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The agents' preference rules (
P
i
) and private beliefs (
B
i
) corre-
spond to their counterparts in Definition 3.1 and Definition 4.1, re-
spectively. However, they become somewhat more complicated in the
new context of NTU-BU games, as will be discussed in the next sec-
tion.
Example 6.2
We can formally define the example illustrated in Ex-
ample 6.1 as an NTU-BU game
g
=
i
)
,P,
(
B
i
)
,s
∗
N,S,
(
as follows:
•
N
=
{
a, b, c
}
.
, with the prevailing state
s
∗
being either
s
1
if 4G LTE
phones are well received, or
s
2
if they are badly received.
•
S
=
{
s
1
,s
2
}
i
) are as listed in Figure 6.1 (for both cases
•
Agents' preferences (
s
=
s
1
and
s
=
s
2
).
•
Agents' environmental beliefs are
S
1
=
S
2
=
{
s
1
,s
2
}
,and
P
=
(
S
1
,S
2
).
•
Agents' private beliefs (
B
i
) are as listed in Figure 6.2.
6.3.1
Preferences and beliefs in NTU-BU Games
We are now ready to define agents' preferences and beliefs in NTU-BU
Games.
To capture the uncertainty in an agent's preferences, we define an
agent's certain preferences and uncertain preferences, as follows:
For any two coalitions
C
1
and
C
2
,wewrite
C
1
i
C
2
if agent
i
certainly prefers
C
1
to
C
2
without uncertainty, and write
C
1
C
2
if the agent may prefer
C
1
to
C
2
, but this is uncertain and may
be conditioned on other factors such as the agent's decision making
strategy amidst his internal uncertainty.
Definition 6.2 (Agent's Certain Preference
)
Given an agent
i
and two coalitions
C
1
and
C
2
,
i
C
2
, we say agent
i
certainly
prefers
C
1
to
C
2
, written
C
1
i
C
2
,if
i
)thereexists
s ∈ S
such that
s ∈ S
i
,and
C
1
∈
C
1
∩
i
C
2
holds, and
ii
) there does not exist any
s
2
∈ S
,
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