Dynamic Pricing with Machine Learning: A Practical Guide to Pricing Models, Demand Forecasting, and Revenue Optimization

ScienceDirect

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

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.

Taylor & Francis Online

Artificial intelligence and dynamic pricing: a systematic literature review

With dynamic pricing becoming more widespread across various industries, artificial intelligence has made it even more sophisticated and widespread. The authors conducted a systematic literature review and analyzed a dataset of 95 peer-reviewed articles from international journals selected in Web of Science and Scopus to better understand artificial intelligence’s impact on dynamic pricing. The authors identified four clusters related to financial modeling, market dynamics, commodity markets, and behavior and decision-making. They also found that China has overtaken the USA in the number of published articles. They identified the themes of market simulation investment, crude oil commodity dependence, and behavior traders’ prices. A systematic literature review is essential to understand the impact of artificial intelligence on dynamic pricing and its implications for businesses, consumers, and society.

INFORMS.org PubsO

Analytics for an Online Retailer: Demand Forecasting and Price Optimization

We present our work with an online retailer, Rue La La, as an example of how a retailer can use its wealth of data to optimize pricing decisions on a daily basis. Rue La La is in the online fashion sample sales industry, where they offer extremely limited-time discounts on designer apparel and accessories. One of the retailer’s main challenges is pricing and predicting demand for products that it has never sold before, which account for the majority of sales and revenue. To tackle this challenge, we use machine learning techniques to estimate historical lost sales and predict future demand of new products. The nonparametric structure of our demand prediction model, along with the dependence of a product’s demand on the price of competing products, pose new challenges on translating the demand forecasts into a pricing policy. We develop an algorithm to efficiently solve the subsequent multiproduct price optimization that incorporates reference price effects, and we create and implement this algorithm into a pricing decision support tool for Rue La La’s daily use. We conduct a field experiment and find that sales does not decrease because of implementing tool recommended price increases for medium and high price point products. Finally, we estimate an increase in revenue of the test group by approximately 9.7% with an associated 90% confidence interval of [2.3%, 17.8%].

Elsevier SSRN

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

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, i.e., 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√T log 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＋log T), 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.

Science Direct

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

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.

informs PubsOnline

Dynamic Pricing Without Knowing the Demand Function: Risk Bounds and Near-Optimal Algorithms

We consider a single-product revenue management problem where, given an initial inventory, the objective is to dynamically adjust prices over a finite sales horizon to maximize expected revenues. Realized demand is observed over time, but the underlying functional relationship between price and mean demand rate that governs these observations (otherwise known as the demand function or demand curve) is not known. We consider two instances of this problem: (i) a setting where the demand function is assumed to belong to a known parametric family with unknown parameter values; and (ii) a setting where the demand function is assumed to belong to a broad class of functions that need not admit any parametric representation. In each case we develop policies that learn the demand function “on the fly,” and optimize prices based on that. The performance of these algorithms is measured in terms of the regret: the revenue loss relative to the maximal revenues that can be extracted when the demand function is known prior to the start of the selling season. We derive lower bounds on the regret that hold for any admissible pricing policy, and then show that our proposed algorithms achieve a regret that is “close” to this lower bound. The magnitude of the regret can be interpreted as the economic value of prior knowledge on the demand function, manifested as the revenue loss due to model uncertainty.

Journal of Computer Science

DYNAMIC PRICING WITH UPDATED DEMAND FOR THE SPORTS AND ENTERTAINMENT TICKET INDUSTRY

Revenue Management (RM) helped increase profitability for many travel industries. Selling perishable products with a fixed event date, the Sports and Entertainment (S&E) ticket industry can potentially benefit from RM ideas but has received less attention in the literature. In this study we develop dynamic pricing models for stochastic S&E demand in a discrete finite time setting, where demand depends not only on ticket prices but also on remaining times until the show dates. We assume the show popularity is uncertain to the seller, but this information can be learned via Bayesian updates as early sales are revealed. We present stochastic dynamic programs for Sports and Entertainment tickets pricing decisions. We test the models using real data obtained from a major performance venue in the U.S. to understand properties of the model solutions and performance under different scenarios. Our results show that demand learning is most beneficial when the initial estimates are incorrect. In addition, we found it is less necessary for the seller to vary price every period if demand variation is low and/or a large amount of demand arrives close to the show dates. Overall, we found that the benefits from having flexibility of price changes and demand learning can complement each other to achieve as much as 8.15% revenue increase on average, as compared to static pricing.

