The Alabama, New States & Population Paradoxes

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  • 0:01 Introduction to Paradoxes
  • 1:13 Hamilton's Method of…
  • 3:56 The Alabama Paradox
  • 5:39 The New State Paradox
  • 7:39 Population Paradox
  • 9:03 Lesson Summary
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

A paradox is a logical procedure that results in illogical outcomes. This lesson will review three paradoxes that are associated with population-based apportionment.

Introduction to Paradoxes

In this lesson, we are going to talk about three famous paradoxes that impact the method of apportionment in the House of Representatives; they are the Alabama, New State, and Population Paradoxes. A paradox is a contradictory situation in which logical procedures seem to result in illogical outcomes - otherwise stated: logical steps leading to an illogical result; that's a paradox! A common paradox can be found in the substance dry-ice. It is so cold that it actually burns the skin. How can something cold burn? It's a paradox - a contradiction that exists in a special situation.

As I mentioned before, we will be discussing three famous paradoxes that relate to apportionment of voting seats in the House of Representatives. Apportionment is the dividing, or portioning out, of voting seats to each state being represented based on population. Over time, paradoxes were found in the number of seats each state received under special circumstances. To evaluate these paradoxes, we'll use the Hamilton method of apportionment. So, what is the Hamilton method of apportionment? Let's review it quickly.

Hamilton's Method of Apportionment

The Hamilton method of apportionment states that each state should be assigned their lower quota of seats, then, any leftover seats should be assigned based on descending order of fractional quotas. To understand all these terms, it is easiest to just work through the calculations, pointing out the terms on the way. In this scenario, let's assume that you are building a local focus group to support your school system. You want 40 teachers representing the high school, middle school, and primary school to be on your board.

First, you find the standard divisor for the group, which is the total population divided by the number of items to be apportioned. In our example, we divide the total population of 20,000 by the number of seats available, 40, and get 500.

Next, we need to find each school's original standard quota. This is a fractional number derived by dividing the individual population by the standard divisor, representing the raw number of seats the school should get. Here, the high school standard quota is 10,170 divided by 500, or 20.34. Using the same calculation, we get the middle school standard quota of 18.3, and the primary school gets 1.36 seats on the board.

Now, it is obvious that we can't send part of a person to the board, thus we must round the numbers. Lower quotas (whole number portion of the standard quota) are assigned at the outset. Thus, the high school would have 20, the middle school 18, and the primary school, 1 representative on the board.

But that only adds up to 39 representatives on the board. Now we get to use those fractional parts. We assign another seat to the school with the largest fractional portion, the primary school. If there were more extra seats, we'd keep assigning one seat in descending order of fractional pieces until we reach the correct number of seats.

So, the final apportionment is 20 seats for the high school, 18 for the middle school, and 2 for the primary school, and we have completed Hamilton's method of apportionment. If you would like more practice with Hamilton's method, please see the lesson on his method. Okay, now that we have a scenario to work with, we can fully examine the three paradoxes that can affect them.

The Alabama Paradox

The Alabama Paradox states that increasing the total number of available voting seats causes a state to lose seats overall. It is called the Alabama Paradox because it was found in 1880 by C.W. Seaton and related to the fact that Alabama would actually lose a seat if the number of total seats available were 300.

To duplicate this paradox in our example, we'll increase our members of the board to 41. Will it have any impact? Let's check the numbers. First, wipe away the old numbers and create a new standard divisor. We get a new standard divisor of 487.8 when we divide 20,000 by 41. Now we proceed to the standard quotas for each school. Dividing each population by the new standard divisor, we get the high school at 20.848, the middle school is 18.757, and the primary school is 1.39.

We have the same lower quota numbers as before, but, interestingly, this time the fractional portion of the high school is higher than that of the primary school. Thus, the high school will receive the first additional seat. Since we now have a board of 41, the second additional seat goes to the middle school (working in decreasing fractional order); that means the primary school will go from having 2 representatives to just 1 representative on the board - all because the board was made bigger! More seats were added, but the primary school lost a seat. That's a paradox!

The New State Paradox

The New State Paradox states that adding a new entity to the population as well as a fair number of additional seats to accommodate the new entity can still impact the existing entities' numbers. This paradox was found in 1907 when Oklahoma was added to the Union. Even though five seats were added to the House of Representatives to accommodate Oklahoma's quota, the re-apportionment altered other states' total apportioned seats.

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