# MATH 1007 Chapter Notes - Chapter 7-8 & 19-20: United States Congressional Apportionment, Highest Averages Method, Monotonicity Criterion

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Samuel Mohebban Exam 2 Notes Math and Politics

Chapter 7- Hamilton’s Method

Apportionment- a method designed to round real numbers into whole numbers.

o Makig a fai shae of seats aodig to a state/outy populatio

Hamilton- a method od apportionment

o Deteie eah state’s fai shae (standard quota) by dividing its population by

the total population and multiplying this by the total number of seats to be

apportioned

o Then round these standard quotas either up to the next largest integer (upper

quota) or down to the next smallest integer (lower quota) in such a way that

(1) the correct number of seats have been doled out,

(2) the benefit of the the extra representative is doled out to the states in

decreasing order of the fractional part of their standard quotas

Fractional part is the decimal after a number (1.33= 33)

n- positive integer which stands for the number of states in the problem

o In the US n=50

h- the number of seats in the house of representatives

o In the U.S h=435

Pk- the population of the kth state

o Population of the first state is denoted P1 and the 17th state is denoted P17

p- the total population of all states

o p= p+p+….+pn

Census- refers to the numbers h, n, and p1,p2,pn collectively

o Output- is to be a list of n whole numbers that we donate by a1,a2,a3,an

ak- as the number of seats accorded to the kth state

o ak might equal 0 for some

a- is for apportionment

Apportionment Method- is a function whose domain is the set of all possible values of

h, n, and p1, p2, pn, where h and n are positive integers and the numbers p1, p2, pn are

positive real numbers, and whose input is a sequence of n nonnegative integers a1, a2,

an satisfying

o a+a+…+an=h

Represents the requirement that the total number of seats accorded to

the states must equal the prescribed house size exactly

o Goal is to make p/h as close to pn/an

o Fraction ak/pk (reciprocal) can be thought of as the measure of the strength in

voice in congress of a single citizen in the kth state

o For perfection, the apportionment number ak must be an integer

Standard quota- the real number qk= h(pk/p) for the kth state

o For Maryland, the standard quota = 435(5,307,886/281,424,177) = 8.2

Also can be shown in the form pk/(p/h)

o Standard Divisor- the denominator in this expression

Designated by the letter s= p/h

o Standard quota of a state- qk=pk/s (dividing its population by standard divisor)

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Samuel Mohebban Exam 2 Notes Math and Politics

Standard divisor- represents the population of an ideal district

o Designated by the letter s= p/h

Standard Quota- represents the ideal number of seats to assign to that state

o The sum of all standard quotas = h

o Represented by qk=pk/s (dividing its population by standard divisor)

Two natural choices for ak is:

o Lower quota- the standard quota rounded down (8.2= 8)

o Upper quota- the standard quota rounded up (8.2= 9)

Ex. Estimates suggests that Montana will have a standard quota of 1.4.

Whether this will be rounded up to 2 or down to 1 is critically important

to the welfare of the Montanans. If Montana is allocated just one seat in

the House, the lone representative from this state will represent 905,316

people, making Montana the largest congressional district in the nation,

much larger than the average size of a district nationwide. On the other

hand, if Montana is allocated two seats, each congressional district in

Montana will contain 453,000 residents, far fewer than the national

average. The only choices for Montana are extremes, making the

residents either the most represented or the least represented

Americans.

Integer part- is the greatest integer less than or equal to the real number (3.141= 3)

Fractional Part- is the difference between that number and its integer part of a real

number (3.141= .141)

Hamilton Method- Assigns every state its lower quota. Then assigns the seats that

remain to the states, at most on per state, in decreasing order of the size of the

fractional parts of their standard quotas

o **example**

o assigns each state either its lower or upper quota

o Any method that does this is called the quota method

o Because the sum of the standard quotas = h, rounding all of them up would be

greater than h

o Hamilton argued that a state should be awarded a number of representatives

that is at least the integer part of its standard quota (the lower quota), and then

it should be considered for one additional seat based on the fractional part of

the standard quota

o Shows major flaws when applying the population, Alabama, and Oklahoma

paradoxes

o Violates population and house monotonicity criterion.

o Satisfies quota criterion (upper and lower quota rule)

Paradoxes

Hamilton method can behave weirdly in the face of changes to the house size, h, the

number of n states, or the populations p1,p2,pn of the states

Alabama Paradox- where a state loses a seat when the house size is increased

o **example**

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Samuel Mohebban Exam 2 Notes Math and Politics

House monotone (monotonicity criterion)- criterion for apportionment that forbids the

Alabama paradox

o If an in increase in h, all other parameters remaining fixed, can never result in a

decrease in any ak

o Hailto’s ethod iolates this

Population Paradox- when one state gains in population while another loses, and yet in

the transition the first state loses a seat while the second gains a seat

o Hailto’s ethod iolates this

Population Monotonicity- a method that avoids the population paradox

o Hailto’s ethod iolates this

New states (Oklahoma) Paradox- the paradox showing the impact of the addition of

another state into the union

Chapter 8- Divisor Methods

h is fixed and given to us in advance, and our ob is to distribute exactly that many seats

In the U.S, the house size was not fixed in advance, but was determined by

congressional apportionment legislation

o First apportionment of the House was based on a census that allotted h=105

members. The number rose over decades, reaching its current value of h=435

If h is not predetermined, then another approach to apportionment is suggested

Jefferson proposed that instead of beginning with a fixed value of h, begin with a fixed

idea of how large a congressional district should be

When we have a fixed value of h, we use the standard divisor of s=p/h to represent this

quantity

When we do not have a fixed value of h, then we choose an arbitrary number d that we

regard as an appropriate size for a congressional district

d- is called the modified divisor

o Modified refers to the fact that the divisor is not the standard one

Modified divisor- Like the standard divisor s, we divide the modified divisor into the

population of each state

o Modified divisor for state k is pk/d

Represents the appropriate number of seats to assign to state k, if each

district is to have a population d

o Not always an integer

So Jefferson proposed that we round these numbers down to the next

whole number

Jefferson Method- choose a modified divisor d. Compute the modified quotas pk/d.

Round each of these numbers down to obtain ak. If a+a+…+a=h, the e hae

Jefferson apportionment. Otherwise, modify the divisor d and try again

o Usually d is smaller than s

o If d is too lage, the the total ue of seats, a+a+…+a ill e salle tha

h

o If d is too small, then the total ue of seats, a+a+…+a ill e lage tha h

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