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On the effect of competition and strategic consumer behavior in revenue management Mantin, Binyamin 2008

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On the Effect of Competition and Strategic Consumer Behavior in Revenue Management by Binyamin Mantin A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Business Administration) THE UNIVERSITY OF BRITISH COLUMBIA February 2008 c© Binyamin Mantin, 2008 ii Abstract In this thesis we investigate important issues in the area of dynamic pricing for revenue management. Studying the effect of competition and strategic consumer behavior, we characterize the dynamic pricing policies for retailers who sell homogeneous goods in multi-period, discrete time, finite horizon settings. In the first essay an impatient consumer visits only one of two competing retailers in each period. If he does not purchase the good, he visits the competing retailer in the ensuing period. Compared to the corresponding single store monopoly, when the consumer’s valuation is uniformly distributed, prices decline exponentially rather than linearly, with a dramatically lower initial price, and a substantially lower system profit. The model is extended to accommodate many consumers, who may be either identical or similar, a more general valuation distribution, and situations wherein capacities are limited. The base case of a centralized two-store monopoly is also examined. In the second essay the consumer may return to the same retailer with some certain probability. This probability is either affected by market structure characteristics, or it may depend on the consumer’s experience at the last store visited. The robustness of the exponential decline of prices is reinforced. It occurs even when a strong retailer faces competition from a relatively much weaker retailer. We investigate the impact of the return probabilities on prices, profits, and consumer surplus. The model is extended to an oligopoly, and to situations with many similar consumers. The effect of strategic consumer behavior on prices and profits is revealed in the third essay. Characterizing the pricing policies arising in a two-period monopoly and duopoly settings, we find that strategic consumer behavior inflicts larger losses to a duopoly than to a monopoly. A lower strategic consumers’ discounting factor, which is beneficial to a monopoly, may be harmful to a duopoly. Ignoring strategic consumer behaviour is costly to a monopoly, but may, on the other hand, be beneficial to a duopoly. An extension to three periods is studied, and with longer horizons the model is analyzed for the case when all the consumers are strategic. iii Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Co-Authorship Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Revenue Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Preliminaries and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Consumer Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Overview of the Essays and Summary of Contributions . . . . . . . . . . . . . . . . . 7 1.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 2 A Dynamic Pricing Model under Duopoly Competition . . . . . . . . . 12 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Uniformly Distributed Valuation . . . . . . . . . . . . . . . . . . . . . . . . . 19 iv 2.2.3 Values at Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.4 N Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Extensions to the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Different Valuation Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2 Limited Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Identical Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.1 Uncapacitated Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.2 Capacitated Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Two-Store Monopoly with N Consumers . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5.1 Two-Store Monopoly with N Similar Consumers . . . . . . . . . . . . . . . . 39 2.5.2 Two-Store Monopoly with N Identical Consumers . . . . . . . . . . . . . . . 39 2.6 Managerial Insight and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 3 Price Skimming in the Presence of Competition: Multi-Period Markov Store Choice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 The Single Consumer Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 Duopoly: The Single Consumer Case . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.2 Oligopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 N Similar Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Structural Duopoly with N Consumers . . . . . . . . . . . . . . . . . . . . . 65 3.3.2 Behavioral Duopoly with N Consumers . . . . . . . . . . . . . . . . . . . . . 71 3.4 Summary and Managerial Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4.1 Managerial Insight and Future Work . . . . . . . . . . . . . . . . . . . . . . . 71 3.4.2 Discussion of Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Chapter 4 Dynamic Pricing of Perishable Goods for a Monopoly and a Duopoly in the Presence of Strategic Consumers . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 v4.2 A Two-Period Selling Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2.2 Monopoly – Two-Period Selling Horizon . . . . . . . . . . . . . . . . . . . . . 88 4.2.3 Duopoly – Two-Period Selling Horizon . . . . . . . . . . . . . . . . . . . . . . 92 4.2.4 Implications of Competition and Consumers’ Strategic Behavior . . . . . . . 95 4.2.5 Obliviousness to Consumers’ Strategic Behavior . . . . . . . . . . . . . . . . . 98 4.2.6 The Case of Perfect Markets: λ = δ . . . . . . . . . . . . . . . . . . . . . . . 100 4.3 Longer Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.1 A Three-Period Selling Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.2 T-Period Selling Horizon: Only Strategic Consumers . . . . . . . . . . . . . . 104 4.4 Summary and Managerial Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Chapter 5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.1 Proofs of Lemmas, Propositions, and Theorems in Chapter 2 . . . . . . . . . . . . . 117 A.2 Single Store Monopoly with a Consumer’s Valuation Drawn from a Uniform [a,b] Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.3 Proofs of Lemmas, Propositions, and Theorems in Chapter 3 . . . . . . . . . . . . . 127 A.4 N Similar Consumers with Demand Updating . . . . . . . . . . . . . . . . . . . . . . 134 A.4.1 Two-Period Structural Duopoly with N Similar Consumers . . . . . . . . . . 134 A.4.2 Two-Period Behavioral Duopoly with N Similar Consumers . . . . . . . . . . 137 A.4.3 Two-Period Structural Duopoly with Limited Inventory . . . . . . . . . . . . 138 A.5 Proofs of Lemmas, Propositions, and Theorems in Chapter 4 . . . . . . . . . . . . . 140 vi List of Tables 2.1 Price ratios, prices, and profits under zigzag competition and monopoly settings . . . . . . 19 2.2 Duopoly and monopoly prices and profits in a two-period selling horizon when consumers are identical and Ri1 ≤ Rj1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1 Structural duopoly: Prices and profits-to-go in the final two periods . . . . . . . . . . . . . 53 3.2 Prices and consumers visit pattern in the first three periods of a structural duopoly when Pricing Policy (1) is employed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 vii List of Figures 2.1 A 20-period model: price, price ratio and profit-to-go . . . . . . . . . . . . . . . . . . . . 22 2.2 Profit, profit loss, and consumer surplus in monopoly and duopoly under zigzag competition 23 2.3 Prices in a 10-period selling horizon when valuation follows a power distribution . . . . . . 27 2.4 System profit for different selling horizons when valuation follows a power distribution . . . 27 2.5 Prices a monopolist sets in a 20-period model when K = 1 for varying values of N . . . . . 29 2.6 Prices a monopolist sets in a 2-period model and the corresponding expected profit when N = 10 for varying values of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 First period prices and profits in a 2-period model when total capacity is 4 . . . . . . . . . 32 2.8 Prices in a five-period model under competition with limited capacity . . . . . . . . . . . . 34 2.9 The centralized system minimal and maximal profit and their ratio . . . . . . . . . . . . . 40 3.1 Structural duopoly: Price ratios and prices . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Structural duopoly: values at convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Behavioral symmetric duopoly: Bt in the final periods, which coincides with the initial price in a horizon with T − t+ 1 periods, and the convergence graph of Bt, B . . . . . . . . . . 57 3.4 Behavioral symmetric duopoly: Prices and consumer’s surplus in a 20-period selling horizon 58 3.5 Structural symmetric oligopoly: Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.6 Behavioral symmetric oligopoly: Prices and profits at convergence . . . . . . . . . . . . . . 65 3.7 Structural duopoly: Retailer 1’s initial price - single vs. N consumers, T = 2, .., 5 . . . . . . 66 3.8 Structural duopoly with N consumers: Prices and profits when T = 5 . . . . . . . . . . . . 70 3.9 Structural duopoly: Prices set by retailers for T = 10: single consumer vs. N consumers . . 70 3.10 Behavioral symmetric duopoly: N -consumer vs. single-consumer cases . . . . . . . . . . . 72 3.11 Prices in a 10-period selling horizon when valuation follows a power distribution . . . . . . 75 viii 4.1 Prices set by a monopolist in a 2-period model . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 System expected profits in a 2-period model with α strategic consumers, λ = 1 . . . . . . . 96 4.3 System absolute profit loss due to consumers’ strategic behavior in a 2-period model (the circle indicates larger monopoly loss than duopoly loss) . . . . . . . . . . . . . . . . . . . 97 4.4 Duopoly total and per period expected profit per consumer: ignorance vs. no ignorance of consumers’ strategic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5 Monopoly’s pricing policy and expected profit under perfect markets, λ = δ . . . . . . . . . 101 4.6 Monopolist’s pricing strategy zones, λ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.7 Retailers’ combined profit at convergence when all consumers are strategic . . . . . . . . . 106 4.8 Prices in a 20-period horizon when all consumers are strategic with λ = δ < 1 . . . . . . . . 107 4.9 Retailers’ combined profit at convergence when all consumers are strategic with λ = δ < 1 . 108 A.1 Structural duopoly: Retailer i’s initial price for different N values, T = 2 . . . . . . . . . . 136 A.2 Structural symmetric duopoly with N similar consumers: Initial price when inventory is limited, T = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 ix Acknowledgements I am deeply grateful to my supervisors Professor Daniel Granot and Professor Frieda Granot, who have continuously supported and encouraged me throughout the entire duration of my research. Their broad support, reflected through their endless effort in directing me through the different steps of the program, and very generous support, has been essential and critical in enabling me reach this ultimate goal. Their tremendous effort in reviewing the research work and insistence on ever improving process have definitely shaped the three essays the compose the thesis. I would like to mention that Daniel and Frieda Granot are also the Godparents of my firstborn son. I am also grateful to Professor Charles Weinberg, who has played an essential role on my thesis committee. His invaluable comments and insights have contributed immensely to the quality of my essays. His marketing perspective has definitely enriched my thesis to its current level. Many thanks go to all the other faculty members at UBC, who were willing to offer their assistance and support in many different ways, and especially to Professor Mahesh Nagarajan, who has always been around to offer his ideas and commentary; to all the graduate students, with whom I have shared my time at UBC; and to the many others who have contributed in one way or another to the achievement of the goal and the final product, called Ph.D thesis. I would like to thank Professor Asher Tishler, who has been my Master’s thesis supervisor in Tel-Aviv. It is his vision and direction which has led me eventually to choose the Ph.D. at UBC. The time I have spent pursuing my Ph.D. at UBC has been completely rewarding and produc- tive. Asides from achieving the main goal of obtaining a Ph.D. degree, I have become a father to two wonderful kids, Bar and Emma, and a husband to a great wife, Nicole. My closest family was always there to provide support, warmth, and love. Finally, deep thanks go to my parents and siblings, who probably greet this thesis with a big sigh of relief. Thank you all! xDedication To my loving family. xi Co-Authorship Statement This is a manuscript-based thesis. Chapters 2 to 4, which can be read independently, have been, or will be, submitted to academically refereed journals for publication. The candidate, Binyamin Mantin, is responsible for conducting the mathematical analysis and writing these three manuscripts. Chapters 2 and 3 were closely supervised by Professor Daniel Granot and Professor Frieda Granot. Chapter 4 was solely supervised by Professor Daniel Granot. Professors Daniel and Frieda Granot have provided a continuous and an invaluable guidance, feedback and support for this research work. All three chapters have benefited immensely from in-depth discussions with Professor Charles Weinberg, who has served as a committee member for this thesis. 1Chapter 1 Introduction Asking consumers how much they have paid for an identical good may reveal different answers depending on the timing of the purchase by the consumers, the location where the goods were purchased, and the type of consumers asked. This thesis studies dynamic pricing policies that result with such differences in prices. Modern economics suggests that goods have no normative right price. Actual prices in the market, which depend on sellers’ willingness to sell and buyers’ willingness to buy, can fluctuate dramatically. The variation in prices can be attributed to many factors, some of which are the base of this thesis work. Prices of homogenous goods can change, quite substantially, over time, across different locations, and from one distribution channel to another. Prices may vary from one store to another due to, for example, differences in locations of the stores or levels of service they provide. In the absence of such differences in attributes between different stores, prices of homogenous goods may still vary. Inter- temporal pricing and price variation between different distribution channels (e.g., between online and brick-and-mortar stores) allow firms to address different segments of consumers by charging them different prices. Firms are becoming increasingly more efficient in their market segmentation capabilities. Employing inter-temporal pricing policies, firms charge different prices at different times for otherwise seemingly completely identical goods. For some goods the price may decrease over time, as is the case for several consumer electronics, while for some other products the price may increase over time. This is usually the case in the travel industry, experienced by many, wherein the price charged well ahead of the travel date may be substantially different than the price charged shortly before that date. Indeed, certain types of consumers tend to purchase earlier, while others can set to make their choice only close to the actual departure date. These differences in prices are the works of pricing and revenue management techniques. 21.1 Revenue Management The three essays in this thesis are in the area of dynamic pricing within the richer context of revenue management. Pricing is a major concern for firms. If they price too high, they may not sell their goods, while if they price too low, potential profit may be lost. Finding the right price is not always a trivial decision for firms. Pricing can take different forms. For example, price can be set strategically to position a product with respect to other competing products in the market, or it can be set on a cost basis, a method also known as a cost-plus (or mark-up) pricing. The former and the latter pricing methods reflect marketing and accounting, respectively, pricing approaches. Revenue management is the operational approach to pricing. In a multi-period selling horizon, firms have the opportunity to change their prices during the selling horizon - a practice known as dynamic pricing. Dynamic pricing can be implemented in different ways. Penetration pricing, for example, prescribes lower initial prices to accelerate product adoption, and once such a momentum is gained, prices can be increased over time. On the other hand, price skimming (or markdown pricing), which is at the core of this work, prescribes ever decreasing prices over the selling horizon to address different segments of consumers in the market. Namely, seeking to price discriminate among a finite population of consumers, high- valuation consumers are targeted earlier than low-valuation consumers. Markdown pricing is a very common practice in retailing. It is employed for many consumer goods with limited life cycles. The ubiquity of markdown pricing in retailing, and especially in the fashion industry, is evident from the fact that, according to a market research from 2002, 78% of all apparel that was sold at American chains such as JCPenny, Sears, and Kohl’s was marked down (Top of the Net (2002)). Dynamic pricing, when perceived as an operational tool, can be classified under the broader umbrella of perishable asset revenue management, or simply revenue management, often referred to as yield management. There is no single unified definition to revenue management. In general, it refers to the set of activities aimed at maximizing firms’ stream of revenues from perishable goods: collection of relevant historical data, assessing and forecasting demand and its characteristics, designing the optimal mix of products to offer, their quantities, and their prices (and sometimes their rationing rules) over the duration of the selling horizon. It is the difference in consumers’ behavior and characteristics that gives the rise to revenue management practices, as revenue management 3is mostly concerned with segmentation: selling the right product to the right customer, at the right time, at the right price. For example, in the airline industry, where revenue management practices have already been employed for more than 20 years, it is well recognized that, in general, prices increase as the departure date approaches. This aims at targeting price sensitive consumers earlier, while reaping large revenues from the less flexible and less financially constrained business passengers.1 Revenue management has a long history of symbiotic implementation in the airline industry - a business which has popularized the use of revenue management and where these practices appear to be crucial, primarily since a huge number of different products as well as a tremendous number of transactions have to be continuously handled. Indeed, ticket prices may be updated several times during the day. Over the years, these practices were adopted by other leisure and travel businesses (such as hotels, car rentals, and concerts). Today, revenue management is not an obscure buzzword anymore. It is well accepted and used in other industries as well. Revenue management practices are very beneficial - American and Delta Airlines estimate that revenue management techniques contribute $500 and $300 million, respectively, per year to their total revenue (Boyd (1998)). Overlooking revenue management can be detrimental - infamous is the case of PeopleExpress Airlines which went bankrupt in 1987, while its competitors improved their revenue management schemes that enabled them to compete more aggressively on prices, and eventually to drive PeopleExpress out of the market. Revenue management has experienced an immense progress over the last years. The advances in this area are best summarized in the recent book by Talluri and van Ryzin (2004). Therein the authors provide a brief history of the revenue management area, explain important concepts relevant for revenue management, elaborate on the different practices employed, and review different industries that make use of these practices. Several other reviews are provided in Bitran and Caldentey (2003), Elmaghraby and Keskinocak (2003), and Chan et al. (2004). Together, they all deliver the current state of the revenue management area. The above mentioned reviews have identified several challenges facing this research area. Chief challenges include pricing of multiple products, incorporation of consumers behavior aspects (such 1In fact, airline pricing is far more complex than this simple example. Airlines, which operate and control a network of interconnected flights, commonly employ different booking limits (usually based on a method known as Expected Marginal Seat Revenue), other fare restrictions, and occasionally they allow for overbooking. 4as consumer choice and consumer rationality), integration with marketing research, as well as the effect of competition. The latter is of central value, as firms do not operate in a vacuum and, in reality, they face competition from a variety of sources. Competition cannot be ignored and need to be modelled explicitly. This thesis aims at providing insights into some of these important chal- lenges. Specifically, this thesis introduces and analyzes multi-period competitive models between retailers selling a homogenous good, and investigates the effect of strategic consumer behavior on both monopoly and duopoly settings. The essays in this thesis employ a game-theoretic approach to model competition and strategic consumer behavior. The next section provides some preliminaries and background to the three essays of the thesis. §1.2.1 summarizes the essentials of game theory, and §1.2.2 briefly reviews the key approaches in the literature to model consumer behavior. 1.2 Preliminaries and Background 1.2.1 Game Theory Game theory provides a formal framework for modelling and understanding the behavior and inter- action of agents (or players) in strategic situations, which are also known as games. Each game has to describe the agents who are involved in this interaction, the information available to each of the agents at the time they make their decision, their action space, which substantiate their strategies, and the payoffs to each of the agents as a function of the actions chosen by the agents. The principal aspect of game theory is the notion of an equilibrium. An equilibrium, if it exists in a game, allows to provide predictions about the strategic interaction in the real world, and an explanation for agents’ behavior that can be rationally sustained. Probably the most important equilibrium concept is the Nash equilibrium. In a joint strategy which constitutes a Nash equilibrium, none of the agents has an incentive to deviate from this equilibrium. This equilibrium may be of pure strategies or mixed strategies. A mixed strategy defines a probability space over an agent’s available strategies, such that he chooses a certain pure strategy with some probability. Games may have more than one possible Nash equilibrium. Some games may not posses a Nash equilibrium in pure strategies at all, but every (finite strategic form) game has at least one Nash equilibrium (in mixed strategies). 5Games are usually represented either in a normal (or strategic) form or in an extensive form. The former is commonly illustrated as a table, while the latter is associated with trees, normally representing dynamic games. In these trees, nodes represent decision epochs, and arcs are the different actions taken by the agents. The optimal strategies for extensive form games, assuming agents have perfect information, are found by backwards induction. That is, starting from the end, the game is solved backwards, revealing the optimal strategy set at each level of the game. Solving backwards, the game can be broken into smaller elements, known as subgames, each representing a game by itself. A subgame is a part of the complete game that has a single initial node, and it contains all successors thereof. The key idea behind the notion of subgame is to find, similar to the principle of optimality in dynamic programming, an equilibrium in the complete game that is simultaneously an equilibrium in each of its subgames. This idea was conceptualized by Selten’s subgame perfection, which is a refinement of Nash equilibrium into dynamic games. When we say that a strategy is subgame perfect Nash equilibrium (SPNE, or SPE), the strategy profile constitutes a Nash equilibrium in every subgame of the complete game. 1.2.2 Consumer Behavior Consumer behavior is a main driver for prices. Assumptions made about consumer behavior are at the core of price choices made by firms. These assumptions, which include, e.g., modelling demand functions, arrival of consumers, as well as their willingness to pay (which may be derived from their utility functions) and to wait, can substantially affect prices set by firms. There is a vast literature on consumer behavior, which studies both internal and external influ- ences. The different models that conceptualize consumer behavior can vary, sometimes dramatically, from each other, as they focus on different aspects and serve for different objectives. The difference in purchase decisions made by consumers can be explained by their diverse back- grounds, the information they posses, their different choice processes, and the different contextual settings they face. Following Lilien et al. (1992), consumer behavior is a process that starts with a need arousal, activated internally (e.g., hunger) or externally (e.g., advertisement), proceeds through a stage of information search, as often the arousal cannot be satisfied immediately, contin- ues with an evaluation stage, then consumers face their purchasing decision, and eventually they posses post-purchase feelings. Another dimension that affects consumers’ purchase behavior is the 6level of involvement in the purchase. For example, buying daily commodities does not require nearly as much (mental) effort as is the purchase of major appliances. As stated earlier, a variety of models exists in the literature to model consumer behavior. Here, we elaborate, briefly, only on stochastic models of brand choice,2 due to their relevance to the models introduced by the essays in this thesis. These stochastic models differ in the way they treat the feedback provided from previous purchase events on the current purchase event. Specifically, how the past affects the probability distribution of different brands in the current purchase event. In general, we can distinct between two types of models: Markov models and learning models. Markov models assume that only a certain amount of previous brand choice decisions affects this probability distribution. In learning models, the entire history bears relevance but more weight is assigned to recent purchases. Zero-order (Markov) models, which state that brand choice is completely independent of past purchases, are sometimes perceived as a separate type of a brand choice model. Markov brand choice models are widely used in the marketing literature, usually to investigate purchase decisions by consumers of goods, typically with low involvement. Zero-order choice models can generate the well known Bernoulli model, which, in exchange, is used in this context to calculate purchase probabilities, which represent market share. In a first-order stationary Markov model, only the immediate previous purchase incidence affects the purchase probabilities in the current purchase decision, and these probabilities are time invariant. Let Xt denote the brand chosen on the tth purchase incidence, then the probability a consumer purchases brand j given his purchase history is P (Xt = j|Xt−1 = i,Xt−2 = k, . . .) = P (Xt = j|Xt−1 = i) = Pij , with 0 ≤ Pij ≤ 1 and ∑ i Pij = 1. That is, Pij is the probability that a consumer who purchases brand i will choose brand j in the next purchase decision. In conjunction with current market shares of the different brands, these Markov models can be employed to estimate future market shares. In learning models, as mentioned earlier, the entire history is considered to evaluate purchase probabilities. These models assume reinforcement of past choices, by assuming that a purchase of 2Closely related to the stochastic models of choice are the purchase incidence models, such as the negative binomial distribution (NBD) and the condensed negative binomial distribution (CNBD), which address timing of purchases by consumers, i.e., the elapsed time from one purchase to another. 7a certain brand increases the likelihood this brand will be purchased again in the next purchase incidence. There are also more complex models (e.g., the multinomial logit model) that incorporate, for example, other marketing decision variables. Yet, these models are beyond the scope of this work. 1.3 Overview of the Essays and Summary of Contributions Competition between retailers selling a perishable goods in a multi-period setting, as mentioned earlier, has been recognized as an important and hardly addressed area in the literature. All three essays in this thesis model competition in a multi-period, discrete-time, finite horizon setting. Contrasting these settings with the corresponding monopolistic setting, we are able to evaluate and quantify the effect of competition on prices and profits. While the first two essays focus on the case of myopic consumers, in the third essay we further consider the presence of strategic consumers. In our models, a myopic consumer is perceived as an impatient consumer who purchases the good at interest as soon as a price which is below his valuation is observed. On the other hand, a strategic consumer takes into account future prices he will encounter along the path of his visit pattern and his discounting factor in order to optimally choose the timing of his purchase, so as to maximize his surplus. This thesis provides a novel approach to resolve the challenge of introducing competition into a multi-period setting with a finite population of consumers. In the first essay we assume that two competing retailers, who offer a homogenous good, face a myopic consumer who arbitrarily visits one of them in the first period. Upon observation of the posted price, if it exceeds the consumer’s valuation, i.e., the consumer’s willingness to pay for that good, he leaves the store, and in the ensuing period he visits the competing retailer. This pattern is maintained throughout the selling horizon, as long as the consumer remains active. In other words, the consumer zigzags between the two competing retailers deterministically as long as he has not purchased the good. We refer to this type of competition as zigzag competition. In the second essay, this visit pattern is relaxed, and the myopic consumer may return to the same store with some certain probability which depends on structural characteristics of the market or the consumer’s experience at the last store visited. We refer to these types of competition as structural and behavioral competition, respectively. In 8the third essay, the zigzag competition model is revisited to allow for the existence of both myopic and strategic consumers in the market. Assuming that the valuation of the consumer for the good is uniformly distributed over [0, 1], we find that under zigzag competition, the price declines exponentially, rather than linearly, as is the case in the corresponding monopolistic setting (which is similar to the model in Lazear (1986)), with a substantially lower initial price. Moreover, the impact of competition on profits is immense, as the loss incurred to retailers may be up to 30%. The zigzag competition model is also extended to accommodate a more general valuation distribution, and it is further analyzed with many consumers who are either similar or identical. When consumers are similar, their valuations may be different from each other, and we find that in the uncapacitated case (i.e., retailers can always satisfy realized demand) the pricing is exactly as in the single consumer case. In the capacitated setting (i.e., retailers have limited inventories) we further distinguish between situations wherein demand is relatively low and retailers update their demand knowledge after every realization, and situations where demand is relatively high and retailers act upon demand expectations. When consumers are identical, they all have the same valuation for the good, and retailers face an opportunity to price differently to learn about the consumers’ common valuation. We find that (in the symmetric case) price dispersion may arise in the market when retailers have sufficiently low or sufficiently high inventory levels with respect to the size of the consumer population. With intermediate levels of inventories a continuum of pure-strategy SPNE exists. This competitive setting with identical consumers necessitates the investigation of the corresponding monopolistic base case with two retail stores, wherein we also find that price dispersion in the market arises with sufficiently low or sufficiently high inventory levels. The structural and behavioral competition, introduced in the second essay, make use of the common and widely used Markov chains. Specifically, we introduce the concept of Markov store- choice models, wherein the probability that a consumer visits each of the retailers in the market may depend on the visit realization in the preceding period. In a zero-order Markov store-choice model, to which we refer as a structural competition, the visit history does not play a role in the probabilities associated with each retailer. In a first-order Markov store-choice model, to which we refer as a behavioral competition, these probabilities depend on the location of the consumer in 9the preceding period. This generalization of the visit pattern allows us to extend the first essay in several important directions, as it allows us to consider a setting wherein retailers are not necessarily symmetric, and to an oligopoly. This essay establishes the robustness of the exponential decline of prices, as it occurs even with the mere threat of losing the consumer due to competition. Explicitly, even the presence of a very weak retailer in the market suppresses prices quite substantially in our model. The behavior of the price decline and profit are characterized, and it is further shown that replacing a duopoly with an oligopoly has a relatively marginal effect on prices. The second essay extends the model to incorporate a market consisting of many similar con- sumers. Essentially, this setting provides the same insights about the effect of competition on prices and corresponding profits as in the single consumer case. Namely, the pricing scheme takes the same form as prescribed for the single consumer case. However, a refined price skimming policy is revealed in that case, wherein only returning consumers are skimmed by the dominant retailer during the early periods of the selling horizon. Finally, we also briefly explore the uncapacitated setting. An appendix to this essay investigates a setting with a relatively low demand, such that retailers update their beliefs after every realization. The third essay sheds light on pricing policies employed by retailers in multi-period settings in the presence of myopic and strategic consumers in both monopoly and duopoly environments. Specifically, this essay enriches the monopoly and the duopoly under zigzag competition settings by assuming that part, or all, of the population of consumers behave strategically, while the other consumers behave myopically as before. In a two-period horizon, the pricing scheme at equilibrium prescribes similar actions in both a monopoly and a duopoly - to skim only high valuation consumers in the first period. However, when there are sufficiently many myopic consumers, or when the discounting factor of the strategic consumers is sufficiently high, retailers skim only myopic high- valuation consumers in the first period and none of the strategic consumers purchases the good in the first period. The third essay also characterizes the impact of strategic consumer behavior on retailers’ profits and the effect of competition on profits in the presence of these consumers. We find that strategic consumer behavior inflicts a larger (relative) profit loss to competing retailers than to a monopolistic retailer. Surprisingly, however, while a monopoly is not advised to ignore strategic consumer behavior, competing retailers may, in fact, benefit by being oblivious to strategic consumer behavior 10 by treating all consumers as myopic. An extension of the model is illustrated for a three-period horizon, and with longer horizons, we characterize the pricing policy and resulting expected profit when all consumers are strategic. Assuming a common discounting factor for both the retailers and the consumers, we find that strategic consumer behavior inflicts a larger (relative) profit loss to a duopoly than to a monopoly. 11 1.4 Bibliography Bitran, G. and Caldentey, R. (2003). An overview of pricing models for revenue management. Manufacturing & Service Operations Management, 5(3):203-229. Boyd, A. (1998). Airline alliance revenue management. OR/MS Today, 25:28-31. Chan, L. M. A., Shen, Z. J. M., Simchi-Levi, D., and Swann, J. L. (2004). Coordination of pricing and inventory decisions: A survey and classification. In Simchi-Levi, D., Wu, S. D., and Shen, Z. J. M., editors, Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era, pages 335-392. Kluwer Academic Publishers. Elmaghraby, W. and Keskinocak, P. (2003). Dynamic pricing in the presence of inventory con- siderations: Research overview, current practices, and future directions. Management Science, 49(10):1287-1309. Lazear, E. P. (1986). Retail pricing and clearance sales. American Economic Review, 76(1):14-32. Lilien, G. L., Kotler, P., and Moorthy, K. S. (1992). Marketing Models. Prentice Hall, New Jersey. Talluri, K. T. and van Ryzin, G. J. (2004). The Theory and Practice of Revenue Management, volume 68 of International Series in Operations Research and Management Science. Springer. Top of the Net (2002). Anchors away for markdown solution at old navy. November 4, available at www.retailsystems.com. 12 Chapter 2 A Dynamic Pricing Model under Duopoly Competition∗ 2.1 Introduction Markdowns, which are used by retailers for various of reasons (Pashigian and Bowen, 1991), became so prevalent over the years that a new coin was termed in the apparel industry - “markdown money”1. Indeed, as Top of the Net (2002) reports, American retailers are losing more than $200 billion a year due to markdowns2 and 78% of all apparel that is currently sold at American chains such as JCPenny, Sears, and Kohl’s is marked down.3 The ubiquity of markdowns is also reflected by the New York Times (2002) article on pricing: “Like all retailers, Saks [an upscale fashion retailer] faces the problem of deciding when to start marking down prices, and at what prices, so the company can glean as much profit as it can, without discounting so early that the company sells out and alienates customers, or so late that the company looks like a museum for unwanted merchandise”. The question, then, is not whether to markdown, but when and how. There is a vast literature which addresses markdowns, and, in general, dynamic pricing and revenue management, in a finite horizon, fixed inventory (i.e., no replenishment) setting. The various models that were introduced to analyze this setting differ mainly in the manner with which they model demand: consumers’ arrival and their valuations. In Lazear (1986) and Besanko and Winston (1990) consumers have valuations which are uniformly distributed. Further, in Lazear all ∗A version of this chapter has been submitted for publication. Granot, D., Granot, F. and Mantin, B. A Dynamic Pricing Model under Duopoly Competition. 1“Markdown money” is the amount stores subtract from suppliers’ checks because, they say, the clothes had to be marked down to sell them (New York Times, 2005ab). 2An analysis of retail and economic trends by the US Census Bureau and the National Retail Federation (NRF), as cited by Top of the Net (2002). 3STS market research study, as cited by Top of the Net (2002). 