The menu complexity of "one-and-a-half-dimensional" mechanism design

We study the menu complexity of optimal and approximately-optimal auctions in the context of the "FedEx" problem, a so-called "one-and-a-half-dimensional" setting where a single bidder has both a value and a deadline for receiving an [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 is $2^n-1$. This matches exactly the upper bound provided by Fiat et al.'s algorithm, and resolves one of their open questions [FGKK16]. - Fully polynomial menu complexity is necessary and sufficient for approximation: For all instances, there exists a mechanism guaranteeing a multiplicative (1-\epsilon)-approximation to the optimal revenue with menu complexity $O(n^{{3/2}\sqrt{\frac{\min{n/\epsilon,\ln(v_{\max})}}{\epsilon}})} = O(n^{2/\epsilon)$,} 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 $\Omega(n^2)$. Our main technique is the polygon approximation of concave functions [Rote19], 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.