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G180/MAT1402 Module 03 Assignment Instructions: Answer all questions completely. Make sure you show all work (when applicable) for full credit. You may include explanations to receive partial credit if your answer is incorrect. Using the table below, answer the first 5 questions. A group of students were asked to vote for their favorite action movies. Their choices were Avengers, Braveheart, Captain America, and Die Hard (A, B, C, D for short). The following table gives their preference schedule. Number of Voters 21 19 13 9 9 1 Choice B A G D G 2"! Choice D B A A D 3" Choice A Cc D e B 4" Choice Cc D B B A 1. How many students voted? How many first place votes are needed for a majority? Show all calculations and final answer here. A B Cc D Total Number of first placed 19 21 22 2) 7 votes There are 71 students who voted Since C has the majority number of vates, then 22 is the number of votes needed for a majority. 2. Use the plurality method to find the winner of the election. Show all calculations, explain how you determined the winner, and your final answer here. A B Cc D Number of first placed 19 21 22 9 votes Using Plurality method, C gets the majority votes hence the winner. 3. Use the Borda count method to find the winner of the election. Show all calculations, explain how you determined the winner, and your final answer here. candidates points TOTAL A 21(2) 19(4) 13(3) 9(3) 9(1) =166 B 21(4) 19(3) 13(1) 9(1) 9(2) =181 Cc 21(1) 19(2) 13(4) 9(2) 9(4) =165 D 21(3) 19(1) 13(2) 9(4) 9(3) =171 Since B has most points, then B is the winner. 4, Use the plurality-with-elimination method to find the winner of the election. Show all calculations, explain how you determined the winner, and your final answer here. A=19 B=21 C=22 D=9 A B iS 52 A c 19 52 A D 19 52 B c 40 31 B D 53 18 Cc D 41 30 The winner is obtained by comparing votes between two movies in the election. Candidate B wins the most pairwise comparisons so the movie B wins ( in fact, the movie is Condorcet winner) 6. Three players (Anna, Bill and Carol) are sharing a pizza. Suppose that the pizza is divided into three slices (s1, Sz, 83). The following table gives the value of each slice in the eyes of each of the players. Sa Sz $3 Anna $5.50 $5.25 $5.00 Bill $4.50 $5.50 $5.00 Carol $5.60 $4.60 $5.00 Using this table, answer the following questions: a) Which of the three slices are fair shares to Anna? b) Which of the three slices are fair shares to Bill? c) Which of the three slices are fair shares to Carol? d) Find a fair division of pizza using S:, S2, and S3 as fair shares. If this is not possible, explain why not. Three players (April, Beth and Clint) are sharing a cake. Suppose that the cake is divided into three slices (s1, Sz, 83). The percentages represent the value of the slice as a percent of the value of the entire cake. Si Sz $3 April 30% 29% 41% Beth 36% 32% 32% Clint 30% 42% 28% Using this table, answer the following questions: a) Which of the three slices are fair shares to April? b) Which of the three slices are fair shares to Beth? c) Which of the three slices are fair shares to Clint? d) Find the fair division of the cake, using sj, Sz and s3as fair shares. If this is nat possible, explain why not. Answer all four parts here. Make sure you show all of your calculations and explain why they are fair shares and how you found the fair division. Si S2 S3 Total Fair share (Total /3) April 30% 29% 41% 100% 33.33% Beth 36% 32% 32% 100% 33.338 Clint 30% 42% 28% 100% 33.335 (a) April’s share = 33.33% is actual fair Those shares less than or equal to actual fair will be April’s share The answer is S; which is less than actual and S2 which is less than fair share. (8) Beth’s share = 33.33% is actual fair Those shares less than or equal to actual fair will be Beth’s share The answer is S2 which is less than actual and Ss which is less than fair share. (c) Clint’s share = 33.33% is actual fair Those shares less than or equal to actual fair will be Carol's share The answer is S; which is less than actual and S3 which is less than fair share. (D) Yes , there is a way the fair division of the pizza, using s1, s2 and s3 as fair shares if we would choose April = si Beth = S, or any other Clint= S3 