Manufacturing & Service Operations Management

Analytics for an Online Retailer: Demand Forecasting and Price Optimization

We present our work with an online retailer, Rue La La, as an example of how a retailer can use its wealth of data to optimize pricing decisions on a daily basis. Rue La La is in the online fashion sample sales industry, where they offer extremely limited-time discounts on designer apparel and accessories. One of the retailer’s main challenges is pricing and predicting demand for products that it has never sold before, which account for the majority of sales and revenue. To tackle this challenge, we use machine learning techniques to estimate historical lost sales and predict future demand of new products. The nonparametric structure of our demand prediction model, along with the dependence of a product’s demand on the price of competing products, pose new challenges on translating the demand forecasts into a pricing policy. We develop an algorithm to efficiently solve the subsequent multiproduct price optimization that incorporates reference price effects, and we create and implement this algorithm into a pricing decision support tool for Rue La La’s daily use. We conduct a field experiment and find that sales does not decrease because of implementing tool recommended price increases for medium and high price point products. Finally, we estimate an increase in revenue of the test group by approximately 9.7% with an associated 90% confidence interval of [2.3%, 17.8%].

The American Statistician

Forecasting at Scale

Forecasting is a common data science task that helps organizations with capacity planning, goal setting, and anomaly detection. Despite its importance, there are serious challenges associated with producing reliable and high-quality forecasts—especially when there are a variety of time series and analysts with expertise in time series modeling are relatively rare. To address these challenges, we describe a practical approach to forecasting “at scale” that combines configurable models with analyst-in-the-loop performance analysis. We propose a modular regression model with interpretable parameters that can be intuitively adjusted by analysts with domain knowledge about the time series. We describe performance analyses to compare and evaluate forecasting procedures, and automatically flag forecasts for manual review and adjustment. Tools that help analysts to use their expertise most effectively enable reliable, practical forecasting of business time series.

International Journal of Forecasting

Temporal Fusion Transformers for interpretable multi-horizon time series forecasting

Multi-horizon forecasting often contains a complex mix of inputs – including static (i.e. time-invariant) covariates, known future inputs, and other exogenous time series that are only observed in the past – without any prior information on how they interact with the target. Several deep learning methods have been proposed, but they are typically ‘black-box’ models that do not shed light on how they use the full range of inputs present in practical scenarios. In this paper, we introduce the Temporal Fusion Transformer (TFT) – a novel attention-based architecture that combines high-performance multi-horizon forecasting with interpretable insights into temporal dynamics. To learn temporal relationships at different scales, TFT uses recurrent layers for local processing and interpretable self-attention layers for long-term dependencies. TFT utilizes specialized components to select relevant features and a series of gating layers to suppress unnecessary components, enabling high performance in a wide range of scenarios. On a variety of real-world datasets, we demonstrate significant performance improvements over existing benchmarks, and highlight three practical interpretability use cases of TFT.

Operations Research

Dynamic Pricing Under a General Parametric Choice Model

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 [Formula: see text], by establishing an [Formula: see text] lower bound on the worst-case regret under an arbitrary policy, and presenting a pricing policy based on maximum-likelihood estimation whose regret is [Formula: see text] 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.

papers.ssrn.com

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

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, i.e., 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√T log 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＋log T), 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.

PMLR

Pricing with Contextual Elasticity and Heteroscedastic Valuation

We study an online contextual dynamic pricing problem, where customers decide whether to purchase a product based on its features and price. We introduce a novel approach to modeling a customer’s e…

arXiv.org

Dynamic Pricing and Learning with Long-term Reference Effects

We consider a dynamic pricing problem where customer response to the current price is impacted by the customer price expectation, aka reference price. We study a simple and novel reference price mechanism where reference price is the average of the past prices offered by the seller. As opposed to the more commonly studied exponential smoothing mechanism, in our reference price mechanism the prices offered by seller have a longer term effect on the future customer expectations. We show that under this mechanism, a markdown policy is near-optimal irrespective of the parameters of the model. This matches the common intuition that a seller may be better off by starting with a higher price and then decreasing it, as the customers feel like they are getting bargains on items that are ordinarily more expensive. For linear demand models, we also provide a detailed characterization of the near-optimal markdown policy along with an efficient way of computing it. We then consider a more challenging dynamic pricing and learning problem, where the demand model parameters are apriori unknown, and the seller needs to learn them online from the customers’ responses to the offered prices while simultaneously optimizing revenue. The objective is to minimize regret, i.e., the $T$-round revenue loss compared to a clairvoyant optimal policy. This task essentially amounts to learning a non-stationary optimal policy in a time-variant Markov Decision Process (MDP). For linear demand models, we provide an efficient learning algorithm with an optimal $\tilde{O}(\sqrt{T})$ regret upper bound.

