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- Question : EG1 - The Math Appreciation Society (MAS) is a student club dedicated to an unsung but worthy cause: that of fostering the enjoyment and appreciation of mathematics among college students. The MAS chapter at Tasmania State University is holding its annual election for club president, and there are four candidates running: Alisha, Boris, Carmen, and Dave (A, B, C, and D for short). Every member of the club is eligible to vote, and the vote takes the form of a preference ballot. Each voter is asked to rank each of the four candidates in order of preference. There are 37 voters who submit their ballots, and the 37 preference ballots submitted are shown in Fig. 1-2.
- Question : EG2 - Suppose now that we have pretty much the same situation as in Example 1.1 (same candidates, same voters, same preference ballots), but in this election we have to choose not only a president but also a vice-president, a treasurer, and a secretary. According to the club bylaws, the president is the candidate who comes in first, the vice-president is the candidate who comes in second, the treasurer is the candidate who comes in third, and the secretary is the candidate who comes in fourth. Given that there are four candidates, each candidate will get to be an officer, but there is a big difference between being elected president and being elected treasurer (the president gets status and perks; the treasurer gets to collect the dues and balance the budget). In this version how you place matters, and the outcome should be a full ranking of the candidates.
- Question : EG3 - The Academy Awards (also known as the Oscars) are given out each year by the Academy of Motion Picture Arts and Sciences for Best Picture, Best Actress, Best Actor, Best Director, and many other, lesser categories (Sound Mixing, Makeup, etc.). The election process is not the same for all awards, so for the sake of simplicity we will just discuss the selection of Best Picture. The voters in this election are all the eligible members of the Academy (a tad over 6000 voting members for the 2016 Academy Awards). After a complicated preliminary round (a process that we won
- Question : EG4 - The Heisman Memorial Trophy Award is given annually to the
- Question : EG5 - American Idol is a popular reality TV singing competition for individuals. Each year, the winner of American Idol gets a big recording contract, and many past winners have gone on to become famous recording artists (Kelly Clarkson, Carrie Underwood, Taylor Hicks). While there is a lot at stake and a big reward for winning, American Idol is not a winner-only competition, and there is indeed a ranking of all the finalists. In fact, some nonwinners (Clay Aiken, Jennifer Hudson) have gone on to become great recording artists in their own right. The 12 (sometimes 13) candidates who reach the final rounds of the competition compete in a weekly televised show. During and immediately after each
- Question : EG6 - Table 1-2 shows the preference schedule summarizing the results of the most recent election for mayor of the city of Kingsburg (there actually is a city by that name, but the election is fictitious). Just by looking at the preference schedule we can answer all of the relevant input questions: ? Candidates: there were five candidates (A, B, C, D, and E, which are just abbreviations for their real names). ? Voters: there were 300 voters that submitted ballots (add the numbers at the head of each column: 93 + 44 + 10 + 30 + 42 + 81 = 300). ? Balloting: the 300 preference ballots were organized into six piles as shown in Table 1-2. The question that still remains unanswered: Who is the winner of the election? In the next four sections we will discuss different ways in which such output questions can be answered
- Question : EG7 - We discussed the Math Club election in Section 1.1. Table 1-3 shows once again the preference schedule for the election. Counting only first-place votes, we can see that A gets 14, B gets 4, C gets 11, and D gets 8. So there you have it: In the case of a winner-only election (see Example 1.1) the winner is A (Headline:
- Question : EG8 - Like many states, Maine chooses its governor using the plurality method. In the 2010 election there were five candidates: Eliot Cutler (Independent), Paul LePage (Republican), Libby Mitchell (Democrat), Shawn Moody (Independent), and Kevin Scott (Independent). Table 1-4 shows the results of the election. Before reading on, take a close look at the numbers in Table 1-4 and draw your own conclusions.
