We study the following Bayesian setting: m items are sold to n selfish bidders in m independent second-price auctions. Each bidder has a private valuation function that expresses complex preferences over all subsets of items. Bidders only have beliefs about the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular valuation functions, then every Bayesian Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game a pure Nash always exists and can be found in time that is polynomial in both m and n.Combinatorial Auctions. In a combinatorial auction m items M = {1, . . . , m} are offered for sale to n bidders N = {1, . . . , n}. Each bidder i has a valuation function (or valuation, in short) v i that assigns a non-negative real number to every subset of the items. v i expresses i's preferences over bundles of items. The value v i (S) can be thought of as specifying i's maximum willingness to pay for S. Two standard assumptions are made on each v i : v i (∅) = 0 (normalization), and v i (S) ≤ v i (T ) for every two bundles S ⊆ T (monotonicity). The objective is to find a partition of the items among the bidders S 1 , . . . , S n (where S i ∩ S j = ∅ for all i = j) such that the social welfare Σ i v i (S i ) is maximized.The interplay between selfishness and computational optimization in combinatorial auctions is well-studied. Each of these aspects alone can be handled in a satisfactory way: The celebrated VCG mechanisms [19,3,10] motivate agents to truthfully report their private information, and optimize the social-welfare. The caveat is that this may take exponential time [15,16] (in the natural parameters of the problem m and n). On the other hand, if we disregard strategic issues, it is possible to obtain good approximations to the optimal social-welfare

Abstract. We study the following Bayesian setting: m items are sold to n selfish bidders in m independent second-price auctions. Each bidder has a private valuation function that expresses complex preferences over all subsets of items. Bidders only have beliefs about the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular valuation functions, then every Bayesian Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game a pure Nash always exists and can be found in time that is polynomial in both m and n.

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Scheduling on related machines (Q||Cmax) is one of the most important problems in the field of Algorithmic Mechanism Design. Each machine is controlled by a selfish agent and her valuation can be expressed via a single parameter, her speed. Archer and Tardos [4] showed that, in contrast to other similar problems, a (non-polynomial) allocation that minimizes the makespan can be truthfully implemented. On the other hand, if we leave out the game-theoretic issues, the complexity of the problem has been completely settled -the problem is strongly NP-hard, while there exists a PTAS [9,8].This problem is the most well-studied in singleparameter Algorithmic Mechanism Design. It gives an excellent ground to explore the boundary between truthfulness and efficient computation. Since the work of Archer and Tardos, quite a lot of deterministic and randomized mechanisms have been suggested. Recently, a breakthrough result [7] showed that a randomized, truthful-in-expectation PTAS exists. On the other hand, for the deterministic case, the best known approximation factor is 2.8 [10,11].It has been a major open question whether there exists a deterministic truthful PTAS, or whether truthfulness has an essential, negative impact on the computational complexity of the problem. In this paper we give a definitive answer to this important question by providing a truthful deterministic PTAS.

Scheduling on related machines (Q||C max ) is one of the most important problems in the field of Algorithmic Mechanism Design. Each machine is controlled by a selfish agent and her valuation can be expressed via a single parameter, her speed. In contrast to other similar problems, Archer and Tardos [4] showed that an algorithm that minimizes the makespan can be truthfully implemented, although in exponential time. On the other hand, if we leave out the game-theoretic issues, the complexity of the problem has been completely settled -the problem is strongly NP-hard, while there exists a PTAS [9,8].This problem is the most well studied in single-parameter algorithmic mechanism design. It gives an excellent ground to explore the boundary between truthfulness and efficient computation. Since the work of Archer and Tardos, quite a lot of deterministic and randomized mechanisms have been suggested. Recently, a breakthrough result [7] showed that a randomized truthful PTAS exists. On the other hand, for the deterministic case, the best known approximation factor is 2.8 [11,12].It has been a major open question whether there exists a deterministic truthful PTAS, or whether truthfulness has an essential, negative impact on the computational complexity of the problem. In this paper we give a definitive answer to this important question by providing a truthful deterministic PTAS.