Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 40

sections.

3.3 NTU Games with Internal Uncertainty

In Section 2.1, we have discussed and formally defined coalition games
with non-transferable utility (NTU games). Intuitively, the set N =
{ 1 ,...,n } of all agents is called the grand coalition, and each subset

S N is called a coalition (or sometimes sub-coalition). Each coalition

S is associated with a set V ( S )

⊆

X of feasible consequences, which
is a subset of the set X of all possible consequences, and includes only
those outcomes that can be achieved as a result of some joint action
of the members of that coalition (or sub-coalition). For example, the
consequence of a buyer coalition game may be the number of goods
received by each member of the buyer coalition and the price each
of them pays, whereas the set of feasible consequences are those that
conform to the selling price of the items. Each agent has a preference
relation on the set of feasible consequences such that for any two fea-
sible consequences x 1 ∈

⊆

X ,wehave x 1 i x 2 if and only
if x 1 is not less preferred than x 2 by that agent.
More formally, we define an NTU game by a tuple

X and x 2 ∈

g = N,X,V, ( i ) ,

where, N = { 1 ,...,n} is the set of agents (the grand coalition). X is
the set of consequences. V :2 N

→ 2 X

is a function that maps each

coalition S ⊆ N to a set of feasible consequence V ( S ) ⊆ X . Finally,

i is the i -th agent's preference relation on X .
The core of an NTU game is then defined as the set of consequences
such that no sub-coalition S N can defect by finding an alternative

consequence where each member of the sub-coalition S would prefer
the alternative consequence, that is, a consequence x

⊆

V ( N )isinthe
core if there does not exist a sub-coalition S ⊆ N andanalternative
consequence y ∈ V ( S ) such that y i x , for all i ∈ S .

∈