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This paper focuses on joint dynamic pricing and demand learning in an oligopolistic market. Each firm seeks to learn the price-demand
relationship for itself and its competitors, and to set optimal prices, taking into account its competitors’ likely moves.
We follow a closed-loop approach to capture the transient aspect of the problem, that is, pricing decisions are updated dynamically
over time, using the data acquired thus far.
We formulate the problem faced at each time period by each firm as a Mathematical Program with Equilibrium Constraints (MPEC).
We utilize variational inequalities to capture the game-theoretic aspect of the problem. We present computational results
that provide insights on the model and illustrate the pricing policies this model gives rise to.

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... In all of the above reviewed papers, demand is either deterministic or stochastic with known distribution. Perakis and Sood (2006), Bertsimas and Perakis (2006), Kachani et al. (2007), and Cooper et al. (2013) study dynamic pricing problems with competition when there is limited demand information. Their work is reviewed in Section 4 together with all other work involving limited demand information. ...

... There are also studies (Sen and Zhang 2009) that assume that both the arrival rate and some parameters of the WtP distribution are unknown. Linear demand functions with unknown parameters are also quite commonly used (Lobo and Boyd 2003, Bertsimas and Perakis 2006, Kachani et al. 2007, Cooper et al. 2013, Keskin and Zeevi 2013). There are also several more complex demand models used to deal with time-varying demand function (Aviv and Pazgal 2005, Besbes and Zeevi 2011, Keskin and Zeevi 2013, Chen and Farias 2013, strategic customer behavior ( Levina et al. 2009), or multiple products ( Gallego and Talebian 2012). ...

... Maximum likelihood estimation is also quite commonly used (Carvalho and Puterman 2005, Broder and Rusmevichientong 2012, den Boer and Zwart 2011). Other approaches used include linear least squares estimation (Bertsimas and Perakis 2006, Kachani et al. 2007, Cooper et al. 2013, Keskin and Zeevi 2013, Besbes and Zeevi 2014, and simple empirical estimation which is quite different from all other approaches and is described in detail below when we review the corresponding papers (Besbes and Zeevi 2009, 2011, Wang et al. 2014, Chen and Farias 2013. Unlike the Bayesian approach, these approaches do not require prior knowledge of an unknown parameter. ...

Dynamic pricing enables a firm to increase revenue by better matching supply with demand, responding to shifting demand patterns, and achieving customer segmentation. In the last twenty years, numerous success stories of dynamic pricing applications have motivated a rapidly growing research interest in a variety of dynamic pricing problems in the academic literature. A large class of problems that arise in various revenue management applications involve selling a given amount of inventory over a finite time horizon without inventory replenishment. In this paper, we identify most recent trends in dynamic pricing research involving such problems. We review existing research on three new classes of problems that have attracted a rapidly growing interest in the last several years, namely, problems with multiple products, problems with competition, and problems with limited demand information. We also identify a number of possible directions for future research.This article is protected by copyright. All rights reserved.

... Gaimon (1988) berücksichtigt erstmals Maximalkapazitäten, die zu einem vorgegebenen Kostensatz auch erweitert werden können. Aktuelle Arbeiten innerhalb des Forschungsfeldes stammen von Adida/ Perakis (2007) und Adida /Perakis (2006), wobei in letzterer Veröffentlichung Prinzipien der robusten Optimierung in das kontrolltheoretische Problem integriert werden. ...

... Auch die in den Abschnitten 5.4 und 5.6 behandelten Aspekte lassen sich in Modelle mit Demand Learning integrieren: Beispielsweise betrachten Bertsimas/ Perakis (2006) sowie Kachani et al. (2007) konkurrierende Anbieter, Levina et al. (2008 bilden strategisches Kundenverhalten ab. ...

Gegenstand des Dynamic Pricing ist die situative Anpassung von Angebotspreisen an im Zeitablauf variierende Rahmenbedingungen. Die hier seit einigen Jahren beobachtbare Intensivierung der wissenschaftlichen Auseinandersetzung hat es heute zu einem der methodisch fortgeschrittensten Forschungsgebiete an der Schnittstelle zwischen Operations Research, Marketing, Economics und E Commerce werden lassen. Der vorliegende Beitrag liefert zunächst eine kompakte inhaltliche Einführung in die Grundlagen des Dynamic Pricing sowie in die elementaren mathematischen Modellierungsansätze. Zu ihrer Einordnung wird ein spezifischer Kriterienkatalog entwickelt. Dieser dient als Grundlage für die anschließende, umfassende Literaturanalyse, welche den Schwerpunkt des Beitrags bildet. Abschließend werden mögliche künftige Forschungspotenziale identifiziert. This paper presents a current review of the academic literature on dynamic pricing, which has evolved to one of the leading research topics at the interface between Operations Research, Marketing, Economics, and E-Commerce during the last decade. The authors begin by delivering a primer on dynamic pricing and explaining the basic model formulations. Subsequently, they develop a set of criteria which allow for the unified classification of relevant publications and serve as a basis for the extensive literature analysis. Possible future research directions are discussed as well.

