Nossi Ch 4.1 and 4.2

A Mathematical View A Mathematical View of Our Worldof Our World

11stst ed. ed.

Parks, Musser, Trimpe, Parks, Musser, Trimpe, Maurer, and MaurerMaurer, and Maurer

Chapter 4Chapter 4

Fair DivisionFair Division

Section 4.1Section 4.1

Divide and Choose MethodsDivide and Choose Methods• GoalsGoals

• Study fair-division problemsStudy fair-division problems• Continuous fair divisionContinuous fair division• Discrete fair divisionDiscrete fair division• Mixed fair divisionMixed fair division

• Study fair-division proceduresStudy fair-division procedures• Divide-and-choose method for 2 playersDivide-and-choose method for 2 players• Divide-and-choose method for 3 playersDivide-and-choose method for 3 players• Last-diminisher method for 3 or more playersLast-diminisher method for 3 or more players

4.1 Initial Problem4.1 Initial Problem• The brothers Drewvan, Oswald, and The brothers Drewvan, Oswald, and

Granger are to share their family’s 3600-Granger are to share their family’s 3600-acre estate.acre estate.

• Drewvan:Drewvan:• Values vineyards three times as much as fields.Values vineyards three times as much as fields.

• Values woodlands twice as much as fields.Values woodlands twice as much as fields.

• Oswald:Oswald:• Values vineyards twice as much as fields.Values vineyards twice as much as fields.

• Values woodlands three times as much as fields.Values woodlands three times as much as fields.

4.1 Initial Problem, cont’d4.1 Initial Problem, cont’d

• Granger:Granger:• values vineyards twice as much as fields.values vineyards twice as much as fields.

• Values fields three times as much as woodlands.Values fields three times as much as woodlands.

• How can the brothers fairly divide the How can the brothers fairly divide the estate?estate?• The solution will be given at the end of the The solution will be given at the end of the

section.section.

Fair-Division ProblemsFair-Division Problems• Fair-division problemsFair-division problems involve fairly dividing involve fairly dividing

something between two or more people, something between two or more people, without the aid of an outside arbitrator.without the aid of an outside arbitrator.

• The people who will share the object are The people who will share the object are called called playersplayers..

• The solution to a problem is called a The solution to a problem is called a fair-fair-division proceduredivision procedure or a or a fair-division schemefair-division scheme. .

Types of Fair-Division ProblemsTypes of Fair-Division Problems

• ContinuousContinuous fair-division problems: fair-division problems: • The object(s) can be divided into pieces of The object(s) can be divided into pieces of

any size with no loss of value.any size with no loss of value.• An example is dividing a cake or brownies An example is dividing a cake or brownies

(like we did in class) or an amount of money (like we did in class) or an amount of money among two or more people.among two or more people.

Types of Fair-Division, cont’dTypes of Fair-Division, cont’d

• DiscreteDiscrete fair-division problems: fair-division problems: • The object(s) will lose value if divided.The object(s) will lose value if divided.

• We assume the players do not want to sell We assume the players do not want to sell everything and divide the proceeds.everything and divide the proceeds.

• However, sometimes money must be used However, sometimes money must be used when no other fair division is possiblewhen no other fair division is possible

• An example is dividing a car, a house, and a An example is dividing a car, a house, and a boat among two or more people.boat among two or more people.


• MixedMixed fair-division problems: fair-division problems: • Some objects to be shared can be divided Some objects to be shared can be divided

and some cannot.and some cannot.

• This type is a combination of continuous This type is a combination of continuous and discrete fair division.and discrete fair division.

• An example is dividing an estate consisting of An example is dividing an estate consisting of money, a house, and a car among two or money, a house, and a car among two or more people.more people.

Question:Question:

Three cousins will share an Three cousins will share an inheritance. The estate includes a inheritance. The estate includes a house, a car, and cash. What type of house, a car, and cash. What type of fair-division problem is this?fair-division problem is this?

a. continuous a. continuous

b. discreteb. discrete

c. mixedc. mixed


• This section will consider only This section will consider only continuous fair-division problems.continuous fair-division problems.

• We make the assumption that the value We make the assumption that the value of a player’s share is determined by his of a player’s share is determined by his or her values.or her values.• Different players may value the same Different players may value the same

share differently. share differently.

Value of a ShareValue of a Share

• In a fair-division problem with In a fair-division problem with nn players, players, a player has received a a player has received a fair sharefair share if that if that player considers his or her share to be player considers his or her share to be worth at least 1/worth at least 1/nn of the total value of the total value being shared. being shared.