13 consumers are myopic and identical, while in Besanko and Winston they are strategic and non- identical; in Gallego and van Ryzin (1994), a price-dependent stochastic demand is modelled as a Poisson process; Bitran and Mondschein (1997) have extended Gallego and van Ryzin’s model to the case where customers have heterogeneous valuations, and Zhao and Zheng (2000) have studied a more general model where both the intensity of the customer arrival process and the reservation price distribution are time dependent. Desiraju and Shugan (1999) consider a two- period model wherein the market consists of price sensitive and insensitive segments which exhibit varying arrival intensities. Aviv and Pazgal (2005) incorporate additional statistical properties about demand uncertainty and adopt a Markov chain modelling to describe demand dynamics over time. For recent extensive reviews of pricing models, the reader is referred, e.g., to Elmaghraby and Keskinocak (2003), Bitran and Caldentey (2003), and Chan et al. (2004). Competition has a major effect on pricing. Indeed, as Bitran and Caldentey (2003) conclude: “Including market competition is another important extension to the model. [...] price competition among retailers is today the main driver in their selection of a particular pricing policy”. Simi- larly, Elmaghraby and Keskinocak (2003) have acknowledged that in the NR-I4 markets literature, competition is one of the missing bricks, and Chan et al. (2004) have admitted that “The reality is that most companies are not monopolies, and many specifically compete on product prices or service differentiation, so these effects need to be considered when establishing coordinated pricing and production policies”. Nevertheless, in spite of the recognition that competition has a significant effect on prices, until quite recently, the effect of competition in the context of multi-period revenue management, where inventory is fixed, has received relatively little attention. Recent papers addressing competition include Perakis and Sood (2006), who employ a robust optimization approach to study a multi- period, fixed inventory, dynamic pricing model under competition; Gallego and Hu (2006), who extend Gallego and van Ryzin (1994) to oligopolistic competition by modifying the demand intensity (non-homogeneous Poisson process) to depend on prices set by all players; Talluri (2003), where prices are pre-determined but the competing retailers choose to offer only a subset of pre-designed products in each period; Lin and Sibdari (2004), who consider a consumer choice model to study competition between firms who sell substitutable products and face a consumer arrival which follows 4No Replenishment (NR) of inventory with Independent (I) demand over time, where consumers are either Myopic (M) or Strategic (S). 14 a Bernoulli process; Friesz et al. (2005), who study a competitive model among service providers on a network which incorporates demand dynamics (differential evolutionary game); and Xu and Hopp (2006), where retailers set inventory levels in the first stage and compete on prices in the second stage. In their model consumers are homogeneous and their arrival is either piecewise deterministic or follows a Brownian motion. Our main objective in this chapter is to incorporate competition in a finite, multi-period, fixed- inventory dynamic pricing setting, and to evaluate the effect of competition on prices and profits. Our modeling approach is new, and the results and insights we derive are substantial. In our base model, two profit-maximizing retailers, each of whom can satisfy the entire market, compete over prices, and face a single consumer whose valuation is uniformly distributed over [0, 1]. The myopic5 consumer, who does not have full price information, has to visit the retail stores in order to observe the prices.6 We neglect search costs, but assume that due to these costs the consumer is limited to a single store visit per period. At the store, the consumer observes the posted price and if it is below his valuation he purchases the good. Otherwise, he leaves the store and in the following period visits the other retailer. Thus, the consumer keeps visiting the stores in a zigzag manner to check if prices have dropped below his valuation, at which point he purchases the good. We refer to this type of competition as zigzag competition, and note that it exposes the two retailers to the difficulties of pricing correctly in a competitive market. If the price is set too low, potential profit may be lost, while if it is set too high, the product may not be sold. The key features of our model, which distinguish it from the above mentioned multi-period competitive setting models, stem from our assumptions about demand. Specifically, we assume that there is a fixed pool of consumers, who lack price information and who shop around, in a zigzag manner and possibly for the entire selling horizon, until, hopefully, prices drop below their valuations. We demonstrate in this paper the dramatic effect of zigzag competition on prices and profits. Indeed, in our base model, wherein there is a single consumer, we show that not only are prices strictly decreasing, as in Lazear7, but they decrease exponentially rather than linearly. The linear 5One may suggest that due to the popularity of markdowns, consumers learn to expect them, and may also postpone their purchasing decisions even if the posted price is below their valuations. However, it is commonly assumed in the literature that consumers are myopic and buy the good as soon as the price drops below their valuations, see, e.g., Bitran and Mondschein (1997). 6This is a common assumption in the literature. Bitran and Mondschein (1997), for example, state that “customers [...] have little information about current prices before going to the store”. 7Since Lazear’s model can be perceived as a monopoly facing a single consumer over T periods, we compare our 15 decline in the monopolistic case is due to the price discovery process employed by the monopolist, and the shift to an exponential decline in prices is due to the competitive forces. The initial price, which is increasing in the selling horizon, is shown to converge very fast to 0.543, as compared to 1 in Lazear’s model, and in every period the price drops by a factor of about 0.5. Thus, within a very few periods the price in our competition model drops nearly to zero, which may explain the short life cycle of some products in competitive markets. The profit loss of the system due to the introduction of zigzag competition, which is measured as the difference between the combined profit of the two competing retailers and the profit of the monopolist, is shown to increase in the length of the selling horizon and to converge to 0.147, which represents a loss of about 29.6% of the profit. The profit loss of a monopolist due to the introduction of such competition is shown to converge to 64.78% of his pre-competition profit. Finally, we find that all results hold when there are N similar consumers, whose valuations are independently drawn from the same uniform distribution. Thus, our model enables us to quantify the significant effect of competition on both prices and profits in a finite multi-period competitive setting. To investigate the robustness of our results, we analyze the effect of zigzag competition when some of our assumptions are modified or extended. More specifically, we consider the following cases: (i) the valuation of the good follows a power, rather than a uniform distribution on [0, 1], (ii) the system is capacitated in the sense that the retailers cannot satisfy the entire demand in the market, and (iii) consumers are identical rather than similar, i.e., all consumers have the same valuation for the product which is uniformly distributed. We note that the proper investigation of cases (i) and (ii) has necessitated the analysis of the corresponding monopoly case. The analysis of the effect of zigzag competition with a power distribution reveals that the nature of the exponential decline of prices depends on the parameter, q, −1 < q ≤ ∞, of the distribution, which, for q = 0, coincides with the uniform distribution. Specifically, the decline of prices is most severe when q → −1, and it is most moderate when q → ∞ (yet exponential, when T → ∞), i.e., when the valuations by the consumers are very likely to be very close to one. The analysis of the capacitated case, when retailers cannot satisfy the entire demand, demonstrates that zigzag competition still has a major impact on prices and profits. We separately investigate situations in which demand is relatively low and retailers update their demand knowledge after every realization, results with those obtained by Lazear. 16 and where demand is relatively high and retailers simply act upon demand expectations. In the former case we show that the effect of competition on prices and profits diminishes as retailers are more constrained, i.e., they can supply a smaller fraction of the market. In the latter case we further prove that, as prescribed by our base model, prices decline rapidly even when retailers are far from being able to satisfy the entire market. However, as retailers are more constrained in terms of capacity, the price decline becomes more moderate. Prices in an uncapacitated duopoly with identical consumers appear to decline a bit faster than when consumers are similar. Further, a qualitative difference between the similar and identical consumers cases is uncovered. Specifically, while there is a unique pure-strategy subgame perfect Nash equilibrium (SPNE) in prices when consumers are similar, in the identical consumers case, there are multiple, and even, in some cases, a continuum of pure-strategy SPNE. In particular, in the two-period case with identical consumers and symmetric retailers it is shown that for sufficiently low or sufficiently high inventory levels there are precisely two pure-strategy SPNE in prices, both of which are non-symmetric, while for intermediate inventory levels there is a continuum of pure- strategy SPNE. The former case arises as the existence of different prices in the market provides an additional learning opportunity about the consumers’ common valuation. Such price dispersion in the market can bolster the zigzag visiting behavior by consumers, as consumers do not know which retailer posts the lower price. To properly evaluate the effect of competition we also study the base case of a two-store monopoly, wherein a central planner is setting possibly different prices at the two stores. We find that a centralized two-store monopoly system does not have any advantage over the single- store monopoly when consumers are similar. However, when consumers are identical, the profit of the two-store monopoly could be strictly higher than that of the single-store monopoly. Moreover, similar to the two-period duopoly case, with identical consumers and symmetric stores (having the same inventory levels and encountering the same number of consumers in the first period), price dispersion arises only if stocking levels at the two locations are sufficiently high or sufficiently low. Our main contribution in this chapter, aside for our model of zigzag competition, is that we show that in our setting the effect of multi-period competition on prices and profits is very significant: prices decline exponentially along the selling horizon and the monopolist profit is dramatically reduced when a competitor enters the market. Further, the effect of such competition is shown to 17 be robust in the sense that it essentially holds (i) for a large class of product valuation distribution, (ii) when the retailers cannot satisfy the entire market, and (iii) for both identical and similar consumers. However, it is further shown that the price trajectory is affected by product valuation and, in the capacitated case, it is affected by the fraction of the market each retailer can satisfy. The rest of the chapter is arranged as follows. In Section 2.2 we introduce and analyze the basic zigzag competition model, first with a single consumer and, subsequently, with N similar consumers. Section 2.3 considers extensions to the model: a more general valuation distribution and a capacitated system. In Section 2.4 we study the behavior of the model when consumers are identical rather than similar, and Section 2.5 investigates the case of a two-store monopoly. Section 2.6 concludes with a brief discussion of managerial insights and future work. 2.2 The Model 2.2.1 The Basic Model Two competing retailers, each of whom has in stock one unit of an identical good, encounter a single myopic consumer whose valuation for that good is V . That is, the consumer is willing to pay up to V for the good, but no more, and he purchases the good once he observes a price which is below his valuation. The retailers do not know V with certainty, but they do have prior knowledge of the density of V , denoted f(V ), with distribution function F (V ). The two retailers are risk-neutral and seek to maximize their expected profits. They have T periods to sell their good to the consumer, and in each of these periods they can post a new price. The consumer does not have any knowledge about the posted prices in the market in each period. To obtain that information he needs to visit each retail store and observe the posted price. We neglect search costs, but assume that the consumer is limited to a single retail store visit per period. We introduce in this chapter a new type of competition between the retailers, to be referred to as zigzag competition. In a zigzag competition, the retailers compete to sell their product to the single consumer who visits only one store per period. In period t he observes a price Rt at the retailer he visits. If this price is below his valuation, he purchases the good. Otherwise he does not 18 purchase the good, and in the following period he visits the other retailer with certainty.8 We refer to the retailer who encounters (resp., does not encounter) the consumer in the first period as Retailer 1 (resp., 2). Thus, in an odd (resp., even) period, if no purchase has occurred before that period, the consumer visits Retailer 1 (resp., 2). We further assume that the periods are well defined in the sense that when a retailer encounters the consumer in period t, t > 1, he is aware that the consumer has visited the competing retailer in the previous period.9 In each period the price posted at the retail store not visited by the consumer is not relevant, and no assumption is needed for a retailer’s knowledge about the other retailer’s posted prices. However, since both retailers solve the same model, each of them can compute the prices that the other retailer is posting. If the good is not sold in period t at price Rt, the retailers can immediately infer that V < Rt. We denote Retailer 1’s (resp., Retailer 2’s) expected profit-to-go function at period t as pi1t (resp., pi2t ). If t is odd, Retailer 1 is visited by the consumer in that period, and we have: pi1t =Rt[1− Ft(Rt)] +Rt+2[1− Ft+2(Rt+2)]Ft+1(Rt+1)Ft(Rt) + · · · = bT−t 2 c∑ i=0 ( Rt+2i[1− Ft+2i(Rt+2i)] t+2i−1∏ j=t Fj(Rj) ) , (2.1) where the first term in pi1t is the expected profit from a sale in period t. It is equal to the price charged in period t times the probability that the good sells in period t, given the information from period t− 1; the second term is the conditional expected profit from a sale in period t+ 2, and so on, and bac denotes the largest integer which is smaller than or equal to a. Similarly, the expected profit-to-go of Retailer 2, who does not encounter the consumer in that 8Consumers’ switching behavior may be traced back to first-order Markov brand choice models that are widely used in the marketing literature (see, e.g., Leeflang et al. (2000) and references therein). In these models, the probability a consumer purchases a particular brand depends on his purchasing decision in the preceding period. Thus, our zigzag competition model resembles an extreme case (wherein the probability is zero) of the first-order Markov brand choice model, except that we consider a single purchase instead of a repeated purchase, and a store visit in our model coincides with a purchase event in the brand choice models. Switching behavior of consumers has been also considered in the operations literature, e.g., by Zhao and Atkins (2007), Netessine and Shumsky (2005), and Netessine et al. (2006). However, in their models switching from one retailer to another may occur due to stockouts. 9In the absence of this assumption, Retailer 2, who encounters the consumer in the second period for the first time, may keep the first period price well into the second period. And, in general, without this assumption, Retailer 2 may post Retailer 1’s prices with a one period delay, and as a result, he never sells. Note, that in more realistic situations when there are many consumers in the market (as is the case, e.g., in §2.2.4), this assumption becomes redundant as both retailers are visited by different sets of consumers in each period. 19 period (since t is odd), can be written as: pi2t =Rt+1[1− Ft+1(Rt+1)]Ft(Rt) +Rt+3[1− Ft+3(Rt+3)]Ft+2(Rt+2)Ft+1(Rt+1)Ft(Rt) + · · · = bT−t−1 2 c∑ i=0 ( Rt+1+2i[1− Ft+1+2i(Rt+1+2i)] t+2i∏ j=t Fj(Rj) ) . (2.2) The profit-to-go functions from (2.1) and (2.2) can be expressed in a recursive form. If t is odd, pi1t = Rt[1− Ft(Rt)] + pi1t+1Ft(Rt) and pi2t = pi2t+1Ft(Rt). (2.3) Thus, Retailer 1 sets prices, at odd periods, so as to Maxpi1t =Max Rt (Rt[1−Ft(Rt)]+pi1t+1Ft(Rt)), and similarly, at even periods, Retailer 2 solves: Maxpi2t =Max Rt (Rt[1−Ft(Rt)]+pi2t+1Ft(Rt)).Maxpi2t = Max Rt (Rt[1− Ft(Rt)] + pi2t+1Ft(Rt)).10 2.2.2 Uniformly Distributed Valuation For simplicity, suppose first that the prior on the valuation V is uniformly distributed between zero and one. By Bayes’ Theorem, this implies that the posterior distribution carried from period t to t+ 1 is uniform between zero and Rt, so that Ft+1(V ) = VRt . To illustrate, we solve the problem, backwards, for the last four periods of the zigzag compe- tition. In Table 2.1 we display the retail prices for both the zigzag competition model and the monopoly model for T = 4. The price ratio at period t is RtRt−1 . Observe from Table 2.1 that, under zigzag competition, prices decline much more rapidly and the initial price is much lower compared to the monopoly model. Zigzag Competition Monopoly Price ratio Retailer 1’s Price Retailer 2’s Price pi1t pi 2 t Price Ratio Price pit Period 1 225 418 R1 ≈ 0.538 Not observed ≈ 0.269 ≈ 0.077 45 0.8 0.4 Period 2 8 15 Not observed R2 ≈ 0.287 ≈ 0.071R1 ≈ 0.266R1 34 0.6 0.375R1 Period 3 1 2 R3 ≈ 0.143 Not observed 0.25R2 0.0625R2 23 0.4 0.33R2 Period 4 1 2 Not observed R4 ≈ 0.0717 0 0.25 12 0.2 0.25R3 Table 2.1: Price ratios, prices, and profits under zigzag competition and monopoly settings 10This setting, with a single consumer, is somewhat reminiscent of models in the economic literature, such as Cyert and DeGroot (1970) and Maskin and Tirole (1988), wherein retailers, in an alternating manner, commit to actions for two consecutive periods. In our model, however, when the N -consumer case is considered and retailers encounter consumers in each period, both retailers modify their prices in each period. 20 Lemma 2.1 The profit-to-go in each period is an affine function of the price that is set in the previous period. That is, piit = S i tRt−1, i = 1, 2, where Sit are scalars, with R0 = 1. Moreover, at odd (resp., even) periods, the values of S1t and S 2 t can be expressed recursively as follows: S 1 t = 1 4(1−S1t+1) (resp., S1t = S1t+1 4(1−S2t+1)2 ) and S2t = S2t+1 4(1−S1t+1)2 (resp., S2t = 1 4(1−S2t+1) ), with SiT+1 = 0, i = 1, 2. Proofs of statements made in this chapter are provided in Appendix A.1. Note that the scalars S1t and S 2 t have a dimension of sales, and will thus be referred to as the normalized sales-to-go in period t of Retailers 1 and 2, respectively, with respect to the price that is set in the previous period, Rt−1. Since R0 = 1, the initial normalized sales-to-go coincide with the retailers total expected profits, pi11 = S 1 1 and pi 2 1 = S 2 1 . Let {s1t } (resp., {s2t }) denote the sequence of normalized sales-to-go of the retailer who encounters (resp., does not encounter) the consumer in period t. That is, for an odd t, s1t = S 1 t = 1 4(1−s2t+1) (resp., s2t = S 2 t = s1t+1 4(1−s2t+1)2 ), and for an even t, s1t = S 2 t (resp., s 2 t = S 1 t ). Thus, for all t, s1t = 1 4(1− s2t+1) and s2t = s1t+1 4(1− s2t+1)2 . (2.4) Note that in both {s1t } and {s2t }, as well as in all other sequences considered in the sequel, the sequences progress from t = T to t = 1. Thus, e.g., the first element in {s1t } is s1T and the last element is s11. Lemma 2.2 The normalized sales-to-go sequences {s1t } and {s2t } are bounded sequences. Specif- ically, {s1t } (resp., {s2t }) is bounded from below by s1 = 14 (resp., s2 = 0) and from above by11 s1 = 32 (19+3 √ 33) 1 3 (19+3 √ 33) 1 3+(19+3 √ 33) 2 3+4 ≈ 0.271 (resp., s2 = 56 − 16(19 + 3 √ 33) 1 3 − 23 1 (19+3 √ 33) 1 3 ≈ 0.0803). Proposition 2.1 The sequences of normalized sales-to-go, {s1t } and {s2t }, are monotone convergent sequences as the number of periods, T , goes to infinity, while t goes to 1. Proposition 2.2 The normalized sales-to-go of the retailer who encounters the consumer in pe- riod t is strictly larger than the normalized sales-to-go of the retailer who does not encounter the consumer in that period. Proposition 2.3 The normalized sales-to-go of a monopolist is at least as large as the combined normalized sales-to-go of both retailers in the zigzag competition model. 11These bounds are obtained by omitting the subscripts from (2.4) and solving for s1 and s2. 21 Note that for t = 1 the normalized sales-to-go of the retailers coincide with their expected profits. Therefore, Proposition 2.3 implies: Corollary 2.1 A monopolist’s profit is at least as large as the combined profits of both retailers in the zigzag competition model. Let us next consider the sequence of price ratios, {Bt}, defined as Bt ≡ Rt Rt−1 = 1 2(1− s2t+1) . (2.5) Since R0 = 1, the initial price ratio and the initial price coincide, R1 = B1. From (2.4) and (2.5) one can observe that s1t = 1 2Bt and s 2 t = 4s 1 t+1(s 1 t ) 2. Thus, we find that Bt = 1 2− (Bt+1)2Bt+2 , (2.6) with BT+1 = BT+1 = 0. Proposition 2.4 The sequence of price ratios, {Bt}, is backwards increasing and each element therein is bounded from below by 0.5 and from above by12 0.55. For every t, Bt = RtRt−1 ≤ B1 ≤ 0.55. Thus, since R0 = 1, Rt ≤ (0.55)t−1. We conclude: Theorem 2.1 Under zigzag competition the prices decrease exponentially. To illustrate the behavior of the model, consider an example with 20 periods. The price ratios and the prices that are set under zigzag competition in each of the 20 periods and the prices in the monopoly setting are plotted in Figure 2.1(a), whereas the profit-to-go of the retailers are plotted in Figure 2.1(b). We observe that: • The price ratio under zigzag competition does not change much over most of the selling season. As it is solved recursively, it slightly increases from a value of 0.5 at T = 20 to a value of about 0.543 at t = 1, which is dramatically lower than the initial price set in the corresponding monopoly model, 2021 . In other words, the fear of losing a sale due to competition significantly suppresses the initial price. Since the price ratio is equal to the initial price had that period been the first period of the selling horizon, we conclude that the initial price is decreasing with perishability (measured by the length of the selling horizon), though only marginally. 12In Proposition 2.5 we show that {Bt} converges to 0.543 from below and thus each element in the sequence is bounded from above by the convergence value. 22 • As proven by Theorem 2.1, and illustrated in Figure 2.1(a), prices under zigzag competition are decreasing exponentially. They approach zero fairly quickly. By contrast, in a monopoly market, prices decline linearly, as is also illustrated in Figure 2.1(a). • The profit-to-go of Retailers 1 and 2 also decrease exponentially with some spikes. As the price declines quickly, the conditional expected valuation of the consumer declines as well, and there is less profit to be made. Thus, the largest part of the profit is achieved in the early periods, and later periods of the selling season contribute very marginally to the total profit. The spikes occur since a retailer who does not encounter the consumer in period t will encounter him only with some probability in period t+ 1.  0 0.2 0.4 0.6 0.8 1 1 3 5 7 9 11 13 15 17 19 t, Period Number Pr ic e,  Pr ic e Ra tio Duopoly Prices Monopoly Prices Price Ratio (a) Prices and price ratio  0 0.05 0.1 0.15 0.2 0.25 0.3 1 3 5 7 9 11 13 15 17 19 t, Period Number Pr o fit - to - go Retailer 1 Retailer 2 (b) Retailers’ profit-to-go Figure 2.1: A 20-period model: price, price ratio and profit-to-go Let us examine the retailers expected profit over a range of selling horizons, as demonstrated in Figure 2.2(a), wherein the selling horizon, T , ranges from T = 1 up to T = 50. Observe that Re- tailer 1’s expected profit equals 0.25 in a single and two-period horizons and then converges quickly to about 0.271. Similarly, Retailer 2’s expected profit equals zero in a single period model (as he does not encounter the consumer at all), rises to 0.0625 in a two-period horizon and then converges quickly to about 0.08. Thus, we may conclude that longer selling horizons do not contribute much to the expected profit of each of the retailers. Consequently, the system’s profit converges quickly to about 0.35. It can be observed from Figure 2.2(b) that the monopolist’s profit ( T2(T+1)) converges slower but to a higher value. Thus, the larger the value of T is, for T > 1, the larger is the profit advantage of a monopoly over the zigzag competitors. 23 Consumer’s surplus measures the difference between the consumer’s valuation and the price charged, if the consumer purchased the good. Thus, under both settings, the consumer’s surplus equals ∑t=T t=1 1 2(Rt−1−Rt)2. Figure 2.2(c) contrasts the consumer’s surplus under zigzag competition and under monopoly. With longer horizons, the monopolist can more efficiently price the good and extract more of the surplus from the consumer and, therefore, the consumer’s surplus in the monopoly setting is decreasing in T . However, in zigzag competition the two competing retailers are mainly engaged in price cutting competition and, therefore, the consumer’s surplus first increases from 0.125 to 0.156 (as a second lower price is introduced), and then it slightly decreases (since the initial price also marginally increases) and quickly converges to 0.147.  0 0.05 0.1 0.15 0.2 0.25 0.3 0 10 20 30 40 50 T, Selling horizon Pr o fit Retailer 1 Retailer 2 (a) Retailers’ profits  0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40 50 T, Selling Horizon Pr o fit Monopoly Profit Loss Duopoly (b) System’s profit and profit loss  0 0.04 0.08 0.12 0.16 0 10 20 30 40 50 T, Selling horizon Co n su m er  Su rp lu s Monopoly Duopoly (c) Consumer’s surplus Figure 2.2: Profit, profit loss, and consumer surplus in monopoly and duopoly under zigzag competition 2.2.3 Values at Convergence We first provide sharp bounds for the initial price. Proposition 2.5 The initial price, R1, is bounded from below by 12 (if T=1 or 2) and monotonically converges to 13(17 + 3 √ 33) 1 3 − 23 1 (17+3 √ 33) 1 3 − 13 ≈ 0.543 as T →∞. Recall from Theorem 2.1 that under zigzag competition prices decrease exponentially. From Proposition 2.5 the price ratio converges to 0.543. Thus, the exponential decline of prices is further bounded as follows: Rt ≤ (0.543)t−1. Proposition 2.6 The expected profit of Retailer 1 (resp., 2) is increasing in T . Further, it is bounded from below by 14 (resp., 0) if T = 1, and monotonically converges to about 0.2718 (resp., 0.0803), as T →∞. 24 To measure the effect of zigzag competition we compare the system’s profit under zigzag com- petition and under monopoly, both of which are bounded from below by 14 when T = 1, increase in T , and converge to 0.3523 and 12 , respectively. Thus, the system’s profit loss due to zigzag competition, at convergence, is 12 − 0.3523 = 0.1477. Proposition 2.7 The system’s expected profit loss due to zigzag competition is bounded from below by zero (when T = 1), and it converges to about 0.1477 (when T → ∞), which represents a profit loss of about 29.6%. If both retailers are equally likely to be visited by the consumer at the first period, we have: Proposition 2.8 The expected profit loss of a monopolist retailer due to the introduction of zigzag competition with a second retailer is bounded from below by 18 when T = 1 and converges to about 0.3239, which represents a loss of about 64.78% of the profit. 2.2.4 N Consumers In this subsection we extend the model to N similar consumers, whose valuations are drawn in- dependently from a uniform distribution on [0,1], and each of the retailers can satisfy the entire demand in the market, i.e., each has in stock N units of the good. The consumers do not all necessarily zigzag as a single group. That is, in the first period N1 (resp., N2) consumers visit Retailer 1 (resp., 2), with N1 + N2 = N , and, as before, the consumers maintain the zigzagging search pattern. Let us denote by N it , i = 1, 2, the number of consumers who visit Retailer i in period t. Thus, N it ≤ N jt−1, i 6= j = 1, 2, and N i1 = N i, i = 1, 2. Let pii(N it ,Njt ) denote the profit-to-go of Retailer i, i = 1, 2, in period t when there are N it +N j t consumers in the system in period t, and let Rit denote the respective price in that period. Since consumers’ valuations may differ, it is possible that some consumers will make a purchase in a certain period while the others will continue to zigzag. Let PN i t n , n = 0, ..., N it , i = 1, 2, denote the probability that Retailer i, who is visited by N it consumers, sells n units in that period. Thus, in period t, PN i t n = N it ! n!(N it−n)! [Ft(Rit)] N it−n[1− Ft(Rit)]n. Proposition 2.9 The profit-to-go of Retailer i, i = 1, 2, in period t when there are N it + N j t , i 6= j = 1, 2, similar consumers in the system, such that N it (resp., N jt ) of them visit Retailer i 25 (resp., j) in that period, can be expressed as pii (N it ,N j t ) = N itpi 1 t +N j t pi 2 t , i 6= j = 1, 2, if t is odd, and pii (N it ,N j t ) = N itpi 2 t +N j t pi 1 t , i 6= j = 1, 2, if t is even, where pi1t and pi2t are the profit-to-go functions of Retailers 1 and 2, respectively, in the model with a single consumer. The resulting pricing policy is the same as in the single consumer case. In other words, the pricing policy is independent of the number of consumers in the system, as long as consumers are similar and each of the retailers can satisfy the entire market demand. 2.3 Extensions to the Model So far we have studied the model under the assumptions that consumers’ valuations are uniformly distributed and that the competing retailers can satisfy the entire market. Relaxing these two as- sumptions, in §2.3.1 we study the basic setting while assuming that the single consumer’s valuation follows a more general distribution, and in §2.3.2 we investigate the capacitated case. 2.3.1 Different Valuation Distributions In this subsection we assume that the single consumer’s valuation, V , follows a power distribution, with a p.d.f. of the form (q+1)V q, q > −1, 0 ≤ V ≤ 1.13 Thus, F (0) = 0 and F (1) = 1. Note that when q = 0, the power distribution coincides with the uniform distribution. To evaluate the effect of competition, we first consider the monopolistic case and subsequently, the duopolistic case. 2.3.1.1 Monopoly under Power Distribution One can show that when the monopolist is facing a single consumer, his profit-to-go in each period is an affine function of the price that is set in the previous period. That is, pit = StRt−1, where St is a scalar, which can be expressed recursively. By induction, it can be shown that the sequence {St} is backwards increasing in t and converges to S∗ = q+1q+2 . Thus, the expected profit of the monopolist converges to q+1q+2 . The price ratio in period t, Bt, can be expressed recursively: Bt = 1 ((1− q+1 q+2 Bt+1)(q+2))1/(q+1) , and the sequence {Bt} is a convergent sequence, which converges, not 13This form of valuation distribution may follow from the power law distribution of personal income, originally proposed by Pareto (1897). See Badger (1980) for a summary of empirical studies. Further, we note that the power distribution was used by Lariviere and Porteus (2001) to model demand in their investigation of the effectiveness of wholesale price only contracts. 26 surprisingly, to 1. It is worth noting that for q = 0 the price decline is linear while for q < 0 (resp., q > 0) the decline is faster (resp., slower) than linear. 2.3.1.2 Duopoly under Power Distribution In a duopoly, it can be shown, as before, that the profit-to-go expressions in each period are affine functions of the price that is set in the previous period, and that the price ratio, Bt, can be expressed recursively: Bt = ( 1 (1− q+1 q+2 (Bt+1)q+2Bt+2)(q+2) )1/(q+1) . By induction, one can show that {Bt} is backwards increasing in t and that it is bounded from below by ( 1q+2)1/(q+1) and from above by B∗, which is a real number that satisfies ( 1q+2) 1/(q+1) ≤ B < 1 and solves B(q + 2− (q + 1)(B)q+3 )1/(q+1) = 1. Thus, based on the Monotone Convergence Theorem, {Bt} is a convergent sequence, which converges to B∗. 2.3.1.3 Effect of Competition under Power Distribution As q increases the center of the valuation distribution shifts to the right. For values of q very close to −1 it is very likely that the consumer has a very low valuation for the product, and therefore the monopolist drops prices fairly quickly to increase the number of search points with low prices. Yet, the monopolist does not give up the opportunity of searching for the chance the consumer has a high valuation. As a consequence, we can see that, for example, when q = −0.5, even though there is a 50% chance that the consumer’s valuation is less than 0.25, the first seven prices the monopolist sets in a 10-period selling horizon are higher than 0.25 (Figure 2.3(a)). So the monopolist is trading the opportunity of a high gain with low probability and a low gain associated with high probability. On the other hand, the competing retailers do not have the luxury of spending search points on high prices and due to competition they mark down prices fairly aggressively. For the same example, when q = −0.5, only the first two prices posted by the retailers exceed 0.25 (Figure 2.3(a)). As q increases and the center of valuation distribution shifts to the right, the consumer is more likely to have a high valuation for the product and, therefore, both the monopolist and the duopoly’s retailers set higher prices in each period. In a sense, both systems slow down the price search, but since the monopolist is not subject to competition, he can focus the search only on the “promising” area and, thus, be more successful. Figure 2.3 demonstrates the price trajectories over a 10-period selling horizon for different q values and Figure 2.4 exhibits the expected system profit (both for 27 a monopoly and a duopoly), as a function of the selling horizon, for different values of q. The difference between the plots in Figure 2.4 is the system profit loss due to competition, which is increasing in T , the length of the selling horizon. Essentially, for sufficiently long selling horizons the profit loss can be shown to be declining percentage-wise in q as q increases from -1 to ∞, and asymptotically, as q →∞, the percentage loss due to competition goes to zero.  0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period number Monopoly Duopoly (a) q = − 1 2  0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period number Monopoly Duopoly (b) q = 0  0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period number Monopoly Duopoly (c) q = 5  0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period number Monopoly Duopoly (d) q = 50 Figure 2.3: Prices in a 10-period selling horizon when valuation follows a power distribution  0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 T, Selling Horizon Monopoly Duopoly (a) q = − 1 2  0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 T, Selling Horizon Monopoly Duopoly (b) q = 0  0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 T, Selling Horizon Monopoly Duopoly (c) q = 5  0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 T, Selling Horizon Monopoly Duopoly (d) q = 50 Figure 2.4: System profit for different selling horizons when valuation follows a power distribution Anecdotal evidence of markdowns observed for fashionable and other seasonable goods is con- sistent with our results. Indeed, for some goods, the selling horizon is terminated after 2-3 major markdowns, since the posted price reaches a very low level (see Figures 2.3(a) and 2.3(b)). For other goods, discounts are fairly moderate and the price does not drop very fast, as is the case in Figure 2.3(d). Evidently, Figures 2.3 and 2.4 reveal the importance, to the retailers, of acquiring better information about consumers’ valuations. They also suggest that both retailers might be better off by jointly attempting to increase consumers’ valuations, say, by joint advertising, as it would allow them to price their goods higher and to offer smaller discounts. 28 2.3.2 Limited Capacity So far we assumed that both retailers can satisfy the entire market. In this section we analyze the capacitated case, wherein each of the retailers cannot satisfy the entire market demand, consumers are similar and their valuations are uniformly distributed. We first develop the monopolistic setting (§2.3.2.1), as a base case, and subsequently, the case of duopolistic competition (§2.3.2.2). In both subsections it is assumed that after each realization retailers update their information about demand, as to fine-tune their prices in subsequent periods. By contrast, in §2.3.2.3 it is assumed that retailers simply act upon demand expectations.14 2.3.2.1 Monopoly under Limited Capacity The single unit case In Lazear’s model a monopolist facing identical consumers has only one unit of the good in stock. Assume that this monopolist is facing N similar, rather than identical, consumers. It can be proven by induction that the profit-to-go in each period can be expressed as an affine function of the preceding price, i.e., pit = StRt−1, where St is a scalar, with R0 = 1. It can be further proven that the sequence {St} is a monotone convergent sequence, which backwards converges to S∗ = NN+1 , which represents the monopolist’s expected profit at convergence. This result is not surprising, as at convergence the monopolist should obtain a profit which equals the expected value of the highest valuation amongst the N consumers. Figure 2.5 illustrates the prices set by a monopolist having a single unit in stock over a selling horizon of 20 periods for varying values of N . Evidently, the price decline is more moderate when there are more consumers in the market. Further, the price decline is linear, as in Lazear, only when there is a single consumer in the market. K units in stock Now assume that the monopolist facing N similar consumers has K units of the good in stock. Let PNtnt denote the probability that nt of the Nt consumers who visit the monopolist in period t have valuations which exceed the posted price, Rt, i.e., PNtnt ≡ Nt!nt!(Nt−nt)!( RtRt−1 )Nt−nt(1 − RtRt−1 )nt . Note, that (i) Nt+1 = Nt − nt, nt ≤ Nt, (ii) N1 = N , and (iii) R0 = 1. The monopolist profit in a 14The use of demand expectations is commonly employed in the literature. See, for example, Liu and van Ryzin (2005) who employ expectations approach in a two-period monopolistic rationing setting. 29  0 0.2 0.4 0.6 0.8 1 1 3 5 7 9 11 13 15 17 19 t, Period Number R t, Pr ic e N=50 N=10 N=3 N=1 Figure 2.5: Prices a monopolist sets in a 20-period model when K = 1 for varying values of N two-period selling horizon is pi = K−1∑ n1=0 PNn1 ( n1R1 + N−n1∑ n2=0 PN−n1n2 min(n2,K − n1)Rn12 ) + N∑ n1=K PNn1KR1, where Rn12 is the price the monopolist sets in period 2 when n1 < K consumers purchase the good in the first period. The optimal prices are found by solving this equation backwards, first for all second period prices, Rn12 , n1 = 0, . . . ,K − 1, and then for the first period price, R1. Due to the complexity of the expressions for the optimal prices, we cannot provide a general closed-form solutions for R1 and R2 for any N and K. However, we can demonstrate their main characteristics. Essentially, as N increases, both R1, R2, and the corresponding profit increase (and this is partially observed from Figure 2.5 for the simple case of K = 1). Figure 2.6(a) illustrates, as K increases, the optimal prices when N = 10, wherein the solid line represents R1 and the triangles below represent the second period prices in a decreasing order of leftover inventory. For example, when K = 5, the uppermost triangle is the second period price when the monopolist has one unit remaining in stock (i.e., he sold 4 units in the first period), and the lowermost triangle corresponds to the case of 5 units in stock. Observe that when K = N = 10, the system is uncapacitated, resulting with Lazear’s monopolist prices of 23 and 1 3 in the first and second period, respectively. Note that as Figure 2.6(b) demonstrates, as K increases the expected profit increases as well, yet, the profit per unit stocked is decreasing. 