Piotr Krysta University of Liverpool, UK Orestis Telelis AUEB, Greece

1. AAMAS 2013 best-paper:“Mechanisms for Multi-Unit Combinatorial Auctions with a Few Distinct Goods” PiotrKrysta University of Liverpool, UK OrestisTelelis AUEB, Greece Carmine Ventre Teesside University, UK

2. Multi-unit Combinatorial Auctions m goods Good j available in supply sj 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$ n bidders Objective: find an allocation of goods to bidders that maximizes the social welfare (sum of the bidders’ valuations)

3. (Multi-unit) CAs: applications

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

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

6. Our results at a glance Greatest improvement over previous result! First deterministic poly-time mechanism even for m=1.

7. 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. Elicit declarations, b. 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.

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

9. Allocation algorithm in input • Demands rounding • Supply adjustment • 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:

10. Truthfulness of the mechanism • THEOREM: The allocation algorithm A is MiR. Proof: The set {x in X : there exists b s.t. A(b)= x} is the range of the algorithm. 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.)

11. 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!

12. 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!