• A division that results in every player A division that results in every player receiving a fair share is called receiving a fair share is called proportionalproportional..

Value of a Share, cont’dValue of a Share, cont’d

• We assume that a player’s values in a We assume that a player’s values in a fair-division problem cannot change fair-division problem cannot change based on the results of the division. based on the results of the division.

• We also assume that no player has any We also assume that no player has any knowledge of any other player’s values.knowledge of any other player’s values.

Fair Division for Two PlayersFair Division for Two Players

• The standard procedure for a The standard procedure for a continuous fair-division problem with continuous fair-division problem with two players is called the two players is called the

divide-and-choose methoddivide-and-choose method..• This method is described as dividing a This method is described as dividing a

cake, but it can be used to fairly divide cake, but it can be used to fairly divide any continuous object. any continuous object.

Divide-And-Choose MethodDivide-And-Choose Method• Two players, X and Y, are to divide a cake.Two players, X and Y, are to divide a cake.

1)1) Player X divides the cake into 2 pieces that he Player X divides the cake into 2 pieces that he or she considers to be of equal value.or she considers to be of equal value.

• Player X is called the Player X is called the dividerdivider..

2)2) Player Y picks the piece he or she considers Player Y picks the piece he or she considers to be of greater value.to be of greater value.

• Player Y is called the Player Y is called the chooserchooser..

3)3) Player X gets the piece that player Y did not Player X gets the piece that player Y did not choose. choose.

Divide-And-Choose Method, cont’dDivide-And-Choose Method, cont’d

• This method produces a proportional This method produces a proportional division.division.

• The divider thinks both pieces are equal, The divider thinks both pieces are equal, so the divider gets a fair share.so the divider gets a fair share.

• The chooser will find at least one of the The chooser will find at least one of the pieces to be a fair share or more than a pieces to be a fair share or more than a fair share. The chooser selects that fair share. The chooser selects that piece, and gets a fair share. piece, and gets a fair share.

Example 1Example 1

• Margo and Steven will share a $4 Margo and Steven will share a $4 pizza that is half pepperoni and half pizza that is half pepperoni and half Hawaiian.Hawaiian.

• Margo likes both kinds of pizza equally.Margo likes both kinds of pizza equally.

• Steven likes pepperoni 4 times as much Steven likes pepperoni 4 times as much as Hawaiian.as Hawaiian.

Example 1, cont’dExample 1, cont’d• Margo cuts the Margo cuts the

pizza into 6 pieces pizza into 6 pieces and arranges them and arranges them as shown.as shown.

a)a) What monetary What monetary value would Margo value would Margo and Steven each and Steven each place on the original place on the original two halves of the two halves of the pizza?pizza?

Example 1, cont’dExample 1, cont’d• Solution: The whole pizza is worth $4.Solution: The whole pizza is worth $4.

• Margo values both kinds of pizza equally. To Margo values both kinds of pizza equally. To her each half is worth half of the total value, or her each half is worth half of the total value, or $2.$2.

• Steven values pepperoni 4 times as much as Steven values pepperoni 4 times as much as Hawaiian. To him the pepperoni half is worth Hawaiian. To him the pepperoni half is worth 4/5 of the total value, or 4/5($4) = $3.20. The 4/5 of the total value, or 4/5($4) = $3.20. The Hawaiian half is worth 1/5($4) of the total, or Hawaiian half is worth 1/5($4) of the total, or $0.80.$0.80.

Example 1, cont’dExample 1, cont’d

b)b) What value What value would each would each person place on person place on each of the two each of the two plates of pizza?plates of pizza?

c)c) What plate will What plate will Steven choose?Steven choose?


b)b) Solution: The whole pizza is worth $4.Solution: The whole pizza is worth $4.• Margo values both kinds of pizza Margo values both kinds of pizza

equally. To her each plate of pizza is equally. To her each plate of pizza is worth half of the total value, or $2.worth half of the total value, or $2.

Example 1, cont’dExample 1, cont’db)b) Solution, cont’d:Solution, cont’d:

• Steven values pepperoni 4 times as Steven values pepperoni 4 times as much as Hawaiian. much as Hawaiian.

• To him each pepperoni slice is worth To him each pepperoni slice is worth $3.20/3 = $1.067 and each Hawaiian slice is $3.20/3 = $1.067 and each Hawaiian slice is worth $0.80/3 = $0.267.worth $0.80/3 = $0.267.

