AAMAS 2013 best-paper: “Mechanisms for Multi-Unit Combinatorial Auctions with a Few Distinct Goods” Piotr KrystaUniversity of Liverpool, UK Orestis TelelisAUEB, Greece Carmine VentreTeesside University, UK

Multi-unit Combinatorial Auctions m goods Good j available in supply s j n bidders Objective: find an allocation of goods to bidders that maximizes the social welfare (sum of the bidders’ valuations) Each bidder has valuation functions for (multi) set of goods expressing his/her complex preferences, e.g., v( blue set ) = 290$ v( green set ) = 305$

(Multi-unit) CAs: applications

CAs: paradigmatic problem in Algorithmic Mechanism Design “We can always return the optimum social welfare truthfully (ie, when bidders lie) using VCG” “CAs is hard to approximate within √m and we have a polynomial-time algorithm that guarantees that” Polynomial-time (deterministic) algorithms and truthfulness? VCG is in general not good to obtain approximate solutions [Nisan&Ronen, JAIR 2007]

Few distinct goods Polynomial-time (deterministic) algorithms and truthfulness for m=O(1) and s j in N? VCG-based mechanisms do the job in this case!

ValuationPrevious best apxNEW APX (m=O(1))Apx lower bound Single-minded2-apx (m=1) [Mualem, Nisan’02] FPTAS (m=1) [BKV’05] (1+ε,1+ε,…,1+ε)-FPTAS (m=O(1)) [GKLV’10] Weakly NP-complete (, 1+ε, …,1+ε) - hard (arbitrary m) [NEW] Multi-mindedPTAS (m=1) [Dobzinski, Nisan’07] (1+ε,1,…,1)-PTAS (1,1+ε,…,1+ε)-FPTAS Strongly NP-hard (m≥2) [ChK’00] No FPTAS (m=1) [DN’07] Weakly NP-complete Submodular1-apx (m=1) [Vickrey’61] Exponential-time (1+ε,1,…,1)-PTAS ? General2-apx (m=1) [DN’07](2, 1,…,1)-apx2-MiR-hard (m=1) [DN’07] First deterministic poly-time mechanism even for m=1. Greatest improvement over previous result! Our results at a glance

VCG-based mechanisms: Maximum-in- Range (MIR) algorithms [NR, JAIR 07] Algorithm is MIR, if it fully optimizes the Social Welfare over a subset of allocations. Truthful (Poly-Time) α-approximate VCG-based mechanism: 1. Commit to a range, R, prior to the bidders’ declarations. 2.Elicit declarations, b. 3.Compute solution in R with best social welfare according to b. 4. Use VCG payments. Tricky: R should be “big” enough to contain good approximations of opt for all b and “small” enough to guarantee step 3 to be quick.

Multi-minded bidders Bidders demand a collection of multi-sets of goods Valuation Function

Allocation algorithm in input 1.Demands rounding 1.Supply adjustment 1.Optimize rounded instance by dynamic programming Optimality (1, 1+ε, …, 1+ε)-FPTAS: Feasible solutions to the original instance are feasible for the “rounded” instance Feasibility (1, 1+ε, …, 1+ε)-FPTAS:

Truthfulness of the mechanism THEOREM: The allocation algorithm A is MiR. THEOREM: There is an economically efficient truthful FPTAS for multi-minded CAs, violating the supplies by (1 + ε), for any ε > 0. (Important: Bidders declare (and can lie about) both demand sets and values.) Proof: The set {x in X : there exists b s.t. A(b)= x} is the range of the algorithm.

Violating the supply? Theoretically needed to obtain an FPTAS – Strongly NP-hardness for m ≥ 2 Common practice in multi-objective optimization literature Sellers do that already!

Conclusions Studied Multi-Unit CAs with constant number of goods and arbitrary supply – most practically relevant CAs setting – dramatically changes the problem to be algorithmically tractable! Designed best possible deterministic poly-time truthful mechanisms for broad classes of bidders: multi- minded, submodular, general. – Mechanism for submodular valuations is the first deterministic poly-time. Our assumptions (m = O(1), relaxing supplies) are provably necessary!