September 2009
Intermediate to advanced
336 pages
9h 37m
English
Content preview from Auction Theory, 2nd Edition
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244 Packages and Positions
On the other hand, the seller’s revenue
π
0
T
=max
S
j∈N
β
j
T
S
j
|x
j
=max
S
j∈N
max
x
j
S
j
−π
j
T
,0
=max
S
max
I⊆N
j∈I
x
j
S
j
−π
j
T
= max
I⊆N
max
S
j∈I
x
j
S
j
−π
j
T
= max
I⊆N
w(
I) −
j∈I
π
j
T
and so, in particular, for
I =N \i, this implies that
π
0
T
+
j=i
π
j
T
≥w(N \i) (17.3)
contradicting (17.2). We have thus argued that truthful reporting leads to the
VCG outcome.
In order to show that truthful reporting constitutes an ex post equilibrium,
consider a bidder i and suppose that all bidders j =i report truthfully. We know
from (17.3) that independent of what bidder i submits as his value vector, the
seller’s revenue π
0
T
≥w(N \i)−
j=i
π
j
T
. The right-hand ...
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