CHAPTER 3 FAIR DIVISION GAMES - TEST REVIEW GUIDE:

You should be familiar with each of the following:

  • How to determine the value fractions.
  • How to calculate how much a player values a given share or piece of the goods.
  • How to divide or cut the goods into fair shares. (Be the Divider in a game)
  • What methods are for continuous and for discrete goods?

3.2 Divider – Chooser
  • Be able to explain how the method works.
  • Be able to perform the method as the divider. (3.1 – 3.3 Worksheet #9 – 11)
  • What is a divider?
  • What is a chooser?
3.3 Lone – Divider
  • Use a chart of players’ share values to determine a fair division.
  • What is a bid list?
  • What happens when 2 or more players want the same share?
/ 3.6 Method of Sealed Bids
  • What are required CONDITIONS about the players involved in this method?
  • How do you calculate the fair dollar share?
  • How are items assigned?
  • What is the first settlement?
  • What is surplus?
  • What is the final settlement?
3.7 Method of Markers
  • What are required CONDITIONS about the players involved in this method?
  • How do players value the items?
  • How are markers placed?
  • What happens to leftovers?

CHAPTER 3 REVIEW PROBLEMS:

1)Alex values chocolate 3 times as much as he values strawberry. He also enjoys vanilla 4 times as much as strawberry. What are the value fractions for Alex’s food preference?

2)Papa John delivers a pizza that is 1/3 Hawaiian, 1/3 Pepperoni, and 1/3 the Works.
3 roommates Matt, Andy, and John want to split the pizza fairly. MATT likes all pizza equally well. ANDY likes Hawaiian twice as much as pepperoni and the works. JOHN likes the works and pepperoni, but hates Hawaiian.

  1. What are the value fractions for each roommate?
  1. Find the value of each piece listed below to each roommate:
  2. Piece One: 300 of H and 450 of W
  3. Piece Two: 900 of H and 600 of P
  4. Piece Three: 750 of W and 600 of P

3)Three players (Bryce, Morgan, Lamont) must divide a pizza among themselves. The values of the entire pizza and of each of the three slices are shown in the following table.

Whole Pizza / S1 / S2 / S3
Bryce / $12 / $2.50 / $4.50 / $5
Morgan / $15 / $5 / $5 / $5
Lamont / $18 / $7 / $5.50 / $5.50

a)Indicate which of the three slices are fair shares to Bryce, Morgan, and Lamont.

b)Describe a fair division of the pizza. (Give each player one slice only!)

4) Four partners (Ashley, Andres, Simone, and Lyndsey) jointly own a piece of land. The following table shows the relative value for 4 parcels of land.

S1 / S2 / S3 / S4

Ashley

/ 45% / 20% / 10% / 25%
Andres / 27% / 33% / 24% / 16%
Simone / 25% / 15% / 40% / 20%
Lyndsey / 23% / 23% / 23% / 31%

a)Indicate which of the four parcels are FAIR SHARES for each partner.

b)Describe a FAIR DIVISION of the land. (Give each partner only one piece!)

S1 / S2 / S3 / S4
Max / 25% / 25% / 25% / 25%
Seamus / 2% / 50% / 32% / 16%
Jack / 10% / 10% / 60% / 20%
Tyler / 9% / 0% / 60% / 31%

5)For the given preference table for the division of a plot of land into 4 parcels,

  1. What are the BID LISTS of each brother?
  1. Which brother is the DIVIDER?
  1. Describe its FAIR DIVISION among 4 brothers.

6)Kerry and Ramona decide to share large sandwich that is 6 inches of Vito and 6 inches of Pepe from Jimmy John’s using the divider chooser method.

Kerry likes the Vito four as much as the Pepe.

Ramona likes the Pepe twice as much as Vito.

  1. Ramona (Divider) initially cuts the sandwich:
  2. Where will Ramona cut the sandwich? (Vito or Pepe Section – Pick One)
  3. What is the actual amount (size) of the two pieces Ramona will create?
  4. What piece will Kerry (Chooser) take and what does she value it at?
  1. Kerry (Divider) initially cuts the sandwich:
  2. Where will Kerry cut the sandwich? (Vito or Pepe Section – Pick One)
  3. What is the actual amount (size) of the two pieces Kerry will create?
  4. What piece will Ramona (Chooser) take and what does she value it at?

