AAMAS 2013 best-paper:
“Mechanisms for Multi-Unit
Combinatorial Auctions with a Few
Distinct Goods”
Piotr Krysta
Orestis Telelis
Carmine Ventre
University of Liverpool, UK
AUEB, Greece
Teesside University, UK
Multi-unit Combinatorial Auctions
m goods
Good j
available in
supply sj
n bidders
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$
Objective: find an allocation of goods to bidders that maximizes the social welfare
(sum of the bidders’ valuations)
(Multi-unit) CAs: applications
CAs: paradigmatic problem in
Algorithmic Mechanism Design
“CAs is algorithms
hard to approximate
within
“We
can always return the (deterministic)
Polynomial-time
and
√m and we have a polynomial-time
optimum social welfare truthfully
truthfulness?
algorithm that guarantees that”
(ie, when bidders lie) using VCG”
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!
Our results at a glance
Valuation
Previous best apx
NEW APX (m=O(1))
Single-minded 2-apx (m=1)
[Mualem, Nisan’02]
FPTAS (m=1)
[BKV’05]
Apx lower bound
Weakly NP-complete
Weakly NP-complete
( m , 1+ε, …,1+ε) hard (arbitrary m)
[NEW]
(1+ε,1+ε,…,1+ε)-FPTAS
(m=O(1))
[GKLV’10]
Multi-minded
PTAS (m=1)
(1+ε,1,…,1)-PTAS
[Dobzinski, Nisan’07]
Strongly NP-hard
(m≥2) [ChK’00]
No FPTAS (m=1)
[DN’07]
(1,1+ε,…,1+ε)-FPTAS Weakly NP-complete
Submodular
1-apx (m=1)
[Vickrey’61] (1+ε,1,…,1)-PTAS
Exponential-time
General
2-apx (m=1)
[DN’07] (2, 1,…,1)-apx
First
deterministic
poly-time
mechanism
even for m=1.
Greatest
improvement
over previous
result!
?
2-MiR-hard (m=1)
[DN’07]
VCG-based mechanisms: Maximum-inRange (MIR) algorithms [NR, JAIR 07]
Algorithm A : b ® X is MIR, if it fully optimizes the
Social Welfare over a subset R Í X of allocations.
Truthful (Poly-Time) α-approximate VCG-based mechanism:
1.
2.
3.
4.
Commit to a range, R, prior to the bidders’ declarations.
Elicit declarations, b.
Compute solution in R with best social welfare according to b.
Use VCG payments.
opt
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.
a
·
·
X R
Multi-minded bidders
Bidders demand a collection of multi-sets of goods
æm
ö
Di = {d1,..., dri } Í ç ´ {0,1,..., s j }÷
è j=1
ø
d1
d2
1 2
ì max{v (d) : d £ x} if {d : d £ x} ¹ Æ
ï dÎD i
i
vi (x) = í
ïî
0
otherwise
1 2
3
1 2
d2
d1
x2
3
v(x1 ) = max{v(d1 ), v(d2 )} = 7
1 2
3
1 2
v(x2 ) = 5
3
v(d2 ) = 5
v(d1 ) = 7
Valuation Function
x1
1 2
3
3
Allocation algorithm in input e > 0
1. Demands rounding
m
æê n × d(l) úö
"i"d Î Di , define d ¢ = ççê
ú÷÷
èë e × sl ûøl=1
1. Supply adjustment
énù
Define sl¢ = ê ú, for l =1, 2,..., m.
êe ú
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:
ê n × xi (l) ú
n × xi (l)
å e × s £ åêë e × s úû + {i : xi Î Di }
l
l
i
i
énù
n
£ ê ú + n £ +1+ n
êe ú
e
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.)
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: multiminded, submodular, general.
– Mechanism for submodular valuations is the first
deterministic poly-time.
• Our assumptions (m = O(1), relaxing supplies) are
provably necessary!