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The essays on the optimal information revelation, and multi-stop shopping
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The essays on the optimal information revelation, and multi-stop shopping
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The Essays On The Optimal Information Revelation, And Multi-Stop Shopping by KyunHwa Kim A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Economics) August 2017 Copyright 2017 KyunHwa Kim Acknowledgements I am deeply indebted to my advisor Yilmaz Kocer for his invaluable guidance and unwavering supports. Without his unconditional support, continuous guidance, and stimulating discussion, this dissertation could not have been completed. It has been an honor to be his student, and I am lucky to have him as my advisor. I would also like to express my deep appreciation to professor Guofu Tan, and Anthony Dukes, for their generosity with their time and valuable comments. I have greatly benefited from their broad vision and intuition in industrial organization, and marketing. During my six years of study at USC, I received an abundance of support and encouragement from my fellow doctoral students. I would like to thank to Mi Hye Lee, Kyung Eun Kim, and Ahram Moon for being reliable and sincere friends. I am also grateful to my colleague, to name a few, Junjie Xia, Bilal Kahn, and Xiao Fu, for their feedback, and cooperation at all times. I wish to say special thanks to my coauthor, Jong Jae Lee, at Johns Hopkins University. I was lucky to know him. From technical comments to broad suggestions, the discussion with him always stimulated me with insightful remarks and I sincerely enjoyed my collaboration with him. My deepest thanks to Sung Ha Park. Getting to know him is a serendipity of my life. And finally, I wish to express my deep appreciation to my family in Korea for their unwavering supports and encouragement. ii Table of Contents Acknowledgements ii List Of Tables v List Of Figures vi Abstract vii Chapter 1: Introduction 1 1.1 The Optimal Information Revelation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Multi-Stop Shopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: Fit, Quality and Optimal Blurring of Advertising: Two Attributes of Goods and Bayesian Persuasion 10 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Bayesian Persuasion and Fit Revelation Activities . . . . . . . . . . . . . . . 16 2.3 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Brick-and-Mortar Store Shopping: Sequential Game . . . . . . . . . . . . . . 23 2.3.2 Online Shopping: Simultaneous Game . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.1 Price Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.2 Quality choice problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.1 Tie-breaking rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.2 Consider heterogenous consumer (θ i ) . . . . . . . . . . . . . . . . . . . . . . 48 2.5.3 Competitive market structure (N firms) . . . . . . . . . . . . . . . . . . . . . 48 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Chapter 3: Consumer Loyalty and Spatial Competition between shopping centers 53 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.2 Profit Maximization Problem and the Optimal Pricing Rules . . . . . . . . . 61 iii 3.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Reference List 72 Appendix A Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.1 Proofs for Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.1.1 Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.1.1.1 The Values of ‘b∗’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.1.1.2 Proofs for Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . 79 A.1.2 Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.1.3 Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 A.1.3.1 At the 2nd stage: given (P A , P B ), optimal signaling structure . . . 84 A.1.3.2 At the 1st stage: the optimal prices . . . . . . . . . . . . . . . . . . 85 A.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 A.2.1 Remarks on Simultaneous Game . . . . . . . . . . . . . . . . . . . . . . . . . 87 A.2.2 Remarks on Mixed Strategy Equilibrium . . . . . . . . . . . . . . . . . . . . . 89 A.2.3 Remarks on Tie-Breaking Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Appendix B Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 B.1 Proof of Lemma 1: One-stop Shopping Equilibrium . . . . . . . . . . . . . . . . . . . 94 B.2 Proof of Theorem 1: Two-stop Shopping Equilibrium . . . . . . . . . . . . . . . . . . 99 iv List Of Tables 2.1 The information structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 v List Of Figures 2.1 The information structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 The timing of the game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 The Partition of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Timing of the game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 The partition of consumers (one-stop shopping) . . . . . . . . . . . . . . . . . . . . . 