Handbook of Corporate Finance Empirical Corporate Finance Volume 1

100 S. Dasgupta and R.G. Hansen

that all bidders’ valuations are drawn from identical distributions, i.e., one can accom-
modate asymmetries among bidders. Asymmetries are important in many real world
situations—for example, in procurement, when both domestic and foreign bidders par-
ticipate, and especially in corporate finance, in the context of takeover bidding.
Before proceeding further, however, we need to introduce some notation. Let
v = (v 1 ,v 2 ,...,vN)denote the set of valuations for bidders 1,...,Nand let
v ∈ V ≡ (×Vi)Ni= 1 , where Vi is some interval [ 0 ,v ̄i]. Likewise, let v−i =
(v 1 ,v 2 ,...,vi− 1 ,vi+ 1 ,...,vN), andv−i ∈ V−i ≡(×Vi)Nj= 1 ,j =i.Letf(v)denote
the joint density of the values; since the values are independently drawn, we have
f(v)=f 1 (v 1 )×f 2 (v 2 )×···×fN(vN), andf−i(v−i)=f 1 (v 1 )×···×fi− 1 (vi− 1 )×
fi+ 1 (vi+ 1 )×···×fN(vN)is similarly defined.
The seller picks a mechanism, i.e., an allocation rule that assigns the object to the
bidders depending on messages sent by the latter. By appealing to the Revelation Prin-
ciple, we can restrict attention todirectmechanisms, i.e., mechanisms that ask the
bidders to report their valuesvi. Thus, the mechanism consists of a pair of functions
〈Qi(v′), Pi(v′)〉Ni= 1 for eachiwhich states the probabilityQiwith which the object
would go to bidderiand the expected paymentPithat bidderiwould have to make
for any vector ofreportedvalues of the bidder valuations. Of course, the mechanism
has to satisfy two conditions: (i) it must be Incentive Compatible, i.e., it must be
(weakly) optimal for each bidder to report her value truthfully given that all others
are doing the same, and (ii) it must be Individually Rational, i.e., the bidders must
be at least as well off participating in the selling process than from not participat-
ing.
Thus, the probability that bidderigets the object when she reports her value to bezi
and all other bidders report truthfully is

qi(zi)=

∫

V−i

Qi(zi,v−i)f−i(v−i)dv−i,

and the expected payment he makes is

pi(zi)=

∫

V−i

Pi(zi,v−i)f−i(v−i)dv−i.

It can be shown^16 that (i) Incentive Compatibility is equivalent to the requirement that
theqi(vi)functions are non-decreasing, i.e., the probability that a bidder gets the object
is non-decreasing in her reported value of the object, and (ii) Individual Rationality is
equivalent to the requirement that thepi(vi)functions satisfypi( 0 )0, i.e., the bidder
with zero value has non-positive expected payment. It can also be shown that in the
optimal selling mechanism, theQi(v)need to be chosen to maximize the following

(^16) For details, please seeMyerson (1981)orKrishna (2002).