... A method of mathematical programming with equilibrium constraints is applied to formulate the model to estimate the competitor parameters. [12] proposed further improvements that adjusted the most accurate parameters of demand function periodically based on the equilibrium demands. [11] considered a firm selling a single product with multiple versions and developed a multinomial logit choice demand model, commonly used in the context of non-cooperative competition between firms selling differentiated goods, each seeking to maximize total revenue. ...

Dynamic pricing is used to maximize the revenue of a firm over a finite-period planning horizon, given that the firm may not know the underlying demand curve "a priori". In emerging markets, in particular, firms constantly adjust pricing strategies to collect adequate demand information, which is a process known as price experimentation. To date, few papers have investigated the pricing decision process in a competitive environment with unknown demand curves, conditions that make analysis more complex. Asynchronous price updating can render the demand information gathered by price experimentation less informative or inaccurate, as it is nearly impossible for firms to remain informed about the latest prices set by competitors. Hence, firms may set prices using available, yet out-of-date, price information of competitors. In this paper, we design an algorithm to facilitate synchronized dynamic pricing, in which competitive firms estimate their demand functions based on observations and adjust their pricing strategies in a prescribed manner. The process is called "learning and earning" elsewhere in the literature. The goal is for the pricing decisions, determined by estimated demand functions, to converge to underlying equilibrium decisions. The main question that we answer is whether such a mechanism of periodically synchronized price updates is optimal for all firms. Furthermore, we ask whether prices converge to a stable state and how much regret firms incur by employing such a data-driven pricing algorithm.

... Several papers have recently appeared where exact knowledge of model parameters is not assumed a priori, but leaned using Bayesian updating [19][20][21][22]. While the Bayesian approach does not assume exact knowledge of all the parameters, it is based nevertheless on stringent assumptions that are hard to be satisfied in real-world applications. ...

Dynamic pricing is commonly adopted in selling perishable products with fixed stock and selling horizon. Because of the technological change and the volatility of customers’ tastes, the probabilistic demand model commonly used in practice cannot be estimated accurately. This paper considers a retailer’s dynamic pricing problem with little information on demand, and proposes a new approach to deal with this problem. We model the uncertain demand as a hybrid variable which describes the quantities with fuzziness and randomness. The dynamic pricing problem is formulated as three types of hybrid programming models—expected revenue maximization model, αα-revenue maximization model and chance maximization model—to meet different goals. To solve the proposed models, a hybrid intelligent algorithm is designed by combining hybrid simulations and genetic algorithm. Numerical examples are also presented to illustrate the modeling idea and to show the effectiveness of the proposed algorithm.

... In the case of models of demand linear with the price, the methods they propose can be applied to estimating the parameters α i (.), β i (.) in our model. For example, Kachani, Perakis and Simon [29] design an approach that enable to achieve dynamic pricing while learning the price-demand linear relationship in an oligopoly. ...

In this paper, we consider a variety of models for dealing with demand uncertainty for a joint dynamic pricing and inventory control problem in a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting, where demand depends linearly on the price. Our goal is to address demand uncertainty using various robust and stochastic opti- mization approaches. For each of these approaches, we first introduce closed-loop formulations (adjustable robust and dynamic programming), where decisions for a given time period are made at the beginning of the time period, and uncertainty unfolds as time evolves. We then describe models in an open-loop setting, where decisions for the entire time horizon must be made at time zero. We conclude that the affine adjustable robust approach performs well (when compared to the other approaches such as dynamic programming, stochastic programming and robust open loop approaches) in terms of realized profits and protection against constraint violation while at the same time it is computationally tractable. Furthermore, we compare the complexity of these models and discuss some insights on a numerical example.

Purpose
The purpose of the study is applying and comparing models that predict optimal time for new product exit based on its demand pattern and survivability. This is to decide whether or not to continue investing in new product development (NPD).
Design/methodology/approach
The study investigates the optimal time for new product exit within the hi-tech sector by applying three models: the dynamic learning demand model (DLDM), the generalized Bass model (GBM) and the hazard model (HM). Further, for inter- and intra-model comparison, the authors conducted a simulation, considering Weiner and exponential price functions to enhance generalizability.
Findings
While higher price volatility signifies an unstable technology, greater investment into research and development (R&D) and marketing results in higher product adoption rates. Imitators have a more prominent role than innovators in determining the longevity of hi-tech products.
Originality/value
The study conducts a comparison of three different models considering time-varying parameters. There are four scenarios, considering variations in advertising intensity and content, word-of-mouth (WOM) effect, price volatility effect and sunk cost effect.

