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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 v i :2 M ->R +. m items, n bidders, each bidder i has a valuation function v i :2 M ->R +. Common assumptions: Normalization: v i ( )=0 Normalization: v i ( )=0 Monotonicity: S T v i (T) v i (S) Monotonicity: S T v i (T) v i (S) Goal: find a partition S 1,…,S n such that the total social welfare v i (S i ) is maximized. Goal: find a partition S 1,…,S n such that the total social welfare v i (S i ) is maximized. Algorithms must run in time polynomial in n and m. Algorithms must run in time polynomial in n and m. In this talk the valuations are subadditive: 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]. A 2 approximation algorithm exists [Feige], and a matching lower bound is also known [Dobzinski-Nisan-Schapira]. What about truthful approximations? What about truthful approximations? The private information of each bidder is his valuation. The private information of each bidder is his valuation.

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Outline A deterministic VCG-based O(m ½ )- approximation mechanism A deterministic VCG-based O(m ½ )- approximation mechanism An (m 1/6 ) lower bound on VCG-based mechanisms. An (m 1/6 ) lower bound on VCG-based mechanisms. A randomized almost-logarithmic approximation mechanism. A randomized almost-logarithmic approximation mechanism.

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Reminder: Maximal in Range Algorithms VCG: Allocate O i to bidder i. Bidder i gets a payment of ki v k (O k ). VCG: Allocate O i to bidder i. Bidder i gets a payment of ki v k (O k ). (O 1,…,O n ) is the optimal solution. (O 1,…,O n ) is the optimal solution. Still truthful if we limit the range. Still truthful if we limit the range. Range := { A=(A 1,…,A n ) | v 1,…,v n : A(v 1,…,v n )=A } Range := { A=(A 1,…,A n ) | v 1,…,v n : A(v 1,…,v n )=A } The Algorithm [Dobzinski-Nisan-Schapira] : The Algorithm [Dobzinski-Nisan-Schapira] : Choose the best allocation where either: Choose the best allocation where either: One bidder gets all items OR One bidder gets all items OR Each bidder gets at most one item. Each bidder gets at most one item. Clearly, the algorithm is maximal-in-range and can be implemented in polynomial time. Clearly, the algorithm is maximal-in-range and can be implemented in polynomial time.

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Case 2: i v i (S i ) i v i (L i ) (small bundles contribute most of the optimal social welfare) i v i (S i ) |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 c i S i, such that: v i ({c i }) > v i (S i ) / m 1/2. Allocate c i to bidder i. Proof of the Approximation Ratio Theorem: If all valuations are subadditive, the algorithm provides an O(m 1/2 )-approximation. Proof: Let OPT=(L 1,..,L l,S 1,...,S k ), where for each L i, |L i |>m 1/2, and for each S i, |S i |m 1/2. |OPT|= i v i (L i ) + i v i (S i ) Case 1: i v i (L i ) > i v i (S i ) (large bundles contribute most of the optimal social welfare) i v i (L i ) > |OPT|/2 At most m 1/2 bidders get at least m 1/2 items in OPT. There is a bidder i s.t.: v i (M) v i (L i ) |OPT|/2m 1/2.

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Outline A deterministic VCG-based O(m ½ )- approximation mechanism A deterministic VCG-based O(m ½ )- approximation mechanism An (m 1/6 ) lower bound for VCG-based mechanisms. An (m 1/6 ) lower bound for VCG-based mechanisms. A randomized almost-logarithmic approximation mechanism. 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)? 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. 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. Even if the conjecture is false, still the only technique that we currently know.

