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Multiagent Systems1 Truth Revelation in Approximately Efficient Combinatorial Auctions Daniel Lehman, Liadan Ita OCallaghan and Yoav Shoham @ Journal of the ACM, 2002 Presenter – Muntasir Raihan Rahman

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Multiagent Systems2 Outline of the Presentation Definition of Combinatorial Auctions Assumptions Main Contribution of the Paper Definition and Notations Single Minded Bidders Hardness of GVA for Single Minded Bidders Greedy Allocation Algorithms Conditions for Ensuring Truthfulness for Single Minded Bidders A Truthful Payment Scheme for the Greedy Allocation Algorithms Extension to Complex Bidders Revenue Considerations Strength and Weaknesses of the Paper Future Work and Open Problems

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Multiagent Systems3 Combinatorial Auction (CA) An Auction where Multiple goods are available Bidders can bid for subsets of goods Single seller, multiple buyers Reverse auction (single buyer, multiple sellers) : similar One stage sealed bid CA

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Multiagent Systems4 Assumptions Normalization: Zero valuation for empty set. Free Disposal: Throw away excess items. No Externalities: An agents valuation is independent of the other agents valuations. Complementarity: Valuation for a set of items can be more than the sum of the valuations for the individual items. Example: a left shoe and a right shoe. Substitutability: Valuation for a set of items is less than the sum of the valuations for the individual items. Example: two tickets to the same football game.

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Multiagent Systems5 Main Contribution of the Paper Present simple approximate allocation algorithms for CAs that posses the following two properties: The algorithm performs a reasonably effective optimization. There is a performance guarantee. There is a payment scheme that converts the approximation algorithm into a truthful mechanism. Not necessarily VCG payments.

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Multiagent Systems6 Definition and Notation P : set of n bidders G : set of k goods A partial allocation function A : G P, P = P U {unallocated} Partial since not all goods are required to be allocated. O = P G is the set of all possible allocations. Type : a bidders evaluation of goods A function v: 2 G R + Θ is the set of bidder types: Θ = R + 2G t is the true type of a bidder t Θ d is the declared type (maybe a lie), d Θ T= : vector of true types. D= : vector of declared types P is a vector of n payments. P R + n

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Multiagent Systems7 Further Definitions Utility: Assume quasi-linear utility. For a bidder of type t, the utility for any bundle s G and payment x, u = t (s) – x. Direct Mechanism for CA: For a declared type vector D, a direct mechanism is a tuple f is the allocation algorithm p is the payment vector: p = p(D) p i (D) denotes the payment for bidder I g i (D) denotes the bundle obtained by i for the allocation f.

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Multiagent Systems8 Recap: Truthful Mechanism Consider a mechanism M = M is truthful iff for every agent i, every type t and any vector D t(g i (D)) – p i (D) t(g i (D)) – p i (D) D is the vector obtained by replacing the ith coordinate of D with t. Generalized Vickrey Auctions (GVA): General method for designing truthful mechanisms. GVA chooses an optimal allocation that maximizes social welfare. GVA payment: the opportunity cost that each agents presence introduces to all other players. GVA ensures individual rationality each agents utility is non-negative.

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Multiagent Systems9 Single Minded Bidders A severe restriction on bidding types. A single minded bidder only cares about a single bundle of items. Zero valuation for any other subset. Formally, a bidder is single minded iff There is a set S and a value v such that t(s) = v if s s t(s) = 0 otherwise A single minded bid is denoted as. Also called an atomic bid.

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Multiagent Systems10 Real Life Examples of Single Minded Bids Authors claim single minded bids not unrealistic and uncommon. Examples: Used for selling timber in New Zealand: single minded bidders A bidder interested in all locations in a specific geographical area. Contiguous bids. Auctions for pollution rights Auto manufacturer selling auto parts Buyers interested only in one set of parts that can be completed into a working vehicle.

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Multiagent Systems11 An Example from Computer Science Consider a communication network where the underlying topology is a tree. So there is a unique path between any pair of nodes. The links are the items to be auctioned off. The players are source sink pairs. Each player interested in building a path from source to sink. So each player is only interested in the set of links that constitute the unique path from its source to sink. So the players are single minded. Green player Red player

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Multiagent Systems12 GVA for Single Minded Bidders Can we apply the GVA mechanism for single minded bidders to obtain an optimal allocation in polynomial time? Unfortunately, NO! Theorem [Hastad, 1999; Sandholm, 2002]: Finding an allocation that maximizes social welfare in the case of single minded bidders is NP-hard. There is no polynomial time (polynomial in k+n) approximation algorithm with ratio better than k ε-1/2 unless NP=ZPP, where k is the number of items and n is the number of bidders.

