Transcript File

Chapter 4—The Mathematics of
Apportionment
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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”.
4.1 Apportionment Problems
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“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.
50 Pieces vs. 5 Children
Child
Minutes
worked
Alan Betty Connie Doug
150
78
173
204
Ellie
295
Total
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.
Preface Terminology
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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)
Number Terminology—numerology? 
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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
4.2 Hamilton’s Method
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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.
Flaws to Hamilton’s Method
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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.
The Quota Rule
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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: 11-21 odd
4.4 Jefferson’s Method
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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!!!
Steps for Jefferson’s Method
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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
4.5 Adam’s Method
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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
4.6 Webster’s Method
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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
Comparisons
Hamilton
Jefferson
Adams
Webster
Quota Rule
No violation
Upper-quota Lower-quota Upper- and
violations
violations
lower-quota
possible
possible
violations
possible
Alabama
Paradox
Possible
Not
Not
Not
Population
Paradox
Possible
Not
Not
Not
New-State
Paradox
Possible
Not
Not
Not
Bias in favor Large states Large states Small states
of
Neutral