- Question : EG9 - Tasmania State University has a superb marching band. They are so good that this coming bowl season they have invitations to perform at five different bowl games: the Rose Bowl (R), the Hula Bowl (H), the Fiesta Bowl (F), the Orange Bowl (O), and the Sugar Bowl (S). An election is held among the 100 band members to decide in which of the five bowl games they will perform. Each band member submits a preference ballot ranking the five choices. The results of the election are shown in Table 1-5 on the next page.
- Question : EG10 - Table 1-6 shows the preference schedule for the Math Club election with the Borda points for the candidates shown in parentheses to the right of their names. For example, the 14 voters in the first column ranked A first (giving A 14 * 4 = 56 points2, B second (14 * 3 = 42 points), and so on.
- Question : EG11 - For general details on the Heisman Award, see Example 1.4. The Heisman is determined using a Borda count, but with truncated preference ballots: each voter chooses a first, second, and third choice out of a large list of candidates, with a first-place vote worth 3 points, a second-place vote worth 2 points, and a third-place vote worth 1 point. Table 1-7 shows a summary of the balloting for the three 2015 finalists. The table shows the number of first-, second-, and third-place votes for each of the three finalists; the last column shows the total point tally for each. Notice that Table 1-7 is not a preference schedule. Because the Heisman uses truncated preference ballots and many candidates get votes, it is easier and more convenient to summarize the balloting this way
- Question : EG12 - The Cy Young Award is an annual award given by Major League baseball for
- Question : EG13 - Let
- Question : EG14 - Table 1-10 shows the preference schedule for the Kingsburg mayoral election first introduced in Example 1.6. To save money Kingsburg has done away with runoff elections and now uses plurality-with-elimination for all local elections. (Notice that since there are 300 voters voting in this election, a candidate needs 151 or more votes to win.)
- Question : EG15 - In 2014 a total of 16 candidates were running for mayor of Oakland, an inordinately large number for an election for political office, and it took 15 rounds of elimination before a winner emerged. But Oakland has been using ranked-choice voting since 2010, so the entire elimination process took place inside a computer, and the final results were known without delay.
- Question : EG16 - We discussed American Idol as an election in Example 1.5. Table 1-11 shows the evolution of the 2016 competition. As noted in Example 1.5, the winner is the big deal, but how the candidates place in the competition is also of some relevance, so we consider American Idol a ranked election. Working our way up from the bottom of Table 1-11, we see how the process of elimination played out: Gianna Isabella and Olivia Rox were eliminated in the first round and tied for 9th-10th place; Lee Jean and Avalon Young were eliminated in the second round and tied for 7th-8th place; Tristan MacIntosh was eliminated in the third round and placed in 6th place . . . and so it went for a total of seven rounds. In the final round it came down to a showdown between La
- Question : EG17 - Table 1-12 shows, once again, the preference schedule for the Math Club election. With four candidates, there are six possible pairwise comparisons to consider (see the first column of Table 1-13). For the sake of brevity, we will go over a couple of these pairwise comparisons in detail and leave the details of the other four to the reader. ? A v B: The first column of Table 1-12 represents 14 votes for A (A is ranked higher than B); the remaining 23 votes are for B (B is ranked higher than A in the last four columns of the table). The winner of this comparison is B. ?? ?C v D: The first, second, and last columns of Table 1-12 represent votes for C (C is ranked higher than D); the third and fourth columns represent votes for D (D is ranked higher than C). Thus, C has 25 votes to D
- Question : EG18 - The Los Angeles LAXers are the newest expansion team in the NFL and are awarded the first pick in the upcoming draft. The team
- Question : EG19 - Table 1-18 shows the preference schedule for a small election. The majority candidate in this election is A with 6 out of 11 first-place votes. However, when we use the Borda count we get A: 29 points, B: 32 points, C: 30 points, D: 19 points, so B is the winner! So here we have a rather messy situation: A has a majority of the first-place votes, and yet A is not the winner under the Borda count method. This is what we mean by
- Question : EG20 - Let
- Question : EG21 - This example comes in two parts
- Question : EG22 - This example is a continuation of Example 1.18 (The NFL Draft). Table 1-21 is a repeat of Table 1-15. We saw in Example 1.18 that the winner of the election under the method of pairwise comparisons is A (you may want to go back and refresh your memory). The LAXers are prepared to make A their number-one draft choice and offer him a big contract. A is happy. End of story? Not quite.
