Answer: Jeff Fran Rudy House $ $ $ Car $ $ $ Boat $ $ $

Transcription

1 Chapter 3 Test 1 1. Jafar and Danielle are planning on dividing up a cake which they jointly won during a dorm wide dance-off. The cake is a third chocolate, a third vanilla, and a third strawberry. They will use the Divider-Chooser Method. After flipping a coin it is determined that Jafar will divide the cake into two pieces and Danielle will choose one. Jafar values chocolate two times as much as he values vanilla; however, he values vanilla and strawberry equally. Help Jafar make a single straight line cut through the center of the cake which results in two fair shares according to his preference system. Answer: 2. A group of friends is going to divide up a plot of land amongst themselves using the Lone-Divider Method. After the divider D divides the land into four parcels s 1, s 2, s 3, and s 4, the choosers C1, C2, and C 3 submit their bids on what they consider a fair share. Suppose that the choosers bids C : s ; C : s ; and C : s. Describe how a fair division would result. are { } { } { } Jeff, Fran, and Rudy are dividing an estate consisting of a house, a car, and a boat, using the Method of Sealed Bids. Their bids on each of the items are given in the following table. Which item(s), if any does Rudy receive, and how much money does he pay/receive, including any surplus? 4. Four players {A, B, C, and D} agree to divide the 16 items shown below by lining them up in order and using the Method of Markers. The players bids are indicated. Which player receives item five? Jeff Fran Rudy House $ $ $ Car $ $ $ Boat $ $ $

2 Chapter 3 Test 2 1. Horace and Pete jointly purchase a pizza which is half pepperoni and half sausage for $24. Suppose that Horace values pepperoni three times as much as he values sausage. According to Horace s preference system, what is the dollar value of the slice shown below? 2. Based on the scenario from problem 1 and knowing Pete values sausage seven times as much as he values pepperoni, what is the dollar value of the slice shown above according to Pete s preference system? 3. Four friends {Joe, Tim, Sam, and Ron} have decided to divide up a multi-flavored cake which was jointly purchased for $24 using the Lone-Divider Method. The cake is divided into four pieces{ s, s, s, and s }. The following table gives the dollar value of each of the slices according to each of the four friends. Which of the four friends is the divider? s 1 s 2 s 3 s 4 Joe $8.00 $7.00 $4.00 $5.00 Tim $6.00 $6.00 $6.00 $6.00 Sam $5.50 $10.00 $4.50 $4.00 Ron $5.50 $9.00 $3.50 $ Based on the scenario from problem 3, find a fair division of the cake using s1, s2, s3, and s 4 as fair shares, and explain why this is the only possible fair division. 5. Suppose a cake which is half chocolate and half vanilla costs $12.00 to buy. The cake is divided into two pieces and one piece is given to Jim and another piece is given to Sally. Sally values her piece as being worth $8.00 while Jim values his piece being worth $6.00. The sum of the two values is $14.00, but the cake only costs $ Clearly explain how this could be possible

3 Chapter 3 Test 2 6. You have decided to live in a converted loft with eleven of your college friends. The loft is a large open room and as a group you have decided to divvy up the floor space using the Last-Diminisher Method. You label yourselves P1, P2, P3,, P12 and the order in which the method is played has P 1 first, P 2 second, and so on. Here is how the method begins: Round 1: P 1 makes the first cut; P 2 passes; P 3 diminishes; P4, P5, P10 pass; P 11 diminishes; P 12 passes. Round 2: After the first claim is made, P 8 is the only diminisher. Round 3: After the first claim is made, all the remaining players pass. After the end of Round 3, which three players have received what they consider a fair share of the loft space? 7. A group of friends is going to divide up a plot of land amongst themselves using the Lone-Divider Method. After the divider D divides the land into four parcels s 1, s 2, s 3, and s 4, the choosers C1, C2, and C 3 submit their bids on what they consider a fair share. Suppose that the choosers bids C : s, s ; C : s, s, s ; and C : s, s. Give two different fair divisions of the land. are { } { } { } Ron, Tim, Sam, and Joe are dividing an estate consisting of a house and a car using the Method of Sealed Bids. Their bids on each of the items are given in the following table. Which item(s), if any, does Tim receive, and how much money does he pay/receive, including any surplus? Ron Tim Sam Joe House $ $ $ $ Car $ $ $ $ Ichabod and Maize jointly purchase a box of Neapolitan Ice-Cream (1/3 chocolate, 1/3 vanilla, and 1/3 strawberry). They will use the Divider-Chooser Method to divide the box. After flipping a coin it is determined that Ichabod will divide the ice-cream into two pieces and Maize will choose one. Ichabod values chocolate two times as much as he values vanilla; however, he values vanilla and strawberry equally. Help Ichabod make a single straight line cut (vertical or horizontal) along the gridlines which results in two fair shares according to his preference system. Answer:

4 Chapter 3 Test Four players {A, B, C, and D} agree to divide the 16 items shown below by lining them up in order and using the Method of Markers. The players bids are indicated. Which player receives item thirteen?

