represents in this problem. c. Use the standard divisor to find each states standard quota. d. Find each state s lower and upper quotas. 2. The Scotia Metropolitan Area Rapid Transit Service (SMARTS) operates 130 buses on six bus routes (A, B, C, D, E and F). The buses are apportioned to the routes based on the average number of daily passengers per route. Route Average Daily Passengers (p i) Upper Quota A 45,300 B 31,070 C 20,490 D 14,160 E 10,260 F 8,720 a. What represents the states in this problem? b. What represents the seats in this problem? c. What represents the populations in this problem? d. Calculate the standard divisor. SD = e. Describe what the standard divisor represents in this problem. f. Use the standard divisor to find each states standard quota. g. Find each state s lower and upper quotas.

3 4.2 Hamilton Method and the Quota Rule Every state gets at least the lower quota. As many states as possible get their upper quota with the one with the highest fraction having first priority. Hamilton s Method 1. Calculate each state s standard quota. 2. Give to each state (for the time being) its lower quota. 3. Give the surplus seats (one at a time) to the states with the largest fractional parts until there are no more surplus seats. State Population SQ LQ TOTAL SEATS A B C D E F The Quota Rule: no state should be apportioned a number of seats smaller than its lower quota or larger than its upper quota. A apportionment method that guarantees that every state will receive either its lower quota or upper quota is said to satisfy the Quota Rule.

4 4.3 The Alabama and Other Paradoxes The Alabama Paradox when an increase in the total number of seats being apportioned forces a state to lose one of its seats. A small country of Calavos consists of three state with 200 seats to be apportioned. M = 200 with a SD = 20,000/200 = 100 State Population Step 1 Step 2 Step 3 Bama 940 Tecos 9030 Ilnos 10,030 Total 20,000 Now imagine that overnight the number of seats is increased to 201 seats while nothing else changes. Now M = 201 and SD = 20,000/201 = State Population Step 1 Step 2 Step 3 Bama 940 Tecos 9030 Ilnos 10,030 Total 20,000 This is known as the Alabama Paradox because it actually happened to Alabama in The Population Paradox a state could potentially lose some seats because its population got too big. State A loses a seat to State B even though A grew at a greater rate than the population of B. See example 4.6 on page 138. The New State Paradox the addition of a new state with its fair share of seats can, in and of itself, affect the apportionments of other states. See example 4.7 on page 140.

5 Discrete Math Lesson 4.3 Hamilton s Method & the Alabama Paradox Exercise 1 Name Period The most serious flaw of Hamilton s Method is known as the Alabama Paradox. This paradox occurs when an increase in the total number of seats being apportioned forces a state to lose one of its seats. In 1882, members of the House of Representatives were debating several different apportionment bills using Hamilton s Method. One bill proposed a House with 299 seats in which Alabama s apportionment would be 8 seats. Another bill proposed a House with 300 seats in which Alabama s apportionment would drop to 7 seats. The additional seat and one of Alabama s seats would be apportioned to Texas and Illinois. The Congress had discovered a flaw in Hamilton s Method! Rather than adopt another method of apportionment, however, Congress finally adopted a bill that proposed a House with 325 seats that satisfied the states. Example 1: Recall the apportionment of 200 congressional seats in the country of Calavos. State State Population (p i ) Total (M) 200 Standard 100 Divisor (SD) Suppose the Congress sets the size of the House of Representatives at 201 seats. Use Hamilton s Method to determine the apportionment of 201 seats in the Calavos Congress. Bama st 10 Tecos Ilnos 10, Population (P) 20, State Bama Tecos Ilnos Population (P) State Population (p i ) Total (M) Standard Divisor (SD) What happens when the seats in Congress are increased? Example 2: The country of Mathland consists of three states: Algebra, Geometry and Trig. Use Hamilton s Method to complete the table and determine the apportionment of 176 seats in the Mathland Congress. State State Population (p i )

