106 S. Dasgupta and R.G. Hansen
strategy, then her expected payoff is
∫bS−^1 (b 1 )0(
v(t, y)−v(y, y))
g(y|t)dy.Differentiating, it is immediate that the first-order condition is satisfied ifb 1 =bS(t)
so thatbS
− 1
(b 1 )=t.
Turning now to the ascending auction, suppose that the bidding is at a stage where all
bidders are still active. Suppose bidder with signalthas the strategy that she will remain
in the bidding until the pricebN(t)=u(t,t,...,t)is established, provided no bidder
has dropped out yet. If the first bidder to drop out does so at the pricepN,lettNbe
implicitly defined bybN(tN)=pN. Then suppose every remaining bidder with signalt
has the strategy of staying until the price reachesbN−^1 (t, pN)=u(t,t,...,t,tN).Let
pN− 1 be the price at which the next bidder drops out. Then lettN− 1 be implicitly defined
bybN−^1 (tN− 1 ,PN)=pN− 1. Now every remaining bidder has a strategy of remaining
in the bidding until the price reachesbN−^2 (t, pN− 1 ,pN)=u(t,t,...,tN− 1 ,tN).Pro-
ceeding in this manner, the bidding strategies of the bidders after each round can be
written down until two bidders remain. Clearly, these strategies entail that each bidder
drops out at that price at which, given the information revealed by the bidding up to
that point, the expected value of the object would be exactly equal to the price if all
remaining bidder except herself were to drop out all at once at that price.
We shall argue that these strategies constitute an equilibrium of the ascending auction.
If bidder 1 wins the auction, thent 1 must exceed all other signals. Now, from the con-
struction of the bidding strategies, it is clear that the bidder with highest signal among
the remaining bidders quits at a priceu(y 1 ,y 1 ,y 2 ,y 3 ,...,yN− 1 ), whereyidenotes the
value of theith highest signal among the rest of the bidders, i.e., excluding bidder 1.
Thus, bidder 1 getsu(t, y 1 ,y 2 ,y 3 ,...,yN− 1 )−u(y 1 ,y 1 ,y 2 ,y 3 ,...,yN− 1 ), which is
strictly positive. Quitting earlier, she would have obtained zero, and any other strategy
that makes her drop out after the bidder with signaly 1 cannot give her any higher pay-
off. Consider now a situation in which bidder 1 does not have the highest draw. For
her to win the auction, she must have to payu(y 1 ,y 1 ,y 2 ,y 3 ,...,yN− 1 ); however, this
exceeds the value of the object to her, which isu(t, y 1 ,y 2 ,y 3 ,...,yN− 1 ). Thus, she
cannot do better than drop out as prescribed by the equilibrium strategy.
To find the equilibrium bid in the first-price auction, assume that each of the other
N−1 bidders follow a bidding strategybF(z), and that bidder 1 bids as though her
private signal werez. Since the bids are increasing, the expected profit for bidder 1
whose signal istis
Π(z,t)=∫z0(
v(t, y)−bF(z))
g(y|t)dy.The derivative of this expression with respect tozis
(
v(t, z)−bF(z))
g(z|t)−bF
′
(z)G(z|t),