New Guarantees for Learning Revenue Maximizing Menus of Lotteries and Two-Part Tariffs
Maria-Florina Balcan, Hedyeh Beyhaghi
TL;DR
The paper investigates the learnability of two revenue-maximizing mechanism families—menus of two-part tariffs and menus of lotteries—under online (adversarial and smoothed) and distributional settings. It develops a data-independent discretization framework that reduces each infinite-parameter problem to a finite set of expert mechanisms, enabling strong no-regret online learning guarantees and improved distributional learning runtimes. For two-part tariffs, the authors establish dispersion under smooth distributions and, in the limited-buyer-type setting, reductions to online linear optimization via barycentric spanners, yielding efficient, no-regret algorithms; for lotteries, they show first online learning guarantees with discretization yet provide evidence of dispersion failure in general. The work highlights both the power and limits of data-driven approaches in mechanism design, showing discretization can be effective where dispersion fails and clarifying when each paradigm is advantageous, with implications for pricing strategies in real-world, multi-item or multi-unit selling contexts. Overall, the results advance data-driven mechanism design by offering tractable online and distributional learning methods for sophisticated menu-based pricing schemes, and by revealing nuanced structural properties that govern learnability.
Abstract
We advance a recently flourishing line of work at the intersection of learning theory and computational economics by studying the learnability of two classes of mechanisms prominent in economics, namely menus of lotteries and two-part tariffs. The former is a family of randomized mechanisms designed for selling multiple items, known to achieve revenue beyond deterministic mechanisms, while the latter is designed for selling multiple units (copies) of a single item with applications in real-world scenarios such as car or bike-sharing services. We focus on learning high-revenue mechanisms of this form from buyer valuation data in both distributional settings, where we have access to buyers' valuation samples up-front, and the more challenging and less-studied online settings, where buyers arrive one-at-a-time and no distributional assumption is made about their values. We provide a suite of results with regard to these two families of mechanisms. We provide the first online learning algorithms for menus of lotteries and two-part tariffs with strong regret-bound guarantees. Since the space of parameters is infinite and the revenue functions have discontinuities, the known techniques do not readily apply. However, we are able to provide a reduction to online learning over a finite number of experts, in our case, a finite number of parameters. Furthermore, in the limited buyers type case, we show a reduction to online linear optimization, which allows us to obtain no-regret guarantees by presenting buyers with menus that correspond to a barycentric spanner. In addition, we provide algorithms with improved running times over prior work for the distributional settings. Finally, we demonstrate how techniques from the recent literature in data-driven algorithm design are insufficient for our studied problems.