- Question : EX1 - The student body at Eureka High School is having an election for Homecoming Queen. The candidates are Alicia, Brandy, Cleo, and Dionne (A, B, C, and D for short). Table 1-26 shows the preference schedule for the election. Number of voters 202 160 153 145 125 110 108 102 55 1st B C A D D C B A A 2nd D B C B A A C B D 3rd A A B A C D A D C 4th C D D C B B D C B
- Question : EX2 - (a) How many students voted in this election? (b) How many first-place votes are needed for a majority? (c) Which candidate had the fewest last-place votes?
- Question : EX3 - An election is held using the
- Question : EX4 - An election is held using the
- Question : EX5 - Table 1-29 shows a conventional preference schedule for an election. Rewrite Table 1-29 using a format like that in Table 1-27 (as if the ballots were
- Question : EX6 - Number of voters 14 10 8 7 4 1st Bob Bob Ana Dee Eli 2nd Ana Dee Bob Cat Bob 3rd Eli Ana Eli Bob Ana 4th Dee Eli Dee Eli Cat 5th Cat Cat Cat Ana Dee
- Question : EX7 - The Demublican Party is holding its annual convention. The 1500 voting delegates are choosing among three possible party platforms: L (a liberal platform), C (a conservative platform), and M (a moderate platform). Seventeen percent of the delegates prefer L to M and M to C. Thirty-two percent of the delegates like C the most and L the least. The rest of the delegates like M the most and C the least. Write out the preference schedule for this election.
- Question : EX8 - The Epicurean Society is holding its annual election for president. The three candidates are A, B, and C. Twenty percent of the voters like A the most and B the least. Forty percent of the voters like B the most and A the least. Of the remaining voters 225 prefer C to B and B to A, and 675 prefer C to A and A to B. Write out the preference schedule for this election.
- Question : EX9 - Table 1-31 shows the preference schedule for an election with four candidates (A, B, C, and D). Use the plurality method to (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX10 - Number of voters 27 15 11 9 8 1 1st C A B D B B 2nd D B D A A A 3rd B D A B C D 4th A C C C D C
- Question : EX11 - Table 1-32 shows the preference schedule for an election with four candidates (A, B, C, and D). Use the plurality method to (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX12 - Number of voters 29 21 18 10 1 1st D A B C C 2nd C C A B B 3rd A B C A D 4th B D D D A
- Question : EX13 - Table 1-25 (see Exercise 3) shows the preference schedule for an election with five candidates (A, B, C, D, and E). In this election ties are not allowed to stand, and the following tie-breaking rule is used: Whenever there is a tie between candidates, the tie is broken in favor of the candidate w
- Question : EX14 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX15 - Table 1-26 (see Exercise 4) shows the preference schedule for an election with four candidates (A, B, C, and D). In this election ties are not allowed to stand, and the following tiebreaking rule is used: Whenever there is a tie between candidates, the tie is broken in favor of the candidate with the fewer last-place votes. Use the plurality method to
- Question : EX16 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX17 - Table 1-25 (see Exercise 3) shows the preference schedule for an election with five candidates (A, B, C, D, and E). In this election ties are not allowed to stand, and the following tie-breaking rule is used: Whenever there is a tie between two candidates, the tie is broken in favor of the winner of a headto-head comparison between the candidates. Use the plurality method to
- Question : EX18 - (a) find the winner of the election. (b) find the complete ranking of the candidates
- Question : EX19 - Table 1-26 (see Exercise 4) shows the preference schedule for an election with four candidates (A, B, C, and D). In this election ties are not allowed to stand, and the following tiebreaking rule is used: Whenever there is a tie between two candidates, the tie is broken in favor of the winner of a headto-head comparison between the candidates. Use the plurality method to
- Question : EX20 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX21 - Table 1-31 (see Exercise 11) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the Borda count method to
- Question : EX22 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX23 - Table 1-32 (see Exercise 12) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the Borda count method to
- Question : EX24 - (a) find the winner of the election. (b) find the complete ranking of the candidate
- Question : EX25 - Table 1-33 (see Exercise 13) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the Borda count method to
- Question : EX26 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX27 - Table 1-34 (see Exercise 14) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the Borda count method to
- Question : EX28 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX29 - Table 1-35 (see Exercise 15) shows the preference schedule for an election with five candidates (A, B, C, D, and E). The total number of people that voted in this election was very large, so the columns of the preference schedule show percentages rather than actual numbers of voters. Use the Borda count method to find the complete ranking of the candidates. (Hint: The ranking is determined by the percentages and does not depend on the number of voters, so you can pick any number to use for the number of voters. Pick a nice round one.)
