Truthful Mechanisms for Combinatorial Auctions with Subadditive Bidders

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Truthful Mechanisms for Combinatorial Auctions
with Subadditive Bidders

• Speaker Shahar Dobzinski
• Based on joint works with Noam Nisan Michael
  Schapira

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Combinatorial Auctions

• m items, n bidders, each bidder i has a valuation
  function vi2M-gtR.
• 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)

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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.

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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.

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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.

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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, Ligtm1/2, and for each Si, Sim1/2.
  OPT Sivi(Li) Sivi(Si)

• Case 2 Sivi(Si) Sivi(Li)
• (small bundles contribute most of the optimal
  social welfare)
• Sivi(Si) OPT/2
• Claim Let v be a subadditive valuation and S a
  bundle. Then there exists an item j?S s.t. v(j)
  v(S)/S.
• Proof immediate from subadditivity.
• Thus, for each bidder i that was assigned a small
  bundle, there is an item ci?Si, such that
  vi(ci) gt vi(Si) / m1/2. Allocate ci to bidder i.

• Case 1 Sivi(Li) gt Sivi(Si)
• (large bundles contribute most of the optimal
  social welfare)
• Sivi(Li) gt OPT/2
• At most m1/2 bidders get at least m1/2 items in
  OPT.
• ? There is a bidder i s.t. vi(M) vi(Li)
  OPT/2m1/2.

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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.

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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.

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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.

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The Cover Number

• Intuitively, the size of the range
• But we dont 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
  i1,2.
• 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.

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The Cover Number

• Lemma Let A be an MIR algorithm with range R. If
  cover(R) lt 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.

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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.

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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 aa1ad,
  Bob holds bb1bd. Is there some t with atbt1?
  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. ai1 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.

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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.

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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.

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Open Questions

• Deterministic mechanisms\lower bounds for
  combinatorial auctions with general valuations?
• Is the gap between randomized and deterministic
  mechanisms essential?

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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.

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Results

• Feige shows a randomized O(logm/loglogm)-truthful
  in expectation mechanism.
• We show that there exists an O(logmloglogm)
  truthful in the universal sense mechanism.

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The Framework

• Two cases
• Case 1 There is a dominant bidder.
• A bidder with v(M) gt 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.

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The Mechanism
A second price auction with a reserve price of
OPT/c
I have an estimate of OPT
SECPRICE group

• Partition the bidders into 3 sets
• STAT with probability e/2, SECPRICE with
  probability 1-e, and FIXED with probability e/2.
• First case there is a dominant bidder.

Statistics Group
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The Mechanism

• Second case there is no dominant bidder.

A second price auction with a reserve price of
OPT/c
FIXED group
I have a (good) estimate of OPT
Statistics Group
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Case 2 No Dominant Bidder

• Assumption For all bidders vi(OPTi) lt 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
Take your most profitable bundle
My price is 2p
Too Expensive!
I paid p
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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?

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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) gt Sj?TTp
• Sp gt v(S)/(100logm)