Dynamic Pricing Problem Research Articles

Contextual Dynamic Pricing with Strategic Buyers

Personalized pricing, which involves tailoring prices based on individual characteristics, is commonly used by firms to implement a consumer-specific pricing policy. In this process, buyers can also strategically manipulate their feature data to obtain a lower price, incurring certain manipulation costs. Such strategic behavior can hinder firms from maximizing their profits. In this article, we study the contextual dynamic pricing problem with strategic buyers. The seller does not observe the buyer’s true feature, but a manipulated feature according to buyers’ strategic behavior. In addition, the seller does not observe the buyers’ valuation of the product, but only a binary response indicating whether a sale happens or not. Recognizing these challenges, we propose a strategic dynamic pricing policy that incorporates the buyers’ strategic behavior into the online learning to maximize the seller’s cumulative revenue. We first prove that existing nonstrategic pricing policies that neglect the buyers’ strategic behavior result in a linear Ω ( T ) regret with T the total time horizon, indicating that these policies are not better than a random pricing policy. We then establish an O ( T ) regret upper bound of our proposed policy and an Ω ( T ) regret lower bound for any pricing policy within our problem setting. This underscores the rate optimality of our policy. Importantly, our policy is not a mere amalgamation of existing dynamic pricing policies and strategic behavior handling algorithms. Our policy can also accommodate the scenario when the marginal cost of manipulation is unknown in advance. To account for it, we simultaneously estimate the valuation parameter and the cost parameter in the online pricing policy, which is shown to also achieve an O ( T ) regret bound. Extensive experiments support our theoretical developments and demonstrate the superior performance of our policy compared to other pricing policies that are unaware of the strategic behaviors. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Dynamic pricing for perishable product in the presence of low-carbon preference

ABSTRACT As countries worldwide strive to implement energy conservation and emission reduction measures, modeling consumers’ low-carbon preferences and purchasing decisions has become a crucial factor in implementing dynamic pricing strategies for firms. This paper investigates the dynamic pricing problems for a single perishable product and substitutable perishable products, respectively, where consumer is affected by low-carbon preference. We first propose the decision framework of consumer choice with low-carbon preference based on the multinomial logit model. On this basis, we establish dynamic pricing models by the dynamic programming method and obtain the optimal pricing strategies. We analytically show that consumer’s low-carbon preference has a positive effect on the optimal price for a single perishable product. Meanwhile, numerical examples illustrate that consumer’s low-carbon preference has a positive effect on the optimal prices for substitutable perishable products. Finally, we give some suggestions to increase firm’s revenue when consumer’s low-carbon preference is present.

Pricing a product with network effects for sale to a fixed population of customers

AbstractWe consider a deterministic dynamic pricing problem for a product that exhibits network effects and that is sold to a fixed heterogeneous population of customers. We begin by introducing a demand model wherein those customers are arrayed over two‐dimensional space according to a bivariate probability distribution. Each customer's location in space provides a description of that customer's intrinsic value for the product as well as the extent to which the customer is influenced by the network effect. In the pricing problem, as sales accumulate over time, the set of customers who have already purchased the product grows, while the set of customers who have not yet purchased the product shrinks. The total customer population remains fixed. Those who have not yet purchased constitute the remaining population of potential buyers of the product. As time moves forward, the mix of customers that remain as potential buyers evolves endogenously. The demand model yields a geometric interpretation of the remaining population of potential buyers, and gives rise to a dynamic program with states that are sets in two‐dimensional space. It is not practical to solve the dynamic pricing problem to optimality, so we present bounds and comparative statics results that help us identify tractable heuristics and obtain rigorous performance guarantees. In numerical experiments, we find that fixed‐price policies may perform poorly, especially when the network effect is strong or the time horizon is long. We also introduce a stochastic version of the problem that uses a spatial Poisson process to describe the customers, and we develop and analyze a heuristic approach for that formulation.

