Problem 9 - Thinking Mathematically

Question

A corporation has two branches, A and B. Each year the company awards 100 promotions within its branches. The table shows the number of employees in each branch. $$ \begin{array}{|l|c|c|c|} \hline \text { Branch } & \text { A } & \text { B } & \text { Total } \\ \hline \text { Employees } & 1045 & 8955 & 10,000 \\ \hline \end{array} $$ a. Use Hamilton's method to apportion the promotions. b. Suppose that a third branch, C, with the number of employees shown in the table below, is added to the corporation. The company adds five new yearly promotions for branch C. Use Hamilton's method to determine if the new-states paradox occurs when the promotions are reapportioned.

Step-by-Step Solution

Answer

Through Hamilton’s method, we initially see that the branches A and B should get around 10 and 90 promotions respectively. Upon the inclusion of branch C, there is a need to recalculate the apportionment for all branches including C. The reapportionment could lead to the new-states paradox if any of the original branches (A or B) loses its share despite the increase in the total promotions.

Step 1: Use of Hamilton's Method for apportionment between branch A and B

Determine the standard divisor (SD) by dividing the total number of promotions by the total number of employees, i.e., SD = \( \frac{100}{10,000} = 0.01 \). Then, calculate quotas for each branch by multiplying the SD with the number of employees for each branch. That will result in \( \frac{1045 × 0.01}{1} =10.45 \) for branch A and \( \frac{8955 × 0.01}{1} = 89.55 \) for branch B. Round the quotas down to get initial apportionment; that's 10 promotions for branch A and 89 for branch B. This accounts for 99 promotions, and since we have 100 promotions to apportion, find the branch with the largest fractional part—in this case, branch A—and award the remaining 1 promotion.

Step 2: Add Branch C and Calculate New Standard Divisor

Add the number of employees in branch C. Suppose it's N. Recalculate the SD by dividing the total promotions, which are now 100+5=105, by the total number of employees, which is 10,000+N. Let's denote the new SD as SD'.

Step 3: Determine New Quotas and Apportionment

Calculate the new quotas by multiplying the SD' with the number of employees for each branch. Then, round down each quota to get the initial apportionment. If any promotions are left, they're awarded to the branch(s) with the largest fractional part.

Step 4: Check for New-States Paradox

Compare the apportionment from Step 1 and Step 3 for branches A and B. If any of them loses promotions despite an increase in total promotions, the new-states paradox occurs.

Key Concepts

Quota Calculation

Quota calculation is a crucial step in apportioning resources, such as promotions or legislative seats, based on proportional representation. In the context of Hamilton's method, also known as the method of largest remainders, this process begins by determining each group's 'fair share' of the resource being distributed. In our example, the corporation's promotions are initially divided between branches A and B before the addition of branch C.

To calculate the quotas, one must first find the standard divisor (SD), which represents the average number of employees per promotion. This is achieved by dividing the total number of promotions by the total number of employees in the corporation. The resulting SD is then used to calculate each branch's quota by multiplying it with the number of employees in each branch. Quotas usually result in fractional numbers, which must be managed carefully to achieve a fair distribution.

Standard Divisor

The standard divisor serves as the benchmark for determining how many units of the resource each group should receive. This figure is calculated by dividing the total number of available promotions or seats by the total population. For example, in our corporation scenario, we initially calculate the SD using the total number of employees and the total promotions available before the entry of branch C.

The SD plays a pivotal role, as any variation in its value can significantly affect the outcome of the apportionment process. It is vital to determine the standard divisor accurately, as it directly influences the initial calculation of quotas. After the introduction of branch C and additional promotions, recalculating the SD is necessary to preserve the fairness and proportionality of the distribution.

New-States Paradox

The new-states paradox is a counterintuitive situation that can arise when applying Hamilton's method of apportionment. It occurs when the addition of new promotions (or legislative seats) and the consequent recalculation of standard divisor and quotas leads to a situation where an existing branch (or state) receives fewer promotions despite the overall increase in promotions.

The paradox is illustrated by adding branch C to our corporation. Even with five additional promotions being made available and supposedly increasing everyone's share, branch A or B might actually end up with fewer promotions than they had before the addition of branch C. This paradox highlights one of the potential quirks of Hamilton's method, which can lead to seemingly illogical results that need careful review to maintain a sense of fairness and proportionality in the apportionment.

Promotions Allocation

Once the quotas for each branch have been calculated, the promotions allocation process begins. Initially, each branch receives a whole number of promotions equal to the whole number part of their quota. The remaining promotions are then allocated one by one to the branches with the largest fractional remainders until no promotions are left.

This approach ensures that each branch receives a number of promotions proportional to its size, although the final promotion allocation might not be exactly equal to the initially calculated quota due to the rounding of fractional remainders. Strategic rounding of the quotas is critical in the allocation process, determining which branches gain additional promotions and which do not. This is why the management of fractional remainders is an essential component of Hamilton's method and one that can significantly impact the distribution of resources among the various groups.