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game called
non-transferable utility games with private beliefs and un-
certainty
(NTU-BU games) which is an extension of the traditional
NTU games, and the concepts presented in Chapter 2 to Chapter 5.
Definition 6.1 (NTU-BU Game)
An NTU-BU game
is defined
as a tuple
g
=
i
)
,P,
(
B
i
)
,s
∗
N,S,
(
, as follows:
•
N
=
{
1
,
2
, ..., n
}
is a set of agents. Any subset
C
⊆
N
is called a
coalition.
•
S
is a set of environmental states, one of which is the prevailing
state
s
∗
.
•
For each state
s ∈ S
, each agent
i
has a strict total-ordered true
preference relation
i
on the set of coalitions of which it is a mem-
ber of, so that for any two coalitions
C
1
and
C
2
,i
∈
C
1
∩
C
2
,we
i
C
2
if agent
i
prefers
C
1
to
C
2
when the current state
is
s
, to be defined in Section 3 (c.f., Definition 3.1).
have
C
1
•
P
=(
S
1
,S
2
, ..., S
n
) is an environmental belief profile, with
Si
⊆
S
being known as the environmental belief of agent
i
.
•
B
i
is the private belief of agent
i
,tobedefinedinSection4(c.f.,
Definition 4.1).
The goal of an NTU-BU game is to partition the set of agents into
a coalition structure (CS) of exhaustive and non-overlapping coali-
tions
CS
=
. Given a coalition structure
CS
,weuse
coal
i
(
CS
) to denote the coalition in
CS
of which the agent
i
is a mem-
ber. We assume that the outcome of a coalition is decided only by the
coalition itself (i.e., who its members are) and which state the current
prevailing state
s
∗
is. It is assumed that the prevailing state is not
observable by the agents. Instead, each agent
i
is associated with a set
of states
S
i
⊆
{
C
1
,C
2
, ..., C
k
}
S
, such that agent
i
has reasons to believe the world is
in one of the states in
S
i
.Each
S
i
is called the agent
i
's environmental
beliefs. The collection of all environmental beliefs of the agent is called
an environmental belief profile
P
=
{
S
1
,S
2
, ..., S
n
}
. In general, we do
not require
s
∗
to be in
S
i
for any
i ∈ N
.
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