Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue
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We study combinatorial auctions with interdependent valuations. In such settings, each agent i has a private signal s_i that captures her private information, and the valuation function of every agent depends on the entire signal profile, =(s_1,...,s_n). Previous results in economics concentrated on identifying (often stringent) assumptions under which optimal solutions can be attained. The computer science literature provided approximation results for simple single-parameter settings (mostly single item auctions, or matroid feasibility constraints). Both bodies of literature considered only valuations satisfying a technical condition termed single crossing (or variants thereof). Indeed, without imposing assumptions on the valuations, strong impossibility results exist. We consider the class of submodular over signals (SOS) valuations (without imposing any single-crossing type assumption), and provide the first welfare approximation guarantees for multi-dimensional combinatorial auctions, achieved by universally ex-post IC-IR mechanisms. Our main results are: (i) 4-approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; (ii) 4-approximation for any combinatorial auction with multi-dimensional signals and separable-SOS valuations; and (iii) (k+3)- and (2(k)+4)-approximation for any combinatorial auction with single-dimensional signals, with k-sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extend to a parameterized version of SOS, d-SOS, while losing a factor that depends on d.
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