30  0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 K, Units in Stock R1 R2 (a) Prices  0 0.5 1 1.5 2 2.5 3 3.5 1 2 3 4 5 6 7 8 9 10 K, Units in Stock Profit Profit K (b) Expected Profit Figure 2.6: Prices a monopolist sets in a 2-period model and the corresponding expected profit when N = 10 for varying values of K 2.3.2.2 Duopoly under Limited Capacity Consider the two-period setting, wherein Retailer i (resp., j) has Ki (resp., Kj) units of good in stock and is visited by N i (resp., N j) consumers in the first period, with Ki,Kj ≤ N i +N j . We assume that both the inventory levels and the distribution of the consumers in the first period are known to both retailers. Additionally, we assume that the retailers cannot observe sales levels at the competing retailer, and as a result, they do not know how many consumers they will encounter in the second period. Let nit denote the number of consumers who visit Retailer i in period t and have valuations in [Rit, R j t−1), with R i 0 = R j 0 = 1. For each realization of sales at his own store, as long as he is not sold out, Retailer i sets a possibly different second period price, Ri,n i 1 2 , n i 1 < K i. Thus, under duopoly, Retailer i’s expected profit is given by15 pii = Ki−1∑ ni1=0 PN i ni1 ( ni1R i 1 +R i,ni1 2 Nj∑ nj1=0 Nj−nj1∑ ni2=0 min ( max(nj1 −Kj , 0) + ni2,Ki − ni1 ) PN j nj1,n i 2 ) + N i∑ ni1=K i PN i ni1 KiRi1, where PN i ni1 denotes the probability that ni1 of the N i consumers who visit Retailer i in period 1 have valuations which exceed the posted price, Ri1, in that period, i.e. P N i ni1 = N i! ni1!(N i−ni1)! (Ri1) N i−ni1(1− 15Essentially, this profit expression assumes that any second period price is lower than a first period price, Rj1 ≥ R i,ni1 2 , n i 1 ≤ Ki, which can be verified to hold. 31 Ri1) ni1 . In a similar way, PN j nj1,n i 2 denotes the probability that out of the N j consumers who visit Retailer j in the first period, nj1 have valuations which exceed the posted price, R j 1, in that period, and as consumers switch in the second period from Retailer j to Retailer i, ni2 have valuations which are in [Ri,n i 1 2 , R j 1), i.e., P Nj nj1,n i 2 = N j ! nj1!n i 2!(N j−nj1−ni2)! (1−Rj1)n j 1(Rj1 −Ri,n i 1 2 ) ni2(Ri,n i 1 2 ) Nj−nj1−ni2 . General closed-form solutions for the capacitated duopoly are hard to obtain as expressions involve polynomial of high degrees. Therefore, we resort to numerical examples to illustrate the behavior of the model. Similar to the monopoly case, hereby we also show that under duopoly, when consumers are similar, pricing depends on the fraction of the market the retailers can satisfy. For simplicity, let us assume that the two retailers are symmetric, in the sense that, in the first period, the inventory levels at both retail stores are the same and each retailer is visited by the same number of consumers. In Figure 2.7 we demonstrate the effect of competition on prices and profits. To allow for a meaningful and system’s perspective comparison, we assume in Figure 2.7 that the total inventory in the monopoly and duopoly systems are equal and that the total number of consumers in the duopoly setting is equally distributed between the two competing retailers. Clearly, by Figure 2.7(a), the first period price16 is increasing in the total number of consumers in the market, when the total inventory in the system is fixed at 4 units (which means that, due to the symmetry assumption, each retailer has 2 units of good in stock). Thus, as N increases, the fraction of the market retailers can satisfy decline, and as a result, the competitive pressure diminishes and the retailers are able to increase their first period prices (contrasted with the lower dashed line, which is the price they set in the uncapacitated setting). Similar to Figure 2.6(a), it can be further illustrated how both the first and second period prices decrease in the available inventory in a capacitated duopoly. By symmetry, the system’s expected profit, which is the combined profits of the two retailers, is exactly equal to twice each retailer’s expected profit. Figure 2.7(b) illustrates the manner with which the system profit is increasing in the total number of consumers in the market, when inventory level is fixed at 4 units. As N increases, the competitive pressure diminishes, and the retailers post higher prices and earn higher profits. Further, as N increases from N = K, the percentage profit loss due to competition diminishes.17 Again, similar to Figure 2.6(b), it can be further illustrated 16Note that in Figure 2.7(a) under the monopoly setting, the system is uncapacitated for both N = 2 and N = 4. 17We note that in the example provided in Figure 2.7(b) when N = K, the percentage profit loss due to competition is almost 10%, while when N = 2 and K = 4, for which case prices and profit per consumer coincide with the single 32  0.4 0.5 0.6 0.7 0.8 2 4 6 8 10 12 N, total number of consumers Monopoly Duopoly (a) 1st period prices 0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 N, total number of consumers Monopoly Duopoly  (b) Expected profits Figure 2.7: First period prices and profits in a 2-period model when total capacity is 4 that as the available inventory, K, increases, prices decline but the expected profit increases. Thus, despite the declining prices, the increased sales compensate for the loss of profit. Yet, since prices decline in K, the profit per unit decreases. To summarize, we observe that, similar to the uncapacitated case, competition in the capacitated case dramatically suppresses prices, which results with a substantial decline in profit. When the number of consumers in the system increases, while inventory levels are fixed, the competitive pressure diminishes. As a result, prices are set closer to the monopolistic case and correspondingly, the relative profit loss is lower. Similarly, when the inventory levels increase, while the number of consumers in the system is fixed, the competitive pressure diminishes as well. Thus, we conclude that the competitive pressure intensifies in the fraction of the market that retailers can satisfy.18 The capacitated duopoly model can be extended to incorporate longer selling horizons. However, such an extension requires some strong behavioral and informational assumptions, which mainly address the behavior of consumers and retailers alike when one of the retailers is out of stock. We leave the full investigation of this extension for future research. We conclude this subsection with a note. When inventory is abundant, as was the case in Subsection 2.2.4, uncertainty about market size or distribution of consumers across the two stores consumer case, the profit loss is 6.25%. 18Due to space limitation, in our numerical examples we have considered only the symmetric case wherein consumers are equally distributed across the two stores and inventory levels are the same at both stores. However, it should be clear that pricing depends on both the distribution of consumers across the two stores and their actual numbers as well as the inventory levels at both stores. 33 does not affect pricing decisions. However, when market size may exceed inventory levels, then, as we have shown above, pricing depends on both inventory levels and number of consumers in the market. Thus, retailers facing uncertainty regarding the market size and its distribution between the two stores, should factor this uncertainty into their pricing decisions. 2.3.2.3 Limited Capacity - Demand Expectations It is quite natural for retailers to set prices along the selling horizon based on demand expectations. In that case retailers need not track any variation in demand realization from expectation, which is, anyway, supposed to be minor if the market size is large. As noted earlier, the expectation approach is common in the literature, and recently was employed by Liu and van Ryzin (2005). To simplify the analysis we consider the symmetric case such that at the beginning of the selling horizon each retailer has K units of the homogenous good in stock and encounters N consumers in the first period (i.e., in total there are 2N consumers in the duopolistic setting). In the duopoly case, when T = 2, Retailer i’s expected profit in the first period is: pii1 =  R11K, if N(1−R11) ≥ K, R11N(1−R11) + pii2, otherwise, where pii2 =  R12(K −N(1−R11)), if NR21(1− R 1 2 R21 ) ≥ K −N(1−R11), R12NR 2 1(1− R 1 2 R21 ), otherwise, Let α ≡ KN , i.e., α is half the fraction of the market size a retailer can satisfy. Thus, when α ≥ 2 the system is uncapacitated, as was covered in §2.2.4. Proposition 2.10 In a two-period selling horizon duopoly with symmetric retailers and similar consumers, if each retailer’s inventory can satisfy a fraction α of the consumers he initially en- counters, then the subgame perfect Nash equilibrium (SPNE) prices are (R11, R 1 2) = (R 2 1, R 2 2) =  (1− 23α, 1− α), if α ≤ 34 , (12 , 1 4), otherwise. Thus, the pricing scheme in a two-period uncapacitated duopoly previously derived (i.e., prices of 12 and 1 4 in the first and second periods, respectively) remains valid even when retailers are far 34 from being able to satisfy the entire demand in the market. Moreover, in this two-period setting, retailers price such that, in expectation, they will never run out of stock in the first period. Extending the analysis to longer horizons, it can be shown that the equilibrium prices in a three period uncapacitated duopoly are (Ri1, R i 2, R i 3) =  (1− 1019α, 1− 1619α, 1− α), if α ≤ 1315 , ( 815 , 4 15 , 2 15), otherwise, for Retailer i, i = 1, 2. In four periods, the equilibrium prices can be shown to be (Ri1, R i 2, R i 3, R i 4) =  (1− 1176224397α, 1− 1841224397α, 1− 2240224397α, 1− α), if α ≤ 194209 , (225418 , 60 209 , 30 209 , 15 209), otherwise. Figure 2.8 further illustrates the price decline for T = 5. The equilibrium prices retailers set coincide with those derived for our zigzag competition model with a single consumer when α > 0.961. However, the lower α is, the more moderate is the decline of prices along the selling horizon. 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 t, Period Number  Figure 2.8: Prices in a five-period model under competition with limited capacity Finally, it can be shown that the corresponding monopolistic case under demand expectations, a monopolist having K units in stock and facing N consumers, with α ≡ KN , would set the following pricing scheme Rt =  1− tT α, if α ≤ TT+1 , T−t+1 T+1 , otherwise, which is clearly linearly declining over the selling horizon. This pricing scheme can be contrasted with the corresponding capacitated duopoly equilibrium pricing scheme to demonstrate the impact 35 that competition has in our model on prices and profits. 2.4 Identical Consumers In this section we analyze the model under the assumptions that consumers are identical with a common valuation uniformly distributed on [0, 1], and the retailers are symmetric, i.e., each has K units of the good in stock and encounters N consumers in the first period. We find that when each retailer can satisfy the entire market, there are two pure-strategy SPNE both of which are non-symmetric and, in general, the price decline exponentially over the selling horizon. In the capacitated case when consumers are identical there are multiple and even a continuum of SPNE. 2.4.1 Uncapacitated Duopoly 2.4.1.1 Two-Period Selling Horizon When retailers can satisfy the entire market (i.e., each has 2N units of the good in stock), we have the following theorem:19 Theorem 2.2 In an uncapacitated two-period selling horizon duopoly with identical consumers, there exists precisely two pure-strategy SPNE both of which are non-symmetric. Specifically, the prices are such that one of the retailers set a price of 12 in the first period while the other sets 37 , and in the second period they (both) set 3 7 if a sale occurs in the first period, and 3 14 otherwise. Recall that in the corresponding case when consumers are similar, the pure-strategy SPNE is unique and symmetric, and is such that each retailer sets a price of 12 in the first period, and 1 4 in the second period. Thus, on average, a two-period duopoly system facing identical consumers reduces the average first period price by 7% and the system’s profit drops by about 1.2% (or 7.4% when contrasted with the corresponding monopoly setting). Apparently, the learning opportunities about the common valuation increase the competitive pressure on prices and reduce profits. Formally, we have: 19It is assumed that retailers observe whether sales occur at the competing retailer or not. When this is not the case or when retailers do not update their demand knowledge after each realization, then the resulting solution is the one previously described in §2.3.2.3, wherein the model is solved via demand expectations. 36 Corollary 2.2 In the uncapacitated zigzag competition model with a two-period selling horizon and symmetric retailers, equilibrium prices and corresponding profits are lower when consumers are identical rather than similar. 2.4.1.2 Longer Selling Horizons We can extend the above analysis to longer selling horizons. However, with longer selling horizons the number of possible SPNE grows quickly. Let us consider two possible such SPNE wherein (i) one of the retailers always sets the lower price while the other sets the higher price and (ii) the retailers alternate in these roles. Retailer i sets the lower price in all periods Defining the price ratios Bkt = Rkt Rit−1 , k = i, j, with Ri0 = 1, it can be shown that B i t = 3 8−3Bit+1 , Bjt = 1 2 , and that {Bit} is a convergent sequence, which converges to 0.451. Therefore, prices decline exponentially over the selling horizon, and at convergence (i.e., T →∞), Ri1 = 0.451, Rj1 = 0.5, pii = 0.338N and pij = 0.3139N . This represents a system profit loss of 34.8% due to competition, when compared with the monopolist’s profit, which converges to N if he holds 2N units in stock. Recall that the price ratio sequence in the corresponding uncapacitated case when consumers are similar is also backwards increasing and bounded from below by 12 , which is greater than or equal to Bjt (which, in turn, is greater than or equal to B i t), for any t, and strictly greater than B i t for t < T − 1. This leads us to the following conclusion: Corollary 2.3 In the uncapacitated zigzag competition model with symmetric retailers, if con- sumers are identical and, in equilibrium, the same retailer constantly sets the lower price at each period, the resulting prices at each period and at each retailer are lower or equal to the prices set when the consumers are similar. Retailers alternate in setting lower prices The analysis for this case is similar to the preced- ing case. Indeed, it can be shown that, at convergence, the lower (resp., higher) price in the first period is 0.445 (resp., 0.5), Corollary 2.3 holds in this case as well, and prices decline exponentially. We also note that when the retailers are not symmetric, in the sense that they are not visited by the same number of consumers in the first period, equilibrium pricing would depend both on 37 the total number of consumers and their distribution across the two stores. 2.4.2 Capacitated Duopoly Recall that α ≡ KN , and that when α = 2, each retailer can satisfy the entire demand in the market. Theorem 2.3 In a capacitated two-period duopoly with identical consumers and symmetric retail- ers, when α > 32 or α < 2 3 , there exist two pure-strategy SPNE, both of which are non-symmetric; when 23 < α < 3 2 there is a continuum of symmetric pure-strategy SPNE; when α = 2 3 or α = 3 2 there exist a unique pure-strategy SPNE. Thus, price dispersion in the market occurs when inventory levels are sufficiently high or suf- ficiently low. The equilibrium prices for the case Ri1 ≤ Rj1, as a function of α, are summarized in Table 2.2.20 As α increases, the competitive pressure increases and the optimal prices exhibit a general declining trend. For example, when α ≤ 12 , each retailer’s inventory is at most a quarter of the market size and the competitive pressure on prices is very low. In this case, in the first period, Retailer i (resp., j) sets the price to be 23 (resp., 5 6). If only Retailer i sells in the first period, the common valuation is uniform on (23 , 5 6). Then, in the second period, it is optimal for Retailer j, who encounters the N(1−α) consumers who switch from Retailer i, to set the price at 23 . At this price, which coincides with the initial price set by Retailer i, Retailer j sells his entire inventory. Similarly, when α ∈ [12 , 23 ], if only Retailer i sells in the first period, then in the second period Retailer j sets the same price set by Retailer i in the first period. However, under the same scenario and when α ∈ [12 , 23 ], Retailer j ends up with some leftover inventory in the second period. As a result, to decrease the chance of a leftover inventory, Retailer j decreases his first period price as α increases, until, for α = 23 , it coincides with Retailer i’s first period price. When α = 2 3 , equilibrium prices are symmetric, unique, and coincide with the corresponding single-store monopolist’s prices. This pricing scheme remains equilibrium pricing up to α = 1. That is, when 23 ≤ α ≤ 1, retailers under competition may achieve the same joint profit as a single store monopoly. On the other hand, when 1 ≤ α ≤ 32 , equilibrium prices may be as low as in the uncapacitated single consumer duopoly case, 20Since Ri1 ≤ Rj1, if Retailer i is the only one who sells in the first period, then in the second period, for α ≤ 1, Retailer j encounters the remaining consumers who haven’t purchased from Retailer i, and he sets Retailer i’s first period price; and for α ≥ 1, only Retailer i encounters consumers, and he sets his first period price again. If none of the retailers sell in the first period, then their second period prices are symmetric and equal half the lower first period price. 38 thereby eliminating any advantage from learning about the consumers’ common valuation. Duopoly Two-Store Monopoly α Ri1 R j 1 pi i pij System’s profit Ri1 R j 1 pi 2 ≤α 3 7 1 2 9 28 N 29 98 N 121 196 N 3 5 4 5 7 10 N 3 2 ≤α≤ 2 1+α 4α−1 1 2 (1+α)2 4(4α−1)N 17α2−6α+2 4(4α−1)2 N 4α3+24α2−4α+1 4(4α−1)2 N −(1+α) 3−6α+α2 2−3α 3−6α+α2 (1−4α)N 2(3−6α+α2) 1 ≤α≤ 3 2 1 2 ≤ Ri1 = Rj1 ≤ 22α+1 516N ≤ pii = pij ≤ (4α−1)N (2α+1)2 5N 8 ≤ pii+j ≤ 2(4α−1)N (2α+1)2 2 3 2 3 2 3 N 2 3 ≤α≤ 1 α 3α−1 ≤ Ri1 = R j 1 ≤ 23 α2(9α−4)N 4(3α−1)2 ≤ pii = pij ≤ αN3 α2(9α−4)N 2(3α−1)2 ≤ pii+j ≤ 2αN3 23 23 23K 1 2 ≤α≤ 2 3 2 3 α+2 6α α 3 N (21α2−12α+4)N 36α 33α2−12α+4 36α N −α(α+1) 3α2−6α+1 α(2α−3) 3α2−6α+1 α2(α−4)N 2(3α2−6α+1) α≤ 1 2 2 3 5 6 1 3 K 13 36 K 25 36 K 3 5 4 5 7 10 K Table 2.2: Duopoly and monopoly prices and profits in a two-period selling horizon when consumers are identical and Ri1 ≤ Rj1 Table 2.2 also provides the retailers’ profit and system profit. When α is sufficiently low, the retailer who sets the higher initial price gains a larger profit, while with sufficiently high α the retailer who sets the lower initial price gains a larger profit. Note that by contrast with capacitated duopoly with similar consumers (Figure 2.7(b)), the total expected profit in a capacitated duopoly with identical consumers may exceed that of a single-store monopoly. However, we further note that the proper effect of competition is obtained by contrasting the capacitated duopoly setting with the corresponding two-store monopoly. Clearly, a two-store monopoly (§2.5.2.2) cannot perform any worse than a single store-monopoly or a duopoly. 2.5 Two-Store Monopoly with N Consumers To properly evaluate the effect of competition we briefly consider in this section the base case in which the two retail stores are owned by a central planner who is setting the prices in both stores to maximize the system profit. Assume that N1 (resp., N2) consumers visit Retail Store 1 (resp., 2) in the first period, with N1 ≥ N2 and N1 +N2 = N , the system is uncapacitated, and the central planner knows N1 and N2. If the central planner restricts prices in both retail stores to be equal, the system reduces to a single-store monopoly. However, allowing for possibly different prices in the two stores, which is a common practice if the stores have different brand names, provides the centralized system with up to twice as many observations on the consumers’ valuations.21 In this case, a centralized system 21Yet, this cannot be viewed as a single-store monopoly with twice as many observations, since at each period two 39 may be able to sell the goods closer to the consumers’ valuations and increase its profit. We show below that a two-store centralized system facing similar consumers can do no better than a single-store monopoly. By contrast, however, it can achieve an expected profit that exceeds the profit obtained by the single-store monopoly when it faces identical consumers. 2.5.1 Two-Store Monopoly with N Similar Consumers Since the valuations of the consumers are independent of each other, information gained about the valuations of consumers who visit Store 1 (resp., 2) bears no relevance to the valuations of the consumers who visit Store 2 (resp., 1) in period t. Therefore, we conclude: Proposition 2.11 A two-store monopoly facing similar consumers who zigzag between the two stores has no advantage over a single-store monopoly, as both post the same prices and yield the same expected profit, regardless of the valuation distribution and the distribution of the consumers between the two stores. 2.5.2 Two-Store Monopoly with N Identical Consumers 2.5.2.1 Uncapacitated Setting Suppose all consumers are identical and all have the same valuation, which is drawn from a uniform distribution over [0, 1]. Then, it follows from Proposition 2.12 below that in each period the best policy for the central planner is to set a higher price at the store which is visited by the larger group of consumers than at the other store. Hereby, we provide some intuition for this result. Let RN 1 t (resp., R N2 t ) denote the price that is observed by the group ofN 1 (resp., N2) consumers in period t, and assume that RN 1 t ≥ RN 2 t . If only the N 2 consumers who observe the lower price, RN 2 t , buy the good, then the common valuation, V , of the remaining N 1 consumers is distributed uniformly on [RN 2 t , R N1 t ]. The central planner can use the additional information on the common valuation to increase the system’s profit. Naturally, the smaller N2 is, the higher is the system’s profit. Specifically, we have: Proposition 2.12 The expected profit of the centralized system is increasing with N1, when N is kept constant, as long as N2 is not zero. prices are set simultaneously. 40 The centralized system’s expected profit is equal to that of the single-store monopoly either when T = 1 (resulting with an expected profit of 14N) or when T → ∞ (with an expected profit of 12N). The centralized system is superior to the single-store monopoly in the intermediate cases, i.e., when 1 < T < ∞. However, for any T , when {N1 = 0, N2 = N}, the centralized system profit coincides with that of single-store monopoly, whereas the highest profit is achieved by the centralized system when {N1 = N − 1, N2 = 1} with N → ∞ (Proposition 2.12). These two bounds are shown in Figure 2.9(a), while Figure 2.9(b) provides their ratio, which measures the maximum advantage of the two-store monopoly versus the single-store monopoly. This can also be viewed as the value of additional observations regarding the consumers’ common valuation. As can be observed, this profit advantage is maximized at T = 3, at which point it reaches a value of 14.28%. Yet, this advantage may be achieved only when there is a very large number of consumers and all but one visit Retail Store 1 in the first period. 0.25 0.3 0.35 0.4 0.45 0.5 1 11 21 31 41 T, Selling Horizon Pr o fit  pe r Co n su m er N1->N, N2->0 N1=0, N2=N  (a) Centralized system profit per consumer 1 1.05 1.1 1.15 1 11 21 31 41 T, Selling Horizon  (b) Profit ratio Figure 2.9: The centralized system minimal and maximal profit and their ratio 2.5.2.2 Capacitated Setting Similar to the corresponding duopolistic case, in a centralized two-store monopoly with symmetric stores, such that each has K units of the good in stock and faces N consumers in the first period, with α ≡ KN , we have the following statement. Proposition 2.13 In a capacitated two-period selling horizon two-store monopoly22 with symmet- 22The term capacitated is slightly abused here, as it means that the total market size exceeds the inventory at each store, while the total inventory may in fact be larger than the market size. 41 ric stores and identical consumers, the first period prices the central planner sets at the two stores are (i) different when α > 32 or α < 2 3 ; (ii) identical when 2 3 ≤ α ≤ 32 . The prices and corresponding profit are provided in Table 2.2, assuming Ri1 ≤ Rj1. Surpris- ingly, in the absence of other costs, the central planner achieves no advantage over the single-store monopoly when 23 ≤ α ≤ 32 . Only when his stocking levels are sufficiently high or sufficiently low at the two stores, he differentiates the prices to gain an advantage from learning about the common valuation. At any case, the profit of the central planner is evidently higher than the combined profit of the two duopolistic retailers. 2.6 Managerial Insight and Future Work In this chapter we have presented and analyzed a multi-period model of competition between two retailers with fixed inventories, under the assumption that each of N similar consumers visits only one of the retailers in any given period. If the posted price is above the consumer’s valuation, which is assumed to be uniformly distributed, in the ensuing period he visits the competing retailer. This model setting was also studied when (i) consumers’ valuations have a power distribution, (ii) retailers may not be able to satisfy the entire demand (capacitated setting), (iii) consumers are identical rather than similar, and (iv) the two retail stores are owned by a central planner, who may elect to post different prices in his two stores. Our main conclusion, which was found to be quite robust, is that competition exerts significant pressure on prices, which drop exponentially rather than linearly, with a dramatically lower initial price. The competitive pressure also reduces, very significantly, the pre-competition profit of the monopolist, a loss which increases in the length of the selling horizon. Our results also lead to several managerial insights. Specifically, a firm which ignores competition, may overprice in the first period, and, as a result, may lose a substantial part of the expected profit. Similarly, retailers under competition should mark down quite substantially. Indeed, in our uncapacitated model, retailers mark down the price roughly by half in consecutive periods, which may explain short life cycles of products in competitive markets. Further, even though our results are found to be robust, additional insights were gained by studying the model in different settings. Specifically, while it was shown that prices decline ex- 42 ponentially when the number of periods is large enough, the price decline over shorter horizons, which is probably a more realistic scenario for most fashion goods, is quite affected by consumers’ valuations. That is, retailers should price higher and offer lower discounts if they believe that the consumers’ valuations are more likely to be higher. Similarly, it is shown that the effect of compe- tition is mollified if the retailers can satisfy a smaller fraction of the market. Equivalently, it may benefit the competing retailers to increase demand, say, by joint advertising, as it would allow them to price their goods higher and to offer lower discounts. It is also shown that prices and profits are affected by consumers’ homogeneity. Specifically, we have shown that while a two-store monopoly system can take advantage of perfect homogeneity of consumers (i.e., identical consumers) and in- crease its profit by up to 14%, a two-store duopoly system with perfectly homogeneous consumers may post lower prices and generate a smaller system profit as compared to the case where con- sumers are completely heterogeneous (i.e., similar consumers). The main managerial insight for a centralized system is that it may be best for the planner to keep a small test market in every period to test a lower price for the good. Our two-period analysis with identical consumers and symmetric capacitated retail stores has revealed several important insights. Under both two-store monopoly and duopoly systems, when inventory levels at the two retail stores are sufficiently low or sufficiently high, the resulting equilib- rium prices at the two locations are different. Thus, we provide an explanation for price dispersion in the market even when consumers are identical and retail stores are symmetric. This price dis- persion may, in return, further induce consumers to zigzag across the two retail stores. Our model sets the ground for a reacher context model wherein retailers may choose initially their inventory levels and then compete in prices over the selling horizon. Clearly, additional vari- ants of the model can be explored, such as allowing for replenishment during the selling horizon, incorporating holding and wholesale costs as well as depreciation of products. Another possible extension is to study the impact of loyal consumers (e.g., frequent flyer programs or captive con- sumers), or to let the two competing retailers engage in marketing efforts in order to raise their initial market share. 43 2.7 Bibliography Aviv, Y. and Pazgal, A. (2005). A partially observed markov decision process for dynamic pricing. 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A commonly used multi-period pricing strategy is price skimming, wherein prices gradually and systematically decline over time. Yet, little is known about the effect of competition in such settings. We introduce multi-period models which allow retailers to employ price skimming strategies in competitive settings. In our models, prices are not common knowledge and consumers, as is true in many real situations, need to actively visit stores to observe the posted prices of the goods.2 Similar to many multi-period monopolistic models, we assume that consumers repeatedly visit the retail stores in each period until they purchase the good. This gives rise to a novel way of modeling competition in multi-period settings. In our model, the visit pattern of consumers at the stores is described in a similar fashion to the very common Markov brand-choice models. Markov brand-choice models, which have long been recognized as an essential building block in the marketing literature (see, e.g., Lilien et al. ∗A version of this chapter has been submitted for publication. Granot, D., Granot, F. and Mantin, B. Price Skimming in the Presence of Competition: Multi-Period Markov Store Choice Models. 1For an excellent review of pricing strategies, the reader is referred to Noble and Gruca (1999). 2It is widely assumed that consumers, or a fraction thereof, lack information about prices (Tellis and Wernerfelt (1987), Simester (1995), Moorthy and Winter (2002), and Kuksov (2006)). Advertisements, such as loss-leader products (Lal and Matutes (1994)), may reveal prices, but these special cases are excluded from our setting. 47 (1992), Leeflang et. al (2000), or Kahn et al. (1986)), investigate repeated purchase behavior of consumers.3 À la Markov brand-choice models, we assume that in a multi-period setting, myopic consumers, if they do not purchase the good, return in the following period to the same store with a certain probability. This return probability could be location-independent (structural com- petition) or location-dependent (behavioral competition). Thus, structural and behavioral types of competition can be thought of as a zero and first-order Markov store-choice models, respectively. We recognize some differences between store-choice and brand-choice models: a store visit in our Markov store-choice models is analogues to a purchase event in the brand-choice models; in our store-choice models we consider (at most) a single purchase by each consumer while brand-choice models consider repeated purchases; and, in the single consumer case, our store-choice models terminate once a purchase occurs (or once the season ends), while brand-choice models terminate once the consumer purchases the good for the last time (e.g., when a toddler no longer requires diapers or when a consumer adopts a different life-style). Structural competition reflects market settings in which the return probability is determined by market conditions such as geography, dominance and other competitive elements. In such a competition, the experience of the consumer at a store does not affect his decision whether to return to that store in the future. On the other hand, in a behavioral competition, the market structure does not play a significant role in determining the return probability. Rather, it is determined by the consumers’ experience at the stores, influenced by factors such as service level, retailer’s friendliness, and store ambience. Models of dynamic choice behavior invoke three notions of temporal dependence: habit persis- tence, state dependence, and unobserved heterogeneity (Roy et al., 1996). All three are addressed in our Markov store-choice models. Habit persistence and state dependence are captured by the return probabilities. Heterogeneity characterizes consumers’ unobserved valuations. Further, when there are many consumers in the market, their visit patterns are independent of each other. In the current analysis, however, we abstract away from the mechanism determining the values of the return probabilities. In other words, employing a theoretical approach, we do not consider the 3Based on a consumer’s previous purchasing realizations, this method assigns a probability for purchasing a particular brand by the consumer in the current (and future) purchasing occasions. The rank of the order indicates the length of history relevant to evaluate these probabilities. Namely, in zero-order Markov models, the purchasing behavior is independent of the past, while in first-order Markov models, only the last purchase has an effect on the current choice. 48 specifications leading consumers to adopt a certain store visit pattern over another. The return probabilities can serve as a proxy for competition intensity in the market. For exam- ple, in a behavioral competition between two symmetric retailers, we are able to study duopolistic competition with the entire range of competition intensities. When the common return probability is one, retailers are able to monopolize consumers once they enter their store. Similarly, when the common return probability is zero, consumers deterministically switch between the two retailers throughout the selling horizon as long as they have not purchased the good, a setting termed zigzag competition by Granot, Granot and Mantin (2007) (Chapter 2, henceforth, GGM). We note that pricing strategies are addressed by a growing body of literature in revenue man- agement. In a traditional revenue management setting, retailers’ objective is to determine an optimal pricing or rationing policy that maximizes revenue from available (limited) inventory over a finite fixed horizon. Many of the papers that study the classic revenue management setting have exclusively focused on the monopoly setting as, e.g., is the case in Lazear (1986), Besanko and Winston (1990), Gallego and van Ryzin (1994), Feng and Gallego (1995), Bitran and Mondschein (1997), Zhao and Zheng (2000), and Aviv and Pazgal (2007). Recent reviews of pricing models within the context of revenue management include Elmaghraby and Keskinocak (2003), Bitran and Caldentey (2003), and Chan et al. (2004). These reviews have recognized the absence of com- petition, and have recommended the pursuance of this direction. Indeed, in recent years efforts have been commenced to investigate this challenging area, which resulted with, among others, the papers by Perakis and Sood (2006), Gallego and Hu (2006), Lin and Sidbari (2004), and Xu and Hopp (2006). The competitive revenue management models in these papers either treat consumers through demand aggregation or assume that each consumer has a single encounter with retailers and then leaves the system. By contrast, in our model the market size is fixed and the consumers remain in the market throughout the entire selling horizon as long as they haven’t purchased the good. Our multi-period model is mostly related to Lazear’s monopoly and GGM’s zigzag competition finite-horizon multi-period models. In Lazear, where a monopolist retailer faces myopic and iden- tical consumers with uniformly distributed valuations, the price was shown to decline linearly over the selling horizon. GGM have assumed that the consumers can visit only one of the two retailers in any period, and that they deterministically switch between the stores as long as the price they 49 observe exceeds their valuations. In their zigzag competition, prices decline exponentially over the selling horizon rather than linearly. The structural and behavioral competition models considered in this chapter significantly gener- alize GGM’s zigzag competition model. Specifically, they allow for a far more general visit pattern by consumers, retailers which are not symmetric, and they can be easily extended to oligopolistic settings. Our analysis starts with the case of a single consumer, who may be perceived as a repre- sentative instance of the entire demand in the market. The single consumer assumption enables a parsimonious game theoretic model with several important insights. We find that under both struc- tural and behavioral types of competition, the price decline follows an exponential shape. When retailers are not symmetric, the price declines within an exponential band, with the lower (resp., upper) bound representing situations wherein the consumer happens to visit only the retailer asso- ciated with the lower (resp., higher) return probability throughout the entire selling horizon. Thus, we demonstrate that even if a retailer is very dominant or has a very high retention rate, the mere fear of losing consumers due to competition, even with a relatively small probability, significantly suppresses prices, as the price decline reverts from a linear shape to an exponential one. Importantly, we find that competition has an immense effect on retailers’ profits. In structural competition, as expected, the dominant retailer earns a larger share of the total profit. However, small variations in a retailer’s dominance, when this level is fairly high, can quite dramatically affect his expected profit. This suggests that dominant retailers should carefully monitor the performance of their weak competitor, since any decrease in their dominance level can substantially decrease the dominant retailer’s profit. Under structural competition the system’s (i.e., the retailers’ combined) profit is U-shaped and is minimized when the retailers are symmetric. Thus, firms who compete in distinct markets are better off by tacitly coordinating away from the symmetric situation wherein both are equally dominant in all markets. This may explain why some markets are dominated by one firm while other markets are dominated by a competing firm. Under behavioral competition with symmetric retailers, the system’s (as well as each retailer’s) expected profit is increasing in the common return probability. Therefore, risk neutral retailers prefer an equilibrium with a high common return probability. Indeed, when the common return probability is low, expected profit is lower. Yet, when symmetric retailers have a high return probability, in expectation, a retailer is equally likely 50 to end up with a huge profit or with almost nothing at all, while with a low return probability, a retailer may end up with a moderate or small expected profit. This suggests that risk averse retailers might prefer an equilibrium with a low return probability over a high return probability. It is also shown that the minimum system’s profit under structural competition is strictly larger than the system’s profit under behavioral competition whenever retailers are symmetric with a return probability smaller than 12 . The latter case is minimized when this return probability is zero, which coincides with GGM’s zigzag competition. The consumer’s surplus, on the other hand, behaves in the opposite manner. That is, the consumer’s surplus is maximized, when consumers ignore retailers’ dominance or stores’ ambience and zigzag deterministically between the two stores, adopting a visit pattern which arises in a behavioral competition model with a zero retention level. Prices are dramatically altered when a second competing retailer enters a monopoly setting, as prices revert from a linear to an exponential decline. Naturally, with additional retailers in the market, the expected profit of each individual retailer declines. Quite surprisingly, though, we find that adding more retailers into the market beyond the duopolistic level, has a marginal effect on prices. This suggests that a regulator probably should not be overly concerned with the lack of competitors in the market as long as they are relatively symmetric under structural competition and not too retentive under behavioral competition. Our model is, then, extended to incorporate a market consisting of many similar consumers, whose valuations are independently drawn from a uniform distribution over [0, 1]. In this setting, the effect of competition on prices and corresponding profits is shown to be essentially the same as in the single consumer case. Namely, the pricing policy for a market with many consumers is very similar to that derived for the single consumer case (and, in fact, the two coincide when the return probability is 0.5), and the corresponding profit per consumer only very marginally differs from that obtained with a single consumer. Yet, it is revealed that when there are many consumers in the market, retailers may choose to employ a more refined skimming policy when the horizon is long enough. Specifically, during the early periods of the selling horizon, the prices set by the retailer with the higher return probability may target only returning consumers (i.e., consumers who have visited him in the preceding period). When this occurs, for example, the initial price at the retailer with the higher return probability (slightly) spikes up and becomes closer to the initial price in the corresponding single consumer case. 51 The chapter is organized as follows. In Section 3.2 we introduce our multi-period store-choice models with a single consumer, both for duopoly and oligopoly settings. These models are extended in Section 3.3 to incorporate many similar consumers. Section 3.4 concludes with managerial insights, future directions for research, and discussion of some of the assumptions. 