Excursions in Modern Mathematics (9th Edition)
9th Edition
ISBN: 9780134468372
Author: Peter Tannenbaum
Publisher: PEARSON
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Textbook Question
Chapter 1, Problem 73E
Consider the following fairness criterion: If a majority of the voters have candidate X ranked last, then candidate X should not be a winner of the election.
a. Give an example to illustrate why the plurality method violates this criterion.
b. Give an example to illustrate why the plurality-with-elimination method violates this criterion. c. Explain why the method of pairwise comparisons satisfies this criterion.
d. Explain why the Borda count method satisfies this criterion.
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A voter who has a weight that is greater than or equal to the quota is called a dictator. In a weighted voting system, the dictator has all the power. A voter who is never a critical voter has no power and is referred to as a dummy. This term is not meant to be a comment on the voter’s intellectual powers; it just indicates that the voter has no ability to influence an election.
a. dictators : v1,v2 ; dummies : v2,v3
b. dictator : v1; dummies : v2,v3,v4,v5
c. dictator : v1; dummy : v2
d. The system is not valid.
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My friends and I are trying to decide where to go on vacation this summer. Paris (P), Italy (I), or Spain (S). Below is a table of our votes:
Rachel
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Toni
Pat
Sam
1st Choice
I
P
I
S
S
2nd Choice
S
I
P
P
I
3rd Choice
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I
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Show how {P, I or S} wins using the Borda count method
Show how {P, I or S} wins using the plurality-with-elimination method
Show how {P, I or S} wins using the pairwise comparison method
Show how {P, I or S} wins using the majority vote method
Chapter 1 Solutions
Excursions in Modern Mathematics (9th Edition)
Ch. 1 - Figure 1-8 shows the preference ballots for an...Ch. 1 - Figure 1-9 shows the preference ballots for an...Ch. 1 - An election is held to choose the Chair of the...Ch. 1 - The student body at Eureka High School is having...Ch. 1 - An election is held using the printed-names format...Ch. 1 - Prob. 6ECh. 1 - Prob. 7ECh. 1 - Table 1-30 shows a conventional preference...Ch. 1 - The Demublican Party is holding its annual...Ch. 1 - The Epicurean Society is holding its annual...
Ch. 1 - Table 1-31 shows the preference schedule for an...Ch. 1 - Table 1-32 shows the preference schedule for an...Ch. 1 - Table 1-33 shows the preference schedule for an...Ch. 1 - Table 1-34 shows the preference schedule for an...Ch. 1 - Table 1-35 shows the preference schedule for an...Ch. 1 - Table1-36 shows the preference schedule for an...Ch. 1 - Table 1-25 see Exercise 3 shows the preference...Ch. 1 - Table 1-26 see Exercise 4 shows the preference...Ch. 1 - Table 1-25 see Exercise 3 shows the preference...Ch. 1 - Table 1-26 see Exercise 4 shows the preference...Ch. 1 - Table 1-31see Exercise 11 shows the preference...Ch. 1 - Table 1-32 see Exercise 12 shows the preference...Ch. 1 - Table 1-33 see Exercise 13 shows the preference...Ch. 1 - Table 1-34 Number of voters 6 6 5 4 3 3 1st A B B...Ch. 1 - Table 1-35 Percent of voters 24 23 19 14 11 9 1st...Ch. 1 - Table 1-36 Percent of voters 25 21 15 12 10 9 8...Ch. 1 - The Heisman Award. Table 1-37 shows the results...Ch. 1 - The 2014 AL Cy Young Award. Table 1-38 shows the...Ch. 1 - An election was held using the conventional Borda...Ch. 1 - Imagine that in the voting for the American League...Ch. 1 - Table 1-31 see Exercise 11 shows the preference...Ch. 1 - Table 1-32 see Exercise 12 shows the preference...Ch. 1 - Table1-33 Number of voters 6 5 4 2 2 2 2 1st C A B...Ch. 1 - Table 1-34 See Exercise 14 shows the preference...Ch. 1 - Table1-39_ shows the preference schedule for an...Ch. 1 - Table1-40_ shows the preference schedule for an...Ch. 1 - Table 1-35 see Exercise 15 shows the preference...Ch. 1 - Table 1-36 see Exercise 16 shows the preference...Ch. 1 - Top-Two Instant-Runoff Voting. Exercises 39 and 40...Ch. 1 - Top-Two Instant-Runoff Voting. Exercises 39 and 40...Ch. 1 - Table 1-31 see Exercise 11 shows the preference...Ch. 1 - Table 1-32 See Exercise 12 shows the preference...Ch. 1 - Table 1-33 see Exercise 13 shows the preference...Ch. 1 - Table 1-34 see Exercise 14 shows the preference...Ch. 1 - Table 1-35 see Exercise 15 shows the preference...Ch. 1 - Table 1-36 see Exercise 16 shows the preference...Ch. 1 - Table 1-39 see Exercise 35 shows the preference...Ch. 1 - Table1-40 see Exercise36 shows the preference...Ch. 1 - An election with five candidates A, B. C, D, and E...Ch. 1 - An election with six candidates A, B, C, D, E, and...Ch. 1 - Use Table 1-41 to