On the Sensitivity of Incremental Algorithms for Combinatorial Auctions

1. On the Sensitivity of Incremental Algorithms for Combinatorial Auctions Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh kastner@cs.ucla.edu Computer Science Department, UCLA WECWIS June 27, 2002

2. Outline • Basics • Combinatorial Auctions (CA) • Integer Linear Programming (ILP) for Winner Determination • Motivating Example: Supply Chains • Incremental Algorithms • Incremental Algorithms for CA • Uses of Incremental CA • ILP for Incremental Winner Determination • Results • Conclusions

3. Combinatorial Auctions Maximize Bids B Objects M $$$ $9 $6 • Given a set of distinct objects M and set of bids B where B is a tuple S v s.t. S  powerSet{M} and v is a positive real number, determine a set of bids W (W  B) s.t.  w·v is maximized

4. Winner Determination Problem • Informal Definition: Auctioneer must figure out who to give the items to in order to make the most money • NP-Hard  need heuristics to quickly solve large instances • Many exact methods to solve winner determination problem • Dynamic Programming – Rothkopf et al. • Optimized Search – Sandholm • CASS, VSA, CA-MUS – Layton-Brown et al. • Integer Linear Program (ILP) We focus on the ILP solution

5. Winner Determination via ILP otherwise Let if bid j is selected as a winner otherwise if item i is in bid j s.t. • Let xjbe a decision variable that determines if bid j is selected as a winner • Let cijbe a decision variable relating item i to bid j • Let vibe the valuation of bid j

6. Supply Chains and CAs • Trend: Supply chains becoming large and dynamic • More complementary companies – larger supply chains • Specialization becoming prevalent – deeper supply chains • Market changes rapidly – need quick reformation • Automated negotiation – CA for supply chains • Supply Chain formation/negotiation through CA • Welsh et al. give an CA approach to solving supply chain problem • Model supply chain through task dependency network Large, dynamic supply chains require automated negotiation/formation

7. Modeling Supply Chains: Task Dependency Graph • Goodslabeled as circles • Producers/consumerslabeled as rectangles • Arrows indicate the goods needed to produce another good • Bids are the number of goods needed/produced and the price to produce e.g. bid(A4) = {$9,(G1,1),(G2,1),(G4,1)} A3 $5 G1 G3 A1 $4 C1 $12.27 A4 $9 G2 G4 A2 $3 C2 $21.68 A5 $5

8. Supply Chains and CA Efficient Allocation A3 $5 A3 $5 G1 G1 G3 G3 A1 $4 A1 $4 C1 $12.27 C1 $12.27 A4 $9 A4 $9 G2 G2 G4 G4 A2 $3 A2 $3 C2 $21.68 C2 $21.68 A5 $5 A5 $5 • “Winning” bidders (companies) are included in supply chain • CA guarantees an optimal supply chain formation • Allocation of goods is efficient – producers get all input goods they need • Maximizes the value of the supply chain – the goods that are produced are done so in the least expensive possible manner

9. Supply Chain Perturbation A3 $5 G1 G3 A1 $4 C1 $12.27 A4 $20 G2 G4 A2 $3 C2 $21.68 Perturbation: A4 changes cost from $9 to $20 Perturbation: A4 changes cost from $9 to $20 A5 $5 A3 $5 G1 G1 G3 A1 $4 A1 $4 C1 $12.27 A4 $9 A4 $9 G2 G2 G4 G4 A2 $3 A2 $3 C2 $21.68 C2 $21.68 A5 $5 • What happens when there is a change in the supply chain? • Want to keep current producer/consumer relationships intact • Want to maximize the efficiency of supply chain • Not always possible to maintain previous relationships when supply chain changes

10. Incremental Algorithms • An original instance I0 of a problem is solved by a full algorithm to give solution S0 • Perturbed instances, I1,I2,,In are generated one by one in sequence • Each instance is solved by an incremental algorithm which uses Si-1 as a starting point find solution Si

11. Perturbations for CA • A bidder retracts their bid. This removes the bid from consideration • A bidder changes the valuation of their bid • A bidder prefers a different set of items • A new bidder enters the bidding process $9 $5 $7 $5 $5

12. Uses for Incremental CA • Supply chain reformation/adjustment • Iterative Combinatorial Auctions • Progressive combinatorial auction – bidding done in rounds • Different protocols governing various aspects • Stopping conditions, price reporting, rules to withdrawal bids • Types of Iterative CA • AkBA – Wurman and Wellman • iBundle – Parkes and Unger • Generalized Vickrey Auction – Varian and MacKie-Mason • Aid development of heuristics for large instances of CA

13. Incremental Winner Determination • Given an original instance I0 of a problem solved by a full algorithm to give solution S0 • S0 is the set of winners which we call the original winners OW • Determined through ILP – exact solution • I0 is perturbed to give a new instance I1 • We wish to find a solution S1 to the instance I1 while: • Maximizing the valuation of the bids in the solution S1 • Maintaining the original winners from solution S0 i.e. maximize |S0 S1| Use ILP to solve incremental winner determination

14. ILP for Incremental Winner Determination • Introduce a new decision variable zicorresponding to each winning bid b S0that corresponds to b also being a winning bid in S1 For each bid bi S0 if bid iis selected as a winner in S1 Let if bid iis not selected as a winner in S1 • Other other variables similar to ILP for winner determination • Let xjbe a decision variable that determines if bid j is selected as a winner • Let cijbe a decision variable relating item i to bid j • Let vibe the valuation of bid j

15. ILP for Incremental Winner Determination • New objective function • Maximize valuation of the winners • Maintain winners from original (unperturbed) solution S0 • wi– propensity for keeping bid as a winner (user assigned) • Original constraint : every item won at most one time s.t. • New constraint : relates original winners to new winners

16. Experimental Flow x Add perturbation (randomly remove x% of winning bids) CATS Winner determination ILP solver S0 # bids I0 # goods Incremental winner determination ILP solver % involuntary dropouts I1 Winner determination ILP solver incremental S1 objective value optimal S1 objective value

17. Benchmarks • Combinatorial Auction Test Suite (CATS) – Leyton-Brown et al. • We focused on three specific distributions • Matching – correspondence of time slices on multiple resources e.g. airport takeoff/landing rights • Regions – adjacency in two dimensional space e.g. drilling rights • Paths – purchase of connection between two points e.g. truck routes

18. Results voluntary dropouts

19. Results – 0% Involuntary Dropout

20. Conclusions • Main Idea: Incremental Combinatorial Auction • Maximize valuation while maintaining solution • Useful in many different contexts • Supply chain reformation/adjustment • Iterative Combinatorial Auctions • Studied incremental tradeoff through incremental CA ILP formulation • Increased perturbation leads to worse solution • Large instances can be solved near-optimally while maintaining solution • Future work • Incremental CA algorithms • Fault tolerant CA solutions

21. On the Sensitivity of Incremental Algorithms for Combinatorial Auctions Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh kastner@cs.ucla.edu Computer Science Department, UCLA WECWIS June 27, 2002

22. Extra Slides

23. Benchmarks • Matching • 35 instances • ~[25 – 20000] bids • ~[50 – 3600] goods • Paths • 21 instances • ~[100 – 20000] bids • ~[30 – 2800] goods • Regions • 18 instances • ~[100 – 10000] bids • ~[40 – 2000] goods

24. Results

25. Results