928 CHAPTER 14 Voting and Apportionment Suppose a college receives a grant to purchase 50 laptops to be distributed to four different libraries on campus. Since the libraries do not all serve the same number of students, the director of libraries decides to apportion the laptops based on the number of students served at each library. In this section, we will discuss some apportionment methods the college can use to distribute the laptops to the four libraries. Apportionment Methods SECTION 14.3 LEARNING GOALS Upon completion of this section, you will be able to: 7 Solve apportionment problems using Hamilton’s method. 7 Solve apportionment problems using Jefferson’s method. 7 Solve apportionment problems using Webster’s method. 7 Solve apportionment problems using Adams’ method. Why This Is Important Apportionment methods can be used by businesses, schools, and government agencies whenever a set of items must be allocated among various groups. Being informed about these methods is an important skill that can be applied to many occupations. When the delegates for the original 13 states met in 1787 to draft a constitution, their most important discussion concerned the representation of the states in the legislature. Some states, particularly the small states, preferred that each state have the same number of representatives. The large states wanted a proportional representation based on population. The delegates resolved this issue by creating a Senate, in which each state received two Senators, and a House of Representatives, in which each state 39. Construct a preference table with three candidates and three rankings of the candidates, similar to the preference table in Exercise 9, such that the Borda count method violates the majority criterion. Have the total number of votes be 19. Answers will vary. 40. Construct a preference table with three candidates and four rankings of the candidates, similar to the preference table in Exercise 11, such that the plurality with elimination method violates the monotonicity criterion. Have the total number of votes be 21. Answers will vary. 41. Construct a preference table with four candidates and three rankings of the candidates, similar to the preference table in Exercise 29, such that the pairwise comparison method violates the irrelevant alternatives criterion. Have the total number of votes be 6. Answers will vary. Not voting according to a voter’s true preference, called insincere voting, is frequently practiced to affect the outcome of an election. Insincere voting is used in Exercise 42. 42. Voting Strategy Consider the following preference table. Assume that a majority is needed to win the election. Since no candidate has a majority but C has the most first-place votes, a runoff election is held between A and B. The winner of the runoff election will run against C. a) Using the plurality method, which candidate will win the runoff election, A or B? A b) Will the candidate determined in part (a) win the election against candidate C? Yes, A c) Suppose that the voters who support candidate C, the candidate with the most first-place votes, learn prior to the vote that C will not win if B is eliminated and C runs against A. How can the voters who support C vote insincerely to enable C to win? The five voters who favor C should vote C, B, A instead of C, A, B. Research Activities 43. Choosing a Leader Choose a country other than the United States and write a research paper on how the president or prime minister is chosen. Describe in detail the voting method used. Include the strengths and weaknesses of the method used. 44. History of Voting Methods Write a research paper on the history of voting methods. Include how the flaws in the voting methods discussed in this section were uncovered. Number of Votes 5 4 2 First C A B Second A C A Third B B C Syda Productions/Shutterstock
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