Handbook of Corporate Finance Empirical Corporate Finance Volume 1

104 S. Dasgupta and R.G. Hansen

3.2. Optimal bidding with a common value

We begin with the illustrative example introduced above, and show how the principles
apply.
Suppose there are only two bidders and the value to each bidder is given as

v=ti+tj, (22)

wheretiandtjare each bidder’s privately known signals. We will suppose that the
signals are independently distributed according to a uniform distribution on[ 0 , 1 ].
Consider first a second-price auction. It is easy to show that in this auction, it is
optimal for each bidder to bid 2ti. Suppose bidderjis following this strategy, and
bidderibidsb. Then bidderiwins the auction if 2tj=bj <b, i.e.,tj<b/2. Her

expected gain is

∫b/ 2

0 (ti+t ̃−^2 t)d ̃ t ̃=ti

b

2 −

1

2

b^2
4. Maximizing with respect tob, one
getsbi= 2 ti, as claimed.
With two bidders, the second-price auction is equivalent to an ascending auction.
Thus, it should be no surprise that the equilibrium bidding strategies in an ascending
auction are identical to the one derived above. To see this, suppose bidderjhas a bid-
ding strategy ofbj= 2 tj. If biddericontinues to be in the auction at a priceb> 2 ti, her
profit ifjended the auction by dropping out would beti+b/ 2 −b=ti−b/ 2 <0, and
thus it cannot be optimal for her to be in the auction at that price. Similarly, ifb< 2 ti
her profit ifjends the auction would beti+b/ 2 −b>0, and thus it cannot be optimal
for her to quit at that price. Consequently, she must stay in the auction until the price
reaches 2ti.
Notice that the bidders do take into account winner’s curse in equilibrium. If the price
reaches a levelb= 2 ti, the value of the object is at leastti+b/ 2 = 2 ti, sincejis still
in the auction. Thus, the expected value is strictly higher than 2ti. However,iwould
still quit at this price, becauseif the auction had endedat this price becausejquit, she
would be breaking even. As we saw above, she would lose if the auction ends at any
higher price and she is the winner.
A first-price auction is more complicated, but similar results hold. One can think
of the optimal bid in a first-price auction as being the result of a two-stage process:
first, adjust one’s expected value for the bias associated with being the highest out of
Nsignals; and second, further lower the bid to account for the strategic nature of an
auction.

3.3.Milgrom and Weber’s(1982a, 1982b)generalized model

3.3.1. Core assumptions

While both the independent private value and the pure common value model capture
many key aspects of real auctions, they are obviously polar cases. Many real auctions
will contain both private value and common value characteristics. In an auction of a
company, for instance, the company’s “core” cash flow will be a common value for all