Mattie Di Giovanni and Julion Cruz
All 3 Investigations are below
Investigation 3A: Alabama Paradox Worksheet
A small country consists of three states: Bama, Tecos and Ilnos with a total population of 20,000 and 200 seats in the House of Representatives. Apportion the seats using Hamilton’s method.
Use divisor: 20000200=100
| State | Population | Quota | Lower Quota | Surplus | Apportionment |
| Bama | 940 | 9.4 | 9 | 0.4 | 10 |
| Tecos | 9030 | 90.3 | 90 | 0.3 | 90 |
| Ilnos | 10,030 | 100.3 | 100 | 0.3 | 100 |
| Total | 20,000 | 200 | 199 | 1 | 200 |
At a later time, a decision is made to add a representative to the House raising the number of seats to 201. Reapportion using Hamilton’s method.
Now, we must change the divisor to: 20000201=99.5
| State | Population | Quota | Lower Quota | Surplus | Apportionment |
| Bama | 940 | 9.45 | 9 | 0.45 | 9 |
| Tecos | 9030 | 90.75 | 90 | .75 | 91 |
| Ilnos | 10,030 | 100.80 | 100 | .80 | 101 |
| Total | 20,000 | 201 | 199 | 2 | 201 |
What do you notice? How many seats were apportioned to the state of Bama initially? What about after an extra seat was added to the house? How does this demonstrate the Alabama paradox?
The surplus in each state increases, but Bama does not get an extra seat like they did before. The other two states have higher surpluses when a seat is added in the equation. Bama was originally apportioned 10 seats but after the seat addition is only reapportioned 9 seats. This demonstrates the Alabama paradox because adding a representative seems like it would be helpful but the Bama state actually loses a representative. So it ends up being counterintuitive to add a seat, because Bama ends up losing a rep. It’s kind of ironic in this situation because Bama gained the extra surplus person in the first scenario but loses in the second even though an extra seat was added.
Mattie Di Giovanni and Julion Cruz
Investigation 3B: Population Paradox Worksheet
In the year 2525 the five planets in the Utopia galaxy finally signed a treaty and agreed to form an Intergalactic Federation governed by an Intergalactic Congress. The population of each planet is given in the table below (in billions). The total population is 900 billion, the total number of seats is 50. Use Hamilton’s method to apportion the seats.
We must use the divisor: 90050=18
| State | Population | Quota | Lower Quota | Surplus | Apportionment |
| A | 150 | 8.33 | 8 | 0.33 | 8 |
| B | 78 | 4.33 | 4 | 0.33 | 4 |
| C | 173 | 9.61 | 9 | 0.61 | 10 |
| D | 204 | 11.33 | 11 | .33 | 11 |
| E | 295 | 16.38 | 16 | .38 | 17 |
| Total | 900 | 50 | 48 | 1 | 50 |
Ten years later, after a new census was conducted, the total population increased; the new numbers are listed below. Reapportion using Hamilton’s method.
Now, we must change the divisor to: 90950=18.18
| State | Population | Quota | Lower Quota | Surplus | Apportionment |
| A | 150 | 8.25 | 8 | 0.25 | 8 |
| B | 78 | 4.29 | 4 | 0.29 | 5 |
| C | 181 | 9.96 | 9 | 0.96 | 10 |
| D | 204 | 11.22 | 11 | .22 | 11 |
| E | 296 | 16.281 | 16 | .28 | 16 |
| Total | 909 | 50 | 48 | 2 | 50 |
What do you notice? What, if anything, happened to the population of Planet B? What happened to the number of seats apportioned to Planet B? What, if anything, happened to the population of Planet E? What happened to the seats apportioned to Planet E? How does this demonstrate the population paradox?
Planet B has the same population in both situations, but in the new apportionment they get an extra seat. They go from 4 seats to 5 seats without having their population change. However, Planet E’s population does increase but yet they lose a seat. This does not properly reflect the populations at hand. Planet E drops from 17 to 16 reps, but their population increased by 1 billion. This demonstrates the Population Paradox because while it seems that the population increase would help them get more representation they lose representation which is counterintuitive because having a larger state should have more representatives.
Mattie Di Giovanni and Julion cruz
Investigation 3C: Jefferson’s Method and the Quota Rule Violation
Consider a country with a total population of 12,500,000, a house with 250 seats, and six states whose populations are given below.
What is the Standard Divisor (SD): 50,000
Describe Jefferson’s method.
Jefferson method is modifying the standard divisor to fit the total number of seats. Basically dividing so that there are no leftover seats to be apportioned and dropping the surplus.
Use it to find the standard quotas for each state and use Jefferson’s method to apportion the seats (fill in the first two blank columns)
| Population | Standard Quota | Lower Quota (Seats Apportioned) | Modified Quota | Modified Lower Quota (Seats Apportioned) | |
| A | 1,646,000 | 32.92 | 32 | 33.25 | 33 |
| B | 6,936,000 | 138.72 | 138 | 140.12 | 140 |
| C | 154,000 | 3.08 | 3 | 3.11 | 3 |
| D | 2,091,000 | 41.82 | 41 | 42.24 | 42 |
| E | 685,000 | 13.7 | 13 | 13.83 | 13 |
| F | 988,000 | 19.77 | 19 | 19.95 | 19 |
| Total | 12,500,000 | 246 | 250 |
Note that there are only 246 seats apportioned before the divisor is modified.
Now, adjust the divisor to 49,500 and calculate the modified quotas and the modified lower quotas which represent the seat apportionment (fill in columns three and four).
Now, the seats do add up to the required 250, but what do you notice about state B?
State B gains not just 1 but 2 seats when using Jeffersons method versus Hamiltons method.
According to the quota rule, the number of seats for state B should be between 138 and 139. What is the actual apportioned number of seats for state B? How does this violate the quota rule? This exceeds the upper quota allotment because the Standard quota is 138.72 which means that the lower level quota is 138 and the upper level quota is 139. This violates the rule because they are exceeding the upper quota, and they can not do that.
