Approval Voting Exercises 1. Ultimate Frisbee is a growing sport at Central City High. The Ultimate Club is sponsoring a weekend event and each participant will receive a tee-shirt. Club members decided to let the participants vote on the color of the tee-shirt using Approval Voting. The possible colors are Fuchsia, Purple, Eggshell, Watermelon, Teal. Here is a summary of the results: 12 participants voted for Fuchsia 7 participants voted for Fuchsia and Teal 20 participants voted for Eggshell and Watermelon 18 participants voted for Purple, Eggshell, and Watermelon 23 participants voted for Teal and Purple 25 participants voted for Watermelon Use Approval Voting to determine the color of the tee-shirt.   2. Tasmania State University marching band has been invited to march in 5 different parades: the Rose Bowl parade, the Hula Bowl parade, the Cotton Bowl parade, the Orange Bowl parade, and the Sugar Bowl parade. The band director decides to use Approval Voting to determine which parade to attend. The assistant in charge of polling the 100 band members misunderstood the instructions and asked each band member to rank the parades from 1st choice to 5th choice and summarized the results in the following preference schedule.
 39 16 45 Rose Hula Cotton Hula Sugar Hula Cotton Orange Sugar Orange Cotton Orange Sugar Rose Rose
 The band director now has to decide what to do with this information since it wasn’t exactly what he needed.a. Suppose the band director decides to change methods and choose the parade using plurality with elimination. Which parade will be chosen? b. Suppose the band director decides to change methods and choose the parade using pairwise comparisons.Which parade will be chosen? c. Suppose the band director decides to change methods and choose the parade using Borda Count. Which parade will be chosen? d. Suppose the band director decides to try Approval Voting anyway. Which parade is chosen if, after re-polling, each band member approved only his/her first, second, and third choices? e. Suppose the band director decides to try Approval Voting anyway. Which parade is chosen if, after re-polling, each band member approved only his/her first and second choices? f. Suppose the band director decides to try Approval Voting anyway. Which parade is chosen if, after re-polling, each band member whose first choice was the Cotton Bowl approved only of the Cotton Bowl but everyone else approved of his/her first three choices? 3. The results of a hypothetical election using Approval Voting are summarized in the table below. An "X" indicates that the voter approves of the candidate.
 VOTERS Candidates Richard Sally Thomas Uma Vera Walter Yvette Zoe ADAMS X X X X X X BARNES X X X COLLINS X X X X
 a. Who is the winner?b. Who wins if COLLINS drops out of the race? Do the vote totals of the remaining candidates change? c. Compare the effect of Sally’s votes (no approvals) to the effect of Vera’s votes (all approvals)?   Do the next two exercises only after you have learned about EXTENDED rankings. In the next two exercises, you will be asked to apply this idea to the Approval Voting method. 4. Go back to exercise #1 above and rank all the tee-shirt colors using extended Approval Voting. 5. Go back to exercise #2d above and rank all the parades using extended Approval Voting.   Do the next two exercises only after you have learned about RECURSIVE rankings. In the next two exercises, you will be asked to apply this idea to the Approval Voting method. 6. Go back to exercise #1 above and rank all the tee-shirt colors using recursive Approval Voting. 7. Go back to exercise #2d above and rank all the parades using recursive Approval Voting.   Do the remaining exercises only after you have learned about EXTENDED rankings and RECURSIVE rankings. Look back at the previous 4 exercises and A Nice Property of Approval Voting. 8. What can you say about the relationship between extended and recursive ranking methods applied to Approval Voting? 9. Go back to exercise #2e above and rank all the parades using extended/recursive Approval Voting. 10. Go back to exercise #2f above and rank all the parades using extended/recursive Approval Voting. 11. Go back to exercise #3 above and rank all the parades using extended/recursive Approval Voting.   The following exercises refer back to problems from the Chapter 1 problem set in your textbook, Excursions in Modern Mathematics. 12. Rank the 4 restaurants in #1 under Approval Voting if we assume that each person approves only their first two choices. 13. Rank the 3 candidates in #2 under Approval Voting if we assume that Sue, Bill, Tom, Pat and Tina approve only their first choice while the others approve both their first and second choice. 14. Rank the 4 possible exam times in #3 under Approval Voting if we assume that each student approves only their first two choices. 15. Rank the 5 candidates in #7 under Approval Voting if we assume that each person whose first choice was candidate A approves their first 3 choices while the others approve only their first two choices. 16. Rank the 5 alternatives in #14 under Approval Voting if we assume that each person whose first choice was alternative H approves their first 3 choices while the others approve only their first two choices. Answers