• The first plate is worth 2($0.267) + The first plate is worth 2($0.267) + 1($1.067) = $1.60.1($1.067) = $1.60.

• The second plate is worth 1($0.267) + The second plate is worth 1($0.267) + 2($1.067) = $2.402($1.067) = $2.40

Example 1, cont’dExample 1, cont’dc)c) Solution:Solution:

• Steven will choose the second plate, Steven will choose the second plate, with one slice of Hawaiian and two slices with one slice of Hawaiian and two slices of pepperoni.of pepperoni.

• Margo gets a plate of pizza that she Margo gets a plate of pizza that she feels is worth half the value. feels is worth half the value.

• Steven gets a plate of pizza that he feels Steven gets a plate of pizza that he feels is worth more than half the value.is worth more than half the value.

Example 2Example 2

• Caleb and Diego will drive 6 hours during Caleb and Diego will drive 6 hours during the day and 4 hours at night.the day and 4 hours at night.

• Caleb prefers night to day driving 2 to 1.Caleb prefers night to day driving 2 to 1.• Diego prefers them equally, or 1 to 1.Diego prefers them equally, or 1 to 1.• How should they divide the driving into 2 How should they divide the driving into 2

shifts if Caleb is the divider and Diego is shifts if Caleb is the divider and Diego is the chooser?the chooser?


• Solution:Solution:• Caleb can assign 2 points to each hour Caleb can assign 2 points to each hour

of night driving and 1 point to each hour of night driving and 1 point to each hour of day driving.of day driving.

• Caleb values the entire drive at 1(6) + 2(4) Caleb values the entire drive at 1(6) + 2(4) = 14 points.= 14 points.

• To Caleb a fair share will be worth half the To Caleb a fair share will be worth half the total value, or 7 points.total value, or 7 points.


• Solution, cont’d:Solution, cont’d:• A possible fair division for Caleb is to create A possible fair division for Caleb is to create

shifts of:shifts of:• 6 hours of daytime driving and 0.5 hours of 6 hours of daytime driving and 0.5 hours of

nighttime driving.nighttime driving.• 3.5 hours of nighttime driving.3.5 hours of nighttime driving.• Both shifts are worth 7 points to Caleb.Both shifts are worth 7 points to Caleb.• AnyAny combination totaling 7points is fair to Caleb. combination totaling 7points is fair to Caleb.


• Solution, cont’d:Solution, cont’d:• Diego can assign 1 point to each hour of Diego can assign 1 point to each hour of

night driving and 1 point to each hour of night driving and 1 point to each hour of day driving.day driving.

• Diego values the entire drive at 1(6) + 1(4) Diego values the entire drive at 1(6) + 1(4) = 10 points.= 10 points.

• To Diego a fair share will be worth half the To Diego a fair share will be worth half the total value, or 5 points.total value, or 5 points.

Example 2, cont’dExample 2, cont’d• Solution, cont’d:Solution, cont’d:

• Diego values the first shift at 1(6) + 1(0.5) = Diego values the first shift at 1(6) + 1(0.5) = 6.5 points.6.5 points.

• Diego values the second shift at 1(3.5) = 3.5 Diego values the second shift at 1(3.5) = 3.5 points.points.

• Diego will choose the first shift, because it is Diego will choose the first shift, because it is worth more to him. Caleb came up with the worth more to him. Caleb came up with the shits and Diego chose the one he valued shits and Diego chose the one he valued most.most.

Two Players, cont’dTwo Players, cont’d• Notice that in both of the previous Notice that in both of the previous

examples:examples:• The divider got a share he or she felt was The divider got a share he or she felt was

equal to exactly half of the total value.equal to exactly half of the total value.

• The chooser got a share he or she felt was The chooser got a share he or she felt was equal to more than half of the total value. equal to more than half of the total value.

• It is often advantageous to be the chooser, It is often advantageous to be the chooser, so the roles should be randomly chosen.so the roles should be randomly chosen.

Fair Division for Three PlayersFair Division for Three Players

• In a continuous fair-division problem In a continuous fair-division problem with 3 players, it is still possible to with 3 players, it is still possible to have one player divide the object and have one player divide the object and the other players choose.the other players choose.

• This method is also called theThis method is also called the

lone-divider methodlone-divider method. .

Divide-And-Choose MethodDivide-And-Choose Method• Three players, X, Y, and Z are to divide a cake.Three players, X, Y, and Z are to divide a cake.

