1. (40 points) Suppose that there is an upcoming election between 5 candidates, Allen, Beth, Cammy, Dennis and Elliott. There are 100 voters with different preference orderings for these candidates. There are 6 different orderings, which we label through VI. The preference orderings (1 for the best to 5 for worst), along with the number of voters with each ordering are shown in the following table (the candidates are identified by the first letter if their last names): 1 2 3 4 5 RANKING GROUPS AND THEIR SIZES 11 25 III 15 IV 10 V6 B B E D E D E B A B D E D A A 1 40 A B D E VI 4 B E D A Assume that all voters vote sincerely and report their preferences truthfully. a. What will be the outcome of the election under the plurality method (the one with the most first preferences)? Explain your answer clearly. b. What will be the outcome of the election under the Borda count method? Assume that the following points are allotted: 5 points for first preference, 4 points for second, 3 points for third, 2 points for fourth and 1 point for fifth. Explain your answer clearly. c. What will be the outcome of the election under the runoff method (the top two first preferences go into a second round)? Explain your answer clearly. d. What will be the outcome of the election under the instant runoff voting? Explain your answer clearly. Note: Instant runoff voting can be explained in five steps: 1. Voters rank all candidates according to their preferences. 2. The votes are counted. 3. If a candidate has earned a majority of the votes, that candidate is the winner. If not, go to step 4. 4. Eliminate candidate(s) with the fewest votes. (Eliminate more than one candidate at the same time only if they tie for the fewest votes.) 5. Redistribute votes from eliminated candidates to the next-ranked choices on those ballots. Once this is done, return to step 2. 1. (40 points) Suppose that there is an upcoming election between 5 candidates, Allen, Beth, Cammy, Dennis and Elliott. There are 100 voters with different preference orderings for these candidates. There are 6 different orderings, which we label through VI. The preference orderings (1 for the best to 5 for worst), along with the number of voters with each ordering are shown in the following table (the candidates are identified by the first letter if their last names): 1 2 3 4 5 RANKING GROUPS AND THEIR SIZES 11 25 III 15 IV 10 V6 B B E D E D E B A B D E D A A 1 40 A B D E VI 4 B E D A Assume that all voters vote sincerely and report their preferences truthfully. a. What will be the outcome of the election under the plurality method (the one with the most first preferences)? Explain your answer clearly. b. What will be the outcome of the election under the Borda count method? Assume that the following points are allotted: 5 points for first preference, 4 points for second, 3 points for third, 2 points for fourth and 1 point for fifth. Explain your answer clearly. c. What will be the outcome of the election under the runoff method (the top two first preferences go into a second round)? Explain your answer clearly. d. What will be the outcome of the election under the instant runoff voting? Explain your answer clearly. Note: Instant runoff voting can be explained in five steps: 1. Voters rank all candidates according to their preferences. 2. The votes are counted. 3. If a candidate has earned a majority of the votes, that candidate is the winner. If not, go to step 4. 4. Eliminate candidate(s) with the fewest votes. (Eliminate more than one candidate at the same time only if they tie for the fewest votes.) 5. Redistribute votes from eliminated candidates to the next-ranked choices on those ballots. Once this is done, return to step 2
