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Combinatorial Auctions with Complement-Free Bidders – An Overview Speaker: Michael Schapira Based on joint works with Shahar Dobzinski & Noam Nisan

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2 Talk Structure Combinatorial auctions with CF bidders Approximating with value queries An incentive compatible O(m 1/2 )-approximation for CF bidders using value queries. Approximating with demand queries. A combinatorial algorithm that obtains a 2-approximation for XOS bidders. A randomized-rounding algorithm that obtains a 2- approximation for XOS bidders using demand queries. Open Questions

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3 Combinatorial Auctions A set M={1,…,m} of items for sale. n bidders, each bidder i has a valuation function v i :2 M ->R +. Common assumptions: Normalization: v i ( )=0 Free disposal: S T v i (T) v i (S) Goal: find a partition S 1,…,S n such that social welfare v i (S i ) is maximized

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4 Combinatorial Auctions Problem 1: finding an optimal allocation is NP- hard. Therefore, we are interested in the possible approximation ratios. Problem 2: the valuations length is exponential in m, while we wish our algorithms to be polynomial in m and n. Problem 3: how can we be certain that the bidders do not lie?

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5 Combinatorial Auctions We are interested in algorithms that based on the reported valuations {v i } i output an allocation which is an approximation to the optimal social welfare. We require the algorithms to be polynomial in m and n. That is, the algorithms must run in sub- linear (polylogarithmic) time.

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6 Access Models How can we access the input ? One possibility: bidding languages. The black box approach: each bidder is represented by an oracle which can answer a certain type of queries.

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7 Access Models Common types of queries: Value: given a bundle S, return v(S). Demand: given a vector of prices (p 1,…, p m ) return the bundle S that maximizes v(S)- j S p j. (demand queries are strictly more powerful than value queries ). General: any possible type of query (the communication model).

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8 Known Results Finding an optimal solution requires exponential communication. Nisan-Segal Finding an O(m 1/2- )-approximation requires exponential communication. Nisan-Segal. ( this result holds for every possible type of oracle) Using demand oracles, a matching upper bound of O(m 1/2 ) exists (Blumrosen-Nisan). Better results might be obtained by restricting the classes of valuations.

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9 The Hierarchy of CF Valuations Complement-Free: v(S T) v(S) + v(T). XOS Submodular: v(S T) + v(S T) v(S) + v(T). Semantic Characterization: Decreasing Marginal Utilities. GS: (Gross) Substitutes: Solvable in polynomial time. OXS GS SM XOS CF Lehmann, Lehmann, Nisan

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10 Talk Structure Combinatorial auctions with CF bidders Approximating with value queries An incentive compatible O(m 1/2 )-approximation for CF bidders using value queries. Approximating with demand queries. A combinatorial algorithm that obtains a 2-approximation for XOS bidders. A randomized-rounding algorithm that obtains a 2- approximation for XOS bidders using demand queries. Open Questions

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11 Value Queries Valuation Class Upper BoundLower Bound Generalm/(log 1/2 m) (Holzman, Kfir-Dahav, Monderer, Tennenholz) (Incentive Compatible) m/(logm) (Nisan-Segal) CFm 1/2 (Incentive Compatible) XOSm 1/2- SM2 (Lehmann,Lehmann,Nisan) e/(e-1)- (Khot, Lipton,Markakis, Mehta) GS1 (Bertelsen, Lehmann) (Incentive Compatible)

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12 Incentive Compatibility & VCG Prices We want an algorithm that is truthful (incentive compatible). I.e. we require that the dominant strategy of each of the bidders would be to reveal true information. VCG is the main general technique known for making auctions incentive compatible (if bidders are not single-minded): Each bidder i pays: ki v k (O -i ) - ki v k (O i ) O i is the optimal allocation, O -i the optimal allocation of the auction without the ith bidder.

