With only the first constraint in mind, on bid vector
v
, the auctioneer should
never allocate if
v
1
< m
1
and
v
2
< m
2
, but otherwise, should allocate to the
bidder with the higher value of
v
i

m
i
. (We can ignore ties since they have zero
probability.)
It turns out that following this prescription (setting the payment of the winner
to be the threshold bid for winning), we obtain a truthful auction, which is therefore
BIC. To see this, consider first the case where
λ
1
=
λ
2
.
The auction is then a
Vickrey auction with a reserve price of
m
1
. If
λ
1
6
=
λ
2
, then the allocation rule is
α
i
(
b
1
, b
2
) =
(
1
b
i

m
i
>
max(0
, b

i

m

i
)
0
otherwise.
(14.23)
If bidder 1 wins he pays
m
1
+ max(0
, b
2

m
2
), with a similar formula for bidder 2.
Exercise
14.9.8
.
Show that this auction is truthful, i.e., it is a dominant
strategy for each bidder to bid truthfully.
Remark
14.9.9
.
Perhaps surprisingly, when
λ
1
6
=
λ
2
, the item might be allo
cated to the lower bidder. See also Exercise 14.14.
14.9.3. The multibidder case.
We now consider the general case of
n
bid
ders. The auctioneer knows that bidder
i
’s value
V
i
is drawn from a strictly increas
ing distribution
F
i
on [0
, h
] with density
f
i
. Let
A
be an auction where truthful
bidding (
β
i
(
v
) =
v
for all
i
) is a BayesNash equilibrium, and suppose that its
allocation rule is
α
:
R
n
7→
R
n
.
Recall that
α
[
v
] := (
α
1
[
v
]
, . . . , α
n
[
v
]), where
α
i
[
v
] is the probability
15
that the item is allocated to bidder
i
on bid
16
vector
v
= (
v
1
, . . . , v
n
), and
a
i
(
v
i
) =
E
[
α
i
(
v
i
, V

i
)].
The goal of the auctioneer is to choose
α
[
·
] to maximize
E
"
X
i
p
i
(
V
i
)
#
.
Fix an allocation rule
α
[
·
] and a specific bidder with value
V
that is drawn from
the density
f
(
·
). As usual, let
a
(
v
),
u
(
v
) and
p
(
v
) denote his allocation probability,
expected utility and expected payment, respectively, given that
V
=
v
and all
bidders are bidding truthfully. Using condition (3) from Theorem 14.6.1, we have
E
[
u
(
V
)] =
Z
∞
0
Z
v
0
a
(
w
)
dwf
(
v
)
dv.
Reversing the order of integration, we get
E
[
u
(
V
)] =
Z
∞
0
a
(
w
)
Z
∞
w
f
(
v
)
dv dw
=
Z
∞
0
a
(
w
)(1

F
(
w
))
dw.
(14.24)
15
The randomness here is in the auction itself.
16
Note that since we are restricting attention to auctions for which truthful bidding is a
BayesNash equilibrium, we are
assuming
that
b
= (
b
1
, . . . , b
n
) =
v
.
14.9. MYERSON’S OPTIMAL AUCTION
259
Thus, since
u
(
v
) =
va
(
v
)

p
(
v
), we obtain
E
[
p
(
V
)] =
Z
∞
0
va
(
v
)
f
(
v
)
dv

Z
∞
0
a
(
w
)(1

F
(
w
))
dw
=
Z
∞
0
a
(
v
)
v

1

F
(
v
)
f
(
v
)
f
(
v
)
dv.
(14.25)
Definition
14.9.10
.
For agent
i
with value
v
i
drawn from distribution
F
i
, the
virtual value
of agent
i
is
ψ
i
(
v
i
) :=
v
i

1

F
i
(
v
i
)
f
i
(
v
i
)
.
In the example of
§
14.9.2,
ψ
i
(
v
i
) =
v
i

1
/λ
i
.
Remark
14.9.11
.
Contrast the expected payment in (14.25), i.e., the expecta
tion of the buyer’s allocated
virtual value
, with the expectation of the
value
allocated
to him using
a
(
·
), that is,
Z
∞
0
a
(
v
)
v f
(
v
)
dv.
The latter would be the revenue of an auctioneer using allocation rule
a
(
·
) in a
scenario where the buyer could be charged his full value. The difference between
the value and the virtual value captures the auctioneer’s loss of revenue that can
be ascribed to a buyer with value
v
, due to the buyer’s value being private.
We have proved the following proposition:
Lemma
14.9.12
.
The expected payment of agent
i