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An (m 1/6 ) lower bound on VCG-based mechanisms [Dobzinski-Nisan] We define two complexity: We define two complexity: Cover Number: (approximately) the range size Cover Number: (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 Number: a lower bound on the communication complexity. Intersection Number: a lower bound on the communication complexity. We therefore want it to be small (polynomial) We therefore want it to be small (polynomial) Lemma (informal): If the cover number is large then the intersection number must be large too. 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. 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 Intuitively, the size of the range But we dont want to count degenerate allocations… 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 T i C i for i={1,2}. 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 T i C i for i={1,2}. cover(R) is the size of the smallest set C that covers R. 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. 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) < e m/400, then A provides an approximation ratio of at most Lemma: Let A be an MIR algorithm with range R. If cover(R) < e m/400, then A provides an approximation ratio of at most Proof: Using the probabilistic method. Proof: Using the probabilistic method. Fix an allocation T=(T 1,T 2 ) from the minimal cover C. Fix an allocation T=(T 1,T 2 ) from the minimal cover C. Construct an instance with additive bidders: v(S) = j S v({j}) Construct an instance with additive bidders: v(S) = j S v({j}) For each item j, set with probability ½ v 1 ({j})=1 and v 2 ({j})=0 (or vice versa with probability ½ ). For each item j, set with probability ½ v 1 ({j})=1 and v 2 ({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 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 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 (A 1,A 2 )(B 1,B 2 ) D we have that A 1 intersects B 2 and A 2 intersects B 1. A set of allocations D is called an intersection set if for each (A 1,A 2 )(B 1,B 2 ) D we have that A 1 intersects B 2 and A 2 intersects B 1. Let intersect(R) be the size of the largest intersection set in R. 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. 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: Proof: Reduction from disjointness: Alice holds a=a 1 …a d, Bob holds b=b 1 …b d. Is there some t with a t =b t =1? Requires t bits of communication. Reduction from disjointness: Alice holds a=a 1 …a d, Bob holds b=b 1 …b d. Is there some t with a t =b t =1? Requires t bits of communication. Given a disjointness instance, construct a combinatorial auction with subadditive bidders: Given a disjointness instance, construct a combinatorial auction with subadditive bidders: Let {(A 1,B 1 ),…,(A d,B d )} be the intersection set. Let {(A 1,B 1 ),…,(A d,B d )} be the intersection set. Set v A (S)=2 if there is an index i s.t. a i =1 and A i S. Otherwise v A (S)=1. Similar valuation for Bob. The valuations are subadditive. 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. 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. 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. 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. 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. Corollary: No polynomial time VCG-based algorithm provides an approximation ratio better than 2.

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Summary A deterministic VCG-based O(m ½ )- approximation mechanism A deterministic VCG-based O(m ½ )- approximation mechanism An (m 1/6 ) lower bound on VCG-based mechanisms. An (m 1/6 ) lower bound on VCG-based mechanisms. A randomized almost-logarithmic approximation mechanism. A randomized almost-logarithmic approximation mechanism.

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

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Randomness and Mechanism Design Randomization might help in mechanism design settings. Randomization might help in mechanism design settings. Two notions of randomization: Two notions of randomization: The universal sense: a distribution over deterministic mechanisms (stronger) The universal sense: a distribution over deterministic mechanisms (stronger) In expectation: truthful behavior maximizes the expectation of the profit (weaker) In expectation: truthful behavior maximizes the 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.

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

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The Framework Two cases: Two cases: Case 1: There is a dominant bidder. Case 1: There is a dominant bidder. A bidder with v(M) > OPT/(100log m loglog m) (denote the denominator by c) A bidder with v(M) > OPT/(100log m loglog m) (denote the denominator by c) We can simply allocate all items to this bidder. We can simply allocate all items to this bidder. Case 2: There is no dominant 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. 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? 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 success to 1-. Next we will see how to increase the probability of success to 1-.

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The Mechanism Partition the bidders into 3 sets: Partition the bidders into 3 sets: STAT with probability /2, SECPRICE with probability 1-, and FIXED with probability /2. First case: there is a dominant bidder. First case: there is a dominant bidder. Statistics Group I have an estimate of OPT SECPRIC E group A second price auction with a reserve price of OPT/c

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The 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

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Case 2: No Dominant Bidder Assumption: For all bidders v i (OPT i ) < OPT / c Assumption: For all bidders v i (OPT i ) < OPT / c In the FIXED group: a fixed-price auction where each item has a price of p (depends on the statistics group) 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 2*p I paid p Too Expensive !

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Still Missing… Why does the fixed price auction (with a good price) provides a good approximation ratio? 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? 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: 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) > j T |T|*p For each T S: v(T) > j T |T|*p |S|*p > v(S)/(100*logm) |S|*p > v(S)/(100*logm)

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