Consider a seller with m heterogeneous items for sale to a single additive buyer whose values for the items are arbitrarily correlated. It was previously shown that, in such settings, distributions exist for which the seller's optimal revenue is infinite, but the best "simple" mechanism achieves revenue at most one (Briest et al. [5], Hart and Nisan [22]), even when m = 2. This result has long served as a cautionary tale discouraging the study of multi-item auctions without some notion of "independent items".In this work we initiate a smoothed analysis of such multi-item auction settings. We consider a buyer whose item values are drawn from an arbitrarily correlated multi-dimensional distribution then randomly perturbed with magnitude δ under several natural perturbation models. On one hand, we prove that the [5,22] construction is surprisingly robust to certain natural perturbations of this form, and the infinite gap remains.On the other hand, we provide a smoothed model such that the approximation guarantee of simple mechanisms is smoothed-finite. We show that when the perturbation has magnitude δ, pricing only the grand bundle guarantees an O(1/δ)-approximation to the optimal revenue. That is, no matter the (worst-case) initially correlated distribution, these tiny perturbations suffice to bring the gap down from infinite to finite. We further show that the same guarantees hold when n buyers have values drawn from an arbitrarily correlated mn-dimensional distribution (without any dependence on n).Taken together, these analyses further pin down key properties of correlated distributions that result in large gaps between simplicity and optimality.

We consider a revenue-maximizing seller with n items facing a single buyer. We introduce the notion of symmetric menu complexity of a mechanism, which counts the number of distinct options the buyer may purchase, up to permutations of the items. Our main result is that a mechanism of quasipolynomial symmetric menu complexity suffices to guarantee a (1 − ε)-approximation when the buyer is unit-demand over independent items, even when the value distribution is unbounded, and that this mechanism can be found in quasi-polynomial time.Our key technical result is a poly-time, (symmetric) menu-complexity-preserving black-box reduction from achieving a (1 − ε)-approximation for unbounded valuations that are subadditive over independent items to achieving a (1 −O(ε))-approximation when the values are bounded (and still subadditive over independent items). We further apply this reduction to deduce approximation schemes for a suite of valuation classes beyond our main result.Finally, we show that selling separately (which has exponential menu complexity) can be approximated up to a (1 − ε) factor with a menu of efficient-linear (f (ε) · n) symmetric menu complexity.

We study the menu complexity of optimal and approximately-optimal auctions in the context of the "FedEx" problem, a so-called "one-and-a-halfdimensional" setting where a single bidder has both a value and a deadline for receiving an item [FGKK16]. The menu complexity of an auction is equal to the number of distinct (allocation, price) pairs that a bidder might receive [HN13]. We show the following when the bidder has n possible deadlines:• Exponential menu complexity is necessary to be exactly optimal: There exist instances where the optimal mechanism has menu complexity ≥ 2 n − 1. This matches exactly the upper bound provided by • Fully polynomial menu complexity is necessary and sufficient for approximation: For all instances, there exists a mechanism guaranteeing a multiplicative (1 − )-approximation to the optimal revenue with menu complexity O(n 3/2 min{n/ ,ln(v max )} ) = O(n 2 / ), where v max denotes the largest value in the support of integral distributions.• There exist instances where any mechanism guaranteeing a multiplicative (1 − O(1/n 2 ))-approximation to the optimal revenue requires menu complexity Ω(n 2 ).Our main technique is the polygon approximation of concave functions [Rot92], and our results here should be of independent interest. We further show how our techniques can be used to resolve an open question of [DW17] on the menu complexity of optimal auctions for a budget-constrained buyer.

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