- Question : EX30 - Table 1-36 (see Exercise 16) shows the preference schedule for an election with five candidates (A, B, C, D, and E). The total number of people that voted in this election was very large, so the columns of the preference schedule show percentages rather than actual numbers of voters. Use the Borda count method to find the complete ranking of the candidates. (Hint: The ranking is determined by the percentages and does not depend on the number of voters, so you can pick any number to use for the number of voters. Pick a nice round one.)
- Question : EX31 - The 2014 Heisman Award. Table 1-37 shows the results of the balloting for the 2014 Heisman Award. Find the ranking of the top three finalists and the number of points each one received (see Example 1.11)
- Question : EX32 - Player School 1st 2nd 3rd Amari Cooper Alabama 49 280 316 Melvin Gordon Wisconsin 37 432 275 Marcus Mariota Oregon 788 74 22
- Question : EX33 - The 2014 AL Cy Young Award. Table 1-38 shows the top 5 finalists for the 2014 American League Cy Young Award. Find the ranking of the top 5 finalists and the number of points each one received (the point values are the same as those used for the National League Cy Young
- Question : EX34 - An election was held using the conventional Borda count method. There were four candidates (A, B, C, and D) and 110 voters. When the points were tallied (using 4 points for first, 3 points for second, 2 points for third, and 1 point for fourth), A had 320 points, B had 290 points, and C had 180 points. Find how many points D had and give the ranking of the candidates. (Hint: Each of the 110 ballots hands out a fixed number of points. Figure out how many, and take it from there.)
- Question : EX35 - Imagine that in the voting for the American League Cy Young Award (7 points for first place, 4 points for second, 3 points for third, 2 points for fourth, and 1 point for fifth) there were five candidates (A, B, C, D, and E) and 50 voters. When the points were tallied A had 152 points, B had 133 points, C had 191 points, and D had 175 points. Find how many points E had and give the ranking of the candidates. (Hint: Each of the 50 ballots hands out a fixed number of points. Figure out how many, and take it from there.)
- Question : EX36 - Table 1-31 (see Exercise 11) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the plurality-with-elimination method to
- Question : EX37 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX38 - Table 1-32 (see Exercise 12) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the plurality-with-elimination method to
- Question : EX39 - (a) find the winner of the election. (b) find the complete ranking of the candidates
- Question : EX40 - Table 1-33 (see Exercise 13) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the plurality-with-elimination method to
- Question : EX41 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX42 - Table 1-34 (see Exercise 14) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the plurality-with-elimination method to
- Question : EX43 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX44 - Table 1-39 shows the preference schedule for an election with five candidates (A, B, C, D, and E). Find the complete ranking of the candidates using the plurality-with-elimination method.
- Question : EX45 - Number of voters 8 7 5 4 3 2 1st B C A D A D 2nd E E B C D B 3rd A D C B E C 4th C A D E C A 5th D B E A B E
- Question : EX46 - Table 1-40 shows the preference schedule for an election with five candidates (A, B, C, D, and E). Find the complete ranking of the candidates using the plurality-with-elimination method.