In this paper, we consider a pricing problem faced by a seller that sells a given inventory of some product over a short selling horizon with limited demand information. The seller knows only that the demand is a linear function of the price, but does not know the parameters involved in the demand function. However, the seller knows that each parameter involved in the demand function belongs to a known interval. The seller's objective is to determine the optimal price for the entire selling season to minimize the maximum regret, where the maximum regret is defined as the maximum possible loss of revenue due to not knowing the precise values of the parameters. We derive closed-form optimal solutions for the problem under all possible cases of input parameters and identify some structural properties of the solution. We conduct computational tests to compare our modeling approach with several benchmark approaches and report related insights.

In the models that we studied thus far, we considered the decisions made by a single firm. The implicit assumption in our development was that the other firms do not react to the decisions of each other. Naturally, this is almost never the case. When a firm decreases its prices, fearing loss of customers, its competitors may also decrease its prices. Both online and brick-and-mortar retail stores consider the assortments offered by the other stores when making planning their assortments. There is vast literature on modeling competition. Nevertheless, despite the fact that competition is the rule rather than an exception and there is vast literature on modeling competition, the development of operational models that can drive real-time decision making under competition is in its infancy. In most operational models, it is often the case that the competition is ignored or modeled rather simplistically. Perhaps, the most important reason for this is that explicitly modeling competition often times results in intractable models. Thus, for the sake of computational tractability, the reactions of the other firms are ignored. Furthermore, the data that drive the operational models are often collected in a competitive environment, and one usually naively hopes that building a noncompetitive model driven by data collected in a competitive environment will take care of the competition itself, but of course, this hope is not based on any scientific evidence.

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.

We address the problem of simultaneous pricing of a line of several products, both complementary products and substitutes, with a number of distinct price differentiation classes for each product (e.g., volume discounts, different distribution channels, and customer segments) in both monopolistic and oligopolistic settings. We provide a generic framework to tackle this problem, consider several families of demand models, and focus on a real-world case-study example. We propose an iterative relaxation algorithm, and state sufficient conditions for convergence of the algorithm. Using historical sales and price data from a retailer, we apply our solution algorithm to suggest optimal pricing, and report on numerical results.

Thesis (S.M.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2008. This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. Includes bibliographical references (p. 143-146). In this thesis, we develop a pricing strategy that enables a firm to learn the behavior of its customers as well as optimize its profit in a monopolistic setting. The single product case as well as the multi product case are considered under different parametric forms of demand, whose parameters are unknown to the manager. For the linear demand case in the single product setting, our main contribution is an algorithm that guarantees almost sure convergence of the estimated demand parameters to the true parameters. Moreover, the pricing strategy is also asymptotically optimal. Simulations are run to study the sensitivity to different parameters.Using our results on the single product case, we extend the approach to the multi product case with linear demand. The pricing strategy we introduce is easy to implement and guarantees not only learning of the demand parameters but also maximization of the profit. Finally, other parametric forms of the demand are considered. A heuristic that can be used for many parametric forms of the demand is introduced, and is shown to have good performance in practice. by Thibault Le Guen. S.M.

In this paper we propose a methodology to set prices of perishable items in the context of a retail chain with coordinated prices among its stores and compare its performance with actual practice in a real case study. We formulate a stochastic dynamic programming problem and develop heuristic solutions that approximate optimal solutions satisfactorily. To compare this methodology with current practices in the industry, we conducted two sets of experiments using the expertise of a product manager of a large retail company in Chile. In the first case, we contrast the performance of the proposed methodology with the revenues obtained during the 1995 autumn-winter season. In the second case, we compare it with the performance of the experienced product manager in a "simulation-game" setting. In both cases, our methodology provides significantly better results than those obtained by current practices.

Abstract Practical policies for the monopolistic pricing problem with uncertain demand are discussed (for discrete time, continuous prices and demand, in a linear and Gaussian setting). With this model, the introduction of price vari- ations is rationally justified, to allow for a better estimate of the elasticity of demand, and increased profits due to better pricing. An approximation of the dynamic programming solution is introduced, exploiting convex op- timization methods for computational tractability. Numerical experiments are described. First draft April 1999 (talk at INFORMS Cincinnati, May 1999), second draft June

Many industries face the problem of selling a fixed stock of items over a finite horizon. These industries include airlines selling seats before planes depart, hotels renting rooms before midnight, theaters selling seats before curtain time, and retailers selling seasonal goods such as air-conditioners or winter coats before the end of the season. Given a fixed number of seats, rooms, or coats, the objective for these industries is to maximize revenues in excess of salvage value. When demand is price sensitive and stochastic, pricing is an effective tool to maximize revenues. In this paper we address the problem of deciding the optimal timing of a single price change from a given initial price to either a given lower or higher second price. Under mild conditions, we show that it is optimal to decrease (resp., to increase) the initial price as soon as the time-to-go falls below (resp., above) a time threshold that depends on the number of yet unsold items.

