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Truthful Mechanisms for
Combinatorial Auctions
with Subadditive Bidders
Speaker: Shahar Dobzinski
Based on joint works with Noam Nisan & Michael Schapira
Combinatorial Auctions
 m items, n bidders, each bidder i has a
valuation function vi:2M->R+.
Common assumptions:
 Normalization: vi()=0
 Monotonicity: ST  vi(T) ≥ vi(S)
 Goal: find a partition S1,…,Sn such that the
total social welfare Svi(Si) is maximized.
 Algorithms must run in time polynomial in n
and m.
 In this talk the valuations are subadditive:
for every S,T  M: v(S)+v(T) ≥ v(ST)
(but all of our results also hold for submodular valuations)
Truthful Approximations?
 A 2 approximation algorithm exists [Feige],
and a matching lower bound is also
known [Dobzinski-Nisan-Schapira].
 What about truthful approximations?
 The private information of each bidder is his
valuation.
Outline
 A deterministic VCG-based O(m½)approximation mechanism
 An W(m1/6) lower bound on VCG-based
mechanisms.
 A randomized almost-logarithmic
approximation mechanism.
Reminder: Maximal in
Range Algorithms
 VCG: Allocate Oi to bidder i. Bidder i gets a
payment of Sk≠ivk(Ok).
 (O1,…,On) is the optimal solution.
 Still truthful if we limit the range.
 Range := { A=(A1,…,An) |v1,…,vn: A(v1,…,vn)=A }
 The Algorithm [Dobzinski-Nisan-Schapira]:
 Choose the best allocation where either:
 One bidder gets all items OR
 Each bidder gets at most one item.
 Clearly, the algorithm is maximal-in-range and
can be implemented in polynomial time.
Proof of the
Approximation Ratio
Theorem: If all valuations are subadditive, the algorithm provides an
O(m1/2)-approximation.
Proof: Let OPT=(L1,..,Ll,S1,...,Sk), where for each Li, |Li|>m1/2, and for
each Si, |Si|≤m1/2. |OPT|= Sivi(Li) + Sivi(Si)
Case 2:
1: Siivvii(S
(Lii)) >≥ Siivi(L
(Si)i)
(“small”
(“large” bundles contribute most of the optimal social welfare)
Sivi(S
(Lii) >≥|OPT|/2
|OPT|/2
Claim:
At mostLet
m1/2
v be
bidders
a subadditive
get at least
valuation
m1/2 items
andinS OPT.
a bundle. Then there
exists an item jS s.t. v({j}) ≥ v(S)/|S|.
Proof:
 There
immediate
is a bidder
from
i s.t.:
subadditivity.
vi(M) ≥ vi(Li) ≥ |OPT|/2m1/2.
Thus, for each bidder i that was assigned a small bundle, there is an
item ciSi, such that: vi({ci}) > vi(Si) / m1/2. Allocate ci to bidder i.
Outline
 A deterministic VCG-based O(m½)approximation mechanism
 An W(m1/6) lower bound for VCG-based
mechanisms.
 A randomized almost-logarithmic
approximation mechanism.
About the Lower Bound
 Why lower bounds on VCG-Based
mechanisms (a.k.a. maximal-in-range
algorithms)?
 Conjectured characterization: All mechanisms
that give a good approximation ratio for
combinatorial auctions with subadditive bidders are
maximal in their range.
 Even if the conjecture is false, still the only
technique that we currently know.
An W(m1/6) lower bound on
VCG-based mechanisms
[Dobzinski-Nisan]
 We define two complexity:
 Cover Number: (approximately) the range size
 must be “large” in order to obtain a good approximation ratio.
 Intersection Number: a lower bound on the communication
complexity.
 We therefore want it to be “small” (polynomial)
 Lemma (informal): If the cover number is large then
the intersection number must be large too.
 From now on, only 2 bidders, thus a lower bound of 2.
The Cover Number
 Intuitively, the size of the range
 But we don’t want to count “degenerate
allocations”…
 A set of allocations C covers a set of
allocations R if for each allocation S in R there
is an allocation T in C such that TiCi for
i={1,2}.
 cover(R) is the size of the smallest set C that covers
R.
 Observation: An MIR on range C provides a
better approximation ratio than on R.
The Cover Number
 Lemma: Let A be an MIR algorithm with range R. If
cover(R) < em/400, then A provides an approximation
ratio of at most 1.99.
 Proof: Using the probabilistic method.
 Fix an allocation T=(T1,T2) from the minimal cover C.