59 3.2 The partition of consumers (two-stop shopping) . . . . . . . . . . . . . . . . . . . . . 59 A.1 The values of b* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.2 Compare: b*(firm A preferred-tie rule) vs b**(share the market-tie rule) . . . . . . . 93 vi Abstract This dissertation consists of two essays on the optimal information revelation, and multi-sop shop- ping model. Chapter 2 entitled “Fit, Quality and Optimal Blurring of Advertising: Two Attributes of Goods and Bayesian Persuasion” examines firms strategic decisions regarding how much information to provide to consumers who are uncertain about the fit between their personal tastes and the products horizontal attributes. Compared to information on product quality which is publicly observable, idiosyncratic fit information is hard to know before the purchase. Thus, firms participate in fit revelation activities such as advertising, free trials, consumer reviews, or return policies in order to help consumers learn. I examine two research questions pertinent to firms fit-revealing strategies. First, what is the role of the inherent quality of the firms product in providing fit information? Second, how do the channels of market affect the means of providing fit information? I find that the results are consistent with the widely held belief that the low quality firm provides more information unless the quality advantage of the high quality firm is large enough, when two firms compete in offline. On the other hand, an interesting result arises in online competition. Compared to the high quality firm who sends out bluffing signals (Type-II error), a low quality firm sends out humble signals (Type-I error) in order to survive in the market. vii Chapter 3 entitled “Consumer Loyalty and Spatial Competition between Shopping Centers,” aims to explore multi-stop shopping behavior of consumers in light of the customer loyalty in two aspects. Firstly, what is the effect of customer loyalty on the existence of a multi-stop shopping equilibrium? Secondly, if the multi-stop shopping equilibrium exists, what is the specific features of such a shopping behavior? As a variant of Hotelling’s celebrated model which account for the model of two types of products, we find that the optimal price strategy of shops which induce consumers’ the multi-stop shopping exists and these two-stop shoppers always enjoy the customer loyalty on their shopping trips when the loyalty level is high enough to compensate the travel costs of consumers. Under this condition (when the consumers’ relative benefit from multi-stop shopping is large enough), it is also profitable for the shop to set the low enough prices to induce consumers’ multi-stop shopping and they always seek for this loyalty. viii Chapter 1 Introduction This dissertation consists of two essays about the optimal information revelation, and multi-stop shopping model. The first essay, “Fit, Quality and Optimal Blurring of Advertising: Two At- tributes of Goods and Bayesian Persuasion,” studies how much information the firm provides to consumers, who are uncertain about the products fit by using Bayesian persuasion model (Ka- menica and Gentzkow (2011)) of persuasion competition. The second essay, “Consumer Loyalty and Spatial Competition between Shopping Centers,” co-worked with Jong Jae Lee 1 , explores the multi-stop shopping behaviors of consumers. We relaxes the single-stop assumption of traditional Hotelling model and investigate the effect of customer loyalty on the existence of multi-stop shop- ping equilibrium and the features of such an equilibrium. 1.1 The Optimal Information Revelation In many product categories such as food, clothes, cosmetics, and even electronics, a consumers pur- chase decision heavily depends on not only how good a products quality is (vertical attributes) but 1 Department of Economics, The Johns Hopkins University, Baltimore, MD 21218, USA. Email: jlee427@jhu.edu 1 also how well a product fits his personal tastes ( horizontal attributes) (Desai, 2001). Compared to the quality information of a product which is publicly observable, the fit information such as match, or taste is often unobservable before the purchase is made, which strongly affects the consumers decision-making. In order to assist consumers in finding the fit between their personal tastes and the products horizontal attributes, almost all firms in the market, regardless of product category, participate in fit revelation activities to facilitate consumer learning (Gu and Xie, 2013). For example, Samsung spends millions of dollars to open retail boutiques called Samsung mini store in 1400 Best Buy lo- cations to provide consumers with opportunities to inspect their products. Amazon and eBay, who have no brick-and-mortar shops, have implemented easy returns and generous money back guar- antee policies to resolve the consumers valuation uncertainty. Similarly, whenever a new version of Mathematica or MS Office is introduced, software vendors provide free trial versions with user