The Apportionment Problem

Question 1: Determine the number of seats each state using the Hamilton Method

Question 2: Determine the average constituency for each state and explain the decision making process for allocating the remaining seats

The average constituency of a state is the state’s population quotient divided by representatives drawn from each state. The average constituency measures to what extent an appointment is fair and it is calculated by dividing the state’s population with the number of representatives. The process of allocating the seats that are surplus has been done by placing the seats remaining in states that have the highest fractional parts left. States with the largest fractional parts started moving downwards until all the seats had been allocated.

Question 3: Calculation of the Absolute and Relative Unfairness of the Apportionment

In these calculations, the two states that will be used will be state 5 and state 2. The absolute unfairness is 5571 as shown in the calculation below:

On the other hand, the relative unfairness is 15.097 as depicted in the calculation below:

Question 4: Using an Example from the Results above, Explain how Changes in state Populations or Boundaries Affects the Congress Balance Representation

In a situation where a state’s population or boundaries changes, the next meeting to be convened at the house of representatives will set an agenda to review the representation and change if there is need to change the representation parameter. Some of the factors that can cause positive changes in population include baby booming, migration, and immigration. On the contrary, factors that can cause population decrease includes deaths, migration and immigration and natural disasters. Factors affecting population change such as deaths and births do not work fast to push for an overnight change in representation. Changing of state boundaries is very rare mainly because of diplomatic reasons. However, a boundary of a state can change in an event a natural disaster such as a landslide breaks off the original boundary or if the boundary is provided by natural occurrences such as rivers and they happen to change route, it results in a natural boundary change.

A numerical example of how changes in population or boundaries can affect the increase or decrease in representation has been provided below.

After modification

When the population of state 1 increases the representatives increase to 7. However, when the population of state 2 is lowered, the number of representatives drops to 2. This shows how a change in boundary or occurrence of massive migration across the states can change the equation of representatives.

Question 5: How and Why Could an Alabama Paradox Occur?

An Alabama Paradox would occur when there is an increase in the total number of seats at the expense of other states particularly the small states. A paradox occurs because when the number of seats increases, it means seats fill faster in larger states while smaller states are shortened. For instance, large state A and B have their seats fill faster compared to small C. Thus, when fractional parts of B and A increases faster than C, the fraction of C is overtaken which results in losing a seat. This is because the Hamilton method is based on the basis that states with largest fraction gets extra seats (Apportionment Paradoxes, n.d.).

Question 6: How does the Huntington-Hill Apportionment Method Helps to Avoid an Alabama Paradox?

Huntington- Hill is an equal proportion method which allows states to remain with their current number of seats and only add a new seat to states which are in more need of them. This method has a fixed house which helps in avoiding conflict (Pirnot, 2010).

Question 7: Do You Feel Apportionment Is the Best Way to Achieve Fair Representation?

My believe is that apportionment is the best-fit method of achieving a fair representation. My believe is drawn from the fact that population will always be changing which necessities a shift in representation. Through this method, the number of representative from each state is kept fair and only changes when there is an adequate reason which is feasible. Through apportionment, the ratio of constituents to representatives is highly taken into consideration. However, the fairness of the apportionment method can be challenged because of the aspect of Alabama Paradox. Nevertheless, this can be controlled using the Huntington-Hill method.

Question 8: Suggestion of a Strategy to Achieve Fair Representation

A strategy that can work to achieve a fair apportionment would have to be the Webster Method. Compared to the Adams and Jefferson methods, the Webster method is best appropriate to provide a fair representation. The Jefferson method rounds the fractional down to the nearest one which negatively affects states with higher fractions. This is unfair in terms of having a fair constituent to representative ratio. On the other hand, the Adams method rounds fractions up to the nearest one which can favor the small states (Lauwers, & Puyenbroeck, 2006). However, the Webster method combines the functioning of these two models in that it implements conventional rounding (Lauwers, & Puyenbroeck, 2006). This makes the representation fair because rounding the fraction part can either go down or up and hence not pre-determined.

In conclusion, this assignment has put into light the fact that there is no perfect to attain a fair representation. However, there are available mechanisms that can be used to rectify problems such as the Alabama Paradox. Different techniques such as Jefferson method, Adam’s and Webster methods all provide a unique approach to answering the fairness questions. The Webster method being a combination of the Jefferson and Adam’s method can be regarded as the best alternative to achieving a fair representation.

References

Apportionment Paradoxes. (n.d.). Apportionment Paradoxes. Retrieved August 18, 2014, from http://www.ctl.ua.edu/math103/apportionment/paradoxs.htm

Lauwers, L., & Puyenbroeck, T. V. (2006). The Hamilton apportionment method is between the Adams method and the Jefferson method. Mathematics of Operations Research, 31(2), 390-397.

Pirnot, T. L. (2010). Apportionment. Mathematics all around 4th ed. Boston: Addison-Wesley. pp. 532-576.