Non-transferable Utility Games and Coalition Formation Mechanisms - Belief-based Stability in Coalition Formation with Uncertainty - page 20

Mechanisms

Here we assume that the smaller scale project that Grand-hill Indus-
tries and Halogen Enterprise can work on together also brings a profit
of $1.5 million, while that of Frankford Corporation and Halogen En-
terprise brings a moderate profit of only $1 million. If each of these
companies work on its own, then each of them cannot make a profit
more than $0.2 million.

Again, individually rationality and Pareto optimality can be de-
fined similarly in TU games as in NTU games. We will not go into the
details here.
The strictest and most commonly used criterion in transferable
utility games, as in NTU games, is also the core. Historically, an idea
similar to the TU core was actually first found in an earlier work in the
19th century. Francis Edgeworth, in his 1881 work that dealt with equi-
librium concepts in an exchange market, proposed one of the earliest
stability criterion which he called ' final settlement ,' which was defined
as 'a settlement which cannot be varied by recontract within the field
of competition' [6]. The main idea is that the final settlement cannot
be altered by recontracting , which, in modern terminology, means that
the final settlement is in the core. Edgeworth's idea did not raise a
lot of attention at his time. It was not until 1953 when this idea was
resurrected in the Ph.D. thesis of Gillies, which was first published in
1959 [7].
As in the NTU counterpart, the idea of the TU core is that there
does not exist any alternative coalition that is not in the current coali-
tion structures that has a higher value than the sum of the original
allocated utility of its members:

Definition 2.6 (The Core in TU Games) The core (TU core)

of a TU game

g = N,v

is the set of all such allocations

x = {x 1 ,x 2 ,...,x n }