manuals. Among these activities, advertising might be the most common. Once the firm delivers fit information via advertising, the consumer has a chance to learn the fit of the product. These fit revelation activities become important part of the marketing strategies of firms today. One distinguishable feature of my framework from the previous studies is, the firm can control the degree of information conveyed to consumers by strategically designing the signaling structure. Gu and Xie (2013), Jing (2015), and Guo and Zhang (2012) considered the model in which an agents uncertainty is fully resolved, once the agent engages in the activities. However, uncertainty is not always fully resolved in reality. Even though a consumer observes the advertisement and has a chance to learn his valuation of the product, he may draw a good impression with one probability or draw a bad impression with another probability. Moreover, full revelation is not always optimal for the firm. If full informative advertising is available, the consumer may decide not to buy the 2 good if he realizes that it does not fit his taste. In some situations, a certain level of blurring is a better strategy for the firm. Thus, in this study, the firm mainly considers whether to engage in these activities and if it decides to do so, how much information to reveal. The strategic decision of the informativeness of signals, which is the choice variable of the firm can vary. For example, when the firm designs advertising she considers how many attributes to deliver. The method of advertising that conveys no specific information about the product, such as the first generation iPhone television advertisement “Hello,”is called non-attribute-focused adver- tising (image advertising). In this case, a consumer cannot learn about his valuation of the product from the firms activity. Thus, this cannot resolve any uncertainty about the products fit. On the other hand, advertisements with some level of information about the product are referred to as attribute-focused advertisements. In this case, the more attributes are contained in the advertising, the more information is delivered to consumers therefore, consumers becomes more certain about their values. Besides advertising, any types of fit revelation activities are carefully designed by the firms optimal signaling strategy. Firms that have offline experience shops such as the Apple Store or the Samsung mini store strategically decide the scales of their shops. The more in-store display capacity the firm has, the more learning possibility the consumer has. For example, Samsung has aggressively invested in experience shops these days, and their boutique shops will be around 460 square feet, even larger than the Apple store. Similarly, the generosity of return policy is also the firms strategic choice. Since a more generous return policy enables consumers to learn more (learn deeply) about the products before making a purchase decision. For example, Nordstrom Racks 90 days return policy is more generous than Saks Fifth Avenues 30 days return policy. In this paper, I construct a general model of signaling competition which has two important 3 features. First, in order to find the optimal informativeness of advertising, I model the persua- sion game in which the firm can freely design any information structure in an effort to persuade consumers to purchase its product. As in Kamenica and Gentzkow (2011), each firms strategy is described as a distribution of signal realizations that may increase or decrease the buyers evaluation of the product. When the firm designs the mechanism, she faces a tradeoff: she wants to provide good signals as much as possible to increase the sales, but sending good signals more often hurts the credibility of these signals, hence makes it less likely that the consumer will purchase the product. Thus, balancing these two conflicting objectives is the key of the firms strategy. Second, I focus on the competition between a firm selling a high quality product and a firm selling a low quality product. Even though a firm with superior quality does not have to provide informative advertising in a monopoly market, she may reveal some level of information in a duopoly market by bifurcating the signals she provides into a good signal and bad signal. That is, the firm chooses a good enough signal to persuade consumers to buy, and, at the same time, the firm chooses a bad enough signal to make the signal look more credible. I examine two research questions pertinent to firms fit-revealing strategy. First, what is the role of the inherent quality in providing the fit information? It comes from the distinct feature of the horizontal attribute. Fit uncertainty is specific to the match between an individual consumer and a particular product, implying that one firms fit revelation cannot resolve consumers fit uncer- tainty regarding another product (Gu and Xie, 2013). In other words, each firm can produce the information only about his own product, not all products under