On the (in)-approximability of Bayesian Revenue Maximization for a Combinatorial Buyer
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We consider a revenue-maximizing single seller with m items for sale to a single buyer whose value v(·) for the items is drawn from a known distribution D of support k. A series of works by Cai et al. establishes that when each v(·) in the support of D is additive or unit-demand (or c-demand), the revenue-optimal auction can be found in poly(m,k) time. We show that going barely beyond this, even to matroid-based valuations (a proper subset of Gross Substitutes), results in strong hardness of approximation. Specifically, even on instances with m items and k ≤ m valuations in the support of D, it is not possible to achieve a 1/m^1-ε-approximation for any ε>0 to the revenue-optimal mechanism for matroid-based valuations in (randomized) poly-time unless NP ⊆ RP (note that a 1/k-approximation is trivial). Cai et al.'s main technical contribution is a black-box reduction from revenue maximization for valuations in class 𝒱 to optimizing the difference between two values in class 𝒱. Our main technical contribution is a black-box reduction in the other direction (for a wide class of valuation classes), establishing that their reduction is essentially tight.
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