1.	After having been roommates for four years at college, Alex and Bob are moving on.  Several items they have accumulated belong jointly to the pair, but now must be divided between the two.  They assign points to the items as follows:

Object	
Alex's points	
Bob's points	

Bicycle	
10	
12	

Textbooks	
20	
16	

Barbells	
5	
2	

Rowing machine	
7	
10	

Music collection	
8	
11	

Computer	
15	
17	

Novels	
20	
22	

Desk	
15	
10	

Use the adjusted winner procedure to determine a fair division of the property. 	

2.	In recent labor-management negotiations, several issues were identified, and points assigned to them indicating relative importance to each side, as follows:

Issue	
Management's points	
Labor's points	

Base pay	
20	
23	

Incentive pay	
10	
5	

Health care benefits	
6	
12	

Worker safety	
8	
15	

Opportunity for promotion	
20	
13	

Retirement package	
20	
20	

Employee accountability	
16	
12	

Use the adjusted winner procedure to determine a fair resolution between labor and management. 	

3.	A husband and wife are getting divorced.  The property to be divided, and the relative value of the items to each, are as follows:

Object	
Husband's points	
Wife's points	

House	
30	
15	

Car	
25	
25	

Cabin	
20	
40	

Boat	
16	
15	

Television	
5	
3	

Stereo	
4	
2	

Use the adjusted winner procedure to determine a fair division of the property.	

4.	The administration and the student body at a local college are at odds over several issues.  The relative importance of each issue to each group is as follows:

Issue	
Administration's points	
Student's points	

Mandatory meal plan	
40	
15	

Mandatory campus living	
5	
30	

Cars on campus	
5	
20	

Campus curfew	
10	
15	

Weekday fraternity parties	
40	
20	

Use the adjusted winner procedure to determine a fair resolution.	

5.	Henry and Lisa inherit a house.  If their monetary bids on the house are $135,000 and $114,000, respectively, what is the fair division arrived at by the Knaster inheritance procedure? 	

6.	Two people inherit a painting.  If their monetary bids on the painting are $12,560 and $9750, what is the fair distribution arrived at by the Knaster inheritance procedure? 	

7.	John, Ken, and Linda inherit a painting.  If their monetary bids on the painting are $25,200, $21,600, and $18,000 respectively, what is the fair distribution arrived at by the Knaster inheritance procedure? 	

8.	John, Ken, and Linda inherit a painting.  Their monetary bids on the painting are $25,200, $21,600, and $18,000 respectively.  If Linda had known of John and Ken's evaluations in advance, should she have changed her own evaluation? 	

9.	A parent leaves a house, a farm, and a piece of property to be divided equally among four children who submit dollar bids on these objects as follows:
	
Children	

Objects	
First	
Second	
Third	
Fourth	

House	
120,000	
125,000	
100,000	
90,000	

Farm	
80,000	
60,000	
70,000	
75,000	

Property	
40,000	
30,000	
25,000	
45,000	

What is the fair division arrived at by the Knaster inheritance procedure? 	

10.	Three indivisible objects A, B, and C are to be shared equally among four people.  Assume that the objects have monetary values to the four people as follows:
	
People	

Object	
First	
Second	
Third	
Fourth	

A	
8600	
3500	
2300	
4800	

B	
5500	
4200	
4400	
2700	

C	
3600	
5100	
3400	
2300	

What is the fair distribution arrived at by the Knaster inheritance procedure? 	

11.	Suppose that Bob and Carol view a cake as shown below:

 
Assume that Bob and Carol use divide-and-choose to divide the cake between them, and that all cuts are made corresponding to vertical lines.  If Bob is the divider, how many units of value will Bob and Carol think he or she is receiving? 	

12.	Suppose that Bob and Carol view a cake as shown below:

 
Bob and Carol will use divide-and-choose to divide the cake between them, and all cuts made correspond to vertical lines.  Will Bob like the results better if he is the divider or the chooser?  Explain your answer. 	