1)1) Player X (the divider) divides the cake into 3 Player X (the divider) divides the cake into 3 pieces that he/she considers to be of equal pieces that he/she considers to be of equal value.value.

2)2) Players Y and Z (the choosers) each decide Players Y and Z (the choosers) each decide which pieces are worth at least 1/3 of the total which pieces are worth at least 1/3 of the total valuevalue..

• These pieces are said to be acceptable.These pieces are said to be acceptable.

3)3) The choosers announce their acceptable The choosers announce their acceptable piecespieces..


3)3) There are 2 possibilities:There are 2 possibilities:

a)a) If at least 1 piece is unacceptable to both Y If at least 1 piece is unacceptable to both Y and Z, Player X gets that piece. and Z, Player X gets that piece.

• If Y and Z can each choose acceptable If Y and Z can each choose acceptable pieces, they do so.pieces, they do so.

• If Y and Z cannot each choose acceptable If Y and Z cannot each choose acceptable pieces, they put the remaining pieces pieces, they put the remaining pieces back together and use the two player back together and use the two player method to re-divide.method to re-divide.


3)3) Cont’d:Cont’d:

b)b) If every piece is acceptable to both Y If every piece is acceptable to both Y and Z, they each take an acceptable and Z, they each take an acceptable piece. Player X gets the leftover piece. piece. Player X gets the leftover piece.

• Note: The divide-and-choose method can Note: The divide-and-choose method can be extended to more than 3 players. The be extended to more than 3 players. The more players, the more complicated the more players, the more complicated the process becomes.process becomes.

Example 4.3Example 4.3• Emma, Fay, and Grace will divide 24 ounces of Emma, Fay, and Grace will divide 24 ounces of

ice cream, which is made up of equal amounts of ice cream, which is made up of equal amounts of vanilla, chocolate, and strawberry.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 Fay prefers chocolate 2 to 1 over either other flavor and prefers vanilla and strawberry flavor and prefers vanilla and strawberry equally well.equally well.

• Grace prefers vanilla to chocolate to Grace prefers vanilla to chocolate to strawberry in the ratio 1 to 2 to 3. strawberry in the ratio 1 to 2 to 3.

• If Emma is the divider, what are the results of the If Emma is the divider, what are the results of the divide-and-choose method for 3 players?divide-and-choose method for 3 players?

Example 4.3, cont’dExample 4.3, cont’d• Solution: Suppose Emma divides the ice cream Solution: Suppose Emma divides the ice cream

into 3 equal parts, each consisting of one of the into 3 equal parts, each consisting of one of the flavors. flavors.

Example 4.3, cont’dExample 4.3, cont’d• Solution, cont’d: Fay is one of the choosers. Solution, cont’d: Fay is one of the choosers.

• Faye finds portions 1 and 3 unacceptable.Faye finds portions 1 and 3 unacceptable.

Example 4.3, cont’dExample 4.3, cont’d• Solution, cont’d: Grace is the other chooser. Solution, cont’d: Grace is the other chooser.

• She finds portion 1 unacceptableShe finds portion 1 unacceptable

Example 4.3, cont’dExample 4.3, cont’d• Solution, cont’d: All of the players’ values are Solution, cont’d: All of the players’ values are

summarized in the table below.summarized in the table below.

Example 4.3, cont’dExample 4.3, cont’d• Solution, cont’d: Solution, cont’d:

• Portion 1 is unacceptable to both Fay and Portion 1 is unacceptable to both Fay and Grace. As the divider, Emma will receive Grace. As the divider, Emma will receive portion 1.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: The division is Emma: portion 1; Fay: portion 2; Grace: portion 3.portion 2; Grace: portion 3.

Last-Diminisher MethodLast-Diminisher Method

• A method for continuous fair-division A method for continuous fair-division problems with 3 or more players is called problems with 3 or more players is called the the last-diminisher methodlast-diminisher method..

• Suppose any number of players X, Y, … Suppose any number of players X, Y, … are dividing a cake. are dividing a cake.

1)1) Player X cuts a piece of cake that he or she Player X cuts a piece of cake that he or she considers to be a fair share.considers to be a fair share.

