Wireless Network Design: Optimization Models and Solution ...

166 Karla Hoffman
often a synergy value associated with the package, any such pricing approach cannot
do this disaggregation so that there are minimal distortions from the true prices and
so that prices do not fluctuate each round.
The reason for wanting such prices relates to a problem unique to package bidding
auctions. The problem is labeled the threshold problem, whereby smaller bidders
may have difficulty overcoming the high bid of a bid on a large package. An
extreme example is when a bidder creates a package of all items and places a high
price on that package bid. Each individual bidder need not make up the total shortfall
between the collection of small bids and the large bidder, but rather they must
collectively overcome the bid price and they need to know how much of the shortfall
each should be paying. Essential to overcoming this threshold problem is providing
good price information and having activity rules that force participation.
A number of pricing algorithms have been suggested that are based on the dualprice
information that arises from solving the linear-programming relaxation to the
integer programming problem. However, these prices are only approximations to
the “true” dual prices of a combinatorial optimization problem, and are often call
pseudo-dual prices. See Bikhchandani and Ostroy [5], Kwasnica et al. [36], Dunford
et al. [21], and Bichler et al. [4] for more on pricing in combinatorial auctions
settings.
The FCC combinatorial software uses the following pseudo-price estimates, although
there are many others suggested in the literature. The pseudo-price estimates
are obtained by solving the following optimization problem, Pseudo-Dual Price Calculation
(PDPC):
[PDPC] min z ∗ δ = subject to
∑
j∈B\W
δi
∑
i∈I j
πi + δ j ≥ b j, ∀ j ∈ B\W, (7.11)
∑
i∈I j
πi = b j, ∀ j ∈ W, (7.12)
δ j ≥ 0, ∀ j ∈ B\W, (7.13)
πi ≥ ri, ∀i ∈ I, (7.14)
where j is the set of bids, I is the set of items, I j is the set of all bids of bidder
j, and ri is the reserve price on item i. Constraint set (7.12) assures that the sum
of the bid prices πi of the items i in the winning bid is equal to the winning bid
amount. Constraint set (7.11) together with the objective function tries to enforce
dual feasibility as much as possible, where the δ j’s represent the deviation from dual
feasibility 7 . Constraint set (7.13) maintains the non-negativity of these violations,
7 In integer linear optimization, there can be a duality gap whereby the primal and the dual problems
do not provide the same values. In this case, if one uses the prices directly from the linear
programming relaxation, one would obtain prices that would overestimate the prices needed. In