# Combinatorial Inference on the Optimal Assortment in Multinomial Logit Models

arXiv: https://arxiv.org/abs/2301.12254 · HTML: https://mlsanity.com/paper/2301.12254/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 63 references that are in the map.
- First posted on arXiv 28 Jan 2023.

## Builds on (most cited first, 10 of 10)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [An Evaluation of Change Point Detection Algorithms](https://mlsanity.com/paper/2003.06222/) (2020)
- [MNL-Bandit: A Dynamic Learning Approach to Assortment Selection](https://mlsanity.com/paper/1706.03880/) (2017)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Thompson Sampling for the MNL-Bandit](https://mlsanity.com/paper/1706.00977/) (2017)
- [Uncertainty quantification in the Bradley-Terry-Luce model](https://mlsanity.com/paper/2110.03874/) (2021)
- [Dynamic Assortment Optimization with Changing Contextual Information](https://mlsanity.com/paper/1810.13069/) (2018)
- [Lagrangian Inference for Ranking Problems](https://mlsanity.com/paper/2110.00151/) (2021)
- [Robust Dynamic Assortment Optimization in the Presence of Outlier Customers](https://mlsanity.com/paper/1910.04183/) (2019)

## Built on by (most cited first, 1 of 1)

- [Doubly High-Dimensional Contextual Bandits: An Interpretable Model for Joint Assortment-Pricing](https://mlsanity.com/paper/2309.08634/) (2023)

## Abstract (arXiv)

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.