Problem 12 - Thinking Mathematically

Question

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. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Person } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Contribution } & \$ 2013 & \$ 187 & \$ 290 & \$ 3862 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

After applying Adam's method of apportionment, each person receives the following number of shares: \(A=?\), \(B=?\), \(C=?\), \(D=?\) (Replace ? with the calculated values). The final allocation will add up to a total of 60 shares.

Step 1: Calculate the standard quotas

First, calculate the standard quotas for each person by taking each person's contribution and dividing it by the divisor \(d=108\). The formulas are as follows: \(Q_A = \dfrac{2013}{108}\), \(Q_B = \dfrac{187}{108}\), \(Q_C = \dfrac{290}{108}\), \(Q_D = \dfrac{3862}{108}\).

Step 2: Allocate based on standard quotas

Allocate each person shares based on their respective standard quota by rounding down to the nearest whole number: \(q_A = \lfloor Q_A \rfloor\), \(q_B = \lfloor Q_B \rfloor\), \(q_C = \lfloor Q_C \rfloor\), \(q_D = \lfloor Q_D \rfloor\).

Step 3: Calculate remaining shares

Calculate the remaining shares to be allocated. This is done by subtracting the sum of the allocated shares from the total number of shares (60): \(r = 60 - (q_A + q_B + q_C + q_D)\).

Step 4: Calculate modified quotas

Calculate modified quotas by using the formula \(MQ_i = Q_i - 0.5\), where \(i\) stands for each person (A, B, C, and D).

Step 5: Allocate remaining shares

The remaining shares are then allocated to those persons whose modified quotas are closest to the next upper whole numbers. This is done until all shares are distributed.

Step 6: Confirm total shares

Finally, confirm that the total number of shares assigned equals the original number of shares (60) to ensure an accurate allocation.

Key Concepts

Adams's Method

When it comes to apportionment, or the division of items among several recipients, various methods exist to execute this task fairly. One such technique is Adams's method, named after President John Quincy Adams. This method is recognized for its emphasis on protecting the interests of smaller groups or entities by my authority using a smaller divisor when calculating quotas.

In Adams's method, the standard quota for each party is computed, but with a twist: when disparities arise between the calculated quotas and the number of items to apportion (such as shares of stock), the method uses modified quotas to allocate the remaining items. This prevents rounding issues which could unfairly benefit larger groups at the expense of smaller ones.

In the provided exercise, Adams's method is applied to determine the fair distribution of stock shares among four individuals based on their monetary contributions. The method ensures that each person receives a number of shares proportional to their contribution, while any remaining shares are allocated in a way that avoids giving undue advantage to those with the largest contributions.

Standard Quota Calculation

The standard quota calculation is a fundamental step in many apportionment methods, including Adams's. This calculation helps to determine the initial distribution of items based on each recipient's share or contribution.

To calculate the standard quota, you would divide each individual's share of the contribution by a predetermined divisor. The divisor represents the value that yields a desired ratio of shares to contribution amounts. In this specific exercise, the divisor is given as 108, a figure that can be determined by various methods based on the context.

After calculating the standard quotas using this divisor, you typically round down to the nearest whole number to find out how many initial shares each person should receive. While the standard quota provides a starting point, adjustments are usually required to ensure all shares are fairly distributed. These adjustments lead to the concept of modified quotas, which address any remaining shares.

Divisor in Apportionment

The divisor in apportionment plays a critical role in determining the fair distribution of items, such as shares of stock in our exercise, across various parties based on their respective contributions. In simple terms, the divisor is a value that, when divided into each party's contribution, results in the standard quota—a measure of how many items each party would receive if items were distributed proportionately.

The choice of the divisor is pivotal because it dictates the scale of the quotas and consequently impacts the initial allocation of shares. While in some apportionment methods the divisor is set to achieve specific goals, such as minimizing the advantage of larger states or groups, in Adams's method the focus is on preventing larger contributors from disproportionately benefiting from the rounding process. Therefore, the selected divisor inherently influences the balance between equity and proportionality in the distribution.

Modified Quotas

Modified quotas come into play when dealing with the allocation of residual shares after the initial distribution based on the standard quotas. Following the rounding down of the standard quotas to the nearest whole numbers, there may be some shares that have not been allocated. This is where modified quotas are utilized to determine who should receive them.

To calculate modified quotas, a small amount—typically 0.5—is subtracted from each person's original quota. The resulting figures are less susceptible to the rounding errors that could favor larger contributors in the initial allocation. The remaining shares are then assigned to individuals based on whose modified quotas are closest to the next whole number, thus ensuring a more equitable distribution of the leftovers.

In the presented exercise, this step is key to guarantee that each person ends up with a fair and proportional number of shares, and that all 60 shares are distributed without bias towards those who contributed more.