The topic of dynamic pricing and learning has received a considerable amount of attention in recent years, from different scientific communities. We survey these literature streams: we provide a brief introduction to the historical origins of quantitative research on pricing and demand estimation, point to different subfields in the area of dynamic pricing, and provide an in-depth overview of the available literature on dynamic pricing and learning. Our focus is on the operations research and management science literature, but we also discuss relevant contributions from marketing, economics, econometrics, and computer science. We discuss relations with methodologically related research areas, and identify directions for future research.

Dynamic pricing and learning: Historical origins, current research, and new directions

We consider a stylized dynamic pricing model in which a monopolist prices a product to a sequence of T customers who independently make purchasing decisions based on the price offered according to a general parametric choice model. The parameters of the model are unknown to the seller, whose objective is to determine a pricing policy that minimizes the regret, which is the expected difference between the seller's revenue and the revenue of a clairvoyant seller who knows the values of the parameters in advance and always offers the revenue-maximizing price. We show that the regret of the optimal pricing policy in this model is $\Theta(\sqrt T)$ , by establishing an $\Omega(\sqrt T)$ lower bound on the worst-case regret under an arbitrary policy, and presenting a pricing policy based on maximum-likelihood estimation whose regret is $\cal{O}(\sqrt T)$ across all problem instances. Furthermore, we show that when the demand curves satisfy a “well-separated” condition, the T-period regret of the optimal policy is Θ(log T). Numerical experiments show that our policies perform well.

Dynamic Pricing Under a General Parametric Choice Model

https://ris.utwente.nl/ws/files/5090800/memo2025.pdf

Dynamic pricing and learning: historical origins, current research, and new directions

We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the "price of non-stationarity," which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one.

Non-stationary Stochastic Optimization

We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: (1) adversarial online convex optimization and (2) the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well-performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the “price of non-stationarity,” which ...

/pdf/non-stationary-stochastic-optimization-4bfqjy516k.pdf

Non-Stationary Stochastic Optimization

Motivated by applications in financial services, we consider a seller who offers prices sequentially to a stream of potential customers, observing either success or failure in each sales attempt. The parameters of the underlying demand model are initially unknown, so each price decision involves a trade-off between learning and earning. Attention is restricted to the simplest kind of model uncertainty, where one of two demand models is known to apply, and we focus initially on performance of the myopic Bayesian policy (MBP), variants of which are commonly used in practice. Because learning is passive under the MBP (that is, learning only takes place as a by-product of actions that have a different purpose), it can lead to incomplete learning and poor profit performance. However, under one additional assumption, a constrained variant of the myopic policy is shown to have the following strong theoretical virtue: the expected performance gap relative to a clairvoyant who knows the underlying demand model is bounded by a constant as the number of sales attempts becomes large.

This paper was accepted by Gerard P. Cachon, stochastic models and simulation.

https://www0.gsb.columbia.edu/faculty/azeevi/PAPERS/hkz-mbp.pdf

Bayesian Dynamic Pricing Policies: Learning and Earning Under a Binary Prior Distribution

We consider a monopolist who sells a set of products over a time horizon of T periods. The seller initially does not know the parameters of the products' linear demand curve, but can estimate them based on demand observations. We first assume that the seller knows nothing about the parameters of the demand curve, and then consider the case where the seller knows the expected demand under an incumbent price. It is shown that the smallest achievable revenue loss in T periods, relative to a clairvoyant who knows the underlying demand model, is of order T in the former case and of order log T in the latter case. To derive pricing policies that are practically implementable, we take as our point of departure the widely used policy called greedy iterated least squares ILS, which combines sequential estimation and myopic price optimization. It is known that the greedy ILS policy itself suffers from incomplete learning, but we show that certain variants of greedy ILS achieve the minimum asymptotic loss rate. To highlight the essential features of well-performing pricing policies, we derive sufficient conditions for asymptotic optimality.

Dynamic Pricing with an Unknown Demand Model: Asymptotically Optimal Semi-Myopic Policies

We consider a seller who can dynamically adjust the price of a product at the individual customer level, by utilizing information about customers’ characteristics encoded as a d-dimensional feature vector We assume a personalized demand model, parameters of which depend on s out of the d features The seller initially does not know the relationship between the customer features and the product demand, but learns this through sales observations over a selling horizon of T periods We prove that the seller’s expected regret, ie, the revenue loss against a clairvoyant who knows the underlying demand relationship, is at least of order s √T under any admissible policy We then design a near-optimal pricing policy for a “semi-clairvoyant” seller (who knows which s of the d features are in the demand model) that achieves an expected regret of order s √Tlog T We extend this policy to a more realistic setting where the seller does not know the true demand predictors, and show that this policy has an expected regret of order s √T(log d＋logT), which is also near-optimal Finally, we test our theory on simulated data and on a data set from an online auto loan company in the United States On both data sets, our experimentation-based pricing policy is superior to intuitive and/or widely-practiced customized pricing methods such as myopic pricing and segment-then- optimize policies Furthermore, our policy improves upon the loan company’s historical pricing decisions by 47% in expected revenue over a six-month period

Personalized Dynamic Pricing with Machine Learning: High Dimensional Features and Heterogeneous Elasticity

We consider a dynamic pricing problem in which a seller faces an unknown demand model that can change over time. The amount of change over a time horizon of T periods is measured using a variation metric that allows for a broad spectrum of temporal behavior. Given a finite variation “budget,” we first derive a lower bound on the expected performance gap between any pricing policy and a clairvoyant who knows a priori the temporal evolution of the underlying demand model, and then we design families of near-optimal pricing policies, the revenue performance of which asymptotically matches said lower bound. We also show that the seller can achieve a substantially better revenue performance in demand environments that change in “bursts” than in demand environments that change “smoothly,” among other things quantifying the net effect of the “volatility” in the demand environment on the seller’s revenue performance.

Chasing Demand: Learning and Earning in a Changing Environment

We consider a dynamic pricing problem in which a seller faces an unknown demand model that can change over time. We measure the amount of change over a time horizon of T periods using a quadratic variation metric, and allow a finite "budget" for such changes. We first derive a lower bound on the expected performance gap between any pricing policy and a clairvoyant who knows a priori the temporal evolution of the underlying demand model, and then design families of near-optimal pricing policies, the revenue performance of which asymptotically matches said lower bound. We also show that the seller can achieve a substantially better revenue performance in demand environments that change in "bursts" than it would in a demand environment that changes "smoothly." Finally, we extend our analysis to the case of rapidly changing demand settings, and obtain a range of results that quantify the net effect of the volatility in the demand environment on the seller’s revenue performance.

N. Bora Keskin

Papers

Bayesian Dynamic Pricing Policies: Learning and Earning Under a Binary Prior Distribution

Dynamic Pricing with an Unknown Demand Model: Asymptotically Optimal Semi-Myopic Policies

Personalized Dynamic Pricing with Machine Learning: High Dimensional Features and Heterogeneous Elasticity

Chasing Demand: Learning and Earning in a Changing Environment

Chasing Demand: Learning and Earning in a Changing Environment