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Chapter 13: Problem 41

Describe the plurality method. Why is ranking not necessary when using this method?

Short Answer

Expert verified
The plurality method is a voting system where each voter votes for one candidate, and the candidate with the most votes wins. Ranking is not necessary in this system because it's solely based on the number of first-place votes a candidate gets; the order or spread of other choices doesn't matter.

Step by step solution

01

Explanation of Plurality Method

In politics, the plurality method, also known as 'first past the post', is a voting system where each voter is allowed to vote for only one candidate, and the candidate who polls the most votes is elected. In mathematical terms, in case of election of a single candidate, a candidate needs more votes than any other candidate to be elected. In the case of multiple candidacies, the highest polling candidates fill the vacancies.
02

Explanation of Absence of Ranking

Because each voter casts a single vote for their top choice, there is no need for a ranking system. Unlike systems such as the Borda count or instant-runoff voting, where voters rank their preferences, in the plurality method ranking is not required as it鈥檚 based solely on the candidate with the highest number of first-place votes.
03

Providing a Simple Example

For example, in an election with 5 candidates, each voter casts one vote for their chosen candidate. The votes are then tallied, and the candidate with the most votes wins. The sequence or preference of the unfavored candidates doesn't have an effect on the final result, thus ranking isn't necessary.

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Most popular questions from this chapter

Fifty-three people are asked to taste-test and rank three different brands of yogurt, \(A, B\), and \(C\). The preference table shows the rankings of the 53 voters. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & 27 & 24 & 2 \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { C } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { B } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Which brand has a majority of first-place votes? b. Suppose that the Borda count method is used to determine the winner. Which brand wins the taste test? c. Is the majority criterion satisfied? Explain your answer.

A university is composed of five schools. The enrollment in each school is given in the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \begin{array}{c} \text { Human- } \\ \text { ities } \end{array} & \begin{array}{c} \text { Social } \\ \text { Science } \end{array} & \begin{array}{c} \text { Engi- } \\ \text { neering } \end{array} & \text { Business } & \begin{array}{c} \text { Educa- } \\ \text { tion } \end{array} \\ \hline \text { Enrollment } & 1050 & 1410 & 1830 & 2540 & 3580 \\ \hline \end{array} $$ There are 300 new computers to be apportioned among the five schools according to their respective enrollments. Use Hamilton's method to find each school's apportionment of computers.

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. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { District } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { Total } \\ \hline \begin{array}{l} \text { Original } \\ \text { Population } \end{array} & 780 & 1500 & 1730 & 2040 & 2950 & 9000 \\ \hline \text { New Population } & 780 & 1500 & 1810 & 2040 & 2960 & 9090 \\ \hline \end{array} $$

A university is composed of five schools. The enrollment in each school is given in the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \begin{array}{c} \text { Liberal } \\ \text { Arts } \end{array} & \begin{array}{c} \text { Educa- } \\ \text { tion } \end{array} & \text { Business } & \begin{array}{c} \text { Engi- } \\ \text { neering } \end{array} & \text { Sciences } \\ \hline \text { Enrollment } & 1180 & 1290 & 2140 & 2930 & 3320 \\ \hline \end{array} $$ There are 300 new computers to be apportioned among the five schools according to their respective enrollments. Use Hamilton's method to find each school's apportionment of computers.

Your class is given the option of choosing a day for the final exam. The students in the class are asked to rank the three available days, Monday (M), Wednesday (W), and Friday (F). The results of the election are shown in the following preference table. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 4} & \mathbf{8} & \mathbf{3} & \mathbf{1} \\ \hline \text { First Choice } & \text { F } & \text { F } & \text { W } & \text { M } \\ \hline \text { Second Choice } & \text { W } & \text { M } & \text { F } & \text { W } \\ \hline \text { Third Choice } & \text { M } & \text { W } & \text { M } & \text { F } \\ \hline \end{array} $$ a. How many students voted in the election? b. How many students selected the days in this order: \(\mathrm{F}, \mathrm{M}, \mathrm{W} ?\) c. How many students selected Friday as their first choice for the final? d. How many students selected Wednesday as their first choice for the final?
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