Information Technology Reference
In-Depth Information
197.3
m=9,n=20
197.2
197.1
197
196.9
196.8
196.7
1
2
3
4
5
6
7
8
9
Auction
Fig. 1.
A varying revenue in a series of auctions for the normal distribution
0.8
m=9, n=20
m=9, n=19
m=9, n=18
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
Auction
Fig. 2.
A bidder's cumulative ex-ante expected profit (
α
j
) for the normal distribution
normal distribution is denoted
(
μ, ν
) where
μ
is the mean and
ν
is the variance. Let
μ
v
,
μ
c
,and
μ
s
denote the mean for the value, cost and surplus respectively. Also, let
ν
v
,
ν
c
,and
ν
s
denote the variance for the value, cost, and surplus respectively. We took
μ
v
= 200,
μ
c
=2,
ν
v
=0
.
5,and
ν
c
=0
.
5. These values ensure that
c
L
>
0 and
v
L
>c
H
for more than 99
.
8 percent of the population. Given this, for the
j
th auc-
tion, we get the mean and variance for the surplus as
μ
s
=
μ
v
/
(
n
N
−
j
+1)
−
μ
c
and
ν
s
=1
.
0 [6].
Let
F
(
x
) and
f
(
x
) denote the distribution and density function for the surplus where:
1
ν
s
√
2
π
e
−
(
x−μ
s
)
2
/
2
ν
s
2
f
(
x
)=
From a continuous distribution with cumulative distribution function
F
(
x
),if
n
ran-
dom samples are drawn, then the expectation of the second highest order statistic of
these
n
samples between limits
x
and
x
is [5]:
1)
x
E
(
s
n
)=
n
(
n
x
[
F
(
x
)]
n−
2
[1
−
−
F
(
x
)]
f
(
x
)
dx
(14)
x