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Multiagent Systems13 Proof Sketch of the Hardness Result Reduction from the Weighted Independent Set (WIS) problem. Given a graph G=(V,E) We will assign a weight w v to each vertex v V. An independent set is a set of vertices such that no two vertices are adjacent. The WIS problem asks for the independent set with maximum weighted sum. Reduction: Consider the combinatorial auction with n players and k items. Construct a graph G CA (V, E) V is the set of players E is the set of items The edges connected to a vertex v correspond to the single bundle of items v is interested in. The weight of the vertex v is the valuation of the corresponding player for his bundle.

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Multiagent Systems14 Reduction (Continued) Agent 1 Agent 3 Agent 4 Agent 2 V 1 : 7 V 1 : 3V 1 : 5 V 1 : 4 Agent 1 bid: Agent 2 bid: Agent 3 bid: Agent 4 bid: Max weighted independent set: vertices corresponding to Agent 1 and Agent 4 Total weight 7+3=10 This corresponds to the optimal allocation Giving a and b to Agent 1, c and d to Agent 2 c db a

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Multiagent Systems15 Greedy Allocation Algorithms Finding optimal allocation for single minded bidders is computationally intractable. So authors propose a family of simple greedy algorithms. The greedy algorithms consist of two phases: Sorting Phase: The bids are sorted according to some criteria (norm). [Defined later] This phase runs in O(nlogn) time. [quick sort] Allocation Phase: Let L be the list of bids obtained in the first phase. For each bid in L If the current bid does not conflict with the previously allocated bids Then accept current bid. Else Reject current bid. This phase requires linear time.

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Multiagent Systems16 Efficiency of the Greedy Algorithm The greedy algorithm is polynomial time computable. So it achieves computational efficiency. What about economic efficiency? In order to increase economic efficiency, The norm must be selected in such a way that bids that are more likely to increase efficiency are pushed forward in the sorted list. A bid can increase economic efficiency if: It has high valuation Has a small number of items [this can help in the later stage by keeping more items free for later bids] A norm that can satisfy these two properties : For a bid b =, norm(b) = (v / |s|), where |s| is the size of the subset.

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Multiagent Systems17 Efficiency (Continued) So the bids should be sorted in decreasing order of the norm value. Infact any norm of the form norm(b) = (v / |s| x ) will do. The case x = 0.5 is especially interesting, since in this case the greedy algorithm can guarantee the best possible approximation ratio. [Recall the hardness result for single minded bidders !] [Theorem]: The greedy allocation scheme with norm (v / |s| 1/2 ) approximates the optimal allocation within a factor of k 1/2. This is the best possible approximation ratio that can be achieved!

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Multiagent Systems18 Greedy Algorithm in Action Consider the case of two goods {a, b} and three bidders {R, G, B}. R:, G:, B: norm(R)=10, norm(G)=9.5, norm(B)=8 give a to R Bid of G conflicts with R, so G gets nothing! Finally B gets b. Social welfare = 18. Notice that this is not optimal, the optimal allocation would give both a and b to G with total social welfare = 19!

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Multiagent Systems19 How to Ensure Truthfulness How can the greedy allocation be made truthful? Intuitive idea: use VCG payments. However a theorem by [Nisan and Ronen, 2000] shows that A truthful mechanism with VCG payments must find optimal allocations. But the greedy algorithms are not optimal! So an approximate allocation is incompatible with VCG payments. Need to search for alternate payment rules that can ensure truthfulness for single minded bidders. Impose further restrictions on the bidders and allocation schemes.

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Multiagent Systems20 Sufficient Conditions for Truthfulness for Single Minded Bidders Properties of allocation algorithms and payment schemes that will ensure truthfulness. Exactness: A single minded bidder gets exactly the set that he desires, nothing more, or he gets nothing. Bid, bidder gets the set s if he wins, not any superset of s, if he loses he gets nothing. A natural condition, since free disposal means the agent doesnt care about the additions items.

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Multiagent Systems21 Sufficient Conditions (Continued) Monotonicity: If is a winning bid, then for s s and v v, is also a winning bid. Proposing more money for fewer goods cannot cause a winning bidder to lose. A very natural assumption. The GVA scheme also satisfies monotonicity.

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Multiagent Systems22 Sufficient Conditions (Continued) Implications of Monotonicity: [Lemma] In a mechanism that satisfies exactness and monotonicity, for any bidder j, there exists a critical value v c such that If j bids below v c, he wins nothing If j bids above v c, he wins the exact set of items he desired. The proof follows immediately from the conditions of exactness and monotonicity. The GVA mechanism also satisfies this property. Recall the Vickrey auction Here the critical value is the second highest bid A bidder bidding below that will lose

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Multiagent Systems23 Sufficient Conditions (continued) Critical: A satisfied bidder pays exactly the critical value. Compare with Vickrey auction. The winner pays the second highest bid [critical value] The payment is independent of the agents own bid. Setting a payment below or above the critical value can encourage lying. Participation: An agent who wins nothing, pays nothing. Very natural restriction! Akin to Individual Rationality All players utility are non-negative.

