1. Fair Division
Fair Division Problem: A problem that involves the dividing up of an object or set of objects among several individuals (players) so that each individual considers the part he or she receives to be a fair portion.
Assumptions:   
Cooperation: players are willing participants
Rationality: players are rational
Privacy: players have no info on other players
Symmetry : players have equal rights
copyright 2005
Dr. Annette M. Burden

2. Fair Division - Objectives
Fair Share: each of the N individuals gets what he/she considers a fair 1/N portion of the whole.
What may be considered a fair share by 1 player may not be considered a fair share by another player
It is possible for a player to get a fair share portion but not the preferred piece.
Envy Free: Each player gets a piece of the whole that he/she considers is the best share
copyright 2005
Dr. Annette M. Burden

3. Fair Division - Types
Origins go back 5,000 years. Modern era of fair division in math began in Poland during WWII
Continuous: The item(s) can be divided many ways & by small amounts (pie, cake, land, etc.)
Discrete: The item(s) consist of objects that cannot be split up (boat, book, etc.)
Mixed: The item(s) are both continuous and discrete
copyright 2005
Dr. Annette M. Burden

4. Continuous Methods - Overview
Divider - Chooser: 2 player game. One player cuts, other chooses.
Lone Divider: 3 player game. One player cuts, two players choose.
Lone Chooser 3 player game. Two players cut, one player chooses.
Last Diminisher: More than 3 players
copyright 2005
Dr. Annette M. Burden

5. Basic Concept
Jerry buys a chocolate-strawberry cake for $20. Jerry values chocolate 4 times as much as he values strawberry.
What is the value of the strawberry ½ of the cake?
4x + x = $20 or 5x = 20 or x = $4
What is the value of the chocolate ½ of the cake?
4($4) = $16
A piece of the cake is cut as shown at left. What is the value of the piece to Jerry?
A piece of the cake is cut as shown at left.
Chocolate
Strawberry
400
What is the value of the piece to Jerry?
($16)(40/180) + ($4)(20/180) = $4.00
200
copyright 2005
Dr. Annette M. Burden

6. Basic Concept Which of the three slices are fair shares to:
Whole Cake s1 s2 s3
Andy
$12.00
$3.00
$5.00
$4.00
Paul
$15.00
$4.50
$6.50
Cheryl
$13.50
Which of the three slices are fair shares to:
Andy: $12/3 players = $4 so any piece over $4 or s2,s3
Paul: $15/3 players = $5 so any piece over $5 or s3
Cheryl:$13.50/3 players = $4.50 so any piece over $4.50 or s1,s2,s3
copyright 2005
Dr. Annette M. Burden

7. Divider – Chooser Method
Player A divides cake into 2 parts in any way he or she desires.
Player B chooses the piece he or she wants.
Envy-free scheme.
copyright 2005
Dr. Annette M. Burden

8. Lone-Divider Procedure: 3 players
By random draw, one of the 3 players is designated to be the divider, D. The other 2 players will be choosers, C1 and C2
D divides the cake into 3 pieces s1,s2,s3 as equal as possible.
C1 declares which of the 3 pieces are a fair share to him. Independently, C2 does the same. These are the choosers’ bids.
How the pieces are divided:
Case I: if c1 and c2 like different pieces then D gets the piece neither of the choosers want
Case II: if c1 and c2 both like the same piece and both disapprove of the same piece, then D gets the piece that the choosers both disapprove of. The remaining pieces are put together and the divide and choose method is used to determine who gets what.
copyright 2005
Dr. Annette M. Burden

9. Lone-Divider Procedure: 3 players
copyright 2005
Dr. Annette M. Burden

10. Steinhaus Lone-Divider Procedure: 3 players
33 1/3 % c1
35%
10%
55% c2
40%
25% s1 s2 s3 D
33 1/3 % c1
30%
40% c2
60%
15%
25%
copyright 2005
Dr. Annette M. Burden

11. Lone Chooser Procedure
3 players
Players A and B divide the cake using the Divide and Choose method
Player A then subdivides her share into thirds while player B subdivides his share into thirds.
Player C then selects a 1/3rd share from player A and a 1/3rd share from player B.
Players A & B keep their remaining shares.
copyright 2005
Dr. Annette M. Burden

12. Lone Chooser
copyright 2005
Dr. Annette M. Burden

13. Last Diminisher (4 player example)
A cuts ¼ of cake and hands it to B
If B feels piece > ¼, B trims piece placing trimmings with remainder of cake and passes cut piece to C. If B feels piece is < to ¼, then B passes piece to C without trimming it.
With the piece given to him by B, C does same procedure as B, passing the trimmed or untrimmed piece onto D.
D does same as other players, but D is the last player, so if D trims the piece, D keeps it and exits the game. Otherwise, the piece goes back to the last player who trimmed it or to A if no one trimmed it and that player exits the game.
Process starts over by cutting another piece of approximately 1/3 of the original from the remaining cake.
When down to 2 players, use divide and choose scheme. 
copyright 2005
Dr. Annette M. Burden

