
352 Green Wireless Cellular Networks in the Smart Grid Environment
are given, the best response function B
n
(p
−n
) of retailer n can be defined as
follows:
B
n
(p
−n
) = arg max
p
n
U
n
(p
n
, p
−n
). (12.30)
Therefore, substituting (12.20) into (12.23), the best response function
B
n
(p
−n
) of retailer n is as follows:
B
n
(p
−n
)=
α
n
2(2α
n
X − 1)
⎛
⎝
j=n
p
j
α
j
⎞
⎠
+ Y
n
, (12.31)
where
Y
n
=
α
n
2(2α
n
X − 1)
⎛
⎝
j=n
β
j
α
j
⎞
⎠
+
c
n
− β
n
2
+
α
n
2α
n
X − 1
U|Ψ
∗
| + V |Ψ
∗
|
2
+ W |Ψ
∗
|
3
. (12.32)
In the following, we will show that an arbitrary retailer n’s best response
function B
n
(p
−n
)isastandard function.
Definition 12.5.2. A function B
n
(p
−n
) is standard if for all p
−n
≥ 0,the
following properties are satisfied