Non-transferable Utility Games and Coalition Formation Mechanisms - Belief-based Stability in Coalition Formation with Uncertainty - page 17

transferable utility game (NTU game)

g = N,X,V, ( i )

is Pareto optimal if and only if there exists one or more consequence
x ∈ V ( N ), for which there is no consequence y ∈ V ( N ) such that
for all agents i ∈ N , y i x . Such consequences are known as Pareto

optimal consequences.

Pareto Optimality is an useful concept. However, it has the draw-
back that individual agents or subset of agents, being absolutely ra-
tional (self-interested), may still deviate if it is beneficial for them to
do so. As a result, a number of other stability concepts have been
proposed in non-transferable utility games, with the most well known,
and the strictest criterion being the NTU core [4].
Intuitively, the core of an NTU game contains all those conse-
quences (outcomes) that no agents have the incentive to break away
from the grand coalition N to form a smaller coalition. Hence a coali-
tion is stable if the the current outcome is in the core of the game.
Generally speaker, the core of an NTU game can contain multiple
outcomes. It can also be a singleton or an empty set.

Definition 2.4 (The Core of an NTU Game) The core of a non-

transferable utility game

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

is the set of all those consequences x ∈ V ( N ), such that there does

not exist any subset S ⊂ N and consequence y ∈ V ( S ), such that for

all i ∈ S , y is a better consequence than x ,thatis, y i x .

Unless specified otherwise, the term core in this topic refers to the
NTU core.