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coalition
. However, each agent makes no further as-
sumptions regarding the others' preferences that are not part of his
external beliefs. For example, agent
c
has no opinion on whether agent
a
prefers
{a, b, c}
to
{a, c}
or not. Note that external beliefs are not
required to be correct.
We are now ready to discuss different types of objections that can
arise out of this coalition formation game. Refer to the graph in Figure
6.3, where all five possible coalition structures are shown as nodes, and
possible objections indicated by labels on the edges. The edges are also
numbered for ease of description.
First, consider edge 2. We see that the singleton coalition
{
a, b
}
to
{
a, c
}
{
b
}
is
certainly an objection to
{
a, b, c
}
since agent
b
certainly prefers the
coalition
no matters what states it is in, thus
resulting in the coalition structure
{
b
}
to coalition
{
a, b
}
.
The same can be said about edge 3 and edge 4, since agent
c
will
certainly break away by similar argument, and so are edge 1 and edge
6. We label these
definite objections
. In Figure 6.3, definite objections
are denoted by solid lines.
{{
a
}
,
{
b
}
,
{
c
}}
Fig. 6.3
Definite objections and potential objections.
However, edge 5, which represents the case that agent
b
breaks its
partnership with agent
c
and teams up with agent
a
instead, is more
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