Introduction to Apportionment: Establishing the Intergalactic Congress

It is 2525 and all the planets in the galaxy have finally signed a peace treaty. Five of the planets (Alanos, Betta, Conii, Dugos, and Ellisium) decide to join forces and form an Intergalactic Federation. The Federation will be ruled by an Intergalactic Congress consisting of 50 seats, and each of the 5 planets will be entitled to a number of seats that is proportional to its population. The population of each planet is:

1. Since the total population of 900 billion people will be represented by 50 delegates (one for each seat), on average, how many people will be represented by each of the 50 delegates?
   
   
2. Alanos has a population of 150 billion people. If it is entitled to a number of seats that is proportional to its population, how many seats is Alanos entitled to? HINT: Use your answer in #1. (Round to 2 decimal places.)
   
   
3. Betta has a population of 78 billion people. If it is entitled to a number of seats that is proportional to its population, how many seats is Betta entitled to? HINT: Use your answer in #1. (Round to 2 decimal places.)
   
   
4. Conii has a population of 173 billion people. If it is entitled to a number of seats that is proportional to its population, how many seats is Conii entitled to? HINT: Use your answer in #1. (Round to 2 decimal places.)
   
   
5. Dugos has a population of 204 billion people. If it is entitled to a number of seats that is proportional to its population, how many seats is Dugos entitled to? HINT: Use answer in #1. (Round to2 decimal places.)
   
   
6. Ellisium has a population of 295 billion people. If it is entitled to a # of seats that is proportional to its population, how many seats is Ellisium entitled to? HINT: Use answer in #1. (Round to 2 decimal places.)
   
   
7. There is a problem with using the numbers obtained in #2-6 above; namely, seats cannot be fractional. That is, we can’t give a planet a half or a third of a seat. Given that fact, figure out a logical way of allocating the 50 seats among the 5 planets. Write down your final allocation of seats to the five planets.
   

   PLANET	Alanos	Betta	Conii	Dugos	Ellisium
   # of Seats

8. Explain your reasoning in #7 above.

In allocating seats to the Intergalactic Congress, you went through many of the same processes and encountered many of the same difficulties as our founding fathers as they attempted to allocate seats among the states in the fledgling House of Representatives.

The basic problem here is an example of a process called apportionment.

APPORTIONMENT is the process of fairly dividing a fixed number of identical and indivisible objects among a certain number of units each of which is entitled to a certain proportion of the total.

9. In the Intergalactic Congress exercise, what are the "identical and indivisible" objects being fairly divided among the units?
   
   
10. In the Intergalactic Congress exercise, who (or what) are the "units" among which something is being fairly divided?

In #1, you divided the TOTAL POPULATION by the NUMBER OF SEATS. In the language of apportionment, this result is called the STANDARD DIVISOR. (The Standard Divisor gives the average number of people/ seat.)

11. What was the Standard Divisor for the Intergalactic Congress exercise?

In #2-6, you divided each PLANET’S POPULATION by the STANDARD DIVISOR. In the language of apportionment, each of these results is called the STANDARD QUOTA for its respective planet. (The Standard Quota for each planet is the number of seats each planet would have if seats could be divided into fractional parts.)

12. What were the Standard Quotas for each of the planets?

There are many ways that a rational person might decide to fairly allocate these 50 seats among the 5 planets. One of the simplest methods was advocated by Alexander Hamilton and is often called Hamilton’s Method. Hamilton’s Method is described below.

HAMILTON’S METHOD

• Compute the STANDARD DIVISOR.
• Compute the STANDARD QUOTA for each unit (planet, state, etc.).
• Round each of these STANDARD QUOTAS down to the nearest integer (e.g., 7.85 rounds down to 7) and give the unit (planet, state, etc.) this number of seats. (Standard Quotas that have been rounded down to the nearest integer are called LOWER QUOTAS.)
• If there are seats remaining to be allocated, give them one at a time to the unit (planet, state, etc.) whose Standard Quota has the largest fractional part (e.g., a planet with a Standard Quota of 5.7 would receive an extra seat before a planet with a Standard Quota of 12.3 since .7 is larger than .3). Stop when all the remaining seats have been allocated.
  

13. Allocate the seats in the Intergalactic Congress by using Hamilton’s Method to complete the table below.
    STANDARD DIVISOR=