papers.ssrn.com

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

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, i.e., 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√T log 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＋log T), 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.

arXiv.org

Dynamic Pricing in High-dimensions

We study the pricing problem faced by a firm that sells a large number of products, described via a wide range of features, to customers that arrive over time. Customers independently make purchasing decisions according to a general choice model that includes products features and customers’ characteristics, encoded as $d$-dimensional numerical vectors, as well as the price offered. The parameters of the choice model are a priori unknown to the firm, but can be learned as the (binary-valued) sales data accrues over time. The firm’s objective is to minimize the regret, i.e., the expected revenue loss against a clairvoyant policy that knows the parameters of the choice model in advance, and always offers the revenue-maximizing price. This setting is motivated in part by the prevalence of online marketplaces that allow for real-time pricing. We assume a structured choice model, parameters of which depend on $s_0$ out of the $d$ product features. We propose a dynamic policy, called Regularized Maximum Likelihood Pricing (RMLP) that leverages the (sparsity) structure of the high-dimensional model and obtains a logarithmic regret in $T$. More specifically, the regret of our algorithm is of $O(s_0 \log d \cdot \log T)$. Furthermore, we show that no policy can obtain regret better than $O(s_0 (\log d + \log T))$.

informs PubsOnline

Utility Fairness in Contextual Dynamic Pricing with Demand Learning

This paper introduces a novel contextual bandit algorithm for personalized pricing under utility fairness constraints in scenarios with uncertain demand, achieving an optimal regret upper bound. Our approach, which incorporates dynamic pricing and demand learning, addresses the critical challenge of fairness in pricing strategies. We first delve into the static full-information setting to formulate an optimal pricing policy as a constrained optimization problem. Here, we propose an approximation algorithm for efficiently and approximately computing the ideal policy. We also use mathematical analysis and computational studies to characterize the structures of optimal contextual pricing policies subject to fairness constraints, deriving simplified policies that lay the foundations of more in-depth research and extensions. Further, we extend our study to dynamic pricing problems with demand learning, establishing a nonstandard regret lower bound that highlights the complexity added by fairness constraints. Our research offers a comprehensive analysis of the cost of fairness and its impact on the balance between utility and revenue maximization. This work represents a step toward integrating ethical considerations into algorithmic efficiency in data-driven dynamic pricing.

Journal of Revenue and Pricing Management

Dynamic pricing and learning with competition: insights from the dynamic pricing challenge at the 2017 INFORMS RM & pricing conference

This paper presents the results of the Dynamic Pricing Challenge, held on the occasion of the 17th INFORMS Revenue Management and Pricing Section Conference on June 29–30, 2017 in Amsterdam, The Netherlands. For this challenge, participants submitted algorithms for pricing and demand learning of which the numerical performance was analyzed in simulated market environments. This allows consideration of market dynamics that are not analytically tractable or can not be empirically analyzed due to practical complications. Our findings implicate that the relative performance of algorithms varies substantially across different market dynamics, which confirms the intrinsic complexity of pricing and learning in the presence of competition.

Journal of Revenue and Pricing Management

Dynamic pricing under competition using reinforcement learning

AbstractDynamic pricing is considered a possibility to gain an advantage over competitors in modern online markets. The past advancements in Reinforcement Learning (RL) provided more capable algorithms that can be used to solve pricing problems. In this paper, we study the performance of Deep Q-Networks (DQN) and Soft Actor Critic (SAC) in different market models. We consider tractable duopoly settings, where optimal solutions derived by dynamic programming techniques can be used for verification, as well as oligopoly settings, which are usually intractable due to the curse of dimensionality. We find that both algorithms provide reasonable results, while SAC performs better than DQN. Moreover, we show that under certain conditions, RL algorithms can be forced into collusion by their competitors without direct communication.