- Question : EX47 - Number of voters 7 6 5 5 5 5 4 2 1 1st D C A C D E B A A 2nd B A B A C A E B C 3rd A E E B A D C D E 4th C B C D E B D E B 5th E D D E B C A C D
- Question : EX48 - Table 1-35 (see Exercise 15) shows the preference schedule for an election with five candidates (A, B, C, D, and E). The number of voters in this election was very large, so the columns of the preference schedule show percentages rather than actual numbers of voters. Use the plurality-withelimination method to
- Question : EX49 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX50 - Table 1-36 (see Exercise 16) shows the preference schedule for an election with five candidates (A, B, C, D, and E). The number of voters in this election was very large, so the columns of the preference schedule show percentages rather than actual numbers of voters. Use the plurality-with-elimination method to
- Question : EX51 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX52 - Find the winner of the election given in Table 1-39 using the top-two IRV method.
- Question : EX53 - Find the winner of the election given in Table 1-40 using the top-two IRV method.
- Question : EX54 - Table 1-31 (see Exercise 11) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the method of pairwise comparisons to
- Question : EX55 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX56 - Table 1-32 (see Exercise 12) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the method of pairwise comparisons to
- Question : EX57 - (a) find the winner of the election. (b) find the complete ranking of the candidates
- Question : EX58 - Table 1-33 (see Exercise 13) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the method of pairwise comparisons to
- Question : EX59 - (a) find the winner of the election. (b) find the complete ranking of the candidates
- Question : EX60 - Table 1-34 (see Exercise 14) shows the preference schedule for an election with four candidates (A, B, C, and D). Use the method of pairwise comparisons to
- Question : EX61 - (a) find the winner of the election. (b) find the complete ranking of the candidates.
- Question : EX62 - Table 1-35 (see Exercise 15) shows the preference schedule for an election with five candidates (A, B, C, D, and E). The number of voters in this election was very large, so the columns of the preference schedule give the percent of voters instead of the number of voters. Find the winner of the election using the method of pairwise comparisons.
- Question : EX63 - Table 1-36 (see Exercise 16) shows the preference schedule for an election with five candidates (A, B, C, D, and E). The number of voters in this election was very large, so the columns of the preference schedule give the percent of voters instead of the number of voters. Find the winner of the election using the method of pairwise comparisons.
- Question : EX64 - Table 1-39 (see Exercise 35) shows the preference schedule for an election with 5 candidates. Find the complete ranking of the candidates using the method of pairwise comparisons. (Assume that ties are broken using the results of the pairwise comparisons between the tying candidates.)
- Question : EX65 - Table 1-40 (see Exercise 36) shows the preference schedule for an election with 5 candidates. Find the complete ranking of the candidates using the method of pairwise comparisons.
- Question : EX66 - An election with five candidates (A, B, C, D, and E) is decided using the method of pairwise comparisons. If B loses two pairwise comparisons, C loses one, D loses one and ties one, and E loses two and ties one
- Question : EX67 - (a) find how many pairwise comparisons A loses. (Hint: First compute the total number of pairwise comparisons for five candidates.) (b) find the winner of the election
- Question : EX68 - An election with six candidates (A, B, C, D, E, and F) is decided using the method of pairwise comparisons. If A loses four pairwise comparisons, B and C both lose three, D loses one and ties one, and E loses two and ties one,
- Question : EX69 - (a) find how many pairwise comparisons F loses. (Hint: First compute the total number of pairwise comparisons for six candidates.) (b) find the winner of the election.
- Question : EX70 - Use Table 1-41 to illustrate why the Borda count method violates the Condorcet criterion. Number of voters 6 2 3 1st A B C 2nd B C D 3rd C D B 4th D A A
- Question : EX71 - Use Table 1-32 to illustrate why the plurality-with-elimination method violates the Condorcet criterion.
- Question : EX72 - Use the Math Club election (Example 1.10) to illustrate why the Borda count method violates the IIA criterion. (Hint: Find the winner, then eliminate D and see what happens.)
