# New Guarantees for Learning Revenue Maximizing Menus of Lotteries and Two-Part Tariffs

## TL;DR

This paper develops new online learning algorithms with strong regret guarantees for revenue-maximizing menus of lotteries and two-part tariffs, addressing challenges posed by infinite parameter spaces and discontinuous revenue functions.

## Contribution

It introduces the first online learning algorithms with regret bounds for these mechanisms, including reductions to finite experts and online linear optimization, and improves running times in distributional settings.

## Key findings

- First online algorithms with regret guarantees for these mechanisms.
- Reductions to finite experts and online linear optimization.
- Improved running times over prior work.

## 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.