American Economic Review

Artificial Intelligence, Algorithmic Pricing, and Collusion

Increasingly, algorithms are supplanting human decision-makers in pricing goods and services. To analyze the possible consequences, we study experimentally the behavior of algorithms powered by Artificial Intelligence (Q-learning) in a workhorse oligopoly model of repeated price competition. We find that the algorithms consistently learn to charge supracompetitive prices, without communicating with one another. The high prices are sustained by collusive strategies with a finite phase of punishment followed by a gradual return to cooperation. This finding is robust to asymmetries in cost or demand, changes in the number of players, and various forms of uncertainty. (JEL D21, D43, D83, L12, L13)

informs PubsOnline

Dynamic Pricing Without Knowing the Demand Function: Risk Bounds and Near-Optimal Algorithms

We consider a single-product revenue management problem where, given an initial inventory, the objective is to dynamically adjust prices over a finite sales horizon to maximize expected revenues. Realized demand is observed over time, but the underlying functional relationship between price and mean demand rate that governs these observations (otherwise known as the demand function or demand curve) is not known. We consider two instances of this problem: (i) a setting where the demand function is assumed to belong to a known parametric family with unknown parameter values; and (ii) a setting where the demand function is assumed to belong to a broad class of functions that need not admit any parametric representation. In each case we develop policies that learn the demand function “on the fly,” and optimize prices based on that. The performance of these algorithms is measured in terms of the regret: the revenue loss relative to the maximal revenues that can be extracted when the demand function is known prior to the start of the selling season. We derive lower bounds on the regret that hold for any admissible pricing policy, and then show that our proposed algorithms achieve a regret that is “close” to this lower bound. The magnitude of the regret can be interpreted as the economic value of prior knowledge on the demand function, manifested as the revenue loss due to model uncertainty.

Journal of Computer Science

DYNAMIC PRICING WITH UPDATED DEMAND FOR THE SPORTS AND ENTERTAINMENT TICKET INDUSTRY

Revenue Management (RM) helped increase profitability for many travel industries. Selling perishable products with a fixed event date, the Sports and Entertainment (S&E) ticket industry can potentially benefit from RM ideas but has received less attention in the literature. In this study we develop dynamic pricing models for stochastic S&E demand in a discrete finite time setting, where demand depends not only on ticket prices but also on remaining times until the show dates. We assume the show popularity is uncertain to the seller, but this information can be learned via Bayesian updates as early sales are revealed. We present stochastic dynamic programs for Sports and Entertainment tickets pricing decisions. We test the models using real data obtained from a major performance venue in the U.S. to understand properties of the model solutions and performance under different scenarios. Our results show that demand learning is most beneficial when the initial estimates are incorrect. In addition, we found it is less necessary for the seller to vary price every period if demand variation is low and/or a large amount of demand arrives close to the show dates. Overall, we found that the benefits from having flexibility of price changes and demand learning can complement each other to achieve as much as 8.15% revenue increase on average, as compared to static pricing.

arXiv.org

Dynamic Pricing and Learning with Long-term Reference Effects

We consider a dynamic pricing problem where customer response to the current price is impacted by the customer price expectation, aka reference price. We study a simple and novel reference price mechanism where reference price is the average of the past prices offered by the seller. As opposed to the more commonly studied exponential smoothing mechanism, in our reference price mechanism the prices offered by seller have a longer term effect on the future customer expectations. We show that under this mechanism, a markdown policy is near-optimal irrespective of the parameters of the model. This matches the common intuition that a seller may be better off by starting with a higher price and then decreasing it, as the customers feel like they are getting bargains on items that are ordinarily more expensive. For linear demand models, we also provide a detailed characterization of the near-optimal markdown policy along with an efficient way of computing it. We then consider a more challenging dynamic pricing and learning problem, where the demand model parameters are apriori unknown, and the seller needs to learn them online from the customers’ responses to the offered prices while simultaneously optimizing revenue. The objective is to minimize regret, i.e., the $T$-round revenue loss compared to a clairvoyant optimal policy. This task essentially amounts to learning a non-stationary optimal policy in a time-variant Markov Decision Process (MDP). For linear demand models, we provide an efficient learning algorithm with an optimal $\tilde{O}(\sqrt{T})$ regret upper bound.

arXiv.org

Dynamic pricing with Bayesian updates from online reviews

When launching new products, firms face uncertainty about market reception. Online reviews provide valuable information not only to consumers but also to firms, allowing firms to adjust the product characteristics, including its selling price. In this paper, we consider a pricing model with online reviews in which the quality of the product is uncertain, and both the seller and the buyers Bayesianly update their beliefs to make purchasing & pricing decisions. We model the seller’s pricing problem as a basic bandits’ problem and show a close connection with the celebrated Catalan numbers, allowing us to efficiently compute the overall future discounted reward of the seller. With this tool, we analyze and compare the optimal static and dynamic pricing strategies in terms of the probability of effectively learning the quality of the product.

