Information Technology Reference
In-Depth Information
=
u
i
(1,
a
−
i
)
−
u
i
(0,
a
−
i
)
+
s
ij
u
j
(1,
a
−
i
)
−
u
j
(0,
a
−
i
)
S
i
+
j
∈
N
=
a
j
−
c
i
+
s
ij
a
j
.
(5.3)
P
i
S
i
+
j
∈
N
j
∈
N
−
It is clear from (
5.3
) that no user participating is always a SNE. We therefore conclude
that at least one SNE exists.
Then we show an important property of the social group utility function. It follows
from (
5.3
) that
−
f
i
(1,
a
−
i
)
−
f
i
(0,
a
−
i
)
f
i
(1,
a
−
i
)
−
f
i
(0,
a
−
i
)
⊛
⊞
⊝
⊠
a
j
−
c
i
+
s
ij
a
j
=
a
j
−
c
i
+
s
ij
a
j
−
i
S
i
+
i
S
i
+
j
∈
N
j
∈
N
j
∈
N
j
∈
N
−
−
a
j
)
a
j
)
=
(
a
j
−
+
s
ij
(
a
j
−
(5.4)
P
i
S
i
j
∈
N
j
∈
N
−
+
a
a
i
,
Let
a
≤
denote element-wise inequality (i.e.,
a
i
≤
∀
i
∈
N
). The property
below follows from (
5.4
).
a
−
i
, then
f
i
(1,
a
−
i
)
Property 5.1 (Supermodularity)
If
a
−
i
≤
−
f
i
(0,
a
−
i
)
≤
f
i
(1,
a
−
i
)
f
i
(0,
a
−
i
).
Property 5.1 implies that if a user's best response strategy is to participate, then it
remains the best response strategy if more users participate; if a user's best response
strategy is to not participate, then it remains the best response strategy if less users
participate.
−
5.5
Computing Pareto-Optimal SNE
Next we turn our attention to finding a SNE with desirable system efficiency.
For the PCG for fully altruistic users (i.e.,
s
ij
e
ij
=
1,
∀
∈
N
S
), it is clear that the
social optimal profile
a
∗
is a SNE, which is the solution to the following problem:
⊛
⊝
⊞
⊠
maximize
a
a
i
a
j
−
c
i
i
∈
N
P
i
j
∈
N
−
subject to
a
i
∈{
0, 1
}
,
∀
i
∈
N
.
(5.5)