7)Use the chart below to find the final settlement using The Method of Sealed Bids.

A / B / C / D
House / 180,000 / 200,000 / 190,000 / 185,000
Cabin / 60,000 / 50,000 / 40,000 / 55,000
Boat / 16,000 / 12,000 / 18,000 / 10,000

8) Use the chart below to find the final settlement using The Method of Sealed Bids.

A / B / C
House / 180,000 / 190,000 / 200,000
Boat / 42,000 / 50,000 / 31,000

9)3 siblings are trying to split up triangle, square, and circle toys. There are a total of 18 toys. Each sibling prefers the toys in different amounts. Your job is to take the given order of toys and perform a Method of Markers based on the siblings preferences.

Sibling #1 likes squares 3 times as much as triangles and circles twice as much as triangles.

Sibling #2 likes circles 3 times as much as squares and triangles twice as much as squares.

Sibling #3 likes all the toy shapes equally having already learned the value of sharing.


CHAPTER 3 FAIR DIVISION GAMES - TEST REVIEW GUIDE:

  • How to determine the value fractions.
  • Assign Numbers to the each item based on preference or value statements (hates, twice, etc)
  • Add up all these values and divide each one by the TOTAL to get the value fractions.
  • How to calculate how much a player values a given share or piece of the goods.
  • SUM OF (VALUE FRACTIONS) times (PHYSICAL FRACTIONS)
  • HINT: Make the TABLE and multiply columns and add answers
  • PHYSICAL FRACTIONS = (How Much in Share inches or degrees)/ (How Much in Total Goods)
  • How to divide or cut the goods into fair shares. (Be the Divider in a game)
  • (Proportion: See 3.1 – 3.3 Worksheet #9 – 11 and Review Problem #6)
  • What methods are for continuous and for discrete goods?
  • Divider- Chooser and Lone – Divider = CONTINUOUS GOODS (always cut smaller)
  • Method of Sealed Bids + Markers = DISCRETE GOODS (smallest size exists)

  • 3.2 Divider – Chooser
  • Be able to explain how the method works.
    1) Divider Cuts into 2 equally valued pieces ( ½ and ½)
    2) Chooser Picks which of the two shares that is valued the most
    3) Divider gets remaining share leftover
  • Be able to perform the method as the divider. (See 3.1 – 3.3 Worksheet #9 – 11)
  • What is a divider?
    Player that breaks the goods into ALL equally valued shares.
  • What is a chooser?
    Player that gets to pick (choose) a share from options.
  • 3.3 Lone – Divider
  • Use a chart of players’ share values to determine a fair division.
    Matching Games of Shares when players are assigned shares based on their bids
  • What is a bid list?
    List of all shares that a player believes are FAIR
  • What happens when 2 or more players want the same share?
    Combine (add) shares back together and divide by the number of players in conflict
/
  • 3.6 Method of Sealed Bids
  • What are required CONDITIONS about the players involved in this method?
    1) Players have ENOUGH MONEY
    2) Players will ACCEPT MONEY as a SUBSTITUTE for an item
  • How do you calculate the fair dollar share? TOTAL PLAYER’S BIDS divided by Total Number of Players
  • How are items assigned?
    Highest Bid in the Row
  • What is the first settlement?
    Item Allocation and GET/ PAY statements for each player
  • What is surplus?
    (All Pays – All Gets) divide by number of players, and then each player receives that amount of the surplus
  • What is the final settlement?
List of Items and combining SURPLUS with original GET/PAY statements
  • 3.7 Method of Markers
  • What are required CONDITIONS about the players involved in this method?
1)More Items than Players
2)Items reasonably close in value
  • How do players value the items?
Same as value fractions, but don’t need to calculate denominator
  • How are markers placed?
    Each Player Places marks so that each segment is EQUALLY VALUED
  • What happens to leftovers?
    Leftovers can be shared by repeating the method or random lottery.