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Multiagent Systems24 Characterization of a Truthful Mechanism [Theorem] If a mechanism satisfies the properties: exactness, monotonicity, participation and critical, then it is a truthful mechanism. The proof follows easily from the definition of the properties!

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Multiagent Systems25 A truthful Mechanism with Greedy Allocation A Greedy Payment scheme: [case 1]: If an agent loses, he pays nothing. Consider a winning bid for agent j. [case 2]: If the items allocated to j do not conflict with any other later bids, then j pays nothing. This is because j is not hurting anyone. Example: j gets {a,b}, and no one else wants a or b. [case 3]: Let k be the first agent in the sorted list whose bid conflicted with the bid of j. This means if j did not participate in the auction, k could have won! So j must have appeared before k in the sorted bid list of the obtained in the first phase of the greedy allocation algorithm. Let be js bid. j pays |s|.norm(k)

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Multiagent Systems26 Greedy Payment in Action Consider two goods {a, b} and three players {R, G, B}. Bids: R:, G:, B: norm(R) = 10, norm(G)=9.5, norm(B)=8 R wins a and B wins b, G loses G pays 0 [did not win] R blocks G from winning So R pays 1. norm(G)=9.5 B does not block anyone So B pays 0

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Multiagent Systems27 The Final Result [Theorem] The payment scheme and the greedy allocation compose a truthful mechanism for single minded bidders. Proof Sketch: To prove truthfulness, we have to show that the four properties are satisfied. Participation: Trivial, since according to the payment rule, a losing bidder pays nothing. Monotonicity: Let be a winning bid with norm n. Consider with norm n, where s s and v > v. So n> n, so the bid will be pushed higher in the sorted list of bids remains a winning bid!

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Multiagent Systems28 Proof Continued Critical: The payment is exactly equal to the critical value. Paying less will make a wining bid to lose. For a winner j, let p (j)= |s|. norm(k), where k is defined as before. If j bids v < p (j), then v < |s|.norm(k) v / |s| < norm(k) norm(j) < norm(k) This means the bid of j will be kicked below the bid of k. j will no longer be a winner.

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Multiagent Systems29 Complex Bidders Single minded bidder model too restrictive! Consider Complex bidders. A Complex bidder can be considered as a collection of single minded bidders. Example: complex bid [,, ] The complex bidder controls three single minded bidders Will the single minded bidders act cooperatively or competitively? The authors show that no payment scheme can make greedy allocation truthful for complex bidders. Proof by showing counter-examples [no formal proof]

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Multiagent Systems30 Revenue Maximization Revenue maximization in combinatorial auctions is an active area of research! The authors show by examples that the revenue generated by the greedy allocation and payment does not always coincide with revenue from GVA! Recent approaches: Virtual Valuation Combinatorial Auctions (VVCA) [Sandholm, 2004] Virtual Valuations: linear transformation of real valuations. Applied to VCG mechanism, can it be used with greedy allocation for single minded bidders? Item Pricing for Revenue Maximization [Balcan, Blum, Mansour, 2008] Pricing single items instead of bundle of items in VCG.

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Multiagent Systems31 Strengths and Limitations (+) Well written paper, many illustrative examples to clarify important ideas. (+) The greedy allocation algorithm and payment schemes are polynomial time computable and easy to understand. (-) Model of single minded bidder too restrictive.

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Multiagent Systems32 Future Work Most computational results for Combinatorial Auctions focus on worst case computational complexity. What about average case complexity? Some voting rules that are hard in the worst case are polynomial time computable in the average case. This could inspire average case analysis of hard combinatorial auction problems using some Junta Distributions. The greedy algorithm is very intuitive. What about other heuristics? Approximation algorithms based on linear programming relaxations of the integer programming formulation of CA. Randomized mechanisms [truthful in expectation] [Lavi, Swamy 2005] Is it possible to get deterministic truthful mechanisms for single minded bidders using relaxation? Other search heuristics like Genetic Algorithms, Simulated Annealing. Complex Bidders: Find truthful mechanisms for other type of approximation schemes in case of complex bidders.

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Multiagent Systems33 References Truth Revelation in Approximately Efficient Combinatorial Auctions, Daniel Lehman, Liadan Ita OCallaghan, Yoav Shoham, 2002. Lectures on Combinatorial Auctions, Tim Roughgarden, 2006. Combinatorial Auctions, chapter 11, Algorithmic Game Theory, edited by Noam Nisan, Tim Roughgarden, Eva Tardos, Vijay V Vajirani, 2007. Combinatorial Auctions, edited by Peter Cramton, Yoav Shoham, Richard Steinberg, 2006.

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Multiagent Systems34 Questions? || /*Comments*/

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