Loss Functions for Discrete Contextual Pricing with Observational Data

Abstract

We study a pricing setting where each customer is offered a contextualized price based on customer and/or product features that are predictive of the customer's valuation for that product. Often only historical sales records are available, where we observe whether each customer purchased a product at the price prescribed rather than the customer's true valuation. As such, the data is influenced by the historical sales policy which introduces difficulties in a) estimating future loss/regret for pricing policies without the possibility of conducting real experiments and b) optimizing new policies for downstream tasks such as revenue management. We study how to formulate loss functions which can be used for optimizing pricing policies directly, rather than going through an intermediate demand estimation stage, which can be biased in practice due to model misspecification, regularization or poor calibration. While existing approaches have been proposed when valuation data is available, we propose loss functions for the observational data setting. To achieve this, we adapt ideas from machine learning with corrupted labels, where we can consider each observed customer's outcome (purchased or not for a prescribed price), as a (known) probabilistic transformation of the customer's valuation. From this transformation we derive a class of suitable unbiased loss functions.Within this class we identify minimum variance estimators, those which are robust to poor demand function estimation, and provide guidance on when the estimated demand function is useful. Furthermore, we also show that when applied to our contextual pricing setting, estimators popular in the off-policy evaluation literature fall within this class of loss functions, and also offer managerial insights on when each estimator is likely to perform well in practice.

Enhancing Counterfactual Classification Performance via Self-Training

“…Counterfactual policy optimization and evaluation has received a lot of attention in the machine learning community in recent years (Swaminathan and Joachims 2015a;Joachims, Swaminathan, and de Rijke 2018;Shalit, Johansson, and Sontag 2017;Lopez et al 2020;Kallus 2019;Kallus and Zhou 2018;Wang et al 2019;Gao et al 2021;Biggs, Gao, and Sun 2021). Most of the proposed algorithms can be divided into two categories: counterfactual risk minimization (CRM) and direct method (DM).…”

Convex Loss Functions for Contextual Pricing with Observational Posted-Price Data

“…The surrogate functions they propose are non-convex, hence challenging to optimize. Similar to our setting, Biggs et al (2021a) propose pricing loss functions for the observational posted-price setting when prices are restricted to a discrete price ladder. Here we focus on continuous prices.…”

Enhancing Counterfactual Classification via Self-Training

“…Counterfactual policy optimization and evaluation has received a lot of attention in the machine learning community in recent years (Swaminathan and Joachims 2015a;Joachims, Swaminathan, and de Rijke 2018;Shalit, Johansson, and Sontag 2017;Lopez et al 2020;Kallus 2019;Kallus and Zhou 2018;Wang et al 2019;Gao et al 2021;Biggs, Gao, and Sun 2021). Most of the proposed algorithms can be divided into two categories: counterfactual risk minimization (CRM) and direct method (DM).…”