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where
E
(
π
w
) is the winner's expected profit. This profit is:
E
(
π
w
)=
E
(
f
1
)
E
(
s
1
)
−
(5)
On the basis of the above equilibrium for a single object, we determine equilibrium
for sequential auctions for the
m
objects defined in Section 2 as follows.
Multiple objects
. We first introduce some notation and them derive the equilibrium.
We will denote the first order statistic of the surplus for the
j
th (for
j
=1
,...,m
)
auction as
f
n−j
+1
j
and the second order statistic as
s
n−j
+
j
. Also, we denote a bidder's
cumulative ex-ante expected profit from auctions
j
to
m
(where 1
m
)as
α
j
.
Finally, we denote the winner's expected profit for the
j
th auction as
E
(
π
wj
).Given
this, the following theorem characterises the equilibrium for
m>
1 objects.
Theorem 1.
Fo r
1
≤
j
≤
m
,let
β
j
and
α
j
be defined as:
≤
j
≤
m
and
j
≤
y
≤
y−
1
β
j
=[
(1
−
1
/
(
n
−
y
+
k
+ 1))][1
/
(
n
−
j
+1)]
(6)
k
=
j
m
(
β
j
[
E
(
f
n−j
+1
)
E
(
s
n−j
+1
)+
α
j
+1
])
α
j
=
−
(7)
y
=
j
where
α
m
+1
=0
. Then the
n
-tuple of strategies
(
B
(
·
)
,...,B
(
·
))
with
B
(
·
)
defined in
Equation 8 constitutes an equilibrium for the
j
th (for
j
=1
,...,
(
m
−
1)
) auction at a
stage where
k
bidders have dropped out:
B
0
(
x
ij
)=
E
(
v
ij
−
c
ij
|
S
ij
=
x
ij
)
−
α
j
+1
B
k
(
x
ij
;
b
1
,...,b
k
)=
n
−
j
+1
−
k
E
(
v
ij
|
S
ij
=
x
ij
)
−
E
(
c
ij
|
S
ij
=
x
ij
)
n
−
z
+1
k−
1
1
+
E
(
v
ij
|
B
y
(
S
ij
;
b
1
,...,b
y
)=
b
y
+1
)
n
−
j
+1
y
=0
−
α
j
+1
(8)
For the last auction, the equilibrium is as given in Equation 3 with
n
replaced with
(
n
m
+1)
.
For the above equilibrium, the winner for the
j
th (for
j
=1
,...,m
) auction is the bid-
der with the highest surplus for that auction (see proof of Theorem 3 in the appendix for
details). The following two theorems characterise the expected revenue and the winner's
expected profit.
Theorem 2.
Fo r t h e
j
th (for
j
=1
,...,m
−
−
1
) auction, the winner's expected profit
(denoted
E
(
π
wj
)
)is:
E
(
π
wj
)=
E
(
f
n−j
+1
E
(
s
n−j
+1
j
)
−
)+
α
j
+1
(9)
j
and for the last auction, the winner's expected profit is:
E
(
π
wm
)=
E
(
f
n−m
+1
)
− E
(
s
n−m
+1
m
)
(10)
m
