Finite Math A, Chapter 4
The Mathematics of Apportionment
The Place:Philadelphia
The Time:Summer 1787
The Players:Delegates from the 13 states
The Problem:Draft a Constitution for our new nation
The Big Argument:
How would the people be represented?
What would the legislature look like?
Small states: ______representation
Large states: ______representation
The Connecticut Plan: A compromise, two houses of legislature
Senate: ____ senators per stateHouse: Representatives per state “shall be apportioned… according to their respective numbers” Article 1, Section 2, U.S. Constitution
The Catch:The founding fathers did not outline a plan for how to divide the seats in the House of Representatives proportionally. It should be a relatively straightforward procedure right?
4.1 Apportionment Problems
- Basic Idea: We are dividing and assigning things on a proportional basis in a planned
and organized fashion.This is a DISCRETE fair division problem where (unlike Ch. 3) each player deserves a different share of the goods.
Example 4.1 Kitchen Capitalism
Mom has 50 pieces of identical candy to split among her 5 children. She decides that each child will earn a proportion of the candy based on how many minutes of chores they did during the week.
How many pieces of candy should go to Alan?By similar math:
Betty = 4.33 pieces
Connie = 9.61 pieces
Doug = 11.33 pieces
Ellie = 16.39 pieces
If Mom gives Alan ____ pieces, he gets more than he deserves, and someone else gets shorted.
If Mom gives Alan _____ pieces, he gets less than he deserves, and someone else gets more.
If Mom does traditional rounding there is candy leftover! How much? Who should get it?
What would you do? What should Mom do? Why is this even important!?!
Most familiar example:
How many of the 435 indivisible seats in the House of Representatives be apportioned to each of our states?
Other important examples:
Apportioning nurses to Shifts at hospitals
Apportioning telephone calls to switchboards in a network
Apportioning teachers to classes of students in a school
Example: Parador is a new republic in Central America and consists of six states, which we will call A, B, C, D, E, and F for simplicity. There are 250 seats in Parador’s Congress. What is the “correct apportionment?
State / Population / StandardQuota / Trad.
Rounding
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
Step 1: Compute the Standard Divisor (SD)
SD = total ÷ seats
(the ratio of total population to seats)
Step 2: Compute each state’s Standard Quota
Standard Quota = State Population ÷ SD
(exact fractional part each state deserves)
Example:
What happens if we apportion by traditional rounding?
Quick Terminology and Symbols:
- The States = parties that deserve a piece of the total
- TheSeats = “M” = the number of things to be apportioned (seats, nurses, candies, etc)
- The Populations = the numbers used as the basis for the apportionment (population, minutes worked, students enrolled, etc.)
- Standard Divisor = “SD” the number of population represented by 1 seat
- Standard Quota = The exact number of seats a state would get if fractional parts were allowed
- Lower Quota = Standard Quota rounded Down
- Upper Quota = Standard Quota rounded UP
4.3 Hamilton’s Method
Alexander Hamilton (1757-1804)
Method used in United States from 1850 – 1900
Method still used today in Costa Rica, Namibia, and Sweden.
State / Population / StandardQuota / Lower
Quota / Extra
Seat? / FINAL
A / 1,646,000 / 32.92
B / 6,936,000 / 138.72
C / 154,000 / 3.08
D / 2,091,000 / 41.82
E / 685,000 / 13.70
F / 988,000 / 19.76
Total / 12,500,000 / 250
After the Lower Quotas are assigned, are there any extra seats left?
Which state has the highest residue?
Problems:
Residues don’t take into account what that fraction represents as a percentage of its population.
Bias to large states Example: B (.72) vs E (.70)
Mathematical Paradoxes (discussed in next section)
Great Things:
Easy to Understand. Satisfies the “Quota Rule”
Quota Rule:
If Betty’s standard quota is 4.33, she should end up with either _____ or _____ pieces of candy.
Practice Examples:
1. A small country consists of four states. The population of state A is 44,800. The population of state B is 52,200. The population of state C is 49,200. The population of state D is 53,800. The total number of seats in the legislature is 100. What is the standard divisor? Find each state’s standard quota.
2. A small country consists of four states. The total population of the country is 200,000. The standard quotas for each state are: A: 64.8, B:89.9, C: 39.6, D: 5.7 What is the standard divisor? What is the population of each state?