3.2 The Single Consumer Case In this section we introduce our competitive multi-period models that employ the Markov store- choice concept. The duopoly setting with a probabilistic visit pattern is analyzed at length in §3.2.1. When the return probability is independent of the location of the store previously visited by the consumer, we say that the competition is structural (§3.2.1.1), and when this probability is location-dependent, we say that it is behavioral (§3.2.1.2). In §3.2.2 we further study the oligopoly case for both types of competition. 3.2.1 Duopoly: The Single Consumer Case Assume that in a multi-period selling horizon setting, a single myopic consumer, who is interested in buying a certain good, has no price information and needs to visit the retail stores who offer this good in order to observe the prices. We do not model search costs, but assume that due to these costs the consumer is limited to a single store visit per period. The two profit-maximizing retailers, who offer this good and who compete over prices, can always satisfy the realized demand. That is, in the single consumer case it is equivalent to assuming that each of the retailers has one unit of an identical good in stock. The good is assumed to have zero marginal cost and at the end of the T -period selling horizon it has no salvage value. The two retailers cannot observe the consumer’s valuation for the good, V , but they know its distribution function, F (V ). Let Rit denote the price set by Retailer i in period t, and let Rt denote the price observed by the consumer in that period. If the consumer visits Retailer i, i = 1, 2, in period t and the posted price is below his valuation, he purchases the good and the season ends. Otherwise, he does not purchase the good. In this case, the retailers infer that V < Rt = Rit and by Bayes’ Theorem, the posterior distribution carried from period t to period t + 1 is given by Ft+1(V ) = F (V ) F (Rt) , 0 ≤ V ≤ Rt, and in the ensuing period the consumer revisits the same retailer 52 with probability (w.p.) Pi, and switches to the competing retailer w.p. 1− Pi. These probabilities are location-independent in the structural competition and location-dependent in the behavioral competition. Let pii,vt (resp., pi i,n t ) denote Retailer i ’s expected profit-to-go if the consumer visits (resp., does not visit) his store in period t, which is given by the following recursive expression: pii,vt = R i t[1− Ft(Rit)] + PiFt(Rit)pii,vt+1 + (1− Pi)Ft(Rit)pii,nt+1. (3.1) The first term is the profit from a sale in period t, and the second (resp., third) term is the conditional profit-to-go in period t + 1 if the consumer does not purchase the good in period t and returns to Retailer i (resp., switches to the competing retailer) in period t + 1. Retailer i’s profit-to-go if he does not encounter the consumer in period t is given by: pii,nt = PkFt(R j t )pi i,v t+1 + (1− Pk)Ft(Rjt )pii,nt+1, (3.2) where Pk = Pi (resp., Pk = 1− Pj) in structural (resp., behavioral) competition. Using dynamic programming, the profit maximizing retailers find the optimal pricing policy by solving equations (3.1)-(3.2) backwards and the resulting solution is, therefore, a subgame perfect Nash equilibrium. For simplicity and tractability, we first assume V is uniformly distributed over [0, 1]. Thus,4 Ft(Rt) =  Rt Rt−1 , if Rt < Rt−1, 1, otherwise, with R0 ≡ 1. 3.2.1.1 Structural Duopoly We will refer to a duopoly under structural competition as a structural duopoly. In a structural duopoly, the visit probabilities are location-independent and sum to one. As long as no purchase has occurred, in every period the consumer visits Retailer 1 with probability P1 ≡ P and Retailer 2 with probability P2 ≡ 1−P . When P 6= 12 , the retailer associated with the higher return probability is the dominant retailer, while his counterpart is the weak retailer. The solution for the last two periods is presented in Table 3.1, where the columns, respectively, display Retailer 1 and Retailer 2’s prices and profits-to-go if they are visited or not visited by the 4In §3.4.2.1 we briefly extend the analysis to a power distribution. 53 consumer in the corresponding period. This table suggests that prices and profits-to-go in each period are functions of the prices set in the preceding period. R1t R 2 t pi 1,v t pi 2,v t pi 1,n t pi 2,n t Period T 12RT−1 1 2RT−1 1 4RT−1 1 4RT−1 0 0 Period T − 1 24−P RT−2 23+P RT−2 14−PRT−2 13+P RT−2 P(3+P )2RT−2 1−P(4−P )2RT−2 Table 3.1: Structural duopoly: Prices and profits-to-go in the final two periods We find that in a structural duopoly the profit-to-go expressions in period t are affine functions of the price set in the previous period. That is, pii,jt = S i,j t Rt−1, i = 1, 2, j = v, n, where S i,j t are scalars which can be expressed recursively (see Appendix A.3, Lemma A.6). The scalars Si,vt and Si,nt have a dimension of sales, and will be referred to as the normalized sales-to-go in period t of Retailer i if he encounters or does not encounter, respectively, the consumer in period t. Note that in the first period, pii,j1 = S i,j 1 , i = 1, 2, j = v, n. Let us define the price ratios, B i t ≡ R i t Rt−1 , i = 1, 2. Since R0 = 1, in the first period Ri1 = B i 1. The price ratios in each period reflect the incremental discount rate offered on the last price observed by the consumer. We find that these price ratios can be expressed recursively (see Appendix A.3): B1t = B1t+1 B1t+1(2− PB1t+1)− (B2t+1)2(1− P )(2B1t+1 − 1) , (3.3) B2t = B2t+1 B2t+1(2− (1− P )B2t+1)− (B1t+1)2P (2B2t+1 − 1) , (3.4) with B1T = B 2 T = 1 2 . These expressions imply that the retailers’ price ratios in each period are independent of the consumer’s location in the preceding period. The price ratios set by Retailer 1 in the final 20 periods of the selling horizon for different P values are illustrated in Figure 3.1(a). Evidently, the more dominant a retailer is, the higher are his price ratios, or, in other words, the lower are the discounts he offers. Proposition 3.1 The price ratios set in all periods by the dominant retailer in a structural duopoly are higher than those set by the weak retailer. That is, ∀t, B1t ≥ B2t if P > 12 and B1t ≤ B2t if P < 12 . 5 This proposition implies that, unless P = 12 , price ratios set by the two retailers are different 5All proofs of statements made in this chapter are provided in Appendix A.3. 54 0.4 0.5 0.6 0.7 0.8 0.9 1 1 3 5 7 9 11 13 15 17 19 t, Period Number P=1 P=0.99 P=0.9 P=0.5 P=0  (a) Price ratios over 20 periods 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 10 t, Period Number Only Retailer 2 Only Retailer 1  (b) Prices over 10 periods, P = 0.2 Figure 3.1: Structural duopoly: Price ratios and prices (as demonstrated in Figure 3.1(a)), and, therefore, actual prices observed by the consumer depend on his realized visit pattern between the two stores. For example, in Figure 3.1(b) we consider a selling horizon of 10 periods with P = 0.2, i.e., Retailer 1 is the weak retailer. The solid lines represent the extreme cases when the consumer happens to visit only either the weak (lower line) or the dominant (upper line) retailer throughout the selling horizon. The dashed lines illustrate four other possible realizations of visit patterns (as long as no purchase occurs). As Figure 3.1(a) shows, regardless of a retailer’s dominance, the price ratios decrease as the selling horizon approaches its terminal period. Perceived backwards, they increase and converge. Proposition 3.2 The sequences of the price ratios, {B1t } and {B2t }, in a structural duopoly are monotone convergent sequences as the number of periods, T , goes to infinity, while t goes to 1. Since in the first period prices and price ratios coincide, the convergence value of the price ratios are also upper bounds on the initial prices set by retailers. These initial prices at convergence are plotted in Figure 3.2(a). Retailer 1’s initial price at convergence is increasing in P from 0.5 to 1, while Retailer 2’s initial price is decreasing in P . When P = 0.5, the retailers are symmetric and set the same initial price, 12( √ 5− 1). We observe that Retailer 1’s initial price is much more sensitive to changes in P when P is large. For example, a small decline of 1% (resp., 10%) in P from 1 to 0.99 (resp., 0.9) results with a large decline of 7.9% (resp., 21.3%) in the initial price. Thus, the mere fear of losing the consumer due to competition, even with a small probability, significantly suppresses prices. In other words, a dominant retailer cannot ignore competition stemming from a 55 Retailer 2 Retailer 1 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 P, Return Probability (a) Initial prices System Retailer 2 Retailer 1 0 0.1 0.2 0.3 0.4 0.5  0.2 0.4 0.6 0.8 1 P, Return Probability (b) Expected profits Figure 3.2: Structural duopoly: values at convergence weak retailer and as a result has to lower prices quite dramatically. Since the sequences of price ratios are backwards increasing and bounded away from 1, prices observed by the consumer, when 0 < P < 1, decrease exponentially within a band. The upper (resp., lower) bound in this band corresponds to the case wherein the consumer happens to visit only the dominant (resp., weak) retailer. These bounds (solid lines) were illustrated in Figure 3.1(b) for T = 10 and P = 0.2. Theorem 3.1 In a structural duopoly, when 0 < P < 1, prices decline within an exponential band over the selling horizon. Similar to the price ratios, the normalized sales-to-go sequences, {Si,jt }, i = 1, 2, j = v, n, are also monotone convergent sequences as T goes to infinity in a structural duopoly (see Appendix A.3, Proposition A.2). Recall that in the first period pii,j1 = S i,j 1 , i = 1, 2, j = v, n. Thus, Retailer 1’s (resp., 2’s) total expected profit is given by pi1 = PS1,v1 +(1−P )S1,n1 (resp., pi2 = (1−P )S2,v1 +PS2,n1 ). The profits at convergence of each of the retailers, as well as their combined profit (i.e., system’s profit), are illustrated in Figure 3.2(b). At convergence, Retailer 1’s expected profit is increasing in P from 0 to 0.5. The system’s profit, which is symmetrically U -shaped, is minimized at P = 0.5, which corresponds to the case of equally dominant retailers, and it is maximized at P = 0 and P = 1, which correspond to monopoly settings (as studied by Lazear (1986)). At P = 0.5 the system’s profit is 0.3819, which represents a profit loss of up to about 23.5% due to (structural) competition. However, this profit is higher than the corresponding profit under zigzag competition 56 (as studied by GGM) which is derived as a special case under behavioral duopoly. Since the system’s profit is U -shaped, the total profit earned by the two retailers is not con- stant. Therefore, if the same retailers interact in several distinct markets, they have an incentive to coordinate deviations, in a reciprocal manner, from the symmetric case (i.e., P = 0.5) to the nonsymmetric case wherein, alternately, one of the two retailers significantly dominates the other retailer. Thus, our model provides another explanation for deviations from the symmetric situa- tion6 when firms compete in several distinct markets. Consider, for example, Delta Airlines and American Airlines, who compete in many overlapping city-pair routes. While Delta Airlines dom- inates the route between New York’s JFK Airport and Seattle with 50% market share (vs. 17% by American Airlines), it is American Airlines which dominates the route between New York and Miami with about 80% market share (vs. 15% by Delta). When firms deviate from the symmetric situation, higher prices, on average, are sustained. These insights are in accordance with the liter- ature concerned with multi-market contact competition. Dresner et al. (2007), e.g., have recently developed a theoretical model and, using data from the U.S. airline market, have provided empirical evidence to tacit-collusive pricing facilitated by multi-market contact between carriers in different routes. Also, Bernheim and Whinston (1990) have shown that with rival firms having dominant positions in different markets, prices are even higher - an effect termed as spheres of influence. 3.2.1.2 Behavioral Duopoly We will refer to a duopoly under behavioral competition as a behavioral duopoly. In a behavioral competition, upon entering the store, the consumer not only observes the price but he is also exposed to other characteristics of the store which affect his return probability. We assume that these characteristics are fixed for the duration of the selling horizon (as they cannot be altered in the short run), and that they do not affect the consumer’s valuation of the good. We first discuss the symmetric case, when retailers have identical retention rates, and later we briefly comment on the non-symmetric case. In the symmetric case retailers have identical retention rates, P1 = P2 ≡ P . Due to symmetry, it can be shown that in every period they set identical prices, R1t = R 2 t ≡ Rt, and, therefore, we suppress retailers’ superscripts from all expressions. 6I.e., when the firms are equally powerful. 57 B_{T–11} B_{T–10} B_{T–9} B_{T–8} B_{T–7} B_{T–6} B_{T–5} B_{T–4} B_{T–3} B_{T–2} B_{T–1} B_{T} B 0.5 0.6 0.7 0.8 0.9 1  0 0.2 0.4 0.6 0.8 1 P, Return Probability Figure 3.3: Behavioral symmetric duopoly: Bt in the final periods, which coincides with the initial price in a horizon with T − t+ 1 periods, and the convergence graph of Bt, B Similar to structural duopoly, one can show that (i) the profit-to-go functions in period t are affine functions of the preceding price, i.e., piit = S i tRt−1, i = v, n, where Sit are recursively expressed scalars, and that (ii) the price ratio, Bt, is recursively expressed as follows: Bt = 1 2 + [Bt+1]2 ( Bt+2(2P − 1)− 2P ) . (3.5) Figure 3.3 illustrates the monotone convergence of {Bt} as a function of t and P . Namely, us- ing (3.5), Figure 3.3 shows the progression of Bt from BT = 12 up to BT−11 (the dashed curves), as a function of P . The solid curve is B, the convergence value of {Bt} (see Appendix A.3). One can observe from Figure 3.3 that B is an increasing convex function of P . It increases from 0.543 when P = 0 (which coincides with GGM’s zigzag competition) and approaches 1 as P goes to 1 (i.e., a monopoly). The similarity of the convergence values of the price ratios to the corresponding values in structural duopoly is discussed in §3.2.1.3. Since in the first period R1 = B1, B is also the convergence value of the initial price set by retailers (i.e., when T goes to infinity). Thus, B is an upper bound on the retailers’ price ratios and initial prices. As {Bt} is backwards increasing and for P < 1 it is bounded away from 1, we conclude that in a behavioral duopoly, unless P = 1, prices decrease exponentially over the selling 58 horizon.7 Consider, for example, a selling horizon of 20 periods. The prices in each of these periods are depicted in Figure 3.4(a). The corresponding price ratios are very similar to those illustrated earlier for the structural duopoly (Figure 3.1(a)). It can be observed that prices decrease faster for smaller values of P . For P = 1 the price is decreasing linearly over the selling horizon, while for P < 1 (and P not too close to 1) the price decreases exponentially. Indeed, the shift in the pricing scheme from linear to exponential occurs even with a small decrease in the value of P .8 However, as the value of P further decreases the impact on the pricing scheme is not as dramatic. We can conclude that, similar to structural competition, the mere fear of losing the consumer in a behavioral competition, even with a small probability, significantly suppresses prices. 0 0.2 0.4 0.6 0.8 1 1 3 5 7 9 11 13 15 17 19 t, Period Number P=1 P=0.99 P=0.9P=0.5P=0  (a) Prices, T = 20 0 0.04 0.08 0.12 0.16 1 3 5 7 9 11 13 15 17 19 T, Selling Horizon P=1 P=0.99 P=0.9 P=0.5 P=0  (b) Consumer’s surplus Figure 3.4: Behavioral symmetric duopoly: Prices and consumer’s surplus in a 20-period selling horizon The convergence of the sequences of normalized sales-to-go, {Svt } and {Snt }, follows immediately from the convergence of {Bt}. Since pii1 = Si1, i = v, n, the retailers’ combined (system) total expected profit is piv1 + pi n 1 (and a retailer’s total expected profit is 1 2(pi v 1 + pi n 1 ), assuming the retailers are equally likely to be visited by the consumer in the first period). The system’s profit at convergence behaves very much in the same way the initial price does, which was shown in Figure 3.3, only that the values are different. When P = 0, the model coincides with GGM’s zigzag competition, and the system profit is about 0.352, while when P = 1 the system profit is 0.5, as 7For short selling horizons and P very close to 1, the price decline is almost linear. However, for any P < 1, the price decline is exponential for a sufficiently long horizon. 8Again, as Figure 3.4(a) illustrates, when P = 0.9 the decline is clearly exponential, while for P = 0.99 it is more moderate and closer to a linear decline. Yet, with longer horizons, say T = 50, even when P = 0.99 the decline can be shown to be exponential. 59 the retailer who encounters the consumer in the first period becomes a monopolist. Similar to the initial price, the system’s profit at convergence is sensitive to changes in P when P is close to 1. For example, at convergence, a decrease in P from 1 to 0.9, results with a 12.9% drop in system’s profit. In expectation, the retailers are better off with a higher common retention rate. Risk neutral retailers clearly prefer high common retention rates, and therefore should, jointly or independently, improve their retention levels. We note also that the profit of the retailer who encounters (resp., doesn’t encounter) the consumer in the first period increases (resp., decreases) in P . That is, when P is high and the retailers are symmetric, a retailer may end up with a huge profit or with almost nothing at all, while when P is low he might end up with a moderate or a smaller profit. This suggests that risk averse symmetric retailers might prefer a low common retention rate over a high retention rate, as, in expectation, it ensures each of them a certain profit level. Since symmetric retailers set identical prices in each period, the prices observed by the consumer are not affected by his realized visit pattern. In that case, we can evaluate the consumer’s surplus for different P values, as illustrated in Figure 3.4(b) for varying selling horizons lengths. As P is increasing, the retailers have higher retention rates and are less engaged in ever slashing prices. Therefore, they can extract more profit from the consumer. However, when P is low, competition is fiercer, and they will extract a lower profit. For example, consider the case where P = 0. Then, the first and second period prices (which are 0.5 and 0.25, respectively, for a two period horizon) only marginally change when the number of periods increases. Moreover, the profit contribution from the additional periods, beyond period 2, which are associated with relatively very low prices, is very marginal. Therefore, for P = 0, the consumer’s surplus in a two period horizon (about 0.156), which is quite high, remains high when the number of periods increases. However, for high P values, e.g., P = 0.9, and longer horizons, initial prices increase considerably (Figures 3.3 and 3.4(a)) and subsequent period prices are fine-tuned accordingly and, thereby, the consumer surplus decreases dramatically. Lastly, we briefly comment on the non-symmetric case. Due to the inherent difference between the two retailers, their price ratios are different and they set different prices in each period. Further, it can be shown that a retailer’s initial price at convergence (which determines the retailers’ pric- ing policies) is increasing in his own return probability and decreasing in the competing retailer’s 60 return probability. However, the effect of a change in the competing retailer return probability is much smaller than a change in a retailer’s own return probability. The magnitude of the effect of the competing retailer return probability further diminishes as the retailer’s own return probability increases. This implies that in a non-symmetric behavioral duopoly, a retailer’s own return prob- ability is the primary driver of his pricing policy. The actual prices posted by retailers depend on the consumer’s visit pattern between the two retail stores, and decline within an exponential band in a similar fashion to that illustrated in Figure 3.1(b) for a structural duopoly. 3.2.1.3 Duopoly: Discussion The previous two subsections have introduced and analyzed multi-period structural and behavioral duopoly models with a single consumer. These models presume fairly realistic visit patterns and allow the consumer to return to the same retail store with a certain probability even if the previously observed price therein has exceeded his valuation. Despite the inherent difference between the structural and behavioral competition,9 the results and insights they provide are similar. The pricing scheme under both structural and behavioral competition is strictly decreasing over the selling horizon and follows an exponential shape. When retailers are associated with different return probabilities, prices decline within an exponentially declining band, of which the upper (resp., lower) bound corresponds to the case where the consumer happens to visit only the retailer associated with the larger (resp., smaller) return probability. Clearly, the introduction of competition into a monopoly setting dramatically affects prices in a multi-period selling horizon. While the actual impact on prices depends on the return probabilities and the realized visit pattern, both in structural and behavioral duopoly prices decline exponentially rather than linearly. The mere fear of losing the consumer due to competition, even with a relatively modest probability, already suppresses prices significantly. The similarity of the initial prices (and price ratios) set by Retailer 1 at convergence under structural duopoly (Figure 3.2(a)) and symmetric behavioral duopoly (Figure 3.3) is fairly evident. Thus, the pricing policy (in terms of price ratios) in these settings is fairly similar, at least at con- vergence. When P = 12 the two models coincide. However, when P < 1 2 (resp., P > 1 2), Retailer 1 sets a marginally lower (resp., higher) initial price in structural duopoly than in behavioral duopoly. 9The two models coincide when Retailer 1 is associated with probability P and Retailer 2 with probability 1− P under both behavioral and structural competitions. 61 Indeed, for example, when P < 12 , Retailer 1 is the weak retailer in a structural competition and he faces a dominant competitor, while in a symmetric behavioral competition he faces an equally retentive retailer. Therefore, his initial price when P < 12 is marginally lower in structural duopoly. Figure 3.2(b) quantifies the advantage of being a dominant retailer in a structural duopoly. Specifically, the profit of a retailer appears to increase linearly with his dominance, as captured by his return probability, P . However, for P values close to 0 or 1, the profit appears to increase much faster with P . This observation is consistent, for example, with Figure 3.2(a), which depicts the initial price as a function of P . Indeed, the initial price appears to increase almost linearly in P as long as P is not too close to 1. But, when P is close to 1, the initial price increases at a very fast rate in P . In other words, the mere introduction of a relatively weak independent retailer to a monopoly setting has a disproportionate effect on prices and profit at the previously monopolistic store. An additional increase in the strength of the new entrant, as measured by P , has a relatively more moderate effect. Finally, as observed earlier, at convergence, the system’s profit in a structural duopoly is min- imized when both retailers are symmetric, i.e., P = 12 , and at that point it is equal to 0.3819. Naturally, this profit is higher than the system’s profit under symmetric behavioral duopoly (which is increasing in P ) when the retailers’ common return probability is less than 12 . The system profit in a behavioral duopoly is minimized when both retailers’ return probabilities are zero, i.e., GGM’s zigzag competition, achieving a profit of 0.355. That is, under behavioral competition, when re- tailers’ common return probability is less than 0.5, the competitive pressure is higher than in a structural competition (with any return probability), and the expected system profit is lower. From the consumer’s point of view, he is best off, i.e., he maximizes his surplus, when he is oblivious to store dominance or store ambience, and alternates deterministically, in a zigzag fashion, between the two stores until the posted price falls below his valuation (Figure 3.4(b)). 3.2.2 Oligopoly Our model has so far assumed that the market consists only of two retailers. In this subsection, we generalize the model to allow for any number, M , of symmetric retailers in the market. If Retailer i is visited by the single consumer in period t, his profit-to-go in that period, pii,vt , is given by the recursive expression in display (3.1). The profit-to-go of Retailer j, j 6= i, who is 62 not visited by the consumer in period t, pij,nt , depends on the competition type. Recall that in a structural competition, the return probability is location-independent, and Retailer j is visited by the (active) consumer at any period with probability Pj . In a behavioral competition, the return probability is location-dependent: if the consumer visits Retailer i in period t and does not purchase the good, then with probability 1− Pi he visits one of the other retailers in period t+ 1. Assume that the consumer visits each of the other M − 1 retailers with equal probabilities, i.e., with probability 1−PiM−1 . Thus, Retailer j’s profit-to-go is pij,nt = PkFt(R i t)pi j,v t+1 + PlFt(R i t)pi j,n t+1, where Pk = Pj (resp., Pk = 1−PiM−1) and Pl = 1−Pj (resp., Pl = Pi) in a structural (resp., behavioral) competition. As we assume that the retailers are symmetric, they are associated with the same return prob- ability. In a structural competition, P is endogenized and is determined by the number of retailers, such that P1 = · · · = PM ≡ P = 1M , and in a behavioral competition P1 = · · · = PM ≡ P , which is provided exogenously. As before, the profit maximizing retailers find the optimal pricing by solving the dynamic programming model. 3.2.2.1 Structural Oligopoly with Symmetric Retailers Due to symmetry, we find that retailers post identical prices, and, therefore, retailers’ superscripts are suppressed. Let Rt denote the “common” price set in period t, and pivt (resp., pi n t ) the profit- to-go of a retailer if he encounters (resp., does not encounter) the consumer in period t. Similar to structural duopoly, we have: Proposition 3.3 In a structural oligopoly, the sequence of price ratios, {Bt}, is a convergent sequence as the number of periods, T , goes to infinity, while t goes to 1. The convergence value of the price ratio sequence, {Bt}, as T →∞ and t→ 1, is: B ≡ B(P = 1 M ) =  1−√9−8P 2(2P−2) , if P < 1, 1, if P = 1. (3.6) 63 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 M, Number of Retailers in the Market  (a) Initial price at convergence 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period Number M=1 M=2 M → ∞  (b) Prices over 10 periods Figure 3.5: Structural symmetric oligopoly: Prices As before, B is also the initial price at convergence, and it also determines the price decline over the selling horizon. At convergence, then, the initial price is an increasing convex function of P ,10 which increases from a value of 0.5, when P = 0, and approaches 1 when P = 1. Since P = 1M , the initial price at convergence is a decreasing convex function in M . Figure 3.5(a) depicts the initial price at convergence as a function of M , wherefrom we can conclude that the existence of two fully competing firms in the market is sufficient to push the prices down. Indeed, a dramatic decline in the initial price occurs once competition is introduced into a monopoly market, and the initial price at convergence drops from 1 to 0.618 (a decline of 38.2%). As the number of retailers in the market further increases, the impact on pricing diminishes. By adding, e.g., a third retailer to the market, the initial price drops to 0.568 (an additional decline of 4.9%). Eventually, as M → ∞, the initial price converges to 0.5. The above analysis is summarized in the following two theorems. Theorem 3.2 In a structural symmetric oligopoly, the initial price at convergence is a decreas- ing convex function in the number of retailers in the market. The introduction of competition (into a monopoly market) dramatically affects prices, while further intensifying the competition by introducing additional symmetric retailers has a relatively marginal impact on prices. Theorem 3.3 In a structural symmetric oligopoly, prices decrease exponentially over the selling 10As, for P < 1, ∂B ∂P = 4P+5− √ 9−8P 4 √ 9−8P (P−1)2 > 0 and ∂2B ∂P2 = 60P−24P 2−37+(9−8P ) 3 2 2(9−8P ) 3 2 (P−1)3 > 0. 64 horizon. The exponential decline of prices is illustrated in Figure 3.5(b). This figure further supports the previous statement, and shows that the pricing shifts from linear, in monopoly, to exponential, in a duopoly. As the number of retailers goes to infinity, the shape of the exponential decline does not change much. When M increases, the negative effect on the retailers is two-fold: the profit pie gets smaller (with intensified competition, retailers set lower prices and, therefore, the total profit shrinks) and has to be divided among an increasing number of retailers. Indeed, the expected profit of a retailer, at convergence, can be shown to be a decreasing convex function in the number of competing retailers in the market, which converges to zero as the number of retailers goes to infinity. 3.2.2.2 Behavioral Oligopoly with Symmetric Retailers Earlier, we have observed that in a symmetric behavioral duopoly (§3.2.1.2), the initial price at convergence, which dictates the pricing scheme at convergence, is increasing and convex in P . How would this initial price behave in an oligopoly? Would many retailers associated with a high value of P maintain a relatively high initial price? Here, we briefly explore the behavior of the initial price in the presence of many symmetric retailers in the market. The analysis of the model is similar to that carried out in §3.2.1.2. We focus our attention on results at convergence. The initial price, for several P values, is illustrated in Figure 3.6(a) as a function of M , wherefrom it is observed that the initial price is (discretely) decreasing and convex in M . It is evident that, for a fixed P , a sharp decline in prices occurs when competition is introduced into the market, while a further increase in M has a surprisingly minor impact on pricing. Additionally, it can be observed from Figure 3.6(a) that when M ≥ 2, the main driver of the initial price is the return probability P . As P is increasing, the impact of the number of retailers in the market on the initial price is weakening. Similar to structural competition, we may conclude that introducing competition into a monopoly market has a dramatic impact on prices, while further intensifying the competition by introducing additional symmetric retailers has a relatively marginal impact. The pricing schemes over the selling horizon are similar in fashion to those illustrated in Figure 3.4(a) for the duopoly case. The expected profit of each retailer at convergence, assuming all retailers are equally likely to be 65 0.5 0.6 0.7 0.8 0.9 1 1.1 1 2 3 4 5 6 7 8 9 10 M, Number of Retailers P=1 P=0.99 P=0.9 P=0.5 P=0  (a) Initial price vs. M M=50 M=10 M=5 M=3 M=2 0.05 0.1 0.15 0.2 0.25  0 0.2 0.4 0.6 0.8 1 P, Return Probability (b) Profit per retailer vs. P Figure 3.6: Behavioral symmetric oligopoly: Prices and profits at convergence visited by the consumer in the first period (i.e., with probability 1M ), is illustrated in Figure 3.6(b). With more retailers in the market, lower prices are set, and the total profit is smaller. Furthermore, an increase in the number of retailers in the market reduces each retailer’s chances to encounter the consumer in the first period. Thus, M dramatically affects retailers’ expected profits, and eventually, as M →∞, each retailer’s expected profit approaches zero for any P . 3.3 N Similar Consumers In this section we assume that the retailers, each of whom can satisfy the entire demand in the market, face N similar consumers, whose valuations are independently drawn from a uniform distribution over [0, 1]. We show that the pricing policy when there are N consumers in the market is very similar to that derived earlier for the single consumer case. Further, additional important insights are obtained and discussed in this section. 3.3.1 Structural Duopoly with N Consumers In each period, in structural duopoly, Retailer 1 (resp., 2) encounters, in expectation, a fraction P (resp., 1 − P ) of the remaining active consumers in the market (i.e., those who have’t purchased the good yet). Thus, in the first period Retailer 1 encounters PN consumers, and as he sets a price R11, PN(1 − R11) of these consumers purchase the good and the other PNR11 consumers, whose valuations are below R11, remain active and proceed to the second period. Similarly, (1−P )NR21 of 66 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 P, Return Probability T=5 T=2 N Consumers Single consumer T=3 T=4  Figure 3.7: Structural duopoly: Retailer 1’s initial price - single vs. N consumers, T = 2, .., 5 the consumers who visit Retailer 2 in the first period, and whose valuations are below R21, remain active and proceed to the second period. In the second period, Retailer 1 encounters P 2NR11 returning consumers and P (1− P )NR21 consumers who switch from Retailer 2. Consider a two-period selling horizon. Solving backwards, Retailer 1’s second period price is R12 = 1 2(PR 1 1 + (1 − P )R21), and in the first period he sets R11 = 2P 2−3P−3 2(2P 2−2P−3) . Recall that in the corresponding single-consumer case, Retailer 1’s first period price is 24−P . Thus, a weak (resp., dominant) retailer, in a structural competition, sets a marginally higher (resp., lower) first period price in the N consumers case than in the single consumer case, as demonstrated by the two lower graphs in Figure 3.7. Specifically, for P ≤ 0.5 (resp., P ≥ 0.5) the graph corresponding to the single consumer is slightly below (resp., above) the graph corresponding to the N consumers case. The difference in initial prices between the single-consumer case and the N -consumer is strik- ingly minor. The corresponding difference in the retailers’ expected profits, per consumer, is even smaller, and thus not graphically illustrated.11 In the symmetric case, when retailers are equally dominant, i.e., P = 12 , we find that, not surprisingly, they set identical prices, R i t = R j t , which coincide with the prices in the single consumer case. 11Later, it is shown in Figure 3.8(b), that even when we consider a selling horizon of 5 periods the difference in retailers’ expected profit (per consumer) is hardly observable. 67 Proposition 3.4 In a symmetric structural duopoly (i.e., P = 12) with N similar consumers, the retailers set the same prices as in the single consumer case. Before we proceed with the analysis of asymmetric retailers and longer horizons, we introduce the following definition. Definition 3.1 We say that retailers employ Pricing Policy 1 (PP(1)) if ∀t, R1t , R2t ≤ min(R1t−1, R2t−1), i.e., in every period they set prices which are lower than prices posted by either retailer in the pre- ceding period. We have the following result. Proposition 3.5 In a structural duopoly with N consumers and T ≤ 3, retailers employ only PP(1). The prices each retailer sets, the number of consumers each retailer encounters, as well as the number of consumers that leave each retailer at the end of each period are provided in Table 3.2 for a three period selling horizon, for which, by Proposition 3.5, retailers employ PP (1). As is illustrated in Table 3.2, in period t, t > 1, Retailer 1 encounters P 2NR1t−1 returning (resp., P (1 − P )NR2t−1 switching) consumers, who have visited Retailer 1 (resp., 2) in the preceding period, and at the end of period t, PNR1t consumers leave Retailer 1 without purchasing the good. Evidently, Retailer 1’s profit-to-go in period t is pi1t =  R11PN(1−R11) + pi12, if t = 1, R1tPN ( PR1t−1 ( 1− R1t R1t−1 ) + (1− P )R2t−1 ( 1− R1t R2t−1 )) + pi1t+1, otherwise. As the number of periods in the selling horizon increases, we find that at some point a sufficiently dominant retailer will start segmenting the market into those consumers who have visited him in the preceding period and, therefore, have observed the higher price posted in the market, and those consumers who have visited the competing weak retailer and, therefore, have observed the lower price posted in the market in the preceding period. Clearly, consumers in the former segment have valuations between zero and the high price (high-valuation segment),12 while consumers in the latter segment have valuations between zero and the lower price (low-valuation segment). Accordingly, 12Our analysis in this section, as explained later, has been performed with horizons of up to 5 periods. With much longer horizons, possibly some of the revisiting consumers at a very dominant retailer have previously observed lower prices at the competing retailer. The corresponding pricing policies are not discussed in this chapter. 68 Retailer 1 Retailer 2 Total Period 1 arriving NP N(1− P ) N price R11 R 2 1 leaving NPR11 N(1− P )R21 N(PR11 + (1− P )R21) Period 2 arriving NP (PR11 + (1− P )R21) N(1− P )(PR11 + (1− P )R21) N(PR11 + (1− P )R21) price R12 R 2 2 leaving NPR12 N(1− P )R22 N(PR12 + (1− P )R22) Period 3 arriving NP (PR12 + (1− P )R22) N(1− P )(PR12 + (1− P )R22) N(PR12 + (1− P )R22) price R13 R 2 3 leaving NPR13 N(1− P )R23 N(PR13 + (1− P )R23) Table 3.2: Prices and consumers visit pattern in the first three periods of a structural duopoly when Pricing Policy (1) is employed we introduce the following definition. Definition 3.2 We say that retailers employ Pricing Policy i (PP(i)), i ≥ 2, if max(R1t , R2t ) > min(R1t−1, R2t−1) for t = 2, .., i. That is, in each period of the first t periods, t = 2, .., i, the dominant retailer sets a price which is higher than the price set by the weak retailer in the previous period. Under PP (i), during the first i periods13 of the selling horizon the dominant retailer addresses consumers from the high-valuation segment only. This form of price skimming is beneficial for retailers facing competition. By properly segmenting the market, the (very) dominant retailer maintains relatively high prices (with respect to the weak retailer) and addresses only the high- valuation segments. Towards the end of the selling horizon his prices get closer to those posted by the weak retailer, and, eventually, in the final period, when he targets all remaining consumers who visit him, they set identical prices. Such price skimming is, seemingly, employed by Apple, unquestionably a dominant player in several consumer electronics markets. Apple regularly sets a very high initial price for its products, such as the iPod, targeting only high-valuation consumers, and over time the price is reduced. Eventually, when the price is sufficiently low, the remaining consumers are targeted. To illustrate the price skimming that PP (i) embodies, consider, for example, a selling horizon with five periods, for which the prices and corresponding profits are illustrated in Figure 3.8. When 0.10775 < P < 0.89225, retailers employ PP (1), and in each period they target all consumers who visit them. When P > 0.89225 (or P < 0.10775), PP (2) is employed, which means that in the 13Excluding the first period, since no previous prices have been observed by the consumers. 