A Mathematical View of Our World

1
A Mathematical View of Our World

• 1st ed.
• Parks, Musser, Trimpe, Maurer, and Maurer

2
Chapter 4

• Fair Division

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Section 4.1Divide and Choose Methods

• Goals
• Study fair-division problems
• Continuous fair division
• Discrete fair division
• Mixed fair division
• Study fair-division procedures
• Divide-and-choose method for 2 players
• Divide-and-choose method for 3 players
• Last-diminisher method for 3 or more players

4
4.1 Initial Problem

• The brothers Drewvan, Oswald, and Granger are to
  share their familys 3600-acre estate.
• Drewvan
• Values vineyards three times as much as fields.
• Values woodlands twice as much as fields.
• Oswald
• Values vineyards twice as much as fields.
• Values woodlands three times as much as fields.

5
4.1 Initial Problem, contd

• Granger
• values vineyards twice as much as fields.
• Values fields three times as much as woodlands.
• How can the brothers fairly divide the estate?
• The solution will be given at the end of the
  section.

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Fair-Division Problems

• Fair-division problems involve fairly dividing
  something between two or more people, without the
  aid of an outside arbitrator.
• The people who will share the object are called
  players.
• The solution to a problem is called a
  fair-division procedure or a fair-division
  scheme.

7
Types of Fair-Division Problems

• Continuous fair-division problems
• The object(s) can be divided into pieces of any
  size with no loss of value.
• An example is dividing a cake or an amount of
  money among two or more people.

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Types of Fair-Division, contd

• Discrete fair-division problems
• The object(s) will lose value if divided.
• We assume the players do not want to sell
  everything and divide the proceeds.
• However, sometimes money must be used when no
  other fair division is possible
• An example is dividing a car, a house, and a boat
  among two or more people.

9
Types of Fair-Division, contd

• Mixed fair-division problems
• Some objects to be shared can be divided and some
  cannot.
• This type is a combination of continuous and
  discrete fair division.
• An example is dividing an estate consisting of
  money, a house, and a car among two or more
  people.

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Question

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

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Types of Fair-Division, contd

• This section will consider only continuous
  fair-division problems.
• We make the assumption that the value of a
  players share is determined by his or her
  values.
• Different players may value the same share
  differently.

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Value of a Share

• In a fair-division problem with n players, a
  player has received a fair share if that player
  considers his or her share to be worth at least
  1/n of the total value being shared.
• A division that results in every player receiving
  a fair share is called proportional.

13
Value of a Share, contd

• We assume that a players values in a
  fair-division problem cannot change based on the
  results of the division.
• We also assume that no player has any knowledge
  of any other players values.

14
Fair Division for Two Players

• The standard procedure for a continuous
  fair-division problem with two players is called
  the divide-and-choose method.
• This method is described as dividing a cake, but
  it can be used to fairly divide any continuous
  object.

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Divide-And-Choose Method

• Two players, X and Y, are to divide a cake.
• Player X divides the cake into 2 pieces that he
  or she considers to be of equal value.
• Player X is called the divider.
• Player Y picks the piece he or she considers to
  be of greater value.
• Player Y is called the chooser.
• Player X gets the piece that player Y did not
  choose.

16
Divide-And-Choose Method, contd

• This method produces a proportional division.
• The divider thinks both pieces are equal, so the
  divider gets a fair share.
• The chooser will find at least one of the pieces
  to be a fair share or more than a fair share.
  The chooser selects that piece, and gets a fair
  share.

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Example 1

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

18
Example 1, contd

• Margo cuts the pizza into 6 pieces and arranges
  them as shown.
• What monetary value would Margo and Steven each
  place on the original two halves of the pizza?

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Example 1, contd

• Solution The whole pizza is worth 4.
• Margo values both kinds of pizza equally. To her
  each half is worth half of the total value, or
  2.
• Steven values pepperoni 4 times as much as
  Hawaiian. To him 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.

20
Example 1, contd

1. What value would each person place on each of the
   two plates of pizza?
2. What plate will Steven choose?

21
Example 1, contd

• Solution 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.

