# Loss Functions for Discrete Contextual Pricing with Observational Data

arXiv: https://arxiv.org/abs/2111.09933 · HTML: https://mlsanity.com/paper/2111.09933/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 129 references that are in the map.
- First posted on arXiv 18 Nov 2021.

## Builds on (most cited first, 30 of 35)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Meta-learners for Estimating Heterogeneous Treatment Effects using Machine Learning](https://mlsanity.com/paper/1706.03461/) (2017)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Learning Representations for Counterfactual Inference](https://mlsanity.com/paper/1605.03661/) (2016)
- [Doubly Robust Policy Evaluation and Optimization](https://mlsanity.com/paper/1503.02834/) (2015)
- [From Predictive to Prescriptive Analytics](https://mlsanity.com/paper/1402.5481/) (2014)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [Confounding-Robust Policy Improvement](https://mlsanity.com/paper/1805.08593/) (2018)
- [Offline Multi-Action Policy Learning: Generalization and Optimization](https://mlsanity.com/paper/1810.04778/) (2018)
- [Off-policy Bandits with Deficient Support](https://mlsanity.com/paper/2006.09438/) (2020)
- [Dynamic Pricing in High-dimensions](https://mlsanity.com/paper/1609.07574/) (2016)
- [Balanced Policy Evaluation and Learning](https://mlsanity.com/paper/1705.07384/) (2017)
- [Residual Weighted Learning for Estimating Individualized Treatment Rules](https://mlsanity.com/paper/1508.03179/) (2015)
- [Human-AI Collaboration with Bandit Feedback](https://mlsanity.com/paper/2105.10614/) (2021)
- [Dynamic Pricing with Demand Covariates](https://mlsanity.com/paper/1604.07463/) (2016)
- [Recursive Partitioning for Personalization using Observational Data](https://mlsanity.com/paper/1608.08925/) (2016)
- [Distributionally Robust Counterfactual Risk Minimization](https://mlsanity.com/paper/1906.06211/) (2019)
- [Model Distillation for Revenue Optimization: Interpretable Personalized Pricing](https://mlsanity.com/paper/2007.01903/) (2020)
- [More Efficient Policy Learning via Optimal Retargeting](https://mlsanity.com/paper/1906.08611/) (2019)
- [Revenue Optimization with Approximate Bid Predictions](https://mlsanity.com/paper/1706.04732/) (2017)
- [The Sample Complexity of Auctions with Side Information](https://mlsanity.com/paper/1511.02296/) (2015)
- [Fairness, Welfare, and Equity in Personalized Pricing](https://mlsanity.com/paper/2012.11066/) (2020)
- [Learning Theory and Algorithms for Revenue Optimization in Second-Price Auctions with Reserve](https://mlsanity.com/paper/1310.5665/) (2013)
- [Interpretable Optimal Stopping](https://mlsanity.com/paper/1812.07211/) (2018)
- [Counterfactual Learning from Bandit Feedback under Deterministic Logging: A Case Study in Statistical Machine Translation](https://mlsanity.com/paper/1707.09118/) (2017)
- [Balanced off-policy evaluation in general action spaces](https://mlsanity.com/paper/1906.03694/) (2019)
- [Contextual Reserve Price Optimization in Auctions via Mixed-Integer Programming](https://mlsanity.com/paper/2002.08841/) (2020)
- [Model-Free Assortment Pricing with Transaction Data](https://mlsanity.com/paper/2101.02251/) (2021)

## Built on by (most cited first, 6 of 6)

- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Enhancing Counterfactual Classification via Self-Training](https://mlsanity.com/paper/2112.04461/) (2021)
- [Balanced Off-Policy Evaluation for Personalized Pricing](https://mlsanity.com/paper/2302.12736/) (2023)
- [Confounding-Robust Policy Improvement with Human-AI Teams](https://mlsanity.com/paper/2310.08824/) (2023)
- [Convex Surrogate Loss Functions for Contextual Pricing with Transaction Data](https://mlsanity.com/paper/2202.10944/) (2022)
- [Learning Prescriptive ReLU Networks](https://mlsanity.com/paper/2306.00651/) (2023)

## Abstract (arXiv)

We study a pricing setting where each customer is offered a contextualized price based on customer and/or product features. Often only historical sales data are available, so we observe whether a customer purchased a product at the price prescribed rather than the customer's true valuation. Such observational data are influenced by historical pricing policies, which introduce difficulties in evaluating the effectiveness of future policies. The goal of this paper is to formulate loss functions that can be used for evaluating pricing policies directly from observational data, rather than going through an intermediate demand estimation stage, which may suffer from bias. To achieve this, we adapt ideas from machine learning with corrupted labels, where we consider each observed purchase decision as a known probabilistic transformation of the customer's valuation. From this transformation, we derive a class of unbiased loss functions. Within this class, we identify minimum variance estimators and estimators robust to poor demand estimation. Furthermore, we show that for contextual pricing, estimators popular in the off-policy evaluation literature fall within this class of loss functions. We offer managerial insights into scenarios under which these estimators are effective.