69 second period the dominant retailer targets only the high-valuation segment of consumers. Due to this focused targeting on the high-valuation segment, prices at the dominant retailer spike up a bit, and, interestingly, they spike up at the weak competitor as well. The spike in prices at the weak retailer occurs because he targets all consumers he encounters, and since the price at the dominant retailer is higher, consumers who switch from the dominant retailer have higher valuations and accordingly the weak retailer increases his price. When P > 0.90891 (or when P < 0.09109), PP (3) is employed. Accordingly, the dominant retailer targets the high-valuation segment of consumers not only in the second period, but in the third period as well. Prices set by the retailers along this five-period selling horizon are shown in Figure 3.8(a). The top solid and dashed lines in Figure 3.7 represent the initial price in the N -consumer and single-consumer cases, respectively. When 0.107 < P < 0.892, PP (1) is employed, as stated earlier, and we note that there is a small difference in the initial prices (except when P = 0.5, when they coincide). When PP (2) is employed instead of PP (1), prices at both retailers spike up. Prices spike upwards again when PP (3) is employed (i.e., P < 0.091 or P > 0.908), which is visually fairly negligible. Interestingly, when PP (2) or PP (3) are employed, initial prices by the dominant retailer (i.e., P > 0.5) are closer to their single-consumer levels, after the spike upwards, but do not exceed them. Segmentation of consumers into high and low-valuation segments does not occur when T ≤ 3 for a simple reason. For T = 3, the prices a monopolist sets in the three periods are 34 , 1 2 , and 1 4 . The initial price set by a virtual competing retailer, whose return probability is 0 (i.e., this retailer prices as if he encounters consumers once only, in the first period, and will never see them again), is 1 2 . With any decline in the monopolist dominance, his prices decline, and in particular, his second period price drops below 12 , while the competing weak retailer’s initial price increases to be above 1 2 . That is, the dominant retailer’s price in the second period is bound to address switching consumers from the weak retailer, since his second period price is lower than the weak retailer’s initial price. And recall that in the third and final period, the dominant retailer prices to address all consumers he encounters. Figure 3.8(b) demonstrates that the difference in expected profit per consumer between the single-consumer and the N -consumer cases is so marginal that it is hardly visible. Thus, we conclude, that not only the pricing policy for the single-consumer case may serve as a good estimate 70 for the pricing policy for the N consumes case, but it also provides a very good approximation for the expected profit, per consumer, when there are N consumers in the market. 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 P, Return Probbility R3 R2 R5 R4 R1  (a) Retailer 1’s prices 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 P, Return Probability Retailer 2 System Retailer 1 Single Consumer N Consumers  (b) Profits per consumer: Single consumer vs. N consumers Figure 3.8: Structural duopoly with N consumers: Prices and profits when T = 5 It is hard to extend the analysis to horizons beyond five periods, as expressions involve poly- nomials of high degrees. However, we can bypass this difficulty and illustrate pricing with longer horizons when either retailer is not overly dominant, in which case, PP (1) is employed. Figure 3.9 illustrates the decline of prices over a 10-period selling horizon in the N -consumer case (dashed lines). This decline clearly follows an exponential shape. The solid lines are the bounds from the single consumer model when the consumer happens to visit the same retailer along the entire selling horizon. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 3 4 5 6 7 8 9 10 t, Period Number Retailer 1, Single Consumer Retailer 2, Single Consumer Retailer 1, N Consumers Retailer 2, N Consumers  (a) P = 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 6 7 8 9 10 t, Period Number Retailer 1, Single Consumer Retailer 2, Single Consumer Retailer 1, N Consumers Retailer 2, N Consumers  (b) P = 0.4 Figure 3.9: Structural duopoly: Prices set by retailers for T = 10: single consumer vs. N consumers 71 The analysis can be extended to the oligopoly case. For example, employing the symmetry assumption from the single consumer case (i.e., Pi ≡ P = 1M , i = 1 . . .M), we find that retailers’ price ratios are identical (see Appendix A.3) and coincide with the solution in the single-consumer case. Thus, we are able to extend the result in Proposition 3.4 to an oligopoly with symmetric retailers as in both the single-consumer and N -consumer cases retailers set the same prices. 3.3.2 Behavioral Duopoly with N Consumers The analysis involved in generalizing the behavioral duopoly with a single consumer toN consumers is similar to that carried out for the structural duopoly. To avoid repetition, we merely highlight the main results for the N -consumer case in a behavioral duopoly. In general, we find that, similar to the structural duopoly, the pricing policy for N similar consumers does not perfectly coincide with the pricing policy previously obtained for the single-consumer case. Specifically, when the retailers are symmetric, i.e., both have identical retention rates, Pi = Pj ≡ P , and T > 2, they set prices which coincide with the prices set in the single consumer case only when P = 0, 12 , or 1. However, the difference in prices is small, and the difference in expected profits per consumer is even smaller. Figure 3.10(a) depicts the ratio of price ratios between the N -consumer and single-consumer cases for the final five periods. Clearly, there is no difference in prices when P = 0, 12 , or 1. It is further observed that when P < 12 (resp., P > 1 2), retailers set marginally lower (resp., higher) prices in the N -consumer case than in the single-consumer case. In this example the difference between the price ratios is less than 0.5%. As a result, as can be seen from Figure 3.10(b), when P < 12 (resp., P > 12) in the N -consumer case, retailers achieve a marginally lower (resp., higher) profit per consumer than in the single-consumer case. Again, the difference in expected profits is extremely small. Thus, we conclude that, for the symmetric retailers case, the pricing policy, as derived in the single consumer case, may be implemented for the N -consumer case. 3.4 Summary and Managerial Insight 3.4.1 Managerial Insight and Future Work In this chapter we have studied price skimming strategies in multi-period duopoly and oligopoly models, wherein the modelling of the store visits by consumers is similar in fashion to the common 72 0.996 0.998 1 1.002 1.004 0 0.2 0.4 0.6 0.8 1 P, Return Probability T-2 T, T-1 T-3 T-4  (a) Ratio of price ratios 0.998 0.999 1 1.001 1.002 0 0.2 0.4 0.6 0.8 1 P, Return Probability T-2 T, T-1 T-3 T-4  (b) Ratio of expected profits Figure 3.10: Behavioral symmetric duopoly: N -consumer vs. single-consumer cases Markov brand-choice models. In our competitive settings, myopic consumers need to visit retail stores to learn about prices. Upon observation of prices, they purchase the good if their valuations exceed the posted price. Otherwise, in the ensuing period they may return to the same store with some probability or switch to a competing one. We have considered return probabilities which are either location-independent (zero order) or location-dependent (first-order), leading to structural and behavioral types of competition, respectively. Our analysis in §3.2 has focused on the single consumer case, while in §3.3 we have enriched the setting to many similar consumers, assuming retailers use demand expectations when they decide upon prices.14 We have demonstrated that the pricing policies and resulting profit per consumer in the single consumer and N -consumer cases are very close, which allow us to treat the single-consumer case, for the most part, as a representative case. Our models, which extend GGM’s zigzag competition, provide robustness to their basic result that replacing a monopoly with a duopoly, transforms the declining price trajectory in the market from linear to exponential. Indeed, it occurs under fairly general visit patterns by consumers and when retailers are far from being symmetric. Our analysis of the duopoly model has allowed us to obtain new and important insights. Specifically, even a duopolistic competition between a very dominant retailer and a relatively weak competitor forces the dominant retailer to revert from linear to exponentially declining pricing. The mere fear of losing consumers to a relatively 14We have also considered the case when retailers face a small market and update their demand knowledge after every realization (see Appendix A.4). However, in that case, results that can be derived are very limited and one needs to refer to numerical solutions to gain insights. 73 weak competitor forces the dominant retailer to dramatically suppress prices. Our results also demonstrate that a retailer is better off by increasing his return probability. At the lower range of return probabilities, an increase in the return probability moderately affects profit, while in the upper range of return probabilities, such an increase can dramatically improve a retailer’s profit. Wal-Mart, for example, has recently announced that it would remodel many of its stores, aiming at improving consumers’ experience at the store (New York Times, 2006). Such activities should improve their consumers retention (increase the return probability) and, thereby, positively affect their revenues. Another important insight gained from our analysis is that firms who compete against each other in several distinct markets are better off by tacitly agreeing to share the markets so as to have one firm dominates some markets while the other firm dominates the others. We have also shown that the level of price competition in a duopolistic setting is only marginally intensified when additional retailers are introduced to the market. Thus, our message for the regulator is that he does not have to be too concerned with market concentration. Duopolistic competition between, e.g., relatively equal retailers may exert sufficient pressure on prices. Finally, we have also shown that with longer horizons and many consumers, retailers may employ a more refined price skimming. In particular, during the early periods of the selling horizon, the prices set by a very dominant retailer may not address switching consumers. This, in turn, can explain why returning consumers may end up paying high prices. Our model has not considered the associated cost of improving the return probabilities. An interesting extension to the model is to let the return probabilities be endogenously determined, as functions of retailers’ investment levels (such as marketing or renovation). Another extension to be pursued is to relax the assumption that all consumers are similar in their visit pattern. For example, one may want to study a setting wherein some consumers are loyal to a certain retailer while others shop around before purchasing the good. 3.4.2 Discussion of Assumptions A main assumption of our model is that the valuation for the good is uniformly distributed over [0, 1]. The uniform distribution is widely used in the literature, and specifically, it was assumed in related papers such as Lazear (1986), Besanko and Winston (1990), and Harris and Raviv (1981). Nevertheless, in §3.4.2.1 we demonstrate the robustness of our results and derive additional insights 74 by extending the analysis to the more general power distribution. §3.4.2.2 briefly investigates the case of limited inventory. We show how limited inventory affects pricing decisions, such that the more constrained retailers are, the higher are the prices they set, and the closer these prices get to the monopolistic level. Relaxations of other assumptions made in the model (myopic behavior of consumers, single store visit per period) are left for future research. 3.4.2.1 Structural Duopoly under Power Distribution with a Single Consumer In this subsection we assume that the single consumer’s valuation, V , follows a power distribution, with density f(V ) = (q + 1)V q, q > −1, 0 ≤ V ≤ 1.15 Thus, F (0) = 0 and F (1) = 1. When q = 0, the power distribution coincides with the uniform distribution, which can serve as the reference case. Hereby, we very briefly discuss the impact of the parameter of the power distribution, q, on prices set by retailers under monopoly and under duopoly. When V follows a power distribution, one can obtain recursive expressions for the price ratios and prove their convergence (see Appendix A.3, Proposition A.3). In particular, we find that when P = 0.5, retailers set identical prices in each period. Figure 3.11 illustrates the price decline over a 10-period selling horizon set by a monopoly, and by the two competing retailers in a structural competition, for different q values. Figures 3.11(a)-3.11(d) illustrate the price decline when the two competing retailers are symmetric, i.e., P = 0.5, and Figures 3.11(e)-3.11(h) illustrate the bounds on prices when one retailer is significantly more powerful than the other, where the weak (resp., dominant) retailer’s return probability is P = 0.2 (resp., 1− P = 0.8). Clearly, as q is increasing, the center of the distribution shifts to the right. Accordingly, one can observe from Figure 3.11 that: (i) as the consumer’s valuation is more likely to be closer to 1, the retailers set higher prices. Consequently, the price decline slows down. Correspondingly, not shown but apparent, retailers’ expected profits increase as well; (ii) despite the difference in retailers’ dominance (Figures 3.11(a)-3.11(d) vs. Figures 3.11(e)-3.11(h)), the difference in speed of markdowns is fairly minor; (iii) similar to the case of uniform distribution, the exponential decline of prices in a structural duopoly is maintained as long as q is not too large;16 (iv) due to the substantial decline of prices under competition, the good is highly likely to be sold within 15The power distribution is a special case of the beta distribution, which is defined on [0, 1], and the results presented below can be extended to several additional specific values of the beta distribution. 16However, for any q, one can find a sufficiently long horizon to demonstrate an exponential decline of prices. 75 the early periods of the selling horizon. Longer selling horizons provide very little contribution to retailers’ expected profits, and, consequently, they are likely to terminate the selling horizon after a few major markdowns. Thus, our model may explain short life cycles of goods in competitive markets. 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period numer Monopoly Duopoly  (a) P = 0.5, q = − 1 2   0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period numer Monopoly Duopoly (b) P = 0.5, q = 0 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period numer Monopoly Duopoly  (c) P = 0.5, q = 5 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period numer Monopoly Duopoly  (d) P = 0.5, q = 50 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period numer Monopoly Duopoly: Duopoly: Weak Dominant  (e) P = 0.2, q = − 1 2 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period numer Monopoly Duopoly: Duopoly: Weak Dominant  (f) P = 0.2, q = 0 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period numer Monopoly Duopoly: Duopoly: Weak Dominant  (g) P = 0.2, q = 5 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 t, Period numer MonopolyDuopoly: Duopoly: Weak Dominant  (h) P = 0.2, q = 50 Figure 3.11: Prices in a 10-period selling horizon when valuation follows a power distribution 3.4.2.2 Two-Period Structural Duopoly with Limited Inventory When retailers have limited inventories (i.e., limited capacities), and demand exceeds supply, they may wish to increase prices so as to increase their expected profits. In this case, they will also price the goods so that, in expectation, depending on their inventories, they will exhaust their stock in the final period. Let us consider first the behavioral symmetric duopoly. Assume that there are N similar consumers in the market17 and each retailer has K units of the good in stock. The corresponding equilibrium pricing of the retailers in a two-period selling horizon is (R1, R2) =  (1− 2K(2−P )N(3−P ) , 1− K2N ), if K < N(3−P )2(4−P ) , ( 24−P , 1 4−P ), otherwise, 17In expectation, each of the retailers encounters 1 2 N consumers in the first period. 76 which implies that (i) for a fixed N , as expected, prices decrease as stocking levels at the two retailers increase, and that (ii) under the equilibrium pricing policy, in expectation, retailers never deplete their entire stocks in the first period. Moreover, (iii) the equilibrium prices for the two period, symmetric uncapacitated setting ((R1, R2) = ( 24−P , 1 4−P )) are employed even when each of the retailers is far from being able to satisfy the entire market. Due to the difference in return probabilities, retailers in a structural duopoly may have different stocking levels, and the resulting equilibrium prices depend on the inventory levels at the two retailers. Similar to the behavioral symmetric duopoly, we find that the same insights hold for the structural duopoly with a two-period selling horizon. Specifically, the equilibrium uncapacitated pricing, derived in §3.3, is employed whenever Retailer 1, whose return probability is P , and Retailer 2’s inventories are larger than (2P 2−P−5)NP 2(2P 2−2P−3) and (2P 2−3P−4)N(1−P ) 2(2P 2−2P−3) , respectively. 18 This suggests that our results are fairly robust and are in particular suitable when the inventory levels at the competing retailers are not too constrained. 18These two expressions are almost linear in P , and can be approximated by 2 3 P and 2 3 (1− P ), respectively. In a two-period structural duopoly, when this condition is not satisfied (i.e., at least one of the retailers has lower inventory levels), other pricing schemes may arise. 77 3.5 Bibliography Aviv, Y. and Pazgal, A. (2007). Optimal pricing of seasonal products in the presence of forward- looking consumers. Manufacturing & Service Operations Management, forthcoming. Bernheim, B. D. and Whinston, M. D. (1990). Multimarket contact and collusive behavior. 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Lal, R. and Matutes, C. (1994). Retail pricing and advertising strategies. The Journal of Business, 67(3):345-370. Lazear, E. P. (1986). Retail pricing and clearance sales. American Economic Review, 76(1):14-32. Leeflang, P. S. H., Wittink, D. R., Wedel, M., and Naert, P. A. (2000). Building Models for Marketing Decisions. Kluwer Academic Publishers, The Netherlands. Lilien, G. L., Kotler, P., and Moorthy, K. S. (1992). Marketing Models. Prentice Hall, New Jersey. Moorthy, S. and Winter, R. A. (2002). Price matching guarantees. RAND Journal of Economics, forthcoming. New York Times (2006). Wal-mart tries to find its customer. February 22. Noble, P. M. and Gruca, T. S. (1999). Industrial pricing: Theory and managerial practice. Mar- keting Science, 18(3):435-454. Perakis, G. and Sood, A. (2006). Competitive multi-period pricing for perishable products: A robust optimization approach. Mathematical Programming, 107(1-2):295-335. Simester, D. (1995). Signalling price image using advertised prices. Marketing Science, 14(2):166- 188. 79 Tellis, G. J. and Wernerfelt, B. (1987). Competitive price and quality under asymmetric informa- tion. Marketing Science, 6(3):24-253. Xu, X. and Hopp, W. J. (2006). A monopolistic and oligopolistic stochastic flow revenue manage- ment model. Operations Research, 54(6):1098-1109. Zhao, W. and Zheng, Y. S. (2000). Optimal dynamic pricing for perishable assets with nonhomo- geneous demand. Management Science, 46(3):375-388. 80 Chapter 4 Dynamic Pricing of Perishable Goods for a Monopoly and a Duopoly in the Presence of Strategic Consumers∗ 4.1 Introduction and Literature Review Revenue management, in general, and dynamic pricing methods, in particular, have gained popu- larity in recent years and are supported by an ever growing body of literature. Much of the advances and progress in this area are summarized in Bitran and Caldentey (2003), Elmaghraby and Ke- skinocak (2003), Talluri and van Ryzin (2004), and Chan et al. (2004). Central challenges still facing this research area, and which were identified by the above reviews, include, among others, modelling competitive effects and incorporating a richer underlying demand context that takes into account, e.g., consumers’ behavior. This chapter aims at bridging an existing gap in the literature by addressing monopoly and duopoly settings in the presence of both myopic and forward-looking consumers – two consumer groups who behave in a completely different manner. Forward-looking consumers optimally time their purchase to maximize their surplus, while myopic consumers act impulsively and purchase the good as soon as their surplus becomes positive. It has long been recognized that consumers differ in their characteristic dimensions. This heterogeneity contributes to the difficulty retailers face in pricing their goods over the selling horizon and in their efforts to properly segment the market. Clearly, consumers react differently to prices and vary in their price recall, price awareness, and expectations for future prices. A major area of research, in this respect, incorporates reference prices (Helson (1964)) and their effect on purchasing ∗A version of this chapter will be submitted for publication. Granot, D. and Mantin, B. Dynamic Pricing of Perishable Goods for a Monopoly and a Duopoly in the Presence of Strategic Consumers. 81 decisions (Kahneman and Tversky (1979) and Kalyanaram and Winer (1995)). Reference prices reflect the internal process undertaken by consumers synthesizing the various beliefs they possess about prices. The research regarding reference prices is relevant for our model, as it provides the empirical evidence that consumers react differently to prices.1 Not only do consumers possess perceptions of past and current prices, as proposed by reference price research, but they do also possess some perceptions about discount frequencies and future prices. There is a paramount evidence that while some consumers remain totally oblivious to retailers’ pricing strategies, others are aware of them and behave accordingly.2 The literature distinguishes between these two types of consumers by referring to the former as myopic and to the latter as strategic, rational, or forward-looking. Shugan (2006) has noticed the increase usage of consumers’ rationality in the literature and the variety of terminologies that has been used to describe this phenomena. The different models that address strategic consumer behavior assume that strategic consumers, after observing either the pricing policy or the current price, decide whether to purchase the good right away, to (keep) wait(ing), or to exit altogether (when this option is allowed). That is, re- tailers are the Stackelberg leaders while the consumers are the followers. In other words, behaving strategically, consumers time their purchase to maximize their surplus. Accordingly, in our model, we refer to the forward-looking consumers as strategic consumers. Modelling strategic consumer behavior can be traced back to Coase (1972) who has conjectured that “with complete durability, the price becomes independent of the number of suppliers and is thus always equal to the competitive price.” In other words, a monopolist facing (very patient) strategic consumers is unable to avoid future price reductions and, subsequently, has to price at the marginal cost and make zero profit. Stokey (1979) has formalized this conjecture and has argued that with pre-announced prices, price discrimination may not be beneficial to the monopolist when consumers are strategic. Intuitively, when consumers are acting strategically, a markdown process may induce too many consumers to delay their purchase, which results in an unprofitable price 1Extensive reviews include Lowengart (2002) and Briesch et al. (1997). For recent work in this area, see, e.g., Popescu and Wu (2007) and references therein. 2In their study on consumers’ perceptions of deal frequency and prices, Krishna et al. (1991) have found that, in general, many consumers reasonably recall their magnitudes. Gönül and Srinivasan (1996) have empirically shown that consumers do form beliefs about future coupons and adjust their purchase decisions accordingly, and recently Sun et al. (2003) have found that consumers appear to have accurate perceptions of promotion frequency. 82 discrimination. In her subsequent paper, Stokey (1981) has found that the arising equilibrium in a sequential pricing setting is such that consumers expect the retailer to saturate the market, which he does since there is zero marginal cost, and the resulting price is continuously zero. Considering a simplified model with different discounting rates for the retailer and consumers, Landsberger and Meilijson (1985) have found that price discrimination may be beneficial, after all, if consumers’ discounting rate is higher than the retailer’s. Other papers in this line of research include, e.g., Gul et al. (1986), Besanko and Winston (1990) (henceforth B&W), and Güth and Ritzberger (1998). In the approach adopted by all the above models, and ours as well, all consumers are present in the market from the very beginning, and the market size, that is, the number of consumers in the market, is a common knowledge. Other approaches concerned with strategic consumers in a monopolistic setting, but which in- corporate consumers’ arrival and demand modelling in a multi-period or continuous time settings include Aviv and Pazgal (2007) (Poisson arrival), Levin et al. (2005, 2006) (valuations are random along the horizon), Su (2007) (continuous deterministic arrival of consumers), and Elmaghraby et al. (2004) (multi-unit demands). Two-period models that incorporate strategic consumer behavior include Cachon and Swinney (2007), who have studied pricing and stocking decisions by a monopo- list facing myopic, bargain-hunters, and strategic consumers, Gallego et al. (2006) and Ovchinnikov and Milner (2005), who have considered repeated versions of a two-period setting, and Liu and van Ryzin (2005), who have focused on capacity decisions and rationing risk. Competition can affect, quite dramatically, pricing decisions. Reviewing the literature, El- maghraby and Keskinocak (2003) and Chan et al. (2004) have stated that competitive effects on prices cannot be waved away, and Bitran and Caldentey (2003) have further concluded that “includ- ing market competition is another important extension to the model. [...] price competition among retailers is today the main driver in their selection of a particular pricing policy”. Recognizing the effect of competition on dynamic pricing in the context of multi-period (fixed-inventory) revenue management, different approaches were proposed recently to address this challenging problem. To mention some, Perakis and Sood (2006) have employed a robust optimization approach; in Gallego and Hu (2006), who have extended Gallego and van Ryzin’s (1994) seminal model, demand intensity (non-homogeneous Poisson process) is a function of prices set by all retailers; Lin and Sidbari (2004) have considered a consumer choice model to study competition between substitutable goods; in Xu 83 and Hopp (2006), arrival of homogeneous consumers follows a Brownian motion and consumers purchase only from retailers offering the lowest price; and Granot et al. (2007) (Chapter 2, hence- forth GGM) have extended Lazear’s (1986) model by assuming that consumers switch between competing retailers as long as they have not purchased the good. However, with the exception of Levin et al. (2006) and Liu and van Ryzin (2005), the literature concerned with dynamic pricing in a competitive setting has remained mostly oblivious to the effect of strategic consumer behavior. In Levin et al., firms selling differentiated goods over a finitely long horizon encounter strategic consumers whose valuations are random over time. In each period only a single purchase may take place, if at all, after consumers either assign probabilities to one good or a set of goods that they want to purchase. In the former case they prove the existence of a Markov perfect equilibrium in mixed strategies, and in the latter case in pure strategies as well. When each firm can satisfy the entire market, price is decreasing over the selling horizon and, in a duopoly, if one of the firms misclassifies strategic consumers as myopic, a loss of up to 6% may occur to firms. In Liu and van Ryzin, a monopoly preannounces prices and chooses its stocking level, such that all first period demand is satisfied and (strategic) risk-averse consumers face a rationing risk in the second period. In their oligopoly setting, retailers share equally the first period demand, and consumers consider the aggregate market rationing risk. They find that if there are sufficiently many high-valuation consumers, the symmetric equilibrium involves segmentation and rationing; otherwise it consists of a second period low-price-only with no rationing risk. Our main objectives in this chapter are to characterize the pricing policies and resulting profits in the presence of consumers who behave strategically, to measure the effect of competition and the corresponding profit loss incurred to retailers due to this strategic behavior, and to assess the profit gain (loss) due to retailers’ ignorance of consumers’ strategic behavior. To achieve these objectives, we study a multi-period, discrete-time model with a market consisting of myopic and strategic consumers in monopoly and duopoly environments. In our model, retailers, each of whom can satisfy the entire demand in the market, face uncertainty about consumers’ valuations. Consumers and retailers may possibly possess different discounting factors so as to account for different considerations and perceptions of the different players. Similar to GGM, we assume that the consumers, who can visit only one of two competing retailers in each period, visit retailers in a zigzag fashion. 84 Initially, we study a two-period model. In this setting we demonstrate that the price skimming policy retailers employ depends on the market composition and consumers’ strategicity. Specifi- cally, both in a monopoly and a duopoly, retailers may price such that both myopic and strategic consumers purchase the good in both the first and second periods. However, when there are suf- ficiently many myopic consumers in the market or, alternatively, when the strategic consumers’ discounting factor is sufficiently high, retailers price such that all strategic consumers defer the possibility of buying the good until the second period. The analysis of the pricing policies suggests that a monopoly should investigate ways to enhance the initial “virtual” value of the good, i.e., without adding any new features to it, while decreasing its future status. Such a strategy would moderate consumers’ strategic behavior, which would result with a higher expected profit. Firms are, apparently, becoming much more efficient these days in creating this kind of sense of urgency to purchase their goods as early as they are introduced. The razzmatazz surrounding the introduction of different Apple’s products or the hysteria associated with any new Harry Potter book are such examples. Those who wait usually experience much lower prices (e.g., Apple dropped the price of the iPhone by $200 about two months after its introduction). The recommendation about changing the inter-temporal value of the good should be handled with care in a duopoly setting. For many instances, moderating consumers’ strategic behavior results with a higher duopoly profit. However, in some situations such a change in consumers’ strategic behavior may decrease the profit of the duopoly. Stated differently, it is possible that with a small increase in the strategic consumers’ discounting factor, competing retailers switch from one price skimming policy (where both types of consumers are encouraged to purchase the good in the first period) to another (which skims only high-valuation myopic consumers in the first period), in which case their corresponding expected combined profit is higher. Thus, in our competitive setting, retailers may be better off when the strategic consumers’ discounting factor is higher. Competition, regardless of the composition of consumers in the market and the different dis- counting factors, diminishes retailers’ profits. Indeed, competition suppresses prices and reduces retailers’ price skimming capabilities, eventually resulting with lower profits. We find that this loss of profit may be as low as 6.25% and as high as 34.3% in our setting. That is, the profit loss due to competition in the presence of strategic consumers may be very substantial. 85 Obliviousness to consumers’ strategic behavior is costly to retailers (see, e.g., B&W, Aviv and Pazgal (2007), and Levin et al. (2005)). Indeed, we show in our model that obliviousness results with a loss to a monopoly. Surprisingly, however, we prove that such a behavior may provide a duopoly with a profit gain. In other words, under competition, retailers may be better off by ignoring or becoming oblivious to consumers’ strategic behavior. Moreover, we find that when retailers are oblivious to consumers’ strategic behavior, the combined profit under competition may exceed that of a monopoly. That is, if the monopoly and duopoly retailers treat all consumers as myopic, for whatever reason, the resulting duopoly profit may be larger than the monopoly profit. The extension of our model to incorporate longer horizons focuses on strategic consumers, and as such, it is mostly related to the monopoly model studied by Güth and Ritzberger (1998). Güth and Ritzberger were interested in the conditions for Coase-Conjecture to hold, and have considered a monopoly facing strategic consumers with uniformly distributed valuations and a discounting factor which may be different than the retailer’s. They have studied both the symmetric (as in Gul et al. (1986)) and non-symmetric discounting factors cases, and have concluded that limiting the analysis to the case of symmetric discounting factors (as, e.g., in B&W) may limit the analysis and can obscure possible important results. Our T -period monopoly setting coincides with Güth and Ritzberger’s. However, as we employ a different solution approach, we are able to shed more light on the characterization of the pricing scheme in this case. We also study a competitive version of this setting, wherein, as stated earlier, consumers zigzag between the two retail stores as long as they remain active. In these settings we express the pricing policy in terms of price ratios. In contrast to Güth and Ritzberger’s result for a monopoly, we demonstrate that in a duopoly, results are not distorted by assuming a common discounting factor for consumers and retailers. Asymptotically, we find that the profit loss due to competition in the presence of strategic consumers increases in this common discounting factor. It is fairly low for most discounting factor levels, but it approaches 100% fairly rapidly as this discounting factor goes to 1. It is further proved that the strategic behavior of consumers in this T -period selling horizon inflicts a larger percentage profit loss to a duopoly than to a monopoly. The rest of the chapter is organized as follows. In §4.2.1 we introduce our two-period base model, which is then analyzed for both the monopoly (§4.2.2) and duopoly (§4.2.3) settings. The 86 remaining of §4.2 studies the effects of competition, strategic consumer behavior and ignorance thereof, and the case of perfect markets (wherein players share a common discounting factor). §4.3 extends the model to longer horizons and §4.4 concludes with a summary of the main results and directions for future research. 4.2 A Two-Period Selling Horizon 4.2.1 The Model In their review paper, Bitran and Caldentey (2003) conclude “that incorporating the strategic be- havior of consumers - extending the models by Besanko and Winston (1990) [...] - is an important topic”. In B&W, an uncapacitated monopoly faces strategic consumers, who share the same dis- counting factor as the retailer, over a multi-period selling horizon. The model developed in this section extends a two-period version of B&W’s model in several important ways: (i) it studies a market consisting of both myopic and strategic consumers, rather than strategic consumers only; (ii) it studies both monopolistic and duopolistic settings; and (iii) it allows for possibly different discounting factors for the strategic consumers and the retailers. In both monopoly and duopoly settings, we assume that the profit-maximizing retailers, each of whom can satisfy the entire market’s demand, are selling homogeneous goods with zero salvage value and, without loss of generality, zero marginal production cost.3 Similar to B&W, we seek a price discrimination mechanism in the absence of product availability considerations. Thus, retailers, who have a discounting factor of λ, set prices over the length of the selling horizon so as to maximize their expected profits. Each of the consumers in the market buys at most one unit of the good, for which he has some private valuation v, satisfying 0 ≤ v ≤ 1. The distribution of these valuations, which is uniform over [0, 1], is common knowledge.4 We say that consumers are active as long as they haven’t purchased the good. All active consumers visit retail stores in each period, as long as they haven’t purchased the good.5 In the base model, there are two periods in the selling horizon, and only 3In the presence of a positive marginal production cost, c, it can be proven that, in both monopoly and duopoly settings, the switching points from one pricing policy to another, which are discussed later, are independent of c, prices are linearly scaled between their values when c = 0 and 1, and the retailers’ profits are factored by (1− c)2. 4This case where the lowest consumer valuation coincides with retailers’ marginal cost, both of which are zero, is also known as the “no-gap” case (see Güth and Ritzberger (1998)). 5In their monopolistic settings, ?, ?, and ? also assume that strategic consumers visit the retailer even though 87 active consumers proceed to the second period. We neglect search or travel costs, but assume that due to these implicit costs, consumers are limited to a single store visit per period. In a monopoly, active consumers visit the same retailer in both periods. In a duopoly, each consumer visits one of the retailers in the first period and switches to the competing retailer in the second period, if he remains active. The store switching assumption is justified, among other reasons, by a variety- seeking consumer behavior (see, e.g., McAlister (1982) and Givon (1984)) or pure dissatisfaction with the most recent store visited. This store visit pattern may also be sustained due to other reasons not considered in our model. The market is composed of two different segments of consumers. A fraction α of these consumers are strategic and the remaining 1− α are myopic. In our model, the myopic consumers, who lack price information, are impatient and purchase the good once they observe a price which is below their valuation. If a myopic consumer doesn’t purchase the good in the first period at price R1, it can be inferred that v < R1, and by Bayes’ Theorem, the posterior distribution carried from period 1 to 2 is F2(v) = F (v) F (R1) = vR1 , 0 ≤ v < R1. We assume that all strategic consumers have the same discounting factor, δ, which may be different than the retailers’ discounting factor, λ, to permit different considerations by the players in the model. The strategic consumers’ discounting factor can represent their strategicity level, as in Levin et al. (2005, 2006), or their patience level, as in the economic literature. The strategic consumers, who know α, δ, and λ, can infer the prices along the path of their visit pattern. In a monopoly setting, they consider both prices set by the monopolist. In a duopoly setting, since they visit one retailer in the first period and switch to the competing retailer in the ensuing period, if they remain active, they consider exclusively only the particular prices they may encounter. Strategic consumers choose to purchase the good in the first period, only if their immediate surplus, v−R1, is positive and is greater than or equal to their discounted surplus obtained in the second period, δ(v −R2), realized at the competing retailer. Thus, if a strategic consumer doesn’t purchase the good in the first period at price R1, then v < V1 ≡ min { R1−δR2 1−δ , 1 } , and the posterior distribution carried from period 1 to 2 is F2(v) = vV1 , 0 ≤ v ≤ V1. That is, strategic consumers with valuations which exceed V1 purchase the good in the first period. Su (2007) has employed a similar market segmentation approach by distinguishing between they may decide to wait with their purchase until a later point in time. Consumers may, e.g., visit the stores to monitor prices posted by retailers. 