 Construct an instance with additive bidders: v(S) = SjS v({j})
 For each item j, set with probability ½ v1({j})=1 and v2({j})=0 (or
vice versa with probability ½ ).
 The optimal welfare in this instance is m, but each item j
contributes 1 to the welfare provided by T only if we hit the
corresponding bundle in T (with probability 1/2).
 The expected welfare that T provides is m/2, and we can get a
better welfare only with exponential small probability.
The Intersection Number
 A set of allocations D is called an
intersection set if for each
(A1,A2)≠(B1,B2)D we have that A1
intersects B2 and A2 intersects B1.
 Let intersect(R) be the size of the largest
intersection set in R.
The Intersection Number
 Lemma: Let A be an MIR algorithm with range R. Let
intersect(R)=d. Then, the communication complexity of
A is at least d.
 Proof:
 Reduction from disjointness: Alice holds a=a1…ad, Bob holds
b=b1…bd. Is there some t with at=bt=1? Requires t bits of
communication.
 Given a disjointness instance, construct a combinatorial
auction with subadditive bidders:
 Let {(A1,B1),…,(Ad,Bd)} be the intersection set.
Set vA(S)=2 if there is an index i s.t. ai=1 and Ai  S. Otherwise
vA(S)=1. Similar valuation for Bob.
 The valuations are subadditive.
 A common 1 bit  optimal welfare of 4. Our algorithm is
maximal in range, and the optimal allocation is in the range, so
our algorithm always return the optimal solution. But this
requires d bits of communication.
Putting it Together
 In order to obtain an approximation ratio better than 2,
the cover number must be exponentially large.
 If the MIR algorithm runs in polynomial time then the
intersection number must be polynomial too.
 Lemma (informal): If the cover number is exponentially
large then the intersection number is exponentially
large too.
 Corollary: No polynomial time VCG-based algorithm
provides an approximation ratio better than 2.
Summary
 A deterministic VCG-based O(m½)approximation mechanism
 An W(m1/6) lower bound on VCG-based
mechanisms.
 A randomized almost-logarithmic
approximation mechanism.
Open Questions
 Deterministic mechanisms\lower bounds
for combinatorial auctions with general
valuations?
 Is the gap between randomized and
deterministic mechanisms essential?
Randomness and
Mechanism Design
 Randomization might help in mechanism
design settings.
 Two notions of randomization:
 “The universal sense”: a distribution over
deterministic mechanisms (stronger)
 “In expectation”: truthful behavior maximizes the
expectation of the profit (weaker)
 Risk-averse bidders might benefit from untruthful behavior.
 The outcomes of the random coins must be kept secret.
Results
 Feige shows a randomized
O(logm/loglogm)-truthful in expectation
mechanism.
 We show that there exists an
O(logm*loglogm) truthful in the universal
sense mechanism.
The Framework
 Two cases:
 Case 1: There is a dominant bidder.
 A bidder with v(M) > OPT/(100log m loglog m)
(denote the denominator by c)
 We can simply allocate all items to this bidder.
 Case 2: There is no dominant bidder.
 In this case we can use random sampling: partition the
bidders into two sets, acquire statistics from one set, and
use it to get an approximate solution with the other set.
 How to put the two cases together?
 Flipping a coin works, but with probability of only ½.
 Next we will see how to increase the probability of
success to 1-e.
The Mechanism
A second price
I have an
 Partition
the bidders
into 3 sets:
auction
with
a
SECPRIC
estimate of
 STAT
with probability
e/2, SECPRICE with
probability 1-e, and FIXED
reserve
price
of
OPT
E
group
with probability
OPT/c e/2.
Statistics
 First case: there is a dominant bidder.
Group
The Mechanism
 Second case: there is no dominant bidder.
FIXED
group
A second price
auction with a
reserve price of
OPT/c
I have a
(good)
estimate of
OPT
Statistics
Group
Case 2: No “Dominant”
Bidder
 Assumption: For all
bidders
vi(OPTi) < OPT / c
 In the FIXED group:
a fixed-price auction
where each item has
a price of p (depends
on the statistics
group)
Everything costs p
My price
is 2*p
Take your
most
profitable
bundle
Too
I paid p
Expensive
!
Still Missing…
 Why does the fixed price auction (with a
“good price”) provides a good
approximation ratio?
 Can we find this “good price” using the
statistics group?
A Combinatorial Property
of Subadditive Valuations
 Lemma: Let v be a subadditive valuation
and S a bundle of items. Then we can
assign each item in S a price in {0,p}
such that:
 For each TS: v(T) > SjT|T|*p
 |S|*p > v(S)/(100*logm)