consideration. The idiosyncratic fit valuation requires the independent dimension of the state of the world, which constitutes a the- oretical contribution to the growing literature on competitive Bayesian persuasion (Kamenica and 4 Gentzkow, 2016 2 ). In this situation, I analyze how the firm with superior quality uses her quality advantage, and how the firm with inferior quality survives in the market via fit revelation activities. In particular, as in the 1st generation iPhone commercial, “Hello” 3 , was totally non-attributed- focused which delivers no specific information about the phone to help consumers uncover their valuation. This is the case of the first generation of iPhone, making its move first when it is the highest quality smart-phone in the market., when the quality advantage of one firm has is large enough, then this high quality firm does not provide any information to capture the market regard- less of the timing of the game. However, when the quality difference is not large enough, then even the high quality firm should participate in the fit revelation activities. In the latter case, in order to survive in the market, a low quality firm reveals more information about the fit than the high quality firm, which is intuitive. Second, how does the channel of market−offline shopping and online shopping −affect the way of providing information? This question is important because the timing of the game is different. The main difference lies in whether or not a consumer can observe all the signal realizations before he enters the store. When the consumer shops offline stores such as the Apple store and the Samsung mini Shop, the consumer can update his belief only after he enters each store and physically in- spects one product at a time. On the contrary, when the consumer shops online, the consumer can observe and compare all the signal realizations simultaneously, and it causes a harsh competition between two firms. In particular, an interesting result arises when the consumer shops online, if the quality gap between two firms is not large enough. Compared to the high quality firm who offers a bluffing signal (Type-II error), the low quality firm starts to offer a humble signal (Type-I 2 Kamenica and Gentzkow (2016) requires the model in which agents produce information about all dimensions. The main insight is that no agent can reduce the information available to the principal, thus an outcome is an equilibrium if no agent can benefit by supplying a more informative signal to the principal. Hence, loosely speaking, adding sender causes the set of equilibrium outcome to shrink toward full revelation. (Boleslavsky and Cotton, 2014, Kamenica and Gentzkow, 2016) 3 The first generation iPhones television adverting, called “ Hello 5 error) to survive in the market. In advertising, it is called bluntvertising. 4 That is, in contrast to the traditional advertising approach of exaggeration, which praises the goods with fancy words and attracts consumers attention with dazzling images, the firm chooses to offer information in an opposite way with brutally honest messages. As in Dominos 2009 television commercial “Mae Culpa ads (turn-around campaign)” 5 which admits that some consumers find Dominos pizza terri- ble, sometimes the firm intentionally reveals her bad aspects in order to build credibility, bearing the risk of giving consumers bad impressions. The main analysis is extended in a few directions. First, I incorporate the firms fit revelation activities into a simple model of price competition between two firms. This leads to some new results, since a no-buy option is available when the consumer is considering the price. The role of the signaling is designed to induce consumers to form non-negative expected utility and buy the product. The price competition at the previous stage mitigates the harsh belief-building competi- tion between two firms. As a result, whether it is offline or online, the optimal advertising strategy of both firms is exaggeration. Specifically, the low quality firm offers more informative advertising than the high quality firm while it leaves zero consumer surplus. Second, I analyze the quality choice problem as a prestage game where a firm incurs less costs of improving the quality compared to its competitor. When each firm has a chance to increase the quality level with a cost, the positive investment in quality occurs only when the given cost advantage of two firms is similar enough, which leaves a positive consumer surplus. 4 Advertisers create transparency with brutally honest advertising and messages. (trendhunter.com) 5 In 2009 television commercial Mae Culpa ads (Turn-around campaign), Domino reveals the harsh critics from consumers, such as Totally devoid of flavor, Dominos pizza crust to me is like cardboard, says a woman in a clip taken from a focus-group panel. Another employee reads another review: The sauce tastes like ketchup. 