Reinforcement learning for Multi-Flight Dynamic Pricing

Dynamic pricing is essential for airline revenue management, requiring quick adaptation to fluctuating market environments and complex customer behaviors. This study addresses the Multi-Flight Dynamic Pricing (MFDP) problem, which presents unique challenges due to interdependent demand between multiple flights and high dimensionality. Traditional studies often assume that the demand function modeling customer behavior is either known in advance or follows a predefined structure, failing to capture the dynamic nature of pricing decisions. To fill this gap, we develop deep reinforcement learning (DRL) algorithms—Deep Q-Network (DQN), Advantage Actor-Critic (A2C), Proximal Policy Optimization (PPO), and Trust Region Policy Optimization (TRPO). By formulating the MFDP problem as a Markov Decision Process (MDP), we design an innovative utility function for the Multinomial Logit (MNL) model that captures realistic features of the airline market, such as competition from high-speed rail, the effect of reference fares, and travel time. We compare the performance of our DRL algorithms with traditional algorithms, including Dynamic Programming (DP), Price Pooling (PP), Inventory Pooling (IP), and Inventory and Price Pooling (IPP). Our experiments demonstrate that DRL algorithms alleviate the curse of dimensionality faced by traditional algorithms, expedite the learning process, and deliver satisfactory performance without relying on predefined demand functions. Among these algorithms, TRPO shows superior performance, achieving 99% of the theoretical optimal revenue, proving its adaptability and stability in dynamic pricing applications. We also highlight the importance of considering the null price in the action space of MFDP problems. The larger the market scale, the more pronounced the effect of the null price in accelerating RL algorithm convergence, leading to more efficient computational resource utilization.

Competitive Pricing for Resource Trading in Sliced Mobile Networks: A Multi-Agent Reinforcement Learning Approach

The emergence of network slicing as a flagship technology in 5G networks has not only enhanced network expansion and flexibility in resource management for service continuity, but also provided an avenue for establishing a viable market for resource sharing. To optimize the network's resource usage, stakeholders are encouraged to take pragmatic steps toward dynamic resource sharing. This paper designs a techno-economic model for the strategic interactions among multiple competing mobile virtual network operators (MVNOs) and their users in a trading marketplace. We formulate the dynamic pricing problem as a two-stage Stackelberg game, where the MVNOs are leaders, and the users are followers. In the first stage, the MVNOs compete to set their differentiated unit prices using a negotiation mechanism while considering system-level network load. Then, the users decide their purchasing volumes to match the prices of the MVNOs. We transform the game-based optimization problem into a stochastic Markov decision process (MDP) problem and propose a multi-agent deep Q-network (MADQN) method that obtains an optimal solution for the formulated game. Simulation results and analysis reveal that the proposed algorithm achieves convergence under the competitive pricing scheme (CPS) and independent pricing scheme (IPS) while enhancing MVNOs and users' utilities at acceptable levels.

Matheuristics vs. metaheuristics for joint lot-sizing and dynamic pricing problem with nonlinear demands

This paper considers a joint dynamic pricing and production planning decisions problem for a profit-maximizing firm that produces and sells multiple products. The objective is to develop a coordinated decision approach for multi-product pricing and lot sizing decisions for a manufacturer considering a limited production capacity. The demand for each product is assumed to be iso-elastic and integrates the complementarity and the substitution effects between the products. First, the problem is formulated as a non-convex mixed integer nonlinear programming model (MINLP) incorporating capacity constraints, setup costs, and nonlinear demand functions. Then, since the model is nonlinear and non-convex, a set of approximate approaches based on the Genetic algorithm, Late Acceptance Hill Climbing and Simulated Annealing methods are designed to solve this problem. Based on this study, the performances of two variants of approximate methods: matheuristics and metaheuristics are discussed and analyzed. The extensive experimental study, performed on real-world inspired instances, shows that matheuristic methods with setup-variables encoding scheme outperform the rest of the methods. The research outcomes show that coordinating a decision-making process by optimizing both prices and production plans simultaneously can result in significant profit for a company. However, one must consider the joint effect of the parameters of the demand function as well as the impact of the production capacity. This comprehensive understanding enables the company to avoid excessive investments in less lucrative products with lower sales potential, thereby ensuring resource allocation aligns with profitability and market demand.

