Deck 3: The Mathematics of Sharing: Fair-Division Games

Question 1

Alex, Bob and Cathy used the Lone-Chooser Method to divide up a cake. Alex and Bob are the Dividers for this method, and each of them cuts their portions into thirds allowing Cathy to make her selections. Based off of this chart, which two pieces will Cathy choose?

Answer 1

Cathy will chose S2 and S6

Question 2

A bride and groom are very torn on what flavor of wedding cake they should have. They decide to make the cake 1/3 Chocolate, 1/3 Vanilla and 1/3 Strawberry. The bride values Chocolate five times as much as she values Vanilla, but she values Vanilla four times as much as she values Strawberry. If the cake costs a total of $480, then in the value system of the bride, what is the cost of the entire
Vanilla piece?

Answer 2

$76.80

Question 3

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?

Answer 3

Rudy gets the Car and Boat and
receives $46000.00 in cash.

Question 4

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?

Question 5

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 and the order in which the method is played has P₁ first, P₂ second, and so on. Here is how the method begins:
Round 1: P₁ makes the first cut; P₂ passes; P₃ diminishes; diminishes; P₁₂ passes.
Round 2: After the first claim is made, P₈ 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?

Question 6

Suppose a cake , which is half chocolate and half vanilla , coast $12.00 to buy. We know that Jim has no preference between chocolate and vanilla and sally prefers chocolate two times as much as she prefers vanilla . Jim divides the cake into two pieces and sally choos one of the pieces . sally values her piece as being worth $8.00 . what is the value , to Jim, of the remaining piece?

Question 7

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 s4 } . The following table gives the dollar value of each of the slices according to e3ach of the four friends .

Which of the four slices are considered fair shares according to Ron?
Based on the above scenario , find a fair division of the cake using as fair shares.

Question 8

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 {s₁ , s₂ , s₃ , and s₄} . the following table gives the dollar walue of each of the slices according to each of the four fiends .

which of the four friends is the divider?

Question 9

Question 10

Clearly explain why it is better to be the chooser compared to the divider when using the DividerChooser Method of fair division.

Question 11

A bride and groom are very torn on what flavor of wedding cake they should have. They decide to make the cake 1/3 Chocolate, 1/3 Vanilla and 1/3 Strawberry. The bride values Chocolate five times as much as she values Vanilla, but she values Vanilla four times as much as she values Strawberry. If the cake costs a total of $480, then in the value system of the bride, what is the cost of the entire
Vanilla piece?
Based on the above scenario , would the cut shown below be considered a rational choice for the bride to make if she needed to divide the cake into two pieces using the Divider-chooser Method? Explain.

Question 12

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?

Question 13

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 S1,S2,S3, AND S4,
C1, C2, AND C3, submit their bids on what they consider a fair share. Suppose that the choosers' bids are https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/. Give two different fair divisions of the land.

Question 14

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 s4 } . The following table gives the dollar value of each of the slices according to e3ach of the four friends .

Which of the four slices are considered fair shares according to Ron?
Based on the above scenario , find a different fair division (different from problem 4) of the cake using as fair shares. If no other fair division is possible, explain why.

Question 15

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?

Based on the scenario above 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?

Question 16

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

submit their bids on what they consider a fair share. Suppose that the choosers' bids
are and . Describe how a fair division would result.

Question 17

Alex, Bob and Cathy used the Lone-Chooser Method to divide up a cake. Alex and Bob are the Dividers for this method, and each of them cuts their portions into thirds allowing Cathy to make her selections. Based off of this chart, who (Alex or Bob) acted as the original Divider when performing the Divider-Chooser Method between the two players. Explain how you know this.

Question 18

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 {s₁ , s₂ , s₃ , and s₄} . the following table gives the dollar walue of each of the slices according to each of the four fiends .

which of the four friends is the divider?
Based on the above scenario , find a fair division of the cake using s₁, s₂, s₃ , and s₄ as fair shares, and explain why this is the only possible fair division.

Question 19

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 s4 } . The following table gives the dollar value of each of the slices according to e3ach of the four friends .

Which of the four slices are considered fair shares according to Ron?

Question 20

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?

Question 21

Jeff, Fran, Rudy are dividing an estate consisting of a house , a car and aboar using the method os sealed bids . their bids on each of the items are given in the following table .

how much dose money including surplus, dose Rudy recive/pay?

A) $33000.00
B) $45000.00
C) $71000.00
D) $213000.00
E) None of the above .

Question 22

Jackson and Sherry jointly purchased a circular 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 pepproni . which of the following statments is true ?

