Ch. 3: Auctions in Corporate Finance 91
values to be independent draws from a distributionF(v). We have therefore introduced
a degree of symmetry in the model, that of symmetric beliefs.^3
Fix a particular bidder, and focus on the highest value among the remainingN− 1
values from the otherN−1 bidders, and denote this value asv 2. Sincev 2 is the highest
amongN−1 independent draws from the same distribution, its probability distribution
G(v 2 )(i.e., the probability thatN−1 independent draws are less than a valuev 2 )is
G(v 2 )=F(v 2 )N−^1. (1)
Notice that the distributionG(v 2 )has a density functiong(v 2 )=(N− 1 )F (v 2 )N−^2 ×
f(v 2 ).IfF(v)is uniform over the unit interval, i.e.,F(v)=vfor 0v1, then note
that
G(v 2 )=F(v 2 )N−^1 =vN 2 −^1. (2)2.2. First-price sealed-bid auctions
We are now in a position to evaluate any specific set of auction rules. Turn first to the
common first-price sealed-bid auction, where bidders submit sealed bids and the highest
bidder wins and pays the amount of her bid (hence the “first-price” qualifier). For now
we assume a zero reserve price (a price below which the seller will keep the asset rather
than sell).
In placing a bidb, bidderihas expected profit of
E(πi)=Pr(win)(vi−b), (3)where one can note that in the case that bidderiloses, her profit is zero. While(3)
does not make it explicit, Pr(win) will be a function ofb, normally increasing. This
creates the essential tension in selecting an optimal bid: increasing one’s bid increases
the chance of winning, but the gain upon winning is less.
To solve this model, we need just a bit more structure. Let us use an intuitive version
of the so-called Revelation Principle. Fix a bid functionb(v), and think of bidderias
choosing thevshe “reports” rather than choosing her actual bid. So long asb(v)is
properly behaved, we have not restricted bidderi’s choice in any way, for she could get
to any bidbdesired by simply “reporting” the requisitev.
Looking ahead, we are searching for a symmetric Nash equilibrium in bidding strate-
gies. In terms of ourb(v)function, symmetry means that all bidders use the sameb(v).
Nash equilibrium requires that, given other bidders’ strategies, bidderi’s bid strategy
is optimal. In terms again of ourb(v)formulation, equilibrium requires each bidder to
reportv=vi, i.e., “honest” reporting. Our requirement for Nash equilibrium will there-
fore be as follows. Suppose that the other bidders are usingb(v)and honestly reporting,
(^3) Several papers examine the effects of asymmetric beliefs, for example,Maskin and Riley (2000a, 2000b).
See alsoKrishna (2002).