Minimal Preference Elicitation in Combinatorial Auctions

Minimal Preference Elicitation in Combinatorial Auctions Wolfram Conen Tuomas Sandholm Xonar GmbH Carnegie Mellon University Computer Science Department

Outline • Combinatorial auctions for multi-item auctions • “The revelation problem” • Previous approaches • Our approach: Elicitor “agent” • Topological observations that motivate elicitation • Different elicitation queries • Policy dependent elicitor algorithms • General policy independent elicitor framework (with data structures & assimilation algorithms) & specific elicitor algorithms • Making the elicitor incentive compatible

Combinatorial auction (CA) • Can bid on combinations of items [Rassenti,Smith & Bulfin 82]... • Bidder’s perspective • Allows bidder to express what she really wants • No need for lookahead / counterspeculationing of items • Auctioneer’s perspective: • Automated optimal bundling • Binary winner determination problem: • Label bids as winning or losing so as to maximize sum of bid prices • Each item can be allocated to at most one bid • NP-complete [Rothkopf et al 98 using Karp 72] • Inapproximable [Sandholm IJCAI-99 using Hastad 99]

Another complex problem in (single-shot) combinatorial auctions: “The revelation problem” • Bidders may need to bid on all 2#items combinations • Need to compute the valuation for each combination • Each valuation computation can be NP-complete • For example if a carrier company bids on trucking tasks: TRACONET [Sandholm AAAI-93] • Need to communicate the bids • Need to reveal the bids

Setting Combinatorial auction: m items for sale • Private values auction, no allocative externalities • Each bidder i has value function, vi: 2m R • Unique valuations (to ease presentation) Can avoid unnecessary computation/revelation/communication of valuations!

Approaches for tackling the revelation problem • Classic single-shot full revelation mechanims (Vickrey-Clarke-Groves) • Exponentially many valuations revealed • (Ascending) mechanisms with price feedback (iBundle, [Parkes et al 1999] , akBa [Wurman et al. 2000] , etc.) • Can save revelation • Need exponential revelation in worst case [Nisan 2001] • Our new approach: an elicitor “agent” • Knows things that individual bidders don’t (others’ bids so far) • Asks non-redundant questions from bidders to focus their revelation • Can save revelation • Thrm. Exponential revelation in worst case if only value and order queries are allowed (even with 1 bidder) • Can be combined with price feedback mechanisms

Elicitor algorithms • Query policy dependent elicitor algorithms • Algorithm & query policy are intertwined • Based on search algorithms where each search step involves asking a bidder a question • Policy independent elicitor algorithms • General framework & specific algorithms • Can support any query policy • Use exponential memory (in worst case) • Note: Query policies are online control policies, i.e. contingency plans

Terminology (X1,...,X#bidders) is a collection • Bundle Xi is earmarked for agent i • An allocation is a feasible collection (i.e., collection where Xi’s don’t overlap in items) Objectives: (1) Find Pareto efficient allocation(s) (2) Find social welfare maximizing allocation(s)

Rank Lattice Rank of Bundle Ø A B AB for Agent 1 4 2 3 1 for Agent 2 4 3 2 1 [1,1] [1,2] [2,1] [1,3] [2,2] [3,1] [1,4] [2,3] [3,2] [4,1] [2,4] [3,3] [4,2] [3,4] [4,3] [4,4] Infeasible Feasible Dominated

Query Policy Independent Elicitation Algorithms: Computing Pareto Optima s=(1,...,1); PAR = []; OPEN = [s]; while OPEN ≠ [] do Remove(c,OPEN); SUC = suc(c); ifFeasible(c) then PAR = PAR  {c}; Remove(SUC,OPEN); elseforeach node є SUC do if node OPEN andUndominated(node,PAR) thenAppend(node,OPEN)

