Mathematics Final - The Apportionment Problem (The calculations and wording is done. You need to format it using the attached template.)

The Apportionment Problem

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An ideal democratic government requires that each and every member of the population should participate equally in the decision making process of the country. However, in a country having a population of more than 300 million, incorporating each citizen in the country’s decision making process is not practically possible as it would not only be excessively time-consuming to gather millions of votes on each and every issue, it would also lead to impractically lengthy discussions on every topic brought before the legislative body.

Therefore representatives are chosen from the population and these representatives take part in the decision making process. The process of choosing people’s representatives is a very important job and this should be done in a rational and fair manner so that people of all regions within the country are appropriately represented in the decision making body. The most widely used means of choosing people’ representatives is by establishing a distribution based on population, where individual states are assigned a certain number of representatives based on the population of that particular state.

One such method is the “Hamilton Method” of apportionment. In this method, a fixed number of seats are distributed among the states based on the population of each state. First the total population of all of the states is divided by the total number of seats to be assigned. This quotient is known as the “standard divisor”, and represents the average number of people to be represented by each seat. Then the population of each state is divided by the standard divisor and the whole number portion of that quotient is the minimum number of seats assigned to the corresponding state. If the total number of seats assigned is less than the desired number of seats, then the remaining seats are distributed one at a time to the states based on the fractional portion from dividing the state’s population by the standard divisor by beginning with the state that had the largest fractional portion and continuing in descending order as long as there are still seats to distribute.

For example, given the following populations of ten states:

State Population

1 15475

2 35644

3 98756

4 88346

5 369

6 85663

7 43427

8 84311

9 54730

10 25467

The Hamilton Method would produce the following distribution of seats:

From the above table we see that, under the given method, State 5 has no representation at all as its population is very small as compared to that of the other states. Thus this apportionment scheme will appear to be extremely unfair for the people residing in this state. On the other hand State 3 has the highest average constituency. The actual unfairness for this apportionment scheme can be calculated as follows:

The relative unfairness for this apportionment scheme can be calculated as follows:

The absolute unfairness and the relative unfairness values reveal a serious flaw in this distribution and tells us that a different method is needed for this particular situation.

This fallacy might be resolved by redrawing the state boundaries in order to distribute the population more uniformly. This will alter the percentages in column 3 of the table shown above, which would further alter the distribution of seats. This is because any change in the population of a state changes the percentage contribution of that state to the total population. If the population of one or more states changes or if the state boundaries are redrawn, these changes get reflected in the redistribution of seats in the representative body.

For example, if the boundary between State 5 and State 6 is adjusted (assuming, for the purposes of this example, that they are adjacent) such that the new populations are 20369 and 65663, then the distribution would be as follows:

This redistribution of 20,000 people from State 6 to State 5 has resulted in State 5 being assigned 4 seats, rather than the previous 0 seats. In addition, the average constituency for each state shows more evenness now than it was with the previous distribution. Unfortunately, while local redistribution is often performed for various political or socioeconomic causes, redrawing the boundaries of states has momentous consequences and is not a viable option once the state boundaries have been defined.

Another issue that can arise in this kind of population based apportionment is what is known as the “Alabama Paradox”. This occurs when the body of legislative seats is expanded and a state gains or loses a seat, even though its population has not changed. This occurs because change in the number of seats changes the standard division, and consequently changes the Hamilton Number for each state. If a relatively large number of seats are added to the legislative body, substantial changes in the distribution of seats for each state would be expected. However, when a relatively small numbers of seats are added, the number of seats attributed to the states can be adjusted by changing the fractional portions of the Hamilton Numbers so that any un-apportioned seats are distributed in such a manner that some state might gain a seat, while another state might lose a seat. This occurred following both the 1870 and 1880 U. S. Census. The paradox takes its name from the 1880 occurrence, whereby it was noted that, based on the population values from the census, if the House of Representatives contained 299 seats, then Alabama would be entitled to 8 seats. However, if the House contained 300 seats, then Alabama would only be entitled to 7 seats.

One approach that can be used to help prevent the occurrence of the Alabama Paradox is the “Hamilton-Hill” apportionment method. This method compares the Hamilton Number with the geometric mean of the Lower Quota and Upper Quota for a given state. If a state’s Hamilton Number is larger than the geometric mean, then that state is initially assigned its Upper Quota of seats, rather than the Lower Quota. For example, with State 2 in the original table, the Hamilton Number was 6.698. This value is between 6 and 7. The geometric mean of 6 and 7 is √42, or 6.481. Since 6.698 is greater than 6.481, State 2 would initially be assigned 7 seats instead of the 6 that it was originally assigned.

The Hamilton-Hill method minimizes the effect of small changes in the fractional part of the Hamilton Number by rounding up values instead of taking only the integer portion of the Hamilton Numbers. This method might however lead to an excessive number of seats being assigned, which have to be adjusted consequently.

Applying the Hamilton-Hill method to the initial apportionment gives the following results:

Here the total number of seats allotted exceeds the total number of available seats. As a result, it would be necessary to find out a “modified divisor” in order to adjust the distribution such that exactly 100 seats are distributed to the states. Once again, the small population of State 5 results in that state being scarcely represented.

Neither the Hamilton method nor the Hamilton-Hill methods are perfect solutions to the apportionment problem. As we can see from our example, an uneven population distribution can result in some states being under-represented, while other states are over-represented. In the extreme case, some states might not be represented at all. Thus these methods do not always lead to an accurate representation of the population.

One method of addressing under-representation is to assign a minimum number of seats to each state, and then use the apportionment techniques to distribute the remaining seats. This is similar to what is done with the U.S. House of Representatives where each state has at least three representatives.

It may further be noted that a population-based distribution of representation is essentially unfair, even when the distribution is close to uniform and the average constituencies are approximately equal. This might lead to a situation where the more populous states have more power in the decision making process of the country. It is for this reason that the United States has two houses of Congress, with representation in the House of Representatives being based on population, while the Senate assigns two seats to every state regardless of population. This two tiered approach is the most appropriate way of addressing the limitations of strictly population-based apportionment methods.

References:

· Methods of Apportionment. Methods of Apportionment - History - U.S. Census Bureau. Retrieved March 18, 2016, from http://www.census.gov/history/www/reference/apportionment/methods_of_apportionment.html

· Mark Beumer, Apportionment in Theory and Practice

Retrieved March 18, 2016, from

http://www.illc.uva.nl/Research/Publications/Reports/MoL-2010-07.text.pdf

· Karsten Schuster, Friedrich Pukelsheim, Mathias Drton & Norman R. Draper, Seat biases of apportionment methods for proportional representation, Electoral Studies 22 (2003) 651–676

https://www.math.uni-augsburg.de/emeriti/pukelsheim/2003b.pdf