Other Paradoxes and Apportionment Methods Section 9.4
Slide 2
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.
Slide 3
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.
Slide 4
Calculating Standard Divisor Standard Divisor = total
population number of allocated items
Slide 5
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
Slide 6
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
Slide 7
Calculating Standard Quota To calculate standard quota, you
must first find the standard divisor. Standard Quota = population
of a particular group standard divisor
Slide 8
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
Slide 9
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
Slide 10
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.
Slide 11
Study Tip: Due to rounding, the sum of the standard quotas can
be slightly above or slightly below the total number of allocated
items.
Slide 12
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.
Slide 13
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
Slide 14
Section 9.4 Assignments Classwork: TB pg. 547/1 12find standard
divisors and standard quotas only! Must write problems and show ALL
work.
Slide 15
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`
Slide 16
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.
Slide 17
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
Slide 18
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
Slide 19
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.
Slide 20
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.
Slide 21
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.)
Slide 22
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.
Slide 23
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
Slide 24
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
Slide 25
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.
Slide 26
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.
Slide 27
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
Slide 28
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
Slide 29
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
Slide 30
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.
Slide 31
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
Slide 32
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.
Slide 33
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
Slide 34
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.)
Slide 35
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
Slide 36
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
Slide 37
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.
Slide 38
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.)
Slide 39
Example 14: Rapid Transit Service RoutePassengers Modified
Quota Modified Upper Final Apportionment A 4360 B5130 C7080 D10,245
E15,535 F22,650 Total65,000
Slide 40
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.