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Math 103, Section 11, Tuesday, February 22, 2011
Chapter 3 Handout
1
The Mathematics of Sharing
Assignment Chapter 3A
Please do Exercises 12,18,30, and 32.
Homework will be submitted to Sakai.
Homework to be discussed in class on Friday February 25.
Exercise 7,9, and 11
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Chapter 3
3.1 FairDivision Games: The Mathematics of Sharing
Basic elements of a fairdivision game:
1. The goods or booty
Examples:
cake, pizza, jewelry, art, land, water rights, drilling rights,
chores
2.
The players
Examples: individuals, clubs, political parties, states, nations
3
The value systems
Examples: Each player can look at the set S or any part of the set S and
assign to it a value. Either in absolute terms (to me, that’s worth $3.00)
or in relative terms (to me, that piece is worth 25% of the total value of
S).
Basic Assumptions:
1. Rationalityeach player wants to maximize his or her share of the booty
S. Player uses reason and no psychology, mind games, etc.
2. Cooperation. Players accept the rules of the game. No judges or
referees.
3. Privacy. Players have no useful information on the other players’ value
systems and thus of what kinds of moves they are going to make in the
game.
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Symmetry. Players have equal rights in sharing the set S. A
consequence of this assumption is that, at a minimum, each player is
entitled to a proportional share of S.
for 2 players, each is entitled to at least onehalf of S
for 3 players, each is entitled to at least onethird of S.
Fair Shares and FairDivision Methods
Suppose that s denotes a share of the booty S and that P is one of the
players in a fairdivision game with N players. We will say that s is a
fair
share
to player P if s is worth at least 1/N th of the total value of S in the
opinion of P.
Given the booty S and players P1,P2,P3,,,,PN,
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 Spring '07
 Berkowitz
 Math

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