Super solutions for combinatorial auctions

1. Super solutions for combinatorial auctions Alan Holland & Barry O’Sullivan {a.holland, b.osullivan}@cs.ucc.ie

2. 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

3. 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’

4. Bidders Exposure Problem • Single-item auctions • Example • Two items (X,Y) are sold in two separate auctions • A Bidder values the pair XY @ $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’

5. 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

6. 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]

7. 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

8. Unreliable Bidders in CA’s • Single unit auction • A winning bid is withdrawn => give the item to 2nd priced bidder • CA • A winning bid is withdrawn =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

9. (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 <0,1><1,0><1,1> • <1,1> is a (1,0)-super solution • <0,1> & <1,0> are (1,1)-super solutions

10. (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 Ropt • Add a sum constraint s.t. revenue > Ropt k% • Find any super solution (Constraint Satisfaction) • Optimize on robustness OR revenue

11. (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 • <1,1,0,0>: 1.20: (1,2)-super solution • <0,0,1,0>: 1.15: (1,1)-super solution • <0,0,0,1>: 1.10: (1,1)-super solution • Solution robustness • 2nd & 3rd solutions are robust, but less revenue • 2nd solution dominates 3rd solution • Trade-off ensues between 1st & 2nd solution

12. 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

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

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

15. 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

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

17. Constraint Satisfaction • Running times • Problems with ~75% success rate = most difficult • Scheduling distribution - most expensive • Hybrid approach may improve performance arbitrary regions scheduling

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

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

20. 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

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

22. 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

23. 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

24. 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