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her happiness so as to give the same amount of happiness to another
player.
In general, a TU game consists of a set
N
of agents as players,
and each coalition
S
N
is associated with a real value, which is
the highest possible
worth
that can be created by the joint actions of
the agents in
S
. This value represents the utility produced by that
particular coalition. For example, in Exampleex 2.3, the worth of the
coalition of Frankford and Grand-hill is $1.5 millions, and the worth of
the grand coalition of all three companies is $2 millions. Then this value
(i.e., the produced utility) will be distributed amongst its members,
such that each member
i
ā
ā
S
receives a utility of
x
i
, subject to the
condition that
x
i
v
(
S
)
.
iāN
Definition 2.5 (Transferable Utility Cooperative Games)
A
transferable utility cooperative game (TU game) is a tuple
N,v
,
where
N
=
{a
1
,a
2
,...,a
n
}
is a set of agents, and
v
is a worth function that associates with every
subset
S
ā
N
arealnumber
v
(
S
).
Example 2.4
The scenario of three companies in Example 2.3 can
be formally presented as a TU game, as follows. Let the three com-
panies Frankford Corporation, Grand-hill Industries and Halogen En-
terprise be represented by their respective initial letters,
F
,
G
and
H
.
Then a formal representation of the problem is
N
=
{
F, G, H
}
v
(
{
F, G, H
}
)=2
v
(
{
F, G
}
)=1
.
5
v
(
)=1
v
(
{G, H}
)=1
.
5
v
(
{F }
)=
v
(
{G}
)=
v
(
{H}
)=0
.
2
.
{
F, H
}
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