THE MATHEMATICS
OF SHARING:
FAIR-DIVISION GAMES
Notes 11 – Section 3.1
Essential Learnings
• Students will understand and be able to
determine the basic elements of fair-division
games.
• Students will understand fair shares and fairdivision methods.
Fair Division
• You have 4 children and 20 pieces of candy. How
should you divide the candy up?
• Is that a fair division if the candy pieces are not the
same (packs of gum, mini-hershey bars, etc.)?
Basic Elements of Fair-Game Division
• The goods (booty) – the item or items being
divided (such as candy, pizza, art, land, etc.). The
symbol S will be used to denote the goods.
• The players – the set of parties with a right to
share S (this can be individuals or institutions).
• The value systems – each player can quantify the
value of the goods or any of its parts (i.e. to me,
that piece is worth 30% of the total value of S).
Basic Assumptions
• Rationality – each player is a thinking, rational
entity seeking to maximize his or her share of the
booty S. A player’s moves are based on reason.
• Cooperation – the players are willing participants
and accept the rules as binding. There are no
outsiders such as judges or referees – just the
players and the rules.
Basic Assumptions
• Privacy – players have no useful information on
the other players’ value system or do not know
what moves the they are going to make.
• Symmetry – players have equal rights in sharing
the set S. Each player is entitled to a proportional
share of S.
Fair Shares
• Suppose that s denotes a share of the booty S and
that P is one of the players in a fair-division game
with N players. We will say that s is a fair share to
player P if s is worth at least 1/Nth of the total
value of S in the opinion of P.
• Such a share is often called a proportional fair
share, but for simplicity we will refer to it just as a
fair share.
Example
• Suppose we have 4 players and that S has been
divided into four shares: s1, s2, s3, s4.
• Paul values s1 at 12% of S, s2 at 27% of S, s3 at
18% of S, and s4 at 43% of S.
• s2 and s4 are both fair shares to Paul, since the
threshold for a fair share when dividing among 4
players is 25%.
Fair-Division Methods
• A fair-division method is a set of rules that define
how the game is to be played.
• Continuous fair-division game – the set S is
divisible infinitely many ways, and shares can be
increased or decreased by arbitrarily small
amounts. Examples – land, cake, pizza)
Fair-Division Methods
• Discrete fair-division game – the set S is made up
of objects that are indivisible like paintings,
houses, cars, boats, jewelry.
• Mixed fair-division game – some of the
components are continuous and some are
discrete. Example – dividing an estate consisting
of jewelry, a house, and a parcel of land.
Example – Sharing Pizza
• Homer and Marge jointly buy a half pepperoni and
half mushroom pizza. Marge values the
mushroom pizza three times as much as she
values the pepperoni pizza. Find the dollar value
to Marge of each of the follow pieces of pizza:
• The pepperoni half of the pizza
$ 4 .5 0
M  3P
M  P  18
3 P  P  18
4 P  18
P  $4 . 5 0
Example – Sharing Pizza
• Homer and Marge jointly buy a half pepperoni and
half mushroom pizza. Marge values the
mushroom pizza three times as much as she
values the pepperoni pizza. Find the dollar value
to Marge of each of the follow pieces of pizza:
• The mushroom half of the pizza
3
4.5  $13.50
M  3P
M  P  18
3 P  P  18
4 P  18
P  $4 . 5 0
Example – Sharing Pizza
• Homer and Marge jointly buy a half pepperoni and
half mushroom pizza. Marge values the
mushroom pizza three times as much as she
values the pepperoni pizza.
• If a slice of mushroom has an angle of 72, find
the dollar value to Marge of one slice of
mushroom.
72
180
13.50  $5.40
Example – Sharing Pizza
• Homer and Marge jointly buy a half pepperoni and
half mushroom pizza. Marge values the
mushroom pizza three times as much as she
values the pepperoni pizza.
• If a slice of pepperoni has an angle of 54, find the
dollar value to Marge of one slice of pepperoni.
54
180
4.50  $1.35
Assignment
p. 103: 1, 3, 7, 8, 9