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

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

Combinatorial Auction Example Two parcels of land for sale – Three bidders valuations AB BidderABAB 1$1m$0 2 $½m$0 3 $2m

Exposure Problem Single-item auctions Consider previous example – Two items (A,B) are sold in two separate auctions – Bidder 3 values the pair $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De-commitment in auctions Single item auction – A winning bid is withdrawn =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 give the item to 2 nd highest bidder Combinatorial auction – A winning bid is withdrawn =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 – is a (1,0)-super solution – & are (1,1)-super solutions

(1,b)-super solution algorithm – MAC-based repair algorithm [Hebrard et al ECAI04] Value assigned to the k th 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 = R opt Add a sum constraint s.t. revenue > R opt X 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)-super solution: $1.2m – : (1,0)-super solution: $1.15m – : (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)-super solution: $1.2m – : (1,0)-super solution: $1.15m – : (1,0)-super solution: $1.1m 2 nd & 3 rd solutions are more robust – Less revenue however – 2 nd solution dominates 3 rd – Trade-off ensues between 1 st & 2 nd 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 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