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

Super solutions for combinatorial auctions

Alan Holland & Barry O’Sullivan

{a.holland, b.osullivan}

  • 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
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
combinatorial auction example



Combinatorial Auction Example
  • Two parcels of land for sale
    • Three bidders valuations
exposure problem
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 auctions1
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
  • 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
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
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
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
(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
(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
(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 s1
(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
  • 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
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 distributions1
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
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 satisfaction1
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
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 optimization1
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 optimization2
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
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
  • 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