14. Last Diminisher Ann – cuts ¼ of cake Bob- trims the piece Carl - pass
Deb- pass
Bob claims the slice and is out
Of the game.
copyright 2005
Dr. Annette M. Burden

15. Last Diminisher Ann – cuts ¼ of cake Carl - pass Deb- pass
Ann gets the slice of cake
copyright 2005
Dr. Annette M. Burden

16. Last Diminsher Randomly select a divider between Carl and Deb.
Use divider-chooser method to finish dividing the cake
copyright 2005
Dr. Annette M. Burden

17. Discrete Methods - Overview
Method of sealed bids: Used primarily to divide up an inheritance
Method of Markers: Used primarily to divide up many items that are similar in value
copyright 2005
Dr. Annette M. Burden

18. Method of Sealed Bids
Each player independently assigns a value all assets to be divided.
Award items to the highest bidders.
Compute each person’s total by adding each person’s bids of the items.
Compute each person’s fair share by dividing his/her total by the total number of bidders.
copyright 2005
Dr. Annette M. Burden

19. Method of Sealed Bids
Subtotal for each bidder = Item awarded – (player’s value of item awarded – fair share)
Surplus = fair share – sum of each player’s value of item awarded
Extra = surplus divided by total number of bidders.
Final Division for each bidder = Item awarded – (player’s value of item awarded – fair share) + extra
copyright 2005
Dr. Annette M. Burden

20. Method of Sealed Bids Object James Art Martha Cleo House $80.000
$75,000
$90,000
$60,000
Car
$10,000
$12,000
$13,000
$15,000
Totals
$87,000
$103,000
Fair Share
(Totals/4)
$22,500
$21,750
$25,750
$18,750
Subtotal
House – (90, ,750)
Car –($15, ,750)
Surplus in estate: $64,250 - $3,750 - $22,500 - $21,750 = $16,250
Extra awarded to each: $16,250/4 = $4,062
Totals
$22,500
$21,750
House - $64,250
Car + $3,750
surplus/4
$4,062.50
Final Division
$26,565.50
$25,812.50
House - $60,187.50
Car -$7,812.50
copyright 2005
Dr. Annette M. Burden

21. Method of Markers
The items are laid out in a row with no particular ordering of the items in mind.
(Bidding) Each player independently divides the row into N fair shares by placing N-1 markers between the items.
(Allocations) Scan the array from left to right until the first 1st marker of any player is located. That player is allocated the items up to his/her 1st marker and the player exits. His/her remaining markers are removed.
copyright 2005
Dr. Annette M. Burden

22. Method of Markers
Continue the Allocation procedure scanning from left to right until the first 2nd marker of any player is located. That player is allocated the items from his/her 1st marker to his/her 2nd marker. His/her remaining markers are removed and that player exits. Continue in this manner until all players have been allocated a fair portion of the items.
(Leftovers) The leftover items can be divided among the players via a lottery procedure.
copyright 2005
Dr. Annette M. Burden

23. Method of Markers A1 A2 A3 C1 C3 D3 C2 D2 B3 D1 B1 B2
4 players so there should be 3 divisions markers per player
Black arrows: player A. Red arrows: player B. Blue arrows: player C.
Green arrows: Player D
copyright 2005
Dr. Annette M. Burden

24. Method of Markers A1 A2 A3 C1 C3 D3 C2 D2 B3 D1 B1 B2
Player A (Black) is the first marker encountered, so player A gets the
Grouping taking him to his 1st marker. Player A exits the game and
The remainder of the black markers are removed.
copyright 2005
Dr. Annette M. Burden

25. Method of Markers C1 C3 C2 D2 D3 D1 B1 B2 B3
Player C (Blue) is the first 2nd marker encountered, so player C gets
the items between the 1st blue and 2nd blue markers. Player C exits the
Game and the remainder of the blue markers are removed.
copyright 2005
Dr. Annette M. Burden

26. Method of Markers D2 D3 B2 B3 D1 B1
Player B (Red) is the first 3rd marker encountered, so player B gets
the items between the 2nd red and 3rd red markers. Player B exits the
Game and the remainder of the red markers are removed.
copyright 2005
Dr. Annette M. Burden

27. Method of Markers D1 D2 D3
Player D (Green) gets the last group of items from the green 3rd marker
To the end of the row. The remaining 2 items are divided evenly
Among the 4 players.
copyright 2005
Dr. Annette M. Burden

28. Credits Tannenbaum, Excursions in Modern Mathematics, 5th ed
copyright 2005
Dr. Annette M. Burden