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generally an agent's preference is unknown to other agents. However,
if the agents have formed an impression of one another (e.g., from pre-
vious interactions), each of them also has a
belief
of the other two's
preferences, which we assume are as shown in Figure 4.4.
Agent
b
's belief of
a
's preference:
(
{a, b, c},
movie)
a
(
{a, b},
movie)
a
(
{a, c},
movie)
a
(
{a},
movie)
Agent
c
's belief of
a
's preference:
(
{a, b, c},
movie)
a
(
{a, b},
movie)
a
(
{a, c},
movie)
a
(
{a},
movie)
Agent
a
's belief of
b
's preference:
(
{a, b, c},
movie)
b
(
{b, c},
movie)
b
(
{a, b},
movie)
b
(
{b},
movie)
Agent
c
's belief of
b
's preference:
(
{a, b, c},
movie)
b
(
{b, c},
movie)
b
(
{a, b},
movie)
b
(
{b},
movie)
Agent
a
's belief of
c
's preference:
(
{a, b, c},
movie)
c
(
{a, c},
movie)
c
(
{b, c},
movie)
c
(
{c},
movie)
Agent
b
's belief of
c
's preference:
(
{a, b, c},
movie)
c
(
{a, c},
movie)
c
(
{b, c},
movie)
c
(
{c},
movie)
Fig. 4.4
Each of three agents has a belief of the other two's preferences.
In other words, each of these agents wrongly believes that the oth-
ers prefer a coalition of size three to a coalition of size two, though it
is very clear to every one of them that
a
prefers watching the movie
with
b
,
b
with
c
,and
c
with
a
, and that no one wants to watch the
movie alone. The situation is one that is depicted in Figure 4.5, which
is almost identical Figure 4.3. However, if the agents make the pro-
posals according to their beliefs in the lack of common knowledge,
that is, if each agent only proposes alternatives such that (1) he is
better off in the alternative coalition structure, and (2) he
believes
the proposal can be accepted by his new partners in the alternative
coalition structure (perhaps in order to avoid the embarrassment of
being rejected and to speed up the coalition formation process), then
there is actually a stable coalition structure for this game. Consider
the original edge in Figure 4.3 that leads from
{
(
{
a, b, c
}
,
movie)
}
to
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