For each problem 3, 4
a. Find the Standard Divisor. What does the Standard Divisor represent in this particular
example?
b. Find each state’sStandard Quota.
c. Use Hamilton’s Method to find the apportionment for the given number of seats, M.
3. A local department store has budgeted for 120 eight-hour retail shifts to be staffed every week. The number of shifts staffed on a single day of the week is apportioned based on the total number of shoppers who visit the store during the day. The following table shows the average daily number of shoppers over a two month period.
a)
Total “population”:Standard Divisor:
Number of “seats”:
4. The Faculty Senate at a university has been delegated the duty of apportioning the 500 university owned laptops to five different programs (Engineering, Social Sciences, Nursing, Arts and Sciences, and Business). The laptops are going to be apportioned to each program based on the number of students enrolled in the program. The table below shows the enrollment numbers for each program.
a)
Total “population”:Standard Divisor:
Number of “seats”:
4.3 The Paradoxes – Alabama, Population, New States
The fatal flaw with Hamilton’s method is the Alabama Paradox.
The Alabama Paradox
In 1882 – different apportionment methods were being debated for the House of Representatives.
Discovery: If Hamilton’s Method is used to apportion a House of 299 seats, Alabama gets 8 seats.
If Hamilton’s Method is used to apportion a House of 300 seats, Alabama gets 7 seats.
In 1901 – House sizes where debated from 350 to 400 seats.
M = 350 to 356Maine = 4 seats
M = 357Maine = 3 seats
M = 358 to 381Maine = 4 seats
M = 382Maine = 3 seats
etc.
Quote from your textbook:
pg. 131 “When a bill with M = 357 was proposed, all hell broke loose on the House floor. Fortunately, cooler heads prevailed and the bill never passed. Hamilton’s method was never to be used again.”
Example:
The small country of Calavos consists of three states: Bama, Tecos, and Ilnoswith a total population of 20,000 and 200 seats in the House of Representatives.
Overnight, a decision is made to ADD A REPRESENTATIVE to the house, raising the number of seats to 201.
What do you think should happen?
The Population Paradox
In the year 2525 the five planets in the Utopia galaxy finally signed a peace treaty and agreed to form an Intergalactic Federation governed by an Intergalactic Congress.
In 2525, 50 seats were apportioned using Hamilton’s method as shown to the right.
What was the standard divisor (SD)?
Ten years later, new census….
Conii up 8 billion
Ellisium up 1 billion
What is the new standard divisor (SD)?
NOTICE:
Elisium______even though its population ______
Betta ______even though its population______
Conii ______even though its population ______
The New States Paradox
In 1907, Oklahoma joined the Union. There were currently 386 seats in the House of Rep.’s.
A fair apportionment of seats (based on population) to OK was 5 seats, so 5 seats were added 391
For no other reason:Maine3 seats 4 seats
New York 38 seats 37 seats
Example:
Metro Garbage Company picks up garbage and recycling in Northtown and Southtown. The company runs 100 trucks.
What is the standard divisor (SD)?
The company expands its services to Newtown’s population is 5,250 so the company and adds 5 additional garbage trucks.
What is the standard divisor (SD) now?
What happens?
Examples: State which paradox is occurring in each of the following situations
The Alabama Paradox
The Population Paradox
The New States Paradox
1.Under a certain apportionment method, a state receives an apportionment of 52 seats when the total number of seats in the legislature is 334, but only 51 seats when the total number of seats in the legislature is 335.
2.A mother wishes to apportion 16 pieces of candy to her three children: Abby, Betty, and Cindy based on the number of hours each child spends doing chores around the house. Using a certain apportionment method, she decides to give Abby 9 pieces of candy, Betty 4 pieces, and Cindy 3 pieces. However, just before she hands out candy, she finds out that the neighbor’s daughter Darla has been helping the children with the chores and has worked the same number of hours as Cindy, so she adds 3 pieces, bringing the total candy to 19 pieces. Now, Abby ends up with 10 pieces, Betty with 3 pieces, Cindy with 3 pieces, and Darla with 3 pieces.
3.Under a certain apportionment method, State X receives 41 seats and State Y receives 29 seats. Ten years later the population of State X has increased by 5% while the population of State Y remains unchanged. The seats are reapportioned and now State X receives 40 seats and State Y receives 30 seats.