6 1.2 Multi-Stop Shopping The second essay, “Consumer Loyalty and Spatial Competition between Shopping Centers,” studies a variant of Hotellings celebrated model, which account for multi-good, multi-stop shopping. As there has been a huge shift in shopping patterns during the past few decades, a more realistic approach is required to consider consumers?store choice behavior. Nevertheless, these behaviors are understudied in the models addressing spatial competition of firms, based on the celebrated Hotelling model. Most previous studies about a store choice only focus on ‘where to stop’, and does not consider the problem regarding ‘how many stops’. Basically, the standard Hotelling model considers a single-stop shopping case with a single item purchase problem. Many consumers in re- ality, however, try to fill up their consumption bundle with many products from multiple firms, by visiting here and there. This model opens up the possibility that some of the consumers buy both goods from both stores; multi-stop shopping is possible. According to the study by Bell, Ho, and Tang (1998), the habitual behavior is interesting with respect to store choice, they found out that about 21% of households ever visit more than two supermarket in their data. Moreover, O’Kelly (1981) reported that 63% of grocery shopping are multi-stop shopping, and 74% of non-grocery shopping are multi-stop shopping. Also, Hanson (1980) found that 61% of all shopping trips are multi-stop shopping. These empirical works show that the assumption, a consumer is a single stop and single purpose shopper, is a highly unrealistic as many shoppers engage in different activities on the same shopping trips. In my attempt to better reflect the store choice behavior of consumers, I extend it to the case which considers multi-stop shopping with multiple goods. In this study, we aim to explain these multi-stop shopping behaviors in light of the customer loyalty. Though the definition of customer loyalty may be under dispute, we define it broadly as 7 a psychological benefit from purchasing a product in addition to its intrinsic value. This psycho- logical benefit may include a concern for branding, reputation, quality, loyalty program, and etc. Specifically, as in the study of Zhang at el. (2013), we view this store loyalty as a category-specific trait of consumers who are loyal to different stores in different categories. The goal in this paper is to explore the effect of customer loyalty on two things, (i) the existence of multi-stop behaviors, and (ii) the patterns of such a shopping behaviors, when two shopping centers compete each other. The shopping centers located at the extremes of a linear city, can be either a department store or shopping malls depending on the retail mode. A department store is a multi-product firm that sells two goods at the same location, while a shopping mall is a shopping center of single-product firms that sell each of two goods at the same location. Consumers are uniformly distributed across the city and have unit demand of each good. They may travel to either shopping centers and buy both goods at there, or travel to both shopping centers and buy each good where the perceived price is cheaper. Accordingly, our main result is two-folded. We firstly show that two-stop shopping equilibrium exists when the relative benefits from two-stop shopping is great enough. In this situation, the profit from the product with loyalty is greater than the profit from the product with loyalty regardless of modes of retail. Secondly, and more importantly, we confirm that if a two-stop shopping equi- librium exists, such an equilibrium exhibits the consumer behaviors of loyalty-seeking. Consumers always purchase the product with loyalty at each store and enjoy the extra benefit from this loyalty, which is intuitive. One of our contributions to multi-stop shopping literature lies in the fact that our model yields to describe the determination of the features of multi-stop shopping in addition to the mere existence of such behaviors by applying the customer loyalty. The intuition behind these are simple: In order for consumers to make travel two shopping centers, the level of customer 8 loyalty should be large enough to compensate transportation cost. At the same time, in order for shopping centers to set the prices which induce two-stop shopping, the perceived prices of two-stop shopping should be large enough not to decrease the pecuniary prices. Particularly, the bundle price of individual shopping mall is higher than those of the department store, since the shopping more does not internalize the positive effect of decreasing its price on the demand of its other good. Therefore, the shopping mall, especially the shopping mall who sells the good without loyalty, has the strongest deviation incentive from the prices which sustain the two-stop shopping equilibrium. 