Airline dynamic pricing with patient customers using deep exploration-based reinforcement learning

This paper addresses a crucial issue in the airline industry by tackling a dynamic pricing problem in the presence of patient customers, a scenario that has gained significance due to the revenue loss of airlines caused by customers’ non-myopic decision-making. To effectively capture this non-myopic characteristic, we propose a Markov decision process (MDP) including a history of offered prices as a state variable. In contrast to previous studies, distributions of customers’ properties are assumed to be unknown in advance for a more realistic representation of real-world scenarios. To deal with the new challenges of the problem, we propose utilizing a specific learning framework (i.e., deep exploration-based RL) that is unexplored in this domain. The numerical experiments demonstrate that its performance can be improved on the MDP we designed and show that it outperforms the benchmark algorithm. The structures of pricing policies generated from the bootstrapped deep Q-network algorithm imply that airlines should offer high and low prices alternately from the beginning of the sales period rather than increasing prices as time goes on. We also ascertain that more frequent consecutive high-priced periods can increase airlines’ revenue in environments with higher customer patience levels.

Treating symmetric buyers asymmetrically

AbstractWe investigate a finite‐horizon dynamic pricing problem of a seller under limited commitment. Even when the buyers are ex ante symmetric to the seller, the seller can charge different prices to different buyers. We show that under the class of posted‐price mechanisms this asymmetric treatment of symmetric buyers strictly revenue‐dominates symmetric treatment. The seller implements this by using a priority‐based deterministic tie‐breaking rule instead of using a random tie‐breaking rule. The effect of asymmetric treatment on revenue increment increases monotonically as we increase the time horizon of the game.

Dynamic pricing for group‐buying strategy with network externality

AbstractWe consider a two‐stage dynamic pricing problem for a seller who employs online group purchasing with strategic customers. Customers who arrive in Periods 1 and 2 can decide whether to purchase the product in the current period, delay their purchases to the next period (for those customers in Period 1), or leave the market. The network externality, a typical characteristic of group buying, is captured in our model. That is, more customers who purchase the product would positively or negatively affect the demand. We investigate the impacts of positive/negative externality on the seller's prices, demands, and profits, respectively. By comparing positive and negative externality cases, we find that customers who purchase the product in Period 2 can enjoy a lower price than in Period 1 for negative externality. Interestingly, however, the price in Period 1 might be higher or lower than in Period 2 for positive externality, even if the corresponding demand is lower. Compared to no externality, our result shows that the seller should reduce the price in Period 1 but increase it in Period 2 because this helps increase the total demand and profit in the presence of a positive externality. Nevertheless, when there exists a negative externality, we confirm that the prices in two periods might both decrease for a lower externality, which leads to lower total demand and profit for the seller. Since the seller can decide whether to disclose their reviews or product historical sales to customers, this result can provide practical implications for sellers who adopt a group purchasing strategy.

Multiproduct Dynamic Pricing with Limited Inventories Under a Cascade Click Model

Problem definition: Designing effective operational strategies requires a good understanding of customer behavior. The classic economic theory of customer choice has long been the paradigm in the operations literature. However, the rise of online marketplaces such as e-commerce has triggered considerable efforts in academia and industry to develop alternative models that not only provide a good approximation of customer behavior but also are easily scalable for large-scale implementations. In this paper, we consider a multiproduct dynamic pricing problem with limited inventories under the so-called cascade click model, which is one of the most popular click models used in practice and has been intensively studied in the computer science literature. Methodology/results: We present some fundamental results. First, we derive a sufficiently general characterization of the optimal pricing policy and show that it has a different structure than the optimal policy under the standard pricing model. Second, we show that the optimal expected total revenue under the cascade click model can be upper bounded by the objective value of an approximate deterministic pricing problem. Third, we show that two policies that are known to have strong performance guarantees in the standard revenue management setting can be properly adapted (in a nontrivial way) to the setting with cascade click model while retaining their strong performance. Finally, we also briefly discuss the joint ranking and pricing problem and provide an iterative heuristic to calculate an approximate ranking. Managerial implications: Taking into account customers’ click-and-search behavior leads to different structures of the optimal pricing policy, and some common insights under the standard pricing models may no longer hold. Moreover, our simulation studies show that pricing under a (misspecified) classic choice model that is oblivious to customers click-and-search behavior can severely impact profitability. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2021.0504 .