88 patient (strategic) and impatient (myopic) consumers, who have either a low or high valuation. Cachon and Swinney (2007) have segmented the consumers into myopic, strategic, and bargain hunters. In their 2-period model, myopic consumers (resp., bargain hunters) consider purchasing the good only in the first (resp., second) period, while strategic consumers optimally time their purchase between the two periods. Next, we describe the pricing policies arising in a monopoly and a duopoly. As we employ a game-theoretic approach, we search for subgame perfect Nash equilibria (SPNE) in these two settings. 4.2.2 Monopoly – Two-Period Selling Horizon Let Rmt denote the monopoly price in period t, t = 1, 2, where the superscript m stands for monopoly, pimt denote his (discounted) profit-to-go in period t, and V m 1 denote the strategic con- sumers’ critical valuation. Myopic consumers purchase the good in the first period if v ≥ Rm1 , and in the second period if Rm1 > v ≥ Rm2 . Strategic consumers with v ≥ V m1 = min { Rm1 −δRm2 1−δ , 1 } purchase the good in the first period, and those with V m1 > v ≥ Rm2 purchase the good in the second period. When the monopolist is facingN consumers, his (discounted) expected profit-to-go in the second period is given by pim2 = λR m 2 ( (1− α)NRm1 ( 1− R m 2 Rm1 ) + αNV m1 ( 1− R m 2 V m1 )) , (4.1) and in the first period by pim1 = R m 1 ((1− α)N (1−Rm1 ) + αN(1− V m1 )) + pim2 , (4.2) which is also his total expected profit. Since pim1 ∣∣ N consumers = Npim1 ∣∣ N=1 , the number of consumers in the market, N , merely factors the retailer’s total profit, and the equilibrium pricing is independent of N . Thus, without loss of generality, we suppress N and we will perceive pim1 as the profit per consumer.6 Solving the model, the monopolist finds that the following two pricing policies are possible. Pricing Policy i (PP(i)): price to skim high-valuation consumers of both types in the first pe- 6Equivalently, as we set N = 1, the problem can be perceived as a monopolist facing a single representative consumer who is strategic with probability α and myopic with probability 1− α. 89 riod. Pricing Policy ii (PP(ii)): price to skim only high-valuation myopic consumers in the first pe- riod. That is, none of the strategic consumers purchases in the first period, and they all wait for the second period. Let pim1,i and pi m 1,ii denote the monopoly profit under PP (i) and PP (ii), respectively. Naturally, the monopolist prefers PP (ii) over PP (i) whenever pim1,ii > pi m 1,i. This occurs when δ > δ m c , where δmc is the value of δ which solves pi m 1,i = pi m 1,ii. 7 That is, at δmc the retailer is switching from PP (i) to PP (ii). We conclude that the equilibrium prices in a monopoly can be expressed as follows. (Rm1 , R m 2 ) =  ( 2+λα 4−λ+λα , 1+α 4−λ+λα ) , if δ > δmc ,( (2−2δ+αδ)2 2(1−δ+αδ)(4−λαδ+2αδ−4δ+λδ−λ) , 2−2δ+αδ 2(4−λαδ+2αδ−4δ+λδ−λ) ) , otherwise, (4.3) and the corresponding strategic consumer behavior is expressed by V m1 =  1, if δ > δmc , (2−δ+αδ)(2−2δ+αδ) 2(1−δ+αδ)(4−λαδ−2αδ+4δ−λδ+λ) , otherwise. That is, when δ > δmc , all strategic consumers wait for the second period (since V m 1 = 1). The equilibrium prices from display (4.3) are illustrated in Figure 4.1, wherein we have selected instances in which a switch from PP (i) to PP (ii) occurs. As is evident from Figure 4.1(a), for fixed values of α and λ, as δ increases, the monopolist decreases the first period price and increases the second period price as long as PP (i) is employed. Specifically, when δ = 0, i.e., all consumers are myopic, the first and second period prices are 24−λ and 1 4−λ , respectively, and as δ increases, the monopolist gradually reduces some of the difference between the two period prices so as to provide strategic consumers with an incentive to buy in the first period. However, once strategic consumers’ discounting factor exceeds δmc , they all wait for the second period. Thus, at δ = δ m c , the monopolist switches from PP (i) to PP (ii), resulting with the spike up in the first period price. Thereafter, for δ > δmc , the monopolist is only addressing the myopic consumers in the first period. Therefore, for δ > δmc , the first period price remains constant. In the second period, the monopolist adjusts the price upwards so as to realize some profit from the strategic consumers as well as from 7The explicit expression of δmc is provided in the proof of Proposition 4.1 in the Appendix. 90 the myopic consumers in this period. Similar to the first period price, the second period price is constant for δ > δmc , since all strategic consumers already wait for the second period. Finally, it can be shown that, for any λ, if there are sufficiently many strategic consumers in the market, then δmc ≥ 1 and PP (ii) is never used by the monopolist. An analogous analysis can be carried out for a fixed δ and λ and varying α. That is, for fixed δ and λ, there exists a critical α value, αmc , at which point there is a switch between PP (i) and PP (ii). However, by contrast to the former case, wherein PP (ii) is employed when δ is larger than the critical value, δmc , for fixed δ and λ, the monopolist employs PP (ii) when α is smaller than the critical value αmc . Thus, the monopolist discourages strategic consumers from buying in the first period if they are overly strategic (i.e., δ is sufficiently high) or, alternatively, if their proportion in the market is sufficiently small. In the latter case, their appeal in the first period is low and the monopolist delays their purchasing epoch to the second period. Similar to the previous analysis, for any λ, there exists a discounting factor δ below which αmc ≤ 0 and PP (ii) is never used by the monopolist. If, indeed, λ and δ are such that αmc ≤ 0, PP (i) is always employed by the retailer, and with an increase in α, the monopolist gradually reduces the difference between the two prices by decreasing (resp., increasing) the first (resp., second) period price so as to provide strategic consumers with an incentive to buy early. However, if λ and δ are such that αmc > 0, then for α < αmc the retailer employs PP (ii). In that case, for given prices, with any increase in α, there are more strategic consumers who, under PP (ii), delay their purchase timing to the second period. To address this growing demand in the second period stemming from strategic consumers, the monopolist increases the second period price and accordingly the first period price as well. Once there are sufficiently many strategic consumers in the market, i.e., α ≥ αmc , prices spike down as the monopolist switches to PP (i), and thereafter prices behave as described earlier under PP (i). It is also possible to derive a critical λ value, λmc , above which PP (ii) is strictly preferred over PP (i), for fixed δ and α. If λ is sufficiently low, the monopolist cares less about the future and prices so as to address both segments of consumers in the first period. Indeed, for any λ < 1 the monopolist prefers to collect his revenue in the first period, rather than in the second period. However, once λ exceeds λmc , the future is sufficiently valuable for the monopolist such that he prefers to use PP (ii) and, thereby, revenue from the strategic consumers is collected only in the second period. 91 Formally, the choice of pricing is stated below.8 Proposition 4.1 When δ ≤ min{δmc , 1}, or α ≥ max{αmc , 0}, or λ ≤ min{λmc , 1}, the monopolist employs PP (i), which encourages both myopic and strategic consumers to purchase in the first period. Otherwise, he employs PP (ii) which ensures that all strategic consumers defer from buying in the first period. Furthermore, if α ≥ 4−3λ 8−11λ+4λ2 (resp., δ ≤ 2(3−α− √ 1−α) α2−5α+8 , λ ≤ 2−3δ1−δ ), then PP (i) is employed for any value of δ (resp., λ, α). The equilibrium pricing policy is illustrated in Figure 4.1. Observe that in Figure 4.1(a), wherein α = 0.5 and λ = 1, the corresponding prices when, e.g., δ = 0.55, are to the left of the spike as δ ≤ δmc ∣∣ α=0.5,λ=1 = 0.623. However, for the same parameters’ values in Figure 4.1(b), the prices are to the right of the spike, as α = 0.5 ≥ αmc ∣∣ δ=0.55,λ=1 = 0.237. On the other hand, for δ = 0.8 in Figure 4.1(a), prices are to the right of the spike, as δ > δmc , while the prices in the corresponding case, in Figure 4.1(c), are to the left of the spike, as α < αmc ∣∣ δ=0.8,λ=1 = 0.866. Indeed, as stated in Proposition 4.1, PP (i) is employed for α larger than αmc or for δ smaller than δ m c . The pricing behavior is further summarized as follows. Proposition 4.2 The equilibrium prices in a monopoly are such that (i) the first (resp., second) period price is decreasing (resp., increasing) in δ up to δ = δmc , at which point both prices spike up and remain constant for δmc < δ ≤ 1; (ii) both prices are increasing in α up to α = αmc , at which point they spike down and thereafter, for αmc ≤ α ≤ 1, the first (resp., second) period price is decreasing (resp., increasing) in α; (iii) both prices are increasing in λ with a spike upwards at λ = λmc . R1 R2 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 delta (a) α = 0.5, λ = 1 R2 R1 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 alpha (b) δ = 0.55, λ = 1 R1 R2 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 alpha (c) δ = 0.8, λ = 1 R2 R1 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 lambda (d) δ = 0.8, α = 0.5 Figure 4.1: Prices set by a monopolist in a 2-period model 8All proofs of statements in this chapter are provided in Appendix A.5. 92 The equilibrium pricing scheme results with the following expected profit expression pim1 =  1−α+λα 4−λ+λα , if δ > δ m c , (2−2δ+αδ)2 4(1−δ+αδ)(4−λαδ−2αδ+4δ−λδ+λ) , otherwise. Not surprisingly, increased strategicity or an increase in the proportion of strategic consumers adversely affects the monopolist profit. Formally, we have: Proposition 4.3 The monopolist’s expected profit is (i) decreasing in δ up to δ = δmc , and there- after it remains constant; (ii) piecewise decreasing in α; (iii) piecewise increasing in λ. Proposition 4.3 complements Proposition 4.2 as it contemplates the interplay between the dif- ferent parameters in the model and their effect on profit. Moreover, we find that when (i) δ ≤ δmc and α > 4−4δ+3λδ−3λδ(3λ−2) or (ii) δ > δ m c and α > 4+λ2−2λ+(λ−2)√λ2+12λ+4 2λ(λ−3) , a larger share of the total expected profit is earned in the second period (see Proposition A.6 in Appendix A.5). That is, under both PP (i) and PP (ii), when there are sufficiently many strategic consumers in the market, a larger share of the expected profit is shifted, in equilibrium, to the second period. 4.2.3 Duopoly – Two-Period Selling Horizon While active consumers in a monopoly visit the same retailer in both periods, in a duopoly con- sumers arbitrarily visit one of the retailers in the first period, and in the second period active consumers switch to the competing retailer. Similar to the monopolistic case, we first show that in a duopoly, the equilibrium pricing policy and the corresponding duopoly profit are independent of the number of consumers in the market or their distribution between the two retailers. Assume Retailer i, i = 1, 2, who sets a price Rd,it in period t, t = 1, 2, where the superscript d stands for duopoly, encounters N i consumers in the first period, with N1 + N2 = N . Let V d,i1 denote the critical valuation of the strategic consumers who visit Retailer i in the first period, and pid,it denote Retailer i’s (discounted) profit-to-go in period t. Then, pid,i2 = λR d,i 2 ( (1− α)Rd,j1 N j ( 1− R d,i 2 Rd,j1 ) + αV d,j1 N j ( 1− R d,i 2 V d,j1 )) , where (1 − α)Rd,j1 N j (resp., αV d,j1 N j) represents the number of active myopic (resp., strategic) consumers who have visited Retailer j, j 6= i = 1, 2, in the first period and switch to Retailer i in 93 the second period. Retailer i’s total expected profit is pid,i1 = R d,i 1 ( (1− α)N i ( 1−Rd,i1 ) + αN i(1− V d,i1 ) ) + pid,i2 . Since pricing decisions by Retailer i in the two periods are completely independent, his total expected profit can be decomposed into two sub-problems: (discounted) profit in the second period, pid,i2 , and profit from the first period, pi d,i 1 − pid,i2 . Note, that N j and N i factor these two sub- problems, respectively. As the equilibrium pricing and total duopoly profit are independent of N1 and N2, we solve, instead, a suppressed problem with N1 = 1 and N2 = 0.9 That is, in the first period Retailer 1 encounters a single representative consumer, who is strategic (resp., myopic) with probability α (resp., 1 − α). If this consumer remains active, in the second period he visits Retailer 2. With N1 = 1 and N2 = 0, Retailer 1’s and Retailer 2’s profits represent the profit per consumer in periods 1 and 2, respectively. Suppressing retailers’ superscripts, the discounted expected profit of Retailer 2 is pid2 = λR d 2 ( (1− α)Rd1 ( 1− R d 2 Rd1 ) + αV d1 (1− Rd2 V d1 ) ) , and the expected profit of Retailer 1 is pid1 = R d 1 ( (1− α) ( 1−Rd1 ) + α(1− V d1 ) ) . As in the monopolistic case, Retailer 1, considers the two options of whether to price so as to encourage (high-valuation) strategic consumers to purchase the goods in the first period (PP (i)), or to completely ignore them in the first period and price optimally by targeting only the myopic consumers (PP (ii)). Solving the model backwards, we have:10 ( Rd1, R d 2 ) =  ( 1 2 , 1+α 4 ) , if δ > δdc ,( 2−2δ+αδ 4(1−δ+αδ) , 1 4 ) otherwise, V d1 =  1, if δ > δdc , 2−δ+αδ 4(1−δ+αδ) , otherwise, ( pid1 , pi d 2 ) =  ( 1−α 4 , λ(1+α)2 16 ) , if δ > δdc ,( 2−2δ+αδ 8(1−δ+αδ) , λ 16 ) , otherwise, pid ≡ pid1 + pid2 =  4(1−α)+λ(1+α)2 16 , if δ > δ d c , (1−δ)(4+λ)+αδ(2+λ) 16(1−δ+αδ) , otherwise, where δdc (resp., α d c) is the critical value of δ (resp., α), above (resp., below) which PP (ii) is 9Note, however, that the distribution of consumers in the first period between the two retailers determines the distribution of profit between them. 10For technical reasons, we exclude the case wherein all consumers are strategic and have a discounting factor of 1, i.e., α = 1 and δ = 1. 94 preferred over PP (i) by Retailer 1. These critical values determine the pricing policies employed by the competing retailers. Proposition 4.4 When δ ≤ δdc , or equivalently when α ≥ αdc , then, in equilibrium, retailers in a duopoly employ PP (i), which encourages both myopic and strategic consumers to purchase in the first period. Otherwise, they employ PP (ii), and all strategic consumers defer from buying in the first period. Moreover, if α ≥ 12 or if δ ≤ 23 , the SPNE prices always follow PP (i). The behavior of the equilibrium pricing policy can be described as follows: Proposition 4.5 The equilibrium prices competing retailers set are such that (i) the first (resp., second) period price is decreasing in (resp., independent of) δ up to δ = δdc , at which point both prices spike up and remain constant for δdc < δ ≤ 1; (ii) both prices are increasing in α up to α = αdc , at which point they spike down and thereafter, for αdc ≤ α ≤ 1, the first (resp., second) period price is decreasing (resp., increasing) in α. Proposition 4.4 and Proposition 4.5 imply that the behavior of the equilibrium monopoly pricing is preserved in a duopoly. In both settings, if strategic consumers have a sufficiently high willingness to wait, i.e., δ exceeds some critical value, or there are sufficiently many myopic consumers in the market, i.e., α is below a certain value, then PP (ii) (none of the strategic consumers purchases in the first period) is employed by retailers. Otherwise, they price according to PP (i), which encourages strategic consumers to purchase in both periods. Moreover, in both settings, as δ increases from sufficiently low values, the first period price decreases and the spread between the prices in the two periods diminishes, and once δ exceeds a critical value, prices in both periods spike up and remain constant in δ thereafter. There is, however, a crucial difference between the two settings. When the monopolist uses PP (ii), he merely discourages strategic consumers from buying in the first period. Indeed, he subsequently encounters them and attempts to sell them the good in the second period. By contrast, in a duopoly, when PP (ii) is employed, retailers completely ignore the strategic consumers who visit them in the first period, since all active consumers visit the competing retailer in the second period. If we assume that consumers arbitrarily visit one of the retailers in the first period, then, in expectation, each retailer initially encounters 12N consumers. 95 Proposition 4.6 The duopoly profit, as well as each retailer’s profit, is piecewise decreasing in α with a spike down at αdc and it is decreasing in δ up at δ d c at which point it spikes up, and thereafter, for δdc < δ ≤ 1, it remains constant. By sharp contrast with a monopoly, wherein profit is never increasing in δ, Proposition 4.6 reveals that a duopoly may be better off with a higher consumers’ discounting factor. This is also evident from Figure 4.2. When α < 12 (Figures 4.2(a) and 4.2(b)), the duopoly profit spikes up at δdc , and this increase in profit can be quite substantial. Clearly, in a duopoly, retailers are never eager to shift profit from the first to the second period, as any profit not realized in the first period might be collected in the second period by the competing retailer. Accordingly, we find that in a duopoly, the expected profit per consumer in the first period is larger than that in the second period, unless δ > δdc and α > 2 √ 1+2λ−2−λ λ , which occurs only if δ > 0.965 and 0.464 < α < 0.5 (see Proposition A.7). Thus, the likelihood that a larger share of the profit will be experienced in the second period is very low, as it occurs only under very restrictive conditions. Contrasting this result with the corresponding one obtained for the monopolistic retailer, ac- cording to which the majority of the profit is mostly realized in the first period, we conclude that under competition, retailers lose their ability to properly shift demand and, consequently, profit between the two periods. That is, while the monopolist’s equilibrium pricing shifts profit back and forth between the two periods, under competition retailers are eager to maintain the principal share of the profit in the first period. In fact, it can be further shown that, for every combination of the parameters, a relatively smaller portion of the expected profit is realized in the first period in a monopoly than in a duopoly. In the subsequent subsection we further dwell into the comparison between the monopolistic and duopolistic settings. 4.2.4 Implications of Competition and Consumers’ Strategic Behavior Retailers in a duopoly are affected negatively in several ways due competition and the existence of strategic consumers. These effects are elaborated on in this subsection. Naturally, competition suppresses prices and diminishes retailers’ segmentation capabilities. Formally, we have: 96 Proposition 4.7 In a duopoly (i) the switch from PP (i) to PP (ii), for fixed α and λ, occurs at a higher δ value than in a monopoly, i.e., δdc ≥ δmc ; (ii) more strategic consumers purchase the good in the first period than in a monopoly, i.e., V m1 ≥ V d1 ; and (iii) retailers’ combined expected profit is smaller than in a monopoly. Figure 4.2 depicts the behavior of the retailers’ combined expected profit in a monopoly and in a duopoly. Recall that, as a function of δ, the monopoly (resp., duopoly) profit levels off for δ > δmc (resp., δ > δdc ). The ability of the monopolist to segment the market earlier can be observed from Figure 4.2, as the monopolist’s profit becomes flat much earlier than the competing retailers’ total profit. For example, when α = 0.25 and λ = 1 (Figure 4.2(b)), the monopolist switches from PP (i) to PP (ii) at δ = 0.553, while under competition the switch occurs only when δ = 0.8. It is further evident from Figure 4.2 that the monopolistic profit dominates the duopolistic profit. Monopoly Duopoly 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 delta (a) α = 0.1 Monopoly Duopoly 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 delta (b) α = 0.25 Monopoly Duopoly 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 delta (c) α = 0.5 Monopoly Duopoly 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 delta (d) α = 1 Figure 4.2: System expected profits in a 2-period model with α strategic consumers, λ = 1 From Figure 4.2 one can also learn about the difference between the monopoly profit and the duopoly profit, pim − pid, which is the absolute profit loss due to competition. It can be proved (see Proposition A.8) that this loss is increasing in δ up to δ = δdc , at which point it spikes down (reflected by the upwards spike in the duopoly profit as shown in Figure 4.2), and remains constant in δ thereafter; the absolute profit loss is maximized, just before the spike, at δ = δdc , when α ≤ 0.5, and at δ = 1 otherwise. The profit loss, due to competition, may be as low as 0.0208 when α = 0 or when δ = 0, which represents a profit loss of about 6.25%, and as large as 0.098 (when δ = 1, α = 0.5 and λ = 1), which represents a loss of up to about 34.3% of the profit. The relative profit loss can be shown to behave in a very similar way. When all consumers are myopic, the (combined) expected profit of the retailers is maximized. The absolute (resp., percentage) profit loss of the retailers due to consumers’ strategic behavior is pii|α=0 − pii (resp., pi i|α=0−pii pii|α=0 ), i = m, d. The former is illustrated in Figure 4.3. Evidently, this 97 loss is always positive. Comparing the profit loss incurred to retailers due to consumers’ strategic behavior in a monopoly and in a duopoly leads to the following result. Proposition 4.8 Consumers’ strategic behavior inflicts (i) a larger percentage profit loss to a duopoly than to a monopoly and (ii) a larger absolute profit loss to a monopoly than to a duopoly only if δ > δdc and 3δ−2 2δ > α > 4−λ−√9λ2−56λ+80 2(λ−4) . The conditions implied by Proposition 4.8(ii) can be satisfied only when λ > 0.8, δ > 6 12−√33 ≈ 0.959 and 0.5 > α > 0.457. In other words, according to Proposition 4.8, only if strategic consumers’ discounting factor is very high (above 0.959), the retailer’s discounting factor is fairly high (above 0.8), and the fraction of strategic consumers is marginally below one half (but more than 0.457), then, and only then, the monopolist’s (absolute) expected profit loss due to consumers’ strategic behavior is larger than the combined profit loss of the retailers under competition, and even then, this difference is small. Nevertheless, the possibility that consumers’ strategic behavior will harm the monopolist more than the competing retailers, despite the rareness of this event and its low magnitude, is quite surprising. This result is illustrated by Figure 4.3. Specifically, in Figure 4.3(b), wherein δ = 1 and λ = 1, when 0.457 < α < 0.5 the duopoly’s profit loss dips below the monopoly’s profit loss (indicated by the circle). Perceived percentagewise though, the profit loss incurred to the competing retailers due to consumers’ strategic behavior is always larger than the corresponding loss incurred to a monopolist and can be shown to behave in a very similar way to that in Figure 4.3, by replacing the vertical axis with a scale of up to 50%. This percentage profit loss may be as low as zero and may exceed 40% for both systems. Duopoly Monopoly 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.2 0.4 0.6 0.8 1 alpha (a) δ = 1, λ = 0.5 Monopoly Duopoly 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.2 0.4 0.6 0.8 1 alpha (b) δ = 1, λ = 1 Duopoly Monopoly 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.2 0.4 0.6 0.8 1 alpha (c) δ = 0.5, λ = 1 Duopoly Monopoly 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.2 0.4 0.6 0.8 1 alpha (d) δ = 0.8, λ = 0.8 Figure 4.3: System absolute profit loss due to consumers’ strategic behavior in a 2-period model (the circle indicates larger monopoly loss than duopoly loss) 98 4.2.5 Obliviousness to Consumers’ Strategic Behavior We have shown that for some instances, retailers should price their goods by targeting only the myopic consumers in the first period, and that the existence of strategic consumers inflicts, at times, quite a substantial loss to retailers. In this section we elaborate on the benefit (or the lack of) to retailers when they are oblivious to consumers’ strategic behavior (or when they can credibly commit to behave in this way), and price as if all consumers were myopic. Strategic consumers are aware of this sub-optimal pricing and react accordingly. 4.2.5.1 The Monopoly Case Can the monopolist be better off (i.e., achieve a larger expected profit) by becoming oblivious to consumers’ strategic behavior? Being oblivious to consumers’ strategic behavior, the prices a monopolist sets are R1 = 24−λ and R2 = 1 4−λ . Ignorance is costly to the monopolist (see Proposi- tion A.9). This cost is especially high when α and δ are large and the retailer’s discounting factor, λ, is relatively small. In the extreme case, for example, when α = δ = 1 and λ = 0 the loss may reach 100%. Our result is in accordance with many, such as Aviv and Pazgal (2007) and B&W, who report on profit losses to a monopolistic retailer due to ignorance of consumers’ strategic be- havior. Also in Levin et al. (2005) the monopolist may suffer substantial profit losses, when, in their setting, supply levels are high. A monopolist in our setting, then, can never achieve a higher profit by ignoring consumers’ strategic behavior. But does the same conclusion hold for competing retailers? Namely, can retailers in a duopoly achieve higher profits by ignoring consumers’ strategic behavior? 4.2.5.2 The Duopoly Case In a duopoly, when both retailers are oblivious to consumers’ strategic behavior,11 the prices they set are R1 = 12 and R2 = 1 4 . In that case, if δ > 2 3 all strategic consumers wait for the second period. Figure 4.4 describes the duopoly profit per consumer when they ignore and do not ignore consumers’ strategic behavior. The difference between these plots (ignorance vs. no ignorance) depicts the profit gain or loss due to ignorance. Specifically, when δ ≤ δdc , the profit per consumer in 11The analysis may be extended to cases where only one of the retailers ignores consumers’ strategic behavior, with different information structures for strategic consumers. 99 the first (resp., second) period is lower (resp., higher) under ignorance, and when δ > δdc there is no difference in the first period profit, while the second period profit is lower (see Proposition A.10).12 To verify whether the gain in one period is greater than the loss in another, the examination of the total profit loss (gain) per consumer in both periods, due to ignorance of strategic consumer behavior, is required. Recall that, in expectation, it is assumed that each retailer initially encounters the same number of consumers in the first period. Theorem 4.1 Under competition, when both retailers are oblivious to consumers’ strategic behav- ior, the duopoly profit, as well as each of the retailers’ profit, may be higher than when consumers’ strategic behavior is not ignored. Specifically, retailers have a combined profit gain due to ignorance of strategic behavior either when δdc ≥ δ > max { 2 3 , 2−λ 3−2α−λ+λα } or when δ < min { 2 3 , λ 2α+λ−λα } . Since max { 2 3 , λ 2α+λ−λα } ≤ δdc , the combined profit may be higher only if PP (i) is employed when retailers are not oblivious to consumers’ strategic behavior. Observe, that PP (i) is employed to the left (resp., right) of the spike in Figure 4.4(a) (resp., 4.4(b)). When strategic behavior is ignored, prices optimally target myopic consumers only. By ignoring strategic consumers, the system achieves a higher profit from myopic consumers. However, the effect on profit from strategic consumers depends on the different parameters. As Figure 4.4 shows, profit can be significantly higher due to ignorance. In particular, in Figure 4.4(b), when α is slightly above αdc (about 0.388 therein) the profit can increase by as much as 22.5%. In fact, not only that a duopoly can realize additional profit when retailers are oblivious to consumers’ strategic behavior, but its profit can exceed the monopoly profit. Formally, we have: Theorem 4.2 If α > λ(λ−4) λ3−12λ2+36λ−32 and retailers are oblivious to consumers’ strategic behavior, then there exists a δ for which the expected duopoly profit exceeds the expected monopoly profit. Theorems 4.1 and 4.2 imply that, at least in some situations, retailers might be much better off by not performing market research to understand consumers’ behavioral types, exemplifying the possible negative benefit of additional information. Theorem 4.2, though, should not be misinterpreted; when a monopoly does not ignore con- sumers’ strategic behavior, its profit always exceeds the duopoly profit, whether the competing 12When the retailers oblivious to consumers’ strategic behavior, all the strategic consumers wait for the second period when δ > δdc ≥ 23 . 100  System Period 1 Period 2    Ignorance    No Ignorance (a) λ = 1, α = 0.4 (δdc = 0.909)  System Period 1 Period 2    Ignorance    No Ignorance (b) λ = 1, δ = 0.9 (αdc = 0.308) Figure 4.4: Duopoly total and per period expected profit per consumer: ignorance vs. no ignorance of consumers’ strategic behavior retailers ignore consumers’ strategic behavior or not. Finally, we note that Levin et al. (2006), who have also studied a multi-period competitive setting, have reported on profit losses to retailers due to ignorance of consumers’ strategic behavior. By contrast, however, we find that competing retailers may actually be much better off by being oblivious to strategic behavior of consumers, and we have provided in Theorem 4.1 the conditions under which this occurs. 4.2.6 The Case of Perfect Markets: λ = δ Similar to B&W, in this subsection, we briefly investigate the case of perfect markets. Specifically, we consider both monopoly and duopoly settings, wherein strategic consumers and retailers share the same discounting factor, δ. Such a common discounting factor arises when players base their actions on information provided by the financial markets.13 In a monopoly with perfect markets, prices and profits expressions are exactly as in §4.2.2 (with λ = δ), with the exception that the critical value corresponding to the transition point between PP (i) and PP (ii) (i.e., the value of δmc ) has to be modified. Let δ m 1 ≤ δm2 ≤ 1 denote the values of δ which solve ∂R m 1 ∂δ ∣∣ PP (i) = 0 and which are always never greater than one.14 Numerically, we find that δm1 ≥ 0 when α ≥ 0.5. 13Specifically, as in B&W, δ = erL, where r is the annual interest rate and L is the length of a period in annual terms. In perfect markets it is more natural to refer to strategic consumers as rational as in B&W. 14Observe that both δm1 and δ m 2 are functions of α. 101  PP(i) PP(ii) positive negative positive positive PP(i) PP(i) 0 0.2 0.4 0.6 0.8 1 de lta 0 0.2 0.4 0.6 0.8 1 alpha m 1δ m 2δ m cδ 02 11 ≥ ∂ ∂ = ∂ ∂ δ pi δ mmR  02 11 ≥ ∂ ∂ = ∂ ∂ δ pi δ mmR   02 11 ≥ ∂ ∂ = ∂ ∂ δ pi δ mmR  02 11 ≤ ∂ ∂ = ∂ ∂ δ pi δ mmR  (a) Pricing policy alpha=0.5 alpha=0.75 alpha=0 alpha=0.25 alpha=1 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 delta (b) Expected profit Figure 4.5: Monopoly’s pricing policy and expected profit under perfect markets, λ = δ Proposition 4.9 In a monopoly with perfect markets, equilibrium prices and corresponding profit behave as follows. When δ ≤ δmc , i.e., PP (i) is employed, for a fixed α, if δm1 ≥ 0, then the first period price is increasing in δ up to min{δmc , δm1 }, and then it is decreasing in δ for δm1 ≤ δ ≤ δmc , if δm1 ≤ δmc ; otherwise, if δm1 < 0, it is decreasing in δ up to min{δmc , δm2 }, and then it is increasing in δ for δm2 ≤ δ ≤ δmc , if δm2 ≤ δmc . At δmc , the monopolist switches from PP (i) to PP (ii), the first period price spikes up, and for δ > δmc it is increasing in δ. The expected profit behaves in a similar way to the first period price without the spike up at δmc . Figure 4.5 illustrates Proposition 4.9, which, in turn, complements Propositions 4.2 and 4.3. Indeed, with different discounting factors, the impact of a simultaneous increase in δ and λ on the monopoly prices and profit at equilibrium were not always trivially determinable. By contrast, Figure 4.5(a) clearly displays the choice of the pricing policy for different combinations of α and δ, and the derivative, with respect to δ, of the initial price and the expected profit. Thus, fixing α, one can follow the change in these values as δ increases. Figure 4.5(b) illustrates the changing behavior of the profit, as a function of δ = λ and α. Evidently, the lower α is, the larger is the expected profit. In a duopoly with perfect markets, prices and profits expressions are exactly as in §4.2.3 (with λ = δ), and δdc remains the same as well. The only expressions that were affected by λ, which now equals to δ, were the second period profit and total profit per consumer. Thus, we find that the second period profit is now piecewise increasing in δ, while the behavior of the total profit per 102 consumer depends on the relationship between δ and α. Specifically, when δ ≤ δdc , the total profit per consumer is increasing in δ if δ < 1− √ 2α 1−α , and decreasing in δ for 1− √ 2(1−α) 2 ≤ δ ≤ δdc . It spikes up at δdc and increases thereafter. 4.3 Longer Horizons Thus far we have focused our attention on the two-period selling horizon. In §4.3.1 we briefly consider an extension of the model to a three period horizon, and in §4.3.2 we study the case of T periods and only strategic consumers. Following GGM, we assume that remaining active consumers zigzag between retailers in every period. 4.3.1 A Three-Period Selling Horizon In a three-period selling horizon, strategic consumers with private valuations satisfying V j1 ≤ v ≤ 1 (resp., V j2 ≤ v < V j1 and R3 ≤ v < V j2 ), j = m, c, purchase the good in period 1 (resp., 2 and 3). Solving the model backwards, retailers find that the following pricing policies are possible. Under Pricing Policy i (PP (i)), retailers price to skim high-valuation consumers of both types earlier. That is, V j2 ≤ V j1 < 1, and strategic consumers possibly purchase the good in either of the three periods. Under PP (ii), V j2 < V j 1 = 1, retailers skim only high-valuation myopic consumers in the first period, and all strategic consumers defer from purchasing in the first period. Under PP (iii), strategic consumers, if they purchase at all, wait for period 3, since V j2 = V j 1 = 1. 4.3.1.1 Monopoly The monopolist sets prices based on the pricing policy that provides him with the largest expected profit. Let pimk denote the monopolist expected profit under PP (k), k = i, ii, iii. Naturally, the monopolist prefers PP (ii) over PP (i) (resp., PP (iii) over PP (ii)) whenever pimii > pi m i (resp., pimiii > pi m ii ). 15 The choice of the pricing policies, at equilibrium, is illustrated in Figure 4.6 as a function of α and δ, when λ = 1.16 As λ decreases, the area of PP (i) expands upwards and leftwards.17 We make the following observation: 15pimii = pi m i (resp., pi m iii = pi m ii ) is a polynomial expression in δ of eighth (resp., fourth) degree. 16In Proposition A.11 we show that when λ = 1 and PP (iii) is employed, the price decline is linear. 17It can be further shown that when λ = δ, pricing is very similar to Figure 4.6. 103 PP(iii) PP(i) PP(ii) 0 0.2 0.4 0.6 0.8 1 d e l t a 0.2 0.4 0.6 0.8 1 alpha Figure 4.6: Monopolist’s pricing strategy zones, λ = 1 Observation 4.1 In a three-period selling horizon: (i) the monopolist, in general, employs PP (i) unless α is small while δ and λ are large, and (ii) when λ = 1 (or λ = δ), ∀α < 1 ∃δ above which the monopolist employs PP (iii). It can be shown that when all consumers are myopic, the monopolist is better off with the addition of a third period, as he can better segment consumers along the selling horizon. When all consumers are strategic with δ = 1, he is indifferent to the addition of a third period as price in both settings is fixed at 12 in all periods. However, when there are relatively many strategic consumers (i.e., α is relatively large) whose discounting factor is relatively high and λ is not too high, the monopolist loses profit when a third period is added. By contrast, from numerical analysis we find that adding a third period to the selling horizon increases the monopolist’s profit when δ and α are small and λ is large. 4.3.1.2 Duopoly With a third period in the selling horizon, the remaining active consumers switch after the second period back to the same retailer they have encountered in the first period. Characterization of the profit in that case is not trivial, so we have conducted a numerical analysis to derive some useful insights. When δ = 0, i.e., all consumers behave myopically, the duopoly is better off with a third period, and, by contrast to a monopoly, when λ = 1 (and δ is not too small) a duopoly could be worse off with a third period. In general, for fixed α and λ, as δ increases, the profit gain due to the third period diminishes, and eventually it becomes negative. Yet, in a three-period horizon, the 104 switch from PP (i) to PP (ii) occurs much earlier than in a two-period horizon, and results with a substantial spike in the profit per consumer in period 2. In other words, retailers in a duopoly are, in expectation, much better off with the addition of a third period when they employ PP (ii) if in the corresponding two-period setting they employed PP (i). 4.3.2 T-Period Selling Horizon: Only Strategic Consumers The previous subsection has demonstrated the difficulty involved in extending the two-period model with a mix of strategic and myopic consumers to longer horizons.18 Accordingly, in the T -period selling horizon considered in this subsection, we limit attention to the case of strategic consumers only. 4.3.2.1 Monopoly – T-Period Selling Horizon Let Bmt ≡ R m t Rmt−1 denote the price ratio between the prices posted by the monopolist in periods t and t− 1, with Rm0 = 1, and let V mt denote the critical valuation in period t, such that consumers with Vt ≤ v < Vt−1 purchase the good in period t. The sequence of price ratios, {Bmt }Tt=1, T > 1, can expressed recursively as follows Bmt =  1 2−δ , if t = T , 1 (2−λBmt+1)(1−δBmt+1)+δ , if 1 < t < T , 1−δ (2−λBm2 )(1−δBm2 ) , if t = 1. Proposition 4.10 (i) {Bmt }2t=T converges from above (resp., below) when λ ≤ δ(2 − δ) (resp., λ ≥ δ(2 − δ)); (ii) when λ = δ(2 − δ), the price decay, Bmt , is constant at a rate of 12−δ , t ≥ 2. When λ < δ(2 − δ) (resp., >) the rate of price decay decreases (increases) throughout the selling horizon. Assuming the horizon is arbitrary long, i.e., T = ∞, we perform an analysis at convergence. To find the convergence value of {Bmt }2t=T , we suppress the period subscripts and solve B = 1 (2−B)(1−δB)+δ , B ∈ [0, 1], and obtain Bm = 1− √ 1−λ λ . This is the convergence value of B m 2 . The initial price at convergence is Rm1 = B m 1 = 1−δ (2−λ 1− √ 1−λ λ )(1−δ 1− √ 1−λ λ ) = 1−δ 1−δ+√1−λ . 18It is possible, though, to obtain insights from other certain instances. For example, when δ = 1 and λ = 1, the prices in a monopoly decline linearly (see Proposition A.12). 105 Proposition 4.11 (Güth and Ritzberger (1998)19) When T = ∞, the monopolist’s total ex- pected profit is pim1 =  1−δ 2(1−δ+√1−λ) , if δλ < 1, 1 4 , otherwise. Moreover, as λ→ 1 and δ → 1 any profit in (0, 12) can be realized. This result is illustrated in Figure 4.7(a), and intuitively, if δλ < 1, the monopolist’s profit is increasing in λ and decreasing in δ. Güth and Ritzberger have noted that by assuming λ = δ, a range of qualitatively different solutions may be ignored. Particularly, while any profit in (0, 12) is feasible as λ → 1 and δ → 1, when λ = δ → 1, the profit goes to zero. We also observe that, in general, the expected profit at convergence is sensitive to changes in the discounting factors when they are very close to 1. Otherwise, assuming a common discounting factor provides a good approximation for the profit. In the duopolistic case, studied below, we show that results are not dramatically altered when it is assumed that all players share a common discounting factor. 4.3.2.2 Duopoly – T-Period Selling Horizon As was stated earlier in §4.2.1, in our duopoly setting, all consumers visit one of the retailers in each period, in a zigzag manner, as long as they remain active.20 The strategic consumers, who can infer the prices along their visit pattern, time their purchase to maximize their surplus. Under zigzag competition with strategic consumers, the price ratios can be expressed recursively: Bdt =  1 2−δ , if t = T , 2−δ 4−2δ−δ2 , if t = T − 1, 1 2(1−δBdt+1)−λ2(Bdt+1)2Bdt+2(1−δBdt+2)+δ , if 1 < t < T − 1, 1−δ 2(1−δBd2 )−λ2(Bd2 )2Bd3 (1−δBd3 ) , if t = 1. (4.4) Using (4.4), it can be graphically illustrated that {Bdt }2t=T is convergent for any λ and δ < 1. The convergence value, Bd, is the value of B which solves (2B − δ2B4 − 1)(1 − δB) = 0 such that 20In the absence of such an assumption, or other “restrictive” assumptions, in a Coase-like setting with strategic consumers, the trivial solution, wherein prices drop to the marginal cost, arises. ?, for example, who study trigger strategies to maintain collusive pricing in a multi-period setting with overlapping cohorts of consumers, restrict themselves to the case wherein the fraction of the new cohort in each period is strictly positive. 