Information Technology Reference
In-Depth Information
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Appendix
We must prove that equations (5) define a payoff division that lies inside the core.
Group Rationality.
First note that always
x
i
≥
0
. Moreover, a coalition
S
such that
p
0
/
∈
S
has
V
(
S
)=0
. So the only thing to prove is that, for every
P
⊆
N
−{
p
0
}
,the
P
has a coalitional value
V
(
S
)
such that
x
(0) +
i∈P
x
(
i
)
coalition
S
=
{
p
0
}∪
≥
,
p
=
b
i
∈P
b
i
i
,
q
=
b
i
∈Q
b
i
i
.Then
V
(
S
)
. Let us define
Q
=
N
−
P
−{
b
0
}
x
(
i
)=
t
0
(
p
+
q
)
2
+
b
i
0
p
(
b
i
0
+
p
+
q
)
2
(11)
i
∈
P
and
p
V
(
S
)=
t
0
(12)
b
i
0
+
p
The difference between (11) and (12) is
t
0
b
i
0
q
≥
0
(13)
Therefore we have proved group rationality.
Global Rationality.
Note that the coalitional value as a function
V
=
v
(
t
0
,b
i
0
,b
i
1
,
...,b
i
n
)
(equation 4) is homogeneous of degree 1. Therefore,
n
x
(
i
)=
t
0
∂V
∂V
∂b
i
0
∂V
∂b
i
n
∂t
0
+
b
i
0
+
...
+
b
i
n
=
V
(14)
0