13.	Suppose that Bob and Carol view a cake as shown below:

 
Assume that Bob and Carol use divide-and-choose to divide the cake between them and that all cuts are vertical.  If Bob is the divider, how many units of value will Bob and Carol think he or she is receiving?  Is this different if Carol is the divider?  Explain.	

14.	Suppose that Bob and Carol view a cake as shown below:

 
Assume that Bob and Carol know how each other values the cake (and that neither is spiteful).  Suppose they are to divide the cake using the rules of divide-and-choose but not necessarily the strategies.  If Bob is the divider, where will he cut the cake and how many units of value will Bob and Carol think he or she is receiving? 	

15.	Suppose that Bob, Carol, and Ted view a cake as shown below:

 
Assume that each player regards a piece as acceptable if and only if it is at least five square units of value. Assume all cuts are vertical.  Provide a total of three drawings to show how each player views a division of the cake, by Bob, into three pieces he or she  considers to be the same size or value.	

16.	Suppose that Bob, Carol, and Ted view a cake as shown below:

 
Assume that each player regards a piece as acceptable if and only if it is at least five square units of value. Assume all cuts are vertical.  Provide a total of three drawings to show how each player views a division of the cake by Bob into three pieces he considers to be the same size or value.  Now use the Steinhaus lone-divider method to get a proportional allocation, indicating who approves of which piece and how large a piece each player thinks he or she is receiving.	

17.	Assume that Bob, Carol, Ted, and Alice view a cake as shown below:

 
Notice that each player views the cake as having 16 units of value.  Assume that the four players use the last-diminisher method to divide the cake among themselves.  Give a step-by-step description of the cuts made and the pieces with which each player exits assuming that the order in which they proceed is Bob, then Carol, then Ted, and then Alice.	

18.	Assume that Bob, Carol, Ted, and Alice view a cake as shown below:

 
Notice that each player views the cake as having 16 units of value.  Assume that the four players use the last-diminisher method to divide the cake among themselves.  Give a step-by-step description of the cuts made and the pieces with which each player exits assuming that the order in which they proceed is Alice, then Ted, then Carol, and then Bob.	

19.	Suppose players 1, 2, and 3 view a cake as shown below, and that all cuts are vertical.

 
Describe the steps in using the last-diminisher method to divide the cake. Assume players will use the order of player 1, then player 2, and then player 3 and that all pieces will be cut from the left side of the cake.  Use drawings to clarify your description.	

20.	Suppose players 1, 2, and 3 view a cake as shown below and that all cuts are vertical.

 
Illustrate the envy-free procedure for n = 3 (yielding an allocation of part of the cake) by following steps a–c below:
a) Provide a total of three drawings to show how each player views a division of the cake by player 1 into three pieces he or she considers to be the same size or value.  Label the pieces A, B, and C.
b) Redraw the picture from player 2's view and illustrate the trimmings of piece B that he or she would do.  Label the trimmed piece b' and the actual trimmings T.
c) Indicate which piece each player would choose (and what he or she thinks its size is) if the players choose in the following order: player 3, player 2, player 1 according to the rules of the procedure.	

21.	Apply the remainder of the Selfridge-Conway procedure to what was obtained in question 20 by completing a–c below:
a) Draw a picture of T from each player's view.
b) The procedure calls for the player (other that player 1) who did not receive the trimmed piece to divide T into three pieces he or she considers to be the same size.  Illustrate this division, and label the pieces X, Y, and Z.
c) Indicate which parts of T (and the sizes or values) the players will choose when they go in the order: player 2, player 1, player 3.	

22.	Suppose Chris and Terry need to make a fair division of a stereo they jointly purchased.  Suppose also, that both agree Chris is entitled to 2/3 of the stereo and Terry to 1/3.  Chris and Terry place monetary bids on the value of the stereo of $500 and $450 respectively.  Modify the Knaster inheritance procedure to make a fair division of the stereo. 	