106 B ∈ [0, 1].21 Thus, with an arbitrarily long horizon, the initial price is Rd1 = 1−δ(2−λ2(Bd)3)(1−δBd) , and the duopoly’s profit is (1−δ)(2−λ 2(Bd)3+λBd) 2(2−λ2(Bd)3)2(1−δBd) . This profit at convergence is illustrated in Figure 4.7(b). Therefrom, we can clearly conclude that, by contrast with the monopoly case, no qualitatively different solutions are lost when it is assumed the λ = δ. That is, when δ → 1 and λ → 1, the profit is not sensitive to changes in the relative patience. Figure 4.7(b) also shows that, similar to a monopoly, in a duopoly the profit is increasing in λ and decreasing in δ. Thus, the profit in a duopoly is maximized when δ = 0 and λ = 1, reaching a value of about 0.352, while in a monopoly it is maximized when λ = 1 for any δ. On the other hand, the duopoly profit is minimized when δ = 1, resulting with a zero profit. Assuming λ = δ < 1, i.e., as in B&W, retailers and strategic consumers share the same dis- counting factor (see §4.2.6), and we have the following result. Proposition 4.12 When λ = δ < 1: (i) {Bdt }2t=T is a convergent sequence; (ii) at convergence, the duopoly’s expected profit is (1−δ)(2−δ 2(Bd)3+δBd) 2(2−δ2(Bd)3)2(1−δBd) , where B d is the convergent value of {Bdt }2t=T . Moreover, as λ = δ → 1, the initial price converges to zero (limλ=δ→1Rd1 = 0) and, consequently, the retailers’ expected profits converge to zero as well. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 lambda delta pr of it (a) Monopoly 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 lambda delta pr of it (b) Duopoly Figure 4.7: Retailers’ combined profit at convergence when all consumers are strategic 21The equation (2B − δ2B4 − 1)(1 − δB) = 0 is obtained by suppressing the superscripts and period subscripts from the expression of Bdt when 1 < t < T − 1. We obtain, Bd ≡ 112λ ` 2 6 √ 2 3 √ 32T − 12 √ 211 12 √ 35 r 12 √ 3λ 3√ M−√6T 3 √ M2−2 6 √ 65Tλ2 3√ MT ´ , where M = λ2(9 + √ 81− 48λ2) and T = r 3√ M2+2 3 √ 6λ2 3√ M . 107 4.3.2.3 Monopoly vs. Duopoly – T-Period Selling Horizon In this section we briefly contrast the monopoly and duopoly cases when all consumers behave strategically, and the discounting factor of the consumers and the retailers coincide, i.e., λ = δ < 1. Naturally, competition suppresses prices, as is evident from Figure 4.8, which illustrates the price decline over a selling horizon of 20 periods in both a monopoly and a duopoly for several values of the common discounting factor. In a duopoly as δ increases, the initial price decreases and prices converge faster to zero. However, this is not the case for a monopoly. Rather, in a monopoly, when δ is not too close to 1, a lower initial price is associated with a much slower price decline, with later prices being higher for larger values of δ. Further, in a monopoly, the length of the selling horizon significantly affects the price decline. For example, while in a two period horizon, the initial price increases in δ for δ > 23 (see Figure 4.5(a)), in a 20-period horizon it can be shown that this occurs only for δ greater than about 0.925, and at convergence, the initial price (which behaves in a similar way to the profit which is depicted in Figure 4.9) only decreases in δ < 1.22 0 0.1 0.2 0.3 0.4 0.5 1 4 7 10 13 16 19 δ =0.75 δ =0.99 δ =0 δ =0.9  (a) Monopoly 0 0.1 0.2 0.3 0.4 0.5 1 4 7 10 13 16 19       =0       =0.75       =0.9       =0.99 δ δ δ δ  (b) Duopoly Figure 4.8: Prices in a 20-period horizon when all consumers are strategic with λ = δ < 1 The retailers’ combined expected profits at convergence in a monopoly and in a duopoly when λ = δ < 1 are exhibited in Figure 4.9. Lower prices in a duopoly result with corresponding lower expected profits. However, the difference in expected profit is not substantial, especially when δ is not too close to 1. This difference is at most 0.057, when δ = 0.93. Nevertheless, in terms of relative profit, the percentage profit loss due to competition in the presence of strategic consumers 22Essentially, in a monopoly, when λ = δ = 1, price is fixed at 1 2 in all periods. 108 can be shown to be increasing in δ; it is fairly small when δ is not too large, but when δ is large, it increases rapidly, and eventually, as δ goes to 1, the profit loss approaches 100%. Finally, similar to §4.2.4, we can evaluate the profit loss incurred to retailers due to consumers’ strategic behavior. Namely, we can compare the profit realized when δ = λ, with the profit achieved with the same discounting factor for retailers when the consumers behave myopically. We obtain: Monopoly Duopoly 0 0.05 0.1 0.15 0.2 0.25 0.2 0.4 0.6 0.8 1 delta Figure 4.9: Retailers’ combined profit at convergence when all consumers are strategic with λ = δ < 1 Proposition 4.13 When λ = δ < 1 and T = ∞, in relative terms, consumers’ strategic behavior always results with a larger profit loss to a duopoly. Yet, in absolute terms, strategic behavior inflicts a larger profit loss to a monopoly than to a duopoly when δ > 0.932. That is, the presence of strategic consumers further amplifies the retailers’ relative loss due to competition. 4.4 Summary and Managerial Insights Pricing is a major concern for firms and retailers. Pricing goods too low or too high may have substantial effects on profits. The difficulty involved in pricing is aggravated when prices have to be updated along the selling horizon, consumers are heterogenic, and retailers face competition. In this chapter we have introduced and analyzed multi-period discrete-time models which address these difficulties. Specifically, we have studied monopoly and duopoly models by distinguishing between two types of consumers: myopic and forward-looking (strategic). Myopic consumers are those impatient consumers who purchase the good as soon as the posted price drops below their 109 valuations. Strategic consumers, on the other hand, act upon the current price and their expectation of future prices as well. Qualitatively, our two-period selling horizon monopoly and duopoly models behave in a sim- ilar way. In both settings, when strategic consumers’ discounting factor is sufficiently high, or, alternatively, when there are sufficiently many myopic consumers in the market, retailers employ a pricing policy which ensures that all strategic consumers will not purchase the good in the first period. Otherwise, the pricing policy encourages both myopic and strategic consumers to purchase in both periods. In general, quite intuitively, retailers are better off when there are fewer strategic consumers in the market whose discounting factor is low. Yet, under competition, we find that an increase in the strategic consumers’ discounting factor may, in fact, be beneficial for a duopoly. Our model can assist managers in choosing the appropriate (multi-period) pricing policy. It can also support other important decisions. For instance, marketing efforts may increase the market size and change consumers behavior (creating the right buzz induces impatient behavior). Factoring these effects and their costs into our model, may aid managers in setting proper investments for these efforts and choosing the corresponding pricing policy. Quite naturally, competition suppresses prices and corresponding retailers’ expected profits. Consumers’ strategic behavior amplifies the impact of competition, as it inflicts a larger (relative) profit loss to competing retailers than to a monopolistic retailer. A monopolistic retailer should be careful with his market research, as wrongly interpreting strategic consumer behavior as myopic, can be very costly. On the other hand, ignorance of strategic consumer behavior may be quite beneficial to a duopoly, as retailers’ expected profit may be larger. In other words, competing retailers can be absent minded regarding consumers’ characteristics and still experience a profit gain. Moreover, when all retailers ignore consumers’ strategic behavior, a duopoly may achieve a larger profit than a monopoly. That is, if retailers in both a monopoly and a duopoly, for whatever reason (e.g., not performing an appropriate market research), treat all consumers as myopic, then a duopoly, despite the competitive forces, may achieve a larger profit than a monopoly. A multi-period setting facilitates an inter-temporal competition. Indeed, the well known Coase- Conjecture (and its later refinements) propose(s) that a monopolist facing strategic consumers in a multi-period selling horizon may have to set prices as if he were facing competition, i.e., at his marginal cost level. In the extension of our model to longer horizons, when all consumers are 110 strategic, we are able to elaborate on this result. We find that when the strategic consumers and the retailers share the same discounting factor, monopoly profit exceeds duopoly profit, and that the profit loss due to competition is increasing in the common discounting factor. At the limit, as the common discounting factor goes to 1, prices and profits in a monopoly and in a duopoly drop to zero. Further, we find that not only do retailers lose profit due to competition, but consumers’ strategic behavior inflicts a higher relative loss to a duopoly than to a monopoly. 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When - all three essays of this thesis explore prices posted in different periods of the selling horizon. Where - in the first two essays we find that different retail stores may post different prices at the same period. Who - the third essay reveals that strategic and myopic consumers who have the same valuation for the good, may end up paying different prices. In this thesis we have investigated several important issues in the area of dynamic pricing for revenue management. Studying the effect of competition and strategic consumer behavior, we have characterized the dynamic pricing policies for retailers who sell homogeneous goods in multi-period, discrete time, finite horizon settings. In the first two essays we have focused on markets that consist of myopic consumers only, and in the third essay we have further considered the presence of strategic consumers. Asides from our modelling contribution, i.e., the approach introduced here to model competition in a multi-period, discrete time, finite horizon settings, the analysis in the three essays have granted us with important insights regarding competition, implications for consumers, retailers, and the regulator in such markets. Below are some of the main insights obtained in this thesis. Competition Competition has a substantial impact on prices and profits in multi-period settings, and this impact is more profound with longer horizons. Competition transforms the decline of prices from a linear shape under monopoly to an exponential shape under competition. The exponential decline of prices was found to be robust, as it occurs even when a strong retailer faces competition from a relatively much weaker retailer. 116 Consumers Consumers are best off when they ignore store ambience or store dominance. They maximize their surplus when they alternate deterministically, in a zigzag fashion, between the two stores. Retailers Retailers are better off by increasing their retention rates. Their profits are in particular sensitive to changes in the retention rates when they are close to one, and therefore, the performance of the competitor, even a very weak one, should be closely monitored. The regulator He should not be too concerned with market concentration as competition be- tween relatively symmetric retailers under structural competition or between retailers whose retention rates are not too hight under behavioral competition can exert sufficient pressure on prices. The essays in this thesis can be extended in several ways. As the retailer who encounters a consumer in the first period is usually awarded with the lion share of the profit, it may be beneficial to investigate an initial stage wherein retailers compete against each other via marketing expenditures in attracting consumers to the stores in the first place. Another important extension is related to the inventory available to retailers. Under such a setting (strategic) consumers will be exposed to a rationing risk and will have to consider the probability of obtaining the good once they decide to postpone their purchase to a later period. Finally, one can also consider different arrival processes, such as arrival of consumers in each of the periods, and possibly also assign uncertainty to these arrivals. 117 Appendices A.1 Proofs of Lemmas, Propositions, and Theorems in Chapter 2 Proof of Lemma 2.1. The proof is by induction. From Table 2.1 it can be verified that the recursions hold for the final periods. Assume the Lemma holds for t + 1, and we will show that it holds for an odd t. Now, Retailer 1’s profit-to-go is pi1t = Rt(1 − RtRt−1 ) + pi1t+1 RtRt−1 = Rt(1− RtRt−1 ) + S1t+1Rt RtRt−1 , where the last equality follows from the induction hypothesis. Setting the derivative of pi1t with respect to Rt to zero results with Rt = 1 2(1− S1t+1) Rt−1. (A.1) pi1t is a concave in Rt if S 1 t+1 < 1, which is shown to hold in Lemma 2.2. Thus, Rt from (A.1) is the maximizer of the profit-to-go function in period t. Using (A.1) we rewrite pi1t and pi 2 t to obtain: pi1t = 1 4(1−S1t+1) Rt−1 and pi2t = S2t+1 4(1−S1t+1)2 Rt−1. Thus, S1t = 1 4(1− S1t+1) and S2t = S2t+1 4(1− S1t+1)2 , and due to the induction hypothesis S1t and S 2 t are scalars as well. The proof for an even t follows in a similar way. Proof of Lemma 2.2. The proof is by induction. The induction base is satisfied for the last two periods as s1T = s 1 T−1 = 1 4 , s 2 T = 0, and s 2 T−1 = 1 16 . Suppose that 1 4 ≤ s1t+1 < s1, 0 ≤ s2t+1 ≤ s2, for some t. As s1t = 1 4(1−s2t+1) , the lowest value for s1t is obtained if the lower bound value of s 2 t is used. Thus, s1t = 1 4(1−s2t+1) ≥ 14(1−0) = 14 = s1, where the last inequality is satisfied by the induction hypothesis. The largest value for s1t is obtained if the upper bound value of the sequence of s2t is used. We have s 1 t = 1 4(1−s2t+1) ≤ 1 4(1−s2) , as follows from the induction hypothesis. Thus, s1t ≤ 1 4 ( 1− ( − 1 6 (19+3 √ 33) 1 3− 2 3 1 (19+3 √ 33) 1 3 + 5 6 )) = 32 (19+3√33) 13 (19+3 √ 33) 1 3+(19+3 √ 33) 2 3+4 = s1t . As required, we have shown that s1 ≤ s1t ≤ s1. The proof for the boundedness of {s2t } follows in a similar way. 118 Proof of Proposition 2.1. The proof is based on the Monotone Convergence Theorem. The two sequences of the normalized sales, {s1t } and {s2t }, are bounded from above as follows from Lemma 2.2. Next, we show that the sequences are increasing. {s1t } is increasing if s 1 t s1t+1 = 1 4s1t+1(1−s2t+1) ≥ 1, which certainly holds (by Lemma 2.2) as 0 < 4s1t+1(1 − s2t+1) ≤ 1. The proof that {s2t } is increasing is by induction. We have that s2T−1 = 1 16 ≥ s2T = 0. Assume s2t+1 s2t+2 ≥ 1, and we will verify that s2t s2t+1 ≥ 1. Notice that s2t = s 1 t s 1 t+1 1−s2t+1 . Thus, s 2 t s2t+1 = s1t s 1 t+1(1−s2t+2) s1t+1s 1 t+2(1−s2t+1) . Since {s1t } is increasing and by the induction hypothesis 1−s 2 t+2 1−s2t+1 ≥ 1, s2t s2t+1 ≥ 1. Proof of Proposition 2.2. By contradiction, assume that s1t = 1 4(1−s2t+1) ≤ s2t = s 1 t+1 4(1−s2t+1)2 . Rearranging yields: 1 ≤ s1t+1 + s2t+1, which is a contradiction by Lemma 2.2. Proof of Proposition 2.3. It can be shown that the normalized sales-to-go of the single retailer in Lazear’s model, sLt , can be expressed in a recursive form as follows: s L t = 1 4(1−sLt+1) . By induction we prove that sLt ≥ s1t + s2t . Observe that sLT = 14 = s1T + s2T , and that sLT−1 = 13 > 5 16 = 1 4+ 1 16 = s 1 T−1+s 2 T−1. Assume that s L t+1 ≥ s1t+1+s2t+1, and we will show that sLt = 14(1−sLt+1) ≥ 1 4(1−s2t+1) + s1t+1 4(1−s2t+1)2 = s1t + s 2 t . The last inequality holds if and only if s L t+1 ≥ s1t+1+s 2 t+1−(s2t+1)2 1+s1t+1−s2t+1 , which holds, since we have sLt+1 ≥ s1t+1 + s2t+1 ≥ s1t+1+s 2 t+1−(s2t+1)2 1+s1t+1−s2t+1 , where the first inequality follows from the induction hypothesis and the second inequality holds since s1t+1 ≥ 0. Proof of Proposition 2.4. Since BT = BT−1 = 12 , and BT−2 = 8 15 , the assertion holds for the first three elements of the sequence, and using (2.6), one can use induction to establish the bounds of 0.5 and 0.55 for every element in the sequence. Thus, it remains to prove that {Bt} is increasing for all t, t ≤ T − 1. The proof of this part is also by induction. We have BT−1BT = 1 2 1 2 = 1, and BT−2BT−1 = 8 15 1 2 = 1 115 > 1. Assume that Bt Bt+1 ≥ 1 and Bt−1Bt ≥ 1. To complete the induction proof, we need to show that Bt−2Bt−1 > 1. By contradiction, assume that Bt−2 Bt−1 < 1. Then, by substituting the expressions of Bt−1 and Bt−2, using (2.6), we obtain: Bt−2 Bt−1 = 2−[Bt]2Bt+1 2−[Bt−1]2Bt < 1. Since the denominator is positive, the last inequality can be rewritten as follows: (Bt−1Bt ) 2 Bt Bt+1 < 1, which contradicts the induction assumption as both ratios are larger or equal to one. Thus we conclude that Bt−2Bt−1 ≥ 1. Proof of Proposition 2.5. Since B1 coincides with R1, it suffices to prove the Proposition for B1. By Proposition 2.4, {Bt} is a monotone convergent sequence bounded from below by 0.5 and from above by 0.55. To find B, the convergence value of {Bt}, we omit the period subscripts 119 in (2.6). That is, B = 1 2−B2B . Rearranging, we solve (B − 1)(B3 +B2 +B − 1) = 0, for which the only relevant root that lies within the bounds of {Bt} is B = (17+3 √ 33) 2 3−2−(17+3√33) 13 3(17+3 √ 33) 1 3 ≈ 0.543. Proof of Proposition 2.6. Since s11 (resp., s 2 1) coincides with the expected profit of Re- tailer 1 (resp., 2), it suffices to prove the Proposition by using {s1t } and {s2t }. By Proposition 2.1, {s1t } and {s2t } are monotone convergent sequences and their bounds are provided in Lemma 2.2. The convergence values of {s1t } and {s2t } can be shown using (2.4) to be 0.2718 and 0.0803, respectively. Proof of Proposition 2.9. The proof is by induction. For the initial step, T may be even or odd, and for the inductive step t may also be even or odd. We prove the Proposition for the case of T even and the inductive t is odd. The proof of the other cases is similar. For t = T , the assertion holds since pii (N iT ,N j T ) = PN i T 1 R i T + 2P N iT 2 R i T + · · · + N iTP N iT N iT RiT = RiT ∑N iT n=1 ( nP N iT n ) = RiTN i T [1−FT (RiT )] = N iTpi2T . Next, since the inductive t is odd, t+1 is even. Thus, assume that pii (N it+1,N j t+1) = N it+1pi 2 t+1 +N j t+1pi 1 t+1. To complete the induction proof we need to show that pii (N it ,N j t ) = N itpi 1 t +N j t pi 2 t holds as well: pii (N it ,N j t ) = N it∑ ni=0 ( P N it ni ( niRit + Njt∑ nj=0 P Njt nj pii (Njt−nj ,N it−ni) )) = N it∑ ni=0 ( P N it ni ( niRit + Njt∑ nj=0 P Njt nj ( (N jt − nj)pi2t+1 + (N it − ni)pi1t+1 ))) = Rit N it∑ ni=0 ( ni N it ! ni!(N it − ni)! [Ft(Rit)] N it−ni [1− Ft(Rit)]n i ) +pi2t+1 N it∑ ni=0 Njt∑ nj=0 ( (N jt − nj) N it ! ni!(N it − ni)! [Ft(Rit)] N it−ni [1− Ft(Rit)]n i · N j t ! nj !(N jt − nj)! [Ft(R j t )] Njt−nj [1− Ft(Rjt )]n j ) +pi1t+1 N it∑ ni=0 Njt∑ nj=0 ( (N it − ni) N it ! ni!(N it − ni)! [Ft(Rit)] N it−ni [1− Ft(Rit)]n i · N j t ! nj !(N jt − nj)! [Ft(R j t )] Njt−nj [1− Ft(Rjt )]n j ) = RitN i t [1− Ft(Rit)] + pi2t+1N jt Ft(Rjt ) + pi1t+1N itFt(Rit) = N it ( Rit[1− Ft(Rit)] + pi1t+1Ft(Rit) ) +N jt pi 2 t+1Ft(R j t ) 120 = N itpi 1 t +N j t pi 2 t , where the second equality follows by the induction step and the last one follows from the recursive expressions in (2.3). Proof of Proposition 2.10. Solving backwards, in the second period R12 =  1− KN − (R11 −R21), if R11 ≤ 1− KN + 12R21, 1 2R 2 1, otherwise. In the first period, if N(1−R11) ≤ K −N(1−R11) then the realized demand exceeds the available inventory, and retailers set a price of 1− KN . Otherwise, inventory satisfies realized demand in the first period and the price is R11 =  3 4 − 2K4N + 14R21, if R11 ≤ 1− KN + 12R21, 1 2 , otherwise. Solving simultaneously for both retailers, while taking into account the conditions, the optimal prices set by retailers are: (R11, R 1 2) = (R 2 1, R 2 2) =  (1− 2K3N , 1− KN ), if KN ≤ 34 , (12 , 1 4), otherwise, as required. Proof of Theorem 2.2. Assume that Ri1 ≤ Rj1, and 2N ≤ K. Then, the profit functions for Retailers i and j can be written1 as pii = NRi1(1 − Ri1) + (Rj1 − Ri1)Ri1N + Ri1NRi2(1 − R i 2 Ri1 ), and pij = NRj1(1 − Rj1) + Ri1NRj2(1 − R j 2 Ri1 ). Solving backwards, we find that Ri2 = R j 2 = 1 2R i 1. Solving for the first period, we have Ri1 = min(R j 1, 2(1+Rj1) 7 ) and R j 1 = max( 1 2 , R i 1). Thus, we have that the pure-strategy SPNE prices are Ri1 = 3 7 , R j 1 = 1 2 , and R i 2 = R j 2 = 3 14 . Obviously, the symmetric case in which Ri1 = 1 2 , R j 1 = 3 7 , R i 2 = R j 2 = 3 14 is also a SPNE. Proof of Theorem 2.3. We separately analyze the three possible cases. Case 1: 1 ≤ α ≤ 2 (i.e., N ≤ K ≤ 2N) The profit expressions are pii = NRi1(1 − Ri1) + (Rj1 − Ri1)Ri1(K − N) + Ri1NRi2(1 − R i 2 Ri1 ) and pij = NRj1(1 − Rj1) + Ri1NRj2(1 − R j 2 Ri1 ). Solving backwards, we find that Ri2 = R j 2 = 1 2R i 1. Solving for the first period, we have Ri1 = min(R j 1, 2(N+Rj1(K−N)) 4K−N ) and R j 1 = max( 1 2 , R i 1). Thus, we have 1By Appendix A.2, if Retailer i sells in the first period, he sets the same price, Ri1, in the second period. 121 that for 32 ≤ α ≤ 2 the pure-strategy SPNE is Ri1 = 1+α4α−1 , Rj1 = 12 , Ri2 = R2j = 1+α2(4α−1) and the symmetric SPNE wherein i and j are exchanged, and for 1 ≤ α ≤ 32 any 12 ≤ Ri1 = Rj1 ≤ 22α+1 and Ri2 = R 2 j = 1 2R i 1 is a SPNE. Case 2: 1 2 ≤ α ≤ 1 (i.e., 1 2 N ≤ K ≤ N) The profit functions are pii = KRi1(1−Ri1) +Ri1KRi2(1− R i 2 Ri1 ) and pij = KRj1(1−Rj1) + (Rj1 − Ri1)R i 1(N − K) + Ri1KRj2(1 − R j 2 Ri1 ). As before, Ri2 = R j 2 = 1 2R i 1. Solving for the first period, we have Ri1 = min(R j 1, 2 3) and R j 1 = max( α+Ri1(1−α) 2α , R i 1). Thus, for 1 2 ≤ α ≤ 23 , the SPNE is Ri1 = 23 , Rj1 = α+2 6α , and R i 2 = R 2 j = 2 3 ; and for 2 3 ≤ α ≤ 1 the SPNE is α3α−1 ≤ Ri1 = Rj1 ≤ 23 , and Ri2 = R 2 j = 1 2R i 1. Case 3: α ≤ 1 2 (i.e., K ≤ 1 2 N) The profit functions are pii = KRi1(1−Ri1) +Ri1KRi2(1− R i 2 Ri1 ) and pij = KRj1(1−Rj1) + (Rj1 − Ri1)R i 1K +R i 1KR j 2(1− R j 2 Ri1 ). As before, Ri2 = R j 2 = 1 2R i 1. That is, the price is the same is in Case 3 when α = 12 . Therefore, R i 1 = 2 3 , R j 1 = 5 6 , and R i 2 = R 2 j = 2 6 . Proof of Proposition 2.12. Preliminaries required for the proof of Proposition 2.12: By Appendix A.2, if in period t only the N2 consumers who observe the lower price, RN 2 t , buy the good and RN 1 t ≤ (T − t+ 1)RN 2 t , (A.2) then the centralized system profit-to-go in period t + 1 is p̄it+1 = N1 RN 1 t (T−t−1)+RN 2 t (T−t+1) 2(T−t) and the price in each of the following periods is Rt = (T−t)RN1t +(t−t)RN 2 t T−t , t = t+ 1, ..., T . The dynamic programming problem of maximizing the expected profit is solved backwards. In period t, the profit-to-go of the system, as long as N2 is not zero2, is: pit = N1RN 1 t ( 1− Ft(RN1t ) ) +N2RN 2 t ( 1− Ft(RN2t ) ) +Ft(RN 2 t )pit+1+ ( Ft(RN 1 t )− Ft(RN 2 t ) ) p̄it+1, (A.3) where the first two terms are the expected profits of the two stores in period t; the third (resp., fourth) term is the conditional expected profit-to-go of the system in period t + 1 if no consumer makes a purchase in period t (resp., if only the group with N2 consumers, who faces a lower price, makes a purchase). Display (A.3) is valid if (A.2) holds, which is verified to be the case in Lemma A.1. 2The case of {N1 = N , N2 = 0} coincides with the single store monopoly case and can be solved by setting {N1 = 0, N2 = N}. 122 Lemma A.1 For each period t, as long as no sale has occurred, RN 1 t ≤ (T − t+ 1)RN2t . Proof. The proof of Lemma A.1 requires the proofs of the following two lemmas. Lemma A.2 The profit-to-go in period t can be expressed as the normalized sales-to-go in period t times the lower price that was set in the preceding period. That is, pit = StRN 2 t−1, where St is a scalar which can be expressed recursively as St = (N)2 + (T − t)((N)2 − 2N1St+1) 2 ( (T − t)(2N2 +N1 − 2St+1) + 2(N − St+1) ) . (A.4) Proof. By induction. It is easily verified that the Lemma holds for the final periods. Assume the Lemma holds for period t + 1 and we will show that it holds also for period t. Setting the derivatives of pit (as given in display (A.3)) with respect to RN 1 t and R N2 t to zero and solving simultaneously, results with: RN 2 t = N + (T − t)N2 (T − t)(2N2 +N1 − 2St+1) + 2(N − St+1)R N2 t−1, (A.5) and RN 1 t = (T − t)(2N2 +N1 − 2St+1) +N (T − t)(2N2 +N1 − 2St+1) + 2(N − St+1)R N2 t−1. (A.6) Plugging (A.5) and (A.6) into (A.3) we find that: pit = (N)2+(T−t)((N)2−2N1St+1) 2 ( (T−t)(2N2+N1−2St+1)+2(N−St+1) )RN2t−1. Thus, St = 12 (N)2+(T−t)((N)2−2N1St+1) (T−t)(2N2+N1−2St+1)+2(N−St+1) ,which is a scalar, and the proof is complete. Lemma A.3 The sequence of normalized sales-to-go, {St}, is bounded from above by 12N . Proof. By induction. In the last period the normalized sales-to-go is ST = 14N < 1 2N . Assume the Lemma holds for period t+1 and it remains to show that St ≤ 12N . By contradiction, assume that St > 12N . Since the denominator of St, as given in (A.4), is positive, then, it can be shown that St − 12N = (2St+1−N) ( N+N2(T−t) ) 2[(T−t+1)(N+N2−2St+1)+N1] ≤ 0, which contradicts the assumption that St > 1 2N . Using (A.5) and (A.6), RN 1 t ≤ (T − t + 1)RN2t holds if (T−t)(2N2+N1−2St+1)+N (T−t)(2N2+N1−2St+1)+2(N−St+1)R N2 t−1 ≤ (T−t+1) ( N+(T−t)N2 ) (T−t)(2N2+N1−2St+1)+2(N−St+1)R N2 t−1. Since the denominator in both sides is positive (follows from Lemma A.3), the last inequality holds only if (T − t)(2N2 + N1 − 2St+1) +N ≤ (T − t+ 1)(N + (T − t)N2), or, equivalently, 0 ≤ (T − t) ( (T − t)N2 + 2St+1 ) , which certainly holds. The proof of Proposition 2.12 follows from the following two lemmas. 123 Lemma A.4 The sequence of the system’s normalized sales-to-go, {St}, is a monotone convergent sequence as the number of periods, T , goes to infinity while t goes to 1. Proof. By Lemma A.3, {St} is bounded from above. It remains to show that for each t, StSt+1 ≥ 1. By contradiction, assume that StSt+1 < 1. Using (A.4), St St+1 = (N) 2+(T−t)((N)2−2St+1N1) 2St+1 ( (T−t)(2N2+N1−2St+1)+2(N−St+1) ) . Since the denominator of the right hand side is positive, StSt+1 < 1 implies that (T − t + 1)(N − 2St+1)2 < 0, which is a contradiction. Lemma A.5 The normalized sales-to-go of the centralized system is increasing with N1, as long as N2 is not zero, when N is kept constant. Proof. ST = 14N and thus ∂ST ∂N1 = 0, since N is constant. Assume that the assertion holds for t+1. That is, ∂St+1 ∂N1 ≥ 0. To complete the proof we need to show that ∂St ∂N1 ≥ 0. Since N2 = N − N1, using (A.4), we can express St as follows: St = 12 (N)2+(T−t)((N)2−2N1St+1) (T−t)(2N−N1−2St+1)+2(N−St+1) . Thus, we want to show that ∂St ∂N1 = − ( (T−t)N1 (T−t)(2N−N1−2St+1)+2(N−St+1) − ((N)2+(T−t)((N)2−2N1St+1))(−2(T−t)−2) 2(((T−t)(2N−N1−2St+1)+2N−2St+1)2) ) ∂St+1 ∂N1 − (T−t)St+1 (T−t)(2N−N1−2St+1)+2(N−St+1) + ((N)2+(T−t)((N)2−2N1St+1))(T−t) 2((T−t)(2N−N1−2St+1)+2N−2St+1)2 ≥ 0. Simplifying, this holds if ∂St+1 ∂N1 ≥ − (T−t)(N−2St+1)2(T−t+1) 2((T−t)N1−N(T−t)−N)2 , which certainly holds since the right hand side is negative. Since pi1 = S1, the proof of Proposition 2.12 has been obtained. Proof of Proposition 2.13. Assume Retail Stores i and j have Ki and Kj units in stock and face N i and N j consumers in the first period, respectively. The centralized two-store monopoly expected profit, assuming Ri1 ≤ Rj1, is pi = Ri1 ( 1−Ri1 ) min ( Ki, N i ) +Rj1 ( 1−Rj1 ) min ( Kj , N j ) + ( Rj1 −Ri1 ) Ri1 ( min ( N j ,Ki −min (Ki, N i))+min (Kj , N i −min (Ki, N i))) +Ri1 ( min ( Ki, N j ) Ri2 ( 1− R i 2 Ri1 ) +min ( Kj , N i ) Rj2 ( 1− R j 2 Ri1 )) , where the first two terms represent the first period profit; the third term is the expected profit in the second period if in the first period a sale occurs only at Retail Store i, where a possibly lower price is posted. In that case the second period price is Ri1; 3 the last term is the expected profit in the second period when neither stores sells in the first period. Set Ki = Kj ≡ K, N i = N j ≡ N , and let α ≡ KN . Solving backwards, we find that the second 3Specifically, given a sale only at Retail Store i, it is inferred that the common valuation is distributed U [Ri1, R j 1], and, by Appendix A.2, the second period price is max( 1 2 Rj1, R i 1), which can be shown to equal R i 1. 124 period price when neither stores sells is 12R i 1, and we obtain that the optimal first period prices and corresponding profit are (Ri1, R j 1) =  (35 , 4 5), α ≤ 12 , ( −α(α+1) 3α2−6α+1 , α(2α−3) 3α2−6α+1), 1 2 ≤ α ≤ 23 , (23 , 2 3), 2 3 ≤ α ≤ 32 , ( −(1+α) 3−6α+α2 , 2−3α 3−6α+α2 ), 3 2 ≤ α ≤ 2, (35 , 4 5), Otherwise, and pi =  7 10K, α < 1 2 , α2(α−4)N 2(3α2−6α+1) , 1 2 ≤ α ≤ 23 , 2 3K, 2 3 ≤ α ≤ 1, 2 3N, 1 ≤ α ≤ 32 , (1−4α)N 2(3−6α+α2) , 3 2 ≤ α ≤ 2, 7 10N, Otherwise, respectively. 125 A.2 Single Store Monopoly with a Consumer’s Valuation Drawn from a Uniform [a,b] Distribution. Consider a single store monopoly which has T periods to sell its good to a single consumer, whose valuation for the good follows a uniform distribution over [a, b]. In that case, pit = Rt (1− Ft(Rt))+∑T i=t+1 (∏i−1 j=t Fj(Rj) ) Ri (1− Fi(Ri)), where Ft(Rt) = F (Rt)F (Rt−1) = Rt−aRt−1−a . Proposition A.1 Under monopoly, when the consumer’s valuation is uniformly distributed over [a, b], both the price, Rt, and the profit-to-go, pit, can be written recursively as follows: Rt =  T−t+1 T−t+2Rt−1, if Rt−1 > (T − t+ 2)a, (T−t)Rt−1+a T−t+1 , Otherwise, and pit =  (T−t+1)(Rt−1)2 2(T−t+2)(Rt−1−a) , if Rt−1 > (T − t+ 2)a, Rt−1(T−t)+a(T−t+2) 2(T−t+1) , Otherwise. Proof. By induction. For t = T , piT = RT ( 1− F (RT )F (RT−1) ) . Taking the derivative w.r.t. RT and equating to zero results with RT = max(a, 12RT−1), or, RT =  1 2RT−1, if RT−1 > 2a, a, Otherwise, and piT =  (RT−1)2 4(RT−1−a) , if RT−1 > 2a, a, Otherwise. Assume the Proposition holds for period t. We have to show it holds for t − 1. When Rt−1 > (T − t + 2)a, we solve Max Rt−1 pit−1 = Max Rt−1 {Rt−1(1 − Ft−1(Rt−1)) + pitFt−1(Rt−1)} = Max Rt−1 {Rt−1Rt−2−Rt−1Rt−2−a + (T−t+1)(Rt−1)2(Rt−1−a) 2(T−t+2)(Rt−1−a)(Rt−2−a)}, which results with Rt−1 = T−t+2T−t+3Rt−2 and pit−1 = (T−t+2)(Rt−2)2 2(T−t+3)(Rt−2−a) as required. When Rt−1 ≤ (T − t + 2)a, we solve MaxRt−1 pit−1 = MaxRt−1 {Rt−1(1 − Ft−1(Rt−1))+pitFt−1(Rt−1)} =Max Rt−1 {Rt−1Rt−2−Rt−1Rt−2−a + Rt−1(T−t)+a(T−t+2) 2(T−t+1) Rt−1−a Rt−2−a}, for which the so- lution is Rt−1 = (T−t+1)Rt−2+a T−t+2 and the corresponding profit-to-go is: pit−1 = Rt−2(T−t+1)+a(T−t+3) 2(T−t+2) , as required. 126 Thus, the monopolist expected profit, which coincides with his profit-to-go in period 1, is given by: pi =  1 2 T T+1 b2 b−a , if b > (T + 1)a, b(T−1)+a(T+1) 2T , Otherwise, and the price he sets in period t is: Rt =  T−t+1 T+1 b, if b > (T + 1)a, (T−t)b+ta T , Otherwise. 127 A.3 Proofs of Lemmas, Propositions, and Theorems in Chapter 3 Lemma A.6 In a structural duopoly the profit-to-go expressions in period t are affine functions of the price set in the previous period. That is, pii,jt = S i,j t Rt−1, i = 1, 2, j = v, n, with R0 = 1, where Si,jt are scalars, which can be expressed recursively as follows: S 1,v t = 1 4(1−PS1,vt+1−(1−P )S1,nt+1) , S1,nt = PS1,vt+1+(1−P )S1,nt+1 4(1−(1−P )S2,vt+1−PS2,vt+1)2 , S2,vt = 1 4(1−(1−P )S2,vt+1−PS2,nt+1) , and S2,nt = (1−P )S2,vt+1+PS2,nt+1 4(1−PS1,vt+1−(1−P )S1,vt+1)2 , with Si,vT = 1 4 and S i,n T = 0, i = 1, 2. Proof. The proof is by induction. Solving for the final period, we have, as shown in Table 3.1, R1T = R 2 T = 1 2RT−1, pi 1,v T = pi 2,v T = 1 4RT−1, and pi 1,n T = pi 2,n T = 0. Then, since S 1,v T+1 = S 1,n T+1 = S2,vT+1 = S 2,n T+1 = 0, S 1,v T = 1 4(1−PS1,vT+1−(1−P )S1,nT+1) = 14 , S 1,n T = PS1,vT+1+(1−P )S1,nT+1 4(1−(1−P )S2,vT+1−PS2,vT+1)2 = 0, S2,vT = 1 4(1−(1−P )S2,vT+1−PS2,nT+1) = 14 , and S 2,n T = (1−P )S2,vT+1+PS2,nT+1 4(1−PS1,vT+1−(1−P )S1,vT+1)2 = 0, the lemma holds for period T . (Table 3.1 can be similarly used to illustrate that the lemma holds for period T−1 as well.) Assume that the lemma holds for period t + 1 and we will show it also holds for period t. Following the induction hypothesis, the profit-to-go functions in period t can be written as: pi1,vt = R 1 t [1− Ft(R1t )] + PFt(R1t )S1,vt+1R1t + (1− P )Ft(R1t )S1,nt+1R1t , (A.7) pi1,nt = PFt(R 2 t )S 1,v t+1R 2 t + (1− P )Ft(R2t )S1,nt+1R2t , (A.8) pi2,vt = R 2 t [1− Ft(R2t )] + (1− P )Ft(R2t )S2,vt+1R2t + PFt(R2t )S2,nt+1R2t , and (A.9) pi2,nt = (1− P )Ft(R1t )S2,vt+1R1t + PFt(R1t )S2,nt+1R1t . (A.10) To find the prices that are set in period t, we set the derivative of pi1,vt (resp., pi 2,v t ) with respect to R1t (resp., R 2 t ) to zero and solve for R 1 t (resp., R 2 t ). 4 Recalling that Ft(Rit) = Rit Rt−1 , i = 1, 2, we obtain that R1t = Rt−1 2(1− PS1,vt+1 − (1− P )S1,nt+1) , and (A.11) R2t = Rt−1 2(1− (1− P )S2,vt+1 − PS2,nt+1) . (A.12) Substituting (A.11) (resp. (A.12)) back into (A.7) and (A.8) (resp., (A.9) and (A.10)) and simpli- fying we obtain pi1,vt = S 1,v t Rt−1, pi 1,n t = S 1,n t Rt−1, pi 2,v t = S 2,v t Rt−1, and pi 2,n t = S 2,n t Rt−1, where S1,vt = 1 4(1− PS1,vt+1 − (1− P )S1,nt+1) , (A.13) 4pi1,vt is concave in R 1 t when PS 1,v t+1 + (1− P )S1,nt+1 < 1, which holds by Proposition A.2 and Lemma A.7. 128 S1,nt = PS1,vt+1 + (1− P )S1,nt+1 4(1− (1− P )S2,vt+1 − PS2,nt+1)2 , (A.14) S2,vt = 1 4(1− (1− P )S2,vt+1 − PS2,nt+1) , and (A.15) S2,nt = (1− P )S2,vt+1 + PS2,nt+1 4(1− PS1,vt+1 − (1− P )S1,nt+1)2 , (A.16) as required. Derivation of (3.3) and (3.4). Using (A.11) and (A.12), the price ratios, {Bit}, i = 1, 2, are defined as: B1t ≡ R1t Rt−1 = 1 2(1− PS1,vt+1 − (1− P )S1,nt+1) and (A.17) B2t ≡ R2t Rt−1 = 1 2(1− (1− P )S2,vt+1 − PS2,nt+1) . (A.18) From (A.17) and (A.13) (resp., (A.18) and (A.15)) we observe that S1,vt = 1 2B 1 t (resp., S 2,v t = 1 2B 2 t ). From (A.17) (resp., (A.18)) we can isolate S1,nt+1 (resp., S 2,n t+1), to obtain: S1,nt+1 = 2B1t (1− PS1,vt+1)− 1 2B1t (1− P ) , (A.19) S2,nt+1 = 2B2t (1− (1− P )S2,vt+1)− 1 2B2t P . (A.20) Combining (A.14) and (A.19) (resp., (A.16) and (A.20)) and solving for B1t (resp., B 2 t ), results with the following recursive expressions for B1t and B 2 t : B1t = B1t+1 B1t+1(2− PB1t+1)− (B2t+1)2(1− P )(2B1t+1 − 1) , B2t = B2t+1 B2t+1(2− (1− P )B2t+1)− (B1t+1)2P (2B2t+1 − 1) . Proof of Proposition 3.2. The proof relies on the Monotone Convergence Principle and it requires the following two lemmas: Lemma A.7 The sequences of the price ratios, {B1t } and {B2t }, are bounded from above by one and from below by one half. Proof. By induction. It can be verified from Table 3.1 that the lemma holds for the last two periods. Assume the lemma holds for period t + 1. To complete the proof, we show that the bounds also holds for period t: 12 ≤ B1t = B1t+1 B1t+1(2−PB1t+1)−(B2t+1)2(1−P )(2B1t+1−1) ≤ 1 and 12 ≤ B2t = 129 B2t+1 P (B1t+1) 2−2B2t+1(P (B1t+1)2−1)−(B2t+1)2(1−P ) ≤ 1. One can verify that ∂B1t ∂B1t+1 ≥ 0, ∂B1t ∂B2t+1 ≥ 0, ∂B2t ∂B1t+1 ≥ 0, and ∂B 2 t ∂B2t+1 ≥ 0. Substituting the lower and upper bounds, which are assumed to hold for B1t+1 and B2t+1, we obtain: B 1 t ≥ 1 2 1 2 (2−P 1 2 )−( 1 2 )2(1−P )(2· 1 2 −1) = 2 4−P ≥ 12 , B2t ≥ 1 2 P ( 1 2 )2−2· 1 2 (P ( 1 2 )2−1)−( 1 2 )2(1−P ) = 2 3+P ≥ 12 for P ∈ [0, 1], and B1t ≤ 11(2−P )−(1)2(1−P )(2−1) ≤ 1 , B2t ≤ 1P (1)2−2(P (1)2−1)−(1)2(1−P ) ≤ 1, completing the proof. Lemma A.8 The sequences of the price ratios, {B1t } and {B2t }, are (backwards) increasing. Proof. To show that the sequence {B1t } is (backwards) increasing we use induction by verifying that the ratio between each two consecutive elements of the sequence is larger than one. The last ratio is: B1T−1 B1T = 2 4−P 1 2 = 44−P ≥ 1 for P ∈ [0, 1], and the ratio between two successive elements in {B1t } is: B 1 t B1t+1 = B1t+1(B 1 t+2(2−PB1t+2)−(B2t+2)2(1−P )(2B1t+2−1)) B1t+2(B 1 t+1(2−PB1t+1)−(B2t+1)2(1−P )(2B1t+1−1)) . It follows from Lemma A.7 that the denominator is strictly positive. Thus, B 1 t B1t+1 ≥ 1 holds if and only if PB1t+1B1t+2(B1t+1 − B1t+2) + (1 − P )B1t+1(B2t+2)2(1 − 2B1t+2) + (1 − P )B1t+2(B2t+1)2(2B1t+1 − 1) ≥ 0. From the inductive step, the first term in the last inequality, PB1t+1B 1 t+2(B 1 t+1 − B1t+2), is positive. The sum of the other two terms, (1 − P )B1t+1(B2t+2)2(1 − 2B1t+2) + (1 − P )B1t+2(B2t+1)2(2B1t+1 − 1) ≥ 0 if and only if( B2t+2 B2t+1 )2B1t+1 B1t+2 1−2B1t+2 1−2B1t+1 ≤ 1. However, from the inductive step, B 2 t+2 B2t+1 ≤ 1 and B 1 t+1 B1t+2 1−2B1t+2 1−2B1t+1 ≤ 1. Similarly, one can show that the sequence {B2t } is (backwards) increasing. Thus, based on Lemma A.7 and Lemma A.8, the proof of Proposition 3.2 is obtained. Finally, we note that the convergence values of the price ratios are found by suppressing period subscripts and solving (3.3) and (3.4) simultaneously. Explicitly, B1, which is real and satisfies 12 ≤ B1 ≤ 1, solves 2P 2(B1)6 − 3P (B1)5 − 3P (B1)4 + (2P + 4)(B1)3 − 3B1 + 1 = 0, and B2 is obtained by replacing B1 by B2 and P by 1− P in the last equation. Proof of Proposition 3.1. By induction. The base step (period T ) can be easily verified. As- sume B1t+1 ≥ B2t+1 when P > 12 . Using (3.3) and (3.4), B1t ≥ B2t iff (suppressing the subscripts t+1) B1((B2)2(1−P )+(B1)2P (2B2−1)) ≤ B2((B1)2P+(B2)2(1−P )(2B1−1)). Let B1 = αB2 and note that 1 ≤ α ≤ 1 B2 . Thus, B1t ≥ B2t iff (B2)3 ( B2(2α(α2P − (1− P )))− (1 + α)(α2P − (1− P ))) ≤ 0. Since α2P ≥ P > 1− P , α2P − (1− P ) > 0. That is, B1t ≥ B2t iff 2αB2 ≤ 1 + α, which clearly holds as 12 ≤ B2 ≤ 1 and 1 ≤ α ≤ 1B2 . Proof of Theorem 3.1. For every t, if 0 < P < 1, Bt = RtRt−1 ≤ max(B11(P ), B21(P )) ≤ max(B1(P ), B2(P )) < 1. Thus, since R0 = 1, Rt(P ) ≤ (max(B1(P ), B2(P )))t−1. For example, if 130 P = 0.2, Bt ≤ max(0.552, 0.722) = 0.722 and Rt < 0.722t−1 Proposition A.2 In a structural duopoly, the normalized sales-to-go sequences, {Si,jt }, i = 1, 2, j = v, n, are monotone convergent sequences as T goes to infinity. Proof. Since S1,vt = 1 2B 1 t and S 1,n t = 2B1t−1(1−P 12B1t )−1 2B1t−1(1−P ) , the convergence of {S1,vt } and {S1,nt } follows immediately from the convergence of {B1t }. Similarly, the convergence of {S2,vt } and {S2,nt } follows from the convergence of {B2t }. Behavioral duopoly with a single consumer: Derivation of B. B, the convergence value of {Bt}, is derived by omitting period subscripts from (3.5). That is, B is the solution of the equation (B − 1)(B3(2P − 1) − B2 − B + 1) = 0 which satisfies B ∈ (0, 1) for P ∈ (0, 1). In order to solve B3(2P − 1)−B2 −B + 1 = 0 we have to consider the polynomial discriminant. Let D denote the polynomial discriminant, D = 27(2P−1) 2−22(2P−1)−5 108(2P−1)4 . When 0 ≤ P < 1127 , D is positive and only one of the roots is real (the other two are complex numbers), so we solve the cubic equation simply by using the cubic formula, which results with a single real rootB = 13 M 2P−1+ 2 3 3P−1 (2P−1)M+ 1 3(2P−1) , where M = ( 63P −17−54P 2+6√3√27P 2 − 38P + 11P −3√3√27P 2 − 38P + 11)1/3.5 For 1127 ≤ P ≤ 1, P 6= 12 , all three roots are real, but only one of them is within the bounds (the other roots are either larger than one or negative). These three real solutions are: Bi = 2 √ 3(2P−1)+1 9(2P−1)2 cos (Θ+2(i−1)pi 3 ) + 1 3(2P−1) , i = 1, 2, 3, where Θ = cos −1 ( −27(2P−1)2+9(2P−1)−2 2 √ (3(2P−1)−1)3 ) . When P < 12 only B1 ∈ [0, 1], and when 12 < P < 1 only B3 ∈ [0, 1]. When P = 12 the cubic equation collapses to a polynomial of degree two, B2 + B − 1 = 0, for which the root that lies within the range [0, 1] is 12( √ 5 − 1). To conclude: B ≡ B(P ) =  1 3 M 2P−1 + 2 3 3P−1 (2P−1)M + 1 3(2P−1) , if 0 ≤ P < 1127 , 2 √ 3(2P−1)+1 9(2P−1)2 cos ( Θ 3 ) + 1 3(2P−1) , if 11 27 ≤ P < 12 , 1 2( √ 5− 1), if P = 12 , 2 √ 3(2P−1)+1 9(2P−1)2 cos ( Θ+4pi 3 ) + 1 3(2P−1) , if 1 2 < P ≤ 1, where M = ( 63P − 17− 54P 2 + 6√3√27P 2 − 38P + 11P − 3√3√27P 2 − 38P + 11 )1/3 and Θ = cos−1 (−27(2P−1)2+9(2P−1)−2 2 √ (3(2P−1)−1)3 ) . 5The other two complex roots are − 1 6 M 2P−1 − 13 3P−1(2P−1)M + 13(2P−1) ± 12I √ 3 ` M 3(2P−1) − 23 3P−1(2P−1)M ´ . 