Scalable reinforcement learning approaches for dynamic pricing in ride-hailing systems

Dynamic pricing is a widely applied strategy by ride-hailing companies, such as Uber and Lyft, to match the trip demand with the availability of drivers. Deciding proper pricing policies is challenging and existing reinforcement learning (RL)-based solutions are restricted in solving small-scale problems. In this study, we contribute to RL-based approaches that can address the dynamic pricing problem in real-world-scale ride-hailing systems. We first characterize the dynamic pricing problem with a clear distinction between historical prices and current prices. We then translate our dynamic pricing problem into Markov Decision Process (MDP) and prove the existence of a deterministic stationary optimal policy. Our solutions are based on an off-policy reinforcement learning algorithm called twin-delayed deep determinant policy gradient (TD3) that performs offline learning of the optimal pricing policy using historical data and applies the learned policy to the next time slot, e.g., one week. We enhance TD3 by creating three mechanisms to reduce our model complexity and enhance training effectiveness. Extensive numerical experiments are conducted on both small grid networks (16 zones) and the NYC network (242 zones) to demonstrate the performance of the proposed algorithm. The results show our algorithm can efficiently find the optimal pricing policy for both the small and large networks, and can significantly enhance the platform profit and service efficiency.

Distributed dynamic pricing of multiple perishable products using multi-agent reinforcement learning

Revenue management (RM) is essential for a wide range of industries such as airlines, hotels, cruise lines, fashion, and seasonal retail. This paper focuses on the multi-perishable-product dynamic pricing (MPPDP) problem, a significant research field in RM, where a company sells multiple interactive and perishable products over a limited selling window without replenishment. Most studies in this field assume customer behavior, which is modeled by demand function, is known in advance. Even when considering uncertainty in customer behavior, most studies still assume the mathematical form or structural properties of the underlying demand function are known in advance. However, these assumptions are usually inconsistent with the actual market situation. Recently, Reinforcement Learning (RL), a potent technique for handling sequential decision-making problems, has been increasingly applied to solve complex dynamic pricing problems without relying on any assumption about demand functions. However, the curse of dimensionality poses a challenge for currently used centralized RL algorithms when solving the MPPDP problem due to the exponential expansion of the joint price space with the number of products. To address this issue, our paper proposes a distributed dynamic pricing framework and innovatively models the MPPDP problem as a Fully Cooperative Markov Game solved by Multi-Agent Reinforcement Learning (MARL). Additionally, we use counterfactual baselines to design appropriate agent-specific reward signals that facilitate faster learning for the agents in our established multi-agent cooperative system. Finally, two MARL-based distributed dynamic pricing algorithms, Counterfactual Q-learning, and Counterfactual DQN, are proposed for the MPPDP problem. Through the case studies on four computer-simulated markets, we show that our algorithms can alleviate the curse of dimensionality faced by centralized RL algorithms, expedite the learning process, and demonstrate satisfactory performance without relying on any assumption about demand functions. In conclusion, our work provides an effective MARL-based distributed dynamic pricing framework and algorithms for companies to efficiently price their multiple perishable products in modern highly uncertain markets.

Heuristic Rules for the Dynamic Pricing Problem

This paper is devoted to the development of heuristics for the dynamic pricing problem. A discrete time model of dynamic pricing on the fixed time horizon is proposed. It is applicable to products that satisfy two properties: 1) product value expires at a certain predetermined date, and 2) consumers demand at most a single unit of the product. This type of demand structure allows deriving a simple system of recursive equations for optimal prices using dynamic programming techniques. Optimal pricing policy is expressed as a function of time to expiration and inventory levels of unsold products. An analytical solution to this problem was obtained for special cases, while for the general case, a numerical algorithm has been developed. Qualitative characteristics of the optimal pricing policy are established, and their implications for dynamics of inventories and prices are discussed. Based on these observations, a simple heuristic rule for dynamic price adjustments is proposed. Performance of this heuristic is evaluated against the optimal dynamic and fixed-price policies using Monte-Carlo experiments. Results demonstrate high efficiency of the proposed heuristic strategy and its even simpler derivatives. Heuristics’ adaptability and ease of implementation should make it suitable and attractive for small and medium businesses.