6 Algebra 9230 Geometry 8231 Trig 139 Population (P) Total (M) Standard Divisor (SD) Suppose Mathland sets the size of its Congress at 177 seats. Use Hamilton s Method to determine the apportionment of 177 seats in the Mathland Congress. State State Population (p i ) Algebra 9230 Geometry 8231 Trig 139 Population (P) Total (M) Standard Divisor (SD) What happens when the seats in Congress are increased? In 1901, Hamilton s Method created even greater controversy when the Alabama Paradox reared its ugly head yet again. The Census Bureau presented to Congress tables showing the possible apportionments for all House sizes from 350 to 400 seats. It showed that two states, Maine and Colorado, were affected by the Alabama Paradox. Maine would receive four seats for all House sizes except 357, 382, 386, 389 and 390; at these House sizes, Maine would receive only three seats. Colorado would receive three seats for all House sizes except 357, at which size it would receive only two seats. It just so happened that the bill submitted before Congress proposed an apportionment of 357 seats. As you can imagine, the debate in Congress became heated and the bill was defeated. Hamilton s Method was subsequently dropped in favor of Webster s Method, and Congress approved an apportionment of 386 seats. Discrete Math Name Lesson 4.3 Hamilton s Method & the Population Paradox Exercise 2 Period

7 In the early 1900 s, it was discovered under Hamilton s Method that a state could potentially lose seats because its population got too big. This population paradox occurs when state A loses a seat to state B even though state A s population grew at a faster rate the state B s. Example 1: Recall the apportionment of course sections to the four math courses being offered by a community college. Math Course Enrollment (p i) Total Sections 25 (M) Standard Divisor 36.6 (SD) Suppose the actual enrollments for Math B and Math D have exceeded projections and are now 99 and 211 respectively; the actual enrollments for Math A and Math C remain as originally projected. Use Hamilton s Method to determine the apportionment of course sections with the increased enrollments. Sections Math A st 11 Math B Math C Math D nd 6 Total Enrollment (P) Math Course Math A Math B Math C Math D Enrollment (p i) Total Enrollment (P) Total Sections (M) Standard Divisor (SD) What happens to the apportionment of course sections to the courses with increased enrollment? Sections Example 2: In the year 2525, five planets in the Utopia Galaxy sign a peace treaty and form the Intergalactic Federation governed by the Intergalactic Congress composed of 50 seats. The populations of the five planets given in the table are measured in billions. Apportion the seats in the Intergalactic Congress using Hamilton s Method. Planet Population (p i) Alanos 150 Betta 78

8 Conii 173 Dugos 204 Ellisium 295 Total Population (P) Total (M) Standard Divisor (SD) After 10 years of peace, the Intergalactic Census shows only a few changes in the population of the Federation. Conii s population increased by 8 billion, Ellisium s population increased by 1 billion. The populations of the other planets remain unchanged. Use Hamilton s Method to determine the apportionment of seats with the increased populations. Planet Alanos Betta Conii Dugos Ellisium Population (p i) Total Population (P) Total (M) Standard Divisor (SD) What happens to the apportionment of seats to the planets with increased enrollment? 4.4 Jefferson s Method method changes the SD to a lower number so there will not be any surplus seats to be apportioned. Jefferson s Method 1. Find a suitable divisor D. A suitable divisor is one that produces an apportionment of exactly M seats when the quotas are rounded down.

