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

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?

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?

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?

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?

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 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.

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?

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?

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

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?

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?

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?

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?

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.

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?

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?

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

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?

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?

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?