Wireless Network Design: Optimization Models and Solution ...

7 Spectrum Auctions 167
and constraint set (7.14) forces the prices on all items to be at least the minimum
price set by the auctioneer.
Since the solution to this problem is not necessarily unique, the algorithm tries
— by solving a second optimization problem — to reduce the fluctuations in prices
from round to round. To accomplish this, an optimization problem with a quadratic
objective function and linear constraints is solved. The objective function that applies
exponential smoothing to choose among alternative pseudo-dual prices with
the additional constraint on the problem that the sum of the slack variables equals
z∗ δ (the optimal value of the Pseudo-Dual Price Calculation). The objective function
minimizes the sum of the squared distances of the resulting pseudo-dual prices in
round t from their respective smoothed prices in round t − 1. Thus, it is the pseudodual
price of item i in round t. The smoothed price for item i in round t, is calculated
using the following exponential smoothing formula: pt i = αt i + (1 − α)pt−1
i ;
where, p t−1
i is the smoothed price in round t −1, 0 ≤ α ≤ 1, and p0 i is the minimum
opening bid amount for item i. The following quadratic program (QP) will find the
pseudo-dual price, for each item i in round t that minimizes the sum of the squared
distances from the respective smoothed prices in round t − 1 while assuring that the
pseudo-dual prices sum up to the provisionally winning bid amounts:
[QP] min∑(π i∈L
t i − p t−1
i ) 2
subject to
∑
i∈I j
π t i + δ j ≥ b j, ∀ j ∈ B i \W t , (7.15)
∑
i∈I j
π t i = b j, ∀ j ∈ W t , (7.16)
∑
j∈Bi \W t
δ j = z ∗ δ , (7.17)
δ j ≥ 0, ∀ j ∈ B i \W t , (7.18)
π t i ≥ ri, ∀i ∈ I. (7.19)
Note that problem (QP) has the same constraints as [PDPC], but has added the
additional restriction (7.17) that the sum of the δ j’s is fixed to the value z∗ δ , the
optimal value from [PDPC].
Experimental testing by Kwasnica et al. [36] has shown — when bidders have
complementary values for packages — that package-bidding auctions that use
pseudo-dual prices result in more efficient outcomes. However, laboratory experiments
also found negatives to such designs. In testing by Goeree et al. [26] and
Brunner et al. [8], results showed that such auctions took longer to complete and
bidders found it difficult to construct packages that fit well with the packages of
the context of this application, overestimating prices could lead to inefficient outcomes. Thus, we
make approximations that assure that the prices are consistent with the revenue obtained from the
integer optimization.