5 Chapter 3 Test 3 1. Jerome and his two friends are planning on fairly dividing up a chocolate-vanilla-strawberry cake. The cake is a third chocolate, a third vanilla, and a third strawberry. Jerome values chocolate two times as much as strawberry, and he values strawberry three times as much as vanilla. Which of the pieces below are considered a fair share according to Jerome s preference system? (a) s 1 (b) s 3 (c) s1 and s 2 (d) s1 and s 3 2. John and Sara have jointly purchased a half pepperoni and half sausage pizza. They use the Divider- Chooser Method to divide the pizza. John is the divider and cuts the pizza into the following two pieces. Based on John s division, he prefers sausage pepperoni. Fill in the blank. (a) equally to (b) twice as much as (c) three times as much as (d) four times as much as 3. Four people are going to fairly divide up a parcel of land worth $240, using the Lone-Divider Method. The divider partitions the land into four different pieces s1, s2, s3, and s 4. According to the divider, what is the dollar value of s 3? (a) $50, (b) $60, (c) $70, (d) $80, Which of the following items would best be divided among five people using a discrete method of division? (a) 1 hour of dog walking (b) multi-flavor cake (c) house and a car (d) large pepperoni-sausage pizza

6 Chapter 3 Test 3 5. Four friends {Joe, Tim, Sam, and Ron} have decided to divide up a multi-flavored cake which was jointly purchased for $24. The cake is divided into four pieces{ s1, s2, s3, and s 4}. The following table gives the dollar value of each of the slices according to each of the four friends. Which of the four slices are considered fair shares according to Ron? (a) s 2 (b) s 3 (c) s2 and s 3 (d) All pieces. (e) None of the above 6. According to the scenario from problem five, which of the following would be a fair division? (a) Joe : s4; Tim : s2; Sam : s3; Ron : s 1 (b) Joe : s4; Tim : s1; Sam : s2; Ron : s 3 (c) Joe : s1; Tim : s3; Sam : s2; Ron : s 4 (d) Joe : s4; Tim : s2; Sam : s1; Ron : s 3 7. You have decided to live in a converted loft with eight of your college friends. The loft is a large open room and as a group you have decided to divvy up the floor space using the Last-Diminisher Method. You label yourselves P1, P2, P3, P8 and the order in which the method is played has P 1 first, P 2 second, and so on. Here is how the method begins: Round 1: P 1 makes the first cut; P 2 passes; P 3 diminishes; P4, P5, P6 pass; P 7 diminishes; P 8 passes Round 2: After the first claim is made, P 3 is the only diminisher. Round 3: After the first claim is made, all the remaining players pass. After the end of Round 3, which three players have received a share of the loft space? (a) P1, P3, and P 7 (b) P1, P6, and P 7 (c) P3, P6, and P 7 (d) P1, P3, and P 6 s 1 s 2 s 3 s 4 Joe $6.00 $6.00 $6.00 $6.00 Tim $4.50 $8.00 $8.00 $3.50 Sam $4.00 $10.00 $4.00 $6.00 Ron $4.00 $8.00 $7.00 $

7 Chapter 3 Test 3 8. Four players {A, B, C, and D} agree to divide the 16 items shown below by lining them up in order and using the Method of Markers. The players bids are indicated. Which player receives item sixteen? (a) A (b) B (c) C (d) D 9. Ron, Tim, Sam, and Joe are dividing an estate consisting of a house and a car using the Method of Sealed Bids. Their bids on each of the items are given in the following table. How much does each person get in surplus money? Ron Tim Sam Joe House $ $ $ $ Car $ $ $ $ (a) $ (b) $ (c) $ (d) $ Tom and Allison jointly purchase the box of Neapolitan Ice-Cream (1/3 chocolate, 1/3 vanilla, and 1/3 strawberry) shown below. They will use the Divider-Chooser Method to divide the box. After flipping a coin it is determined that Allison will divide the ice-cream into two pieces and Tom will choose one. Allison values chocolate three times as much as she values vanilla; however, she values vanilla and strawberry equally. As the divider, which of the divisions could Allison make? Ice-Cream (a) (b) (c) (d) (e) None of the above