23.	Aunt Sally dies and leaves her estate to be shared between Ariel, Binky, and Chris.  The estate consists of a painting and a sculpture by a modern artist.  Aunt Sally had decreed that Ariel was to receive half her estate and that Binky and Chris were to share equally the remaining half.  The values that each heir placed on the painting and sculpture are given below.  Modify the Knaster inheritance procedure to make a fair division of the estate.

Object	
Ariel	
Binky	
Chris	

Painting	
4000	
6300	
6000	

Sculpture	
2300	
1800	
2400	

24.	Explain why the divide-and-choose method of cake division is considered an envy-free procedure for dividing a cake between two participants. 	

25.	Tom and Jerry must rake all the leaves in a yard.  They each view the yard as shown below and agree to divide the chore by the divide-and-choose method.  Tom will divide the yard in two parts and Jerry will choose the side he has to rake.  Describe the results of this fair division.

26.	Suppose Kelly and Jan want to take turns, using the bottom-up strategy, to allocate several textbooks currently held jointly.  Their ranked preferences are listed below:

 
If Kelly chooses first, how are books allocated? 	

27.	Suppose Kelly and Jan want to take turns, using the bottom-up strategy, to allocate several textbooks currently held jointly.  Their ranked preferences are listed below:

 
If Jan chooses first, how are books allocated? 	

28.	Suppose Tony and Jess want to take turns, using the bottom-up strategy, to allocate several items.  If Tony and Jess rank the same item as first on their preference lists, who gets this item? 	

29.	Suppose Tony and Jess want to take turns, using the bottom-up strategy, to allocate several items.  If Tony and Jess have the same last choice on their preference lists, who gets this item?	

30.	If Jack and Jill want to take turns, using the bottom-up strategy, to allocate ten items, what situation would make it possible for each person to get their top five choices? 	

31.	Abby and Annie must make a fair division of four objects inherited from their grandmother. If Abby and Annie use the adjusted winner procedure, what is the division of the objects? 
 
Object	
Abby's points	
Annie's points	

Brass bed	
30	
45	

Cedar chest	
50	
10	

Hand-sewn quilt	
15	
25	

Photo albums	
5	
20	

32.	Paul and Henry must make a fair division of three objects. If Paul and Henry use the adjusted winner procedure, what is the division of the objects?
 
Object	
Paul's points	
Henry's points	

Wooden toy collection	
60	
20	

Antique desk	
25	
45	

China	
15	
35	

33.	Four people must make a fair division of two classic cars. They use the Knaster inheritance procedure.  Their bids are shown below: 

A	
B	
C	
D	

T-Bird	
60,000	
90,000	
50,000	
50,000	

Mustang	
70,000	
45,000	
100,000	
80,000	

What is the division of the objects?	

34.	Three people must make a fair division of four objects. Using the Knaster inheritance procedure, the value each person places on the objects is shown below.

A	
B	
C	

Object 1	
160	
320	
360	

Object 2	
280	
370	
30	

Object 3	
430	
260	
160	

Object 4	
420	
270	
130	

What is the division of the objects?	

35.	Suppose Dave and Merrie want to take turns, using the bottom-up strategy, to allocate the contents of a box of computer software they received as a gift. Their ranked preferences are listed below:

1st	
2nd	
3rd	
4th	
5th	
6th	

Dave	
Flight	
Action	
Role-playing 	
Strategy	
Solitaire	
Word-processing	

simulator	
game	
game	
game	
 suite	

 Merrie	
Solitaire	
Strategy	
Action	
Role-playing	
Word-processing	
Flight	

 game	
 game 	
game	
suite	
 simulator	

If Merrie chooses first, what is the division of the objects? 	

