Truthful Mechanisms for Truthful Mechanisms for Combinatorial Auctions Combinatorial Auctions with Subadditive Bidderswith Subadditive BiddersSpeaker: Shahar DobzinskiSpeaker: Shahar Dobzinski

Based on joint works with Noam Nisan & Michael SchapiraBased on joint works with Noam Nisan & Michael Schapira

Combinatorial AuctionsCombinatorial Auctions

m items, n bidders, each bidder i has a valuation m items, n bidders, each bidder i has a valuation function vfunction vii:2:2MM->R->R++..Common assumptions:Common assumptions:

NormalizationNormalization: v: vii(()=0)=0 MonotonicityMonotonicity: S: ST T v vii(T) ≥ v(T) ≥ vii(S)(S)

Goal: find a partition SGoal: find a partition S11,…,S,…,Snn such that the total such that the total social welfaresocial welfare vvii(S(Sii) is maximized.) is maximized.

Algorithms must run in time polynomial in n and Algorithms must run in time polynomial in n and m.m.

In this talk the valuations are In this talk the valuations are subadditivesubadditive::

for every S,T for every S,T M: v(S)+v(T) M: v(S)+v(T) ≥ v(S ≥ v(ST)T)(but all of our results also hold for submodular valuations)(but all of our results also hold for submodular valuations)

Truthful Approximations?Truthful Approximations?

A A 22 approximation algorithm exists approximation algorithm exists [Feige][Feige], , and a matching lower bound is also and a matching lower bound is also known known [Dobzinski-Nisan-Schapira][Dobzinski-Nisan-Schapira]..

What about What about truthfultruthful approximations? approximations? The private information of each bidder is his The private information of each bidder is his

valuation.valuation.

OutlineOutline

A deterministic VCG-based O(mA deterministic VCG-based O(m½½)-)-approximation mechanismapproximation mechanism

An An (m(m1/61/6) lower bound on VCG-based ) lower bound on VCG-based mechanisms.mechanisms.

A randomized almost-logarithmic A randomized almost-logarithmic approximation mechanism.approximation mechanism.

Reminder: Maximal in Reminder: Maximal in Range AlgorithmsRange Algorithms

VCGVCG: Allocate O: Allocate Oii to bidder i. Bidder i gets a payment of to bidder i. Bidder i gets a payment of

k≠ik≠ivvkk(O(Okk).). (O(O11,…,O,…,Onn) is the optimal solution.) is the optimal solution.

Still truthful if we limit the range.Still truthful if we limit the range. Range Range := { A=(A:= { A=(A11,…,A,…,Ann) |) |vv11,…,v,…,vnn: A(v: A(v11,…,v,…,vnn)=A })=A }

The AlgorithmThe Algorithm [Dobzinski-Nisan-Schapira][Dobzinski-Nisan-Schapira]:: Choose the best allocation where either:Choose the best allocation where either:

One bidder gets all items One bidder gets all items OROR Each bidder gets at most one item.Each bidder gets at most one item.

Clearly, the algorithm is Clearly, the algorithm is maximal-in-rangemaximal-in-range and can be and can be implemented in polynomial time.implemented in polynomial time.

Case 2: ivi(Si) ≥ ivi(Li)(“small” bundles contribute most of the optimal social welfare)

ivi(Si) ≥ |OPT|/2

Claim: Let v be a subadditive valuation and S a bundle. Then there exists an item jS 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 ciSi, such that: vi({ci}) > vi(Si) / m1/2. Allocate ci to bidder i.

Proof of the Proof of the Approximation RatioApproximation RatioTheoremTheorem:: If all valuations are subadditive, the algorithm provides an If all valuations are subadditive, the algorithm provides an

O(mO(m1/21/2)-approximation.)-approximation.

Proof: Proof: Let OPT=(LLet OPT=(L11,..,L,..,Lll,S,S11,...,S,...,Skk), where for each L), where for each Lii, |L, |Lii|>m|>m1/21/2, and for , and for

each Seach Sii, |S, |Sii|≤m|≤m1/21/2. |OPT|= . |OPT|= iivvii(L(Lii) + ) + iivvii(S(Sii))

Case 1: ivi(Li) > ivi(Si)(“large” bundles contribute most of the optimal social welfare)

ivi(Li) > |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.

OutlineOutline

A deterministic VCG-based O(mA deterministic VCG-based O(m½½)-)-approximation mechanismapproximation mechanism

An An (m(m1/61/6) lower bound for VCG-based ) lower bound for VCG-based mechanisms.mechanisms.

A randomized almost-logarithmic A randomized almost-logarithmic approximation mechanism.approximation mechanism.

About the Lower BoundAbout the Lower Bound

Why lower bounds on VCG-Based Why lower bounds on VCG-Based mechanisms (a.k.a. maximal-in-range mechanisms (a.k.a. maximal-in-range algorithms)?algorithms)? Conjectured characterizationConjectured characterization: All mechanisms : All mechanisms

that give a good approximation ratio for that give a good approximation ratio for combinatorial auctions with subadditive bidders are combinatorial auctions with subadditive bidders are maximal in their range.maximal in their range.

