Problemsets 4 & 5
1. Consider the firstprice privatevalue auction where there are
n
bidders,
each with a value drawn from a uniform distribution on [0
,
ω
]. If everyone
but player
i
follows a symmetric bidding strategy
β
(
ω
i
), the expected
utility of a bid
b
i
is given by:
Pr
braceleftbig
β
−
1
(
b
i
)
≥
ω
j
bracerightbig
n
−
1
[
ω
i
−
b
i
]
,
the probability
i
’s bid is the highest multiplied by the payoff if they win.
Assume that
β
(
ω
j
) =
aω
j
.
(a) Find the bestresponse bid
b
⋆
for bidder
i
when their value is
ω
i
.
(b) Find the value of
a
such that
β
(
ω
i
) is a symmetric Bayesian Nash
equilibrium.
(c) Find the expected payment of bidder
i
.
(d) What is the expected payment of bidder
i
in an allpay commonvalue
auction?
2. Consider a secondprice commonvalue auction where the value to each
player
i
is given by
V
i
(
ω
) =
αω
i
+ (1
−
α
)
1
n
−
1
∑
j
negationslash
=
i
(
ω
j
). Each individual
only knows their only component of the value,
ω
i
, and makes a bid
b
i
, and
pays the secondhighest bid.
Again, assume that the individual signals
ω
i
are distributed with a uniform distribution over [0
,
ω
]. The expected
utility is therefore
Pr
braceleftbig
β
−
1
(
b
i
)
≥
ω
j
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 Spring '08
 Staff
 Game Theory, Utility, vi, Bayesian Nash equilibrium, symmetric Bayesian Nash, sequential equilibrium

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