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 • Improves economic efficiency • Removes ‘exposure problem’ from multiple single-item auctions

4. A B Combinatorial Auction Example • Two parcels of land for sale • Three bidders valuations

5. Exposure Problem • Single-item auctions • Consider previous example • Two items (A,B) are sold in two separate auctions • Bidder 3 values the pair AB @ $2m • But either X or Y on its own is valueless ($0) • If she bids $1m for each and wins only one item she has lost $1m • This induces depressed bidding • Solution: Allow bids on XY – ‘combinatorial bids’

6. 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, Home-Base, London Transport Authority

7. Complexity • Potentially 2#items bids to consider • Winner Determination • NP-Complete [Rothkopf ‘98] • Inapproximable • State of the art algorithms work well in practice • CABOB - 1,000’s of bids for 100’s of items in seconds [Sandholm ‘03]

8. Full commitment contracts • Auction solutions assume binding contracts (full commitment) • ‘…a contract might be profitable to an agent when viewed ex ante, it need not be profitable when viewed ex post’ [Sandholm&Lesser02] • The converse is also true • De-committing • Bidders receive better offers/renege on unprofitable agreements/go bankrupt/disqualified • Levelled-commitment contracts offer de-commitment penalties

9. De-commitment in auctions • Single item auction • A winning bid is withdrawn => give the item to 2nd highest bidder • Combinatorial auction • A winning bid is withdrawn => next best solution may require changing all winning bids • Highly undesirable in many circumstances (e.g. SCM) • Auctioneer may be left with a bundle of items that are valueless (Auctioneer’s exposure problem) • ‘Prevention is better than cure’ • Robust solutions => a small break can be repaired with a small number of changes

10. CA solution robustness • Solution robustness • Unreliable bidders are present • Solution stability paramount • E.g. Supply chain formation • Bid withdrawal/disqualification • Next best solution may require changing all winning bids (infeasible in many situations) • Potentially severe implications for revenue

11. (a,b)-super solutions [Hebrard,Hnich&Walsh 04] • An (a,b)-super solution • Guarantees that when ‘a’ variables are broken in a solution, only ‘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

12. (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 => unrepairable assignment • Our approach • Solve the problem optimally using any ILP solver (CPLEX etc…) & optimal revenue = Ropt • Add a sum constraint s.t. revenue > RoptX k%

13. (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>: (1,1)-super solution: $1.2m • <0,0,1,0>: (1,0)-super solution: $1.15m • <0,0,0,1>: (1,0)-super solution: $1.1m

14. (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>: (1,1)-super solution: $1.2m • <0,0,1,0>: (1,0)-super solution: $1.15m • <0,0,0,1>: (1,0)-super solution: $1.1m • 2nd & 3rd solutions are more robust • Less revenue however • 2nd solution dominates 3rd • Trade-off ensues between 1st & 2nd solution

15. Experiments • Aim • Examine trade-off between revenue & robustness • Different economically motivated scenarios • Auctions • Generated by bid simulation tool (CATS) [Leyton-Brown et al] • Scenarios exhibit differing complementarity effects

16. Bid distributions • arbitrary • arbitrary complementarity between items for different bidders, (Simulates electronic component auctions) • regions • complementarity between items in 2-D space (e.g. spectrum auctions, property) • scheduling • Auctions for airport landing/take-off slots

17. Bid distributions • arbitrary • Random synergies => more varied series of items in bids => more overlap constraints • More pruning => lower search times • regions • More mutually exclusive bids • Less pruning => higher search times • scheduling • Bids contain few items => less constraints • More pruning => longer search times

18. Constraint Satisfaction • Is a super solution possible? • (given b & min revenue) – • Sample auctions - 20 items & 100 bids (v. small) • Robust solutions – • arbitrary: super soln’s unlikely - unless min revenue < 85% of optimum & b>2 • regions: super soln’s more likely than for arbitrary- unless tolerable revenue ~ 85% of optimum • scheduling: super soln’s likely - unless min revenue > 95% of optimum or b=0 (See paper for full set of results)

19. Constraint Satisfaction • Running times • Distributions least likely to have a super soln are quickest to solve • Dense solution space implies deeper tree search

20. Constraint Optimization • If no (1,b)-super solution • Optimize robustness & maintain revenue constraint • Minimize number of variables that do not have a repair • Else if many (1,b)-super solutions • Find super soln with optimal revenue

21. Constraint Optimization • Optimizing Robustness • BnB search • Find a solution with the minimum number of irreparable bids • Results • For sched. distribution, no repairs are allowed (b=0), min revenue for a solution is 86% of opt, on average 2.2 bids are irreparable in the most robust solution • Scheduling distribution most difficult to find repairs for all bids (more bids in solution)

22. Constraint Optimization • Optimizing Revenue • Many super solutions – find revenue maximizing SS • Guarantees a robust solution with maximum revenue • Optimal/Near optimal solutions achievable for scheduling • Computationally expensive (esp. scheduling) • Pure CP approach needs to be augmented with hybrid techniques to improve performance • Continuous (LP) relaxation provides tighter bounds

23. Proposed Extensions to Super Solutions • 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 • Introduce a metric for the cost of repair • Break-dependant cost of repair • E.g. If an agent withdraws a bid, changing his other winning bids may be considered a cheap operation • Variable values may have degrees of brittleness • E.g. Various bidders may have differing probabilities of failure

24. Conclusion • Combinatorial Auctions • Improve economic efficiency • NP-complete (although very efficient tailored algorithms exist in practise) • 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 to the framework