) is the expected utility from entering for a bidder with value
v
. Thus,
rF
(
r
)
n

1
=
δ.
Now, let us compare the secondprice auction with entry fee of
δ
=
rF
(
r
)
n

1
to a secondprice auction with a reserve price of
r
. Clearly in both cases,
a
(
v
) =
F
(
v
)
n

1
{
v
≥
r
}
.
Moreover, the expected payment for a bidder in both cases is the same: If
v < r
,
the expected payment is 0 for both. For
v
≥
r
, and the auction with entry fee,
p
(
v
) =
δ
+
F
(
v
)
n

1

F
(
r
)
n

1
E
h
max
i
≤
n

1
V
i
r
≤
max
i
≤
n

1
V
i
≤
v
i
,
whereas with the reserve price,
p
(
v
) =
rF
(
r
)
n

1
+
F
(
v
)
n

1

F
(
r
)
n

1
E
h
max
i
≤
n

1
V
i
r
≤
max
i
≤
n

1
V
i
≤
v
i
.
This means that
u
(
w

v
) =
va
(
w
)

p
(
w
) is the same in both auctions. Hence the
threshold strategy is a BayesNash equilibrium in the entry fee auction, since a
bidder with value below
r
has negative expected utility from entering.
Notice, however, that the payment of a bidder with value
v
can differ in the
two auctions. For example, if bidder 1 has value
v > r
and all other bidders have
value less than
r
, then in the entry fee auction, bidder 1’s payment is
δ
=
rF
(
r
)
n

1
,
whereas in the reserve price auction it is
r
. Moreover, if bidder 1 has value
v > r
,
but there is another bidder with a higher value, then in the entry fee auction, bidder
1 loses the auction, but still pays
δ
, whereas in the reserve price auction he pays
nothing.
Thus, when the entryfee auction is over, a bidder may regret having
10
since
u
E
(
v
)
≥
0 for
v > r
and
u
E
(
v
)
≤
0 for
v < r
, and
u
E
(
·
) is weakly increasing.
248
14. AUCTIONS
participated. This means that this auction is
exinterim individually rational
, but
not
expost individually rational
. See the definitions in
§
14.5.4.
14.5.3. Evaluation fee.
Example
14.5.2
.
The queen is running a secondprice auction to sell her crown
jewels. However, she plans to charge an evaluation fee: A bidder must pay
φ
in order
to examine the jewels and determine how much he values them prior to bidding in
the auction.
Figure 14.5.
The queen has fallen on hard times and must sell her
crown jewels.
Assume that bidder
i
’s value
V
i
for the jewels is a random variable and that
the
V
i
’s are i.i.d. Prior to the evaluation, he only knows the distribution of
V
i
. He
will learn the realization of
V
i
only if he pays the evaluation fee. In this situation,
as long as the fee is below the bidder’s expected utility in the secondprice auction,
i.e.
φ <
E
[
u
(
V
i

V
i
)]
,
the bidder has an incentive to pay the evaluation fee. Thus, the seller can charge
an evaluation fee equal to the bidder’s expected utility, minus some
>
0.
The
expected auctioneer revenue from bidder
i
’s evaluation fee and his payment in the
ensuing secondprice auction is
E
[
u
(
V
i

V
i
)]

+
E
[
p
(
V
i
)] =
E
[
a
i
(
V
i
)
·
V
i
]

.
Since, in a secondprice auction, the allocation is to the bidder with the highest
value, the seller’s expected revenue is
E
h
max
i
(
V
1
, . . . , V
n
)
i

n,
which is essentially best possible.
14.6. CHARACTERIZATION OF BAYESNASH EQUILIBRIUM
249
14.5.4. Exante versus exinterim versus expost.
An auction, with an
associated equilibrium
β
, is called
individually rational
(IR) if each bidder’s ex
pected utility is nonnegative. The examples given above illustrate three different