Consider a firm that owns a fixed capacity of a resource that is consumed in the production or delivery of multiple products. The firm strives to maximize its total expected revenues over a finite horizon, either by choosing a dynamic pricing strategy for each product or, if prices are fixed, by selecting a dynamic rule that controls the allocation of capacity to requests for the different products. This paper shows how these well-studied revenue management problems can be reduced to a common formulation in which the firm controls the aggregate rate at which all products jointly consume resource capacity, highlighting their common structure, and in some cases leading to algorithmic simplifications through the reduction in the control dimension of the associated optimization problems. In the context of their associated deterministic (fluid) formulations, this reduction leads to a closed-form characterization of the optimal controls, and suggests several natural static and dynamic pricing heuristics. These are analyzed asymptotically and through an extensive numerical study. In the context of the former, we show that "resolving" the fluid heuristic achieves asymptotically optimal performance under fluid scaling.

In this paper, we present a robust optimization formulation for dealing with demand uncertainty in a dynamic pricing and inventory
control problem for a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce
a demand-based fluid model where the demand is a linear function of the price, the inventory cost is linear, the production
cost is an increasing strictly convex function of the production rate and all coefficients are time-dependent. A key part
of the model is that no backorders are allowed. We show that the robust formulation is of the same order of complexity as
the nominal problem and demonstrate how to adapt the nominal (deterministic) solution algorithm to the robust problem.

The benefits of dynamic pricing methods have long been known in industries, such as airlines, hotels, and electric utilities, where the capacity is fixed in the short-term and perishable. In recent years, there has been an increasing adoption of dynamic pricing policies in retail and other industries, where the sellers have the ability to store inventory. Three factors contributed to this phenomenon: (1) the increased availability of demand data, (2) the ease of changing prices due to new technologies, and (3) the availability of decision-support tools for analyzing demand data and for dynamic pricing. This paper constitutes a review of the literature and current practices in dynamic pricing. Given its applicability in most markets and its increasing adoption in practice, our focus is on dynamic (intertemporal) pricing in the presence of inventory considerations.

This survey reviews the forty-year history of research on transportation revenue management (also known as yield management). We cover developments in forecasting, overbooking, seat inventory control, and pricing, as they relate to revenue management, and suggest future research directions. The survey includes a glossary of revenue management terminology and a bibliography of over 190 references. In the forty years since the first publication on overbooking control, passenger reservations systems have evolved from low level inventory control processes to major strategic information systems. Today, airlines and other transportation companies view revenue management systems and related information technologies as critical determinants of future success. Indeed, expectations of revenue gains that are possible with expanded revenue management capabilities are now driving the acquisition

We present an optimization approach for jointly learning the demand as a functionof price, and dynamically setting prices of products in an oligopoly environment in order to maximize expected revenue. The models we consider do not assume that the demand as a function of price is known in advance, but rather assume parametric families of demand functions that are learned over time. We first consider the noncompetitive case and present dynamic programming algorithms of increasing computational intensity with incomplete state information for jointly estimating the demand and setting prices as time evolves. Our computational results suggest that dynamic programming based methods outperform myopic policies often significantly. We then extend our analysis in a competitive environment with two firms. We introduce a more sophisticated model of demand learning, in which the price elasticities are slowly varying functions of time, and allows for increased flexibility in the modeling of the demand. We propose methods based on optimization for jointly estimating the Firm's own demand, its competitor's demand, and setting prices. In preliminary computational work, we found that optimization based pricing methods offer increased expected revenue for a firm independently of the policy the competitor firm is following.

In this paper, we introduce a fluid model of dynamic pricing and inventory management for make-to-stock manufacturing systems. Instead of considering a traditional model that is based on how price affects demand, we consider a model that relies on how price and level of inventory affect the time a unit of product remains in inventory. Our motivation is based on the observation that in inventory systems, a unit of product incurs a delay before being sold. This delay depends on the unit price of the product, prices of competitors, and the level of inventory of this product. Moreover, delay data is not hard to acquire and is internally controlled and monitored by the manufacturer. It is interesting to notice that this delay is similar to travel times incurred in a transportation network. The model of this paper includes joint pricing, production and inventory decisions in a competitive, capacitated multi-product dynamic environment. In particular, in this paper we (i) introduce a model for dynamic pricing and inventory control that uses delay rather then demand data and establish connections with traditional demand models, (ii) study analytical properties of this model, (iii) establish conditions under which the model has a solution and finally, (iv) establish an algorithm that solves efficiently a discretized version of the model.

This paper considers the problem of changing prices over time to maximize expectedrevenues in the presence of unknown demand distribution parameters. It providesand compares several methods that use the sequence of past prices and observeddemands to set price in the current period. A Taylor series expansion of the futurereward function explicitly illustrates the tradeoff between short term revenuemaximization and future information gain and suggests a promising pricing policyreferred to as a one-step look-ahead rule. An in-depth Monte Carlo study comparesseveral different pricing strategies and shows that the one-step look-ahead rulesdominate other heuristic policies and produce good short term performance. Thereasons for the observed bias of parameter estimates are also investigated.