Even if the conjecture is false, still the only Even if the conjecture is false, still the only technique that we currently know.technique that we currently know.

An An (m(m1/61/6) lower bound on ) lower bound on VCG-based mechanisms VCG-based mechanisms [Dobzinski-Nisan][Dobzinski-Nisan]

We define two complexity:We define two complexity: Cover NumberCover Number: (approximately) the range size: (approximately) the range size

must be “large” in order to obtain a good approximation ratio.must be “large” in order to obtain a good approximation ratio. Intersection NumberIntersection Number: a lower bound on the communication : a lower bound on the communication

complexity.complexity. We therefore want it to be “small” (polynomial)We therefore want it to be “small” (polynomial)

LemmaLemma (informal): If the cover number is large then (informal): If the cover number is large then the intersection number must be large too.the intersection number must be large too.

From now on, only 2 bidders, thus a lower bound of 2.From now on, only 2 bidders, thus a lower bound of 2.

The Cover NumberThe Cover Number

Intuitively, the size of the rangeIntuitively, the size of the range But we don’t want to count “degenerate But we don’t want to count “degenerate

allocations”…allocations”… A set of allocations C A set of allocations C coverscovers a set of a set of

allocations R if for each allocation S in R there allocations R if for each allocation S in R there is an allocation T in C such that Tis an allocation T in C such that T iiCCii for for i={1,2}.i={1,2}. cover(R)cover(R) is the size of the smallest set C that covers is the size of the smallest set C that covers

R.R. ObservationObservation: An MIR on range C provides a : An MIR on range C provides a

better approximation ratio than on R.better approximation ratio than on R.

The Cover NumberThe Cover Number

LemmaLemma: Let A be an MIR algorithm with range R. If : Let A be an MIR algorithm with range R. If cover(R) < ecover(R) < em/400m/400, then A provides an approximation , then A provides an approximation ratio of at most 1.99.ratio of at most 1.99.

ProofProof: Using the probabilistic method.: Using the probabilistic method. Fix an allocation T=(TFix an allocation T=(T11,T,T22) from the minimal cover C.) from the minimal cover C. Construct an instance with Construct an instance with additiveadditive bidders: v(S) = bidders: v(S) = jjSS v({j}) v({j}) For each item j, set with probability ½ vFor each item j, set with probability ½ v11({j})=1 and v({j})=1 and v22({j})=0 (or ({j})=0 (or

vice versa with probability ½ ).vice versa with probability ½ ). The optimal welfare in this instance is m, but each item j 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 contributes 1 to the welfare provided by T only if we hit the corresponding bundle in T (with probability 1/2).corresponding bundle in T (with probability 1/2).

The expected welfare that T provides is m/2, and we can get a The expected welfare that T provides is m/2, and we can get a better welfare only with exponential small probability.better welfare only with exponential small probability.

The Intersection NumberThe Intersection Number

A set of allocations D is called an A set of allocations D is called an intersection setintersection set if for each if for each (A(A11,A,A22)≠(B)≠(B11,B,B22))D we have that AD we have that A11

intersects Bintersects B22 and A and A22 intersects B intersects B11..

Let Let intersect(R)intersect(R) be the size of the largest be the size of the largest intersection set in R.intersection set in R.

The Intersection NumberThe Intersection Number

LemmaLemma: Let A be an MIR algorithm with range R. Let : Let A be an MIR algorithm with range R. Let intersect(R)=d. Then, the communication complexity of intersect(R)=d. Then, the communication complexity of A is at least d.A is at least d.

ProofProof: : Reduction from Reduction from disjointnessdisjointness: Alice holds a=a: Alice holds a=a11…a…add, Bob holds , Bob holds

b=bb=b11…b…bdd. Is there some t with a. Is there some t with att=b=btt=1? Requires t bits of =1? Requires t bits of communication.communication.

Given a disjointness instance, construct a combinatorial Given a disjointness instance, construct a combinatorial auction with subadditive bidders:auction with subadditive bidders: Let {(ALet {(A11,B,B11),…,(A),…,(Add,B,Bdd)} be the intersection set. )} be the intersection set.

Set vSet vAA(S)=2 if there is an index i s.t. a(S)=2 if there is an index i s.t. a ii=1 and A=1 and Ai i S. Otherwise S. Otherwise vvAA(S)=1. Similar valuation for Bob.(S)=1. Similar valuation for Bob.

The valuations are subadditive.The valuations are subadditive. A common 1 bit A common 1 bit optimal welfare of 4. Our algorithm is optimal welfare of 4. Our algorithm is

maximal in range, and the optimal allocation is in the range, so maximal in range, and the optimal allocation is in the range, so our algorithm our algorithm alwaysalways return the optimal solution. But this return the optimal solution. But this requires d bits of communication.requires d bits of communication.

Putting it TogetherPutting it Together

In order to obtain an approximation ratio better than 2, In order to obtain an approximation ratio better than 2, the cover number must be the cover number must be exponentiallyexponentially large. large.