Last-Diminisher Method, cont’dLast-Diminisher Method, cont’d

2)2) Each player, in turn, judges the fairness Each player, in turn, judges the fairness of the piece.of the piece.

a)a) If a player considers the piece fair or less If a player considers the piece fair or less than fair, it passes to the next player.than fair, it passes to the next player.

b)b) If a player considers the piece more than If a player considers the piece more than fair, the player trims the piece to make it fair, the player trims the piece to make it fair, returning the trimming to the undivided fair, returning the trimming to the undivided portion and passing the trimmed piece to portion and passing the trimmed piece to the next player.the next player.

Last-Diminisher Method, cont’dLast-Diminisher Method, cont’d

3)3) The last player to trim the piece, gets the The last player to trim the piece, gets the piece as his or her share.piece as his or her share.

• If no player trimmed the piece, player X gets If no player trimmed the piece, player X gets the piece.the piece.

4)4) After one player gets a piece of cake, After one player gets a piece of cake, the process begins again without that the process begins again without that player and that piece.player and that piece.

• When only 2 players remain, they use the When only 2 players remain, they use the divide-and-choose method.divide-and-choose method.

Example 4.4Example 4.4• Hector, Isaac, and James will divide 24 Hector, Isaac, and James will divide 24

ounces of ice cream, which is equal parts ounces of ice cream, which is equal parts vanilla, chocolate, and strawberry.vanilla, chocolate, and strawberry.

• Hector values vanilla to chocolate to Hector values vanilla to chocolate to strawberry 1 to 2 to 3.strawberry 1 to 2 to 3.

• Isaac likes the 3 flavors equally.Isaac likes the 3 flavors equally.

• James values vanilla to chocolate to James values vanilla to chocolate to strawberry 1 to 2 to 1.strawberry 1 to 2 to 1.

Example 4.4, cont’dExample 4.4, cont’d

• Using the last-diminisher method with Using the last-diminisher method with Hector as the first divider and Isaac as the Hector as the first divider and Isaac as the first judge, find the results of the division.first judge, find the results of the division.

• Solution: Solution: • Hector assigns 1 point to each ounce of Hector assigns 1 point to each ounce of

vanilla, 2 points to each ounce of chocolate, vanilla, 2 points to each ounce of chocolate, and 3 points to each ounce of strawberry. and 3 points to each ounce of strawberry.

Example 4.4, cont’dExample 4.4, cont’d• Solution, cont’d: A fair share of ice cream, Solution, cont’d: A fair share of ice cream,

to Hector, is worth 48/3 = 16 points.to Hector, is worth 48/3 = 16 points.

Example 4.4, cont’dExample 4.4, cont’d• Solution, cont’d:Solution, cont’d:

• One possible fair share for Hector would One possible fair share for Hector would be all 8 ounces of vanilla plus 4 ounces be all 8 ounces of vanilla plus 4 ounces of chocolate.of chocolate.

• This share is worth 1(8) + 2(4) = 16 This share is worth 1(8) + 2(4) = 16 points to Hector, so he would be happy points to Hector, so he would be happy with this share. with this share.

• Next, Isaac must decide whether the Next, Isaac must decide whether the share is fair, according to his values.share is fair, according to his values.


• Isaac assigns 1 point to each ounce Isaac assigns 1 point to each ounce of vanilla, 1 point to each ounce of of vanilla, 1 point to each ounce of chocolate, and 1 point to each chocolate, and 1 point to each ounce of strawberry. ounce of strawberry.

• Isaac values all of the ice cream at Isaac values all of the ice cream at 1(8) + 1(8) + 1(8) = 24 points. 1(8) + 1(8) + 1(8) = 24 points.

• A fair share to Isaac is 8 points.A fair share to Isaac is 8 points.


• Isaac’s value for Hector’s serving is 1(8) Isaac’s value for Hector’s serving is 1(8) + 1(4) = 12 points.+ 1(4) = 12 points.

• Isaac thinks it is more than a fair share.Isaac thinks it is more than a fair share.

• Isaac trims off 4 points worth of ice Isaac trims off 4 points worth of ice cream.cream.

• Suppose he trims off the 4 ounces of Suppose he trims off the 4 ounces of chocolate.chocolate.


• Next, James must judge the share.Next, James must judge the share.

• James assigns 1 point to each ounce of James assigns 1 point to each ounce of vanilla, 2 points to each ounce of vanilla, 2 points to each ounce of chocolate, and 1 point to each ounce of chocolate, and 1 point to each ounce of strawberry. strawberry.

• James values all of the ice cream at 1(8) James values all of the ice cream at 1(8) + 2(8) + 1(8) = 32 points.+ 2(8) + 1(8) = 32 points.

