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The remainder of the paper is organised as follows. Section 2 describes the auction
setting. Section 3 determines equilibrium bidding strategies. In Section 4, we present
an example auction scenario to illustrate a decline in the revenue of later auctions. Sec-
tion 5 provides a discussion of how our result relates with existing work on sequential
auctions. Section 6 concludes. Appendix A to C provide proofs of theorems.
2
The Sequential Auctions Model
Single object auctions that have both private and common value elements have been
studied in [9]. We therefore adopt this basic model and extend it to cover the multiple
objects case. Before doing so, however, we give an overview of the basic model.
Single object.
A single object auction is modelled in [9] as follows. There are
n
3
risk neutral bidders. The
common value
(
V
1
) of the object to the
n
bidders is equal,
but initially the bidders do not know this value. However, each bidder receives a signal
that gives an estimate of this common value. Bidder
i
=1
,...,n
draws an estimate
(
v
i
1
) of the object's true value (
V
1
) from the probability distribution function
Q
(
v
)
with support [
v
L
,v
H
]. Although different bidders may have different estimates, the true
value (
V
1
) is the same for all bidders and is modelled as the average of the bidders'
signals:
V
1
=
≥
n
i
=1
v
i
1
. Furthermore, each bidder has a
cost
which is different for
different bidders and this cost is its
private value
.For
i
=1
,...,n
,let
c
i
1
denote bidder
i
's signal for this private value which is drawn from the distribution function
G
(
c
) with
support [
c
L
,c
H
] where
c
L
≥
1
c
H
. Cost and value signals are independently
and identically distributed across bidders. Henceforth, we will use the term
value
to
refer to common value and
cost
to refer to private value.
If bidder
i
wins the object and pays
b
, it gets a utility of
V
1
−
0 and
v
L
≥
c
i
1
is
i
's surplus. Each bidder bids so as to maximize its utility. Note that bidder
i
receives two
signals (
v
i
1
and
c
i
1
) but its bid has to be a single number. Hence, in order to determine
their bids, bidders need to combine the two signals into a
summary
statistic. This is done
as follows. For
i
, a one-dimensional summary signal, called
i
's surplus
2
,isdefinedas:
c
i
1
−
b
,where
V
1
−
S
i
1
=
v
i
1
/n
−
c
i
1
(1)
which allows
i
's optimal bids to be determined in terms of
S
i
1
(see [9] for more details
about the problems with two signals and why a one-dimensional surplus is required).
In order to rank bidders from low to high valuations,
Q
(
v
) and
G
(
c
) are assumed to
be log concave
3
. Under this assumption, the conditional expectations
E
(
v
|
S
=
x
) and
E
(
v
x
) are
non-increasing in
x
. In other words, the bidders can be ranked from low to high values
on the basis of their surplus. We now extend this model to
m>
1 objects.
|
S
≤
x
) are non-decreasing in
x
.Furthermore,
E
(
c
|
S
=
x
) and
E
(
c
|
S
≤
Note that
i
's true surplus is
V
1
− c
i
1
which is equal to
v
i
1
/n − c
i
1
+
j
=
i
v
j
1
/n
. But since
2
v
i
1
/n − c
i
1
depends on
i
's signals while
j
=
i
v
j
1
/n
depends on the other bidders' signals,
the term '
i
's surplus' is also used to mean
v
i
1
/n − c
i
1
.
3
Log concavity means that the natural log of the densities is concave. This restriction is met
by many commonly used densities like uniform, normal, chi-square, and exponential, and it
ensures that optimal bids are increasing in surplus. Again see [9] for more details.
