Handbook of Corporate Finance Empirical Corporate Finance Volume 1

Ch. 3: Auctions in Corporate Finance 101

expression:

∑N

i= 1

pi( 0 )+

∑N

i= 1

∫

V

(

vi−

1 −Fi(vi)

fi(vi)

)

Qi(v)f (v)dv

= (18)

∑N

i= 1

pi( 0 )+

∫

V

(N

∑

i= 1

Ji(vi)Qi(vi)

)

f(v)dv

and the payment made by bidderineeds to satisfy

Pi(v)=Qi(v)vi− (19)

∫vi

0

Qi(zi,v−i)dzi.

The quantitiesJ(vi)=vi−^1 −fiF(vi(vi)i)are known as “virtual valuations” for reasons that

will become clear below. Notice that^1 −fiF(vi(vi)i)is the inverse of the hazard rate 1 −f(vFi(vi)i).
If the hazard rate is increasing, then the virtual valuations are increasing invi.Thisis
known as the “regular case” in the literature.
Ignoring the Incentive Compatibility and Individual Rationality constraints for the
moment, it is clear that the objective function(18)is maximized pointwise ifQi(v)
is set equal to the maximum value (i.e. 1, since it is a probability) whenJi(vi)is the
highest for any realizedv, and zero otherwise. Two implications immediately follow.
First, notice that the allocation rule implies that if the bidders are symmetric (i.e.,
the private values are drawn from the same distributionF(vi)for alli), then the bidder
with the highest value gets the object with probability one. Moreover, from(19),any
two selling procedures that have the same allocation rule must also result in the same
expected payment made by the bidders and thus result in the same expected revenue
for the seller. In particular, when the bidders are symmetric, all the standard auctions—
since they result in the highest value bidder getting the object with probability 1—are
optimal selling mechanisms and result in the same expected revenue for the seller.
Second, if the bidders are not symmetric, then the object need not go to the bidder
with the highestvi. For example, supposefi(vi)=bi−^1 ai. ThenJi(vi)= 2 vi−bi.
Thus,vi >vj ⇒Ji(vi)>Jj(vj)if and only ifvi−vj >(bi−bj)/2. In other
words, the high-value bidder may not get the object if the upper bound on her value
for the object is sufficiently high. The intuition is that the potential for such a bidder
to under-represent her value is high; thus, by discriminating against her in terms of the
likelihood of being awarded the object, the seller induces her to report truthfully when
her valuation is high. The basic message here is of considerable importance, as we will
see in more detail later: when bidders are asymmetric, it may pay to discriminate against
the stronger bidder.^17

(^17) Notice that in the regular case, since the virtual valuations are non-decreasing, theqi’s are non-decreasing
as well. Moreover, it is easily checked thatPi( 0 ,v−i)=0forallv−i; hencepi( 0 )=0foralli. Thus,
incentive compatibility and individual rationality conditions are satisfied.