9 2. Each state is apportioned to its lower quota. Republic of Parador State Population Standard Lower Modified Jefferson A 1,646,000 B 6,936,000 C 154,000 D 2,091,000 E 685,000 F 988,000 Total 12,500,000 SD = 50,000 quota D = 49,500 apportionment The serious flaw in the Jefferson Method is it can produce upper quota violations. Discrete Math Lesson 4.4 Jefferson s Method Exercise 1 Name Period Following the 1790 census, Congress passed the first bill of apportionment in The bill, sponsored by Alexander Hamilton, established a House of Representatives with 120 seats apportioned under the method we now call Hamilton s Method. When the bill came to President George Washington for his signature, he vetoed the bill at the urging of Secretary of State Thomas Jefferson (This was the first presidential veto in U.S. history!). Unable to

10 override the veto, Congress passed another bill sponsored by Jefferson which established a House of Representatives with 105 seats apportioned under the method we now call Jefferson s Method. Washington signed this bill into law and so the first method used to apportion the House of Representatives was Jefferson s Method. 1. A company is apportioning 175 newly trained employees to its four manufacturing plants according to how many units are produced at each plant per day. Find the apportionment of employees using Hamilton s Method. Units Plant Produced (pi) Atlanta 3001 Birmingham 1558 Columbia 1049 Durham 517 Total Production (P) Total Employees (M) Standard Quota Lower Quota In Jefferson s Method, the objective is to find a suitable modified divisor (MD) that produces an exact Standard Divisor (SD) apportionment when we assign the lower quotas to the states. We use the standard divisor (SD) as our starting point. Since the assignment of lower quotas results in an under-apportionment of seats, we need to find an MD that causes the standard quotas to increase. This means that the MD must be less than the SD. 2. For the apportionment problem above, SD =. Now find a suitable modified divisor (MD) that produces an exact apportionment using Jefferson s Method. Plant Apportionmen t Units Produced (pi) Atlanta 3001 Birmingham 1558 Columbia 1049 Durham 517 Total Production (P) Total Employees (M) Modified Quota Modified Divisor (MD) 3. Compare the results of the apportionment of employees using Jefferson s Method with that of Hamilton s Method. Does the apportionment between methods differ? How? Apportionmen t Employees Employee s 4. Recall that the apportionment using Hamilton s Method of 176 congressional seats in the country of Mathland gave 92 seats to Algebra, 82 seats to Geometry and 2 seats to Trig. a. What is the standard divisor in the in the Mathland apportionment problem? SD = b. Now find a suitable MD and the apportionment of seats in Mathland using Jefferson s Method.

11 State Population (pi) Algebra 9230 Geometry 8231 Trig 139 Total Population (P) Total (M) Modified Divisor (MD) Modified Quota c. Compare the results of apportionment using Jefferson s Method with that of Hamilton s Method. Does the apportionment between methods differ? How? 7. A college math department wants to apportion 30 teaching assistants among five courses. a. What is the standard divisor in the in this apportionment problem? SD = b. Now find a suitable MD and the apportionment of teaching assistants using Jefferson s Method. Math Course Enrollment (pi) College Algebra 218 Calculus I 142 Calculus II 138 Calculus III 64 Statistics 188 Total Enrollment (P) Total TA s (M) Modified Divisor (MD) Modified Quota Teaching Assistants 4.5 Adam s Method method changes the SD to a higher number D. This makes the quotas smaller and then each is rounded up to exactly fill the number of seats. Adam s Method 1. Find a suitable divisor D. A suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas are rounded up. 2. Each state is apportioned to its upper quota.

12 Republic of Parador State Population Standard Lower Modified Adam s A 1,646,000 B 6,936,000 C 154,000 D 2,091,000 E 685,000 F 988,000 Total 12,500,000 SD = 50,000 quota D = 50,500 apportionment Republic of Parador State Population Standard Lower Modified Adam s A 1,646,000 B 6,936,000 C 154,000 D 2,091,000 E 685,000 F 988,000 Total 12,500,000 SD = 50,000 quota D = 50,700 apportionment The Adam s Method can produce lower quota violations. Discrete Math Lesson 4.5 Adam Method Exercise 1 Name Period Adam s Method is similar to Jefferson s Method in that we use modified divisors (MD) to apportion unassigned seats. The difference between the two methods lies in the initial assignment of seats and in how the modified divisors are calculated. Rather than initially assigning each state its lower quota (as we do in Jefferson s Method), Adam s Method initially apportions to each state its upper quota. 1. Recall the problem in which we apportioned 175 newly trained employees to four manufacturing plants according to how many units are produced at each plant per day.