22
Example 1, contd

• Solution, contd
• Steven values pepperoni 4 times as much as
  Hawaiian.
• To him each pepperoni slice is worth 3.20/3
  1.067 and each Hawaiian slice is worth 0.80/3
  0.267.
• The first plate is worth 2(0.267) 1(1.067)
  1.60.
• The second plate is worth 1(0.267) 2(1.067)
  2.40

23
Example 1, contd

• Solution
• Steven will choose the second plate, with one
  slice of Hawaiian and two slices of pepperoni.
• Margo gets a plate of pizza that she feels is
  worth half the value.
• Steven gets a plate of pizza that he feels is
  worth more than half the value.

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Example 2

• Caleb and Diego will drive 6 hours during the day
  and 4 hours at night.
• Caleb prefers night to day driving 2 to 1.
• Diego prefers them equally, or 1 to 1.
• How should they divide the driving into 2 shifts
  if Caleb is the divider and Diego is the chooser?

25
Example 2, contd

• Solution
• Caleb can assign 2 points to each hour of night
  driving and 1 point to each hour of day driving.
• Caleb values the entire drive at 1(6) 2(4) 14
  points.
• To Caleb a fair share will be worth half the
  total value, or 7 points.

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Example 2, contd

• Solution, contd
• A possible fair division for Caleb is to create
  shifts of
• 6 hours of daytime driving and 0.5 hours of
  nighttime driving.
• 3.5 hours of nighttime driving.
• Both shifts are worth 7 points to Caleb.

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Example 2, contd

• Solution, contd
• Diego can assign 1 point to each hour of night
  driving and 1 point to each hour of day driving.
• Diego values the entire drive at 1(6) 1(4) 10
  points.
• To Diego a fair share will be worth half the
  total value, or 5 points.

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Example 2, contd

• Solution, contd
• Diego values the first shift at 1(6) 1(0.5)
  6.5 points.
• Diego values the second shift at 1(3.5) 3.5
  points.
• Diego will choose the first shift, because it is
  worth more to him.

29
Two Players, contd

• Notice that in both of the previous examples
• The divider got a share he or she felt was equal
  to exactly half of the total value.
• The chooser got a share he or she felt was equal
  to more than half of the total value.
• It is often advantageous to be the chooser, so
  the roles should be randomly chosen.

30
Fair Division for Three Players

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

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Divide-And-Choose Method

• Three players, X, Y, and Z are to divide a cake.
• Player X (the divider) divides the cake into 3
  pieces that he/she considers to be of equal
  value.
• Players Y and Z (the choosers) each decide which
  pieces are worth at least 1/3 of the total value.
• These pieces are said to be acceptable.
• The choosers announce their acceptable pieces.

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Divide-And-Choose Method, contd

• There are 2 possibilities
• If at least 1 piece is unacceptable to both Y and
  Z, Player X gets that piece.
• If Y and Z can each choose acceptable pieces,
  they do so.
• If Y and Z cannot each choose acceptable pieces,
  they put the remaining pieces back together and
  use the two player method to re-divide.

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Divide-And-Choose Method, contd

• Contd
• If every piece is acceptable to both Y and Z,
  they each take an acceptable piece. Player X gets
  the leftover piece.
• Note The divide-and-choose method can be
  extended to more than 3 players. The more
  players, the more complicated the process becomes.

34
Question
The divide-and-choose method for 3 players is
being used to divide a pizza. Player A has cut a
pizza into what she views as 3 equal shares.
Player B thinks that only shares 2 and 3 are
acceptable. Player C thinks that only share 2 is
acceptable. What is the fair division? a.
Player A gets share 3, Player B gets share 1, and
Player C gets share 2. b. Player A gets share 1,
Player B gets share 3, and Player C gets share
2. c. Player A gets share 2, Player B gets share
3, and Player C gets share 1. d. Player A gets
share 1, Player B gets share 2, and Player C gets
share 3.
35
Example 3

• Emma, Fay, and Grace will divide 24 ounces of ice
  cream, which is made up of equal amounts of
  vanilla, chocolate, and strawberry.
• Emma likes the 3 flavors equally well.
• Fay prefers chocolate 2 to 1 over either other
  flavor and prefers vanilla and strawberry equally
  well.
• 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?

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Example 3, contd

• Solution Suppose Emma divides the ice cream into
  3 equal parts, each consisting of one of the
  flavors.

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Example 3, contd

• Solution, contd Fay is one of the choosers.
• Faye finds portions 1 and 3 unacceptable.

