Fair-Division Vocab Fair-division problems: fairly dividing something between 2 or more people.Fair-division problems: fairly dividing something between 2 or more people. Players: The people who will share the object.Players: The people who will share the object. Fair-division procedure or scheme: solution to a fair-division problemFair-division procedure or scheme: solution to a fair-division problem

3 Types of Fair-Division Problems Continuous:Continuous: Object(s) can be divided into pieces of any size with no loss of value. Object(s) can be divided into pieces of any size with no loss of value. Ex: dividing a cake or an amount of money among 2 or more people.Ex: dividing a cake or an amount of money among 2 or more people.

Types of Fair-Division, cont’d Discrete:Discrete: Object(s) will lose value if divided. Object(s) will lose value if divided. Ex: dividing a car, a house, and a boat among two or more people. Ex: dividing a car, a house, and a boat among two or more people.

Types of Fair-Division, cont’d Mixed:Mixed: Some objects can be divided and some cannot. Some objects can be divided and some cannot. Combination of continuous and discrete division. Combination of continuous and discrete division. Ex: dividing an estate consisting of money, a house, and a car among two or more people.Ex: dividing an estate consisting of money, a house, and a car among two or more people.

Question: 3 cousins will share an inheritance. It includes a house, a car, and cash. What type of fair-division problem is this? a. continuous b. discrete c. mixed

Answer: 3 cousins will share an inheritance. It includes a house, a car, and cash. What type of fair-division problem is this? c. mixed

Value of a Share In a problem with n players, a player has a fair share if he considers his share to be worth at least 1/n of the total value being shared.In a problem with n players, a player has a fair share if he considers his share to be worth at least 1/n of the total value being shared. Proportional: When every player receives a fair shareProportional: When every player receives a fair share

Assumptions For solving a division problem, assume: a player’s values cannot change based on the results of the division.a player’s values cannot change based on the results of the division. no player has any knowledge of any other player’s values.no player has any knowledge of any other player’s values.

Fair Division for 2 Players Standard procedure: divide-and-choose method.Standard procedure: divide-and-choose method. described as dividing a cake, but can be used to fairly divide any continuous object.described as dividing a cake, but can be used to fairly divide any continuous object.

Divide-And-Choose Two players, X and Y, will divide a cake.Two players, X and Y, will divide a cake. 1) Player X divides the cake into 2 pieces that he considers to be of equal value. Player X is the divider.Player X is the divider. 2) Player Y picks the piece he considers to be of greater value. Player Y is the chooser.Player Y is the chooser. 3) Player X gets the piece that’s left.

Example 1 Margo and Steven share a $4 pizza that is half pepperoni and half Hawaiian.Margo and Steven share a $4 pizza that is half pepperoni and half Hawaiian. Margo likes both kinds of pizza equally. Margo likes both kinds of pizza equally. Steven likes pepperoni 4 times as much as Hawaiian. Steven likes pepperoni 4 times as much as Hawaiian.

Example 1, cont’d What cash value would Margo and Steven each place on the original two halves of the pizza?

Example 1, cont’d The whole pizza is worth $4.The whole pizza is worth $4. Steven values pepperoni 4 times as much as Hawaiian. Steven values pepperoni 4 times as much as Hawaiian. Let x = value of Hawaiian pizza Let x = value of Hawaiian pizza x + 4x = $4 x + 4x = $4 The pepperoni half is worth 4/5 of the total value, or $3.20. The Hawaiian half is worth 1/5 of the total, or $0.80. The pepperoni half is worth 4/5 of the total value, or $3.20. The Hawaiian half is worth 1/5 of the total, or $0.80.

Example 1, cont’d Margo cuts the pizza into 6 pieces. What value would each person place on each of the two plates of pizza?

Example 1, cont’d b)The whole pizza is worth $4. Margo values both kinds of pizza equally. To her each plate of pizza is worth half of the total value, or $2. Margo values both kinds of pizza equally. To her each plate of pizza is worth half of the total value, or $2.