Operations Research

Dynamic Pricing Under a General Parametric Choice Model

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 [Formula: see text], by establishing an [Formula: see text] lower bound on the worst-case regret under an arbitrary policy, and presenting a pricing policy based on maximum-likelihood estimation whose regret is [Formula: see text] 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.

informs PubsOnline

Utility Fairness in Contextual Dynamic Pricing with Demand Learning

This paper introduces a novel contextual bandit algorithm for personalized pricing under utility fairness constraints in scenarios with uncertain demand, achieving an optimal regret upper bound. Our approach, which incorporates dynamic pricing and demand learning, addresses the critical challenge of fairness in pricing strategies. We first delve into the static full-information setting to formulate an optimal pricing policy as a constrained optimization problem. Here, we propose an approximation algorithm for efficiently and approximately computing the ideal policy. We also use mathematical analysis and computational studies to characterize the structures of optimal contextual pricing policies subject to fairness constraints, deriving simplified policies that lay the foundations of more in-depth research and extensions. Further, we extend our study to dynamic pricing problems with demand learning, establishing a nonstandard regret lower bound that highlights the complexity added by fairness constraints. Our research offers a comprehensive analysis of the cost of fairness and its impact on the balance between utility and revenue maximization. This work represents a step toward integrating ethical considerations into algorithmic efficiency in data-driven dynamic pricing.

arXiv.org

Contextual Dynamic Pricing: Algorithms, Optimality, and Local Differential Privacy Constraints

We study contextual dynamic pricing problems where a firm sells products to $T$ sequentially-arriving consumers, behaving according to an unknown demand model. The firm aims to minimize its regret over a clairvoyant that knows the model in advance. The demand follows a generalized linear model (GLM), allowing for stochastic feature vectors in $\mathbb R^d$ encoding product and consumer information. We first show the optimal regret is of order $\sqrt{dT}$, up to logarithmic factors, improving existing upper bounds by a $\sqrt{d}$ factor. This optimal rate is materialized by two algorithms: a confidence bound-type algorithm and an explore-then-commit (ETC) algorithm. A key insight is an intrinsic connection between dynamic pricing and contextual multi-armed bandit problems with many arms with a careful discretization. We further study contextual dynamic pricing under local differential privacy (LDP) constraints. We propose a stochastic gradient descent-based ETC algorithm achieving regret upper bounds of order $d\sqrt{T}/ε$, up to logarithmic factors, where $ε>0$ is the privacy parameter. The upper bounds with and without LDP constraints are matched by newly constructed minimax lower bounds, characterizing costs of privacy. Moreover, we extend our study to dynamic pricing under mixed privacy constraints, improving the privacy-utility tradeoff by leveraging public data. This is the first time such setting is studied in the dynamic pricing literature and our theoretical results seamlessly bridge dynamic pricing with and without LDP. Extensive numerical experiments and real data applications are conducted to illustrate the efficiency and practical value of our algorithms.

Data Mining and Knowledge Discovery

Controlled experiments on the web: survey and practical guide

The web provides an unprecedented opportunity to evaluate ideas quickly using controlled experiments, also called randomized experiments, A/B tests (and their generalizations), split tests, Control/Treatment tests, MultiVariable Tests (MVT) and parallel flights. Controlled experiments embody the best scientific design for establishing a causal relationship between changes and their influence on user-observable behavior. We provide a practical guide to conducting online experiments, where end-users can help guide the development of features. Our experience indicates that significant learning and return-on-investment (ROI) are seen when development teams listen to their customers, not to the Highest Paid Person’s Opinion (HiPPO). We provide several examples of controlled experiments with surprising results. We review the important ingredients of running controlled experiments, and discuss their limitations (both technical and organizational). We focus on several areas that are critical to experimentation, including statistical power, sample size, and techniques for variance reduction. We describe common architectures for experimentation systems and analyze their advantages and disadvantages. We evaluate randomization and hashing techniques, which we show are not as simple in practice as is often assumed. Controlled experiments typically generate large amounts of data, which can be analyzed using data mining techniques to gain deeper understanding of the factors influencing the outcome of interest, leading to new hypotheses and creating a virtuous cycle of improvements. Organizations that embrace controlled experiments with clear evaluation criteria can evolve their systems with automated optimizations and real-time analyses. Based on our extensive practical experience with multiple systems and organizations, we share key lessons that will help practitioners in running trustworthy controlled experiments.

