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the movie on his own, which is also the default coalition structure
without any negotiation, and the bottom right node
{
(
{
a, c
}
, movie) ,
(
represents that agents a and c form a coalition of size two
and go to see the movie together. In game theoretic terms, each suc-
cessful proposal in a propose-and-evaluate type of coalition formation
mechanism is called an objection to the original coalition structure.
Objections are shown by edges. The left most edge, for example, says
that the coalition structure
{
b
}
, movie)
}
{
(
{
a
}
, movie) , (
{
b
}
, movie) , (
{
c
}
, movie)
}
is objected by the coalition
{
a, b
}
in the coalition structure
{
(
{
a, b
}
,
movie) , (
in the
latter coalition structure, each of its members (agent a and agent b )
really prefers the latter to the former. That is, according to the real
preferences of both a and b , they prefer going out together to going
alone. (Note that the preference of c is irrelevant to this objection, it
is only the preferences of those agents involved in the objection that
count). The readers can check that, as shown in Figure 4.3, each coali-
tion structure is objected by at least one real objection, so the core is
empty in this case according to the classical game theoretic definitions.
{
c
}
, movie)
}
. This is because in the coalition
{
a, b
}
({ a },movie),
({ b },movie),
({ c },movie)
({ a , b , c },movie),
({ a , b },movie),
({ c },movie)
({ a },movie),
({ b , c },movie)
({ a , c },movie),
({ b },movie)
Fig. 4.3 A belief-based dating game.
Example 4.4 (Abelief-based dating game with wrong beliefs)
Let us reconsider the game in Example 4.3. In real life, agents' pref-
erences are private information, not common knowledge. Therefore,
 
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