13 Hamilton s Method: 86 to Atlanta, 44 to Birmingham, 30 to Columbia, 15 to Durham Jefferson s Method: 86 to Atlanta, 45 to Birmingham, 30 to Columbia, 14 to Durham a. In Adam s Method we assign each state its upper quota. In the table below calculate each states standard quota and assign each its upper quota. Units Plant Produced (pi) Atlanta 3001 Birmingham 1558 Columbia 1049 Durham 517 Total Production (P) Total Employees (M) Standard Quota Apportionmen t Upper Quota (UQ) Modified Quota Apportionmen t Employee s Standard Divisor (SD) apportionment to the actual number of employees to be assigned. What do you notice about the initial apportionment? b. Total the upper quotas and compare the initial As in Jefferson s Method, the objective in Adam s Method is to find a suitable modified divisor (MD) that produces an exact apportionment, except that in Adam s Method we assign upper quotas to the states. We use the standard divisor (SD) as our starting point. c. Since the assignment of upper quotas results in an over-apportionment of seats, we need to find an MD that causes the standard quotas to. This means that the MD must be than the SD. d. For the apportionment problem above, SD =. Now find a suitable modified divisor (MD) that produces an exact apportionment using Adam s Method. MD = e. Complete the table with the modified quotas and the final apportionment using Adam s Method. f. Compare the results of the apportionment of employees using Adam s Method with that of Jefferson s Method and Hamilton s Method. Does the apportionment between methods differ? How? 2. Recall that the apportionment of 176 congressional seats in the country of Mathland. Hamilton s Method: 92 seats to Algebra, 82 seats to Geometry, 2 seats to Trig. Jefferson s Method: 93 seats to Algebra, 82 seats to Geometry, 1 seat to Trig. a. What is the standard divisor in the in the Mathland apportionment problem? SD = b. Now find a suitable MD and the apportionment of seats in Mathland using Adam s Method. State Population (pi) Algebra 9230 Geometry 8231 Modified Quota

14 Trig 139 Total Population (P) Total (M) Modified Divisor (MD) c. Compare the results of the apportionment of employees using Adam s Method with that of Jefferson s Method and Hamilton s Method. Does the apportionment between methods differ? How? 3. Recall the apportionment of 30 teaching assistants among five courses. Hamilton s Method: 9 to College Algebra, 6 to Calc I, 5 to Calc II, 3 to Calc III, 7 to Statistics Jefferson s Method: 9 to College Algebra, 6 to Calc I, 5 to Calc II, 2 to Calc III, 8 to Statistics a. What is the standard divisor in the in this apportionment problem? SD = b. Now find a suitable MD and the apportionment of teaching assistants using Adam s Method. Math Course Enrollment (pi) College Algebra 218 Calculus I 142 Calculus II 138 Calculus III 64 Statistics 188 Total Enrollment (P) Total TA s (M) Modified Divisor (MD) Modified Quota Teaching Assistants c. Compare the results of the apportionment of employees using Adam s Method with that of Jefferson s Method and Hamilton s Method. Does the apportionment between methods differ? How? 4.6 Webster s Method method changes the SD to a divisor where conventional rounding exactly fills all the available seats. Webster s Method 1. Find a suitable divisor D. A suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas are rounded the conventional way.

15 2. Find the apportionment of each state by rounding it quota the conventional way. Republic of Parador State Population Standard quota Nearest Quota Webster s SD = 50,000 Integer D = 50,100 apportionment A 1,646, B 6,936, C 154, D 2,091, E 685, F 988, Total 12,500,000 With Webster s Method, we always start with the SD. If we are lucky and the SD happens to work, we are done. If the SD does not work, then we move the D divisor up or down until we find a divisor that works. The Webster Method can violate the quota rule, but such violations are rare in reallife apportionments.