Ch. 3: Auctions in Corporate Finance 105
bidders, but synergies will likely differ across bidders and therefore contribute an el-
ement of independent private values. In a seminal paper,Milgrom and Weber (1982a,
1982b)developed analysis of a generalized valuation model for auctions. The key valu-
ation assumption in Milgrom and Weber’s generalsymmetricmodel is that the value of
the item to bidderiis given by
vi=u(ti,t−i). (23)In(23),tiis the signal privately observed by bidderi, andt−idenotes the vector
of signals(ti,t 2 ,...,ti− 1 ,ti+ 1 ,...,tN). The functionu(·,···)is non-decreasing in all
its variables. The model is symmetric in the sense that interchanging the values of the
components oft−idoes not change the value of the object to bidderi. In this symmetric
model, note that both the private and pure common value models are special cases: if
vi=u(ti)for alli, we have the private value model, and ifvi=u(t 1 ,t 2 ,...,tN)for
alli(i.e.,u(·,···)is symmetric inallthe signals, then the model is a common value
model. The interdependent values model with independent signals discussed earlier is
also obviously a special case, in which the signals are i.i.d.
The symmetric model assumes that the joint density of the signals, denoted by
f(·,···,·)is defined on[ 0 , ̄t]N, and is a symmetric function of its arguments. The
density functions are also assumed to have a statistical property known as “affiliation”,
which is a generalized notion of positive correlation among the signals.
It will be convenient to work in terms of the expected value of the object to bidderi
conditional on her own signaltiand the highest among the remainingN−1 signals.
Without any loss of generality, we will focus on bidder 1, and accordingly, let us define
v(t, y)=E (24)[
vi(···)|t 1 =t,Y 1 =y]
,
whereY 1 is the highest signal among the remainingN−1 signals of bidders 2,...,N.
We will denote the distribution function ofY 1 byG(y)and its density byg(y). Notice
that because of symmetry, it does not matter who among the remaining bidders has the
highest signal, and moreover, by virtue of symmetry with respect to the way in which a
bidder’s own signal affects the value of the object to the bidder, the function is the same
for all bidders. Because of affiliation, it follows thatv(·,·)is non-decreasing function
intandy.
3.3.2. Equilibrium bidding
It is convenient to begin with the second-price auction. Generalizing the example in
Section3.2, we shall show that the symmetric equilibrium bid function is given by
v(t, t). Recall that the functionv(t, t)is the expected value of the bidder’s valuation,
conditional upon the bidder having signaltand on the bidder with the second-highest
signal also having signalt.
To see thatv(t, t)is the symmetric equilibrium bid function, notice that if bidder 1
bidsb 1 assuming that all other bidders are following the proposed equilibrium bidding