Fair Division Games
Every game has:
Players
Players 1, 2, 3, …, N
Every player have the same set of assumptions of playing the game.
Assets
A bunch of stuff. Mathematically we consider it as a set of objects that may be the same
or different or any number of them. The objects themselves have no inherent value.
Value Systems
For any subset of the set S, Players place a numerical value representing their value of
that portion. Different players may place different values on different objects.
● Players 1,2,...,N we say have total values V_1, V_2,...,V_N for the set S.
● Different players have different preferences (e.g. Neapolitan Ice cream)
● Practically, we refer to percentages of total values each player holds. 30%, 100%
Assumptions of our games:
1) Rationality
● All players have a given, fixed value system.
● Only goal is to maximize value at conclusion of method.
● Other players can assume everyone else is rational as well.
2) Cooperation
● All players are bound to rules of method
● All methods must end after a finite number of moves by players.
3) Privacy
● All value systems should be private.
● All other players assume every player is maximizing some value system
4) Symmetry
● All methods must have : “Players have equal rights in sharing assets”
● All methods must guarantee at least 1/N of values V_1, V_2, …, V_N for
players P_1 to P_N respectively.
Definition
Fair Share
Assume N Players.
Let for any Player k appreciate the set S with value V_k. The subset of S, s, given to
Player k is called a fair share to Player k, if P_k values s at least 1/N of V_k.
Fair Division
Let S be divided completely into shares (s1,s2,..., sN) and assign them to the respective
N players. Then this division of assets is a fair division if each player values their share
as a fair share.
Example
Draw Lots, then each one chooses.
a) Alice, then Bob, then Cheryl
b) Bob, then Alice, then Cheryl
s1
s2
s3
Alice
32%
31%
37%
Bob
34%
31%
35%
Cheryl
33.3%
33.3%
33.3%
Definition:
Fair Division Method
Set of rules that when properly used by players guarantees that at the end of the game
each player will have received a fair share of the assets.
DIVIDER­CHOOSER METHOD
2 players
Player 1 ­ Divider
Player 2 ­ Chooser
Example: Can of coke. 2 children, both who hate each other. Both value the coke the
uniformly (the liquid is uniform/same throughout).
Divider: Better not make it unequal.
Chooser: Chooses the bigger half if it exists (PROTIP: use the little cup markings to check)
Result: is if hands are steady the distribution is exactly 50­50.
Example: Neapolitan ice cream. Chocolate ­ Vanilla ­ Strawberry
Preferences
Divider : Chocolate = Strawberry = Vanilla
Chooser : Chocolate = Chocolate = Chocolate. Strawberry = Vanilla = sad face
1) Divider: has no preference. Randomly divides right down the middle, one half
Chocolate/Vanilla, one half Vanilla/Strawberry. Both are worth 50% to them.
2) Chooser: Picks the chocolate/vanilla piece. Gets 100% (!) of his value. Divider gets
50% of his total value of the ice cream. This is a fair share.
What happens if we switch Divider/Chooser at start?
Rationality: Both want the most possible, and also minimize any lost.
Privacy: No way to know each others preferences.
Result: Method guarantees at least 50% of V_1 and V_2 to be distributed to players P_1
and P_2 respectively
LONE­DIVIDER METHOD
N Players
1 Divider ­ Player 1 (D)
(N­1) Choosers ­ Player 2 (C_1), Player 3 (C_2), …. , Player N (C_N­1)
Lone Divider Method for 3 Players
Step 0) Randomly pick the Divider
Step 1) Divide cake into 3 shares (s1,s2,s3)
Step 2) C1, C2 bids which 3 pieces are fair shares for them (at least one must be)
Step 3) Divide pieces into two types: Chosen (C) and Unwanted (U). Unwanted are
those pieces no one listed from above, and Wanted means that at least one Chooser
wants them.
“C­pieces” and “U­pieces”
Step 4)
Case 1: # of C pieces = 1. There are 2 unwanted (U) pieces. The Divider gets pick this
time and must choose a U­piece. The remaining two pieces are recombined (remember,
these 2 pieces combined must be over Two­Thirds of value for both choosers) and
becomes a 2­player Divider­Chooser game.