The authors provide game theoretic foundations for the classic kinke d demand curve and Edgeworth cycle. In their alternating-move model, there are multiple Markov perfect equilibria of both the kinked deman d curve and Edgeworth cycle variety. In any Markov perfect equilibria , profit is bounded away from the Bertrand equilibria level. A kinked demand curve at the monopoly price is the unique symmetric "renegot iation proof" equilibrium when there is little discounting. The auth ors then endogenize the timing by allowing firms to move at any time. They find that firms end up alternating, thus vindicating the fixed timing assumption of the simpler model. Copyright 1988 by The Econometric Society.

In this paper, we develop a stylized partially observed Markov decision process (POMDP) framework, to study a dynamic pricing problem faced by sellers of fashion-like goods. We consider a retailer that plans to sell a given stock of items during a finite sales season. The objective of the retailer is to dynamically price the product in a way that maximizes expected revenues. Our model brings together various types of uncertainties about the demand, some of which are resolvable through sales observations. We develop a rigorous upper bound for the seller’s optimal dynamic decision problem and use it to propose an active-learning heuristic pricing policy. We conduct a numerical study to test the performance of four different heuristic dynamic pricing policies, in order to gain insights into several important managerial questions that arise in the context of revenue management.

We consider the general traffic equilibrium network model where the travel cost on each link of the transportation network may depend on the flow on this as well as other links of the network. The model has been designed in order to handle situations where there is interaction between traffic on different links e.g., two-way streets, intersections or between different modes of transportation on the same link. For this model, we use the techniques of the theory of variational inequalities to establish existence of a traffic equilibrium pattern, to design an algorithm for the construction of this pattern and to derive estimates on the speed of convergence of the algorithm.

This paper addresses the simultaneous determination of pricing and inventory replenishment strategies in the face of demand uncertainty. More specifically, we analyze the following single item, periodic review model. Demands in consecutive periods are independent, but their distributions depend on the item's price in accordance with general stochastic demand functions. The price charged in any given period can be specified dynamically as a function of the state of the system. A replenishment order may be placed at the beginning of some or all of the periods. Stockouts are fully backlogged. We address both finite and infinite horizon models, with the objective of maximizing total expected discounted profit or its time average value, assuming that prices can either be adjusted arbitrarily (upward or downward) or that they can only be decreased. We characterize the structure of an optimal combined pricing and inventory strategy for all of the above types of models. We also develop an efficient value iteration method to compute these optimal strategies. Finally, we report on an extensive numerical study that characterizes various qualitative properties of the optimal strategies and corresponding optimal profit values.

It is a common practice for industries to price the same products at different levels. For example, airlines charge various fares for a common pool of seats. Seasonal products are sold at full or discount prices during different phases of the season. This article presents a model that reflects this yield management problem. The model assumes that (1) products are offered at multiple predetermined prices over time; (2) demand is price sensitive and obeys the Poisson process; and (3) price is allowed to change monotonically, i.e., either the markup or markdown policy is implemented. To maximize the expected revenue, management needs to determine the optimal times to switch between prices based on the remaining season and inventory. Major results in this research include (1) an exact solution for the continuous-time model; (2) piecewise concavity of the value function with respect to time and inventory; and (3) monotonicity of the optimal policy. The implementation of optimal policies is fairly facile because of the existence of threshold points embedded in the value function. The value function and time thresholds can be solved with a reasonable computation effort. Numerical examples are provided.

A firm has inventories of a set of components that are used to produce a set of products. There is a finite horizon over which the firm can sell its products. Demand for each product is a stochastic point process with an intensity that is a function of the vector of prices for the products and the time at which these prices are offered. The problem is to price the finished products so as to maximize total expected revenue over the finite sales horizon. An upper bound on the optimal expected revenue is established by analyzing a deterministic version of the problem. The solution to the deterministic problem suggests two heuristics for the stochastic problem that are shown to be asymptotically optimal as the expected sales volume tends to infinity. Several applications of the model to network yield management are given. Numerical examples illustrate both the range of problems that can be modeled under this framework and the effectiveness of the proposed heuristics. The results provide several fundamental insights into the performance of yield management systems.

James Friedman provides a thorough survey of oligopoly theory using numerical examples and careful verbal explanations to make the ideas clear and accessible. While the earlier ideas of Cournot, Hotelling, and Chamberlin are presented, the larger part of the book is devoted to the modern work on oligopoly that has resulted from the application of dynamic techniques and game theory to this area of economics. The book begins with static oligopoly theory. Cournot's model and its more recent elaborations are covered in the first substantive chapter. Then the Chamberlinian analysis of product differentiation, spatial competition, and characteristics space is set out. The subsequent chapters on modern work deal with reaction functions, advertising, oligopoly with capital, entry, and oligopoly using noncooperative game theory. A large bibliography is provided.

