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Model II
Common values can also be modelled as a special case of
interdependent
values
.
In the interdependent values model
v
1
=
α
s
1
+
γ
s
2
v
2
=
α
s
2
+
γ
s
1
where
s
1
and
s
2
are private signals of bidders 1 and 2,
α
≥
0 is the weight
a bidder puts on her own signal and
γ
≥
0 is the weight she puts on her
opponent’s signal.
Rod Garratt
ECON 177: Auction Theory With Experiments
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View Full Document We consider the case where
α
=
γ
= 1
.
In this case,
v
1
=
v
2
,
which is a case of common values.
The oil lease example ﬁts this model when each signal
s
i
is determined
independently and
s
i
= 0 or 3 with equal probability.
In what follows we will consider a more general treatment of the private
signals.
We will assume throughout that the signals
s
i
are drawn independently
form the uniform distribution on [0
,
100]
.
Rod Garratt
ECON 177: Auction Theory With Experiments
Claim:
The ﬁrstprice auction has a symmetric Nash equilibrium in which
each bidder bids
s
i
.
Proof.
Suppose bidder 2 bids
s
2
and consider an arbitrary bid
b
1
for bidder 1.
We need to write down bidder 1’s expected payoﬀ as a function of her bid
b
1
and show that this is maximized at
b
1
=
s
1
.
We derive bidder 1’s expected payoﬀ as a function of her bid in three steps.
Rod Garratt
ECON 177: Auction Theory With Experiments
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View Full Document Step 1.
Compute the probability that bidder 1 wins with bid
b
1
.
Bidder 1 wins only if her bid is higher than bidder 2’s bid, i.e.,
b
1
>
s
2
.
Since we assume that signals are uniform on [0
,
100]
,
this happens with
probability
b
1
100
.
Thus, bidder 1 wins the auction with bid
b
1
with probability
b
1
100
.
Rod Garratt
ECON 177: Auction Theory With Experiments
Step 2.
Compute bidder 1’s expected value of the item if she wins at bid
b
1
.
Remember that each bidder’s common value for the good it equal to the
sum of the private signals.
Bidder 1 knows
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This note was uploaded on 12/26/2011 for the course ECON 177 taught by Professor Garratt during the Fall '09 term at UCSB.
 Fall '09
 GARRATT

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