Handbook of Corporate Finance Empirical Corporate Finance Volume 1

94 S. Dasgupta and R.G. Hansen

Multiplying this byNgives the seller’s expected revenue as

E(Revenue to seller)= (11)

N− 1

N+ 1

.

Intuitively, since each bidder is bidding her expectation of the highest value among
the remainingN−1 bidders, conditional on her value being the highest, the ex-
pected payment received by the seller should be the unconditional expected value of
the second-highest value. In general, the density for the second-highest value is, from
the theory of order statistics,^6

f 2 (y 2 )=N(N− 1 ) (12)

(

1 −F(y 2 )

)

F(y 2 )N−^2 f(y 2 ).

In the case of the uniform[ 0 , 1 ]distribution, the expected value of the second-highest
value is then

E(y 2 )=

∫ 1

0

xN(N− 1 )( 1 −x)xN−^2 dx

= (13)

N− 1

N+ 1

as expected.^7 To reiterate and emphasize, the seller’s expected revenue from the auction
is exactly the expected value of the second-highest value. This result, of course, extends
beyond the uniform distribution.

2.3. Open and second-price sealed-bid auctions

As compared to the first-price sealed-bid auction, the open auction and second-price
sealed-bid auctions are considerably easier to solve. For this reason, they are often
chosen to model any kind of auction mechanism; the Revenue Equivalence Theorem
discussed below ensures that, in many cases, the results for one auction form extend to
others.
In an open auction, bidders cry out higher and higher bids until only one bidder, the
winner, remains. It is easy to see that “staying in the auction” until the bid exceeds one’s

(^6) To see this, first note that the distribution functionF 2 (y 2 )of the second-highest valuey 2 is the probability
that either: (a) allNvalues are less than or equal toy 2 or (b)anyN−1 values are less thany 2 and the
remaining value is greater thany 2. Note that this latter event can happen inNpossible ways. Thus, the
probability isF 2 (y 2 )=FN(y 2 )+NFN−^1 (y 2 )( 1 −F(y 2 )). Differentiating this expression with respect to
y 2 , we get the expression forf 2 (y 2 ).
(^7) This interpretation of the expected revenue holds for any distribution. Notice that the expected payment
from any bidder is
∫v ̄
0 [Prob(win)·Amount Bid]f(v)dv=
∫v ̄
0 G(v)b(v)f (v) dv=
∫v ̄
0 (
∫v
0 yg(y) dy) dF (v)
from(8). Integrating by parts, this expression becomes
∫ ̄v
0 yg(y) dy−
∫v ̄
0 F(v)vg(v)dv=
∫v ̄
∫v ̄^0 yg(y) dy−
0 F(y)yg(y)dy=
∫v ̄
0 y(^1 −F (y))g(y) dy=
∫v ̄
0 y(^1 −F (y))(N−^1 )F
N− (^2) (y)f (y) dy.Ntimes this
expression is the expected revenue to the seller in the auction, and is exactly the expected value of the second-
highest valuation.