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Example 3, contd

• Solution, contd Grace is the other chooser.
• She finds portion 1 unacceptable

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Example 3, contd

• Solution, contd All of the players values are
  summarized in the table below.

40
Example 3, contd

• Solution, contd
• Portion 1 is unacceptable to both Fay and Grace.
  As the divider, Emma will receive portion 1.
• Only portion 2 is acceptable to Faye.
• Portions 2 and 3 are acceptable to Grace.
• The division is Emma portion 1 Fay portion 2
  Grace portion 3.

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Last-Diminisher Method

• A method for continuous fair-division problems
  with 3 or more players is called the
  last-diminisher method.
• Suppose any number of players X, Y, are
  dividing a cake.
• Player X cuts a piece of cake that he or she
  considers to be a fair share.

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Last-Diminisher Method, contd

• Each player, in turn, judges the fairness of the
  piece.
• If a player considers the piece fair or less than
  fair, it passes to the next player.
• If a player considers the piece more than fair,
  the player trims the piece to make it fair,
  returning the trimming to the undivided portion
  and passing the trimmed piece to the next player.

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Last-Diminisher Method, contd

• The last player to trim the piece, gets the piece
  as his or her share.
• If no player trimmed the piece, player X gets the
  piece.
• After one player gets a piece of cake, the
  process begins again without that player and that
  piece.
• When only 2 players remain, they use the
  divide-and-choose method.

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Example 4

• Hector, Isaac, and James will divide 24 ounces of
  ice cream, which is equal parts vanilla,
  chocolate, and strawberry.
• Hector values vanilla to chocolate to strawberry
  1 to 2 to 3.
• Isaac likes the 3 flavors equally.
• James values vanilla to chocolate to strawberry 1
  to 2 to 1.

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Example 4, contd

• Using the last-diminisher method with Hector as
  the first divider and Isaac as the first judge,
  find the results of the division.
• Solution
• Hector assigns 1 point to each ounce of vanilla,
  2 points to each ounce of chocolate, and 3 points
  to each ounce of strawberry.

46
Example 4, contd

• Solution, contd A fair share of ice cream, to
  Hector, is worth 48/3 16 points.

47
Example 4, contd

• Solution, contd
• One possible fair share for Hector would be all 8
  ounces of vanilla plus 4 ounces of chocolate.
• This share is worth 1(8) 2(4) 16 points to
  Hector, so he would be happy with this share.
• Next, Isaac must decide whether the share is
  fair, according to his values.

48
Example 4, contd

• Solution, contd
• Isaac assigns 1 point to each ounce of vanilla, 1
  point to each ounce of chocolate, and 1 point to
  each ounce of strawberry.
• Isaac values all of the ice cream at 1(8) 1(8)
  1(8) 24 points.
• A fair share to Isaac is 8 points.

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Example 4, contd

• Solution, contd
• Isaacs value for Hectors serving is 1(8) 1(4)
  12 points.
• Isaac thinks it is more than a fair share.
• Isaac trims off 4 points worth of ice cream.
• Suppose he trims off the 4 ounces of chocolate.

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Example 4, contd

• Solution, contd
• Next, James must judge the share.
• James assigns 1 point to each ounce of vanilla, 2
  points to each ounce of chocolate, and 1 point to
  each ounce of strawberry.
• James values all of the ice cream at 1(8) 2(8)
  1(8) 32 points.
• A fair share to James is worth 32/3 points.

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Example 4, contd

• Solution, contd
• The existing share is now just 8 ounce of
  vanilla.
• To James, the share is worth 1(8) 8 points.
• James thinks this is less than a fair share.
• James will not trim the share.

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Example 4, contd

• Solution, contd
• Isaac was the last-diminisher, and gets the share
  of ice cream.
• Hector and James will divide the remaining ice
  cream using the divide-and-choose method.
• Note This is only one of many different possible
  solutions.

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4.1 Initial Problem Solution

• The brothers Drewvan, Oswald, and Granger are to
  share their familys estate, which is 1200 acres
  each of vineyards, woodlands, and fields.
• Drewvan prefers vineyards to woodlands to fields
  3 to 2 to 1.
• Oswald prefers vineyards to woodlands to fields 2
  to 3 to 1.
• Granger prefers vineyards to woodlands to fields
  2 to 1 to 3.

