ET - Apportionment & Paradoxes Lesson

Apportionment & Paradoxes

Many goods or objects cannot be distributed in fractional shares, something shares are rounded for apportionment. An apportionment problem is rounding each number in a set of numbers that add up to a whole number. The sum of the rounded numbers must preserve the sum of the original numbers. The Apportionment Method is most commonly used in the House of Representatives. The Senate consists of two senators from each state, which means each state is represented equally. The House of Representatives is designed to represent each state in proportion to the population. Politicians have struggled with the problem of determining a fair method for apportioning representatives from each state since the ratification of the constitution.

Let's explore...

The administration of several city high schools wants to create a City High student council to unify the city public schools. They felt each school should have the same number of representatives to have equal representation. The population of each school is given in the table below

Apportionment Methods

General terminology is used when dealing with apportionment methods. States refer to the objects being represented, in our case schools. The population is the number of people or items contained in a particular state. The house size is the number of allocations.

The first step to solving apportionment problems is identifying the states, population, and house size.

The second step is to determine the standard divisor. The standard divisor is the total population, divided by the house size.

The third step is to calculate the quota for each state. The ideal quota is the exact share that would be allocated to the state if a whole number were not required. To determine the ideal quota, divide the state's population by the standard divisor.  

All apportionment methods start with these three initial steps leading to the determination of the quotas. The next step, rounding to obtain whole numbers whose sum is the house size, is where the various methods differ.

An apportionment method that always allocates only lower and/or upper bounds is called a quota rule. Some methods may violate the quota rule.

 

Hamilton Method

Alexander Hamilton first introduced the Hamilton Method in 1792. It is known as the method of greatest remainders.

After completing steps 1-3, assign each state a number equal to the integer part of its ideal quota. The integer part is considered the lower quota. If the ideal quota of a state is less than one, assign it one representative.

Next find the sum of the number of assigned seats. If the sum is not equal to the allocated number of seats, allot the remaining seats, one each, to the states whose quotas have the largest fractional part, until the house is filled. If a state was given a seat because its ideal quota was less than one, it doesn't qualify for an additional seat.

The administration has researched the historical background of apportionment in the process of establishing a city student council, and they have decided to experiment with the methods proposed by early politicians in the United States.

Assuming that that there will be 30 seats on the council, how many seats will each school receive using Hamilton's Method?

 

The Hamilton Method created some paradoxes, which led to another method. The Alabama Paradox happens when increase in the total number of seats to be apportioned causes a state to lose a seat. The Population Paradox happens when an increase in a state's population can cause it to lose a seat. The New States Paradox happens when adding a new state with its fair share of seats can affect the number of seats due other states.

The Jefferson Method

 Thomas Jefferson favored a method for apportionment biased in favor of states with large populations. The Jefferson method is one of a class of apportionment methods called the divisor method.

                                     

Jefferson specified the population of the smallest district in the nation. The result of dividing a state's population by a divisor that is not the standard divisor is known as the adjusted quota.

After completing steps 1-3, choose a divisor that is slightly less than the standard divisor. Divide each state's population by the chosen divisor, to get its adjusted quota that is rounded down to the whole number, its lower quota.

Add the lower quotas. If the sum is equal to the house size you are done. If it is more seats, choose a large divisor and repeat the process of finding an adjusted quota. If it is less, choose a smaller divisor and repeat the process of finding an adjusted quota.

Assuming that there will be 30 newly formed members, how many seats will each school receive using Jefferson's Method? 

We need to fill 3 seats. We must use our largest critical divisor. Let's try d=496

We need to fill 2 seats. We must use our largest critical divisor. Let's try d=472.74

We need to fill 1 seat. We must use our largest critical divisor. Let's try d=469.143

Remembering the Constitution states that every state must have a representative. The administration decides to give the extra seat to the school with 0 representatives.

The Webster Method

 The Webster method is the divisor method that rounds the quota (adjusted if necessary) to the nearest whole number.

• Round up when the fractional part is greater than or equal to ½.
• Round down when the fractional part is less than ½.

Based on rounding fraction the usual way, so that the apportionment for state i is < q_i >, where q_i is the adjusted quota for state i.

The Webster method minimizes differences of representative shares between states (it does not favor the large states, as does the Jefferson method).

Assuming that there will be 30 newly formed members, how many seats will each school receive using Webster's Method? 

Steps for the Webster Method:

1. First add up the total population, p, of all the states ∑p_i.
2. Calculate the standard divisor, s = p /h (total population ÷ house).
3. Calculate each state's (or school) quota, q_i = p_i /s.
4. We will call the tentative apportionment   n_i = < q_i > . When the fractional part is greater than or equal to ½, round up; when the fraction part is less than ½, round down (the usual way of rounding).
5. If the sum of the tentative apportionments is enough to fill the house, we are finished, and the tentative become the actual apportionments.
6. If not, we adjust the tentative apportionments. (see Adjusting the Divisor below)             
7. Re-compute the apportionments a_i = p_i / d using this critical divisor, d.
8. Continue this process until the house is filled.

Adjusting the Divisor:                                                                                          

If the tentative apportionments do not fill the house, find which state has the largest critical divisor where   d_i⁺= p_i /(n_i + ½) will receive a seat.                      

If the tentative apportionments overfill the house, find which state has the smallest critical divisor where d_i⁻ = p_i /(n_i − ½) will lose a seat.

We have two more representatives than we have seats! If the tentative apportionments overfill the house, find which state has the smallest critical divisor where d_i^-=p_i/(n_i-1/2) will lose a seat.

The Hill-Huntington Method

The Hill-Huntington method is a divisor method related to the geometric mean that has been used to apportion the U.S. House of Representatives since 1940.  

Assuming that there will be 30 newly formed members, how many seats will each school receive using Hill-Huntington's Method? 

Calculating the critical divisors, if needed ...

• If the tentative apportionments do not fill the house, then the critical divisor for state i with population p_i and tentative apportionment n_i for each state is:

The state with the largest critical divisor is first in line to receive a seat.

• If the tentative apportionments overfill the house, then the critical divisor for state i with population pi and tentative apportionment ni for each state is:

The state with the smallest critical divisor is first in line to lose a seat.

Here, we have overfilled the house by 2 seats, the state with the smallest critical divisor is first in line to lose a seat.

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