WECWIS, June 27, 2002 On the Sensitivity of Incremental Algorithms for Combinatorial Auctions Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh Computer Science Department, UCLA WECWIS June 27, 2002 Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh Computer Science Department, UCLA WECWIS June 27, 2002

WECWIS, June 27, 2002OutlineOutline 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 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

WECWIS, June 27, 2002 Combinatorial Auctions 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 $$$Maximize Objects M Bids B $9 $6

WECWIS, June 27, 2002 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) 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

WECWIS, June 27, 2002 Winner Determination via ILP Let if bid j is selected as a winner otherwise if item i is in bid j s.t. Let x j be a decision variable that determines if bid j is selected as a winner Let c ij be a decision variable relating item i to bid j Let v i be the valuation of bid j Let x j be a decision variable that determines if bid j is selected as a winner Let c ij be a decision variable relating item i to bid j Let v i be the valuation of bid j

WECWIS, June 27, 2002 Supply Chains and CAs 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 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

WECWIS, June 27, 2002 Modeling Supply Chains: Task Dependency Graph A1 $4 A2 $3 G1 G2 A4 $9 A3 $5 A5 $5 G3 G4 C1 $12.27 C2 $21.68 Goods labeled as circles Producers/consumers labeled 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)} Goods labeled as circles Producers/consumers labeled 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)}

WECWIS, June 27, 2002 Supply Chains and CA “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 “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 A3 $5 A5 $5 G3 C1 $12.27 A1 $4 A2 $3 G1 G2 A4 $9 G4 C2 $21.68 A3 $5 A5 $5 G3 C1 $12.27 A1 $4 A2 $3 G1 G2 A4 $9 G4 C2 $21.68 Efficient Allocation

WECWIS, June 27, 2002 Supply Chain Perturbation 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 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 Perturbation: A4 changes cost from $9 to $20 A1 $4 A2 $3 G1 G2 A4 $9 G4 C2 $21.68 A3 $5 A5 $5 G3 C1 $12.27 A1 $4 A2 $3 G1 G2 A4 $9 G4 C2 $21.68 A1 $4 A2 $3 G1 G2 A4 $20 A3 $5 A5 $5 G3 G4 C1 $12.27 C2 $21.68 Perturbation: A4 changes cost from $9 to $20

WECWIS, June 27, 2002 Incremental Algorithms An original instance I 0 of a problem is solved by a full algorithm to give solution S 0 Perturbed instances, I 1,I 2, ,I n are generated one by one in sequence Each instance is solved by an incremental algorithm which uses S i-1 as a starting point find solution S i An original instance I 0 of a problem is solved by a full algorithm to give solution S 0 Perturbed instances, I 1,I 2, ,I n are generated one by one in sequence Each instance is solved by an incremental algorithm which uses S i-1 as a starting point find solution S i

WECWIS, June 27, 2002 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 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

WECWIS, June 27, 2002 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 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

WECWIS, June 27, 2002 Incremental Winner Determination Given an original instance I 0 of a problem solved by a full algorithm to give solution S 0 S 0 is the set of winners which we call the original winners OW Determined through ILP – exact solution I 0 is perturbed to give a new instance I 1 We wish to find a solution S 1 to the instance I 1 while: Maximizing the valuation of the bids in the solution S 1 Maintaining the original winners from solution S 0 i.e. maximize |S 0  S 1 | Given an original instance I 0 of a problem solved by a full algorithm to give solution S 0 S 0 is the set of winners which we call the original winners OW Determined through ILP – exact solution I 0 is perturbed to give a new instance I 1 We wish to find a solution S 1 to the instance I 1 while: Maximizing the valuation of the bids in the solution S 1 Maintaining the original winners from solution S 0 i.e. maximize |S 0  S 1 | Use ILP to solve incremental winner determination

WECWIS, June 27, 2002 ILP for Incremental Winner Determination Introduce a new decision variable z i corresponding to each winning bid b  S 0 that corresponds to b also being a winning bid in S 1 Let if bid i is not selected as a winner in S 1 if bid i is selected as a winner in S 1 For each bid b i  S 0 Other other variables similar to ILP for winner determination Let x j be a decision variable that determines if bid j is selected as a winner Let c ij be a decision variable relating item i to bid j Let v i be the valuation of bid j Other other variables similar to ILP for winner determination Let x j be a decision variable that determines if bid j is selected as a winner Let c ij be a decision variable relating item i to bid j Let v i be the valuation of bid j

WECWIS, June 27, 2002 ILP for Incremental Winner Determination New objective function Maximize valuation of the winners Maintain winners from original (unperturbed) solution S 0 New objective function Maximize valuation of the winners Maintain winners from original (unperturbed) solution S 0 s.t. Original constraint : every item won at most one time New constraint : relates original winners to new winners w i – propensity for keeping bid as a winner (user assigned)

WECWIS, June 27, 2002 Experimental Flow # bids # goods CATS Winner determination ILP solver I0I0I0I0 S0S0S0S0 Add perturbation (randomly remove x% of winning bids) x Winner determination ILP solver I1I1I1I1 optimal S 1 objective value Incremental winner determination ILP solver incremental S 1 objective value % involuntary dropouts

WECWIS, June 27, 2002BenchmarksBenchmarks 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 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

WECWIS, June 27, 2002ResultsResults voluntary dropouts

WECWIS, June 27, 2002 Results – 0% Involuntary Dropout

WECWIS, June 27, 2002ConclusionsConclusions 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 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

WECWIS, June 27, 2002 On the Sensitivity of Incremental Algorithms for Combinatorial Auctions Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh Computer Science Department, UCLA WECWIS June 27, 2002 Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh Computer Science Department, UCLA WECWIS June 27, 2002

WECWIS, June 27, 2002 Extra Slides

WECWIS, June 27, 2002BenchmarksBenchmarks 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 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

WECWIS, June 27, 2002ResultsResults

ResultsResults