Practice Problems on Fair Division

  1. Consider a fair division problem involving 5 players. The phrase "a player receives a fair share" describes the fact that
    1. the player receives a share that is at least as valuable as that of any other player.
    2. the player receives a share that is more valuable than that of any other player.
    3. the player receives a share that, in the player's opinion, has value that is equal to 20% or more of the total.
    4. the player receives a share that, in the player's opinion, has value that is exactly equal to 20% of the total.
    5. None of the above

  2. An estate consisting of a car, a boat a house and a diamond ring must be divided fairly among five heirs. This type of problem is called
    1. a continuous fair division problem.
    2. a discrete fair division problem.
    3. a mixed fair division problem.
    4. the method of sealed bids.
    5. None of the above

  3. Which of the following is a discrete fair division problem?
    1. dividing a gallon of ice cream
    2. dividing a tropical island
    3. dividing a cheese pizza
    4. dividing an antique car collection
    5. None of the above

    Questions 4 through 6 refer to the following situation: Three players (Tiffany, Paxton and Sandra; Paxton divider and Tiffany and Sandra choosers) are going to divide a cookie (a large one) fairly using the lone divider method. Paxton cuts the cake into three slices (s1, s2, and s3).

  4. If the choosers declarations are Sandra: (s2) and Tiffany: (s3}, which of the following is a fair division of the cookie?
    1. Sandra gets s1; Tiffany gets s2; Paxton gets s3.
    2. Sandra gets s3; Tiffany gets s2; Paxton gets s1.
    3. Sandra gets s2; Tiffany gets s3; Paxton gets s1.
    4. Sandra gets s2; Tiffany gets s1; Paxton gets s3.
    5. None of the above

  5. If the choosers declarations are Sandra: (S2, S3) and Tiffany : (S1, S3), which of the following is not a fair division of the cookie?
    1. Sandra gets S2; Tiffany gets S3; Paxton gets S1.
    2. Sandra gets S1; Tiffany gets S3; Paxton gets S2.
    3. Sandra gets S3; Tiffany gets S1; Paxton gets S2.
    4. Sandra gets S2; Tiffany gets S1; Paxton gets S3.
    5. None of the above

  6. If the choosers declarations are Sandra: (s2) and Tiffany: (s1, s2), which of the following is a fair division of the cookie?
    1. Sandra gets S2; Tiffany gets s1; Paxton gets S3.
    2. Sandra gets S2; Tiffany gets S3; Paxton gets s1.
    3. Sandra gets S3; Tiffany gets s1; Paxton gets S2.
    4. Sandra and Tiffany split s2; Paxton gets s1 and S3.
    5. None of the above

  7. Three players (Missy and Tiffany dividers and Paxton chooser) are going to divide a cake fairly using the lone chooser method. Using this method
    1. the first division consists of dividing the cake into 2 pieces, the second division consists of dividing each of these pieces into 2 pieces.
    2. the first division consists of dividing the cake into 3 pieces, the second division consists of dividing each of these pieces into 2 pieces.
    3. the first division consists of dividing the cake into 2 pieces, the second division consists of dividing each of these pieces into 3 pieces.
    4. the first division consists of dividing the cake into 3 pieces, the second division consists of dividing each of these pieces into 3 pieces.
    5. None of the above

  8. Paula and Ginger won a computer in a MAT 131 joint competition. Except for the computer, they own very little of value. As a result they have agreed to divide the computer fairly using the method of sealed bids. Paula bids 3,000 and Ginger bids 2600. After all is said and done, the final outcome is
    1. Paula gets the computer and pays Ginger 2600.
    2. Paula gets the computer and pays Ginger 1500.
    3. Paula gets the computer and pays Ginger 1400.
    4. Paula gets the computer and pays Ginger 1300.
    5. None of the above

    Questions 9 through 12 refer to the following example: Three heirs (Natasha, Taurus, and Ginger) must divide fairly an estate consisting of two items--a house and a painting--using the method of sealed bids. The players' bids (in dollars) are: 

      		Natasha	 	   Taurus  	 Ginger
House		 180,000           200,000         210,000
Painting	  30,000            40,000          33,000
  9. The original fair share of Ginger is worth
    1. $243,000.
    2. $81,000.
    3. $84,000.
    4. $33,000.
    5. None of the above

  10. After the initial allocation to each player is made there is a surplus of
    1. $24,000.
    2. $32,000.
    3. $19,000.
    4. $129,000
    5. None of the above

  11. After all is said and done, the final allocation to Natasha is:
    1. $76,333 in cash.
    2. $70,000 in cash.
    3. the painting and $46,333 in cash.
    4. the house and Natasha must pay $70,000.
    5. None of the above

  12. After all is said and done, the final allocation to Ginger is:

    1. $81,000 in cash.
    2. $129,000 in cash.
    3. the painting and $40,000 in cash.
    4. the house and Ginger must pay $122,667.
    5. None of the above

    Questions 13 through 16 refer to the following: Four players (Michelle, Jeniere, Melicent, Lolitta) agree to divide 15 items using the method of markers. The items are numbered 1, 2, ..., 15.


    Figure 1

    Each of the player's three markers are placed as follows:

    Michelle: immediately to the right of items 1, 6, 12
    Jeniere: immediately to the right of items 3, 8, 14
    Melicent: immediately to the right of items 2, 9, 13
    Lolitta: immediately to the right of items 2, 7, 12.

  13. Item 5
    1. goes to Michelle.
    2. goes to Jeniere
    3. goes to Melicent.
    4. goes to Lolitta.
    5. is left over.