Example 1, cont’d b)Steven values the pepporoni half as $3.20, and the Hawaiian half as $.80 Each pepperoni slice is worth $3.20/3 = $1.067 and each Hawaiian slice is worth $0.80/3 = $0.267.Each pepperoni slice is worth $3.20/3 = $1.067 and each Hawaiian slice is worth $0.80/3 = $ The first plate is worth 2($0.267) + 1($1.067) = $1.60.The first plate is worth 2($0.267) + 1($1.067) = $1.60. The second plate is worth 1($0.267) + 2($1.067) = $2.40The second plate is worth 1($0.267) + 2($1.067) = $2.40

Fair Division for 3 Players lone-divider method. lone-divider method. The divider divides the cake into 3 pieces of equal value. The divider divides the cake into 3 pieces of equal value. The 2 choosers each decide which pieces are worth at least 1/3 of the total value. The 2 choosers each decide which pieces are worth at least 1/3 of the total value. These pieces are called acceptable. These pieces are called acceptable.

Lone-Divider Method, cont’d 3) 2 possibilities: a)If at least 1 piece is unacceptable to both choosers, the divider gets that piece. If the choosers can each choose acceptable pieces, they do. If the choosers can each choose acceptable pieces, they do. If they can’t, they put the 2 pieces back together and use the two player method to re-divide. If they can’t, they put the 2 pieces back together and use the two player method to re-divide.

Divide-And-Choose Method, cont’d b)If all 3 pieces are acceptable to both choosers, they each take a piece. The divider gets the other one.

Question: The divide-and-choose method for 3 players is being used to divide a pizza. Player A cut a pizza into 3 equal shares. Player B thinks shares 2 and 3 are acceptable. Player C thinks that only share 2 is acceptable. What is the fair division?

Answer: The divide-and-choose method for 3 players is being used to divide a pizza. Player A cut a pizza into 3 equal shares. Player B thinks shares 2 and 3 are acceptable. Player C thinks that only share 2 is acceptable. Player C gets share 2, Player B gets share 3, Player A gets share 1.

Example 3 Emma, Fay, and Grace will divide 24oz of ice cream, made up of equal amounts of vanilla, chocolate, and strawberry.Emma, Fay, and Grace will divide 24oz of ice cream, made up of equal amounts of vanilla, chocolate, and strawberry. Emma likes the 3 flavors equally well. Emma likes the 3 flavors equally well. Fay prefers chocolate 2 to 1 over either other flavor, but likes vanilla and strawberry equally well. Fay prefers chocolate 2 to 1 over either other flavor, but likes vanilla and strawberry equally well. Grace prefers vanilla to chocolate to strawberry in the ratio 1 to 2 to 3. Grace prefers vanilla to chocolate to strawberry in the ratio 1 to 2 to 3. If Emma is the divider, what are the results of the divide-and-choose method for 3 players?If Emma is the divider, what are the results of the divide-and-choose method for 3 players?

Example 3, cont’d Suppose Emma divides the ice cream into 3 equal parts, each consisting of one of the flavors.Suppose Emma divides the ice cream into 3 equal parts, each consisting of one of the flavors.

Example 3, cont’d Faye finds portions 1 and 3 unacceptable.Faye finds portions 1 and 3 unacceptable.

Example 3, cont’d Grace finds portion 1 unacceptableGrace finds portion 1 unacceptable

Example 3, cont’d All of the players’ values are summarized in the table below.All of the players’ values are summarized in the table below.

Example 3, cont’d Solution, cont’d:Solution, cont’d: Portion 1 is unacceptable to both Fay and Grace. As the divider, Emma will receive portion 1. Portion 1 is unacceptable to both Fay and Grace. As the divider, Emma will receive portion 1. Only portion 2 is acceptable to Faye. Only portion 2 is acceptable to Faye. Portions 2 and 3 are acceptable to Grace. Portions 2 and 3 are acceptable to Grace. The division is Emma: portion 1; Fay: portion 2; Grace: portion 3.The division is Emma: portion 1; Fay: portion 2; Grace: portion 3.

Homework Pg. 223 Pg , 8, 20, 28 2, 8, 20, 28