A) the piece shown is not a fair share for jackson.
B) the piece shown is not fair share foe Sherry.
C) the piece shown is a fair share for sherry.
D) both (a) and (b)
E) None of the above.

Question 23

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 P1 first, P2 second, and so on. Here is how the method begins:
Round 1: P1 makes the first cut; P2 passes; P3 diminishes; P4, P5, ... P6 pass; P7 diminishes; P8 passes
Round 2: After the first claim is made, P3 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 P7
B) P1, P6, and P7
C) P3, P6, and P7
D) P1, P3, and P6
E) None of the above.

Question 24

According to the scenario from problem five, which of the following would be a fair division?

A)
B)
C)
D)
E) None of the above.

Question 25

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?

A)
B)
C)
D)
E) None of the above.

Question 26

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
pieces {s1 2 3 4 5 , , , , and s s s s } . The following table gives the dollar value of each of the slices according
to each of the five friends

Which of the five slices are considered fair shares according to Bob?
According to the scenario above, which player is the divider?

A) Alice
B) Bob
C) Cathy
D) Dave
E) Earl

Question 27

John and Sara have jointly purchased a half pepperoni and half sausage pizza. They use the DividerChooser 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
E) None of the above.

Question 28

Alex, Bob and Cathy used the Lone-Chooser Method to divide up a cake. Alex and Bob are the Dividers for this method, and each of them cuts their portions into thirds allowing Cathy to make her selections. Based off of this chart, which two pieces will Bob be left with?

A) s₄ and s₅
B) s₄ and s₆
C) s₅ and s₆
D) s₄ and s₂
E) None of the above.

Question 29

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 twelve?

A) A
B) B
C) C
D) D
E) None of the above.

Question 30

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 five times as much as strawberry, and he values strawberry four times as much as vanilla. According to Jerome, what percentage of the entire cake is represented by the piece shown below?

A) 12 %
B) 30 %
C) 50 %
D) 64 %
E) None of the above.

Question 31

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}
E) None of the above.

Question 32

John and Sara have jointly purchased a half pepperoni and half sausage pizza. They use the DividerChooser 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
E) None of the above.

Question 33

Ron , Tim , Sam , and Joe are dividing an estate consisting of a house and a car using the method of fealed bides their bids on each of the items are given in the following table . how much dose each person gets in surplus money?

A) $2000.00
B) $3000.00
C) $4000.00
D) $5000.00
E) N one of the above .

Question 34

Which of the following items would best be divided among five people using a continuous method of division?

A) 1 hour of dog walking
B) multi-flavor cake
C) parcel of land in Hawaii
D) All of the above.
E) None of the above.

Question 35

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
pieces {s1 2 3 4 5 , , , , and s s s s } . The following table gives the dollar value of each of the slices according
to each of the five friends

Which of the five slices are considered fair shares according to Bob?
According to the scenario above , which of the following would be a fair division?

A) Alice : s₄ ; Bob : s₃ ; Cathy : s₂ ; Dave : s₁ ; Earl : s₅
B) Alice : s₁ ; Bob : s₃ ; Cathy : s₂ ; Dave : s₅ ; Earl : s₄
C) Alice : s₄ ; Bob : s₃ ; Cathy : s₁ ; Dave : s₅ ; Earl : s₂
D) Alice : s₂ ; Bob : s₃ ; Cathy : s₄ ; Dave : s₅ ; Earl : s₁
E) None of the above.

Question 36

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 and the order in which the game is played has P₁ first, P₂ second, and so on. Here is how the method begins:

Round 1: P₁ makes the first cut; P₂ passes; P₃ passes; P₄ diminishes; P₅ diminishes; P₆ passes
Round 2: After the first claim is made, P₃ 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)
B)
C)
D)
E) None of the above.

Question 37

Four people are going to fairly divide up a pareel of land worth $240,000.00 using the lone-divider method .
the divider partitions the land into four different pieces s1 , s2 , s3 , and s4 . accourding to the divider what is the doolar value of s3?

A) $50,000.00
B) $60,000.00
C) $70,000.00
D) $80,000.00
E) None of the above

Question 38

Four friends { Joe , Tim , Sam , and Ron } have decided to divide up a multi0flavored cake which was jointly purchased for $24 . the cake is divided into four pieces { s₁ , s₂ , s₃ , and s₄ } . the following table gives the dollar value of each of the slices according to ech of the four friends.

which of the four slices are concedered fair shares according to Ron?

A) s₁
B) s₃ and s₄
C) s₁ , s₃ and s₄
D) All pieces .
E) None of the above .