Homework 3  Solutions
Q1
Suppose valuations are distributed along some interval,
v
i
2
[0
;
v
]
with
v > r:
1
If
i
0
s
valuation is above
r;
and
he bids below
r
then he gets zero for sure. Let
b
±
i
be the highest bid among all bidders other than
r:
Suppose
instead
i
bids
b
i
where
b
i
> r
and
b
i
v
i
:
Then, if
b
±
i
< b
i
;
v
i
> r
bidding below
r
is weakly dominated by bidding anything between
r
and
v
i
:
On the other hand, if
v
i
r;
the agent would have to bid above his valuation if he were to bid more than
r:
We
showed in class that this is dominated by bidding below your valuation, hence in this case, bidding below
r
is
not dominated
Q2
We need to show that bidding anything else is dominated. If the agent bids
b
i
=
v
i
;
and
b
±
i
< b
i
=
v
i
, he makes
v
i
±
b
3
where
b
3
denotes the third highest bid, otherwise he gets zero. If he bids
b
i
< v
i
;
in the case
b
±
i
< b
i
;
b
±
i
is greater than
b
i
but smaller than
v
i
he loses, and gets
zero whereas if he had bid
b
i
=
v
i
v
i
±
b
3
:
If he instead bids some
b
0
i
> v
i
;
and
b
±
i
> v
i
but
b
0
i
> b
±
i
;
so that
i
wins the auction even though he is not the one with the highest valuation, it
i
has the second highest valuation, and wins
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 Spring '08
 Witte
 Macroeconomics, Game Theory, Auction, Vickrey auction, Vickery auction

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