September 2009
Intermediate to advanced
336 pages
9h 37m
English
Content preview from Auction Theory, 2nd Edition
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9.6 Miscellany 141
so bidder N is the first to drop out.
3
Now (9.6) implies that for all i,
v
i
(
φ
1
( p
N
),φ
2
( p
N
),..., φ
N
(
p
N
))
=p
N
that is, all the values at φ
(
p
N
)
are the same. Thus, they all equal the average
value, so, in particular,
v
N
(
φ
1
( p
N
),φ
2
( p
N
),..., φ
N
( p
N
)
)
=v
(
φ
1
( p
N
),φ
2
( p
N
),..., φ
N
( p
N
)
)
Since φ
N
(
p
N
)
=x
N
and for all i =N, φ
i
(
p
i
)
>φ
i
(
p
N
)
, the average crossing
condition implies that
v
N
(
x
1
,x
2
,..., x
N
)
<
v
(
x
1
,x
2
,..., x
N
)
Since the ex post value of bidder N is less than the average ex post value of
all the bidders, it must be that
v
N
(
x
1
,x
2
,..., x
N
)
< max
i
v
i
(
x
1
,x
2
,..., x
N
)
Thus, the person who is the first to drop out does not have the highest value.
The same ...
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