Value-Augmented Rank Lattice Value of Bundle Ø A B AB for Agent 1 0 4 3 8 for Agent 2 0 1 6 9 17 [1,1] 14 13 [1,2] [2,1] 10 12 9 [1,3] [2,2] [3,1] 8 9 [1,4] [2,3] [3,2] [4,1] [2,4] [3,3] [4,2] [3,4] [4,3] [4,4]

Query Policy Independent Elicitation Algorithms: Computing Welfare Maxima s=(1,...,1); OPEN = {s}; CLOSED = Ø; while OPEN ≠ Ødo c = arg maxc є OPEN Σi є N vi(ci) OPEN = OPEN \ {c}; ifFeasible(c) then return(c); CLOSED = CLOSED  {c}; SUC = suc(c); foreach n є SUC do if node OPEN and node CLOSED then OPEN = OPEN  {node}

Policy independent elicitor algorithms

Our Query Types for Agent Interrogation • Order information: Which bundle do you prefer, A or B? • Value information: What is your valuation for bundle A? (Answer: Exact or Bounds) • Rank information: • What is the rank of bundle b? • What bundle is at rank x? • Given bundle b, what is the next lower (higher) ranked bundle?

General Algorithmic Framework for Elicitation Algorithm Solve(Y,G) whilenotDone(Y,G) do o = SelectOp(Y,G)  Choose Question I = PerformOp(o,N)  Ask bidder G = Propagate(I,G)  Update Graph Y = Candidates(Y,G)  Curtail set of candidate collections / allocations Input: Y – set of collections or allocations G – partially augmented order graph Output: Y – set of optimal solutions

(Partially) Augmented Order Graph ∞ ∞ ∞ ∞ ∞ ∞ Agent1 Ø B A AB A Ø 0 0 0 0 > Allocations B B 4 0 3 6 2 6 1 9 Ø A B AB Agent2 1 1 0 0 1 6 [1,1] [1,2] [2,1] Rank Upper Bound [1,3] [2,2] [3,1] 1 9 [1,4] [2,3] [3,2] [1,4] AB [2,4] [3,3] [4,2] 6 [3,4] [4,3] Lower Bound [4,4] Some interesting procedures for combining different types of info

We present algorithms that use any combination of value, order & rank queries • If value queries are used, all social welfare maximizing allocations are guaranteed to be found • Otherwise, all Pareto efficient allocation are guaranteed to be found • We propose several query policies that are geared toward reducing the number of queries needed

Incentive compatibility of the different approaches • Classic single-shot full revelation mechanims (Vickrey-Clarke-Groves) • Can be made dominant strategy incentive compatible • (Ascending) mechanisms with price feedback (iBundle, akBa, etc.) • Can be made incentive compatible in weaker equilibrium notions • Our new approach: an elicitor “agent” • Elicitor’s questions leak information about others’ preferences • Can be made incentive compatible in weaker equilibrium notions • Ask enough questions to determine VCG prices • Could interleave these “extra” questions with real questions • To avoid lazyness; Not necessary from an incentive perspective

Conclusions • Combinatorial auctions are desirable & winner determination algorithms now scale to the large • Another problem: “The Revelation Problem” • Valuation computation / revelation / communication • Presented the design for an elicitor for combinatorial auctions that focuses revelation • Can save revelation • Provably find the welfare maximizing or Pareto efficient allocations • Policy dependent search algorithms for elicitation • Based on topological observation • Policy independent general elicitation framework • Any combination of value, order & rank queries • Several algorithm instantiations in the paper • Several query policies • Presented a way to make the elicitor incentive compatible • Elicitor can be combined with price feedback mechanisms

Future research • Evaluating the elicitor • Savings in revelation (how many queries needed ?) • In general case / in cases with special preference structure • Worst / average case • Generalizing the elicitor • To (combinatorial) exchanges • To (combinatorial) markets with side constraints • To (combinatorial) markets with multiattribute features

For combinatorial auctions

Minimal Preference Elicitation in Combinatorial Auctions