HONORS DISCRETE CHAPTER 3 TEST REVIEW PROBLEMS: SOLUTIONS

1)Alex values chocolate 3 times as much as he values strawberry. He also enjoys vanilla 4 times as much as strawberry. What are the value fractions for Alex’s food preference?

Chocolate = 3Strawberry = 1Vanilla = 44 + 3 + 1 = 8

= 3/8= 1/8= 4/8

2)Papa John’s delivers a pizza that is 1/3 Hawaiian, 1/3 Pepperoni, and 1/3 the Works. 3 roommates Matt, Andy, and John want to split the pizza fairly. MATT likes all pizza equally well. ANDY likes Hawaiian twice as much as pepperoni and the works. JOHN likes the works and pepperoni, but hates Hawaiian.

  1. What are the value fractions for each roommate?

Matt: H = 1/3P = 1/3 W = 1/3

Andy: H = 2/4P = 1/4W = 1/4

John: H = 0P = 1/2W = 1/2

  1. Find the value of each piece listed below to each brother:
  2. Piece One: 300 of H and 450 of W

Matt: 1/3(30/120) + 1/3 (45/120) = 5/24

Andy: 2/4(30/120) + 1/4(45/120) = 1/4

John: 0(30/120) + 1/2(45/120) = 3/16

  1. Piece Two: 900 of H and 600 of P

Matt: 1/3(90/120) + 1/3 (60/120) = 5/12

Andy: 2/4(90/120) + 1/4(60/120) = 1/2

John: 0(90/120) + 1/2(60/120) = 1/4

  1. Piece Three: 750 of W and 600 of P

Matt: 1/3(75/120) + 1/3 (60/120) = 9/24

Andy: 1/4(75/120) + 1/4(60/120) = 1/4

John: 1/2(75/120) + 1/2(60/120) = 9/16

3)Three players (Bryce, Morgan, Lamont) must divide a pizza among themselves. The values of the entire pizza and of each of the three slices are shown in the following table.

Whole Pizza / S1 / S2 / S3
Bryce / $12 / $2.50 / $4.50 / $5
Morgan / $15 / $5 / $5 / $5
Lamont / $18 / $7 / $5.50 / $5.50

a)Indicate which of the three slices are fair shares to Bryce, Morgan, and Lamont.

Bryce = {s2, s3}Morgan = {s1, s2, s3} Lamont = { s1}

b)Describe a fair division of the pizza. (Give each player one slice only!)

Bryce = s3Morgan = s2Lamont = s1

4) Four partners (Ashley, Andres, Simone, and Lyndsey) jointly own a piece of land. The following table shows the relative value for 4 parcels of land.

S1 / S2 / S3 / S4

Ashley

/ 45% / 20% / 10% / 25%
Andres / 27% / 33% / 24% / 16%
Simone / 25% / 15% / 40% / 20%
Lyndsey / 23% / 23% / 23% / 31%

a)Indicate which of the four parcels are fair shares to Ashley, Andres, Simone, and Lyndsey.

Ashley = {s1,s4} Andres = {s1, s2} Simone = {s1, s3}Lyndsey. = {s4}

b)Describe a fair division of the land. (Give each partner only one piece!)

Ashley = s1 Andres = s2Simone = s3Lyndsey. = s4

5)For the given preference table for the division of a plot of land into 4 parcels,

S1 / S2 / S3 / S4
Max / 25% / 25% / 25% / 25%
Seamus / 2% / 50% / 32% / 16%
Jack / 10% / 10% / 60% / 20%
Tyler / 9% / 0% / 60% / 31%
  1. What are the bid lists of each brother?

Max = {s1,s2, s3, s4} Seamus = {s2, s3}
Jack = {s3}Tyler. = {s3, s4}

  1. Which brother is the divider?

Max

  1. Describe its fair division among 4 brothers.

Max = s1, Seamus = s2, Jack = s3, Tyler. = s4,

6) Kerry and Ramona decide to share large sandwich that is 6 inches of Vito and 6 inches of Pepe from Jimmy John’s using the divider chooser method.

