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Problem with Beliefs
tuple
g
=
N,G,
(
i
)
,B
defined as follows.
•
N
=
{
1
,
2
,...,n
}
is a set of agents and any subset
C
⊆
N
is called
a buyer coalition.
•
G
=
{
g
1
,g
2
,...,g
m
}
is a set of goods being sold.
•
The preference of each agent
i
is represented by a total ordered
preference relation
i
on the set of possible bids, so that for any
two bids
b
1
=(
g
1
,s
1
)and
b
2
=(
g
2
,s
2
), we have
b
1
i
b
2
if agent
i
prefers
b
1
to
b
1
.
•
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
(
b
1
j
b
2
) if agent
i
believes that agent
j
considers
his bid
b
1
as no less preferred than
b
2
. The set of all beliefs of all
agents in a NTU-buyer game is represented by a belief profile
B
=
{
bel
1
,bel
2
,...,bel
n
}
where
bel
i
is the private beliefs of the
i
-th agent.
Intuitively, the goal of the game is to partition the set of agents
(buyers) into a coalition structure of exhaustive and non-overlapping
coalitions so that the
i
-th coalition places a bid for the
i
-th product.
For this reason, we define a bid by a couple
b
=(
g
i
,s
)
,
where
g
i
∈
. Given a coalition structure
CS
,we
use
C
j
(
CS
) to denote the coalition in
CS
of which the agent
j
is a
member, and
b
j|CS
to denote its corresponding bid. That is,
G
and 0
<s
|
N
|
b
j|CS
=(
g
i
,
|
C
i
|
)
such that
j ∈ C
i
and
C
i
∈ CS
.
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