The Complete Picture: sb-core and wb-core - Belief-based Stability in Coalition Formation with Uncertainty - page 149

For instance, for the case that should agent b believe the phones
will be well received by the public, then his most preferred choice is
to enter the market on his own in order to fully take advantage of the
untapped market (represented by the singleton coalition {b} ), and his
second most preferred choice would be to team up with the company
of agent a (represented by

), and his next most preferred choices
is to team up with agent c (represented by

{ a, b }

), and so on. However,
should he expect them to be badly received, his most preferred choice
would then form a coalition will all three companies for cost-saving
and risk minimization purposes.
In this example, all three companies are unsure of whether their
new product phones will be well received, which in turn implies that
they may or may not be certain about their preferences. For example,
although agent a certainly knows that he prefers the coalition

{ b, c }

{ a, b } to

the coalition , as his preference order for these two coalitions would

not be affected by the environmental states (sales performance of the
phone), he is actually indecisive regarding the two coalitions

{ a }

{ a, b } and

{ , since it would depend on something that he is unsure about.

Apart from their own individual preferences, each agent also main-
tain belief regarding other agents' preferences, which are represented
by partial order relations as in Figure 6.2. These partial preferences
are called the agents' external beliefs.

a, c }

Fig. 6.2 Agents' mutual beliefs in Example 6.1.

For example, the second last entry in Figure 6.2 says that agent c
believes agent a certainly prefers the coalition {a} ,where a is alone,
to the three agents coalition {a, b, c} and that the also prefers the