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the
j
th (for
j
=1
,...,m
−
1) auctions depends on how much profit it expects to get
from the (
m
j
) auctions yet to be conducted. However, since there are only
m
objects
there are no more auctions after the
m
th one. Thus, a bidder's strategic behaviour during
the last auction is the same as that for a single object English auction. Equilibrium bid-
ding strategies for a single object of the type described in Section 2 have been obtained
in [9]. We therefore briefly summarize these strategies and then determine equilibrium
for our
m
objects case.
−
Single object.
For a single object with value
V
1
, the equilibrium obtained in [9] is as
follows. A bidder's strategy is described in terms of its surplus and indicates how high
the bidder should go before dropping out. Since
n
3, the prices at which some bidders
drop out convey information (about the common value) to those who remain active.
Suppose
k
bidders have dropped out at bid levels
b
1
≥
b
k
. A bidder's (say
i
's)
strategy is described by functions
B
k
(
S
i
;
b
1
...b
k
), which specify how high it must bid
given that
k
bidders have dropped out at levels
b
1
...b
k
and given that its surplus is
S
i
.
The
n
-tuple of strategies (
B
(
≤
...
≤
·
)
,...,B
(
·
)) with
B
(
·
)) defined in Equation 3, constitutes
a symmetric equilibrium of the English auction.
B
0
(
x
i
1
)=
E
(
v
i
1
− c
i
1
|S
i
1
=
x
i
1
)
B
k
(
x
i
1
;
b
1
...b
k
)=
n − k
n
k
−
1
E
(
v
i
1
|S
i
1
=
x
i
1
)+
1
n
E
(
v
i
1
|B
y
(
S
i
1
;
b
1
,...,b
y
)=
b
y
+1
)
y
=0
−E
(
c
i
1
|S
i
1
=
x
i
1
)
(3)
where
x
i
1
is
i
's surplus. The intuition for Equation 3 is as follows. Given its surplus and
the information conveyed in others' drop out levels, the highest a bidder is willing to go
is given by the expected value of the object, assuming that all other active bidders have
the same surplus. For instance, consider the bid function
B
0
(
S
i
1
) which pertains to the
case when no bidder has dropped out yet. If all other bidders were to drop out at level
B
0
(
S
0
),then
i
's expected payoff (
ep
=
V
1
−
c
i
1
−
B
0
(
S
0
)) would be:
ep
=
S
i
1
+
n
−
1
E
(
v
|
S
=
S
0
)
−
B
0
(
S
0
)
n
=
S
i
1
+
n
−
1
E
(
v
|
S
=
S
0
)
−
E
(
v
−
c
|
S
=
S
0
)
n
=
S
i
1
−
S
0
Using strategy
B
0
,
i
remains active until it is indifferent between winning and quitting.
Similar interpretations are given to
B
k
for
k
1; the only difference is that these
functions take into account the information conveyed in others' drop out levels.
Let
f
1
denote the first order statistic of the surplus for the
n
bidders and let
s
1
denote
the second order statistic.
f
1
and
s
1
are obtained from the distribution functions
Q
1
and
G
1
. For the above equilibrium, it has been shown that the bidder with the highest surplus
wins and the expected revenue (denoted
ER
)is[9]:
≥
s
=
f
1
)
ER
=
E
(
V
)
−
E
(
c
|
−
E
(
π
w
)
(4)
