Super solutions for combinatorial auctions

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# Super solutions for combinatorial auctions - PowerPoint PPT Presentation

Super solutions for combinatorial auctions. Alan Holland & Barry O’Sullivan {a.holland, b.osullivan}@cs.ucc.ie. Overview. Combinatorial Auctions (CA’s) Motivation Auction scenarios Implications of unreliable bidders Super solutions (SS) Solution robustness – simple example SS & CA’s

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## Super solutions for combinatorial auctions

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### Super solutions for combinatorial auctions

Alan Holland & Barry O’Sullivan

{a.holland, b.osullivan}@cs.ucc.ie

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
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

A

B

Combinatorial Auction Example
• Two parcels of land for sale
• Three bidders valuations
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’
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
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]
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
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
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
(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
(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%
(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
(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
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
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
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
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)

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