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Multiple objects 4 . For each of the m> 1 objects, the bidders' values are independently
and identically distributed and so are their costs. There are m distribution functions for
the common values, one for each object. Likewise, there are m distribution functions
for the costs, one for each object. For j =1 ,...,m ,let Q j : R +
[0 , 1] denote the
distribution function for the value of the j th object and G j : R +
[0 , 1] that for its
cost. Thus, each bidder receives its value signal for the j th object from Q j and its cost
signal from G j .
The m objects are sold one after another in m auctions that are conducted using
English auction rules. Furthermore, each bidder receives the cost and value signals for
an auction just before that auction begins. The signals for the j th object are received
only after the ( j
1) previous auctions have been conducted. Consequently, although
the bidders know the distribution functions from which the signals are drawn, they do
not know the actual signals for the j th object until the previous ( j
1) auctions are
over.
Each bidder can win at most one object. The winner for the j th object cannot partic-
ipate in the remaining m
j auctions. Thus, if n agents participate in the first auction,
the number of agents for the j th auction is ( n
j +1). For objects j =1 ,...,m and
bidders i =1 ,...,n ,let v ij and c ij denote the common and private values respectively.
The true common value of the j th object (denoted V j )is:
n−j +1
1
V j =
v ij
(2)
n
j +1
i =1
For objects j =1 ,...,m and bidders i =1 ,...,n ,let S ij = v ij /n
c ij denote i 's
surplus for object j .
Note that the values/costs for our model are not correlated. Such correlations oc-
cur across objects, if for a bidder (say i ) the value/cost of object j =2 ,...,m can
be determined on the basis of i 's value/cost signal for the first object. However, in
many cases such a direct relation between the objects may not exist. Hence, we focus
on the case where different objects have different distribution functions. Furthermore,
although each bidder knows the distribution functions from which the values/costs are
drawn before the first auction begins, it receives its signals for an object only just before
the auction for that object begins. In the following section, we determine equilibrium
bidding strategies for this multi-object model.
3
Equilibrium Bidding Strategies
The m objects are auctioned in m separate English auctions that are conducted sequen-
tially. The English auction rules are as follows. The auctioneer continuously raises the
price, and bidders publicly reveal when they withdraw from the auction. Bidders who
drop out from an auction are not allowed to re-enter that auction. A bidder's strategy for
4
Our model for multiple objects is a generalisation of [3]. While [3] studies sequential auctions
for two private value objects, we study sequential auctions for m ≥ 2 objects that have both
private and common values.
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