Optimal operating policies and corresponding managerial insight are developed for the decision problem of coordinating supply and demand when (i) both supply and demand can be influenced by the decision maker and (ii) learning is pursued. In particular, we determine optimal stocking and pricing policies over time when a given market parameter of the demand process, though fixed, initially is unknown. Because of the initially unknown market parameter, the decision maker begins the problem horizon with a subjective probability distribution associated with demand. Learning occurs as the firm monitors the market's response to its decisions and then updates its characterization of the demand function. Of primary interest is the effect of censored data since a firm's observations often are restricted to sales. We find that the first-period optimal selling price increases with the length of the problem horizon. However, for a given problem horizon, prices can rise or fall over time, depending on how the scale parameter influences demand. Further results include the characterization of the optimal stocking quantity decision and a computationally viable algorithm. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 303–325, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10013

Revenue management techniques, practiced for many years in the airline and hotel industries, have gained popularity with retailers. One such technique – dynamic pricing – refers to the sophisticated process of controlling prices over a course of a sales season in a way that maximizes expected revenues. In this paper, we consider a retailer who sells a fashionable good during a short sales season, but is uncertain about how successful the product will be in the market. Will it attract customer attention or not? and to what extent? We propose a continuous-time revenue-maximization model in which the seller continuously learns about the market reaction to his product from sales observations. Customers arrive randomly over the course of the season, and each customer has an individual reservation price, based on which he makes his own purchasing decision. We combine theoretical results with numerical experimentation to study the value of dynamic pricing for a retailer faced with market uncertainty. We compare optimal expected revenues to those obtained under two alternative pricing policies: …xed pricing schemes, and certainty-equivalent policies. The comparative study allows us to explore the value of proactively setting prices to impact the revenue as well as the learning process itself ("active learning") versus "passive learning," where learning is performed continuously, but the impact of prices on learning is ignored. We also consider the value of obtaining perfect information about the market conditions. We wish to thank the Boeing Center for Technology, Information, and Manufacturing (BCTIM), for providing funding to support this research.

In this paper, we present a continuous time optimal control model for studying a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multiproduct capacitated, dynamic setting. We introduce a demand-based model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate, and all coefficients are time-dependent. A key part of the model is that no backorders are allowed. We introduce and study an algorithm that computes the optimal production and pricing policy as a function of the time on a finite time horizon, and discuss some insights. Our results illustrate the role of capacity and the effects of the dynamic nature of demand in the model. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007

In this paper, we examine the research and results of dynamic pricing policies and their relation to revenue management. The survey is based on a generic revenue management problem in which a perishable and nonrenewable set of resources satisfy stochastic price sensitive demand processes over a finite period of time. In this class of problems, the owner (or the seller) of these resources uses them to produce and offer a menu of final products to the end customers. Within this context, we formulate the stochastic control problem of capacity that the seller faces: How to dynamically set the menu and the quantity of products and their corresponding prices to maximize the total revenue over the selling horizon.

Clearance pricing and end of season inventory management are challenging and important problems in retailing. Sales rates depend upon price, seasonal effects, and the remaining assortment of items available to customers. There is little time to react to observed sales, and pricing errors result in either loss of potential revenue or excess inventory to be liquidated. This paper develops optimal clearance prices and inventory management policies that take into account the impact of reduced assortment and seasonal changes on sales rates. Versions of these policies have been tested and implemented at three major retail chains and these applications are summarized and discussed.

This paper studies intertemporal pricing policies when selling seasonal products in retail stores. We first present a continuous time model where a seller faces a stochastic arrival of customers with different valuations of the product. For this model, we characterize the optimal pricing policies as functions of time and inventory. We use this model as a benchmark against which we compare more realistic models that consider periodic pricing reviews. We show that the structure of the optimal pricing policies in this case is consistent with the procedures observed in practice; retail stores successively discount the product during the season and promote a liquidation sale at the end of the planning horizon. We also show that the loss experienced when implementing periodic pricing reviews instead of continuous policies is small when the appropriate number of reviews is chosen. Several interesting economic insights emerge from our analysis. For example, uncertainty in the demand for new products leads to higher prices, larger discounts, and more unsold inventory. Finally, we study the effect of announced discount policies on prices and profits. We show that stores that have adopted this type of strategy usually set prices such that with high probability the merchandise is sold during the first periods and the largest discounts rarely take place.

This paper considers the relationship between pricing and ordering decisions for a monopolistic retailer facing a known demand function where, over the inventory cycle, the product may exhibit: (i) physical decay or deterioration of inventory called wastage; and (ii) decrease in market value called value drop associated with each unit of inventory on hand. The retailer is allowed to continuously vary the selling price of the product over the cycle. We introduce a notion of instantaneous margin, and use it to derive profit maximizing conditions for the retailer.
The model explains the markdown of retail goods subject to decay. It also provides guidance in determining when price changes during the cycle are worthwhile due to product aging, how often such changes should be made, and how such changes affect ordering intervals and quantities.

