The Complete Picture: sb-core and wb-core - Belief-based Stability in Coalition Formation with Uncertainty - page 157

6.3.2 NTU-BU Games Stability

We are now ready to define our stability concepts. First, we note again
that in conventional game theory, the core of a coalitional game is a set
of consequences with which no agent will be motivated to break away
to form a smaller coalition for a better consequence (see Definition
2.4). However, as in Definition 5.4 in which we define the core of
an NTU-Buyer Game, we shall once again abuse the term 'core' and
define the core of a NTU-BU game to be the set of stable coalition
structures. We note again that if we strictly follow Definition 2.4,
then we will have generally different cores (in the conventional sense,
as subsets of coalition stabilising consequences) for different coalitions
in the coalition structure.

Definition 6.5 (Core of NTU-BU Game) A coalition structure

CS = { C 1 ,C 2 ,...,C k }

is in the core of a NTU-BU game if there does not exist a coalition C
such that for all agents i ∈ C ,wehave

s ∗
i

C coal i ( CS )

where s ∗ is the prevailing state.

The problem here is, of course, that the prevailing state is not ob-
servable to the agents, which limits the applicability of core in NTU-
BU games. Moreover, the agents' beliefs, which will affect the decisions
of the agents, are not considered by the core, so that a coalition struc-
ture that is in the core may not be stable in practice, and vice versa.
So instead, we need to define the belief-based stability concepts as
follows. First, we will deal with the more simpler case where uncer-
tain opinions, but not private beliefs, are considered. As in previous
chapters, let us label the objections that involve uncertain preference
as potential objections, and objections that involve only certain pref-
erence as definite objections.