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of utility that there does not exist any coalition
C
⊂
N
such that
v
(
C
)
>
i∈C
x
i
.
Example 2.5
In the scenario of three companies in Example 2.3,
the core contains a single element of (
x
F
=0
.
5
,x
G
=1
,x
H
=0
.
5). It
is easy to verify that no two companies can obtain a higher total utility
by forming a smaller coalition of themselves, and the utility that each
company obtains is higher than what they would have if they work on
a smaller project alone. It is also not dicult to verify that the core
does not contain any other allocations.
Example 2.6
In the scenario of three companies in Example 2.3, if
v
(
{
F, G, H
}
)=1
.
6
while the worth of all other coalitions remain unchanged, then the
core will become empty. In other words, for any feasible allocation
(
x
F
,x
G
,x
H
), there must be a subset of agents that is motivated to
break away. For example, to keep Grand-hill Industries in the grand
coalition, we must have
x
G
0
.
2
,
otherwise it would rather work alone, and
x
F
+
x
H
1 (hence
x
G
<
0
.
6)
,
or these companies will form a smaller coalition for the profit of $1
million. However, the consequence is that
1
x
F
+
x
H
1
.
4
,
which means that between Frankford Corporation and Halogen En-
terprise, one of them must be allocated a utility of not more than
0
.
7. Knowing that the profit Grand-hill Industries must be less than
$0.6 million, the company that receives the lesser portion of utility
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