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Initial Problem Solution, contd

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

55
Initial Problem Solution, contd

• Drewvan values the entire estate at 7200 points.

56
Initial Problem Solution, contd

• To Drewvan, a fair share is worth 7200/3 2400
  points.
• One possible fair division is shown below.

57
Initial Problem Solution, contd

• Next, the two choosers will consider this
  division.
• Granger and Oswald both value the entire estate
  at 7200 points also.
• To Oswald, a fair share is worth 2400 points.
• To Granger, a fair share is worth 2400 points.

58
Initial Problem Solution, contd

• Oswald considers piece 1 to be unacceptable.

59
Initial Problem Solution, contd

• Granger considers pieces 1 and 2 to be
  unacceptable.

60
Initial Problem Solution, contd

• Both choosers think piece 1 is unacceptable, so
  Drewvan gets piece 1.
• Granger thinks only piece 3 is acceptable, so he
  gets that piece.
• Oswald thinks pieces 2 and 3 are acceptable, so
  Oswald gets piece 2.

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Section 4.2Discrete and Mixed Division Problems

• Goals
• Study discrete fair-division problems
• The method of sealed bids
• The method of points
• Study mixed fair-division problems
• The adjusted winner procedure

62
4.2 Initial Problem

• When twins Zack and Zeke turned 16 they received
  
• A pickup truck
• A horse
• A cow
• How can they share these three things?
• The solution will be given at the end of the
  section.

63
Discrete Fair Division

• Recall that discrete fair division problems
  involve sharing objects that cannot be divided
  without losing value.
• Two methods for solving discrete fair-division
  problems are
• The method of sealed bids.
• The method of points.

64
Method of Sealed Bids

• Any number of players, N, are to share any number
  of items.
• If necessary, money will be used to insure
  fairness.
• All players submit sealed bids, stating monetary
  values for each item.
• Each item goes to the highest bidder.
• The highest bidder places the dollar amount of
  his or her bid into a compensation fund.

65
Method of Sealed Bids, contd

• From the compensation fund, each player receives
  1/N of his or her bid on each item.
• Any money leftover in the fund is distributed
  equally to all players.
• Note This method is also called the Knaster
  Inheritance Procedure.

66
Example 1

• Three sisters Maura, Nessa, and Odelia will share
  a house and a cottage.
• Apply the method of sealed bids to divide the
  property, using the bids shown below.

67
Example 1, contd

• Solution Each piece of property goes to the
  highest bidder.
• Odelia gets the family home and places 301,000
  into the compensation fund.
• Nessa gets the cottage and places 203,000 into
  the compensation fund.

68
Example 1, contd

• Solution, contd The compensation fund now
  contains a total of 203,000 301,000
  504,000.

69
Example 1, contd

• Solution, contd Each sister receives 1/3 of her
  total bids from the compensation fund.
• Maura receives
• Nessa receives
• Odelia receives

70
Example 1, contd

• Solution, contd After the distributions, there
  is 504,000 (159,000 160,000 161,000)
  21,000 left in the fund.
• The leftover money is distributed equally to the
  three sisters in the amount of 7000 each.

71
Example 1, contd

• Solution, contd The final shares are
• Maura receives 166,000 and no property.
• Nessa receives the cottage, for which she paid a
  net amount of 33,000.
• Odelia receives the family home, for which she
  paid a net amount of 133,000.

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Example 1, contd

• Solution, contd Note that the division is
  proportional because each sister receives what
  she considers to be a fair share.

73
Method of Points

• Three players will share three items.
• Each player assigns points to each item, so that
  the points for each player total to 100.
• List all 6 possible arrangements of players and
  items, along with the point values.

74
Method of Points, contd

• Note the smallest number of points assigned to an
  item in each arrangement.
• Choose the arrangement with the largest value of
  the smallest number.
• If there is not only one such arrangement, keep
  all the arrangements with that smallest point
  value and go to step 4.

75
Method of Points, contd

• For each arrangement kept in step 3, note the
  middle point value.
• Choose the arrangement with the largest value of
  the middle number.
• If there is not only one such arrangement, keep
  all the arrangements with that middle point value
  and go to step 5.