This paper deals with the determination of optimal pricing policies for firms in oligopolistic markets. The problem is studied as a differential game and optimal pricing policies are established as Nash open-loop controls. Cost learning effects are assumed such that unit costs are decreasing with cumulative output. Discounting of future profits is also taken into consideration.
Initially, the problem is addressed in a general framework, and we proceed to study some specific cases that are related to models presented in recent literature. Three basic classes of sales dynamics are analyzed: competition with price effects only, competition with price as well as adoption effects, and competition with adoption effects only. In some cases it turns out that results which hold for the monopoly case, carry over to the multi-firm case, in the sense that the qualitative structure of optimal pricing strategies is the same in the monopoly and the oligopoly cases. However, due to competitive interdependencies, differences certainly exist in the levels as well as the rates of change of optimal prices.

The advent of optical scanning devices and decreases in the cost of computing power have made it possible to assemble databases with sales and marketing mix information in an accurate and timely manner. These databases enable the estimation of demand functions and pricing/promotion decisions in “real” time. Commercial suppliers of marketing research like A. C. Nielsen and IRI are embedding estimated demand functions in promotion planning and pricing tools for brand managers and retailers.
This explosion in the estimation and use of demand functions makes it timely and appropriate to re-examine several fundamental issues. In particular, demand functions are latent theoretical constructs whose exact parametric form is unknown. Estimates of price elasticities, profit maximizing prices, inter-brand competition and other policy implications are conditional on the parametric form assumed in estimation. In practice, many forms may be found that are not only theoretically plausible but also consistent with the data. The different forms could suggest different profit maximizing prices leaving it unclear as to what is the appropriate pricing action. Specification tests may lack the power to resolve this uncertainty, particularly for non-nested comparisons. Also, the structure of these tests does not permit seamless integration of estimation, specification analysis and optimal pricing into a unified framework.
As an alternative to the existing approaches, I propose a Bayesian mixture model (BMM) that draws on Bayesian estimation, inference, and decision theory, thereby providing a unified framework. The BMM approach consists of input, estimation, diagnostic and optimal pricing modules. In the input module, alternate parametric models of demand are specified along with priors. Utility structures representing the decision maker's attitude towards risk can be explicitly specified. In the estimation module, the inputs are combined with data to compute parameter estimates and posterior probabilities for the models. The diagnostic module involves testing the statistical assumptions underlying the models. In the optimal pricing module the estimates and posterior probabilities are combined with the utility structure to arrive at optimal pricing decisions.
Formalizing demand uncertainty in this manner has many important payoffs. While the classical approaches emphasize choosing a demand specification, the BMM approach emphasizes constructing an objective function that represents a mixture of the specifications. Hence, pricing decisions can be arrived at even when there is no consensus among the different parametric specifications. The pricing decisions will reflect parametric demand uncertainty, and hence be more robust than those based on a single demand model.
The BMM approach was empirically evaluated using store level scanner data. The decision context was the determination of equilibrium wholesale prices in a noncooperative game between several leading national brands. Retail demand was parametrized as semilog and doublelog with diffuse priors for the models and the parameters. Wholesale demand functions were derived by incorporating the retailers' pricing behavior in the retail demand function. Utility functions reflecting risk averse and risk neutral decision makers were specified. The diagnostic module confirms that face validity measures, residual analysis, classical tests or holdout predictions were unable to resolve the uncertainty about the parametric form and by implication the uncertainty with regard to pricing decisions. In contrast, the posterior probabilities were more conclusive and favored the specification that predicted better in a holdout analysis. However, across the brands, they lacked a systematic pattern of updating towards any one specification. Also, none of the priors updated to zero or one, and there was considerable residual uncertainty about the parametric specification.
Despite the residual uncertainty, the BMM approach was able to determine the equilibrium wholesale prices. As expected, specifications influence the BMM pricing solutions in accordance with their posterior probabilities which act as weights. In addition, differing attitudes towards risk lead to considerable divergence in the pricing actions of the risk averse and the risk neutral decision maker. Finally, results from a Monte Carlo experiment suggest that the BMM approach performs well in terms of recovering potential improvements in profits.

This paper studies the problem of a monopoly who is uncertain about the demand it faces and learns about it over time through its pricing experience. The demand curve facing the monopoly is not constant — it changes over time in a Markovian fashion. We characterize the monopoly's optimal policy and inquire how it differs from an informed monopoly's policy. It turns out that, even when the rate at which the demand varies is negligible, the stationary probability with which the monopoly's policy deviates from its informed counterpart is nonnegligible, as long as the discount factor is below 1.

