Other Paradoxes and Apportionment Methods Section 9.4

Objectives: 1. Find the standard divisors and standard quotas. 2. Understand the Apportionment problem. 3. Use Hamiltons method with quotas. 4. Understand the Population Paradox and New-states Paradox. 5. Understand the quota rule. 6. Use Jeffersons method. 7. Use Adams method. 8. Use Websters method.

Terms: 1. Standard Divisor found by dividing the total population under consideration by the number of items to be allocated. 2. Standard Quota (for a particular group) found by dividing that groups population by the standard divisor. 3. Lower Quota standard quota rounded down to the nearest whole number. 4. Upper Quota standard quota rounded up to the nearest whole number.

Calculating Standard Divisor Standard Divisor = total population number of allocated items

Example 1: Calculate the Standard Divisor According to the constitution, of Margaritaville, the congress will have 30 seats, divided among the 4 states. Population of Margaritaville by State StateABCDTOTAL Population (in thousands)2753834657671890

Example 2: Calculate the Standard Divisor According to the countrys constitution, the congress will have 200 seats, divided among the 5 states. Population of Amador by State StateABCDETotal Population (in thousands) 1112111813201515493510,000

Calculating Standard Quota To calculate standard quota, you must first find the standard divisor. Standard Quota = population of a particular group standard divisor

Example 3: Calculate the Standard Quota Standard quotas are obtained by dividing each states population by the standard divisisor. Population of Margaritaville by State StateABCDTOTAL Population (in thousands)2753834657671890 Standard Quota

Example 4: Calculate the Standard Quota According to the countrys constitution, congress will have 200 seats. Population of Amador by State StateABCDETOTAL Population (in thousands)1112111813201515493510,000 Standard Quota

Some Good Advice: Keep in mind that the standard divisor is a single number that we calculate once and then use for the entire apportionment process. However, we must compute the standard quota individually for each state.

Study Tip: Due to rounding, the sum of the standard quotas can be slightly above or slightly below the total number of allocated items.

The Apportionment Problem: The apportionment problem is to determine a method for rounding standard quotas into whole numbers so that the sum of the numbers is the total number of allocated items.

Example 5: Finding lower and upper quotas. Population of Margaritaville by State StateABCDTOTAL Population (in thousands)2753834657671890 Standard Quota 4.36516.07947.381012.174630.0001 Lower Quotas Upper Quotas

Section 9.4 Assignments Classwork: TB pg. 547/1 12find standard divisors and standard quotas only! Must write problems and show ALL work.

4 Methods There are 4 different apportionment methods, which we will look at to solve the apportionment problem. 1. Hamiltons Method (already talked about) 2. Jeffersons Method 3. Adams Method 4. Websters Method`

Method 1 Hamiltons Method 1. Calculate each groups standard quota 2. Round each standard quota down to the nearest whole number, thereby finding the lower quota. Initially, give to each group its lower quota. 3. Give the surplus items, one at a time, to the groups with the largest decimal parts in their standard quotas until there are no more surplus items.

Example 6: A rapid transit service operates 130 buses along six routes A, B, C, D, E, and F. The number of buses assigned to each route is based on the average number of daily passengers per route, given in the table. Use Hamiltons method to apportion the buses. Rapid Transit Service RouteABCDEFTotal Avg Number of Passengers 43605130708010,24515,53522,65065,000

Example 6: Rapid Transit Service RoutePassengers Standard Quota Lower Quota Decimal Part Surplus Buses Final Apportionment A 4360 B5130 C7080 D10,245 E15,535 F22,650 Total65,000

The Quota Rule: A groups apportionment should be either its upper quota or its lower quota. An apportionment method that guarantees that this will always occur is said to satisfy the quota rule, such as the Hamilton Method.

Hamiltons Method: This would be the best method for apportionment, if its only problem was the Alabama paradox, but there are other paradoxes that occur from this method.

Population Paradox: The population paradox occurs when state As population is growing faster than state Bs population, yet A loses a representative to state B. (We are assuming that the total number of representatives in the legislature is not changing.)

Population Paradox and the Hamilton Method The Graduate school at Great Eastern University used the Hamilton method to apportion 15 graduate assistantships among the colleges of education, liberal arts, and business based on their undergraduate enrollments. a) Use Hamiltons method to allocate the graduate assistantships to the three colleges b) Assume that after the allocation was made in part a) that education gains 30 students, liberal arts gains 46, and the business enrollment stays the same. Reapportion the graduate assistantships again using the Hamilton method. c) Explain how this illustrates the population paradox.

