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The Apportionment Problem

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The Apportionment Problem
The apportionment process has generally been motivated by the need to divide resources of a similar nature fairly amongst people who are entitled to different percentages of the total resource.
Several methods of apportionment exist, including Adam's Method, Hamilton's Method, Jefferson's Method, Webster's Method, just to mention a few. For the problem at hand, Hamilton’s method is employed.
The Hamilton Method
The Hamilton Method is an apportionment style named after Alexander Hamilton. It was first used to decide the initial apportionment of the seats in the House of Representatives in 1790.
We shall define the following parameters for Hamilton’s apportionment: Or (this is also known as the average constituency) or So now and then quota being the number of seats each state should receive. For each state the quota is
State 1: q== 2.907807 State 2: q== 6.697633
State 3: q== 18.5566 State 4: q== 16.60052
State 5: q== 0.069336 State 6: q== 16.09638
State 7: q== 8.160086 State 8: q== 15.84233
State 9: q== 10.28396 State 10: q== 4.785339 | Population | Average constituencyS=p/h | Quota | Lower quota | Final Apportionment | State 1 | 15475 | 154.75 | 2.907807 | 2 | 3 | State 2 | 35644 | 356.44 | 6.697633 | 6 | 7 | State 3 | 98756 | 987.56 | 18.5566 | 18 | 19 | State 4 | 88346 | 883.46 | 16.60052 | 16 | 16 | State 5 | 369 | 3.69 | 0.069336 | 0 | 0 | State 6 | 85663 | 856.63 | 16.09638 | 16 | 16 | State 7 | 43427 | 434.27 | 8.160086 | 8 | 8 | State 8 | 84311 | 843.11 | 15.84233 | 15 | 16 | State 9 | 54730 | 547.3 | 10.28396 | 10 | 10 | State 10 | 25467 | 254.67 | 4.785339 | 4 | 5 | Total | 532188 | | 100 | 95 | 100 |

The problem of how to actually deal with the ‘fraction’ seats arises, because seats are discrete objects. Hamilton’s method solves this fraction problem by assigning a lower quota to each state; a value achieved by rounding down the calculated quota to the nearest lower integer. The fractional part of each calculated quota is studied, and starting with the quota with the largest fractional part, seats are then distributed in order from the larges to the smallest. Here we first consider state one for an additional seat, then State 8, 10,2,4 then 3 in that order till all the seats are taken.
Absolute Unfairness between two states is the difference between the average numbers of people per representative in each state. Absolute unfairness in this problem between the best and the worst represented states:
Abs. Unfairness=Average Constituency of Best state – Average constituency of worst state
987.56-3.69= 983.87
Relative unfairness is the relative unfairness divided by the smaller average constituency between the two average constituencies. So between the best represented and the worst represented states:
Relative Unfairness= Absolute unfairness/ smaller Average Constituency
= 983.87/3.69= 266.63
Changes in state boundaries or populations could affect this apportionment. For instance if the population were to increase or the boundary to enlarge causing a population surge, a state could easily lose seats if it is small and growing much faster relative to a bigger state that is growing much slower. This is called the population paradox. It arises due to the rounding down procedure in the apportionment process. For instance, in the table, if state 1 were growing faster than state 2 it could easily lose a seat in the house.
An Alabama Paradox could occur if the total number of seats to be allotted is increased, causing a state to lose a seat. For instance in the table, State 1 could lose a seat if the total number of seats is increased, making the average constituency figure smaller. Increasing number of seats increases the fair share better for larger states than for smaller ones.
The Huntington- Hill method overcomes this Alabama paradox by calculating the geometric mean (the square-root of the product) of the upper and lower quota for each state. If the standard quota equals or exceeds the geometric mean, the standard quota is rounded up, otherwise it is rounded down.
Apportionment is the best way of achieving fair representation. It ensures that everyone receives proportionate representation based on their population size. Bigger populations get to have bigger say in governance since they are the majority. It has measures to that take into account the sizes of the populations in each state, making it fair for everyone.
Alternative Representation Strategy
Another strategy that can be considered for fair representation is the Huntington-Hill apportionment method. This method is also called the ‘Method of Equal Proportions’. It was developed in 1911 by Edward Huntington and Joseph Hill.
The method requires one to calculate the standard divisor, and then acquire each state’s standard quota. Calculate the geometric mean for each state. It then assigns each state an upper quota if rounds up if the standard quota exceeds or equals the fractional part of the geometric mean, or a rounds down the standard quota otherwise. If the sum of the upper/lower quotas equals the house seats, apportion each state its upper/lower quota as determined. It the sum is not equal, obtain a modified divisor in place of the standard divisor and re-do the procedure. This strategy is better than the Hamilton method. It may only violate the quota rule compared to Hamilton’s Method which has more paradoxes.
Conclusion
The Hamilton method of apportion is quite useful as a way of apportioning, but due to the number of errors that arise with the change in population size and house size, this method has undergone improvements resulting in better methods such as the Huntington-Hill Method, which though may not be absolutely perfect, achieve better consistency in representation under various change in variables.

References
Caulfield ,Michael J. Apportioning Representatives in the United States Congress - Hamilton's Method of Apportionment. Gannon University , Mathematical Association of America, 2014. Web . Nov. 28 2014 http://www.maa.org/publications/periodicals/convergence/apportioning-representatives-in-the-united-states-congress-hamiltons-method-of-apportionment
Hodge, Jonathan and Richard Kilmer. The Mathematics of Voting and Elections: A Hands-on Approach. American Mathematical Society, 2005. Electronic. Nov. 28th 2014 Shapiro, Roman. Methods of Apportionment. University of Maryland, Web. Nov. 28 2014. http://honors.cs.umd.edu/reports/shapiro.pdf

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