• A fair share to James is worth 32/3 points.A fair share to James is worth 32/3 points.


• The existing share is now just 8 The existing share is now just 8 ounce of vanilla.ounce of vanilla.

• To James, the share is worth 1(8) = To James, the share is worth 1(8) = 8 points.8 points.

• James thinks this is less than a fair James thinks this is less than a fair share.share.

• James will not trim the share.James will not trim the share.


• Isaac was the last-diminisher, and gets Isaac was the last-diminisher, and gets the share of ice cream.the share of ice cream.

• Hector and James will divide the Hector and James will divide the remaining ice cream using the divide-remaining ice cream using the divide-and-choose method. and-choose method.

• Note: This is only one of many different Note: This is only one of many different

possible solutions.possible solutions.

4.1 Initial Problem Solution4.1 Initial Problem Solution• The brothers Drewvan, Oswald, and Granger are The brothers Drewvan, Oswald, and Granger are

to share their family’s estate, which is 1200 acres to share their family’s estate, which is 1200 acres each of vineyards, woodlands, and fields.each of vineyards, woodlands, and fields.

• Drewvan prefers vineyards to woodlands to fields Drewvan prefers vineyards to woodlands to fields 3 to 2 to 1.3 to 2 to 1.

• Oswald prefers vineyards to woodlands to fields Oswald prefers vineyards to woodlands to fields 2 to 3 to 1.2 to 3 to 1.

• Granger prefers vineyards to woodlands to fields Granger prefers vineyards to woodlands to fields 2 to 1 to 3.2 to 1 to 3.

Initial Problem Solution, cont’dInitial Problem Solution, cont’d

• Use the divide-and-choose method for 3 Use the divide-and-choose method for 3 players.players.• Let Drewvan be the divider.Let Drewvan be the divider.

Initial Problem Solution, cont’dInitial Problem Solution, cont’d• Drewvan values the entire estate at 7200 points.Drewvan values the entire estate at 7200 points.


• To Drewvan, a fair share is worth 7200/3 = To Drewvan, a fair share is worth 7200/3 = 2400 points.2400 points.

• One possible fair division is shown below.One possible fair division is shown below.


• Next, the two choosers will consider this Next, the two choosers will consider this division.division.

• Granger and Oswald both value the Granger and Oswald both value the entire estate at 7200 points also.entire estate at 7200 points also.• To Oswald, a fair share is worth 2400 To Oswald, a fair share is worth 2400

points.points.

• To Granger, a fair share is worth 2400 To Granger, a fair share is worth 2400 points.points.


• Oswald considers piece 1 to be unacceptable. Oswald considers piece 1 to be unacceptable.


• Granger considers pieces 1 and 2 to be Granger considers pieces 1 and 2 to be unacceptable. unacceptable.


• Both choosers think piece 1 is Both choosers think piece 1 is unacceptable, so Drewvan gets piece 1.unacceptable, so Drewvan gets piece 1.

• Granger thinks only piece 3 is Granger thinks only piece 3 is acceptable, so he gets that piece.acceptable, so he gets that piece.

• Oswald thinks pieces 2 and 3 are Oswald thinks pieces 2 and 3 are acceptable, so Oswald gets piece 2.acceptable, so Oswald gets piece 2.

Section 4.1 Assignment

• Pg 223 (1,5,6,7,9,19,31,37)

Section 4.2Section 4.2

Discrete and Mixed Discrete and Mixed

Division ProblemsDivision Problems• GoalsGoals

• Study discrete fair-division problemsStudy discrete fair-division problems• The method of sealed bids (We will only study this The method of sealed bids (We will only study this

one)one)• The method of pointsThe method of points

• Study mixed fair-division problemsStudy mixed fair-division problems• The adjusted winner procedureThe adjusted winner procedure

Discrete Fair DivisionDiscrete Fair Division

• Recall that discrete fair division problems Recall that discrete fair division problems involve sharing objects that cannot be involve sharing objects that cannot be divided without losing value. (Like cars, divided without losing value. (Like cars, boats, houses, etc.)boats, houses, etc.)

• Two methods for solving discrete fair-Two methods for solving discrete fair-division problems are:division problems are:• The method of sealed bids. (This is the one we The method of sealed bids. (This is the one we

will study.)will study.)

• The method of points.The method of points.

Method of Sealed BidsMethod of Sealed Bids• Any number of players, Any number of players, NN, are to share , are to share

any number of items.any number of items.• If necessary, money will be used to insure If necessary, money will be used to insure

fairness.fairness.

