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Chapter 13: Problem 10

A corporation has three branches, A, B, and C. Each year the company awards 60 promotions within its branches. The table shows the number of employees in each branch. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Branch } & \text { A } & \text { B } & \text { C } & \text { Total } \\ \hline \text { Employees } & 209 & 769 & 2022 & 3000 \\ \hline \end{array} $$ a. Use Hamilton's method to apportion the promotions. b. Suppose that a fourth branch, D, with the number of employees shown in the table below, is added to the corporation. The company adds five new yearly promotions for branch D. Use Hamilton's method to determine if the new-states paradox occurs when the promotions are reapportioned. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Branch } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \text { Employees } & 209 & 769 & 2022 & 260 & 3260 \\ \hline \end{array} $$

Short Answer

Expert verified
Promotions are distributed as follows: Branch A: 4, Branch B: 15, Branch C: 41 in the first scenario. After the addition of the Branch D: Branch A retains 4, Branch B also retains 15, Branch C goes down to 40, and Branch D receives 5. New-states paradox occurs in this case since Branch C loses a promotion despite the overall number of promotions increasing.

Step by step solution

01

Calculate Standard Divisors & Quotas

The standard divisor is calculated by dividing the total number of items to be apportioned (promotions) by the total number of units (employees). For the first scenario with branches A, B, and C, the standard divisor will be 60 promotions / 3000 employees = 0.02. Using this divisor, the quotas for A, B, and C can be calculated: quota_A = 209*0.02 = 4.18, quota_B = 769*0.02 = 15.38 and quota_C = 2022*0.02 = 40.44.
02

Apportion Promotions (Hamilton's Method)

Using Hamilton's Method, each branch gets the whole number part of its quota rounded down. Therefore: promotions_A = 4, promotions_B = 15, promotions_C = 40. At this point, there is 1 promotion remaining; this extra promotion will go to the branch with the highest fractional part of the quota, which is branch C. So promotions_C = 41, and the apportionment of promotions is done.
03

Addition of Branch D

Now, with the addition of branch D, repeat the process. The new standard divisor becomes 65 promotions (60 promotions + 5 new promotions for branch D) / 3260 employees = 0.01994. Using this divisor, the quotas for A, B, C and D can be calculated: quota_A = 209*0.01994 鈮� 4.17, quota_B = 769*0.01994 鈮� 15.34, quota_C = 2022*0.01994 鈮� 40.31 and quota_D = 260*0.01994 鈮� 5.18.
04

Apportion Promotions for Four Branches (Hamilton's Method)

Again using Hamilton's Method, each branch gets the whole number part of its quota: promotions_A = 4, promotions_B = 15, promotions_C = 40, promotions_D = 5. There are no extra promotions left this time. Comparing to step 2, Branch D is the only one that gained promotions (5), the number of promotions for branch A and B stayed the same, and Branch C lost one promotion. This is the New States Paradox: when a new state (or branch) is added, an existing state may lose an apportionment, even if the number of total apportionments increase.

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Key Concepts

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

Apportionment
In the context of distributing resources or representation among different entities, apportionment refers to the process of fair and equitable division based on certain criteria. It's a fundamental concept in diverse fields, including politics, where it pertains to the allocation of legislative seats, and in business, concerning resource management among various departments or branches.

Using Hamilton's Method, a historical apportionment strategy named after Alexander Hamilton, it involves straightforward mathematical calculations to ensure a proportional division of shared resources. In the given exercise, the apportionment problem required dividing promotions among different branches of a corporation proportionally to the number of employees in each branch. The objective is to assign the promotions so that each branch receives promotions in relation to its size, providing a fair and equitable distribution of opportunities.
Quota Calculation
Quota calculation is the mathematical heart of apportionment. It involves determining the 'fair share' for each entity or group, based on the proportion of the total they represent. Using a standard divisor鈥攚hich is the total number of items (such as promotions) to be divided by the total number of units (such as employees)鈥攅ach group's quota is calculated.