76
Method of Points, contd

• For each arrangement kept in step 4, note the
  largest point value.
• Choose any arrangement with the largest value of
  the largest number.

77
Example 2

• Three couples, A, B, and C, need to decide who
  gets which room in a hotel.
• The couples assign points to each room as shown
  below.

78
Example 2, contd

• Solution

79
Example 2, contd

• Solution, contd The largest minimum point value
  is 40, in row 4.

80
Example 2, contd

• Solution, contd The arrangement selected is
• Couple A, Room 2
• Couple B, Room 3
• Couple C, Room 1
• Two couples got their first choice and one got
  their second choice.

81
Example 3

• Three couples, A, B, and C, need to decide who
  gets which room in a cabin.
• The couples assign points to each room as shown
  below.

82
Example 3, contd
83
Example 3, contd

• Table from previous slide

• Solution, contd The largest minimum point value
  is 12, which occurs in 4 arrangements.

84
Example 3, contd

• Solution, contd Keep those 4 arrangements and
  look at the middle point values.

85
Example 3, contd

• Solution, contd The largest middle point value
  is 20, which occurs in 3 arrangements.

86
Example 3, contd

• Solution, contd Keep those 3 arrangements and
  look at the largest point values.
• The maximum largest number is 71.

87
Example 3, contd

• Solution, contd The arrangement selected is
• Couple A, Room 2
• Couple B, Room 1
• Couple C, Room 3

88
Question

• The assignments were Couple A, Room 2 Couple B,
  Room 1 and Couple C, Room 3. Considering the
  point values each couple assigned to the three
  rooms, is this division proportional?
• a. yes
• b. no

89
Example 3, contd

• Solution, contd Sometimes the method of points
  produces a division that is not proportional.
  However it is still the best division that can be
  made.

90
Mixed Fair Division

• Recall that mixed fair division problems involve
  sharing a mixture of discrete and continuous
  objects.
• A method for solving mixed fair-division problems
  is the adjusted winner procedure.

91
Adjusted Winner Procedure

• Two players are to fairly divide any number of
  items.
• The items may be discrete and/or continuous.
• Ownership of some items may be shared.
• Each player assigns points to each item, for a
  total of 100 points.

92
Adjusted Winner Procedure, contd

• Each player tentatively receives items to which
  he or she assigned the most points.
• The points are added to the players total.
• If 2 players tie for an item, the item goes to
  the player with the lowest point total so far.

93
Adjusted Winner Procedure, contd

• Look at the players points.
• If the point totals are equal, the process is
  done.
• If Player X has more points than Player Y, then
  give Player Y the item for which the ratio of the
  number of points assigned by X to the number of
  points assigned by Y is the smallest.

94
Adjusted Winner Procedure, contd

• Re-examine the players points.
• If the point totals are equal, the process is
  done.
• If Player X still has more points than Player Y,
  repeat step 3.
• If Player Y now has more points than Player X,
  move a fraction of the last item moved back to X.

95
Adjusted Winner Procedure, contd

1. Contd The formula for case C is as follows,
   where q fraction of item to be moved, TX
   Player Xs point total without this item, TY
   Player Ys point total without this item, PX
   number of points Player X assigned to this item,
   PY number of points Player Y assigned to this
   item.

96
Question

• Three players will divide a car, a boat, and an
  RV. Their point assignments are shown in the
  table below. Complete step 2 in the adjusted
  winner procedure.
• a. Player 1 boat Player 2 RV Player 3 car
• b. Player 1 boat, RV Player 2 nothing Player
  3 car
• c. Player 1 boat Player 2 car, RV Player 3
  nothing

Player 1 Player 2 Player 3
car 15 30 35
boat 35 20 20
RV 50 50 45
97
Example 4

• Two players, A and B, need to divide a house, a
  boat, a cabin, and a condominium.
• Both have assigned points as shown below.

98
Example 4, contd

• Solution The tentative assignment of items,
  along with the current point totals, is shown in
  the table below.

99
Example 4, contd

• Solution, contd Player A has more points than
  Player B.
• Determine which item to move from A to B by
  considering the ratios of both items currently
  assigned to Player A.

100
Example 4, contd

• Solution, contd Move the house, which has the
  smaller ratio, from Player A to Player B.

