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Chapter 13: Problem 12

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

Short Answer

Expert verified
The final distribution of shares using Adams's method will be: A gets 20 shares, B gets 1 share, C gets 2 shares, and D gets 37 shares.

Step by step solution

01

Calculate total contribution

First, calculate the total contribution by adding all the contributions together. In this case, it would be $2013 + $187 + $290 + $3862 = $6352.
02

Initial Apportionment

Carry out the initial apportionment by dividing each individual's contribution by \(d\), then rounding down to the nearest whole number. For person A, B, C, and D this would be \(\lfloor \frac{2013}{108} \rfloor = 18\), \(\lfloor \frac{187}{108} \rfloor = 1\), \(\lfloor \frac{290}{108} \rfloor = 2\), and \(\lfloor \frac{3862}{108} \rfloor = 35\) respectively.
03

Count Initial shares

Next, count their initial shares, which is the sum of the initial apportionment. In this case it would be \(18+1+2+35 = 56 \) shares are distributed initially.
04

Distribute remaining shares

There are \(60-56 = 4\) shares remaining. Distribute the remaining shares individually from the person with the highest remainder to the lowest. In this scenario, Person A gets 2 and person D gets 2 (as they have the highest remainders in the initial apportionment).
05

Final share count

The final shares are thus: A=20, B=1, C=2, D=37. These must be the distributed final shares based on Adams's method.

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Most popular questions from this chapter

In Exercises 27-30, 72 voters are asked to rank four brands of soup: \(A, B, C\), and \(D\). The votes are summarized in the following preference table. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & 34 & 30 & 6 & 2 \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { D } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { D } & \text { B } & \text { C } \\ \hline \text { Fourth Choice } & \text { D } & \text { A } & \text { A } & \text { A } \\ \hline \end{array} $$ Determine the winner using the Borda count method.

Three candidates, A, B, and \(\mathrm{C}\), are running for mayor. Election rules stipulate that the pairwise comparison method will determine the winner. In the event that the pairwise comparison method leads to a tie, the Borda count method will decide the winner. The election results are summarized in the following preference table. Under these rules, which candidate becomes the new mayor? $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{6 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} \\ \hline \text { First Choice } & \text { A } & \text { C } & \text { B } & \text { A } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { A } & \text { C } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { B } & \text { A } & \text { B } & \text { A } \\ \hline \end{array} $$

The preference table shows the results of an election among three candidates, A, B, and C. $$ \begin{array}{|l|l|l|l|} \hline \text { Number of Votes } & \mathbf{7} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Using the plurality method, who is the winner? b. Is the majority criterion satisfied? Explain your answer. c. Is the head-to-head criterion satisfied? Explain your answer. d. The two voters on the right both move candidate A from last place on their preference lists to first place on their preference lists. Construct a new preference table for the election. Using the table and the plurality method, who is the winner? e. Suppose that candidate C drops out, but the winner is still chosen by the plurality method. Is the irrelevant alternatives criterion satisfied? Explain your answer. f. Do your results from parts (b) through (e) contradict Arrow鈥檚 Impossibility Theorem? Explain your answer.

Describe the Borda count method. Is it possible to use this method without ranking the candidates? Explain.

Students at your college are given the option of choosing a topic for which a speaker will be selected. Students are asked to rank three topics: Technology (T), Environmental Issues (E), and Terrorism in the Name of Religion (R). The results of the election are shown in the following preference table. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{7 0} & \mathbf{3 0} & \mathbf{1 0} & \mathbf{5} \\ \hline \text { First Choice } & \mathrm{R} & \mathrm{T} & \mathrm{T} & \mathrm{E} \\ \hline \text { Second Choice } & \mathrm{E} & \mathrm{R} & \mathrm{E} & \mathrm{T} \\ \hline \text { Third Choice } & \mathrm{T} & \mathrm{E} & \mathrm{R} & \mathrm{R} \\ \hline \end{array} $$ a. How many students voted? b. How many students selected the topics in this order: \(\mathrm{T}, \mathrm{E}, \mathrm{R}\) ? c. How many students selected technology as their first choice for a speaker's topic? d. How many students selected environmental issues as their second choice for a speaker's topic?
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