1)1) All players submit sealed bids, stating All players submit sealed bids, stating monetary values for each item.monetary values for each item.

2)2) Each item goes to the highest bidder.Each item goes to the highest bidder.• The highest bidder places the dollar amount of The highest bidder places the dollar amount of

his or her bid into a compensation fund.his or her bid into a compensation fund.

Method of Sealed Bids, cont’dMethod of Sealed Bids, cont’d

3)3) From the compensation fund, each From the compensation fund, each player receives 1/player receives 1/NN of his or her bid of his or her bid on each item.on each item.

4)4) Any money leftover in the fund is Any money leftover in the fund is distributed equally to all players. distributed equally to all players.

• Note: This method is also called the Note: This method is also called the Knaster Inheritance Procedure.Knaster Inheritance Procedure.

Method of Sealed BidsMethod of Sealed Bids

We will look at the example from class: We will look at the example from class: (The corrected procedure is here. The (The corrected procedure is here. The problem was that the 2 bids were too far problem was that the 2 bids were too far apart. Following this example is a simper apart. Following this example is a simper one.)one.)

Jason and Jose bid on a car.Jason and Jose bid on a car. Jason bid $200,000Jason bid $200,000 Jose bid $70,000Jose bid $70,000


We will look at the example from class:We will look at the example from class:

Jason and Jose bid on a car.Jason and Jose bid on a car. Jason bid $200,000Jason bid $200,000 Jose bid $70,000Jose bid $70,000

Jason gets the car and puts $200,000 in a Jason gets the car and puts $200,000 in a Compensation fund.Compensation fund.


Loser (Jose) gets 1/2 his bid = $35,000 Loser (Jose) gets 1/2 his bid = $35,000 paid out of the Compensation fund.paid out of the Compensation fund.


Winner (Jason) gets 1/2 of his bid back out of the Winner (Jason) gets 1/2 of his bid back out of the Compensation fund = $100,000.Compensation fund = $100,000.


Total paid out of the Compensation fund isTotal paid out of the Compensation fund is$100,000 + $35,000 = $135,000.$100,000 + $35,000 = $135,000.

There is $200,000- $135,000 = $70,000 left in the There is $200,000- $135,000 = $70,000 left in the Compensation fund.Compensation fund.

This amount is split evenly between the two.This amount is split evenly between the two.


Jason bid $200,000 and gets backJason bid $200,000 and gets back 1/2 ($200,000) + 1/2 ($65,000) = $132,500 from 1/2 ($200,000) + 1/2 ($65,000) = $132,500 from

the compensation fund the compensation fund andand gets the car. gets the car.

Jose bid $70,000 and gets paid 1/2 ($70,000) + Jose bid $70,000 and gets paid 1/2 ($70,000) + 1/2 ($65,000) = $67,500.1/2 ($65,000) = $67,500.

(Jason buys out Jose’s half of the car for $67,500)(Jason buys out Jose’s half of the car for $67,500)


This was not a very good example because the two This was not a very good example because the two bids were far apart. The incentive behind this bids were far apart. The incentive behind this method is to encourage the two to make fair bids method is to encourage the two to make fair bids by researching the value of the item.by researching the value of the item.

Here is a better example:Here is a better example:Amanda and Blake inherit a car. After carefully Amanda and Blake inherit a car. After carefully

studying the issue, Amanda bids $2900. Blake studying the issue, Amanda bids $2900. Blake also studies and bids $3100.also studies and bids $3100.

Who wins the car according to the Sealed Bids Who wins the car according to the Sealed Bids method?method?


The car goes to the highest bidder: The car goes to the highest bidder: BlakeBlake


Blake also put $3100 in a Compensation Fund.Blake also put $3100 in a Compensation Fund.

Blake gets back 1/2 his bid:Blake gets back 1/2 his bid:1/2 of $3100 = $15501/2 of $3100 = $1550

Amanda gets 1/2 of her bid:Amanda gets 1/2 of her bid:1/2 of $2900 = $1450 to be paid out of the 1/2 of $2900 = $1450 to be paid out of the compensation fund.compensation fund.

Total paid out of the Compensation Fund:Total paid out of the Compensation Fund:$1550 + $1450 = $3000$1550 + $1450 = $3000


That leaves $100 left in the Compensation Fund That leaves $100 left in the Compensation Fund

The $100 is split evenly between Blake and The $100 is split evenly between Blake and Amanda.Amanda.

