Wireless Network Design: Optimization Models and Solution ...

7 Spectrum Auctions 165
prior round. The set of constraints (7.10) assures that the price paid by each bidder
is bounded below by its VCG price and above by its maximum bid price (value).
Based on the prices generated in the above linear optimization problem, one calculates
the discount for bidder j as the difference between the value of the winning
bid and the price at this step of the algorithm. All bids of winning bidders are discounted
by this amount and the winner determination problem (WDP Core(τ)) is again
re-solved. The process continues until one can find no coalition that can increase
revenue to the seller. This algorithm therefore finds prices for each winning bidder
that are in the core. Since there may be many such minimum core prices, Day and
Milgrom [18] suggest that one choose - in order to encourage sincere bidding - the
minimum core prices that are closest in Euclidean distance from the VCG prices.
Alternatively, Erdil and Klemperer [24] argue for a different set of minimum core
prices that are based “on a class of ’reference rules’ in which bidders’ payments are,
roughly speaking, determined independently of their own bids as far as possible”.
These core-selecting second-price sealed-bid mechanisms are relatively new.
They have the following nice properties: They are in the core; they eliminate the
exposure problem; and, they encourage bidders to bid sincerely. As with all sealedbid
auctions, they make collusion and punishment for not adhering to tacit agreements
extremely difficult. In 2010, Portugal is scheduled to hold a core-selecting
second-price auction (see announcement on Portuguese website anacom.pt).
There are, however, negatives associated with this auction design in that it puts
a significant burden on the bidders. Each bidder needs to assess for each possible
combination of items whether it is a package that it would like to acquire and then,
for those that make sense for the bidder, provide a value for the package. In addition,
such mechanisms do not provide any information about how the packages submitted
might “fit” with packages submitted by other bidders. To overcome these problems,
a number of authors have suggested new package-bidding designs that replicate
many of the features of the simultaneous ascending auction.
7.5.2 Two Combinatorial Ascending Auction Designs
7.5.2.1 General Ascending Package-bidding Design:
In 2000, the FCC considered a package-bidding auction that resembled the SAA
auction in many respects. It had most of the same rules as the SAA except that it allowed
bidders to submit package bids. Each round a winner determination problem
was solved to determine the winners with the constraint that at most one bid of any
bidder could be in an optimal solution.
In an ascending auction, the auctioneer must announce prices for the next round.
Since packages, rather than individual items may be winning, one needs to be able to
allocate the package price among the individual elements that make up that package.
If one has such individual item prices, then one can determine the price of any new
package as the sum of the items that make up the package. However, since there is