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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 state House: 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 and Apportionment Methods 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 each player deserves a different share of the goods. “Discrete division” means that objects are not easily or practically split into smaller pieces.

Example 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!?!

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Quick Terminology and Symbols:

The States = the “N” parties that deserve a piece of the total

The Seats = the set of “M” identical, indivisible objects that are being divided.

The Populations = the numbers used as the basis for the apportionment (population, minutes worked, students enrolled, etc.)

Apportionment Method: A systematic procedure that guarantees the division of exactly “M” seats to the “N” states using a formula based off of the state population.

Standard Divisor = “SD” = the ratio of population to seats = the number of population represented by 1 seat

Standard Quota = the ratio of state population to standard divisor = The exact “fair share” number of seats a state would get if fractional parts were allowed

Lower Quota = Standard Quota rounded Down

Upper Quota = Standard Quota rounded UP Example 2: Your college campus is broken into five sections. The board of trustees has recently approved the installation of 70 new emergency blue lights. The lights will be apportioned based on the area of each section. That is, the larger the section, the more lights that it will receive. The table below gives the area, in acres, of each section of campus.

Identify: The states: ______________________________________________________________ The seats: ______________________________________________________________ The populations: __________________________________________________________ a) Find the standard divisor. What does it represent in the context of this question?

𝑆𝐷 =𝑇𝑜𝑡𝑎𝑙 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆𝑒𝑎𝑡𝑠

b) Find the standard quotas.

𝑆𝑄 =𝑆𝑡𝑎𝑡𝑒 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛

𝑆𝐷

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Example 3: 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?

Step 1: Compute the Standard Divisor (SD) SD = total ÷ seats Step 2: Compute each state’s Standard Quota Standard Quota = State Population ÷ SD What happens if we apportion by traditional rounding? Traditional Rounding is sometimes called “Nearest Integer.” It does not work as an apportionment method because it doesn’t ALWAYS give away EXACTLY “M” seats.

State Population Standard Quota

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

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4.2 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.

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. A good apportionment method should be “population neutral.”

Bias to large states Example: B (.72) vs E (.70)

State Population Standard Quota

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

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Mathematical Paradoxes (discussed in section 4.6 ) 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:

a. Find the Standard Divisor. What does the Standard Divisor represent in this particular example?

b. Find each state’s Standard Quota. c. Use Hamilton’s Method to find the apportionment for the given number of seats, M. 1. 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.

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2. 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.

4.3 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 states’ quota is rounded down, there are no surplus seats. How do you get the “modified divisor?” Mostly by guess & check and a little bit of strategy.

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Seems pretty great at first… but there is a major flaw: Jefferson’s Method causes __________________________ violations.

4.4A-- Adam’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.

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4.4B -- 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!

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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.

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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.

4.6 The Quota Rule & The Paradoxes Every method discussed so far has advantages and disadvantages. Hamilton’s Method: Cannot violate the quota rule – good Bias to large states, produces mathematical paradoxes – bad Jefferson’s Method Doesn’t produce mathematical paradoxes – good

Bias to large states, causes upper quota violations –bad Adams’s Method Doesn’t produce mathematical paradoxes – good Bias to small states, causes lower quota violation – bad Webster’s Method Doesn’t produce mathematical paradoxes, population neutral – good Causes upper and lower quota violations - bad (although rarely - good) *If Webster had been used from 1790 to 2000, not a single violation would have occurred…* Example 1: Suppose you know an apportionment problem was solved using Hamilton’s method. You know the standard quota for state X is 47.21 . You are provided the choices at the right. You can answer the question without doing any work. What is it?

Chandler Ross Rachel Joey Joey

The final apportionment to state X is: A) 49 B) 46 C) 47 D) 50

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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 356 Maine = 4 seats M = 357 Maine = 3 seats M = 358 to 381 Maine = 4 seats M = 382 Maine = 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 Ilnos with a total population of 20,000 and 200 seats in the House of Representatives. Apportion using Hamilton’s Method. 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 Alabama Paradox occurs when: An increase in the number of seats being apportioned, in and of itself, forces a state to lose one of its seats.

“In Maine comes and out Maine goes – God help the State of Maine when mathematics reach for her to strike her down.” --Charles E. Littlefield, Maine Congressional Representative 1901

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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_________________________________

The Population Paradox occurs when: state A loses a seat to state B even though the population of A grew at a higher rate than the population of B.

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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: Maine 3 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?

The New States Paradox occurs when: the addition of a new state with its fair share of seats can, in and of itself, affect the apportionments of other states.

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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.

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4. Which method or methods do not violate the quota rule? Hamilton Jefferson Adams Webster None of these 5. Which method or methods cause upper quota violations?

Hamilton Jefferson Adams Webster None of these 6. Which method or methods can produce the population paradox? Hamilton Jefferson Adams Webster None of these 7. Which method or methods does not violate the quota rule and does not produce any paradoxes? Hamilton Jefferson Adams Webster None of these Currently, the House of Representatives is apportioned using a method called the Huntington-Hill Method . 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.

4.5 The Huntington-Hill Method

Up until 1940, the method used to apportion the house of representatives was voted on every 10 years after the census. Choices were made for political reasons and not for logical mathematical reasons. In 1941, the 1941 Apportionment Act was passed. It did the following:

Fixed the House of Rep.’s to 435 seats forever Permanently set the apportionment method to the Huntington-Hill Method The Huntington-Hill method and Webster often produce the same apportionment. Webster: Traditional Rounding Huntington-Hill uses the “Geometric Means” SQ = 10.458 Rounding Rule. (next page) SQ = 10.531

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Geometric Mean: The Geometric Mean of any two numbers, say “A” and “B,” is the square root of their product: √𝐴𝐵 Example 1: Find the Geometric Mean of the following a) 5 and 6 b) 10 and 11

Huntington-Hill Rounding Rule: Example 2: Use the Huntington-Hill Rounding Rule to round the following quotas. a) q = 2.513 b) q = 2.415 c) q = 2.462 d) q = 8.499 e) q = 5.463 f) q = 46.482

Let q be quota with a Lower Quota “L” and an Upper Quota “U”

Then the cutoff for rounding q is given by: c = √𝐿𝑈

That is: If q < c round down

If q > c round up

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Huntington-Hill behaves just like Webster’s in that we follow the following steps: 1. Start with the SD and find each states Standard Quota. 2. Use geometric means 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 Example 3: Apportion Parador’s Congress using the Huntington Hill Method. A modified divisor has been used of MD = 50,100.

NOTE: The Test over Chapter 4 will include some extra credit questions that will

include some information not discussed in class, but covered in the READING in

Chapter 4 pages 125-127

State Population Modified Quota

Cutoff Point Round Quota to:

A

1,646,000 32.85

B

6,936,000 138.44

C

154,000 3.07

D

2,091,000 41.74

E

685,000 13.67

F

988,000 19.72

Total 12,500,000