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Definition 4.1 (NTU-PB Games)
A Non-transferable Uti lity
Games with Private Beliefs
(NTU-PB games) is a tuple
g
=
N,A,
(
i
)
,B
defined as follows. Let
N
=
{
1
,
2
,...,m
}
be a set of
m
agents and let any subset
C
N
be called a coalition.
There is a set
A
of possible actions that are available to the agents so
that each member of the same coalition jointly choose an action
a
⊆
A
.
The preference of each agent
i
is represented by a preference relation
i
on the set 2
N
∈
×A
of
coalitional acts
. Each agent
i
maintains beliefs
regarding other agents' preferences, which is represented by a relation
bel
i
, so that for two agents
i
and
j
,wewrite
bel
i
(
α
1
j
α
2
) if agent
i
believes that agent
j
prefers coalitional act
α
1
to coalitional act
α
2
,
α
1
,α
2
∈
2
N
A
. The set of all beliefs of all agents in an NTU-PB
game is represented by a belief profile
×
B
=
{
bel
1
,bel
2
,...,bel
m
}
where
bel
i
is the private beliefs of agent
i
.
Example 4.7
To illustrate the above concept, let's refer to the
dating example in Example 4.3 again, which can be modelled as a
NTU-PB game
. In this case, the set
N
consists of
the three agents
a
,
b
,and
c
. There is only one action available to
each coalition, which is to see the movie. That is,
A
=
{
movie
}
.
Each coalitional act consists of a set of one to three agents, together
with the only available action movie. For example, (
N,A,
(
i
)
,B
,movie
)
is a coalitional act, and its intuitive meaning is that agents
a
and
b
go to see the movie together. The preference of each agent is a
personal ranking on each of the possible coalitional act. For exam-
ple, the agent
a
may prefer (
{
a, b
}
,movie
), de-
noted (
{a, b},movie
)
a
(
{a, b, c},movie
). Each agent also has a be-
lief regarding the other agents' preferences on the coalitional acts.
{
a, b
}
,movie
)to(
{
a, b, c
}
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