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s
2
i
s
2
=
s
such that both
s
2
∈
S
i
and
C
2
C
1
hold.
Definition 6.3 (Agent's Uncertain Preference
)
Given an
agent
i
and two coalitions
C
1
and
C
2
,
i
C
2
, we say agent
i
is un-
certain about his preferences regarding
C
1
and
C
2
, written
C
1
i
C
2
,
if there exists
S
⊆
∈
C
1
∩
s
1
i
S
, such that
S
i
=
S
and both
C
1
S
,
s
1
,s
2
∈
C
2
s
2
i
and
C
2
C
1
hold.
The agents' external beliefs regarding other agents' preferences are
represented by a relation
B
i
, so that for two agents
i
and
j
,wewrite
B
i
(
C
1
j
C
2
) if agent
i
believes that agent
j
certainly prefers coalition
C
1
to coalition
C
2
.
Definition 6.4 (Agent's External Beliefs)
Given two agents
i
and
j
, and two coalitions
C
1
and
C
2
,
j
C
2
,wewrite
B
i
(
C
1
j
C
2
) if agent
i
believes agent
j
certainly prefers
C
1
to
C
2
and we write
B
i
(
C
1
i
∈
C
1
∩
C
2
) if agent
i
believes agent
j
is uncertain regarding
C
1
and
C
2
.
The set of external beliefs of all agents is represented by an external
belief profile
B
=
{
B
1
,B
2
,...,B
n
}
.
We assume the following consistency requirements for the agents'
external beliefs:
First, each of an agent's certain preferences is represented by a
corresponding external belief. That is
C
1
i
C
2
⇔
B
i
(
C
1
i
C
2
)
.
Second, conflicting external beliefs are not allowed. That is, for any
two agents
i
and
j
, and two coalitions
C
1
and
C
2
,
B
i
(
C
1
j
C
2
) implies
that both
B
i
(
C
2
j
C
1
)and
B
i
(
C
1
j
C
2
)doesnothold.
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