Transcript 2.4 - 2.5 The Shapley

§ 3.1 Fair-Division
Fair-Division Games
 Every fair-division game is made up of:
1. A set of goods to be divided. We will
refer to this set as S.
2. A set of players : P1 , P2, P3 , . . . , ,
PN . Each player has his or her own
value system.
Fair-Division Games
 Given the goods and the players, our
goal is to end up with a fair division of
S--in other words, we wish to divide S
so that each player gets a fair share.
 In this chapter we will consider different
sets of rules called fair-division
methods.
Fair-Division Games
In each fair-division game we consider we will
make the following assumptions about the
players:
 Cooperation: Players agree to follow and
accept the rules of the game. The game will
end with a division of S after a finite number
of moves.
 Rationality: Players act rationally--their
value systems conform to the basic arithmetic
laws.
 Privacy: Players have no ‘inside’
Fair-Division Games
If these assumptions hold then a fairdivision method is guaranteed to give
every player an opportunity to get a fair
share of S.
Fair-Division Games
 Consider a share s of S and a player
P. If there are N players then a fair
share to player P is a share that is worth
at least (1/N )th of the total value of S to
player P.
Fair-Division Games
 There are three types of fair-division
game:
1. Continuous: The set S is divisible in
infinitely many ways.
2. Discrete: The set S is made up of
indivisible objects.
3. Mixed: Some objects in S are
continuous while others are discrete.
Example: (exercise 1, pg 112)
Alex buys a
chocolate-strawberry mousse cake for $12.
Alex values chocolate 3 times as much as he
values strawberry.
(a) What is the value of the chocolate half of
the cake to Alex?
(b) What is the value of the strawberry half of
the cake to Alex?
(c ) A piece of the cake is cut as shown in (ii).
What is the value of the piece to Alex?
Example: (exercise 5 pg 112) Three players
(Ana, Ben and Cara) must divide a cake among
themselves. Suppose the cake is divided into 3 slices
s1, s2 and s3. The values of the entire cake and of each
of the 3 slices in the eyes of each of the players are
shown in the following table.
(a) Indicate which of the three slices are fair shares to
Ana.
Whole
s1
s
s3
(b) Indicate which of the three
slices are2 fair shares
to
Cake
Ben.
Ana which
$12.00
$3.00
$5.00
$4.00
(c) Indicate
of the three
slices are
fair shares
to
Cara.
Ben
$15.00
$4.00
$4.50
$6.50
Cara
$13.50
$4.50
$4.50
$4.50
§ 3.2 The Divider-Chooser
Method
The Divider-Chooser Method
 The Idea: “You cut -- I choose.”
 Given two players, one player is
designated as the divider and the other
is said to be the chooser.
 The divider divides the set S into two
pieces.
 The chooser selects the piece he or
she wants.
§ 3.3 The Lone-Divider Method
The Lone-Divider Method
(for three players)
 Preliminaries: One of the players is
designated as the divider, D. The other
two players will be choosers C1 and C2.
These assignments will be made
randomly.
The Lone-Divider Method
(for three players)
 Step 1. The divider D divides S into
three pieces: s1, s2 and s3.
 Step 2. C1 declares which of the three
pieces are fair to him/her. C2 does the
same independently. Each player must
bid for a piece that they feel is worth 1/3
or more of S.
The Lone-Divider Method
(for three players)
 Step 3. The pieces are distributed.
How? This depends on the bids. We
will separate the pieces into groups: Cpieces (“chosen” pieces that are listed
in one of the chooser’s bids) and Upieces (unwanted pieces that neither
chooser wants). There are two cases to
consider. . .
The Lone-Divider Method
(for three players)
 Step 3.
Case 1. There are at least two Cpieces. Give each chooser one of the
pieces they bid for and give the divider
the remaining piece. (Players may
swap pieces after this if they desire.)
The Lone-Divider Method
(for three players)
 Step 3.
Case 2. There is only one C-piece.
This means that there are two U-pieces-one is given to the divider. The other
U-piece is recombined with the C-piece
to make a single big piece called the Bpiece. The B-piece is now divided by
the two chooser using the dividerchooser method.
Antigonus
is randomly323
picked
to be theAlexander
Example:
Macedonia,
BCE.
divider. He cuts the empire into three
the Greatpieces:
has just
died and his empire
is to be
Greece/Macedonia
(s1), Egypt
Asia generals
(s3).
divided by(sthe
three
Antigonus, Ptolemy
2) and
and Salauceus.
Antigonus
s1
s2
s3
33 1/3 %
33 1/3 %
33 1/3 %
25%
55%
20%
5%
60%
35%
Rome
Ptolemy
Carthage
Salaceus
Athens
Antigonus
is randomly323
picked
to be theAlexander
Example:
Macedonia,
BCE.
divider. He cuts the empire into three
the Greatpieces:
has just
died and his empire
is to be
Greece/Macedonia
(s1), Egypt
Asia generals
(s3).
divided by(sthe
three
Antigonus, Ptolemy
2) and
The following bids would be entered:
and Salauceus.
s
s2
s3
33 1/3 %
33 1/3 %
33 1/3 %
25%
55%
20%
5%
60%
35%
Ptolemy: { s2 } 1
Salaceus: { s2 , s3 }
Antigonus
Rome
Ptolemy
Carthage
Salaceus
Athens
The Lone-Divider Method
(for more than three players)
 Step 3.
Case 1. There is a way to give each
chooser one of the shares listed in his
or her bid. The divider gets the last
unassigned share. Players may swap
pieces after this is done.
The Lone-Divider Method
(for more than three players)
 Step 3.
Case 2. There is a standoff. A standoff
occurs when more than one chooser is
bidding on the same share, or three
chooser are bidding on just two shares
or if K choosers are bidding on less than
K shares. When this happens we first
set aside the shares and players
involved in the standoff from those that
The Lone-Divider Method
(for more than three players)
 Step 3.
Case 2. (cont’d) The players not in the
standoff are assigned their shares and
quit playing. All the shares that are left
are recombined into the whole group
and the process is done all over again.
Example: (exercise 16, pg 117)
Four
partners (Childs, Choate, Chou and DiPalma)
want to divide a piece of land fairly using the
lone-divider method. Using a map, DiPalma
divides the land nto four parcels (s1 ,s2 ,s3 ,s4
)and the choosers make the following
declarations:
Childs: { s2 ,s3 }
Choate: { s1 ,s3 }
Chou: { s1 ,s2 }
(a) Describe a fair division of the land.
Example: (exercise 21, pg 118)
Six
players want to divide cake fairly using the lonedivider method. The divider cuts the cake into
six slices (s1 ,s2 ,s3 ,s4 , s5 ,s6 ) and the choosers
make the following declarations:
Childs: {s2 ,s3 }
Choate: {s1 ,s3 }
Chou: {s1 ,s2 }
(a) Describe a fair division of the cake.