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13 Incentive Compatibility & VCG Prices Problem: VCG requires an optimal allocation! Finding an optimal allocation requires exponential communication and is computationally intractable. Approximations do not suffice (Nisan-Ronen).

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14 VCG on a Subset of the Range Our solution: limit the set of possible allocations. We will let each bidder to get at most one item, or well allocate all items to a single bidder. Optimal solution in the set can be found in polynomial time VCG prices can be computed incentive compatibility. We still need to prove that we achieve an approximation.

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15 The Algorithm Ask each bidder i for v i (M), and for v i (j), for each item j. (We have used only value queries) Construct a bipartite graph and find the maximum weighted matching P. can be done in polynomial time (Tarjan) A B Items Bidders v 1 (A) v 3 (B)

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16 The Algorithm (Cont.) Let i be the bidder that maximizes v i (M). If v i (M)>|P| Allocate all items to i. else Allocate according to P. Let each bidder pay his VCG price (with respect to the restricted set).

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17 Proof of the Approximation Ratio Theorem: If all valuations are CF, the algorithm provides an O(m 1/2 )-approximation. Proof: Let OPT=(T 1,..,T k,Q 1,...,Q l ), where for each T i, |T i |>m 1/2, and for each Q i, |Q i |m 1/2. |OPT|= i v i (T i ) + i v i (Q i ) Case 1: i v i (T i ) > i v i (Q i ) (large bundles contribute most of the social welfare) i v i (T i ) > |OPT|/2 At most m 1/2 bidders get at least m 1/2 items in OPT. For the bidder i the bidder i that maximizes v i (M), v i (M) > |OPT|/2m 1/2. Case 2: i v i (Q i ) i v i (T i ) (small bundles contribute most of the social welfare) i v i (Q i ) |OPT|/2 For each bidder i, there is an item c i, such that: v i (c i ) > v i (Q i ) / m 1/2. (The CF property ensures that the sum of the values is larger than the value of the whole bundle) {c i } i is an allocation which assigns at most one item to each bidder: |P| i v i (c i ) |OPT|/2m 1/2.

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18 Talk Structure Combinatorial auctions with CF bidders Approximating with value queries An incentive compatible O(m 1/2 )-approximation for CF bidders using value queries. Approximating with demand queries. A combinatorial algorithm that obtains a 2-approximation for XOS bidders. A randomized-rounding algorithm that obtains a 2- approximation for XOS bidders using demand queries. Open Questions

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19 Demand Queries Valuation Class Upper BoundLower Bound Generalm 1/2 (Blumrosen-Nisan) m 1/2- (Nisan-Segal) CF2 (Feige) 2- XOSe/(e-1) (Dobzinski-Schapira) e/(e-1)- SM e/(e-1)- (Feige) <1 (Feige) GS1 (Bertelsen, Lehmann) (Incentive Compatible)

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20 XOS The maximum over additive valuations: (a:1 b:2 c:3) (a:2) v({a}) = 2 v({a,b}) = 3 v({a,b,c}) = 6 Examples:

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21 Algorithm I -Example Items: {A, B, C, D, E}. 3 bidders. Price vector: p 0 =(0,0,0,0,0) v 1 : (A:1 OR B:1 OR C:1) XOR (C:2) Bidder 1 gets his demand: {A,B,C}.

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22 Algorithm I -Example Items: {A, B, C, D, E}. 3 bidders. Price vector: p 0 =(0,0,0,0,0) v 1 : (A:1 OR B:1 OR C:1) XOR (C:2) Bidder 1 gets his demand: {A,B,C}. Price vector: p 1 =(1,1,1,0,0) v 2 : (A:1 OR B:1 OR C:9) XOR (D:2 OR E:2) Bidder 2 gets his demand: {C}

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23 Algorithm I -Example Items: {A, B, C, D, E}. 3 bidders. Price vector: p 0 =(0,0,0,0,0) v 1 : (A:1 OR B:1 OR C:1) XOR (C:2) Bidder 1 gets his demand: {A,B,C}. Price vector: p 1 =(1,1,1,0,0) v 2 : (A:1 OR B:1 OR C:9) XOR (D:2 OR E:2) Bidder 2 gets his demand: {C} Price vector: p 2 =(1,1,9,0,0) v 3 : (C:10 OR D:1 OR E:2) Bidder 3 gets his demand: {C,D,E} Final allocation: {A,B} to bidder 1, {C,D,E} to bidder 3.