Kerry likes the Vito four as much as the Pepe.

Ramona likes the Pepe twice as much as Vito.

VALUE FRACTIONS: Ramona: V = 1/3P = 2/3

Kerry: V = 4/5P = 1/5

  1. Ramona (Divider) initially cuts the sandwich:
  2. Where will Ramona cut the sandwich? (Vito or Pepe Section)

Cuts in Pepe because it has value greater than ½ at 2/3

  1. What is the actual amount of the two pieces Ramona will create?

Piece #1: ; P = 4.5 inches of Pepe

Piece #2: 1.5 inches of Pepe + 6 inches of Vito

  1. What piece will Kerry (Chooser) take and what does she value it at?

#1: (1/5)(4.5/6) = 3/20 value

#2: (1/5)(1.5/6) + (4/5)(6/6) = 17/20 value Kerry will select piece #2

  1. Kerry (Divider) initially cuts the sandwich:
  2. Where will Kerry cut the sandwich? (Vito or Pepe Section)

Cuts in Vito because it have value greater than ½ at 4/5

  1. What is the actual size of the two pieces Kerry will create?

Piece #1: ; V = 3.75 inches of Vito

Piece #2:1.25 inches of Vito + 6 inches of Pepe

  1. What piece will Ramona (Chooser) take and what does she value it at?

Piece #1: (1/3) (3.75/6) = 5/24

Piece #2: (1/3) (1.25/6 Vito) + (2/3)(6/6) = 19/24 Ramona will select piece #2,

7)Use the chart below to find the final settlement using The Method of Sealed Bids.

A / B / C / D
House / 180,000 / 200,000 / 190,000 / 185,000
Cabin / 60,000 / 50,000 / 40,000 / 55,000
Boat / 16,000 / 12,000 / 18,000 / 10,000
Fair Dollar Share / 256000/4 =
$64,000 / 262000/4 =
$65,500 / 248000/4 =
$62,000 / 250000/4 =
$62,500
Items Allocated / Cabin @
$60,000 / House @
$200,000 /
Boat @
$18,000 / Nothing @
$0
Owe “Pay”/
Owed “Get”
Money / Get
$4,000 / Pay
$134,500 /
Get
$44,000 /
Get
$62,500
Surplus /
[134500 – (4000 + 44000 + 62500)]/4 = 6,000
Final Settlement / Cabin
Get 10,000 / House
Pay 128,500 /
Boat
Get 50000
/
Get
68,500

8) Use the chart below to find the final settlement using The Method of Sealed Bids.

A / B / C
House / 180,000 / 190,000 / 200,000
Boat / 42,000 / 50,000 / 31,000
Fair Dollar Share / 222000/3 =
$74,000 / 240000/3 =
$80,000 / 231000/3 =
$77,000
Items Allocated / Nothing
$0 / Boat @
$50,000 / House @
$200,000
Owe “Pay”/
Owed “Get” Money / Get
$74,000 / Get
$30,000 / Pay
$123,000
Surplus / [123000- (74000 + 30000)]/3 = $6333.33
Final Settlement / Get
$80,333.33 / Boat
Get $36,333.33 / House
Pay $116,666.67

9)3 siblings are trying to split up triangle, square, and circle toys. There are a total of 18 toys. Each sibling prefers the toys in different amounts. Your job is to take the given order of toys and perform a Method of Markers based on the siblings preferences.

Sibling #1 likes squares 3 times as much as triangles and likes circles twice as much as triangles.

Sibling #2 likes circles 3 times as much as squares and likes triangles twice as much as squares.

Sibling #3 likes all the toy shapes equally having already learned the value of sharing.

Values:Total:Fair Share:

Sibling #1: Squares = 3, Triangles = 1, Circles = 236 pts12 pts

Sibling #2: Squares = 1, Triangles = 2, Circles = 336 pts12 pts

Sibling #3: Squares = 1, Triangles = 1, Circles = 1 18 pts6 pts

Option #1: Sibling 1 Gets First Section

Leftover =

Option #2: Sibling 2 Gets First Section

Leftover =

Option #3: Sibling 3 Gets First Section

Leftover =