arXiv.org

Dynamic Pricing on E-commerce Platform with Deep Reinforcement Learning: A Field Experiment

In this paper we present an end-to-end framework for addressing the problem of dynamic pricing (DP) on E-commerce platform using methods based on deep reinforcement learning (DRL). By using four groups of different business data to represent the states of each time period, we model the dynamic pricing problem as a Markov Decision Process (MDP). Compared with the state-of-the-art DRL-based dynamic pricing algorithms, our approaches make the following three contributions. First, we extend the discrete set problem to the continuous price set. Second, instead of using revenue as the reward function directly, we define a new function named difference of revenue conversion rates (DRCR). Third, the cold-start problem of MDP is tackled by pre-training and evaluation using some carefully chosen historical sales data. Our approaches are evaluated by both offline evaluation method using real dataset of Alibaba Inc., and online field experiments starting from July 2018 with thousands of items, lasting for months on Tmall.com. To our knowledge, there is no other DP field experiment using DRL before. Field experiment results suggest that DRCR is a more appropriate reward function than revenue, which is widely used by current literature. Also, continuous price sets have better performance than discrete sets and our approaches significantly outperformed the manual pricing by operation experts.

Journal of Revenue and Pricing Management

Dynamic pricing under competition using reinforcement learning

AbstractDynamic pricing is considered a possibility to gain an advantage over competitors in modern online markets. The past advancements in Reinforcement Learning (RL) provided more capable algorithms that can be used to solve pricing problems. In this paper, we study the performance of Deep Q-Networks (DQN) and Soft Actor Critic (SAC) in different market models. We consider tractable duopoly settings, where optimal solutions derived by dynamic programming techniques can be used for verification, as well as oligopoly settings, which are usually intractable due to the curse of dimensionality. We find that both algorithms provide reasonable results, while SAC performs better than DQN. Moreover, we show that under certain conditions, RL algorithms can be forced into collusion by their competitors without direct communication.

American Economic Review

Artificial Intelligence, Algorithmic Pricing, and Collusion

Increasingly, algorithms are supplanting human decision-makers in pricing goods and services. To analyze the possible consequences, we study experimentally the behavior of algorithms powered by Artificial Intelligence (Q-learning) in a workhorse oligopoly model of repeated price competition. We find that the algorithms consistently learn to charge supracompetitive prices, without communicating with one another. The high prices are sustained by collusive strategies with a finite phase of punishment followed by a gradual return to cooperation. This finding is robust to asymmetries in cost or demand, changes in the number of players, and various forms of uncertainty. (JEL D21, D43, D83, L12, L13)

arXiv.org

Algorithmic Collusion in Dynamic Pricing with Deep Reinforcement Learning

Nowadays, a significant share of the Business-to-Consumer sector is based on online platforms like Amazon and Alibaba and uses Artificial Intelligence for pricing strategies. This has sparked debate on whether pricing algorithms may tacitly collude to set supra-competitive prices without being explicitly designed to do so. Our study addresses these concerns by examining the risk of collusion when Reinforcement Learning algorithms are used to decide on pricing strategies in competitive markets. Prior research in this field focused on Tabular Q-learning (TQL) and led to opposing views on whether learning-based algorithms can lead to supra-competitive prices. Our work contributes to this ongoing discussion by providing a more nuanced numerical study that goes beyond TQL by additionally capturing off- and on-policy Deep Reinforcement Learning (DRL) algorithms. We study multiple Bertrand oligopoly variants and show that algorithmic collusion depends on the algorithm used. In our experiments, TQL exhibits higher collusion and price dispersion phenomena compared to DRL algorithms. We show that the severity of collusion depends not only on the algorithm used but also on the characteristics of the market environment. We further find that Proximal Policy Optimization appears to be less sensitive to collusive outcomes compared to other state-of-the-art DRL algorithms.

papers.ssrn.com

Artificial Intelligence, Communication, and Strategic Pricing: Evidence from Large Language Models