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24 Algorithm I Input: n bidders, for each we are given a demand oracle and an XOS oracle Init: p 1 =…=p m =0. For each bidder i=1..n Let S i be the demand of the ith bidder at prices p 1,…,p m. For all i < i take away from S i any items from S i. Let q 1,…,q m be the item values in the maximizing clause for S i in v i. For all j S i update p j = q j.

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25 Proof To prove the approximation ratio, we will need these two simple lemmas: Lemma: The total social welfare generated by the algorithm is at least p j. Lemma: The optimal social welfare is at most 2 p j.

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26 Proof – Lemma 1 Lemma: The total social welfare generated by the algorithm is at least p j. Proof: Each bidder i got a bundle T i at stage i. At the end of the algorithm, he holds A i T i. The definition of the prices guarantees that: v i (A i ) j Ai p j

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27 Proof – Lemma 2 Lemma: The optimal social welfare is at most 2 p j. Proof: Let O 1,...,O n be the optimal allocation. Let p i,j be the price of the jth item at the ith stage. Each bidder i asks for the bundle that maximizes his demand at the ith stage: v i (O i )- j Oi p i,j j p i,j – j p (i-1),j Since the prices are non-decreasing: v i (O i )- j Oi p n,j j p i,j – j p (i-1),j Summing up on both sides: i v i (O i )- i j Oi p n,j i ( j p i,j – j p (i-1),j ) i v i (O i )- j p n,j j p n,j i v i (O i ) 2 j p n,j

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28 Algorithm II (Feige) – Step 1 Solve the linear relaxation of the problem: Maximize: i,S x i,S v i (S) Subject To: For each item j: i,S|j S x i,S 1 For each bidder i: S x i,S 1 For each i,S: x i,S 0 Despite the exponential number of variables, the LP relaxation may still be solved in polynomial time using demand oracles. (Nisan-Segal). OPT*= i,S x i,S v i (S) is an upper bound for the value of the optimal integral allocation.

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29 Algorithm II (Feige) – Step 2 Use randomized rounding to build a pre-allocation S 1,..,S n : Randomized Rounding: For each bidder i, let S i be the bundle S with probability x i,S, and the empty set with probability 1- S x i,S. The expected value of v i (S i ) is S x i,S v i (S)

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30 Algorithm II (Feige) – Step 3 Assign every item j, uniformly at random, to one of the bidders i, such that j is in S i. Consider a bidder i such that j is in S i. Bidder i gets j with probability 1/(n j +1), where n j is the number of all bidders who got j in Step 2. Since E(n j )=1 we have reached a 2 approximation.

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31 Talk Structure Combinatorial auctions with CF bidders Approximating with value queries An incentive compatible O(m 1/2 )-approximation for CF bidders using value queries. Approximating with demand queries. A combinatorial algorithm that obtains a 2-approximation for XOS bidders. A randomized-rounding algorithm that obtains a 2- approximation for XOS bidders using demand queries. Open Questions

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32 The Approximability of Submodular Bidders Type of Queries Upper BoundLower Bound Value2 (Lehmann,Lehmann,Nisan) e/(e-1)- (Khot, Lipton,Markakis, Mehta) Demand e/(e-1)- (Feige) <1 (Feige)

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33 Incentive Compatibility Valuation Class Upper BoundLower Bound CFO((log 2 m)/e 3 ) 2- XOS e/(e-1)- SM <1 (Feige)

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