8 Chapter 3 Test 4 1. Five friends {Alice, Bob, Cathy, Dave, and Earl} have decided to divide up a multi-flavored cake which was jointly purchased for $30 using the Lone-Divider Method. The cake is divided into five s, s, s, s, and s. The following table gives the dollar value of each of the slices according pieces { } to each of the five friends. Which of the five slices are considered fair shares according to Bob? (a) s1, s2, s3, and s 4 (b) s3 and s 4 (c) s1 and s 2 (d) Only s According to the scenario from problem one, which of the following would be a fair division? (a) Alice : s4; Bob : s3; Cathy : s2; Dave : s1; Ear l : s 5 (b) Alice : s1; Bob : s3; Cathy : s2; Dave : s5; Ear l : s 4 (c) Alice : s4; Bob : s3; Cathy : s1; Dave : s5; Ear l : s 2 (d) Alice : s2; Bob : s3; Cathy : s4; Dave : s5; Ear l : s 1 3. According to the scenario from problem one, which player is the divider? (a) Alice (b) Bob (c) Cathy (d) Dave (e) Earl 4. Jackson and Sherry jointly purchase a pizza which is half pepperoni and half sausage for $18. Suppose that Jackson values pepperoni five times as much as he values sausage, while Sherry values sausage two times as much as she values pepperoni. According to Jackson s preference system, what is the dollar value of the slice shown below? (a) $5.00 (b) $5.50 (c) $6.00 (d) $6.50 s 1 s 2 s 3 s 4 s 5 Alice $6.00 $6.00 $6.00 $6.00 $6.00 Bob $5.50 $5.50 $8.00 $6.00 $5.00 Cathy $3.00 $10.00 $5.00 $6.00 $6.00 Dave $7.00 $8.50 $3.50 $5.00 $6.00 Earl $7.00 $8.50 $4.50 $5.00 $

9 Chapter 3 Test 4 5. Jeff and Kyle jointly purchase the box of Neapolitan Ice-Cream (1/3 chocolate, 1/3 vanilla, and 1/3 strawberry) shown below. They will use the Divider-Chooser Method to divide the box. After flipping a coin it is determined that Kyle will divide the ice-cream into two pieces and Jeff will choose one. Kyle values chocolate five times as much as he values vanilla; however, he values strawberry four times as much as he values vanilla. As the divider, which of the divisions could Kyle make? Ice Cream (a) (b) (c) (d) (e) None of the above. 6. Four players {A, B, C, and D} agree to divide the 16 items shown below by lining them up in order and using the Method of Markers. The players bids are indicated. Which player receives item three? (a) A (b) B (c) C (d) D 7. Jeff, Fran, and Rudy are dividing an estate consisting of a house, a car, and a boat using the Method of Sealed Bids. Their bids on each of the items are given in the following table. How much does money, including surplus, does Rudy receive/pay? Jeff Fran Rudy House $ $ $ Car $ $ $ Boat $ $ $ (a) $ (b) $ (c) $ (d) $

10 Chapter 3 Test 4 8. Which of the following is an example of an item which could be divided using a continuous method of fair division? (a) Car (b) House (c) Cake (d) Painting 9. You have decided to live in a converted loft with five of your college friends. The loft is a large open room and as a group you have decided to divvy up the floor space using the Last-Diminisher Method. You label yourselves P1, P2, P3, P6 and the order in which the game is played has P 1 first, P 2 second, and so on. Here is how the method begins: Round 1: P 1 makes the first cut; P 2 passes; P 3 passes; P 4 diminishes; P 5 diminishes; P 6 passes Round 2: After the first claim is made, P 3 is the only diminisher. Round 3: After the first claim is made, all the remaining players pass. After the end of Round 3, which three players are still playing the game? (a) P1, P4, and P 5 (b) P2, P4, and P 6 (c) P2, P3, and P 6 (d) P4, P5, and P John and Sara have jointly purchased a half pepperoni and half sausage pizza. They use the Divider- Chooser Method to divide the pizza. John is the divider and cuts the pizza into the following two pieces. Based on John s division, he prefers pepperoni sausage. Fill in the blank. (a) equally to (b) twice as likely as (c) three times as much as (d) four times as much as