We analyze the behavior of large language model (LLM) agents as strategic actors in a price-setting environment. In an experimental market based on a repeated Bertrand pricing game, we find that LLM agents generate coordinated pricing outcomes. Allowing agents to communicate increases prices and profits while reducing variation in both outcomes. Moreover, communication increases the likelihood that agents coordinate on identical prices. Analyzing communication patterns shows that more frequent, longer, and more explicit exchanges reinforce these outcomes. A detailed analysis of agents' decision rationales suggests that communication affects pricing less through explicit game-theoretic reasoning than through reinforcement by cooperative language and pattern completion. Further tests involving different LLM temperature settings, varying the number of interacting agents, and various LLM architectures show that the results hold across these conditions. This consistency can be attributed mainly to language adjustments across different conditions. Overall, these findings highlight how language-based artificial intelligence systems can generate coordinated market outcomes even without specialized training in pricing or strategic interaction.

ojs.aaai.org

Pricing Online LLM Services with Data-Calibrated Stackelberg Routing Game

The proliferation of Large Language Models (LLMs) has established LLM routing as a standard service delivery mechanism, where users select models based on cost, Quality of Service (QoS), among other things. However, optimal pricing in LLM routing platforms requires precise modeling for dynamic service markets, and solving this problem in real time at scale is computationally intractable. In this paper, we propose PriLLM, a novel practical and scalable solution for real-time dynamic pricing in competitive LLM routing. PriLLM models the service market as a Stackelberg game, where providers set prices and users select services based on multiple criteria. To capture real-world market dynamics, we incorporate both objective factors (eg cost, QoS) and subjective user preferences into the model. For scalability, we employ a deep aggregation network to learn provider abstraction that preserve user-side equilibrium behavior across pricing strategies. Moreover, PriLLM offers interpretability by explaining its pricing decisions. Empirical evaluation on real-world data shows that PriLLM achieves over 95% of the optimal profit while only requiring less than 5% of the optimal solution's computation time.

arXiv.org

Menu Pricing of Large Language Models

We develop a framework for the optimal pricing and product design of LLMs in which a provider sells menus of token budgets to users who differ in their valuations across a continuum of tasks. Under a homogeneous production technology, we show that users’ high-dimensional type profiles are summarized by a scalar index, reducing the seller’s problem to one-dimensional screening. The optimal mechanism takes the form of committed-spend contracts: buyers pay for a budget that they allocate across token classes priced at marginal cost. We extend the analysis to environments with multiple differentiated models and to competition between a proprietary leader and an open-source fringe, showing that competitive pressure reshapes both the intensive and extensive margins of compute provision. Each element of our theory (token-budget menus, maximum- and minimum-spend plans, multi-model versioning, and linear API pricing) has a direct counterpart in the observed pricing practices of providers such as Anthropic, OpenAI, and GitHub.

American Economic Review

Artificial Intelligence, Algorithmic Pricing, and Collusion

Increasingly, algorithms are supplanting human decision-makers in pricing goods and services. To analyze the possible consequences, we study experimentally the behavior of algorithms powered by Artificial Intelligence (Q-learning) in a workhorse oligopoly model of repeated price competition. We find that the algorithms consistently learn to charge supracompetitive prices, without communicating with one another. The high prices are sustained by collusive strategies with a finite phase of punishment followed by a gradual return to cooperation. This finding is robust to asymmetries in cost or demand, changes in the number of players, and various forms of uncertainty. (JEL D21, D43, D83, L12, L13)

Manufacturing & Service Operations Management

Analytics for an Online Retailer: Demand Forecasting and Price Optimization

We present our work with an online retailer, Rue La La, as an example of how a retailer can use its wealth of data to optimize pricing decisions on a daily basis. Rue La La is in the online fashion sample sales industry, where they offer extremely limited-time discounts on designer apparel and accessories. One of the retailer’s main challenges is pricing and predicting demand for products that it has never sold before, which account for the majority of sales and revenue. To tackle this challenge, we use machine learning techniques to estimate historical lost sales and predict future demand of new products. The nonparametric structure of our demand prediction model, along with the dependence of a product’s demand on the price of competing products, pose new challenges on translating the demand forecasts into a pricing policy. We develop an algorithm to efficiently solve the subsequent multiproduct price optimization that incorporates reference price effects, and we create and implement this algorithm into a pricing decision support tool for Rue La La’s daily use. We conduct a field experiment and find that sales does not decrease because of implementing tool recommended price increases for medium and high price point products. Finally, we estimate an increase in revenue of the test group by approximately 9.7% with an associated 90% confidence interval of [2.3%, 17.8%].

