# Markdown Pricing Under an Unknown Parametric Demand Model

arXiv: https://arxiv.org/abs/2312.15286 · HTML: https://mlsanity.com/paper/2312.15286/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 22 references that are in the map.
- First posted on arXiv 23 Dec 2023.

## Builds on (most cited first, 4 of 4)

- [Introduction to Multi-Armed Bandits](https://mlsanity.com/paper/1904.07272/) (2019)
- [Individual Fairness in Hindsight](https://mlsanity.com/paper/1812.04069/) (2018)
- [Thresholding Bandit for Dose-ranging: The Impact of Monotonicity](https://mlsanity.com/paper/1711.04454/) (2017)
- [Multi-armed Bandit Requiring Monotone Arm Sequences](https://mlsanity.com/paper/2106.03790/) (2021)

## Abstract (arXiv)

Consider a single-product revenue-maximization problem where the seller monotonically decreases the price in n rounds with an unknown demand model coming from a given family. Without monotonicity, the minimax regret is tilde O(n2/3) for the Lipschitz demand family and tilde O(n1/2) for a general class of parametric demand models. With monotonicity, the minimax regret is tilde O(n3/4) if the revenue function is Lipschitz and unimodal. However, the minimax regret for parametric families remained open. In this work, we provide a complete settlement for this fundamental problem. We introduce the crossing number to measure the complexity of a family of demand functions. In particular, the family of degree-k polynomials has a crossing number k. Based on conservatism under uncertainty, we present (i) a policy with an optimal Theta(log2 n) regret for families with crossing number k=0, and (ii) another policy with an optimal tilde Theta(nk/(k+1)) regret when k≥ 1. These bounds are asymptotically higher than the tilde O(log n) and tilde Theta(√ n) minimax regret for the same families without the monotonicity constraint.