9 Chapter 2 Fit, Quality and Optimal Blurring of Advertising: Two Attributes of Goods and Bayesian Persuasion 2.1 Overview I examine firms optimal decisions on signaling in both online and offline markets where two firms differ in their qualities. This study investigates two research questions pertinent to firms fit-revealing strategies. First, what is the role of the inherent quality of the product in providing fit information? Second, how do the market channel−offline of online −affects the means of providing fit information? The distinct feature of my research from previous studies is that my model allows the firm to consider not only whether or not to engage in fit revealing activities, but also how much information to reveal. The results of this research suggests several tactics to survive in the market for the firm who has a disadvantage in quality: the lower quality firm has to provide more informative advertising than the high quality firm does when the two firms compete offline. On the other hand, in the online competition, compared to the high quality firm who sends out exaggeratedly positive messages, 10 the low quality firm sends out deliberately negative messages about her product in order to build credibility. Related Literature My study contributes to three streams of literature. First, I incorporate two types of information on a firms product into an analysis of the firms information revealing strategy. Information on horizontal attributes (fit) and vertical attributes (quality) were first distinguished by Lal and Sarvary (1999). They define two types of product attributes as digital- and non-digital attributes and considered how the internet is likely to affect the consumer search of these attributes. So far, a significant literature has considered firms activities to reveal information on fit or qual- ity, coupled with consumers discretionary search. Jing (2015) examines the firms efforts to help customer learning by lowering search costs, when fit uncertainty is resolved by costly evaluation. Guo and Zhang (2012) demonstrate firms strategic product line designs and consumers preference learning about their valuations for quality, which is referred to as deliberation. Mayzlin and Shin (2011) also investigate the optimal choice of advertising contents, which invites the consumer to engage in search for learning. The study of Gu and Xie (2013), which highlights interaction between two attributes, is most closely related to my study. They emphasize the role of product quality in decision-making on participating fit revelation activities. A unique feature of my research is that unlike the previous studies, my model allows firm to consider not only whether to engage in fit revealing activities, but also how much information to reveal. In other words, previous studies (Gu and Xie, 2013, Guo and Zhang, 2012, Jing, 2015) only considered the extreme framework in which uncertainty is fully resolved once the agent engages in the activities. However, in my framework, the firm considers the optimal level of informativeness of signals, which expands the set of firms strategies and brings rich managerial implications. Second, from a theoretical point of view, from a theoretical point of view, this study contributes 11 to a growing literature on Bayesian persuasion (Kamenica and Genztkow, 2011 and 2016). Ka- menica and Genztkow (2011) provide a general model to investigate what the best a principal can do is, if she can choose any information structure. In their recent study (2016) they consider the impact of competition on strategic information revealing when there are multiple senders. I deal with two subtle differences from their study. First, their former study (2011) deals with the case of only a single sender, but my model examines two senders competing in either offline or online. Second, compared to their latter study (2016) in which multiple senders share a single state of the world, my model adopts two independent dimensions of fit uncertainty. It comes from the distinct nature of idiosyncratic fit values, which requires each firm to reveal information only about her product, not the rivals. Following the framework of Kamenica and Genztkows study, a number of papers consider the optimal information disclosure with competition (Au H., 2015, Board and Lu, 2015, Boleslavsky and Christopher, 2014, 2015, and 2016, Kolotilin, 2014, Li and Norman, 2015). Among these, my model is most closely related to Boleslavsky and Cottons studies (2014, 2015, and 2016). In their studies (2014, and 2015) of evidence production with limited capacity and strategic choices of grading policy, they consider the competition in revealing the horizontal information between two agents who are vertically differentiated. The main distinction between my model and their study lies in how vertical heterogeneity is defined. Boleslavsky and Cotton assume heterogeneity in prior beliefs, which brings deliberate analysis, whereas I assume heterogeneity in quality valuations, which represents quality differences between firms in a more visible way. Third, this model is also related to a broader literature on signaling and the firms optimal information provision. Early literature on information provision largely focuses on asymmetric in- formation between agents. Thus, it considers the case in