What is the sum of the following
infinite series 1+x+x2+x3+…xn…
where 0<x<1?
A)
B)
C)
D)
E)
Infinity
1+2x
1+3x
1/x
1/(1-x)
Cooperative Game Theory
Coalitional Games
• Focus on what groups can accomplish if they
work together.
• Contrast to Nash equilibrium which focuses on
what individuals can do acting alone.
(sometimes known as non-cooperative game
theory)
Coalitional Game with transferable
payoffs
• A set of players N.
• A coalition S is a subset of N.
– Grand coalition is N itself.
• Coalitional game with transferable payoffs assigns
a value v(S) to every subset of S.
• An action for the coalition S is a distribution of
Its total value to its members.
Think of v(S) as an amount of “money” that the coalition
can earn on its own and can divide this money in any
way that adds to v(S).
Two Player unanimity game:
An Almost Trivial Example
•
•
•
•
Players 1 and 2.
Non-empty subsets are {1},{2},{1,2}.
Let v({1})=v({2})=0 and v({12})=1.
Set of actions available to coalition {1,2}, are
distributions of payoffs (x1,x2) such that x1≥0,
x2≥0, and x1+x2=1.
Three player majority game:
A (slightly) more interesting example
• Players 1, 2, and 3.
• Non-empty subsets of N={1,2,3} are N,
{12}, {13}, {23},{1},{2},{3}.
• A cake whose total value is 1 is to be divided. Any
coalition that is a majority can choose how to divide it.
• Then v({12})=v({23})=v({13})=v({123})=1 and
v({1})=v({2})=v({3})=0.
• Actions available to any coalition are possible divisions
of the cake. For example, coalition{12} can choose
any division such that x1≥0, x2≥0, and x1+x2=1.
Landowner-workers example
• Player 1 owns all of the land. There are k
landless workers. Total food produced is given
by a production function f(x) where x is the
number of people who work on the land.
• Landless workers can produce nothing on
their own. Landowner can only produce f(1)
with his own labor.
Coalition values for landowner worker
model
• Player 1 is the landowner. By himself he gets
v({1})=f(1).
Workers get nothing without land, so any
coalition S to which the landowner does not
belong has v(S)=0.
A coalition S to which the landowner does
belong gets V(S)=f(k+1) where k is the number
of workers in the coalition.
The Core
• The core of a coalitional game is the set of
outcomes x (actions by the grand coalition)
such that no coalition has an action that all of
its members prefer to x.
Examples:
• Two person unanimity game. Core consists of
all possible divisions. Why?
• Three person majority game. Core is the
empty set. Why?
Five people must decide how to
divide a pie of fixed size. Any 4 of
them can agree to overturn the status
quo agreement and redistribute the
pie among themselves.
A) One outcome in the core of this game is that
in which each player gets 1/5 of the pie.
B) One outcome in the core of this game is that
in which 4 players each get ¼ of the pie and
one player gets nothing.
C) The core of this game is empty.
Landowner-worker game, 2 workers
• Let x1,x2,x3 be an allocation of the output f(3)
from 3 people working on landowner’s land.
• If this in the core, then it must be that
• x1≥f(1)—otherwise coalition {1} could do better
by self.
• x1+x2 ≥f(2)—otherwise {12} could do better by
themselves
• x1+x3 ≥f(2)—otherwise {13} could do better by
themselves.
• x1+x2 +x3=f(3).
Conclusions
• x1+x3 ≥f(2) and x1+x2 +x3=f(3)
imply that x2≤f(3)-f(2)
Similarly x1+x2 ≥f(2) and x1+x2 +x3=f(3)
imply that x3≤f(3)-f(2)
Thus no worker gets more than the “marginal
product” of a third worker.
Example
• Suppose f(1)=3, f(2)=5, f(3)=6.
• Then we must have x1≥3, x2≤1, and x3≤1.
Any allocation where 2 and 3 each gets a nonnegative amount and 1 gets more than 3 units
will be in the core.
One owner two possible buyers
• Owner (person 1) has an object that is
worthless to him, worth $1 to either of two
possible buyers (persons 2 and 3). Persons 2
and 3 each start out with more than $1. Trade
is possible.
• Two outcomes are in the core. Person 1 sells
object to 2 for $1. Person 1 sells object to 3
for $1.
• Why is nothing else in the core?
Previous example except that
• Person 2 values object at 1. Person 3 values it
at $v<1.
What is in core?
Person 2 gets the object and pays person 1 a
price p that is between v and 1.