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which, unlike monetary utility, is obviously not transferable.
We formally define non-transferable utility cooperative games (NTU
Game) as follows. The goal of the game is to partition the set of agents
into a coalition structure of exhaustive and non-overlapping coalitions.
Definition 2.1 (Non-transferable Utility Cooperative Games)
A
non-transferable utility cooperative game
is a tuple
g
=
N,X,V,
(
i
)
,
where
N
=
{
a
1
,a
2
,...,a
n
}
is a set of
n
agents and any subset
S
N
is called a
coalition
,
X
is a set of consequences, each representing a possible outcomes, and
V
is a function which assigns to every nonempty subset
S ⊆ N
a
set
V
(
S
)
⊆ X
of feasible consequences. Finally,
i
is the preference
relation of agent
a
i
over
X
.
⊂
Example 2.2
The scenario described before can be formally repre-
sented as an NTU game. If each player of Alice, Bob, Cindy, David,
and Emily is represented by the first letter of his or her given name,
then
N
=
{
A, B, C, D, E
}
.
The set
X
consists of all possible consequences, such as
x
1
='David
and Emily become good friends',
x
2
= 'Everyone is bored',
x
3
= 'Alice
is frustrated',
x
4
= 'Alice is happy', and so on. If we assume that Alice
prefers being happy to being frustrated, then we shall formally denote
this as
x
4
A
x
3
. Other components of the preference
A
of Alice, and
preferences
B
,
C
,
D
,and
E
of all other people, can be defined
accordingly.
Cooperative games, like other branches of game theory, deal with
finding meaningful 'solution concepts' for coalition formations prob-
lems, which are criteria for evaluating the stability of the coalitions.
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