Ch 4: Apportionment
4.1 Apportionment Problems
Basic Elements
• States: the players involved in the
apportionment
• Seats: the set of M identical, indivisible
objects that are being divided among N states;
M>N but not every state is guaranteed a seat
• Populations: set of positive numbers which
are used as the basis for the apportionment of
the seats to the states (actual population,
minutes studied, number of patients, etc)
Divisors & Quotas
• Standard Divisor (SD): the ratio of population
to seats (total population/total # of seats)
• Standard Quotas: exact fractional number of
seats each state would get if fractional seats
were allowed (population/SD)
• Lower Quota: the quota rounded down (the
LQ of 2.62 would be 2)
• Upper Quota: the quota rounded up (the UQ
of 2.62 would be 3)
Pg. 150 #1
• The Bandana Republic is a small country consisting of
four states (Apure, Barinas, Carabobo, and Dolores).
There are M=160 seats in the Bandana Congress. The
populations of each state (in millions) are given in the
following table. Find the standard divisor, standard
quota, and lower and upper quotas.
State
A
B
C
D
Population
3.31
2.67
1.33
0.69
SQ
Lower Quota
Upper Quota
Total
Pg. 150 #3
• The SMARTS operates 6 bus routes (A, B, C, D, E, and F)
and 130 buses. The buses are apportioned among the
routes based on the average number of daily passengers
per route, given in the following table. Describe the
“states” and the “seats”. Find the SD and explain what it
represents in this problem.
Route
A
B
C
D
E
F
Average # of
passengers
45,300
31,070
20,490
14,160
10,260
8,720
Pg. 151 #5
• The total population is 23.8 million. The
following table shows each state’s standard
quota. Find the number of seats, the SD, and
each state’s population.
State
A
B
C
D
E
SQ
40.50
29.70
23.65
14.60
10.55
Pg. 151 #7 & #9
• 7) Texas had 7.43% of the US population in
2000. Find their standard quota of the 435 seats
in the House of Representatives.
• 9) If there are 200 seats apportioned among six
planets, find the SD and SQ for each planet
listed in the following table.
Planet
A
B
C
D
E
% of Population
40.50
29.70
23.65
14.60
10.55
SQ
F
4.2 Hamilton’s Method
Hamilton’s Method
• Step 1: Calculate each states standard
quota.
• Step 2: Give each state its lower quota.
• Step 3: Give the surplus seats to the
states with the largest decimal.
Hamilton’s Method
• Example: State A has a population of 940, state
B has a population of 9,030, and state C has a
population of 10,030. There are 200 seats to be
divided among the three states using Hamilton’s
Method.
State Pop
A
940
B
9030
C
10030
Total
SQ
LQ
HQ
Hamilton’s Method
• Example: There are 50 seats to divide among 5
states. A has a population of 150, B has 78, C
has 173, D has 204, and E has 295. Find each
states apportionment of seats using Hamilton’s
Method.
State Pop
SQ
LQ
HQ
A
150
B
78
C
173
D
204
E
295
Total
Quota Violations
The Quota Rule
• The Quota Rule: Each state must receive
either their lower quota or their upper quota.
Lower Quota Violation
• When a state ends up with an apportionment
smaller than it’s lower quota.
Upper Quota Violation
• When a state ends up with an apportionment
larger than it’s upper quota.
4.3 Paradoxes
Alabama Paradox
• This occurs when an increase in the total
number of seats being apportioned forces a
state to lose one of its seats.
Population Paradox
• This occurs when State A has a higher
percentage of increase in population than
State B, but State A loses one of its seats to
State B.
New-States Paradox
• This occurs when the addition of a new state
with its fair share of seats affects the
apportionments of the other states.
4.4 Jefferson’s Method
Jefferson’s Method
• Find a modified divisor (D) so that each states
lower quotas add to be the exact number of
seats.
Jefferson’s Method
• Example: State A has a population of 940, state
B has a population of 9,030, and state C has a
population of 10,030. There are 200 seats to be
divided among the three states using Jefferson’s
Method.
State
Pop
A
940
B
9030
C
10030
Total
SQ
LQ
JQ
Jefferson’s Method
• Example: There are 50 seats to divide among 5
states. A has a population of 150, B has 78, C
has 173, D has 204, and E has 295. Find each
states apportionment of seats using Jefferson’s
Method.
State
Pop
SQ
LQ
JQ
A
150
B
78
C
173
D
204
E
295
Total
4.5 Adam’s Method
Adam’s Method
• Find a modified divisor (D) so that each states
upper quotas add to be the exact number of
seats.
• Start by using a modified divisor that is bigger
than the standard divisor.
Adam’s Method
• Example: State A has a population of 940, state B has
a population of 9,030, and state C has a population
of 10,030. There are 200 seats to be divided among
the three states using Adam’s Method.
State
Pop
A
940
B
9030
C
10030
Total
SQ
UQ
AQ
Adam’s Method
• Example: There are 50 seats to divide among 5
states. A has a population of 150, B has 78, C has
173, D has 204, and E has 295. Find each states
apportionment of seats using Adam’s Method.
State
Pop
A
150
B
78
C
173
D
204
E
295
Total
SQ
UQ
AQ
4.6 Webster’s Method
Webster’s Method
• Find a modified divisor (D) so that each states
standard quotas rounded conventionally add
to be the exact number of seats.
Webster’s Method
• Example: State A has a population of 940, state B has
a population of 9,030, and state C has a population
of 10,030. There are 200 seats to be divided among
the three states using Webster’s Method.
Webster’s Method
• Example: There are 50 seats to divide among 5
states. A has a population of 150, B has 78, C has
173, D has 204, and E has 295. Find each states
apportionment of seats using Webster’s Method.