If the MIR algorithm runs in polynomial time then the If the MIR algorithm runs in polynomial time then the intersection number must be intersection number must be polynomialpolynomial too. too.

LemmaLemma (informal): If the cover number is exponentially (informal): If the cover number is exponentially large then the intersection number is exponentially large then the intersection number is exponentially large too.large too.

CorollaryCorollary: No polynomial time VCG-based algorithm : No polynomial time VCG-based algorithm provides an approximation ratio better than 2.provides an approximation ratio better than 2.

SummarySummary

A deterministic VCG-based O(mA deterministic VCG-based O(m½½)-)-approximation mechanismapproximation mechanism

An An (m(m1/61/6) lower bound on VCG-based ) lower bound on VCG-based mechanisms.mechanisms.

A randomized almost-logarithmic A randomized almost-logarithmic approximation mechanism.approximation mechanism.

Open QuestionsOpen Questions

Deterministic mechanisms\lower bounds Deterministic mechanisms\lower bounds for combinatorial auctions with general for combinatorial auctions with general valuations?valuations?

Is the gap between Is the gap between randomizedrandomized and and deterministicdeterministic mechanisms essential? mechanisms essential?

Randomness and Randomness and Mechanism DesignMechanism Design

Randomization might help in mechanism Randomization might help in mechanism design settings.design settings.

Two notions of randomization:Two notions of randomization: ““The universal sense”:The universal sense”: a distribution over a distribution over

deterministic mechanisms (stronger)deterministic mechanisms (stronger) ““In expectation”:In expectation”: truthful behavior maximizes the truthful behavior maximizes the

expectation of the profit (weaker)expectation of the profit (weaker) Risk-averse bidders might benefit from untruthful behavior.Risk-averse bidders might benefit from untruthful behavior. The outcomes of the random coins must be kept secret.The outcomes of the random coins must be kept secret.

ResultsResults

Feige shows a randomized Feige shows a randomized O(logm/loglogm)-truthful O(logm/loglogm)-truthful in expectationin expectation mechanism.mechanism.

We show that there exists an We show that there exists an O(logm*loglogm) truthful O(logm*loglogm) truthful in the universal in the universal sensesense mechanism. mechanism.

The FrameworkThe Framework

Two cases:Two cases: Case 1Case 1: There is a : There is a dominantdominant bidder. bidder.

A bidder with v(M) > OPT/(100log m loglog m) A bidder with v(M) > OPT/(100log m loglog m) (denote the denominator by (denote the denominator by cc))

We can simply allocate all items to this bidder.We can simply allocate all items to this bidder. Case 2Case 2: There is : There is nono dominant bidder. dominant bidder.

In this case we can use random sampling: partition the In this case we can use random sampling: partition the bidders into two sets, acquire statistics from one set, and bidders into two sets, acquire statistics from one set, and use it to get an approximate solution with the other set.use it to get an approximate solution with the other set.

How to put the two cases together?How to put the two cases together? Flipping a coin works, but with probability of only ½. Flipping a coin works, but with probability of only ½. Next we will see how to increase the probability of Next we will see how to increase the probability of

success to 1-success to 1-..

The MechanismThe Mechanism

Partition the bidders into 3 sets:Partition the bidders into 3 sets: STAT with probability STAT with probability /2, SECPRICE with probability 1-/2, SECPRICE with probability 1-, and FIXED , and FIXED

with probability with probability /2./2. First case: there is a dominant bidder.First case: there is a dominant bidder.

Statistics Group

I have an estimate of

OPTSECPRICE group

A second price auction with a

reserve price of OPT/c

A second price auction with a

reserve price of OPT/c

The MechanismThe Mechanism Second case: there is no dominant bidder.Second case: there is no dominant bidder.

Statistics Group

I have a (good)

estimate of OPT

FIXED group

Case 2: No “Dominant” Case 2: No “Dominant” BidderBidder

AssumptionAssumption: For all : For all bidders bidders vvii(OPT(OPTii) < OPT / c) < OPT / c

In the FIXED groupIn the FIXED group: : a fixed-price auction a fixed-price auction where each item has where each item has a price of p (depends a price of p (depends on the statistics on the statistics group)group)

Everything costs p Take your most

profitable bundle

My price is 2*p I paid p

Too Expensive

!

Still Missing…Still Missing…

Why does the fixed price auction (with a Why does the fixed price auction (with a “good price”) provides a good “good price”) provides a good approximation ratio?approximation ratio?

Can we find this “good price” using the Can we find this “good price” using the statistics group?statistics group?

A Combinatorial Property A Combinatorial Property of Subadditive Valuationsof Subadditive Valuations

LemmaLemma: Let v be a subadditive valuation : Let v be a subadditive valuation and S a bundle of items. Then we can and S a bundle of items. Then we can assign each item in S a price in {0,p} assign each item in S a price in {0,p} such that:such that: For each TFor each TS: v(T) > S: v(T) > jjTT|T|*p|T|*p

|S|*p > v(S)/(100*logm)|S|*p > v(S)/(100*logm)