101
Example 4, contd

• Solution, contd Now Player B has more points
  than Player A.
• A fraction of the house must be given back to
  Player A.
• The values for the formula are
• TX TA 30
• TY TB 45
• PX PA 45
• PY PB 35

102
Example 4, contd

• Solution, contd The calculation is
• Player B keeps 3/8 of the house and 5/8 of the
  house goes back to Player A.

103
Example 4, contd

• Solution, contd Re-examine the points totals.
• Player A has 30 45(5/8) 58.125 points.
• Player B has 25 20 35(3/8) 58.125 points.
• The division is now proportional.

104
4.2 Initial Problem Solution

• Zack and Zeke need to share a pickup truck, a
  horse, and a cow.
• Solution They could use the adjusted winner
  procedure to share the items. Suppose they
  assign points as shown.

105
Initial Problem Solution, contd

• Tentatively, the assignments are
• Zack truck and cow, 73 points
• Zeke horse, 35 points.
• Zack has more points, so something must be given
  to Zeke.
• Check the ratios for the truck and the cow.

106
Initial Problem Solution, contd

• The ratios are shown in the table.
• The ratio for the truck is smaller.
• Move the truck from Zack to Zeke.

107
Initial Problem Solution, contd

• Now
• Zack has 40 points.
• Zeke has 65 points.
• Since the points are not equal, a fraction of the
  truck must be given back to Zack.
• The values for the formula are
• TX TZack 40 PX PZack 33
• TY TZeke 35 PY PZeke 30

108
Initial Problem Solution, contd

• Solution, contd The calculation is
• Zeke keeps 60.3 of the truck and 39.7 of the
  truck goes back to Zack.

109
Initial Problem Solution, contd

• Solution, contd Re-examine the points totals.
• Zack has 40 33(0.397) 53.095 points.
• Zeke has 35 30(0.603) 53.095 points.
• For a proportional division to be created, the
  truck must be shared.

110
Section 4.3Envy-Free Division

• Goals
• Study envy-free division
• Continuous envy-free division for 3 players

111
4.3 Initial Problem

• Dylan, Emery, and Fordel will share a cake that
  is half chocolate and half yellow.
• Dylan likes chocolate cake twice as much as
  yellow.
• Emery likes chocolate and yellow cake equally
  well.
• Fordel likes yellow cake twice as much as
  chocolate.
• How should they divide the cake?
• The solution will be given at the end of the
  section.

112
Envy-Free Division

• A division, among n players, is considered
  envy-free if each player feels that
• He or she has received at least 1/n of the total
  value
• No other players share is more valuable than his
  or her own.
• Note A proportional division is not always
  envy-free.

113
Question

• Ice cream was divided among three players. The
  players values of the shares are shown in the
  table below.
• The division was Emma, Portion 1 Fay, Portion
  2 Grace, Portion 3.
• Is the division envy free? a. yes b. no

114
Continuous Envy-Free Division

• This section covers envy-free divisions for
  continuous fair-division problems involving three
  players.
• The procedure is split into two parts
• Part 1 distributes the majority of the item.
• Part 2 distributes the excess.

115
Envy-Free Division Part 1

• Players A, B, and C are to share a cake (or some
  other item).
• Player A (the divider) divides the cake into 3
  pieces he or she considers to be of equal value.
• Player B chooses the one most valuable piece of
  these 3 pieces.

116
Envy-Free Division Part 1, contd

• Player B (the trimmer) trims the most valuable
  piece so it is equal to the second most valuable
  piece.
• The excess is set aside.
• Player C (the chooser) chooses the piece he or
  she considers to have the greatest value.

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Envy-Free Division Part 1, contd

• Player B chooses next.
• Player B gets the trimmed piece if it is
  available.
• If the trimmed piece is gone, B chooses the most
  valuable piece from the 2 remaining pieces.
• Player A gets the 1 remaining piece.

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Example 1

• Gabi, Holly, and Izzy will share a cake that is ¼
  chocolate, ¼ white, ¼ yellow, and ¼ spice cake.

119
Example 1, contd

• The girls preference ratios are given below

120
Example 1, contd

• Solution Let Gabi be the divider, Holly the
  trimmer, and Izzy the chooser.
• Gabi assigns point values to slices of cake.

121
Example 1, contd

• Solution, contd Gabi could divide the cake as
  shown below into 3 pieces of equal value to her.