informs PubsOnline

Utility Fairness in Contextual Dynamic Pricing with Demand Learning

This paper introduces a novel contextual bandit algorithm for personalized pricing under utility fairness constraints in scenarios with uncertain demand, achieving an optimal regret upper bound. Our approach, which incorporates dynamic pricing and demand learning, addresses the critical challenge of fairness in pricing strategies. We first delve into the static full-information setting to formulate an optimal pricing policy as a constrained optimization problem. Here, we propose an approximation algorithm for efficiently and approximately computing the ideal policy. We also use mathematical analysis and computational studies to characterize the structures of optimal contextual pricing policies subject to fairness constraints, deriving simplified policies that lay the foundations of more in-depth research and extensions. Further, we extend our study to dynamic pricing problems with demand learning, establishing a nonstandard regret lower bound that highlights the complexity added by fairness constraints. Our research offers a comprehensive analysis of the cost of fairness and its impact on the balance between utility and revenue maximization. This work represents a step toward integrating ethical considerations into algorithmic efficiency in data-driven dynamic pricing.

Cambridge Core

Trustworthy Online Controlled Experiments

Cambridge Core - Knowledge Management, Databases and Data Mining - Trustworthy Online Controlled Experiments

thescipub.com

https://thescipub.com/abstract/10.3844/jcssp.2014.2240.2252

MDPI Data

Capacity Allocation of Game Tickets Using Dynamic Pricing

This study examines a pricing approach that is applicable in the field of online ticket sales for game tickets. The mathematical principle of dynamic programing is combined with empirical data analysis to determine demand functions for university football game tickets. Based on the calculated demand functions, the application of DP strategies is found to generate more revenues than a fixed price strategy. The other important result is the capacity distribution of tickets according to the football game intensity. Prior studies have shown that it is sometimes more profitable or football clubs to allocate a share of tickets to a retailer and earn a commission based on the sales, rather than selling the entire capacity of tickets by itself. This paper finds that in a high intensity game, where the demand is generally high, it is optimal for the club to sell all tickets by itself. Whereas, for less popular games, where there is considerable fluctuation in demand, the capacity allocation problem for maximized revenues from ticket sales, becomes a harder optimization challenge for the club. According to DP optimization, when the demand for tickets is relatively low, it is optimal for the club to retain 20–40% of the tickets and the rest of the capacity should be sold to online retailers. In the real world, this pricing technique has been used by football clubs and thus the secondary market online retailers like Ticketmaster and Vivid Seats have become popular in the last decade.

American Economic Review

Artificial Intelligence, Algorithmic Pricing, and Collusion

Increasingly, algorithms are supplanting human decision-makers in pricing goods and services. To analyze the possible consequences, we study experimentally the behavior of algorithms powered by Artificial Intelligence (Q-learning) in a workhorse oligopoly model of repeated price competition. We find that the algorithms consistently learn to charge supracompetitive prices, without communicating with one another. The high prices are sustained by collusive strategies with a finite phase of punishment followed by a gradual return to cooperation. This finding is robust to asymmetries in cost or demand, changes in the number of players, and various forms of uncertainty.

informs PubsOnline

Utility Fairness in Contextual Dynamic Pricing with Demand Learning

This paper introduces a novel contextual bandit algorithm for personalized pricing under utility fairness constraints in scenarios with uncertain demand, achieving an optimal regret upper bound. Our approach, which incorporates dynamic pricing and demand learning, addresses the critical challenge of fairness in pricing strategies. We first delve into the static full-information setting to formulate an optimal pricing policy as a constrained optimization problem. Here, we propose an approximation algorithm for efficiently and approximately computing the ideal policy. We also use mathematical analysis and computational studies to characterize the structures of optimal contextual pricing policies subject to fairness constraints, deriving simplified policies that lay the foundations of more in-depth research and extensions. Further, we extend our study to dynamic pricing problems with demand learning, establishing a nonstandard regret lower bound that highlights the complexity added by fairness constraints. Our research offers a comprehensive analysis of the cost of fairness and its impact on the balance between utility and revenue maximization. This work represents a step toward integrating ethical considerations into algorithmic efficiency in data-driven dynamic pricing.

New York State Senate

Senate Bill S7033, Enacts the preventing algorithmic pricing discrimination act

Establishes the Preventing Algorithmic Pricing Discrimination Act. This bill prohibits the use of surveillance pricing practices that adjust the cost of goods and services based on consumer data in a discriminatory manner. It ensures that businesses cannot use consumer data to infer personal characteristics such as race, gender, or disability status to alter prices unfairly. Additionally, the bill mandates that businesses utilizing surveillance pricing must notify consumers when prices are determined algorithmically based on their personal data.