Ch. 3: Auctions in Corporate Finance 99
It remains to characterize the reserve prices in different auction settings. Suppose the
reserve price isr. Notice that in both the first-price and the second-price auctions, no
bidder with a value less thanrcan make any positive profit, as they have to bid at leastr
to win the object. On the other hand, the profit of a bidder with value greater thanrmust
be strictly positive (in the second price auction, if no other bidder bids higher thanr,
the bidder paysr). Thus, by continuity, the marginal bidder must have a valuev∗=r.
Note from(17)that the optimal reserve price exceeds the seller’s own valuation and
is independent of the number of bidders. This latter point makes sense given that the
optimal reserve price is only aimed at making the high bidder pay more in the instance
when all other valuations are beneath the reserve price. Note also that a reserve price
destroys the assurance of an efficient allocation; in the case where the highest valuation
among the bidders is less thanv∗but greater thanv 0 , the seller will retain possession
even though one of the bidders has a valuation greater than the seller.
Notice also thatentry feesare an alternative way of implementing a positive reserve
price. By setting an entry fee equal to the expected profit of a bidder with valuerwhen
the reserve price is 0,^14 the seller can ensure that a bidder participates if and only if her
value exceedsr.
2.6. Optimal selling mechanisms
Auctions are best thought of as “selling mechanisms”—ways to sell an object when
the seller does not know exactly how the potential buyers value the object. There is
obviously a very large number of ways in which an object could be sold in such a
situation: for example, the seller could simply post a price and pick one bidder randomly
if more than one buyer is willing to pay that price; post a price and then negotiate;
use any one of the common auctions; use any of thelesscommon forms of auction
such as an “all pay” auction in which all bidders pay their bids but only the highest
bidder gets the object; impose non-refundable entry fees; use a “matching auction” in
which one bidder bids first and the other bidder is given the object if he matches the
first bidder’s bid, and so on. The search for an optimal selling scheme in a possibly
infinite class of selling schemes would indeed seem like a daunting task. The major
breakthrough, however, was the insight that without loss of generality, one could restrict
attention to selling mechanisms in which each buyer is induced to report her valuation
(often called “type”) truthfully. This is the so-called “Revelation Principle” (Myerson,
1981; Dasgupta, Hammond and Maskin, 1979; Harris and Raviv, 1981), and it greatly
simplified the formulation of the problem.
Armed with the Revelation Principle, one can attack the problem in a more general
setting than we have discussed so far. While we will still remain within the confines
of the independent private values framework,^15 we can dispense with the assumption
(^14) From(15),thisis∫ 0 rG(y) dy.
(^15) Myerson’s (1981)framework is slightly more general in that he allows the value estimate of a bidder as
well as the seller to depend on the signals of all other bidders, i.e., his model is one in which the signals are
independent and private, but the valuations are interdependent.