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Chapter 13: Problem 7

The theater society members are voting for the kind of play they will perform next semester: a comedy (C), a drama (D), or a musical (M). Their votes are summarized in the following preference table. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Number of Votes } & 10 & 6 & 6 & 4 & 2 & 2 \\ \hline \text { First Choice } & \mathrm{M} & \mathrm{C} & \mathrm{D} & \mathrm{C} & \mathrm{D} & \mathrm{M} \\ \hline \text { Second Choice } & \mathrm{C} & \mathrm{M} & \mathrm{C} & \mathrm{D} & \mathrm{M} & \mathrm{D} \\ \hline \text { Third Choice } & \mathrm{D} & \mathrm{D} & \mathrm{M} & \mathrm{M} & \mathrm{C} & \mathrm{C} \\ \hline \end{array} $$ \(\text { Which type of play is selected using the plurality method? }\)

Short Answer

Expert verified
The play type selected using the plurality method is the Musical.

Step by step solution

01

Identify First Choice Votes for Each Play

Look at the preference table and identify the first-choice votes for each type of play.
02

Sum Up the Votes

Count the votes for each play according to the number of votes they were given as a first choice. For the Musicals (M), there were 10 votes from the first group and 2 votes from the last, totaling 12 votes. The Comedies (C) were first choice for the second group with 6 votes and the fourth group with 4 votes, totaling 10 votes. The Dramas (D) were first choice for the third group with 6 votes and the fifth group with 2 votes, totaling 8 votes.
03

Compare the Totals

Compare the totals of each play. The Musicals got 12 votes, the Comedies got 10 votes, and the Dramas got 8 votes.
04

Determine the Winner

The play type with the most first-choice votes according to the plurality method is the one that wins. In this case, it's the Musicals with 12 votes, more than any other play type.

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Key Concepts

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

Preference Table
A preference table is an important tool used in voting systems to summarize the choices of voters. It organizes the information about how many votes each option received as the first choice, second choice, and so on. In our voting scenario, we had a theater society deciding between three types of plays: comedy (C), drama (D), or musical (M). The preference table tells us how each group of society members ranked their preferences. In this specific example, the preference table shows us:
  • Number of votes in each column
  • What their first, second, and third play preferences were
This table is laid out with rows indicating the level of preference (first, second, or third) and columns showing the number of votes associated with each preference order. By analyzing this table, we can understand how popular each type of play is among the voters.
First Choice Votes
The concept of first choice votes is central to understanding the plurality method. It involves looking at only the first-choice preferences of each voter or group of voters and tallying those votes. In our example, the first choice votes are extracted directly from the preference table. The key is to count only the votes listed in the first-choice row:
  • Musical (M): 10 votes from one group, and 2 from another, totaling 12 votes
  • Comedy (C): 6 votes from one group, and 4 from another, totaling 10 votes
  • Drama (D): 6 votes from one group, and 2 from another, totaling 8 votes
These numbers reflect the immediate preference of each voter group without considering any secondary or tertiary preferences. It鈥檚 a straightforward approach but might not always convey the complete picture of overall voter satisfaction.
Counts and Comparisons
Once we have the first-choice votes, the next step is to count and compare these numbers to determine which option is preferred by the majority. This takes us to the heart of the plurality method, where the candidate with the most first-choice votes is declared the winner. In this voting exercise, we compared the totals:
  • Musicals (M) received 12 first-choice votes
  • Comedies (C) garnered 10 first-choice votes
  • Dramas (D) accumulated 8 first-choice votes
Through this simple comparison, it's clear that Musicals stand out with the highest number of votes, making it the selected type of play. The plurality method is easy to implement and understand, but sometimes it may not reflect the majority's overall satisfaction if preferences are closer than they appear on the surface.

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Most popular questions from this chapter

What is the quota rule?

A school district has 57 new laptop computers to be divided among four schools, according to their respective enrollments. The table shows the number of students enrolled in each school $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \text { Enrollment } & 5040 & 4560 & 4040 & 610 & 14,250 \\ \hline \end{array} $$a. Apportion the laptop computers using Hamilton鈥檚 method. b. Use Hamilton鈥檚 method to determine if the Alabama paradox occurs if the number of laptop computers is increased from 57 to 58. Explain your answer

A candidate has a majority of the vote, yet lost the election using the plurality-with-elimination method.

Playwright Tom Stoppard wrote, "It's not the voting that's democracy; it's the counting." Explain what he meant by this.

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? $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{6 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} \\ \hline \text { First Choice } & \text { A } & \text { C } & \text { B } & \text { A } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { A } & \text { C } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { B } & \text { A } & \text { B } & \text { A } \\ \hline \end{array} $$
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