- Question : EX73 - Use Table 1-43 to illustrate why the plurality-with-elimination method violates the IIA criterion. (Hint: Find the winner, then eliminate C and see what happens.)
- Question : EX74 - Number of voters 5 5 3 3 3 2 1st A C A D B D 2nd B E D C E C 3rd C D B B A B 4th D B C E C A 5th E A E A D E
- Question : EX75 - Explain why the method of pairwise comparisons satisfies the majority criterion
- Question : EX76 - Explain why the method of pairwise comparisons satisfies the Condorcet criterion.
- Question : EX77 - Explain why the plurality method satisfies the monotonicity criterion.
- Question : EX78 - Explain why the Borda count method satisfies the monotonicity criterion
- Question : EX79 - Explain why the method of pairwise comparisons satisfies the monotonicity criterion.
- Question : EX80 - Alternative version of the Borda count. The following simple variation of the conventional Borda count method is sometimes used: last place is worth 0 points, second to last is worth 1 point, . . . , first place is worth N - 1 points (where N is the number of candidates). Explain why this variation is equivalent to the conventional Borda count described in this chapter (i.e., it produces exactly the same winner and the same ranking of the candidates).
- Question : EX81 - Two-candidate elections. Explain why when there are only two candidates, the four voting methods we discussed in this chapter give the same winner and the winner is determined by straight majority. (Assume that there are no ties.)
- Question : EX82 - Reverse Borda count. Another commonly used variation of the conventional Borda count method is the following: A first place is worth 1 point, second place is worth 2 points, . . . , last place is worth N points (where N is the number of candidates). The candidate with the fewest points is the winner, second fewest points is second, and so on. Explain why this variation is equivalent to the original Borda count described in this chapter (i.e., it produces exactly the same winner and the same ranking of the candidates).
- Question : EX83 - The average ranking. The average ranking of a candidate is obtained by taking the place of the candidate on each of the ballots, adding these numbers, and dividing by the number of ballots. Explain why the candidate with the best (lowest) average ranking is the Borda winner.
- Question : EX84 - The 2006 Associated Press college football poll. The AP college football poll is a ranking of the top 25 college football teams in the country. The voters in the AP poll are a group of sportswriters and broadcasters chosen from across the country. The top 25 teams are ranked using a conventional Borda count: a first-place vote is worth 25 points,
- Question : EX85 - (a) Given that Ohio State was the unanimous first-place choice of all the voters, find the number of voters that participated in the poll.
- Question : EX86 - (b) Given that all the voters had Florida in either second or third place, find the number of second-place and the number of third-place votes for Florida.
- Question : EX87 - (c) Given that all the voters had Michigan in either second or third place, find the number of second-place and the number of third-place votes for Michigan.
- Question : EX88 - The Pareto criterion. The following fairness criterion was proposed by Italian economist Vilfredo Pareto (1848
- Question : EX89 - (a) Explain why the Borda count method satisfies the Pareto criterion.
- Question : EX90 - (b) Explain why the pairwise-comparisons method satisfies the Pareto criterion.
- Question : EX91 - The 2003
- Question : EX92 - Player 1st place 2nd place 3rd place Total points LeBron James 78 39 1 508 Carmelo Anthony 40 76 2 430 Dwayne Wade 0 3 108 117
- Question : EX93 - Top-two IRV is a variation of the plurality-with-elimination method in which all the candidates except the top two are eliminated in the first round and their votes transferred to the top two. (see Exercises 39 and 40).
- Question : EX94 - (a) Use the Math Club election to show that top-two IRV can produce a different outcome than plurality-withelimination.
- Question : EX95 - (b) Give an example that illustrates why top-two IRV violates the monotonicity criterion
- Question : EX96 - (c) Give an example that illustrates why top-two IRV violates the Condorcet criterion.
- Question : EX97 - The Coombs method. This method is just like the plurality-with-elimination method except that in each round we eliminate the candidate with the largest number of lastplace votes (instead of the one with the fewest first-place votes)
- Question : EX98 - (a) Find the winner of the Math Club election using the Coombs method.
