IPO Auctions: English, Dutch, ... French, and Internet

30 BIAIS AND FAUGERON-CROUZET
Substituting this value of q(g; l) in the incentive compatibility condition becomes
N−1
l=0
πl
S − qu(l + 1) − (N − (l + 1))q(b; l + 1)
(vl+1 − p(l + 1))
l + 1
N−1
≥ πlq(b; l)(vl+1 − p(l)).
l=0
To relax this condition, minimize qu(l + 1), for l = 0,...N − 1, by setting it
equal to 0, and minimize q(b; l), l = 0,...N, by setting: q(b; l) = 0, for l > 0,
and q(b;0)= Sk
. Thus the incentive compatibility condition simplifies to
N
N−1
l=0
πl
S
l + 1 (vl+1
Sk
− p(l + 1) ≥ π0
N (v1 − p(0)).
Treating this constraint as an equality, p(N) can be written as a function of the
other prices
πN−1 p(N) = πN−1vN +
N−2
l=0
− π0k(v1 − p(0)).
N πl
l + 1 (vl+1 − p(l + 1))
Now, the objective of the mechanism designer is to maximize (under the incentive
compatibility and individual rationality conditions) the expected proceeds, which
can be written as
N−1
n=0
N−2
µn p(n) = µN p(N) +
µn p(n),
where µn denotes the probability that n signals out of N are good. Noting that µN =
ππN , and substituting the incentive compatibility condition into the objective, the
latter becomes
π
πN−1vN +
N−1
+
n=1
N−2
l=0
µn p(n) + µ0 p(0).
N πl
l + 1 vl+1
N−2
−
l=0
n=0
N πl
l + 1 p(l + 1) − π0k(v1 − p(0))
Noting that µn+1 = ππn N
, and that because of the participation constraint:
n+1