Answer Key

1.	Alex gets the textbooks, the barbells, the desk, and 16/21 of the novels.  Bob gets the rest.	
2.	Management gets its way with regard to incentive pay, opportunity for promotion, and employee accountability, and 60% of its way on the issue of the retirement package.  Labor gets its way on the rest of the issues.	
3.	The husband gets the house, the boat, the television, the stereo, and 20% use of the car.  The wife gets the cabin and 80% use of the car.	
4.	The administration gets its way on the mandatory meal plan, and 11/12 of its way with regard to weekday fraternity parties.  The student body gets its way on the rest of the issues.	
5.	Henry receives the house and gives Lisa $62,250.	
6.	The first person receives the painting and gives the second person $5577.50.	
7.	John receives the painting and pays $15,600.  Ken receives $8400 and Linda receives $7200.	
8.	Yes.  Her bid should be just slightly lower than John's and higher than Ken's.  Her new bid is $25,194.	
9.	The first child receives the farm and pays $11,250.  The second child receives the house and pays $62,500.  The third child receives $57,500 and the fourth child receives the piece of property and $16,250.	
10.	The first person receives objects A and B and pays $8025.  The second person receives object C and pays $250.  The third person receives $4175, and the fourth person receives $4100.	
11.	Bob makes a vertical cut in the middle of the fourth column.  Carol chooses the piece on the left.  Bob thinks he gets 8 square units of value.  Carol thinks she gets 8.5 square units of value.	
12.	Bob will like the results better if he is the chooser.  If he is the divider, he will divide the cake into two pieces he thinks are both worth 8 units and so the piece he is left with is worth 8 units in his mind.  If Carol divides, Bob can choose a piece Carol thinks is worth 8 units but Bob believes is worth 10 units.	
13.	If Bob cuts, he makes a vertical cut between the fourth and fifth column.  Carol chooses the piece on the right.  Bob thinks he gets 8 square units of value.  Carol thinks she gets 12 square units of value. 
If Carol cuts, she makes a vertical cut two-thirds of the way across the fifth column.  Bob chooses the piece on the left.  Carol thinks she gets 8 square units of value.  Bob thinks he gets 10 square units of value.	
14.	Bob makes a vertical cut almost two-thirds of the way across the fifth column.  Carol chooses the piece on the right.  Bob thinks he gets almost 10 square units of value.  Carol thinks she gets slightly more than 8 square units of value.	
15.	
	
16.	