122
Example 1, contd

• Solution, contd Holly is the trimmer.

123
Example 1, contd

• Solution, contd Holly trims piece 3 to make it
  equal in value to pieces 1 and 2.

124
Example 1, contd
125
Example 1, contd

• Solution, contd
• Izzy chooses piece 2.
• Holly gets the trimmed piece, piece 3.
• Gabi gets the last piece, piece 1.

126
Example 1, contd
127
Envy-Free Division Part 2

• Players A, B, and C are to share a cake (or some
  other item) and have completed Part 1.
• Of Players B and C, the player who received the
  trimmed piece becomes the second chooser.
• The other player becomes the second divider.
• The second divider divides the excess into 3
  pieces of equal value.

128
Envy-Free Division Part 2

1. The second chooser takes the piece he or she
   considers to be of the greatest value.
2. Player A chooses the remaining piece that he or
   she considers to be of the greatest value.
3. The second divider gets the last remaining piece.

129
Question

• In Part 1 of an envy-free division, the results
  were as follows
• Player A (the divider) got share 2 Player B
  (the trimmer) got share 1 and Player C (the
  chooser) got share 3, which had been trimmed.
• Who is the second divider for Part 2?
• a. Player A b. Player B c. Player C

130
Example 2

• Gabi, Holly, and Izzy will complete the division
  of the cake.
• The excess portion is shown below.

131
Example 2, contd

• Solution Recall that Gabi was the first divider
  and Holly received the trimmed piece.
• Holly will be the second chooser and Izzy will be
  the second divider.

132
Example 2, contd

• Solution, contd Izzy divides the excess into 3
  pieces.

133
Example 2, contd
134
Example 2, contd

• Solution, contd
• Since the pieces are all of equal value to Holly,
  she arbitrarily chooses a piece, say piece 3.
• Gabi, the first divider, chooses either equally
  valuable piece, say piece 1.
• Izzy receives the last piece, piece 2.

135
Example 2, contd
136
Example 2, contd
137
Example 2, contd
138
Example 2, contd

• Solution, contd The division is envy-free
  because each player feels she received a share
  worth as much or more as every other players
  share.

139
Example 3

• Jenny, Kara, and Lindsey need to divide 10 yards
  each of beige linen, red silk, and yellow
  gingham.
• Each assigns points per yard as shown below.

140
Example 3, contd

• Solution Jenny divides the fabric.

141
Example 3, contd
142
Example 3, contd
143
Example 3, contd
144
Example 3, contd

• Solution, contd Lindsey took the trimmed share,
  so Kara cannot have it.
• Kara chooses the most valuable remaining share,
  share 2.
• The remaining piece, share 1, goes to Jenny.
• Note This completes Part 1.

145
Example 3, contd

• Solution, contd The excess is divided into 3
  equal pieces.
• Since the excess pieces are all the same, there
  is no real choosing to do in Part 2.
• Each sister gets 2/3 yard of silk as her share of
  the excess.

146
Example 3, contd
147
4.3 Initial Problem Solution

• Dylan, Emery, and Fordel will share a cake that
  is half chocolate and half yellow.
• Dylan likes chocolate cake twice as much as
  yellow.
• Emery likes chocolate and yellow cake equally
  well.
• Fordel likes yellow cake twice as much as
  chocolate.
• How should they divide the cake?

148
Initial Problem Solution, contd

• Let Dylan be the divider, Emery the trimmer, and
  Fordel the chooser.

149
Initial Problem Solution, contd

• Part 1 Dylan evaluates the cake

150
Initial Problem Solution, contd

• Dylan creates 3 equal shares

151
Initial Problem Solution, contd
152
Initial Problem Solution, contd

• Emery trims piece 3 so that it is equal in value
  to pieces 1 and 2

153
Initial Problem Solution, contd
154
Initial Problem Solution, contd

• Fordel chooses piece 3, the trimmed piece.
• The remaining pieces are identical, so Emery and
  Dylan each take one.
• Part 2 The excess is all yellow cake, so it can
  merely be divided into 3 equal-sized pieces and
  shared among the players.
• The final division is shown on the next slides.

155
Initial Problem Solution, contd
156
Initial Problem Solution, contd
157
Initial Problem Solution, contd