  14. Item 9
    1. goes to Michelle.
    2. goes to Jeniere
    3. goes to Melicent.
    4. goes to Lolitta.
    5. is left over.

  15. Item 11
    1. goes to Michelle.
    2. goes to Jeniere
    3. goes to Melicent.
    4. goes to Lolitta.
    5. is left over.

  16. Item 15
    1. goes to Michelle.
    2. goes to Jeniere.
    3. goes to Melicent.
    4. goes to Lolitta.
    5. is left over.

    Questions 17 through 20 refer to the following situation: Missy and Sandra want to divide fairly the chocolate-strawberry cake shown below using the divider-chooser method. The upper half is chocolate the lower half strawberry.

    The total cost of the cake was $12.00. Missy values strawberry three times as much as she values chocolate, while Sandra values chocolate twice as much as she values strawberry.


    Figure 2

  17. In Sandra's eyes, the piece in Figure 3 is worth

    Figure 3
    1. $8.00
    2. $10.00
    3. $9.50
    4. $9.00
    5. None of the above

  18. In Missy's eyes, the piece in Figure 4 is worth

    Figure 4
    1. $4.50
    2. $7.50
    3. $10.50
    4. $10.00
    5. None of the above

  19. In Sandra's eyes, the piece in Figure 5 is worth

    Figure 5
    1. $8.00
    2. $6.00
    3. $7.00
    4. $5.00
    5. None of the above
  20. In Missy's eyes, the piece in Figure 6 is worth


    Figure 6
    1. $4.50
    2. $6.00
    3. $7.00
    4. $7.50
    5. None of the above

    Questions 21 through 24 refer to the following situation: Five players agree to divide a cake fairly using the last diminisher method. The players play in the following order: Amber first, Demetrius second, Shanni third, Sherri fourth, and Stacee last. Suppose that there are no diminishers in round 1 and Shanni and Sherri are the only diminishers in round 2.

  21. Which player gets her fair share at the end of round 1?
    1. Amber
    2. Demetrius
    3. Shanni
    4. Sherrie
    5. Stacee

  22. Which player is the first to cut the cake at the beginning of round 2?
    1. Amber
    2. Demetrius
    3. Shanni
    4. Sherrie
    5. Stacee
  23. Which player gets her fair share at the end of round 2?
    1. Amber
    2. Demetrius
    3. Shanni
    4. Sherrie
    5. Stacee

  24. Which player is the first to cut the cake at the beginning of round 3?
    1. Amber
    2. Demetrius
    3. Shanni
    4. Sherrie
    5. Stacee

    ADDITIONAL EXERCISES

  25. If John bids $50.00 and Joan bids $65.00 for a math 131 book, how would you reach a fair division of this object?

  26. Almaz, Eden, and Hiwet inherit a house by the Red Sea. Assume that their evaluations of the house are $305,000, $360,000, and $380,000 respectively. Describe a fair division.

  27. Hiwet and Lydia inherited a plot of land. If their monetary bids on the house are $205, 000 and $214,000, respectively,

    1. Describe a fair division for the two heirs when they receive equal shares.
    2. Describe a fair division for the two heirs if their shares are 2/5 and 3/5, respectively.

  28. Two people inherit a painting. Assume that their evaluations of the painting are $10,680 and $8,880, respectively.

    1. Describe a fair division for the two heirs when they receive equal shares.
    2. Describe a fair division for the two heirs if their shares are 1/5 and 4/5, respectively.

  29. Stacee, Paxton, and Ginger inherit a painting. Assume that their evaluations of the painting are $28,800, $22,600, and $20,000, respectively.

    1. Describe a fair division for the three heirs when they receive equal shares.

    2. Describe a fair division for the heirs if their shares are 1/2, 1/3, and 1/6, respectively.

  30. A parent leaves a house, a grand piano, and plot of land to be divided equally among four children who submit dollar bids on these objects as follows. Describe a fair division for the four heirs. 

    				Children
----------|-----------------------------------------------------------
Objects   |	First 		Second 		Third 		Fourth
----------|------------------------------------------------------------
House	  |	$240,000  	$225,000  	$200,000  	$290,000
----------|------------------------------------------------------------
Piano	  |	$30,000   	$20,000   	$25,000   	$10,000
----------|------------------------------------------------------------
Land      |	$100,000   	$140,000	$125,000   	$145,000
----------|------------------------------------------------------------

  31. In Exercise 30 describe a fair division for the four heirs when their shares are 2/10, 3/10, 1/10, and 4/10, respectively.

    Three indivisible objects X, Y, and Z are to be shared equally among four people. Assume that these three objects have dollar values to the four as follows: 

    				Objects
--------------|-----------------------------------------------------
People        |		  X		  Y		  Z
--------------|-----------------------------------------------------
First         |		$3,600		$4,500		$5,100
--------------|-----------------------------------------------------
Second        |		$3,900   	$4,800		$5,100
--------------|-----------------------------------------------------
Third         |		$2,700   	$5,700		$4,500
--------------|-----------------------------------------------------
Fourth        |		$4,200   	$3,900		$2,400
--------------|-----------------------------------------------------

  32. Describe a fair division for the four people when they receive equal shares.

  33. In Exercise 32 describe a fair division for the four people when their shares are 2/5, 1/5, 1/5, and 1/5, respectively.

  34. Describe an envy-free method of dividing a cake among four people.

  35. Is the divider-chooser method an envy-free method? If yes why, if not, why not?