For example, if a company has 60 promotions and 3000 employees, the standard divisor is 0.02. The quota for each branch鈥攁 ratio of promotions per employee鈥攊s then calculated by multiplying the number of employees in that branch by the standard divisor. Quotas directly instruct on the initial count of promotions for each branch, while considering the need to round to whole numbers for practical allotment.
Mathematical Reasoning
Mathematical reasoning expands beyond just calculating quotas; it entails the logical process that guides the decision-making in resources allocation. After calculating quotas, Hamilton's Method uses mathematical reasoning to apportion the whole number part of quotas. If there are leftovers鈥攁 surplus of promotions not yet allocated鈥攎athematical reasoning dictates they should be awarded starting with the entity having the largest fraction behind.

Through this practice, it ensures that small fractions don't accumulate to disadvantage smaller entities disproportionately. This meticulous reasoning ensures fairness, embodying the axiom that details matter in mathematical processes, thereby making it a compelling and transparent decision-making tool.
New States Paradox
The addition of a new entity, or state, can present an intriguing paradox in the realm of apportionment. The new states paradox occurs when, after the addition of an entity and an increase in the overall number of items to apportion, an existing entity receives fewer items than it did before.

In our corporate example, after adding branch D and increasing the promotion pool, branch C ends up with fewer promotions than it initially had. This counterintuitive outcome is a noteworthy characteristic of Hamilton's Method and can raise questions about the method's fairness and practicality in certain situations. This paradox serves as an excellent reminder of the complexity inherent in apportionment problems and the potential for unexpected results within seemingly straightforward mathematical procedures.

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Most popular questions from this chapter

Four people pool their money to buy 60 shares of stock. The amount that each person contributes is shown in the following table. Use Adams's method with \(d=108\) to apportion the shares of stock. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Person } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Contribution } & \$ 2013 & \$ 187 & \$ 290 & \$ 3862 \\ \hline \end{array} $$

Members of the Student Activity Committee at a college are considering three actors to speak at a campus festival on women in the arts: Whoopi Goldberg \((\mathrm{G})\), Julia Roberts \((\mathrm{R})\), and Meryl Streep (S). Committee members vote for their preferred speaker. The winner is to be selected by the pairwise comparison method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2} & \mathbf{8} & \mathbf{6} \\ \hline \text { First Choice } & \text { S } & \text { R } & \text { G } \\ \hline \text { Second Choice } & \text { G } & \text { G } & \text { R } \\ \hline \text { Third Choice } & \text { R } & \text { S } & \text { S } \\ \hline \end{array} $$ a. Using the pairwise comparison method, who is selected as the speaker? b. Prior to the announcement of the speaker, Meryl Streep informs the committee that she will not be able to participate due to other commitments. Construct a new preference table for the election with \(S\) eliminated. Using the new table and the pairwise comparison method, who is selected as the speaker? c. Is the irrelevant alternatives criterion satisfied? Explain your answer.

A small country is comprised of four states, \(A, B, C\), and \(D\). The population of each state, in thousands, is given in the following table. Use this information to solve. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 138 & 266 & 534 & 662 & 1600 \\ \hline \end{array} $$ According to the country's constitution, the congress will have 200 seats, divided among the four states according to their respective populations. a. Find the standard divisor, in thousands. How many people are there for each seat in congress? b. Find each state's standard quota. c. Find each state's lower quota and upper quota.

The programmers at the Theater Channel need to select a live musical to introduce their new network. The five choices are Cabaret \((C)\), The Producers \((P)\), Rent \((R)\), Sweeney Todd (S), or West Side Story \((W)\). The 22 programmers rank their choices, summarized in the following preference table. Use the table to solve.$ $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{5} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { C } & \text { S } & \text { C } & \text { W } & \text { W } & \text { P } \\ \hline \text { Second Choice } & \text { R } & \text { R } & \text { P } & \text { P } & \text { R } & \text { S } \\ \hline \text { Third Choice } & \text { P } & \text { W } & \text { R } & \text { R } & \text { S } & \text { C } \\ \hline \text { Fourth Choice } & \text { W } & \text { P } & \text { S } & \text { S } & \text { C } & \text { R } \\ \hline \text { Fifth Choice } & \text { S } & \text { C } & \text { W } & \text { C } & \text { P } & \text { W } \\ \hline \end{array} $$ Determine which musical is selected using the pairwise comparison method.

Suppose that the pairwise comparison method is used to determine the winner in an election. If there are nine candidates, how many comparisons must be made?
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