Testing Linear Pricing Algorithms for use in Ascending Combinatorial Auctions (A5) Giro Cavallo David Johnson Emrah Kost

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# Testing Linear Pricing Algorithms for use in Ascending Combinatorial Auctions (A5) Giro Cavallo David Johnson Emrah Kost - PowerPoint PPT Presentation

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

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Testing Linear Pricing Algorithms for use in Ascending Combinatorial Auctions(A5)Giro CavalloDavid JohnsonEmrah 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.
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.
• 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
• 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