4.4 Jefferson’s Method
Thomas Jefferson
(1743 – 1826)
Method used in U.S. from 1792 to 1840
Still used in Austria, Brazil, Finland, Germany, and the Netherlands.
Jefferson’s Idea: Let’s tweak our standard divisor, so that when every state’s quota is rounded down, there are no surplus seats!
How do you get the “modified divisor”? Mostly by guess & check + a little bit of strategy.
Seems pretty great at first… but there is a major flaw:
Jefferson’s Method causes ______violations.
4.5Adam’s Method
John Quincy Adams
(1767 - 1848)
Adam’s Idea: Let’s tweak our standard divisor, so that when every state’s quota is rounded up, there are no surplus seats!
Slightly different method, but essentially same problem as Jefferson’s Method.
Adam’s method causes ______violations.
Summary:Jeffersonvs.Adams
Find SQsFind SQs
Give LQsGive UQs
OK? Yes = Done, No = ModifyOK? Yes = Done, No = Modify
Find MQsFind MQs
Give LQsGive UQs
Causes ______quota violations Causes ______quota violations
Example. A small country consists of four states. The population of state A is 44,800. The population of state B is 52,200. The population of state C is 49,200. The population of state D is 53,800. The total number of seats in the legislature is 100.
Look back to Example 1, page 4.
What was the standard divisor?What was each state’s standard quota?
SD: ______A = ______B= ______
C = ______D = ______
a.Use a modified divisor of D = 1950 to find each state’s Modified Quota.
Next, apportion using Jefferson’s method.
Are there any quota violations?
b.Use a modified divisor of D = 2045 to find each state’s Modified Quota.
Next, apportion using Adams’ method.
Are there any quota violations?
4.6 Webster’s Method
Daniel Webster (1782 – 1852)
Lawyer, Statesman, Senator from Massachusetts
Method used in 1842, 1901, 1911, 1931
Basically a compromise between Jefferson and Adams
Example:
How do we make Webster’s Method work?
1. Start with the SD and find each states Standard Quota.
2. Use traditional rounding. Does the number of seats apportioned = the number of seats available?
- If yes, you’re done!
- If the number of seats apportioned = too many make your divisor a little BIGGER
and try again
- If the number of seats apportioned = too few make your divisor a little SMALLER
and try again
It may take several attempts to do this successfully!
Examples:
1. A certain country has five states and 240 seats in the legislature, and the populations of the states are: A: 427,000 B: 754,000 C: 4,389,000 D : 3,873,000 E: 157,000
Use a modified divisor of D = 40,100 to find each state’s modified quota and apportion using Webster’s method:
2. A grandmother is going to distribute 225 pieces of candy to her four grandchildren based on how many minutes of housework they’ve completed over the past week. The table below gives the number of minutes each child spent doing housework during the past week. Use Webster’s Method to Apportion the candy.
3. Four friends are lost on a tropical island. Luckily the friends find a stash of 75 coconuts. The coconuts will be apportioned based on the weight of each person (i.e. the heavier a person is, the more he gets). The table below shows the weight of each of the four friends. Find a modified divisor and apportion the 75 coconuts among the four friends.
4. Which method or methods do not violate the quota rule?
HamiltonJeffersonAdamsWebsterNone of these
5. Which method or methods cause upper quota violations?
HamiltonJeffersonAdamsWebsterNone of these
6. Which method or methods can produce the population paradox?
HamiltonJeffersonAdamsWebsterNone of these
7. Which method or methods does not violate the quota rule and does not produce any paradoxes?
HamiltonJeffersonAdamsWebsterNone of these
In conclusion
It seems like every method has its problems. Either the method violates the quota rule or creates mathematical paradoxes. Some favor large states, some favor small states.
Webster’s Method does not suffer from paradoxes and it does not show large/small state bias
However, it can potentially (and rarely) cause violations of the upper and lower quota.
If Webster had been used from 1790 to 2000, not a single violation would have occurred…
Currently, the House of Representatives is apportioned using a method called the Huntington-Hill Method which is discussed in detail on pages 152 – 163 and involves geometric means rounding instead of arithmetic means rounding…
The Huntington Hill method was created by a mathematician instead of a politician.
The method is almost identical to Webster’s and most of the time produces exactly the same results.
NOTE: The Test over Chapter 4 will include some extra credit questions that will come from the READING in Chapter 4 pages 122 – 143.
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