131 Proof of Proposition 3.3. Follows by the same arguments as those used in the proof of Proposition 3.2. Proof of Theorem 3.2. Follows immediately from (3.6). Proof of Theorem 3.3. Follows by the same arguments as those in the proof of Theorem 3.1. Proof of Proposition 3.4. With N consumers in the market and symmetric retailers, the price ratio can be expressed recursively by Bt = 12−(Bt+1)2 , which is obtained by the same arguments as in the derivation of (3.3) and (3.4). In the single consumer case, when P = 12 , it can be verified that B1t = B 2 t ≡ Bt = 12−(Bt+1)2 , as required. Proof of Proposition 3.5. For T = 2 or 3 it can be easily verified that retailers optimally set prices such that Rit, R j t ≤ Rit−1, Rjt−1, ∀t. Structural Oligopoly with symmetric retailers - derivation of the price ratio. Retailer i’s expected profit-to-go in period t is given by piit = R i t ( P 2NRit−1(1− Rit Rit−1 ) + P (1− P )NRjt−1(1− Rit Rjt−1 ) ) + piit+1, where Rjt is the “common” price set by all other symmetric retailers. By induction it can be shown that piit = S i t(R i t−1)2 + S ij t R i t−1R j t−1 + S j t (R j t−1) 2, where Sit , S ij t , and S j t are scalars (and have the flavor of normalized sales-to-go considered previously). As the model is solved backwards we find the following recursive expressions: Rit = P 2NRit−1 + (1− P )PNRjt−1 2PN − 2Sit+1 − Sijt = BitR i t−1 +B j tR j t−1 = (B i t +B j t )Rt−1 = BtRt−1, Bit = PBt, B j t = (1− P )Bt, St = Sit + S ij t + S j t , St = N2P 2(PN − Sit + Sjt ) 2PN − 2Sit+1 − Sijt = B2t PN = Bt(St+1Bt + PN(1−Bt)), Sit = P 2St, S ij t = 2P (1− P )St, Sjt = (1− P )2St, and after several manipulations we find that Bt = 12+2B2t+1P−2B2t+1+Bt+1−2PBt+1 . We note that this price ratio coincides with the price ratio in the single-consumer case. Proposition A.3 (i) Symmetric retailers (i.e., P = 0.5) in a structural duopoly with a single consumer, whose valuation’s density is f(V ) = (q + 1)V q, q > −1, 0 ≤ V ≤ 1, set identical prices; 132 (ii) the sequences of price ratios, {B1t } and {B2t }, are monotone convergent sequences as the number of periods, T , goes to infinity, while t goes to 1. Proof. The proof requires the following lemma, which can be proved by induction (see Lemma A.6). Lemma A.9 Under structural duopoly, when the consumer’s valuation has density f(V ) = (q + 1)V q, q > −1, 0 ≤ V ≤ 1, the profit-to-go of each of the retailers in period t is an affine function of the price that is set in the previous period. That is, pii,jt = S i,j t Rt−1, i = 1, 2, j = v, n, where S i,j t are scalars. Moreover, the values of Si,jt can be expressed recursively as follows: S1,vt = (q + 1)/(q + 2) [(q + 2)(1− PS1,vt+1 − (1− P )S1,nt+1)]1/(q+1) , (A.21) S1,nt = PS1,vt+1 + (1− P )S1,nt+1 [(q + 2)(1− (1− P )S2,vt+1 − PS2,nt+1)](q+2)/(q+1) , (A.22) S2,vt = (q + 1)/(q + 2) [(q + 2)(1− (1− P )S2,vt+1 − PS2,nt+1)]1/(q+1) , and (A.23) S2,nt = (1− P )S2,vt+1 + PS2,nt+1 [(q + 2)(1− PS1,vt+1 − (1− P )S1,nt+1)](q+2)/(q+1) , (A.24) with Si,jT+1 = 0, i = 1, 2, j = v, n. As the model is solved backwards, we find that (A.17) and (A.18) hold. From (A.22) and (A.24) we have S1,nt+1 = (q + 2)(B1t ) q+1(1− PS1,vt+1)− 1 (q + 2)(1− P )(B1t )q+1 , (A.25) S2,nt+1 = (q + 2)(B2t ) q+1(1− (1− P )S2,vt+1)− 1 (q + 2)P (B2t )q+1 . (A.26) Using (A.25) and (A.26), and observing that S1,vt = q+1 q+2B 1 t and S 2,v t = q+1 q+2B 2 t , after several manipulations we can express the price ratios recursively as follows: B1t = B1t+1 [(B1t+1)q+1(q + 2− P (q + 1)B1t+1)− (1− P )(B2t+1)q+2((q + 2)(B1t+1)q+1 − 1)]1/(q+1) , (A.27) B2t = B2t+1 [(B2t+1)q+1(q + 2− (1− P )(q + 1)B2t+1)− P (B1t+1)q+2((q + 2)(B2t+1)q+1 − 1)]1/(q+1) . (A.28) Part (i) of the proposition, that Rit = R j t , follows from the recursive expressions of B 1 t and B 2 t , while Part (ii) follows from the next two lemmas. 133 Lemma A.10 Under structural duopoly, when the consumer’s valuation has density f(V ) = (q + 1)V q, q > −1, 0 ≤ V ≤ 1, the sequences of the price ratios, {B1t } and {B2t }, are bounded from above by one and from below by 1 (q+2)1/(q+2) . Proof. Assume first that the expressions in [ ] in the denominators of Bit and B j t (displays (A.27) and (A.28)) are positive, and later it can be verified to be true. Now, it can be shown that the following derivatives are positive: ∂B 1 t ∂B1t+1 , ∂B 1 t ∂B2t+1 , ∂B 2 t ∂B1t+1 , and ∂B 2 t ∂B2t+1 . By setting Bit+1 = Bjt+1 = 1 (q+2)1/(q+2) , we find the lower bound Bit = 1 (q+2)1/(q+2)[ 1 q+2 (q+2−P (q+1) 1 (q+2)1/(q+1) )]1/(q+1) = 1 (q+2)1/(q+2) 1 [1−P (q+1) 1 (q+2)(q+2)/(q+1) ]1/(q+1) ≥ 1 (q+2)1/(q+2) , where the last inequality follows since the expression in [ ] of the denominator of the LHS is positive, as (q+2) q+2 (q+1)q+1 ≥ 1 ≥ P , and is bounded from above by 1. Setting Bit+1 = B j t+1 = 1, we find the upper bound B i t = 1 holds. Similarly, the proof can be obtained for Bjt . Lemma A.11 Under structural duopoly, when the consumer’s valuation has density f(V ) = (q + 1)V q, q > −1, 0 ≤ V ≤ 1, the sequences of the price ratios, {B1t } and {B2t }, are backwards increasing in t, as the number of periods in the selling horizon, T , goes to infinity. Proof. We prove that the sequence {B1t } is (backwards) increasing by induction. The base step can be easily verified to hold. Assume B1t+1 B1t+2 ≥ 1 and we will show B1t B1t+1 ≥ 1. Using display (A.27), B 1 t B1t+1 ≥ 1 iff P (q + 1)(B1t+1B1t+2)q+1(B1t+1 − B1t+2) + (1 − P )(B1t+1)q+1(B2t+2)q+2(1 − (q + 2)(B1t+2) q+1) + (1−P )(B1t+2)q+1(B2t+1)q+2((q+2)(B1t+1)q+1− 1) ≥ 0. From the inductive step, the first term in the last inequality, P (q + 1)(B1t+1B 1 t+2) q+1(B1t+1 − B1t+2), is positive. The sum of the other two terms is positive if and only if ( B2t+2 B2t+1 )q+2(B1t+1 B1t+2 )q+1 1−(q+2)(B1t+2)q+1 1−(q+2)(B1t+1)q+1 ≤ 1. However, from the inductive step, B2t+2 B2t+1 ≤ 1 and it can be shown that ( B1t+1 B1t+2 )q+1 1−(q+2)(B1t+2)q+1 1−(q+2)(B1t+1)q+1 ≤ 1. A similar proof follows for {B2t }. By Lemmas A.10 and A.11, relying on the Monotone Convergence Principle, the proof of Part (ii) of Proposition A.3 has been obtained. 134 A.4 N Similar Consumers with Demand Updating In §3.3 it is assumed that when the retailers face N similar consumers, whose valuations are independently drawn from a uniform distribution over [0, 1], they set prices along the selling horizon using demand expectations. Retailers can also take advantage of actual realizations and update their demand knowledge after every period. That is, after observing the sales levels in each period, retailers infer the number of remaining active consumers, and update their prices accordingly. Nevertheless, updating demand knowledge requires retailers to plan ahead their prices for different sales scenarios that might occur, and planning such pricing schemes can be very demanding. With small markets and short horizons, this strategy can be readily implemented. In this section we develop the pricing scheme for a two-period horizon, when retailers update their demand knowledge, and we demonstrate the difficulty involved with these pricing schemes. In §A.4.1 (resp., §A.4.2) we present the structural (resp., behavioral) duopoly case, and §A.4.3 briefly examines the case of limited inventories. A.4.1 Two-Period Structural Duopoly with N Similar Consumers In a structural duopoly we set Pi = P and Pj = 1 − P . Let N i (resp., N j), with N i + N j = N , denote the number of consumers who visit Retailer i (resp., j) in the first period, and let ni ≤ N i (resp., nj ≤ N j) denote the number of consumers who purchase the good from Retailer i (resp., j) in the first period. Assume Retailer i (resp., j) doesn’t observe nj (resp., ni), and he believes that nj = (1 − Rj1)N j (resp., ni = (1 − Ri1)N i). Decomposing each consumer’s behavior, Retailer i’s profit in the second period is given by pii2 = PR i 2 ( (N i − ni) ( 1− R i 2 Ri1 ) +N jRj1 ( 1− R i 2 Rj1 )) , where N i − ni is the number of consumes who have not purchased the good from Retailer i, and N jRj1 is the expected number of consumes who have not purchased the good from Retailer j. The optimal profit-maximizing price is Ri2 = (N i−ni+NjRj1)Ri1 2(N i−ni+NjRi1) with pii2 = P (N i−ni+NjRj1)2Ri1 4(N i−ni+NjRi1) .6 Let PN N i = ( N N i ) PN i (1− P )N−N i denote the probability that out of the N consumers, N i visit Retailer i in the first period, and let QN i ni = ( N i ni ) (Ri1) N i−ni(1−Ri1)n i denote the probability that out 6This excludes the trivial instances in which all demand is satisfied in the first period by the same retailer. That is, N i = ni = N or N j = nj = N , as the season terminates after the first period. 135 of these N i consumers, ni purchase the good in the first period, as their valuations exceed Ri1. Re- tailer i’s profit expression in the first period is given by pii1 = ∑N N i=0 P N N i (∑N i ni=0Q N i ni ( niRi1 + pi i 2 )) . We find that: Proposition A.4 In a structural duopoly, when T = 2 and N = 2, there exists a unique sub-game perfect Nash equilibrium. Proof. Let Πi2[N i, ni] denote Retailer i’s expected profit-to-go in the second period when he is visited by N i consumers in the first period and n1 of them purchase the good. Retailer i’s profit is then: pii1 = P 2 (( 2 0 ) (Ri1) 2(0 + Πi2[2, 0]) + ( 2 1 ) Ri1(1−Ri1)(Ri1 +Πi2[2, 1]) + ( 2 2 ) (1−Ri1)2(2Ri1 + 0) ) + 2P (1 − P ) ((10)Ri1(0 + Πi2[1, 0]) + (11)(1−Ri1)(Ri1 +Πi2[1, 1])) + (1 − P )2Πi2[0, 0], which, after sub- stituting the expressions and slightly simplifying, becomes pii1 = P 2(12(R i 1) 3P + 2Ri1(1− Ri1)(Ri1 + 1 4R i 1P ) + 2(1−Ri1)2Ri1) + 2P (1−P )( (1+R j 1) 2P (Ri1) 2 4(1+Ri1) + (1−Ri1)(Ri1+ 14(Rj1)2P )) + 12(1−P )2(Rj1)2P . Given Retailer j’s first period price, Rj1, Retailer i sets his first period price, R i 1, so as to max- imize his expected profit, pii1. Using the implicit relation ρ i(R1) ≡ ∂pi i 1 ∂Ri1 = 0, where R1 = (Ri1, R j 1), the reaction function of Retailer i can be obtained and written as Γi = {(Ri1, Rj1) : ρi(R1) = 0}. The second derivative of pii1, given R j 1, defined on [0, 1] × [0, 1], is ∂2pii1 ∂(Ri1) 2 = P(−(4−P 2)Ri1((Ri1)2+3Ri1+3)+P (1−P )Rj1(Rj1+2)−4+P) (1+Ri1) 3 < 0, since (1 + Ri1) 3 > 0, −(4 − P 2)Ri1((Ri1)2 + 3Ri1 + 3) < 0, and P (1 − P )Rj1(Rj1 + 2) − 4 + P < 0.7 Thus, since the expected profit is concave in Ri1, it is sufficient for Retailer i to solve the implicit relation ρi(R1) to obtain the optimal price, which is uniquely defined due to this concavity. Further, since ∂pii1 ∂Ri1 ∣∣ Ri1= 1 2 = 89P 2((1−P )Rj1(10−2Rj1)+(4P +5)) ≥ 0, retailers never post an initial price below 12 . Next, we prove that Retailer i’s reaction function, Γi, is monotonically increasing in Rj1. Using implicit differentiation, ( dRi1 dRj1 ) ρi = P (1−P )[R j 1(1+R j 1)−Ri1((Ri1)2+2Ri1+2)] −(4−P 2)Ri1((Ri1)2+3Ri1+3)+P (1−P )Rj1(Rj1+2)−4+P is positive when- ever Ri1 > √ 1 +Rj1 − 1, which holds true when Rit ≥ 12 and Rjt ≥ 12 . Also, ( dRi1 dRj1 ) ρi ∣∣∣ ( 1 2 , 1 2 ) = P (1−P ) 12−2P−P 2 > 0, for P ∈ (0, 1). Additionally, ρi(Ri)|Rj1= 12 (resp., ρ i(Ri)| Rj1=1 ), we find that the corresponding optimal Ri1 ≥ 12 (resp., Ri1 ≤ 23) for 0 < P < 1. That is Ri∗1 ∈ [12 , 23 ]. Using similar arguments, we show that Retailer j’s reaction function is also monotonically increasing and concave. Since both Ri∗1 and R j∗ 1 are ∈ [12 , 23 ], and the two reaction functions are increasing, there is a unique intersection point. This point is the unique Nash equilibrium. 7Specifically, P (1−P )Rj1(Rj1+2)−4+P is increasing in Rj1, and evaluated at Rj1 = 1 it equals −3P 2+4P −4 < 0. 136 0.5 0.67 0 0.2 0.4 0.6 0.8 1 P, Return Probability N=1 N=2 N=4 N=10 Exp Exp N=1 0.571  Figure A.1: Structural duopoly: Retailer i’s initial price for different N values, T = 2 When retailers are symmetric (i.e., P = 12) and N = 2 or 3, the problem can be analytically solved, resulting with a solution which coincides with the single consumer prices (i.e., Ri1 = R j 1 = 4 7 ≈ 0.571). However, a general closed-form solution for this problem for any N and P is hard to obtain, as the term QN i ni leads to expressions with polynomials of high degrees. Resorting to numerical solutions, Figure A.1 illustrates the initial price set by Retailer i as a function of P for different N values. Interestingly, as can be observed from that figure, with similar consumers, unless P = 0, 12 , or 1, Retailer i’s initial price does not coincide with his corresponding initial price from the single consumer case. Relying on Figure A.1 we make the following observation. Observation A.1 In a structural duopoly with N similar consumers, when T = 2 the dominant (resp., weak) retailer’s initial price decreases (resp., increases) in N . This difference in the initial price is, however, very small. The intuition for this changing behavior of prices in N is as follows. When there is a single consumer in the market, he visits either the dominant or the weak retailer in the first period, and the price retailers set in the second period is half the price observed by the consumer in the first period. As N increases, both retailers are more likely to encounter some consumers in the first period, and the second price they set becomes some weighted average of prices observed by consumers in the first period. From the dominant retailer’s perspective, since there are consumers 137 who observe the weak retailer’s lower price in the first period, the expected second period price is lower than had they all visited the dominant retailer in the first period. Due to this decrease in his second period price, the dominant retailer decreases his initial price accordingly. Similarly, for the same (counter) argument, the weak retailer increases his first period price. In expectation, more consumers visit the dominant retailer in the first period and observe a higher price. Consequently, the majority of consumers who proceed to the second period, have visited the dominant retailer in the first period. Setting the second period price, the dominant retailer gives the majority of the weight to the consumers who have observed his own price. Therefore, the change in the pricing is relatively small. A.4.2 Two-Period Behavioral Duopoly with N Similar Consumers Using the same assumptions as for the structural duopoly we have that, by decomposing for each consumer, Retailer i’s profit in period 2 is given by pii2 = R i 2 ( Pi(N i − ni) ( 1− R i 2 Ri1 ) + (1− Pj)N jRj1 ( 1− R i 2 Rj1 )) . Solving for the final period’s price, we find that Ri2 = (Pi(N i−ni)+(1−Pj)NjRj1)Ri1 2(Pi(N i−ni)+(1−Pj)NjRi1) , with pii2 = (Pi(N i−ni)+(1−Pj)NjRj1)2Ri1 4(Pi(N i−ni)+(1−Pj)NjRi1) . Assuming that retailers are equally likely to be visited by con- sumers in the first period, Retailer i’s profit expression is pii1 = ∑N N i=0..N P N N i (∑N i ni=0..N i Q N i ni ( niRi1 + pi i 2 )) , where PN N i = 1 2N ( N N i ) and QN i ni is as before in structural duopoly. When the retailers are symmetric (i.e., Pi = Pj ≡ P ), this problem can be solved analytically for N = 2, resulting with Ri1 = R j 1 = 2 4−P , which is the same solution we have obtained for the single consumer case.8 With N > 2 it is harder to obtain an analytical solution, and resorting to numerical solutions, we find that in general Ri1 = R j 1 = 2 4−P . 9 That is, it appears that when retailers are symmetric, for T = 2, they set the same initial prices as in the single consumer case. When retailers are not symmetric, the results are similar in fashion to those obtained in a structural duopoly. Specifically, we can prove the following result. 8When N = 2 this solution does not require the assumption that retailers are equally likely to be visited by consumers in the first period. 9For some instances, it can be further shown that Ri1 = R j 1 = 2 4−P is a local optima. 138 Proposition A.5 In a behavioral duopoly, when T = 2 and N = 2, there exists a unique sub-game perfect Nash equilibrium in pricing. In a behavioral duopoly with non-symmetric retailers, prices depend on Pi and Pj . Moreover, as in a structural duopoly, we also find that prices are changing in N . Essentially, as N increases, the initial price set by the more retentive retailer decreases. A plot similar to Figure A.1 can be drawn, with a main difference: the cross-over point is not 12 anymore but the point where retailers are symmetric. Thus, we repeat the observation from the structural competition. Observation A.2 In a behavioral duopoly with N similar consumer, when T = 2, the more (resp., less) retentive retailer’s initial price decreases (resp., increases) in N . A.4.3 Two-Period Structural Duopoly with Limited Inventory Assume Retailer i, who has a stock of Ki units, encounters N i consumers in the first period, and that ni of these consumers have valuations above the posted price, Ri1. If n i ≤ Ki then all ni consumers purchase the good. Otherwise, there are Ki − ni unsatisfied consumers with valuations above Ri1. Retailer i cannot observe n j , and expects nj to equal N j(1 − Rj1). Assume second period prices are lower than first period prices (which can later be verified to hold), and to avoid complexity, assume that even if a stock-out occurs, consumers may still return to the same store, then Retailer i’s second period profit is pii2 =  0, if Ki ≤ ni, (N i−ni+NjRj1)2PRi1 4(N i−ni+NjRi1) , if Ki > ni and Kj > N j(1−Rj1), (N i−ni+Nj−Kj)2PRi1 4(N i−ni+NjRi1) , otherwise, and his first period profit is ∑N N i=0 1 2N ( N N i )(∑N i ni=0 ( N i ni ) (Ri1) N i−ni(1−Ri1)n i (min(Ki, ni)Ri1+pi i 2) ) . To simplify the demonstration, assume retailers are symmetric, such thatKi = Kj ≡ K and P = 12 . Solving the model (numerically), in Figure A.2 we depict retailers’ first period prices as a function of the number of consumers in the market for fixed inventory levels. It is apparent that prices increase in N , the total number of consumers in the market, and decrease in inventory levels. In other words, prices decrease in the share of market retailers can satisfy. As the market share retailers can satisfy decrease, the competitive pressure diminishes, and prices approach the monopolistic level (refer to 139 GGM for the corresponding monopolistic setting under limited inventory). 0.5 0.6 0.7 0.8 0.9 1 1 3 5 7 9 11 13 15 N, Number of Consumers K=2 K=3 K=4 K=5 K=1  Figure A.2: Structural symmetric duopoly withN similar consumers: Initial price when inventory is limited, T = 2 140 A.5 Proofs of Lemmas, Propositions, and Theorems in Chapter 4 Proof of Proposition 4.1. Under PP (i): V m1 = Rm1 −δRm2 1−δ < 1. Solving backwards, in the second period Rm2 = 1 2((1−α)Rm1 +αV m1 ), and since V m1 = Rm1 −δRm2 1−δ , resolving for R m 2 , the optimal second period price is Rm2 = (1−δ+αδ)Rm1 2−2δ+αδ . Solving for the first period R m 1 = (2−2δ+αδ)2 2(1−δ+αδ)(4−λαδ+2αδ−4δ+λδ−λ) . Under PP (ii): V m1 = Rm1 −δRm2 1−δ ≥ 1. We solve the model by setting V m1 = 1 in (4.1) and (4.2) to obtain Rm2 = 1 2((1− α)Rm1 + α) and Rm1 = 2+λα4−λ+λα .10 The conditions δ ≤ δmc , α ≥ αmc and λ ≤ λmc are obtained by solving pim1,i ≥ pim1,ii for δ, α and λ re- spectively, where δmc ≡ 2(λ−2)(−5+2λ−2λα+3α+ √ (1−α)(1−α+λα)) α2(8+4λ2−11λ)+α(31λ−28−8λ2)+4(6−5λ+λ2) , α m c ≡ −24+8λ2δ+28λ+28δ−31λδ−8λ2+ √ (16δ−16)λ3+(−88δ+80+17δ2)λ2+(−40δ2−128+144δ)λ+64+16δ2−64δ 2δ(−11λ+8+4λ2) , and λmc = −31αδ2+11α2δ2+28αδ+20δ2+16−36δ− √ δ2(1−α)(7α3δ2+24α2δ−15α2δ2−8αδ+16α−8αδ2−32δ+16+16δ2) 8(1−δ+αδ)2 . The conditions α ≥ 4−3λ 8−11λ+4λ2 , δ ≤ 2−λ3−λ , and λ ≤ 2−3δ1−δ are obtained by solving δmc ≥ 1, αmc ≤ 0, and λmc ≤ 0, respectively. Proof of Proposition 4.2. We first construct the proof for δ. The initial price is piecewise non-increasing in δ since ∂Rm1 ∂δ =  0, if δ > δmc , (2−2δ+αδ)α(−αδ+λαδ+2δ−λδ+λ−2) (1−δ+αδ)2(λαδ−2αδ+4δ−λδ+λ−4)2 , otherwise, where (2−2δ+αδ)α(−αδ+λαδ+2δ−λδ+λ−2) (1−δ+αδ)2(λαδ−2αδ+4δ−λδ+λ−4)2 is negative when −αδ+λαδ+2δ−λδ+λ−2 < 0, which holds when δ < 2−λ2−α+λα−λ . However, since δ m c < 2−λ 2−α+λα−λ , we have that (2−2δ+αδ)α(−αδ+λαδ+2δ−λδ+λ−2) (1−δ+αδ)2(λαδ−2αδ+4δ−λδ+λ−4)2 is always negative when δ < δmc . Since ∂Rm1 ∂δ ≤ 0, when δ ≤ δmc the initial price, R1, is maximized at δ = 0, reaching a value of 24−λ which is less or equal to 2+λα 4−λ+λα . The second period price is piecewise non-decreasing in δ since ∂Rm2 ∂δ =  0, if δ > δmc , λα 2(λαδ−2αδ+4δ−λδ+λ−4)2 ≥ 0, otherwise. Therefore, when δ ≤ δmc , the second period price, Rm2 , is maximized at δ = δmc , and Rm2 ∣∣ δ=δmc ≤ 10One needs to verify that Rm1 −δRm2 1−δ ≥ 1, which holds when δ ≥ 2−λ3−α−λ+λα . PP (ii) is employed when δ > δmc and δmc ≥ 2−λ3−α−λ+λα . Therefore, Rm1 −δRm2 1−δ ≥ 1. 141 1+α 4−λ+λα . Taking the derivatives of prices with respect to α we have ∂Rm1 ∂α =  λ(2−λ) (4−λ+λα)2 ≥ 0, if α < αmc , (2−2δ+αδ)δ(1−δ)(−αδ+λαδ+2δ−λδ+λ−2) 2(1−δ+αδ)2(λαδ−2αδ+4δ−λδ+λ−4)2 ≤ 0, otherwise, and ∂Rm2 ∂α =  λ(2−λ) (4−λ+λα)2 ≥ 0, if α < αmc , δλ(1−δ) 2(λαδ−2αδ+4δ−λδ+λ−4)2 ≥ 0, otherwise. The derivatives of prices with respect to λ are ∂Rm1 ∂λ =  2(1+α) (4−λ+λα)2 > 0, if λ > λ m c , (2−2δ+αδ)2 2(λαδ−2αδ+4δ−λδ+λ−4)2 ≥, otherwise, and ∂Rm1 ∂λ =  (1+α)(1−α) (4−λ+λα)2 ≥ 0, if λ > λmc , (2−2δ+αδ)(1−δ+αδ) 2(λαδ−2αδ+4δ−λδ+λ−4)2 ≥ 0, otherwise. Finally, due to the equivalence between δmc , α m c and λ m c , the spike in prices is maintained. Proof of Proposition 4.3. The proof follows immediately from ∂pim1 ∂α =  − (λ−2)2 (4−λ+λα)2 < 0, if δ > δ m c , (2−2δ+αδ)δ(1−δ)(−αδ+λαδ+2δ−λδ+λ−2) 2(1−δ+αδ)2(λαδ−2αδ+4δ−λδ+λ−4)2 ≤ 0, otherwise, where the expression −αδ+λαδ+2δ−λδ+λ−2 is negative as shown in the proof of Proposition 4.2, ∂pim1 ∂δ =  0, if δ > δmc , (2−2δ+αδ)α(−αδ+λαδ+2δ−λδ+λ−2) 2(1−δ+αδ)2(λαδ−2αδ+4δ−λδ+λ−4)2 ≤ 0, otherwise, and from ∂pim1 ∂λ =  (1+α)2 (4−λ+λα)2 ≥ 0, if δ > δmc , (2−2δ+αδ)2 4(λαδ−2αδ+4δ−λδ+λ−4)2 > 0, otherwise, as required. Proposition A.6 When (i) δ ≤ δmc and α > 4−4δ+3λδ−3λδ(3λ−2) or (ii) δ > δmc and α > 4+λ2−2λ+(λ−2)√λ2+12λ+4 2λ(λ−3) , a larger share of the total expected profit is earned in the second period. Proof. The net expected profit in the first period is pim1 − pim2 . When δ ≤ δmc , pim1 − pim2 > pim2 ⇔ α > 4−4δ+3λδ−3λδ(3λ−2) . When δ > δmc , pim1 − pim2 > pim2 ⇔ α > 4+λ 2−2λ+(λ−2)√λ2+12λ+4 2λ(λ−3) . 142 Proof of Proposition 4.4. Assume N1 = 1 and N2 = 0. Let pid1,i and pi d 1,ii denote Retailer 1’s expected profit under PP (i) and PP (ii), respectively. The conditions δ ≤ δdc and α ≥ αdc are obtained by solving pid1,i ≥ pid1,ii for δ and α, respectively, where δdc ≡ 2 3− 2α, and α d c ≡ 3δ − 2 2δ . The conditions α ≥ 12 and δ ≤ 23 are obtained by solving δdc ≥ 1 and αdc ≤ 0, respectively. Proof of Proposition 4.5. The initial price is piecewise non-increasing in δ since ∂Rd1 ∂δ =  0, if δ > δdc , − α 4(1−δ+αδ)2 ≤ 0, otherwise, and the second period price is independent of δ. Therefore, when δ ≤ δdc , the initial price, Rd1, is maximized at δ = 0, achieving a value of 12 , the same value obtained when δ > δ d c , and as for R d 2, it is obvious that 14 ≤ 1+α4 . Taking derivatives with respect to α ∂Rd1 ∂α =  0, if δ > δdc , − δ(1−δ) 4(1−δ+αδ)2 ≤ 0, otherwise, ∂Rd2 ∂α =  1 4 , if δ > δ d c , 0, otherwise, and due to the equivalence between δdc and α d c , the spike in prices is maintained. Proof of Proposition 4.6. Noting the equivalence between the conditions δ > δdc and α < αdc , the proof follows from the following derivatives ∂(pid1 + pi d 2) ∂α =  λ(1+α)−2 8 ≤ 0, if δ > δdc , δ(δ−1) 8(1−δ+αδ)2 ≤ 0, otherwise, and ∂(pid1 + pi d 2) ∂δ =  0, if δ > δdc , − α 8(1−δ+αδ)2 ≤ 0, otherwise, and since (pid1 + pi d 2)|PP (i),δ=δdc ≤ (pid1 + pid2)|PP (ii),δ=δdc . Proposition A.7 The expected profit per consumer in the first period is larger than the expected profit per consumer in the second period, unless δ > δdc and α > 2 √ 1+2λ−2−λ λ , for which necessary conditions are δ > 0.965 and 0.464 < α < 0.5. Proof. Let N1 = 1 and N2 = 0. When δ ≤ δdc , Retailer 1’s profit always exceeds that of Retailer 2 since pid1 ≥ pid2 ⇔ δ ≤ 4−λ4−λ+α(λ−2) which is always satisfied since 4−λ2+α(2−λ) ≥ 4−λ4−λ = 1 ≥ δ. When δ > δdc , pi d 1 < pi d 2 ⇔ α > −2+λ−2 √ 1+2λ λ . Recalling that δ > δ d c ⇔ α < αdc = 3δ−22δ , we find that pid1 < pi d 2 is achievable only when δ > 2λ 5λ+4−4√1+2λ ≥ 0.965 (or, alternatively, when 143 0.464 < α < αdc ≤ 12). Proof of Proposition 4.7. (i) We have that δdc − δmc = (6λ−8)α2+(20+4λ2−16λ)α−4λ2−12+14λ−(6λ−4λα−12+α) √ (1−α)(1−α+λα) (3−2α)((−11λ+4λ2+8)α2+(31λ−28−8λ2)α+4λ2+24−20λ) . The denominator of the last term is positive, and the numerator, to be denoted as n, can be shown to be increasing in λ (specifi- cally, ∂n∂λ = (6α2−8λ−16α+14+8λα) √ (1−α)(1−α+λα)−15λα2+9λα−22α+24α2+6+6α3λ−8α3√ (1−α)(1−α+λα) and the denomina- tor thereof is positive and the numerator thereof, when equated to zero and solved for λ, has three roots: root1 ≥ 1, root2 ≤ 0 and root3 > 1 in the range, and it is positive when root2 ≤ λ ≤ root1.) Since n is increasing in λ we evaluate it at zero to verify it is still positive: n ∣∣ λ=0 = 0, which completes the argument. (ii) Since δdc ≥ δmc and V m1 =  1, if δ > δmc , (2−δ+αδ)(2−2δ+αδ) 2(1−δ+αδ)(4−λαδ−2αδ+4δ−λδ+λ) , otherwise, V d1 =  1, if δ > δdc , αδ−2 4(αδ−1) , otherwise, to show that V m1 ≥ V d1 it remains to verify that (2−δ+αδ)(2−2δ+αδ)2(1−δ+αδ)(4−λαδ−2αδ+4δ−λδ+λ) ≥ αδ−24(αδ−1) , which certainly holds (as by rearranging the last inequality, one obtain δ(1− α) ≤ 1). (iii) To show that the monopolistic retailer’s expected profit is larger than the combined profit of retailers under competition, we consider the difference between them, pim − pid =  λ2(1−α)(1+α)2 16(4−λ+λα) , if δ > δ d c , (−(1−α)2λ2+14α(α−1)λ+8α(3−2α))δ+(1−α)λ2+12λα−16α 16(4−λ+λα)(1−δ+αδ) , if δ d c ≥ δ > δmc , (1−δ+αδ)λ2 16(4+αδ(2−λ)−4δ−λ) , otherwise, (A.29) and we study the three different cases to show that they all result with a non-negative expression. Case i: when δ > δdc , it follows immediately that λ2(1−α)(1+α)2 16(4−λ+λα) ≥ 0. Case ii: when δdc ≥ δ > δmc , the denominator, 16(4 − λ + λα)(1 − δ + αδ), is positive and we need to verify that the numerator is positive as well. The numerator, denoted n, is decreasing (resp., increasing) in δ if α > 2λ 2−14λ+24−2√144−168λ+57λ2 2(16+λ2−14λ) (resp., α < 2λ2−14λ+24−2√144−168λ+57λ2 2(16+λ2−14λ) ). When n is decreasing in δ, it is minimized at δdc , and thus n ∣∣ δ=δdc = λ(8α+4α 2+λ−λα) 3−2α ≥ 0. When n is increasing in δ, it is minimized at δmc , and we have n ∣∣ δ=δmc = k 4(1−α)2λ2+(31α−11α2−20)λ+8α2−28α+24 , where the denominator is positive (since it is decreasing in λ and, therefore, greater than 8(1 − α)+α(3+α) (evaluated at λ = 1), which is positive), and the numerator, k = (20α2−11α3−7α− 144 2)λ3 + (4− 116α2 + 68α3 + 44α)λ2 + (240α2 − 104α− 120α3)λ+ 96α+ 64α3 − 160α2 + ((−2α2 − 2+4α)λ3+(4− 36α+32α2)λ2+(−88α2+104α)λ− 96α+64α2)√(1− α)(1− α+ λα), is positive when 0 ≤ λ < 2 ≤ 20−11α− √ 16+8α−7α2 8(1−α) . Case iii: when δ ≤ δmc , (1−δ+αδ)λ 2 16(4+αδ(2−λ)−4δ−λ) is positive if 4 + αδ(2 − λ) − 4δ − λ ≥ 0 ⇔ δ ≤ λ−4 2α−λα+λ−4 . The last inequality is always satisfied, since δ ≤ 1 ≤ λ−42α−λα+λ−4 , which completes the proof. Proposition A.8 The absolute profit loss due to competition, pim − pid, is increasing in δ up to δdc , with ∂(pim−pid) ∂δ ∣∣ δdc≥δ>δmc ≥ ∂(pim−pid) ∂δ ∣∣ δmc ≥δ. Thereafter, it spikes down and remains constant in δ. Thus, the absolute profit loss is minimized when δ = 0 (or α = 0) at which point it equals 148 , which represents a profit loss of 6.25%, it is maximized at δ = min{δdc , 1}, and it may be as large as 0.098, when δ = 1, α = 0.5 and λ = 1, which represents a loss of up to about 34.3% of profit due to competition. Proof. Considering the derivative of the absolute profit loss with respect to δ, we have ∂(pim − pid) ∂δ =  0, if δ > δdc , α 8(1−δ+αδ)2 , if δ d c ≥ δ > δmc , αλ2 8(λ−4+(4−2α+λα−λ)δ)2 , otherwise. ∂(pim−pid) ∂δ ∣∣ δdc≥δ>δmc ≥ ∂(pim−pid) ∂δ ∣∣ δmc ≥δ ⇔ α 8(1−δ+αδ)2 > αλ2 8(λ−4+(4−2α+λα−λ)δ)2 which holds when δ < 2−λ 2−α+λα−λ . Since 2−λ 2−α+λα−λ ≥ 22−α ≥ 1, the derivative when δdc ≥ δ > δmc is always greater than the derivative when δ ≤ δmc . At δ = δmc , the monopoly profit is continuous (so the profit loss is continuous as well) while at δ = δdc the duooly profit spikes up (so the profit loss spikes down). Thus, the profit loss is maximized at δ = min{δdc , 1}. Formally, using the expressions from (A.29), we have (−(1−α)2λ2+14α(α−1)λ+8α(3−2α))δ+(1−α)λ2+12λα−16α 16(4−λ+λα)(1−δ+αδ) ∣∣ δ=δdc = λ(4α 2+8α−λα+λ) 16(4−λ+λα) ≥ λ 2(1−α)(1+α)2 16(4−λ+λα) , as re- arranging the last inequality results with λ ≤ 41−α , which certainly holds. We conclude that the profit loss reaches its largest value when δdc = 2 3−2α = 1, which occurs when α = 0.5. The resulting profit loss is also increasing in λ. Thus, the profit loss, evaluated at α = 0.5, δ = 1, and λ = 1, is bounded from above by 0.098, which represents a loss of about 34.3%, when measured percentage-wise. 145 Proof of Proposition 4.8. (i) The difference in percentage profit loss is given by pid,pl − pim,pl =  (2−α−α2−λ)λ2α (4−λ+λα)(4+λ) , if δ > δ d c , α(24δ−14λδ+14λαδ+12λ−16(1+αδ)−λ3(1−δ+αδ)) (4−λ+λα)(4+λ)(1−δ+αδ) , if δ d c ≥ δ > δmc , (4δ−4−3αδ)αλ2δ 4(λαδ−2αδ+4δ−δλ−4+λ)(λ+4)(1−δ+αδ) , otherwise. For each of the cases, we show below that this difference is positive. δ > δdc : The difference in percentage profit loss is positive for α < 1 2( √ 9− 4λ− 1), which holds since δ > δdc ⇔ α < αdc and αdc = 3δ−22δ ≤ 12 < √ 5−1 2 ≤ 12( √ 9− 4λ− 1). δdc ≥ δ > δmc : The difference in percentage profit loss is positive for δ > 16−12λ+λ 3 24−14λ+14λα−16α+λ3−λ3α , which holds since δ > δmc ≥ 16−12λ+λ 3 24−14λ+14λα−16α+λ3−λ3α . δmc ≥ δ > 0: the difference in profit loss, (4δ−4−3αδ)αλ 2δ 4(λαδ−2αδ+4δ−δλ−4+λ)(λ+4)(1−δ+αδ) is positive since 4δ − 4 − 3αδ ≤ 0 and λαδ − 2αδ + 4δ − δλ − 4 + λ ≤ 0 (since λαδ − 2αδ + 4δ − δλ − 4 + λ ≤ 4δ − δλ− 4 + λ = (1− δ)(λ− 4) ≤ 0) and all other expressions are positive. (ii) The difference in profit loss is given by pid,al − pim,al =  λ2α(4−4α−2λ+λα+λα2−4α2) 16(4−λ)(4−λ+λα) , if δ > δ d c , α(−48δ+32−32λ+40δλ+8λ2−9λ2δ+32αδ−36λαδ+9λ2αδ) 8(4−λ+λα)(4−λ)(1−δ+αδ) , if δ d c ≥ δ > δmc , αλ2δ 8(λαδ−2αδ+4δ−δλ−4+λ)(λ−4) , otherwise. For each of the cases, we show below that this difference is positive. δ > δdc : recall that δ > δ d c ⇔ α < αdc . The difference in profit loss, λ 2α(4−4α−2λ+λα+λα2−4α2) 16(4−λ)(4−λ+λα) ≥ 0 when α ≤ 4−λ− √ 9λ2−56λ+80 2(λ−4) . Thus, the difference in profit loss is negative when 3δ−2 2δ > α > 4−λ−√9λ2−56λ+80 2(λ−4) . Noting that 4−λ−√9λ2−56λ+80 2(λ−4) ≥ √ 33 6 − 12 ≈ 0.457 and αdc = 3δ−22δ ≤ 12 the difference in profit loss may be negative only if λ > 45 and δ > 6 12−√33 ≈ 0.959. δdc ≥ δ > δmc : the difference in profit loss is positive if δ ≥ 8(4−4λ+λ 2) 48−40λ+9λ2−32α+36λα−9λ2α . Since δmc ≥ 8(4−4λ+λ 2) 48−40λ+9λ2−32α+36λα−9λ2α , it implies that difference in profit loss is positive in the range δdc ≥ δ > δmc . δ ≤ δdc : the difference in profit loss, αλ 2δ 8(λαδ−2αδ+4δ−δλ−4+λ)(λ−4) is positive since in the denomi- nator λαδ − 2αδ + 4δ − δλ− 4 + λ ≤ 0 and λ− 4 is negative as well. Proposition A.9 A monopolist is never better off by being oblivious to consumers’ strategicity. 146 Proof. Given the prices R1 = 24−λ and R2 = 1 4−λ , strategic consumers purchase the good in the first period whenever V m,i1 − R1 ≥ δ(V m,i1 − R2). Thus, when the monopolist ignores consumers’ strategicity, the strategic consumers’ critical values are given by V m,i1 =  1, if δ > 2−λ3−λ , 2−δ (4−λ)(1−δ) , otherwise, and the monopolist retailer respective profit is pim,i1 =  4(1−α+αλ)−λ(1+αλ) (4−λ)2 , if δ > 2−λ 3−λ , 4−λ−2δ(2+α)+λδ(1+α) (1−δ)(4−λ)2 , otherwise. As δmc ≡ 2(λ−2)(−5+2λ−2λα+3α+ √ (1−α)(1−α+λα)) α2(8+4λ2−11λ)+α(31λ−28−8λ2)+4(6−5λ+λ2) ≥ 2−λ3−λ for any λ, α ∈ (0, 1), the monopolist’s profit loss due to ignorance is pim1 − pim,i1 =  1−α+αλ 4−λ+λα − 4(1−α+αλ)−λ(1+αλ)(4−λ)2 = λα 2(2−λ)2 (4−λ+αλ)(4−λ)2 ≥ 0, if δ > δmc , (2−2δ+αδ)2 4(4−αλδ+2αδ−4δ+λδ−λ)(1−δ+αδ) − 4(1−α+αλ)−λ(1+αλ)(4−λ)2 , if δmc ≥ δ > 2−λ3−λ , (2−2δ+αδ)2 4(4−αλδ+2αδ−4δ+λδ−λ)(1−δ+αδ) − 4−λ−2δ(2+α)+λδ(1+α)(1−δ)(4−λ)2 , otherwise. When δmc ≥ δ > 2−λ3−λ , the profit loss due to ignorance, pim1 − pim,i1 , is negative (i.e., there is a profit gain) whenever δ > δ ≡ 2(λ−2)(2αλ2−2λ2−10αλ−20+12α+13λ+ √ (λ+4α−4)(−4+λ+4α−4αλ+αλ2)) 104λ+48λα2−32α2−24λ2α2−144αλ−36λ2−8αλ3+4λ3+4α2λ3−96+112α+59αλ2 . However, we find that δ ≥ δmc . That is, when δmc ≥ δ > 2−λ3−λ there is always a profit loss due to ignorance. When δ ≤ 2−λ3−λ , we find that pim1 −pim,i1 is positive whenever δ ≤ 32−24λ+5λ 2 5λ2−24λ−4αλ2+16αλ+32−16α . Since 32−24λ+5λ2 5λ2−24λ−4αλ2+16αλ+32−16α ≥ 1 ≥ δ, there is always a profit loss due to ignorance when δ ≤ 2−λ3−λ . Proposition A.10 Under competition, if both retailers are oblivious to consumers’ strategicity, then when δ ≤ δdc , the profit per consumer in the first (resp., second) period is lower (resp., higher) than when retailers are not oblivious to consumers’ strategicity, and when δ > δdc , there is no difference in the first period profit, while the second period profit is lower. Proof. Assume N1 = 1 and N2 = 0. When retailers ignore consumers’ strategicity, they set R1 = 12 and R2 = 1 4 . Strategic consumers purchase in the first period only if V d,i 1 − 12 ≥ δ(V d,i1 − 14). 147 Thus, we have V d,i1 =  1, if δ > 23 , 2−δ 4(1−δ) , otherwise, pid,i1 =  1−α 4 , if δ > 2 3 , 2−2δ−αδ 8(1−δ) , otherwise, and pid,i2 =  λ(1+2α) 16 , if δ > 2 3 , λ(1−δ+αδ) 16(1−δ) , otherwise. Since δdc = 2 3−2α ≥ 23 , Retailer 1’s profit loss due to ignorance is given by pid1 − pid,i1 =  1−α 4 − 1−α4 = 0, if δ > δdc , 2−2δ+αδ 8(1−δ+αδ) − 1−α4 = α(2−3δ+2αδ)8(1−δ+αδ) ≥ 0, if δdc ≥ δ > 23 , 2−2δ+αδ 8(1−δ+αδ) − 2−2δ−αδ8(1−δ) = δ 2α2 8(1−δ+αδ)(1−δ) ≥ 0, otherwise, where α(2−3δ+2αδ)8(1−δ+αδ) is positive since 2− 3δ + 2αδ ≥ 0 ⇔ δ ≤ 23−2α = δdc . Considering Retailer 2, his profit loss (gain) due to ignorance is pid2 − pid,i2 =  λ(1+α)2 16 − λ(1+2α)16 = λ(α) 2 16 ≥ 0, if δ > δdc , λ 16 − λ(1+2α)16 = −λα8 ≤ 0, if δdc ≥ δ > 23 , λ 16 − λ(1−δ+αδ)16(1−δ) = − λδα16(1−δ) ≤ 0, otherwise, and it is evident that when δ < δdc Retailer 2’s expected profit under ignorance exceeds his expected profit when they are not ignorant to consumers’ strategicity. Proof of Theorem 4.1. In that case, the combined profit loss of both retailers due to ignorance is pid1 − pid,i1 + pid2 − pid,i2 =  0 + λ(α) 2 16 ≥ 0, if δ > δdc , α(2−3δ+2αδ) 8(1−δ+αδ) − λα8 = α(2−3δ+2αδ+λδ−λ−λαδ)8(1−δ+αδ) , if δdc ≥ δ > 23 , δ2α2 8(1−δ+αδ)(1−δ) − λδα16(1−δ) = δα(2αδ−λ+λδ−λαδ)16(1−δ+αδ)(1−δ) , otherwise. When δdc ≥ δ > 23 , the profit loss of both retailers combined due to strategicity ignorance, α(2−3δ+2αδ+λδ−λ−λαδ) 8(1−δ+αδ) , is negative if δ > 2−λ 3−2α−λ+λα . That is, a profit gain occurs when δ d c ≥ δ > max(23 , 2−λ 3−2α−λ+λα). When δ ≤ 23 , retailers’ combined profit loss due to strategicity ignorance, δα(2αδ−λ+λδ−λαδ)16(1−δ+αδ)(1−δ) , is negative if δ < λ2α+λ−λα . That is, a profit gain occurs when δ < min( 2 3 , λ 2α+λ−λα). 148 Proof of Theorem 4.2. We have the following derivatives of profit with respect to δ ∂pim,i1 ∂δ =  0, if δ > 2−λ3−λ , α(λ−2) (1−δ)2(4−λ)2 , otherwise, and ∂(pid,i1 + pi d,i 2 ) ∂δ =  0, if δ > 23 , 16α(λ−2) (1−δ)2 , otherwise. We note that pim,i1 ∣∣ δ=0 = 14−λ ≥ (pid,i1 + pid,i2 ) ∣∣ δ=0 = 4+λ16 , i.e., when all consumers are myopic, the monopoly profit is larger than the duopoly profit, and that when retailers are oblivious to consumers’ strategicity, then in a monopoly a smaller δ value is required for strategic consumers to defer their purchase to the second period, since 2−λ3−λ ≤ 23 . Thus, if δ ≤ 2−λ3−λ , then ∂pim,i1 ∂δ ≤ ∂(pid,i1 +pi d,i 2 ) ∂δ , i.e., the monopoly profit declines faster in δ than the duopoly profit. At δ = 0 the monopoly profit is larger than the duopoly profit, but declines faster. If at δ = 2−λ3−λ (i.e., when the monopoly profit flattens) the monopoly profit is lower, then there exists a value of δ for which the duopoly is better off. Evaluating the profit when δ = 2−λ3−λ , we have pi m,i 1 ∣∣ δ= 2−λ 3−λ < (pid,i1 + pi d,i 2 ) ∣∣ δ= 2−λ 3−λ ⇒ α > λ(λ−4) λ3−12λ2+36λ−32 . Proof of Proposition 4.9. When λ = δ, δmc is the solution to (4− 8α+4α2)(δmc )4+ (−28+ 39α− 11α2)(δmc )3 + (64− 56α+8α2)(δmc )2 + (−56+ 24α)δmc +16 = 0. The derivative of the initial price with respect to δ is ∂Rm1 ∂δ =  2(1+α) (4−δ+αδ)2 > 0, if δ > δ m c , (2−2δ+αδ)(2+11αδ−4α−6δ+6δ2−12αδ2−2α2δ+6α2δ2−2δ3+5αδ3−4α2δ3+α3δ3) 2(−δ2+αδ2+5δ−2αδ−4)2(1−δ+αδ)2 , otherwise. Let δm1 ≤ δm2 ≤ 1 denote the values of δ which solve ∂R m 1 ∂δ ∣∣ PP (i) = 0 (i.e., they solve 2+11αδ− 4α− 6δ + 6δ2 − 12αδ2 − 2α2δ + 6α2δ2 − 2δ3 + 5αδ3 − 4α2δ3 + α3δ3 = 0) and which are always never greater than one. Thus, if δ < δm1 or δ > δ m 2 , ∂Rm1 ∂δ ∣∣ PP (i) > 0, and if δm1 < δ < δ m 2 , ∂Rm1 ∂δ ∣∣ PP (i) < 0. The second period price is piecewise non-decreasing in δ since ∂Rm2 ∂δ =  (1+α)(1−α) (4−δ+αδ)2 ≥ 0, if δ > δmc , 2δ2−3αδ2−4δ+4αδ+2+α2δ2 2(−δ2+αδ2+5δ−2αδ−4)2 ≥ 0, otherwise. where 2δ2 − 3αδ2 − 4δ + 4αδ + 2 + α2δ2 is positive for α ≥ 3δ−4+ √ δ2−8δ+8 2δ . However, we have α ≥ 0 ≥ 3δ−4+ √ δ2−8δ+8 2δ . To demonstrate the spike in prices we use to the critical value of α, αmc = −24− 39δ2 + 56δ + 8δ3 +√64 + 240δ2 − 192δ − 144δ3 + 33δ4 2δ(−11δ + 4δ2 + 8) . 149 Rm1 ∣∣ PP (ii),α=αmc > Rm1 ∣∣ PP (i),α=αmc and Rm2 ∣∣ PP (ii),α=αmc > Rm2 ∣∣ PP (i),α=αmc ⇒ α > 2 − √2. Since αmc > 0 if α > 2− √ 2, we conclude that a spike in prices occurs once the monopolist switches from PP (i) to PP (ii). The monopolist’s expected profit behaves in a similar way to his first period profit since ∂pi m 1 ∂δ = 1 2 ∂Rm1 ∂δ , and it doesn’t spike up at δ m c as the transition from PP (i) to PP (ii) occurs when pi m 1,i|λ=δ = pim1,ii|λ=δ. Proposition A.11 In a three-period selling horizon, when the monopolist employs PP (iii), price decline linearly only if λ = 1. Proof. When PP (iii) is employed, the resulting first, second, and third period prices are αλ2+λα−λ+4 2(λα−2λ+4) , λα+1 λα−2λ+4 , 1−λα+3α 2(λα−2λ+4) , respectively. When λ = 1 prices decline in a constant decrement of 12(α+2) . When λ < 1, a constant decrement requires α = 1 λ−3 , which is impossible as 1 λ−3 < 0. Proposition A.12 When δ = λ = 1, the monopolist’s prices are linear along the selling horizon with Rt = T−t+1+α(T−1) T+1+α(T−1) , and an expected profit of T 2(T+1+α(T−1)) . Thus, (i) as T increases, the spread of prices increases (as the initial (resp., final) price increases (resp., decreases)) and the expected profit increases; (ii) as α increases prices along the equilibrium path increase (while their spread decreases) and the expected profit decreases. Proof. Since strategic consumers simply purchase the good at the lowest price offered, the monopolist can either price the good such that the price is the same across all periods, R1 = R2 = · · · = RT , or strictly decreasing across periods, R1 > R2 > · · · > RT . In the former case, the problem collapses into a single period setting with an expected profit of 0.25. In the latter case, all strategic consumers wait until the final period, and the problem yields a system of first order conditions: ∂pi ∂R1 = (1− α)(1− 2R1 +R2) = 0 ... ∂pi ∂RT−1 = (1− α)(RT−2 − 2RT−1 +RT ) = 0 ∂pi ∂RT = (1− α)(RT−1 − 2RT−) = 0, 150 for which the solution is Rt = T−t+1+α(T−1) T+1+α(T−1) with an expected profit of T 2(T+1+α(T−1)) . Since T 2(T+1+α(T−1)) ≥ 0.25, strictly decreasing pricing is the preferred pricing strategy. From the above, it follows immediately that ∂R1∂T = 1+α (T+1+α(T−1))2 > 0, ∂RT ∂T = α−1 (T+1+α(T−1))2 ≤ 0, ∂pi∂T = 1−α 2(T+1+α(T−1))2 ≥ 0, ∂Rt∂α = t(T−1)(T+1+α(T−1))2 > 0, ∂(RT−R1)∂α = − (T−1) 2 (T+1+α(T−1))2 < 0, and ∂pi ∂α = − T (T−1) 2(T+1+α(T−1))2 < 0. Proof of Proposition 4.10. The proof of the proposition follows from the following two lemmas. Lemma A.12 When λ ≤ δ(2 − δ) (resp., ≥) {Bmt }Tt=2 is bounded from below by Bm ≡ 1− √ 1−λ λ (resp., 12) and from above by 1 (resp., B m). Proof. Omitting the base step, in period t, ∂B m t ∂Bmt+1 = 2δ+λ−2δλBmt+1 ((2−λBmt+1)(1−δBmt+1)+δ)2 ≥ 0, since 2δ+λ 2δλ ≥ 2δ+12δ ≥ 32 ≥ 1 ≥ Bmt+1, so to verify that Bmt satisfies the limits, it suffices to evaluate Bmt at the extremes values of Bmt+1. By the hypothesis, when λ ≤ δ(2− δ) (resp., ≥), Bm ≤ Bmt+1 ≤ 1 (resp., 12 ≤ Bmt+1 ≤ Bm). Now, Bmt ∣∣ Bmt+1=1 = 12−λ−δ+λδ ≤ 1, Bmt ∣∣ Bmt+1= 1 2 = 1 2− 1 2 λ+ 1 4 λδ ≥ 12 , and Bmt ∣∣ Bmt+1=B m= 1− √ 1−λ λ = 1− √ 1−λ λ = B m. Thus, when λ ≤ δ(2− δ) (resp., ≥), Bm ≤ Bmt ≤ 1 (resp., 1 2 ≤ Bmt ≤ Bm). Lemma A.13 When λ ≤ δ(2− δ) (resp., ≥) {Bmt }Tt=2 is backwards decreasing (resp., increasing). Proof. When λ ≤ δ(2 − δ) (resp., ≥) B m t−1 Bmt = (2−Bmt+1)(1−δBmt+1)+δ (2−Bmt )(1−δBmt )+δ ≤ 1 (resp., ≥) ⇒ (2−Bmt+1)(1−δBmt+1) (2−Bmt )(1−δBmt ) ≤ 1 (resp., ≥) which holds by the induction hypothesis. By Lemma A.12 and Lemma A.13, relying on the Monotone Convergence Theorem, the proof of Proposition 4.10 has been obtained. Proof of Proposition 4.12. (i) To prove the convergence of {Bdt }Tt=2 we show, by induction, that when λ = δ, {Bdt }Tt=2 is backwards decreasing and bounded from below (resp., above) by 12 (resp., 1). It can be verified that BdT = 1 2−δ , B d T−1 = 2−δ 4−2δ−δ2 , and B d T−2 = (4−2δ−δ2)2 32−32δ−10δ2+6δ3+4δ4+δ5 , which satisfy the assertion. Assume that 0.5 ≤ Bdt+1 ≤ Bdt+2 ≤ 1, t = 2, . . . , T − 2, and we will prove that (a) Bdt ≤ Bdt+1 and (b) 0.5 ≤ Bdt ≤ 1. (a) Bdt ≤ Bdt+1: Using (4.4), Bdt ≤ Bdt+1 ⇔ (1− δBdt+1)(1− 2Bdt+1) ≤ δ2(Bdt+1)3Bdt+2(1− δBdt+2), which is certainly true, since 1− 2Bdt+1 ≤ 0 and all other terms are positive. 151 (b) 0.5 ≤ Bdt ≤ 1: Bdt is increasing in Bdt+1 since ∂B d t ∂Bdt+1 = 2δ(1+δBdt+1B d t+2(1−δBdt+2)) (2(1−δBdt+1)−λ2(Bdt+1)2Bdt+2(1−δBdt+2)+δ)2 > 0, but it is decreasing (resp., increasing) in Bdt+2 if 1 − 2δBdt+2 ≤ 0 (resp., >) since ∂B d t ∂Bdt+2 = δ2(Bdt+1) 2(1−2δBdt+2) (2(1−δBdt+1)−λ2(Bdt+1)2Bt+2(1−δBdt+2)+δ)2 . If 1 − 2δBdt+2 ≤ 0, then Bdt is bounded from above by Bdt ∣∣ Bdt+1=B d t+2=1 = 1 2−δ−δ2+δ3 , which is bounded between 0.5 and 1, and B d t is bounded from below by Bdt ∣∣ Bdt+1=B d t+2= 1 2 = 16 32−2δ2+δ3 ≥ 0.5. If 1 − 2δBdt+2 > 0, then since Bdt+1 ≤ Bdt+2, Bdt is bounded from above by Bdt = 1 2−δ−δ2+δ3 , which is bounded between 0.5 and 1, and B d t is bounded from below by Bdt = 4 8−δ2+δ3 ≥ 0.5. Since {Bdt }Tt=2 is bounded and backwards decreasing, by the Monotone Convergence Theorem, it is a convergent sequence. (ii) Assume there is only one consumer in the market who visits Retailer 1 in the first period. At convergence, the profits of Retailer 1 and Retailer 2 are 1−δ 2(2−δ2(Bd)3)(1−δBd) and λBd(1−δ) 2(2−λ2(Bd)3)2(1−δBd) , respectively. The summation of the two provides the desired expression. When λ = δ → 1, it is immediate to show that the initial price, Rd1 = 1−δ(2−δ2(Bd)3)(1−δBd) , converges to zero by applying l’Hospital rule twice. Proof of Proposition 4.13. When δ < 1 and T = ∞, the monopoly profit is pim1 = 1−δ 2(1−δ+√1−λ) and the duopoly profit is pi d 1 + pi d 2 = (1−δ)(2−λ2(Bd)3+λBd) 2(2−λ2(Bd)3)2(1−δBd) . The absolute profit loss incurred to monopoly (resp., duopoly) due to consumers’ strategicity is defined as pim1 ∣∣ δ=0 −pim1 ∣∣ δ=λ (resp., (pid1 + pi d 2) ∣∣ δ=0 − (pid1 + pid2) ∣∣ δ=λ ). Solving pim1 ∣∣ δ=0 − pim1 ∣∣ δ=λ > (pid1 + pi d 2) ∣∣ δ=0 − (pid1 + pid2) ∣∣ δ=λ ⇒ λ > 0.9326. The relative profit loss incurred to monopoly (resp., duopoly) due to consumers’ strategicity is defined as pi m 1 |δ=0−pim1 |δ=λ pim1 |δ=0 (resp., (pid1+pi d 2)|δ=0−(pid1+pid2)|δ=λ (pid1+pi d 2)|δ=0 ). Solving pi m 1 |δ=0−pim1 |δ=λ pim1 |δ=0 > (pid1+pi d 2)|δ=0−(pid1+pid2)|δ=λ (pid1+pi d 2)|δ=0 ⇒ λ < 0.

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