Amanda ends up with $1450 + 50 = $1500Amanda ends up with $1450 + 50 = $1500

Blake ends up with the car and he only paid $1500 Blake ends up with the car and he only paid $1500 which is less than 1/2 of his bid.which is less than 1/2 of his bid.


An alternate method when there are only 2 bidding on one An alternate method when there are only 2 bidding on one item:item:

Blake bids $3100Blake bids $3100Amanda bids $2900Amanda bids $2900

The average of the two: ($3100+$2900)/2 = The average of the two: ($3100+$2900)/2 = ($6,000)/2 =($6,000)/2 = $3000$3000

Blake pays Amanda 1/2 (3,000) = $1,500 which is the same as Blake pays Amanda 1/2 (3,000) = $1,500 which is the same as we got by the Sealed Bids method.we got by the Sealed Bids method.

Example 4.6 from pg 237Example 4.6 from pg 237• Three sisters Maura, Nessa, and Odelia Three sisters Maura, Nessa, and Odelia

will share a house and a cottage. will share a house and a cottage. • Apply the method of sealed bids to divide Apply the method of sealed bids to divide

the property, using the bids shown below.the property, using the bids shown below.


• Solution: Each piece of property goes to Solution: Each piece of property goes to the highest bidder.the highest bidder.

• Odelia gets the family home and places Odelia gets the family home and places $301,000 into the compensation fund.$301,000 into the compensation fund.

• Nessa gets the cottage and places $203,000 Nessa gets the cottage and places $203,000 into the compensation fund.into the compensation fund.


• Solution, cont’d: The compensation Solution, cont’d: The compensation fund now contains a total of fund now contains a total of

• $203,000 + $301,000 = $504,000.$203,000 + $301,000 = $504,000.


• Solution, cont’d: Each sister receives 1/3 Solution, cont’d: Each sister receives 1/3 of her total bids from the compensation of her total bids from the compensation fund.fund.

• Maura receivesMaura receives

• Nessa receivesNessa receives

• Odelia receivesOdelia receives

286000 203000$163,000

3

+=

289000 188000$159,000

3

+=

301000 182000$161,000

3

+=


• Solution, cont’d: After the Solution, cont’d: After the distributions, there is distributions, there is

$504,000 – ($159,000 + $160,000 + $504,000 – ($159,000 + $160,000 + $161,000) = $21,000 left in the fund.$161,000) = $21,000 left in the fund.

• The leftover money is distributed The leftover money is distributed equally to the three sisters in the equally to the three sisters in the amount of $7000 each.amount of $7000 each.

Example 4.6, cont’dExample 4.6, cont’d• Solution, cont’d: The final shares Solution, cont’d: The final shares

are:are:

• Maura receives $166,000 and no Maura receives $166,000 and no property.property.

• ($159,000+$7,000=$166,000)($159,000+$7,000=$166,000)

Example 4.6, cont’dExample 4.6, cont’d• Solution, cont’d: The final shares Solution, cont’d: The final shares

are:are:

Nessa receives the cottage, for which she Nessa receives the cottage, for which she paid a net amount of $33,000. paid a net amount of $33,000.

(Nessa paid in $203,000 and got back (Nessa paid in $203,000 and got back $163,000+$7,000 = $170,000$163,000+$7,000 = $170,000

Net: $203,000 - $170,000 = $33,000)Net: $203,000 - $170,000 = $33,000)

Example 4.6, cont’dExample 4.6, cont’d• Solution, cont’d: The final shares are:Solution, cont’d: The final shares are:

• Odelia receives the family home, for which Odelia receives the family home, for which she paid a net amount of $133,000.she paid a net amount of $133,000.

She paid in $301,000 and got back:She paid in $301,000 and got back:

$161,000 + $7,000 = $168,000.$161,000 + $7,000 = $168,000.

$301,000 - $168,000 = $133,000$301,000 - $168,000 = $133,000

Example 4.6, cont’dExample 4.6, cont’d• Solution, cont’d: Note that the division is Solution, cont’d: Note that the division is

proportional because each sister receives what proportional because each sister receives what she considers to be a fair share. she considers to be a fair share.


• Pg 249 (9,11,12)


• Pg 249 (9,11,12,15,19,21,27)


•Pg 223 (1,5,6,7,9,19,31,37)

Nossi Ch 4.1 and 4.2

Technology

Transcript of Nossi Ch 4.1 and 4.2