- Question : EX99 - (b) Give an example that illustrates why the Coombs method violates the Condorcet criterion.
- Question : EX100 - (c) Give an example that illustrates why the Coombs method violates the monotonicity criterion.
- Question : EX101 - Bucklin voting. (This method was used in the early part of the 20th century to determine winners of many elections for political office in the United States.) The method proceeds in rounds. Round 1: Count first-place votes only. If a candidate has a majority of the first-place votes, that candidate wins. Otherwise, go to the next round. Round 2: Count firstand second-place votes only. If there are any candidates with a majority of votes, the candidate with the most votes wins. Otherwise, go to the next round. Round 3: Count first-, second-, and third-place votes only. If there are any candidates with a majority of votes, the candidate with the most votes wins. Otherwise, go to the next round. Repeat for as many rounds as necessary.
- Question : EX102 - (a) Find the winner of the Math Club election using the Bucklin method.
- Question : EX103 - (b) Give an example that illustrates why the Bucklin method violates the Condorcet criterion.
- Question : EX104 - (c) Explain why the Bucklin method satisfies the monotonicity criterion.
- Question : EX105 - The 2016 NBA MVP vote. The National Basketball Association Most Valuable Player is chosen using a modified Borda count. Each of the 131 voters (130 sportswriters from the U.S and Canada plus one aggregate vote from the fans) submits ballots ranking the top five players from 1st through 5th place. Table 1-46 shows the results of the 2016 vote. (For the first time in NBA history a single player
- Question : EX106 - The Condorcet loser criterion. If there is a candidate who loses in a one-to-one comparison to each of the other candidates, then that candidate should not be the winner of the election. (This fairness criterion is a sort of mirror image of the regular Condorcet criterion.)
- Question : EX107 - (a) Give an example that illustrates why the plurality method violates the Condorcet loser criterion.
- Question : EX108 - (b) Give an example that illustrates why the pluralitywith-elimination method violates the Condorcet loser criterion.
- Question : EX109 - (c) Explain why the Borda count method satisfies the Condorcet loser criterion.
- Question : EX110 - 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
- Question : EX111 - (a) Give an example to illustrate why the plurality method violates this criterion.
- Question : EX112 - (b) Give an example to illustrate why the plurality-withelimination method violates this criterion.
- Question : EX113 - (c) Explain why the method of pairwise comparisons satisfies this criterion.
- Question : EX114 - (d) Explain why the Borda count method satisfies this criterion.
- Question : EX115 - Suppose that the following was proposed as a fairness criterion: If a majority of the voters rank X above Y, then the results of the election should have X ranked above Y. Give an example to illustrate why all four voting methods discussed in the chapter can violate this criterion. (Hint: Consider an example with no Condorcet candidate.)
- Question : EX116 - Consider a modified Borda count where a first-place vote is worth F points (F 7 N where N denotes the number of candidates) and all other places in the ballot are the same as in the ordinary Borda count: N - 1 points for second place, N - 2 points for third place, . . . , 1 point for last place. By choosing F large enough, we can make this variation of the Borda count method satisfy the majority criterion. Find the smallest value of F (expressed in terms of N) for which this happens.
- Question : EX117 - . Consider the election given by the preference schedule shown in Table 1-47.
- Question : EX118 - (a) Using the Voting Methods applet verify that in this election different methods give different results (sometimes the winner is A, sometimes the winner is B and sometimes there is a tie for first-place between A and B).
- Question : EX119 - (b) Imagine that you are an
- Question : EX120 - 49 40 40 51 43 30 1st A A B B C C 2nd B C A C A B 3rd C B C A B A
- Question : EX121 - Consider once again the election given by the preference schedule shown in Table 1-47. Manipulate the results of this election (changing as few ballots as possible) so that each of the three candidates is a winner under one of the methods.
- Question : EX122 - Consider the following fairness criterion: If a majority of the voters rank candidate X above candidate Y, then the results of the election should rank X above Y. Use the Voting Methods applet to find an election in which all four of the voting methods violate this criterion.

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