Abstract

We give a general technique to obtain approximation mechanisms that are truthful in expectation. We show that for packing domains, any α-approximation algorithm that also bounds the integrality gap of the LP relaxation of the problem by α can be used to construct an α-approximation mechanism that is truthful in expectation. This immediately yields a variety of new and significantly improved results for various problem domains and furthermore, yields truthful (in expectation) mechanisms with guarantees that match the best known approximation guarantees when truthfulness is not required. In particular, we obtain the first truthful mechanisms with approximation guarantees for a variety of multiparameter domains. We obtain truthful (in expectation) mechanisms achieving approximation guarantees of O ( √ m) for combinatorial auctions (CAs), (1 + ɛ) for multi-unit CAs with B = Ω(log m) copies of each item, and 2 for multi-parameter knapsack problems (multi-unit auctions). Our construction is based on considering an LP relaxation of the problem and using the classic VCG [33, 9, 18] mechanism to obtain a truthful mechanism in this fractional domain. We argue that the (fractional) optimal solution scaled down by α, where α is the integrality gap of the problem, can be represented as a convex combination of integer solutions, and by viewing this convex combination as specifying a probability distribution over integer solutions, we get a randomized, truthful in expectation mechanism. Our construction can be seen as a way of exploiting VCG in a computational tractable way even when the underlying social-welfare maximization problem is NP-hard. 1

Citations