which the informed sender reveals its type (Spence, 1973) or the less informed agent sets the optimal mechanism (Akerlof, 1970). However, 12 in more recent studies with models based on Bayesian persuasion, both sender and receiver do not recognize the states. The sender does not benefit from information advantage, but does benefit from persuasion by designing an optimal information structure. Also, my study is similar to cheap talk (Crawford and Sobel, 1982) in that the cost of signals is zero. However, my study is different from cheap talk, since there is a commitment in the signaling structure. Moreover, Peter and Szentes (2007) discuss a model of advising where the advisor discloses to the client the clues that refine the clients original private valuation of a project. Their model is similar to mine, because the principal sends signals which affect the valuations of the agent. However, in their model, the advisor may not be perfectly aware of the impact of her signal on the clients value estimates, whereas the sender in Bayesian models is perfectly aware of it and strategically designs the distribution over the values. The rest of paper proceeds as follows. Section 2.2 describes the model with firms competition in strategic fit revelation activities and its preliminary features. Section 2.3 analyzes the model, which consists of two parts: offline shopping (sequential game) and online shopping (simultaneous game). Section 2.4 incorporates price competition and the quality choice problem into the analysis, extending the main results. Section 2.5 discusses some of the assumptions of the model in more detail. Section 2.6 concludes with a brief discussion of the models implications and summarizes the results. 2.2 Model 2.2.1 Setup I consider a duopoly market where vertically heterogeneous firms sell two products to the consumer who is uncertain about the horizontal attribute. The two firms are denoted by A and B, respectively, 13 firm A (high quality firm) gives α (α>1) times higher utility than firm B (low quality firm) when it fits well to a consumer 1 . A continuum of consumers exist whose mass is normalized to one, each consumer shares the common value of quality for both products. The consumer utility, U =X, comes from two types of product attributes of goods (as in Desai 2001): quality (vertical attribute) and fit (horizontal attribute). 2 In contrast to costlessly observ- able quality information, the fit information is hard to know before purchasing the goods. Thus, each consumer is uncertain about the fit between his personal taste and the product’s horizontal attribute. The value ofX can be either V j (j = A, B) or zero, depending on whether it fits well or not. The high quality firm A provides weakly better utility than the low quality firm B, as follows (α > 1). X A = αV with probability. μ 0 , 0 with probability. 1-μ 0 . and X B = V with probability. μ 0 , 0 with probability.1-μ 0 . It implies the complimentary between two attributes−fit and quality which specifying the utility two levels−positive or zero. As long as the product gives non-negative utility and as long as the high quality product gives a physically higher utility than the low quality firm’s, the specification of utility function does not hurt the general results. There are binary states of the world, w∈{Good, Bad}, and the prior probability to be a good match (μ 0 ) and bad match (1-μ 0 ) is common knowledge. In other words, the consume knows whether the physical level of quality is high (V A =αV , α>1) or low (V B =V), but does not know whether each product fits well or not. Each consumer has a single unit demand and buys the product from either firm A or B, which gives higher expected utility. His ex-ante expected utility 1 There are two female firms and a male consumer. 2 These two attributes also can be defined as digital attribute (which can be communicated on the Web at very low cost) and non-digital attribute (for which physical inspection of the product is necessary), according to Lal and Sarvary (1999). 14 is u A =αμ 0 V or u B =μ 0 V , thus the consumer buys from firm A without fit revelation activities. Without a loss of generality, assume that the consumer buys from firm A, which gives a weakly better utility, if he is indifferent between two firms to simplify the problem. 3 In terms of setup, the important feature of this model is that the belief about the value of one firm is not affected by the information revealed by the other firm. It comes from the distinct nature of idiosyncratic fit values, which requires the independent dimension of the state of the world for each firm. This is the main distinction from the study of Kamenica and Genztkow (2016), which deals with the single state with multiple senders. As in a criminal trial case in their study, when both senders share a single state and the beliefs are intersected in the court, all senders provide information about a single state of the world. The main intuition that no agent can benefit by supplying a more informative signal to the principal in the equilibrium simplifies the way of obtaining equilibria, which brings the full revelation as the equilibrium outcome. However, as in the fit revelation activity in this model, when there is an independent fit dimension for each firm, she can reveal the information only about her state not the rival’s, thus full-revelation may not arise. I solve for the Perfect Bayesian Equilibrium of this game. Consumer’s beliefs following the observation of signal realizations are consistent with the optimal signals by firms, and the behaviors of firms are consistent with equilibrium consumer’s updated beliefs. 