An Automated Deep Reinforcement Learning Pipeline for Dynamic Pricing

Dynamic pricing problem is difficult due to the highly dynamic environment and unknown demand distributions. In this paper, we propose a Deep Reinforcement Learning (DRL) framework, which is a pipeline that automatically defines the DRL components for solving a Dynamic Pricing problem. The automated DRL pipeline is necessary because the DRL framework can be designed in numerous ways, and manually finding optimal configurations is tedious. The levels of automation make non-experts capable of using DRL for dynamic pricing. Our DRL pipeline contains three steps of DRL design, including MDP modeling, algorithm selection, and hyper-parameter optimization. It starts with transforming available information to state representation and defining reward function using a reward shaping approach. Then, the hyper-parameters are tuned using a novel hyper-parameters optimization method that integrates Bayesian Optimization and the selection operator of the Genetic algorithm. We employ our DRL pipeline on reserve price optimization problems in online advertising as a case study. We show that using the DRL configuration obtained by our DRL pipeline, a pricing policy is obtained whose revenue is significantly higher than the benchmark methods. The evaluation is performed by developing a simulation for the RTB environment that makes exploration possible for the RL agent.

The Value of Observing the Buyers’ Arrival Time in Dynamic Pricing

We consider a dynamic pricing problem in which a firm sells one item to a single buyer to maximize expected revenue. The firm commits to a price function over an infinite horizon. The buyer arrives at some random time with a private value for the item. He is more impatient than the seller and strategizes over the timing of the purchase in order to maximize his expected utility, which implies either buying immediately, waiting to benefit from a lower price, or not buying. We study the value of the seller’s ability to observe the buyer’s arrival time in terms of her expected revenue. When the seller can observe the buyer’s arrival, she can make the price function contingent on the buyer’s arrival time. On the contrary, when the seller can’t, her price function is fixed at time zero for the whole horizon. The value of observability (VO) is defined as the worst-case ratio between the expected revenue of the seller when she observes the buyer’s arrival and that when she does not. First, we show that, for the particular case in which the buyer’s valuation follows a monotone hazard rate distribution, the upper bound of VO is [Formula: see text]. Next, we show our main result: in a setting very general on valuation and arrival time distributions: VO is at most 4.911. To obtain this bound, we fully characterize the solution to the observable arrival problem and use this solution to construct a random and periodic price function for the unobservable case. Finally, we show by solving a particular example to optimality that VO has a lower bound of 1.136. This paper was accepted by Itai Ashlagi, revenue management and market analytics. Funding: This work was partially supported by the Agencia Nacional de Investigación y Desarrollo Chile [Grants ACT210005 and FONDECYT 1220054], and by the Center for Mathematical Modeling at the University of Chile [Grant FB210005]. The work of D. Pizarro has benefited from the Artificial and Natural Intelligence Toulouse Institute, which is funded by the French “Investing for the Future – PIA3” program [Grant ANR-19-P3IA-0004]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2023.4794 .

A differential game model research on dynamic pricing and coordination of fresh agricultural products supply chain based on freshness

For the problem of optimal dynamic pricing and coordination of fresh agricultural products supply chain, the differential equations related to freshness and market needs are constructed, and the optimal control theory is used to solve the optimal strategy under centralized and decentralized decision-making, i.e., optimal selling price and optimal preservation input. The equilibrium results under the two scenarios are also compared. The comparison finds that the decentralized scenario leads to low overall supply chain profits, based on which a two-part pricing contract is proposed to coordinate the supply chain operating companies. The results show that freshness effectiveness directly impacts pricing, and the centralized scenario does not necessarily lead to high prices for fresh produce due to the ‘double marginal effect’. the two pricing contracts can successfully coordinate the supply chain. The freshness effectiveness increases the supply chain coordination ability of fresh produce suppliers, while operational inefficiencies, on the contrary, decrease the coordination ability.