illustrate why the Borda count...Ch. 1 - Use Table 1-32 to illustrate why the...Ch. 1 - Use Table 1-42 to illustrate why the plurality...Ch. 1 - Use the Math Club election Example 1.10 to...Ch. 1 - Use Table 1-43 to illustrate why the...Ch. 1 - Explain why the method of pair wise comparisons...Ch. 1 - Prob. 57ECh. 1 - Explain why the plurality method satisfies the...Ch. 1 - Explain why the Borda count method satisfies the...Ch. 1 - Explain why the method of pairwise comparisons...Ch. 1 - Two-candidate elections. Explain why when there...Ch. 1 - Alternative version of the Borda count. The...Ch. 1 - Reverse Borda count. Another commonly used...Ch. 1 - The average ranking. The average ranking of a...Ch. 1 - The 2006 Associated Press college football poll....Ch. 1 - The Pareto criterion. The following fairness...Ch. 1 - The 2003-2004 NBA Rookie of the Year vote. Each...Ch. 1 - Top-two IRV is a variation of the...Ch. 1 - The Coombs method. This method is just like the...Ch. 1 - Bucklin voting. This method was used in the early...Ch. 1 - The 2016 NBA MVP vote. The National Basketball...Ch. 1 - The Condorcet loser criterion. If there is a...Ch. 1 - Consider the following fairness criterion: If a...Ch. 1 - Suppose that the following was proposed as a...Ch. 1 - Consider a modified Borda count where a...
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- 4arrow_forwardYour Question: Number of voters 21 21 10 34 38 16 1st choice C D B A C A 2nd choice B A C B A D 3rd choice A B A D B B 4th choice D C D C D C Find the winner of the election using the Pairwise Comparison method. For each comparison, enter the number of times each candidate was preferred to the other.A vs. BVotes where A is preferred to B : Votes where B is preferred to A : A vs. CVotes where A is preferred to C : Votes where C is preferred to A : A vs. DVotes where A is preferred to D : Votes where D is preferred to A : B vs. CVotes where B is preferred to C : Votes where C is preferred to B : B vs. DVotes where B is preferred to D : Votes where D is preferred to B : C vs. DVotes where C is preferred to D : Votes where D is preferred to C : Tally the results:Points for A: Points for B: Points for C: Points for D: Who is the winner? A B C D There is a tiearrow_forwardWho wins by plurality? A BBM (B) LENI C MANNY D) No winner Number of Votes 1st Rank 2nd Rank 3rd Rank 14 BBM LENI MANNY 12 LENI MANNY BBM 8 BBM LENI MANNY 10 MANNY LENI BBM ...arrow_forward
- Use the plurality with elimination method to determine the winner and compare it to the plurality method. Who wins the preferential vote and the single candidate vote? There are 74 voters and the candidates need 37 votes to win. There are 5 candidates for the house of representatives and you have to rank them by preference. Votes 20 16 12 15 11 1st Emilio Howard Richard Melissa Hannah 2nd Howard Melissa Hannah Hannah Emilio 3rd Melissa Emilio Melissa Richard Richard 4th Hannah Richard Emilio Howard Howard 5th Richard Hannah Howard Emilio Melissa a Plurality with elimination: Richard wins with 43, Plurality: Emilio wins with 20 Plurality with elimination: Melissa wins with 43, Plurality: Emilio wins with 20 Plurality with elimination: Hannah wins with 43, Plurality: Emilio wins with 20 d. Plurality with elimination: Emilio wins with 37, Plurality: Emilio wins with 20arrow_forwardA campus club has money left over in its budget and must spend it beforethe school year ends. The members arrive at five different possibilities, and each memberranks them in order of preference. The results are shown in the table below. 1. Using the Plurality method, how should they spend the money? 2. Use Plurality with elimination. to know how the money should be spent? 3. Using Pairwise Comparison, how should they spend the money? 4. Using the Borda count method of voting, how should the money be spent?5. In your opinion, which of the previous three methods seems most appropriate in thissituation? Why?arrow_forwardThe following statements are true except (a) The plurality method of voting satisfies the majority criterion. (b) The Borda count method of voting always violates the majority criterion. (c) The plurality-with-elimination method of voting satisfies the majority criterion. (d) The pairwise comparison method of voting satisfies the head-to-head criterionarrow_forward
- Please show solutions.arrow_forwardHere are the vote tallies from an election. Determine the winner using plurality, runoff, and eliminate the loser methods. Number of Students 18 votes 12 votes 10 votes 9 votes 4 votes 2 votes First Choice B Second A E D E B B Choice Third E D E D D Choice Fourth B B Choice Fifth Choice A B A A A A Determine the winner using a) Plurality method b) Runoff method c) Eliminate the loser method.arrow_forwardHelp me pleasearrow_forward
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