1. Chapter 4—The Mathematics of Apportionment
Each state has two senators.
Each state has representatives based on the state’s population.
Constitution does NOT state the equation to use for finding the number of representatives.
This is now called “The apportionment problem”.

2. 4.1 Apportionment Problems
“apportion”—to divide and assign in due and proper proportion or according to some plan.
In apportionment problems: i) We are dividing and assigning things and ii) we are doing this on a proportional basis and in a planned, organized fashion.
What if a mom had 50 pieces of the same candy and has 5 children—many of us would believe this to be an easy “fair division” problem—give each child 10 pieces.
Let’s say that mom wants to divide the candy up based on how many hours each child helps with the chores.

3. 50 Pieces vs. 5 Children Child Alan Betty Connie Doug Ellie Total
Minutes worked
150 78
173
204
295
900
Alan worked 150 of the 900 minutes. However, he cannot get 16 2/3% of the candy since that would amount to 8 1/3 pieces. The candy is indivisible. He would feel shorted if he only got 8 pieces and he would get more than his fair share if mom gave him 9. What is mom to do?
This problem shows all the elements of an apportionment problem.

4. Preface Terminology
The “states”—term used to describe all players involved in the apportionment. (if no names are given, we will use A1, A2, …,AN) (children)
The “seats”—term which describes the set of M (identical, indivisible objects) that are being divided among N states. (candy)
The “populations”—set of N positive numbers which are the basis for the apportionment of the seats. (minutes)

5. Number Terminology—numerology? 
The standard divisor—this is the ratio of total population to seats—SD=P/M. (900 minutes/ 50 pieces = 18 minutes per piece)
The standard quotas—fractional (2-3 decimal places) number of seats a state would get—use q1, q2, …qN.—
quota= (state’s) population/SD (Alan worked 150 minutes…so, 150/18 = 8 1/3)
Lower quota(L1, L2, …LN)—quota rounded down (8)
Upper quota(U1, U2, …UN)—quota rounded up (9)
OUR GOAL this chapter—to use a procedure that
i) Will always produce a valid apportionment (exactly M seats are apportioned, and ii) Will always produce “fair” apportionment.
Ex. “Turtles, Turtles, Who Gets the Turtles?” Page 1
Classwork/Homework Pg. 150: 1-9 odd

6. 4.2 Hamilton’s Method
Also known as “Vinton’s method” or the “method of largest remainders”—used in the US between 1850 and 1900.
Every state gets “at least” its lower quota.
Step 1 —Calculate each state’s standard quota.
Step 2 —Give to each state (for the time being) its lower quota.
Step 3 –Give the surplus seats (one at a time) to the states with the largest fractional parts until there are no more surplus seats.

7. Flaws to Hamilton’s Method
A state with a fractional part of 0.72 may end up with one more seat than a state with a fractional part of 0.70—major flaw in the way it relies entirely on the size of the fractional parts without consideration of what those fractional parts represent as a percent of the state’s population—creates a bias in favor of larger states. Should be population neutral.
Alabama Paradox—occurs when an increase in the total number of seats being apportioned, in and of itself, forces a state to lose one of its seats.
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.
The New-States Paradox—the addition of a new state with its fair share of seats can, in and of itself, affect the apportionments of other states.
When using Hamilton’s method, all three paradoxes can occur—definitely not a good thing! (more detail on paradoxes in section 4.3)
Good points of Hamilton’s method—1) it is easy to understand, and 2) it satisfies an extremely important requirement for fairness called the quota rule.

8. The Quota Rule
Definition—a state should not be apportioned a number of seats smaller than its lower quota (lower-quota violation) or larger than its upper quota (upper-quota violation).
Remember, Step 2 of Hamilton’s Method satisfies the quota rule.
Ex. “Apportionment” Page 1
Classwork/Homework Pg. 152: odd

9. 4.4 Jefferson’s Method
Jefferson’s Method was the first apportionment method used in US House of Reps. (terminated in 1832)
Since in Hamilton’s Method there is always a surplus, Jefferson’s Method involves “tweaking” the standard divisor. If you lower the standard divisor (call this the modified divisor D), the quotas increase. Likewise, if you increase the standard divisor, the quotas decrease.
Our Goal—to apportion M seats without any surplus!!!

10. Steps for Jefferson’s Method
Step 1—find a “suitable” divisor D—a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded down.
Step 2—Each state is apportioned its lower quota (using the “suitable” divisor D).
Biggest problem with Jefferson’s Method—It can produce upper-quota violations!!! The upper-quota violations tend to favor the larger states.
Ex. “Turtles” Page 2
Classwork/Homework Pg. 152: 23, 25

11. 4.5 Adam’s Method
Step 1—find a “suitable” divisor D—a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded up. (opposite of Jefferson’s) This means that quotas have to be made smaller by using a larger divisor (larger than the standard divisor).
Step 2—Each state is apportioned its upper quota (using the “suitable” divisor D).
Biggest problem with Adam’s Method—It can produce lower-quota violations!!! Adam’s Method was never passed to apportion the House of Reps.
Ex. “Turtles” Page 3
Classwork/Homework Pg. 153: 33, 35

12. 4.6 Webster’s Method
Compromise between Jefferson’s Method (rounding down) and Adam’s Method (rounding up)—rounding up if 0.5 or higher and rounding down if less than 0.5—BUT, using a modified divisor.
Step 1—find a “suitable” divisor D—a suitable divisor is a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded the conventional way. (a suitable divisor CAN be the standard divisor…always check the SD first!!!!)
Step 2—Find the apportionment of each state by rounding its quota the conventional way (using the “suitable” divisor D).
Ex. “Apportionment” Page 2
Classwork/Homework Pg. 153: 43, 45 (important chart next slide)
Test this Friday

13. Comparisons Hamilton Jefferson Adams Webster Quota Rule No violation
Upper-quota violations possible
Lower-quota violations possible
Upper- and lower-quota violations possible
Alabama Paradox
Possible
Not
Population Paradox
New-State Paradox
Bias in favor of
Large states
Small states
Neutral