Risk-averse dynamic pricing using mean-semivariance optimization

In many revenue management applications risk-averse decision-making is crucial. In dynamic settings, however, it is challenging to find the right balance between maximizing expected rewards and avoiding poor performances. In this paper, we consider time-consistent mean-semivariance (MSV) optimization for dynamic pricing problems within a discrete MDP framework, which are shown to be NP hard. We present a novel fixpoint-based dynamic programming approach to compute risk-sensitive feedback policies with Pareto-optimal combinations of mean and semivariance. We illustrate the effectiveness and the applicability of our concepts compared to state-of-the-art heuristics. For various numerical examples the results show that our approach clearly outperforms all other heuristics and obtains a performance guarantee with less then 0.2% optimality gap. Our approach is general and can be applied to MDPs beyond dynamic pricing.

Real-Time Spatial–Intertemporal Pricing and Relocation in a Ride-Hailing Network: Near-Optimal Policies and the Value of Dynamic Pricing

In “Real-Time Spatial–Intertemporal Pricing and Relocation in a Ride-Hailing Network: Near-Optimal Policies and The Value of Dynamic Pricing,” Chen, Lei, and Jasin consider a dynamic pricing problem faced by a ride-hailing service provider who manages a fixed number of servers and serves price-sensitive customers within a network. Servers serve arriving customers by relocating from the requested origins to destinations within a certain travel time. The authors first propose a static pricing policy based on the optimal solution to a deterministic relaxation of the original stochastic problem. They show that the proposed static policy matches the best possible asymptotic performance of any static policy. The authors further propose a dynamic pricing policy that adaptively changes the prices in a way that reduces the impact of past demand randomness on the balance of future distributions of servers and customers across the network. They show that the dynamic pricing policy achieves significantly better asymptotical performance. The proposed policies and their performance guarantees are further extended to a case where the firm jointly decides the relocation of vacant servers

Demand management in time-slotted last-mile delivery via dynamic routing with forecast orders

In this paper, we propose a partially time-windowed dynamic routing approach with forecast orders to tackle the dynamic pricing problem of attended home delivery, one of the challenging problems in last-mile logistics. The purpose of forecast orders is to find a cost-effective route map for delivery with forecast orders to guide the delivery system to accept real orders in potentially better time slots for servicing. Initially, we build and optimise virtual routes using forecast orders without time windows. Upon the arrival of real orders, we replace the forecast orders with incoming real orders to help the delivery system estimate the opportunity cost of making the delivery in each time slot. Dynamic pricing is then used to influence the customer’s choices of servicing slots, and based on what the customer selects, the real order replaces the forecast order, and the route map is re-optimised. This strategy leads to better scheduling for the overall route map and more slot availability for incoming customers. Through computational study, we demonstrate the benefit of our approach compared to static pricing and previous dynamic pricing policies, with or without forecast orders. In particular, by employing forecast orders without time windows, we witness a decrease in delivery costs and an increase in the number of accepted orders, leading to higher profits.

An Omnichannel Retailing Operation for Solving Joint Inventory Replenishment Control and Dynamic Pricing Problems From the Perspective of Customer Experience

Customer experience, a key element in the domain of omnichannel retailing, plays a vital role in attracting consumers. Implementing a new BOPS (buy online, pick up in store) strategy will not only affect the inventory structure and pricing strategy of retail enterprises, but will also impact the original sales channel. In accordance with the consumer experience, a dynamic optimization model of a retailer inventory system was established in this research. Based on the optimal control theory, analyze the optimal control strategies under four sales modes: online, offline, BOPS, and BOPS-PLUS (BOPS and buy offline online-logistics distribution). Through numerical experiments, the inventory control strategies and optimal discounted profits under the four modes are compared and analyzed. The results show that the optimal experience investment and dynamic pricing levels gradually decrease with time, and the inventory state has a vital influence on the discounted profit of the system. The traditional dual-channel retail model is optimal when the channel inventory is not out of stock. In situations where channel inventory is in short supply (both online and offline), a deeply integrated omnichannel retail mode BOPS-PLUS is best. The numerical analysis also reveals that price sensitivity coefficient, experience service coefficient and online return rate also have significant influence on the selection of optimal channel strategy.