Example 7: Apportion the 15 graduate assistantships before enrollments increase. Graduate School Great Eastern University College # of Students Standard Quota Integer Parts Fractional Parts Assign 2 Additonal Assistantships Education 940 Liberal Arts 1470 Business1600 Total4010

Example 8: Apportion the 15 graduate assistantships with the increased enrollment. Graduate School Great Eastern University College # of Students Standard Quota Integer Parts Fractional Parts Assign 2 Additonal Assistantships Education 970 Liberal Arts 1516 Business1600 Total4086

New-states Paradox The new-states paradox occurs when a new state is added, and its share of seats is added to the legislature causing a change in the allocation of seats previously given to another state.

Example 9: New-States Paradox A small country, Namania, consists of three states A, B, and C with populations given in the following table. Namanias legislature has 37 representatives that are to be apportioned to these states using the Hamilton method. a) Apportion these representatives using the Hamilton method. b) Assume that Namania annexes the country Darelia whose population is 3,000 (in thousands). Give Darelia its current share of representatives using the current standard divisor and add the number to the total number of representatives of Namania. Reapportion Namania again using the Hamilton method. c) Explain how the new-states paradox occurred.

Example 9: (a) Country of Namania 37 Seats State Pop. (in thousands) Standard Quota Integer Parts Fractional Parts Assign Additonal Reps A 2750 B6040 C3350 Total12,140

Example 9: (b) Country of Namania State Pop. (in thousands) Standard Quota Integer Parts Fractional Parts Assign Additonal Reps A 2750 B6040 C3350 D(arelia)3000 Total15,140

Example 9: (c) Country of Namania 46 Representatives State Pop. (in thousands ) Standar d Quota Integer Parts Fractional Parts Assign Additonal Reps A27508.3680.369 B604018.35180.3518 C335010.18100.1810 D(arelia)30009.1190.119 Total15,1404546 Country of Namania 37 Representatives State Pop. (in thousands) Standard Quota Intege r Parts Fractional Parts Assign Additonal Reps A27508.3880.388 B604018.41180.4119 C335010.21100.2110 Total12,1403637

Method 2: Jeffersons Method Use trial and error to find a modified divisor which is smaller than the standard divisor for the apportionment. Calculate the modifed quota (states population/modified divisor) for each state and round it down. Assign that number of representatives to each state. (Keep varying the modifed divisor until the sum of these assignments is equal to the total number being apportioned.) Note: This method was adopted in 1791 and used until the apportionment of 1832, when NY received 40 seats, with a standard quota of 38.59. Due to this violation the Jefferson Method was never used again. After this the Hamilton method was resurrected by Congress.

Example 10: Find modified divisor, should be lower than standard divisor. Rapid Transit Service RoutePassengers Modified Quota Modified Lower Final Apportionment A 4360 B5130 C7080 D10,245 E15,535 F22,650 Total65,000

Note: If the total numbers assigned is too small, then we need larger modified quotas. In order to have larger modified quotas, you will need to find smaller modified divisors.

Example 11: Jefferson Method Rapid Transit Service RoutePassengers Modified Quota Modified Lower Final Apportionment A 4360 B5130 C7080 D10,245 E15,535 F22,650 Total65,000

Method 3: Adams Method Use trial and error to find a modified divisor which is larger than the standard divisor for the apportionment. Calculate the modified quota (states population/modified divisor) for each state and round it up. Assign that number of representatives to each state. (Keep varying the modified divisor until the sum of these assignments is equal to the total number being apportioned.)

Example 12: Find Modified Divisor, should be larger than standard divisor Rapid Transit Service RoutePassengers Modified Quota Modified Upper Final Apportionment A 4360 B5130 C7080 D10,245 E15,535 F22,650 Total65,000

Example 13: Find Modified Divisor, should be larger than standard divisor Rapid Transit Service RoutePassengers Modified Quota Modified Upper Final Apportionment A 4360 B5130 C7080 D10,245 E15,535 F22,650 Total65,000

Background: In 1832, Daniel Webster suggested an apportionment method that sounds like a compromise between Jeffersons and Adams. He suggested if the decimal part was greater than 0.5, then we round up to the next whole number, whereas if the fractional part is less than 0.5, then we round to the whole number.

Method 4: Websters Method Use trial and error to find a modified divisor. Calculate the modified quota for each state and round it in the usual way. Assign that number of representatives to each state. (Keep varying the modified divisor until the sum of these assignments is equal to the total number being apportioned.)

Example 14: Rapid Transit Service RoutePassengers Modified Quota Modified Upper Final Apportionment A 4360 B5130 C7080 D10,245 E15,535 F22,650 Total65,000

Section 9.4 Assignment Classwork: TB pg. 547/23 26, 39 42, and 54 Remember you must write problems and show ALL work to receive credit for this assignment. Due Friday, Nov. 04, 2011 Reminder: If the assignment is not turned in by due date, then 10 points are minus for each day late, up to 3 days. On the 4 th day it is an automatic 0.

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Other Paradoxes and Apportionment Methods Section 9.4