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Appendix
A
Proof of Theorem 1
Proof. We consider each of the m auctions by reasoning backwards.
- m th auction. To begin, consider the m th auction for which there are ( n
m +1)
bidders. Since this is the last auction, an agent's bidding behaviour is the same as
that for the single object case. Hence, the equilibrium for this auction is the same
as that in Equation 3 with n replaced with ( n
m +1).For j =1 ,...,m ,let
α ij denote an agent's cumulative ex-ante expected profit from auctions j to m .
Recall that although the bidders know the distribution (from which the cost and
value signals are drawn) before the first auction begins, they draw the signals for
the j th auction only after the ( j
1) earlier auctions end. Since α ij is the ex-ante
expected profit (i.e., it is computed before the bidders draw their signals for the j th
auction), it is the same for all participating bidders. Thus, we will simplify notation
by dropping the subscript i and denote α ij simply as α j We know from Equation 5
that:
1
m +1 ( E ( f n−m +1
E ( s n−m +1
m
α m =
)
))
(16)
m
n
This is because all the ( n
m +1)agents that participate in the m th auction have
ex-ante identical chances of winning it. Note that the right hand side of Equation 16
does not depend on i . In other words, since bidders receive their signals for the m th
auction after the ( m
1)th auction, the ex-ante expected profit for the m th auction
(before the ( m
1)th auction ends) is the same for all the ( n
m +1)bidders.
- ( m
1) th auction. Consider the ( m
1)th auction. During this auction, a bidder
bids b if ( V m− 1
c m− 1
b
α m )or( b
V m− 1
c m− 1
α m ). Hence, a symmetric
equilibrium for the ( m
1)th auction is obtained by substituting j = m
1 in
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