Bounds on the Menu-Size of Approximately Optimal Auctions via Optimal-Transport Duality

The question of the minimum menu-size for approximate (i.e., up-to-ε) Bayesian revenue maximization when selling two goods to an additive risk-neutral quasilinear buyer was introduced by Hart and Nisan (2013), who give an upper bound of O(1/ε4) for this problem. Using the optimal-transport duality framework of Daskalakis et al. (2013, 2015), we derive the first lower bound for this problem — of Ω(1/ 4 √ ε), even when the values for the two goods are drawn i.i.d. from a bounded-support distribution having a polynomial for a density function, establishing how to reason about approximately optimal mechanisms via this duality framework. For distributions having a density functions with bounded partial derivatives, even when the values for the two goods are correlated, we strengthen the upper bound of Hart and Nisan (2013) to O(1/ε2). Our lower bound implies a tight bound of Θ(log 1/ε) on the minimum deterministic communication complexity guaranteed to suffice for running some approximately revenue-maximizing mechanism for any fixed number of goods, thereby completely resolving this problem.