Case 2: # of C pieces = 2.
In this case its always possible for the two Choosers to get a piece (they both picked two
different ones, or one Chooser picked one and the other picked two, or they both picked
both).
The divider gets the remaining piece. Thats why the Divider divided very carefully.
Note: When Bid Lists have identical bids, serve the list with only one (1) bid. Then serve
based on mutual preference ({s1,s3}, {s3,s1} is easy to split). Finally, if bid lists are
identical, split randomly.
Example: Case 2
Divider: Dale (you can tell because he split evenly)
s1
s2
s3
Dale
33.3%
33.3%
33.3%
Cindy
35%
10%
55%
Cher
40%
25%
35%
Bid Lists:
Cindy ­ {s3,s1}
Cher ­ {s1,s3}
C = 2
s1
s2
s3
Dale
33.3%
33.3%
33.3%
Cindy
20%
30%
50%
Cher
60%
15%
25%
s1
s2
s3
Dale
33.3%
33.3%
33.3%
Cindy
20%
30%
50%
Cher
20%
15%
65%
Bid Lists:
Cindy ­ {s3}
Cher ­ {s1}
C = 2
Example: Case 1
Cindy ­ {s3}
Cher ­ {s3}
C = 1
Lone Divider for 4 players
1) Divider divides S into 4 pieces
2) The 3 Choosers submit lists of what they believe is a fair share.
3)
Case 1: Everything is good. We can serve shares in a way all 3 choosers are happy and
the Divider gets the last piece.
Case 2: Everything is bad. There are more bidders than than shares bidded The
crybabies are put in a corner and everyone else takes a share. We’re left with 2 players or
3 players. The rest of the shares are recombined and divided again with less players.
Example:
s1
s2
s3
s4
Demi
25
25
25
25
Carol
30
20
35
15
Clark
20
20
40
20
Cheryl
25
20
20
35
Bid Lists:
Example:
s1
s2
s3
s4
Demi
25
25
25
25
Carol
20
20
20
40
Clark
15
35
30
20
Cheryl
22
23
20
35
Bid Lists:
Example: Case 2
s1
s2
s3
Dale
33.3%
33.3%
33.3%
Cindy
20%
30%
50%
Cher
20%
15%
65%
Lone Chooser Method
1 Chooser ­ C
2 Dividers ­ D1, D2
1) D1 and D2 perform a fair division of S into 2 shares between them (Divider­Chooser)
2) Each divider splits their share into three pieces. D1, D2 splits their shares s1, s2 into
(s1_a,s1_b, s1_c), (s2_a,s2_b,s2_c) respectively.
3) The chooser C selects one of D1’s pieces and one of D2’s pieces ( ⅙+⅙ = ⅓ of S). The
dividers get to keep the remaining pieces.
Method of Sealed Bids
Or “silent auction” / ”sealed bid auction”
Step 1) Bidding: Everyone makes a bid with their money.
Step 2) Allocate items based on highest bid.
Step 3) First Settlement
Calculate the fair share value based on total money bid. This is the total value each
player should end this step with. Players must receive money from the estate if they don’t
receive fair share value in items. Or else they pay the estate to pay back the extra value
they received.
Step 4) Division of surplus.
The surplus is divided up equally.
Step 5) Count up everything after surplus.
Example:
Adele
Bev
House
200,000
100,000
Island
300,000
100,000
Total Value
500,000
200,000
Fair Share
250,000
100,000
Example:
Alice
Boffo
Crudler
Dan
House
400
450
410
390
Car
80
70
88
90
Stamp Coll.
600
620
630
580
Total Value
1080
1140
1128
1060
Fair Share
270
285
282
265
Method of Markers
N Players
Easiest method here:
1) Line up the objects (randomly).
2) Each player picks N intervals by placing (N­1) markers that separate the intervals.
3) Go from left to right.
Pick up the first 1­marker from a player (randomly pick if markers are the same place). This
player gets the interval from beginning to 1­marker. Remove the players’ other markers.
Find the closest 2­marker. Pick this players interval from 1­marker to 2­marker. Remove
player.
Keep repeating for 3,4,.... N­marker.