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Super solutions for combinatorial auctions Alan Holland & Barry O’Sullivan {a.holland,

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Overview Combinatorial Auctions (CA’s) Motivation Auction scenarios Implications of unreliable bidders Super solutions (SS) Solution robustness – simple example SS & CA’s SS for different types of auctions Experimental Results Extensions to framework

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Combinatorial Auctions Motivation Multiple distinguishable items Bidders have preferences over combinations of items Facilitates expression of complementarities / substitutabilities Improve economic efficiency Removes bidders ‘exposure problem’

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Bidders Exposure Problem Single-item auctions Example Two items (X,Y) are sold in two separate auctions A Bidder values the pair $100 But either X or Y on its own is valueless ($0) If she bids $50 for each and wins only one item she has lost $50 This induces depressed bidding Solution: Allow bids on XY – ‘combinatorial bids’

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Combinatorial Auctions Bids on all combinations of items are allowed Forward Auction – selling items Maximize revenue Weighted Set Packing problem Reverse Auction – buying items Minimize cost Set Covering Problem No Free Disposal => Set Partitioning Problem Gaining in popularity FCC spectrum auctions, Mars, GE, Home-Base, London Transport Authority

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Complexity Potentially 2 #items bids Winner Determination NP-Complete & Inapproximable [Rothkopf 98] State-of-the-art algorithms work very well in practice 1,000’s of bids for 100’s of items solved optimally in seconds [Sandholm 03]

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CA solution robustness Solution robustness Necessary when unreliable/untrustworthy bidders are present E.g. Supply chain formation, procurement Spectrum auctions are less suitable Bid withdrawal/disqualification Potentially dire consequences for revenue

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Unreliable Bidders in CA’s Single unit auction A winning bid is withdrawn => give the item to 2 nd priced bidder CA A winning bid is withdrawn => next best solution (in terms of revenue) may require changing the status of many other bids Undesirable in many circumstances (e.g. SCM) Auctioneer may be left with a bundle of items that are valueless (Auctioneer’s exposure problem) Preventative action -> robust solutions

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(a,b)-super solutions An (a,b)-super solution Guarantees that when ‘a’ variables are broken in a solution, at most ‘b’ other changes are required to find a new solution Thus providing solution robustness Example Solutions to a CSP are is a (1,0)-super solution & are (1,1)-super solutions

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(1,b)-super solution algorithm MAC-based repair algorithm [Hebrard et al ECAI04] Value assigned to the kth variable AC & Repairability check on the first k-1 variables If more than b changes are required => irreparable Our approach Solve the problem optimally using any ILP solver (CPLEX etc…) Get optimal revenue R opt Add a sum constraint s.t. revenue > R opt k% Find any super solution (Constraint Satisfaction) Optimize on robustness OR revenue

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(1,b)-super solutions for CA’s Zero values may be considered ‘robust’ Withdrawal of losing bids is immaterial (when a=1) Example CA - Valid solutions : 1.20: (1,2)-super solution : 1.15: (1,1)-super solution : 1.10: (1,1)-super solution Solution robustness 2 nd & 3 rd solutions are robust, but less revenue 2 nd solution dominates 3 rd solution Trade-off ensues between 1 st & 2 nd solution

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Experiments Aim: Examine the trade-off between revenue & robustness in different economically motivated CA scenarios Exhibiting different complementarity/ substitutability effects among items Auction distributions Arbitrary - Simulates component auctions (arbitrary complementarity between items for different bidders) Regions - Complementarity between items in 2-D space (e.g. spectrum auctions, property) Scheduling - Auctions for airport landing/take-off slots

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Experiments Arbitrary-npv distribution Random synergies => more varied series of items in bids => more constraints More pruning => lower search times

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Experiments Regions-npv distribution More mutually exclusive bids Less pruning => higher search times

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Experiments Scheduling distribution Bids contain few items => less constraints More pruning => longer search times N.B. poly-time matching can be used if bids are very short

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Constraint Satisfaction Find any super solution b & min %revenue: 20 items+100 bids Robust solutions – more likely for regions & scheduling arbitrary regionsscheduling

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Constraint Satisfaction Running times Problems with ~75% success rate = most difficult Scheduling distribution - most expensive Hybrid approach may improve performance arbitraryregionsscheduling

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Constraint Optimization Optimizing robustness BnB search; 20items+100bids Minimizing number of irreparable variables scheduling solutions are most robust

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Constraint Optimization Optimizing Robustness; b=0 When no super solution exists regions provides the most robust solutions (revenue > 95½% max)

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Constraint Optimization Optimizing Revenue: 20 items+100bids Many super solutions – find revenue maximizing SS Revenue constraint disallows potential repair solutions Results show avg opt revenue SS as % of overall optimal solution

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Constraint Optimization Optimizing Revenue Near optimal solutions achievable Computationally expensive (esp. scheduling) Hybrid techniques req’d to improve performance

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Proposed Extensions More flexibility required True cost of repair may not just be measured by number of variables changed E.g. Changing a winning bid to a losing one is more costly than vice versa Cost of repair may depend on the break E.g. If an agent withdraws a bid, changing his other bids may be considered a cheap operation Variables may have probabilistic failure E.g. Various bids may have probabilities of failure over time

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Current & Future Work Extend the SS framework Introduce a metric for the cost of repairing each variable value Generate repair solutions for sets of variables whose probability of breaking is above a certain threshold Improve performance

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Conclusion Combinatorial Auctions Improve economic efficiency NP-complete (very effective algorithms) Application domains are expanding Some applications require robustness Potential exposure problem for the auctioneer Super solutions for CA’s Framework for establishing robust solutions CA’s motivate useful extensions

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