Lecture XVI: Auctions
Markus M. M¨obius
April 14, 2002
Readings for this class:
P. Klemperer  Auction Theory:
A
Guide to the Literature (especially parts of the appendix  the
main text provides an excellent introduction to auction theory but
is optional)
1
Introduction
Auctions are extremely common. Natural resources such as wireless spec
trum, oil and minerals etc. are auctioned of.
There are several important types oF auctions:
•
ascending price auction
•
secondprice auction
•
sealed bid (±rst price) auction
•
allpay auction (good model For legislative lobbying, war oF attrition)
There are many possible scenarios For agents’ types. The two most im
portant ones are:
•
Private Value environment: each agent’s valuation oF the good is drawn
i.i.d. From some distribution
F
on [
v
,
v
].
•
Common value auction: agents’ valuations are correlated (oil leases).
Easy Formulation:
v
i
=
V
(
t
1
, ..., t
n
)where
t
i
are agents’ signals.
(1)
1
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Some Simple Solved Examples
We frequently work with values (signals) drawn from the uniform distribu
tion. In that case it is useful to know that the expected value of the
ith
order
statistic is
v
+
n
+1
−
k
n
(
v
−
v
)(
2
)
2.1
Sealed Bid Auction (Private Value)
In the sealed bid auction where the players’ valuations are independently
uniformly distributed on [0
,
1] the unique BNE is:
f
∗
1
(
v
1
)=
v
1
2
f
∗
2
(
v
2
v
2
2
Proof:
To verify that this is a BNE is easy. We just show that each type of
each player is using a BR:
E
v
2
(
u
1
,f
∗
2
;
v
1
,v
2
)=(
v
1
−
b
1
)
Prob
(
f
∗
2
(
v
2
)
<b
1
)+
1
2
(
v
1
−
b
1
)
(
f
∗
2
(
v
2
b
1
)
We assume
b
1
∈
£
0
,
1
2
¤
. No larger bid makes sense given
f
∗
2
. Hence:
E
v
2
(
u
1
∗
2
;
v
1
2
v
1
−
b
1
)2
b
1
This is a quadratic equation which we maximize by the FOC:
0=2
v
1
−
4
b
1
(3)
Hence
b
1
=
v
1
2
.
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 Fall '10
 ding
 Auction, price auction, First Price Auction, private value

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