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

AAMAS 2013 best-paper:“Mechanisms for Multi-Unit Combinatorial

Auctions with a Few Distinct Goods”

Piotr Krysta University of Liverpool, UKOrestis Telelis AUEB, GreeceCarmine Ventre Teesside University, UK

Multi-unit Combinatorial Auctionsm goods

Good j available in supply sj

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 sj in N?

VCG-based mechanisms do the job in this case!

Valuation Previous best apx NEW APX (m=O(1)) Apx lower bound

Single-minded 2-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

Weakly NP-complete

( , 1+ε, …,1+ε) -hard (arbitrary m) [NEW]

Multi-minded PTAS (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

Submodular 1-apx (m=1) [Vickrey’61]Exponential-time

(1+ε,1,…,1)-PTAS ?

General 2-apx (m=1) [DN’07] (2, 1,…,1)-apx 2-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 biddersBidders demand a collection of multi-sets of goods

Valuation Function

Allocation algorithm in input 1. Demands rounding

2. Supply adjustment

3. 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!