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Chapter 13: Problem 8

A town has five districts in which mail is distributed and 50 mail trucks. The trucks are to be apportioned according to each district鈥檚 population. The table shows these populations before and after the town鈥檚 population increase. Use Hamilton鈥檚 method to show that the population paradox occurs. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { District } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { Total } \\ \hline \begin{array}{l} \text { Original } \\ \text { Population } \end{array} & 780 & 1500 & 1730 & 2040 & 2950 & 9000 \\ \hline \text { New Population } & 780 & 1500 & 1810 & 2040 & 2960 & 9090 \\ \hline \end{array} $$

Short Answer

Expert verified
By applying Hamilton's method, it can be found that, despite an increase in the total population, and an increase in District C's population, District C receives fewer trucks in the new allocation. This is a demonstration of the population paradox.

Step by step solution

01

Original Apportionment

First we need to compute the standard divisor (SD) by dividing the total population by the total number of trucks, that is \( SD = \frac{9000}{50} = 180 \). Then let's find the initial standard quotas for each district by dividing their population by the SD. For example, for district A this is \( \frac{780}{180} = 4.33 \). Since we cannot allocate 'partial trucks,' we apply Hamilton's method by rounding down this number, ending with 4 trucks for district A. Repeat this for all districts and tally the total.
02

New Apportionment

Following population increase, recompute the standard divisor \( SD' = \frac{9090}{50} = 181.8 \). Once again find the standard quotas for each district and round down as per Hamilton's method. For example, district C's quota is now \( \frac{1810}{181.8} = 9.96 \), or 9 trucks when rounded down.
03

Identify the Paradox

Now, compare both allocations. Although the population has increased, District C will get fewer trucks in the new allocation compared to the original one, even though its population increased. This is the population paradox.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hamilton's Method
Hamilton's Method, also known as the method of largest remainders, is a step-by-step process designed to distribute resources proportionally across different areas based on certain quotas like population. It's commonly used for apportioning seats in legislative bodies or, as in our example, dividing resources such as mail trucks among districts.

The process begins by calculating a standard divisor (SD), which is the total amount of the resource (in our example, the total population) divided by the number of units to distribute (mail trucks). Each district is allocated a number of trucks equal to their quota鈥攆ound by dividing the district's population by the SD鈥攔ounded down to the nearest whole number. Any remaining trucks are then given out starting with the district with the highest decimal remainder until all trucks are allocated.
Apportionment
Apportionment is the process of fairly dividing and distributing resources or representation among various groups or regions, like states, districts, or organizations. Major applications include determining the number of representatives for electoral districts or allocating financial resources.

The principle behind apportionment is to ensure a distribution that is proportional to the size or need of each group. The challenge arises in finding a method of apportionment that is fair and minimizes discrepancies between the proportions of the resource received and the proportions of the determining factor, such as the population.
Standard Divisor
The standard divisor (SD) is a crucial element in apportionment processes. It signifies the average number of people or items per apportioned unit. Calculated by dividing the total population by the total number of units (mail trucks in our exercise), the SD serves as a yardstick to determine how many units each group should receive.

In our example, determining SD is the starting point. By assessing the changes in SD between the original and new population, we can see how subtle shifts in population size impact the apportionment of mail trucks. This calculation plays a pivotal role in identifying how resources are initially allocated and how they're redistributed after demographic changes.
Population Distribution
Population distribution refers to the spread of people across various areas, such as districts in a city. Understanding this distribution is key for effectively apportioning resources. When the population grows or migrates, it can lead to a change in how resources are allocated.

In the context of our exercise, analyzing the before and after populations in each district illustrates the potential for a population paradox when using Hamilton's Method. It becomes evident that despite an overall population increase in District C, the reallocation of mail trucks actually resulted in fewer trucks for them, which is counterintuitive to the increase. This highlights the delicate balance in resource distribution based on shifting population demographics.

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Most popular questions from this chapter

In your own words, state Arrow's Impossibility Theorem.

Research and present a group report on how voting is conducted for the Academy Awards. Describe the single transferable voting method, a variation of the pluralitywith-elimination method, in the nomination stage for best picture. (Members of the Irish Senate are also elected by this method.) Be sure to describe some of the more bizarre occurrences at the Oscar ceremonies.

Describe the difference between the modified divisor, \(d\), in terms of the standard divisor using Jefferson's method and Adams's method.

Citizen-initiated ballot measures often present voters with controversial issues over which they do not think alike. Here's one your author would like to initiate: Please rank each of the following options regarding permitting dogs on national park trails. a. Unleashed dogs accompanied by their caregivers should be permitted on designated national park trails. b. Leashed dogs accompanied by their caregivers should be permitted on designated national park trails. c. No dogs should be permitted on any national park trails. Your author was not happy with the fact that he could not take his dog running with him on the park trails at Point Reyes National Seashore. Of course, that is his issue. For this project, group members should write a ballot measure, perhaps controversial, like the sample above, but dealing with an issue of relevance to your campus and community. Rather than holding an election, use a random sample of students on your campus, administer the ballot, and have them rank their choices. a. Use each of the four voting methods to determine the winning option for your ballot measure. b. Check to see if any of the four fairness criteria are violated.

A country has 200 seats in the congress, divided among the five states according to their respective populations. The table shows each state鈥檚 population, in thousands, before and after the country鈥檚 population increase. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { Total } \\ \hline \begin{array}{l} \text { Original } \\ \text { Population } \\ \text { (in thousands) } \end{array} & 2224 & 2236 & 2640 & 3030 & 9870 & 20,000 \\ \hline \begin{array}{l} \text { New Population } \\ \text { (in thousands) } \end{array} & 2424 & 2436 & 2740 & 3130 & 10,070 & 20,800 \\ \hline \end{array} $$ n thousands) 2424 2436 2740 3130 10,070 20,800 a. Use Hamilton鈥檚 method to apportion the 200 congressional seats using the original population. b. Find the percent increase, to the nearest tenth of a percent, in the population of each state. c. Use Hamilton鈥檚 method to apportion the 200 congressional seats using the new population. What do you observe about the percent increases for states A and B and their respective changes in apportioned seats? Is this the population paradox?
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