Transcript Motivations for Linear Pricing

Testing Linear Pricing
Algorithms for use in Ascending
Combinatorial Auctions
(A5)
Giro Cavallo
David Johnson
Emrah Kostem
Motivations for Linear Pricing
• Combinatorial ascending proxy auctions translate to
non-linear and non-anonymous pricing
• While a non-linear auction achieves an efficient
outcome at minimum competitive equilibrium
prices, it is not necessarily the most time efficient
• Price feedback in ascending proxy auctions is highly
specific, making determination of individual items
in a combinatorial setting difficult
Further Motivations
• Since bundles could be coupled together to create a
“winning” set, determining minimal cost partnering
for a given bidder is a complex problem
• In cases where items can be both substitutes and
complements for bidders, providing complete price
information is unsolved problem
• Ascending proxy auctions have proven to be
computationally inefficient
Price feedback 1
• Provide prices for all bundles
– Can’t even enumerate them all in many cases (2100
possible bundles over 100 items).
– Many bundles have no bids / are irrelevant.
Price feedback 2
• Provide highest bid price for every bundle that’s: a) been bid
on, and b) would be allocated
– Easy to do
– Clearly indicates how to win bundles that satisfy these
conditions
– Gives little or no information regarding bundles that
don’t
Price feedback 3
• Linear prices: prices for individual items s.t.
sum of prices for items in bundle B maps in
some way to a “price” for B
– Motivation: allows bidders to extrapolate prices
for arbitrary bundles, in a simple way
– Problem: bundle bids are often not linear!
(substitutes/compliments)
– Paper A5
Linear pricing algorithms
• Dealing with combination of
substitutes/compliments: unsolved problem.
• Providing exact pricing info is intractable.
• Use approximate strategies – different ones
do better depending on setting.
Basic Theory of Linear Pricing
• Bidders can only bid in the form of a set linear function
y=aX+b, where a is determined and called the bidding
increment, X is the variable the bidder can control(possibly
contingent on the round), and b is some reservation price,
occasionally set by the auction
• a, or the bidding increment, determines the time efficiency
of the auction, the larger the increment, the quicker the
solution of the winner determination program, albeit at the
cost of efficiency of the auction compared to the ascending
proxy auction
• Linear pricing also partially solves the anonymity problem by
creating a range for the valuation functions of the bidders
Pseudo-dual prices
• For winning bids: force sum of constituent
item prices to equal bid for bundle.
• Non-winning bids: allow sum of item prices
to exceed bid.
• Ensures sum of pseudo-dual prices = max
revenue for round.
Pseudo-dual price constraints

i
  j  bj

i
 bW
iI
iI
j
j
j 0
 i  ri
j  B \ W
j  W
j  B \ W
i  I
Choosing prices
• Many solutions that satisfy constraints.
– Test “quality of prices” produced by various
methods based on: auction length, computational
effort, efficiency, prices paid (closeness to VCG).
• Smoothed anchoring
• Nucleolus
Duality Theory
• Every linear optimization problem has an equivalent dual
one (variables and constraints are reversed)
• Dual variables provide pricing information
• bj(A) + bj(B) < bj(AB) (super-additive)
• bj(A) + bj(B) > bj(AB) (sub-additive)  Revenue problem!
(solution XOR use a phantom good D, bid for AD, BD,
ABD)
Linear Pricing Algorithms
• All algorithms are based on the dual of the winner
determination problem
• Pseudo-Dual Prices:
–
–
–
–
–
Resulting prices might not exists.
Estimate the prices from the maximal revenue of the round.
Define a slack variable for non-winning bids. (Infeasibility)
Minimize the total infeasibility [CP].
Solution is not unique! Creates fluctuations in the prices between
rounds…
– Confused bidders 
Linear Pricing Algorithms
• Smooth Anchoring Method
– Idea is to choose a solution that reduces the price fluctuations
between rounds.
– Add a linear quadratic program to smooth the price [QP].
– Solve [CP] & fix the optimum infeasibility…
– Solve [QP] …
– Not unique but less confused bidders.
Linear Pricing Algorithms
• Nucleolus Method
–
–
–
–
–
Treat the items as “agents”. Allocate the maximum revenue among the items.
Cost allocation game, agent compete for a “fair” allocation.
Minimize the maximum derivation from ideal prices. (duality, linearity)
Find the optimum infeasibility (slack variable) for each item iteratively.
Unique allocation. Dual feasibility…
• Constrained Nucleolus
– Same above but the sum of the prices in a winning bid is forced to be equal
to the winning bid amount.
– Different convergence properties.
Linear Pricing Algorithms
• The RAD Algorithm
– Same as nucleolus except bids on all packages, rather than the highest
non-winning bid, are considered.
– Once dual feasibility is obtained the smallest item price is maximized.
• The Smoothed Nucleolus Algorithm
– Start with nucleolus
– Stop when RAD achieves dual feasibility
– Perform smoothed anchoring
Results
• Studied the impact of increment size by running the same
auctions for increments of $5,000, $30,000, and $60,000
• The benchmark for efficiency was the ascending proxy
auction and they compared the results of these auctions for
different valuation functions to this benchmark
• The size of the increment determined both how quickly the
winner determination problem is solved and how close the
final prices come to exact second prices
• Smoothed anchoring method currently used by FCC comes
converges to revenue and bidder payments that are on
average close to optimal
Applications to Our Project
• In solving the winner determination, we must give price feedback to both
airlines and wireless competitors without disclosing too much
information, linear pricing helps add anonymity to the process
• Another problem to consider applies after each round, given the
allocation, how do we allow bids to be placed on “non-winning
packages” that could reallocate the current allotment and what price
information to we provide about the current allocation
• How can the price feedback help prevent bidders from misrepresenting
their true valuation functions or lead to a quicker convergence to their
true valuation?
• What are we most concerned about, quick determination of the winner
determination problem or the most efficient outcome that could be
achieved through an ascending proxy auction?