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

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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

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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

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Price feedback 1 Provide prices for all bundles –Cant even enumerate them all in many cases (2 100 possible bundles over 100 items). –Many bundles have no bids / are irrelevant.

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Price feedback 2 Provide highest bid price for every bundle thats: 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 dont

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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

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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.

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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

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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.

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Pseudo-dual price constraints

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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

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Duality Theory Every linear optimization problem has an equivalent dual one (variables and constraints are reversed) Dual variables provide pricing information b j (A) + b j (B) < b j (AB) (super-additive) b j (A) + b j (B) > b j (AB) (sub-additive) Revenue problem! (solution XOR use a phantom good D, bid for AD, BD, ABD)

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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

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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.

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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.

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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

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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

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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?

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