We study a multi-period oligopolistic market for a single perishable product with fixed inventory. Our goal is to address
the competitive aspect of the problem together with demand uncertainty using ideas from robust optimization and variational
inequalities. The demand function for each seller has some associated uncertainty and we assume that the sellers would like
to adopt a policy that is robust to adverse uncertain circumstances. We believe this is the first paper that uses robust optimization
for dynamic pricing under competition. In particular, starting with a given fixed inventory, each seller competes over a multi-period
time horizon in the market by setting prices and protection levels for each period at the beginning of the time horizon. Any
unsold inventory at the end of the horizon is worthless. The sellers do not have the option of periodically reviewing and
replenishing their inventory. We study non-cooperative Nash equilibrium policies for sellers under such a model. This kind
of a setup can be used to model pricing of air fares, hotel reservations, bandwidth in communication networks, etc. In this
paper we demonstrate our results through some numerical examples.

We consider a two-echelon distribution system in which a supplier distributes a product toN competing retailers. The demand rate of each retailer depends on all of the retailers' prices, or alternatively, the price each retailer can charge for its product depends on the sales volumes targeted by all of the retailers. The supplier replenishes his inventory through orders (purchases, production runs) from an outside source with ample supply. From there, the goods are transferred to the retailers. Carrying costs are incurred for all inventories, while all supplier orders and transfers to the retailers incur fixed and variable costs. We first characterize the solution to the centralized system in which all retailer prices, sales quantities and the complete chain-wide replenishment strategy are determined by a single decision maker, e.g., the supplier. We then proceed with the decentralized system. Here, the supplier chooses a wholesale pricing scheme; the retailers respond to this scheme by each choosing all of his policy variables. We distinguish systematically between the case of Bertrand and Cournot competition. In the former, each retailer independently chooses his retail price as well as a replenishment strategy; in the latter, each of the retailers selects a sales target, again in combination with a replenishment strategy. Finally, the supplier responds to the retailers' choices by implementing his own cost-minimizing replenishment strategy. We construct a perfect coordination mechanism. In the case of Cournot competition, the mechanism applies a discount from a basic wholesale price, based on thesum of three discount components, which are a function of (1) annual sales volume, (2) order quantity, and (3) order frequency, respectively.

We examine a monopoly facing an uncertain demand and maximizing profits over a two-period horizon. Conditions are developed under which the firm will find it optimal to "experiment," or adjust initial prices or quantities away from their myopically optimal level in order to increase the informativeness of observed market outcomes and hence increase future profits. We establish conditions under which experimentation will lead a quantity-setting firm to increase or decrease quantity. Finally, we develop conditions under which experimenting firms will choose to be either price-setters or quantity-setters. Copyright 1993 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

The problem of controlling a stochastic process, with unknown parameters over an infinite horizon, with discounting is considered. Agents express beliefs about unknown parameters in terms of distributions. Under general conditions, the sequence of beliefs converges to a limit distribution. The limit distribution may or may not be concentrated at the true parameter value. In some cases, complete learning is optimal; in others, the optimal strategy does not imply complete learning. The paper concludes with examination of some special cases and a discussion of a procedure for generating examples in which incomplete learning is optimal. Copyright 1988 by The Econometric Society.

By observing the quantity demanded at particular prices, a firm may learn about the parameters of its demand curve. In such an environment, price changes obstruct the learning process by inducing additional noise. The authors' paper constructs a dynamic model where a price-setting firm endogenously controls the speed of learning. The model provides a possible explanation for price inertia, as a stable pricing policy allows the firm to learn more rapidly, which improves future expected profits. Furthermore, even in the long run, learning continues to affect the firm's optimal price. Copyright 1990 by Royal Economic Society.

Sellers of new products are faced with having to guess demand conditions to set price appropriately. But sellers are able to adjustprice over time and to learn from past mistakes. Additionally, it is not necessary that all goods be sold with certainty. It is sometimes better to set a high price and to risk no sale. This process is modeledto explain retail pricing behavior and the time distribution of transactions. Prices start high and fall as a function of time on theshelf. The initial price and rate of decline can be predicted and depends on thinness of the market, the proportion of customers who are"window shoppers," and other observable characteristics. Copyright 1986 by American Economic Association.

Optimal Markdown Mechanisms in the Presence of Rational Customers with Multi-unit Demand

- W Elmaghraby
- A Guleu
- P Keskinocak

Dynamic Pricing and Ordering Decisions by a Monopolist View Article

- A Rajan
- Rakash
- R Steinberg

Dynamic Pricing Strategies Under Repeated Interactions. Working Paper, INSEAD

- Y Popescu
- Y Wu

Revenue Management: Research Overview and Prospects. Transportation Science: Focused Issue on Yield Management in Transportation

- J I Mcgill
- Ryzin
- G Van

Revenue Management: Research Overview and Prospects. Transportation Science: Focused Issue on Yield Management in

- J I Mcgill
- Ryzin