Chapter 3: The Mathematics
of Sharing
Fair-Division Games
Basic Elements
• The goods: the item(s) being divided.
Notation: S
• The players: the set of parties amongst which
S is divided
• The value systems: Each player has an
internalizes value system that gives the player
the ability to quantify the value of the goods
or any of its parts, ie To me, that’s worth $37
Basic Assumptions
• Rationality: Each player is a rational entity
seeking to maximize his/her share of S. A
player’s moves are based on reason alone (no
mind games).
• Cooperation: The players are willing
participants and accept the rules of the game
as binding. (No outside judges)
More Assumptions
• Privacy: Players have no useful info on the
other players’ value system and what kinds of
moves others will make in the game.
• Symmetry: Players have equal rights in sharing
S. So, each player is entitled to at least a
proportional share of S.
• The ultimate goal is to end up with a fair
division of S, that is, to divide S into N shares
and assign shares to players in such a way that
each player gets a fair share.
• Suppose that s denotes a share of S and that P
is one of N players. We say that s is a fair share
to player P if s is worth at least 1/N of the
total value of S in the opinion of P.
Jif Peanut Butter Commercial
http://www.youtube.com/watch?v=AdYFVN35h5w
2 Players: Divider-Chooser Method
• You cut and I choose.
• It’s always better to be the chooser!
Dividing a Cake
• We have a cake which is half chocolate, half
lemon
• You like both kinds of cake equally. I like
chocolate but detest lemon. (And we don’t
know what the other person’s preferences
are.)
Is this a fair cake division?
• You cut. I will choose.
YES!
• To you, the chocolate is worth 50% and the
lemon is worth 50%.
• To me, the chocolate is worth 100%, and the
lemon is worth 0%.
• Your piece is worth 50%. My piece, in my eyes,
is worth 66.66%!
• Since we both get a 50% or better share, this
is a fair division!
Other methods for more players
• Lone-Divider
• Lone-Chooser
• Last Diminisher
Method of Sealed Bids
• Parents leave a cabin, a vintage car, and a
Picasso to their children Art, Betty, Carla, and
Dave.
• Assumptions:
1. Each player must have enough money to play the
game. Each must be prepared to buy some or all
of the items, in order to make honest bids.
2. Each player must accept money as a substitute
for any item.
• Step 1: Bidding
Each of the players makes a bid for each of the
items in the estate. Bids are secret.
Art
Betty
Carla
Dave
Cabin
220,000
250,000
211,000
198,000
Car
40,000
30,000
47,000
52,000
Picasso
280,000
240,000
234,000
190,000
• Step 2: Allocation
Each item goes to the highest bidder for that
item.
- Cabin: Betty
- Car: Dave
- Picasso: Art
- It is possible that one player gets multiple items,
or that another player gets none.
• Step 3: First Settlement
Depending on what items a player gets in the
previous step, he/she will owe money to or be
owed money by the estate.
Calculate each player’s fair-dollar share: For
each player, add their bids and divide by the
number of players.
Art
Betty
Carla
Dave
Cabin
220,000
250,000
211,000
198,000
Car
40,000
30,000
47,000
52,000
Picasso
280,000
240,000
234,000
190,000
Bid Total
540,000
520,000
492,000
440,000
Fair-dollar share
135,000
130,000
123,000
110,000
If the total value of the items the player gets is more than
his/her fair-dollar share, then the players pays estate the
difference.
If the total value of the items is less than the fair share,
then the player gets the difference in cash.
• Art: Fair-dollar share is 135,000 and gets the
painting worth 280,000. He owes the estate
145,000.
• Betty: Fair-dollar share is 130,000 and gets the
cabin worth 250,000. She owes 120,000.
• Carla: Fair-dollar share is 123,000. She gets no
items, so the estate owes her 123,000 in cash.
• Dave: Fair-dollar share is 110,000 and gets car
worth 52,000. The estate owes him 58,000.
• Step 4: Surplus
Art and Betty contributed 265,000 to the
estate, and Carla and Dave got 181,000. So,
there’s a surplus of 84,000!
The surplus is common money that belongs to
the estate, and is thus divided equally among
the players.
84,000/4 = 21,000
• Step 5: Final Settlement
Add the surplus money to the first settlement.
-Art: Gets painting and pays the estate
145,000-21,000 = 124,000
-Betty: Gets cabin and pays the estate
120,000-21,000 = 99,000
-Carla: Gets 123,000+21,000 = 144,000
-Dave: Gets car plus 58,000+21,000
Homework
• Read Chapter 4
• Finish Sealed Bids Worksheet
• In Chapter 3 Exercises: 1, 3, 53