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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
āŠ†
āˆˆ
S receives a utility of x i , subject to the
condition that
x i v ( S ) .
iāˆˆN
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
āŠ†
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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