Carol approves of only piece C, and Ted approves of only piece C.  Thus, Bob is the only one to approve of piece A.  Bob is given piece A.  Carol and Ted use divide-and-choose on pieces B and C recombined.  If Carol is the divider, she makes a vertical cut in the middle of the fifth column.  Ted chooses the piece on the right.  Bob thinks he gets 5 square units of value.  Carol thinks she gets 6 square units of value.  Ted thinks he gets 7 square units of value.	
17.	Bob makes a vertical cut between the second and third column.  The piece made up of the first and second columns is, in Bob's eyes, 4 out of the 16 units of value, and thus one-fourth of the cake.
Carol sees this piece as 7 out of 16 square units of value, so she trims it by making a vertical cut between the first and second column.  The piece made up of column one is, in Carol's eyes, one-fourth of the cake.
Ted passes, as does Alice, because they both think column one represents no more than one-fourth of the cake.
Carol takes this piece (column one) and exits the game.
Bob now makes a vertical cut between column three and four.  The piece made up of column two and three is, in Bob's eyes, 4 square units of value.
Ted passes, as does Alice, because both think the piece represents no more than one-fourth of the cake.
Bob takes this piece, column two and three, and exits the game. 
Ted and Alice now use divide-and-choose on the rest of the cake (the fourth, fifth, and sixth columns).
Ted, as divider, makes a vertical cut between the fifth and sixth column.
Alice chooses the piece made up of columns four and five (she thinks there are seven square units of value).
Ted receives the piece made up of column six (he thinks there are 5 square units of value).	
18.	Alice makes a vertical cut between the fourth and fifth column.  The piece made up of the first four columns is, in Alice's eyes, 4 out of 16 square units of value, and thus is one-fourth of the cake.
Ted sees this piece as 8 out of 16 units of value, so he trims it by making a vertical cut between the first and second column.  The piece made up of column one is, in Ted's eyes, one-fourth of the cake.
Carol passes, as does Bob, because they both think column one represents no more than one-fourth of the cake.
Ted takes this piece (column one) and exits the game.
Alice now makes a vertical cut one-sixth of the way across column five.  The piece made up of columns two, three, four, and one-sixth of column five is, in Alice's eyes, 4 square units of value.
Carol sees this piece as 5.5 out of 16 square units of value, so she trims it by making a vertical cut between the third and fourth column.  The piece made up of columns two and three is, in Carol's eyes, one-fourth of the cake.
Bob passes because he thinks this piece represents no more than one-fourth of the cake.
Carol takes this piece (columns two and three) and exits the game.
Alice and Bob now use divide-and-choose on the rest of the cake (the fourth, fifth, and sixth columns).
Alice, as divider, makes a vertical cut 11/12 of the way across the fifth column.
Bob chooses the piece on the right (1/12 of column five plus column six).
Alice receives the piece on the left (column four plus 11/12 of column five).	
19.	Player 1 cuts between the third and fourth column, viewing the piece made of column one, two, and three as worth 5 square units of value and so one-third of the cake.
Player 2 views the piece made of the first three columns as worth 7 square units of value and so trims between the first and second column. The piece made up of the first column is, in Player 2's eyes, worth one-third of the cake.
Player 3 views this as worth no more than one-third of the cake and so passes.
Player 2 receives the piece made up of column one and exits the game.
Player 1 and Player 3 now use divide-and-choose on the remainder of the cake (columns two through six).
Player 1 cuts the cake halfway through the fifth column.  Player 3 chooses the part made of columns two, three, four, and half of five, for a total (in Player 3's eyes) of 8 square units of value.  Player 1 is left with the portion made of half of column 5 and column six for a total, in Player 1's eyes, of 6 square units of value.	
20.	a) 

b) 

c) Player 3 will choose A (which he thinks is worth 4 square units).
Player 2 will choose B (which he thinks is worth 3 square units).
Player 1 will receive C (which he thinks is worth 5 square units).	
21.	a) 

b) 

c) Player 2 chooses first and will take Z.  Player 1 chooses next and takes either X or Y (he considers them equal).  Player 3 receives the remaining piece.	
22.	Chris gets the stereo and pays Terry $158.33.	
23.	Ariel gets $3862.50.
Binky gets the painting and pays $3918.75.
Chris gets the sculpture and $56.25.	
24.	Divide-and-choose is envy free because the divider believes both pieces are of equal worth and the chooser gets to pick a piece he believes is worth at least half.	
25.	Tom divides the yard in the middle of the third column.  Jerry chooses the left side to rake.	
26.	Kelly selects Painting, Sculpture, Music.  Jan selects Architecture, Design, Textiles.	
27.	Jan selects Architecture, Sculpture, Design.  Kelly selects Painting, Music, Textiles.	
28.	Whoever goes first.	
29.	Whoever goes second.	
30.	If Jack's top five choices are the same as Jill's bottom five choices.	
31.	Annie gets the quilt, the photo albums, and 7/15 of the bed. Abby gets the rest.	
32.	Paul gets the toys and 2/7 of the desk. Henry gets the rest.	
33.	A gets $45,937.50.
B gets the T-Bird and pays $42,812.50.
C gets the Mustang and pays $49,062.50.
D gets $45,937.50.	
34.	A gets Object 3 and Object 4, and pays $140.
B gets Object 2 plus $156.68.
C gets Object 1 and pays $16.68.	
35.	Merrie gets the solitaire game, the action game, and the role-playing game. Dave gets the rest.