3 This tie-breaking rule (the firm A-preferred tie-breaking rule ) can be relaxed into a market-sharing tie rule− if the belief about the values of both products is the same, then he randomizes fairly between two firms. Still, the same results hold, only thing different is that the value is included in each firm’s strategy. Thus, I assume firm A-preferred tie-breaking rule to simplify the notation. When it comes to simultaneous game, the change in tie-breaking rule slightly shifts the values of the critical point of μ o and b∗ in the equilibrium. (See Section 5 for the relax of this assumption.) 15 2.2.2 Bayesian Persuasion and Fit Revelation Activities Two firms compete in fit revelation activities such as advertising, experience shops, or free trials, which allow a consumer to learn the fit values between her taste and the product. Each firm decides the degree of information conveyed to consumers by strategically designing the signaling structure. Formally, when there are binary states of the world of each firm j (=A, B), w∈{Good j , Bad j } with the prior probability μ 0 and 1-μ 0 , the firm’s optimal fit revelation activity such as advertising, is modeled as a binary random variable, s j i ∈{s j g , s j b } (j=A, B): each consumer draws either a real- ization of good signal (s g ) or realization of bad signal (s b ), corresponding to his impression of each product after observing the advertising from both firms. There are two firms in the market, thus the consumer draws two signal realizations−one for each firm-after observing the advertisements designed by each firm 4 . The consumer updates his beliefs of each product after observing the signal realization{s A i } and{s B i } (i=g,b) according to Bayes’ Rule, denoted by μ A i and μ B i . This posterior belief is defined as μ A i =probability{w=Good|s A i } and μ B i =probability{w=Good|s B i }, respectively. Following the representation of signaling by Kamenica and Genztkow (2011), I will represent the firm’s choice of how much information to provide as the choice of the distributions over the posteriors, φ j (μ j i=g,b , q j ). The strategy profile of firm j, φ j (μ j i=g,b , q j ), consists of two parts: the posterior beliefs after observing either a good or bad signal (μ j i=g , μ j i=b ) and the probability of observing this signal realization (q j , 1-q j ). When the firm designs the distributions over the posteriors, she faces the tradeoff 5 : she wants to provide good signals as much as possible to attract consumers and increase the sale, but sending good signals more often huts the credibility of these signals, hence makes it less likely that the consumer will purchase the product. Thus, balancing 4 Thus, there are four possibility of signals to be realized: {s A g , s B g },{s A g , s B b },{s A b , s B g }, and{s A b , s B b }. 5 Kolotilin (2014) said, the firm should consider the trade-off between quality of signals ( μ j ) versus quantity of signals (q j ). 16 these two conflicting objectives is the key of the firm’s strategy. This is closely related to ‘ Bayes- plausibility’. The firms’ problem is to determine how much information to provide in order to maximize the expected profits. When the firm chooses the distribution over the expected values, one rationality condition is required, which is called ‘Bayes-plausibility’. 6 The expected value of the posterior belief generated by any distributions should be equal to the prior probability, μ 0 . μ j g ∗ q j + μ j b ∗ (1-q j ) = μ 0 , where μ j g , μ j b , and q j ∈ [0, 1]. This Bayesian rationality constrains each firm from making her product appear to be better, on average, than she truly is. The firm wants to deliver the good signals as much as possible, or may want to deliver the good signals only, to increase the sales. However, this is not feasible,due to Bayes-plausibility constraint. Any probability mass generating greater belief than prior must be offset by a probability mass generating lower belief than the prior (Boleslavsky and Cotton, 2014). In order words, the study of Bayesian persuasion answers to the question, “What is the optimal information structure to increase the principal’s profit, without hurting the given prior distribution”. It does not just answer to the question of whether the specific signal benefits the sender or not, it does answer to a more general question of how to be better off if the sender can design any types of information structures. Table 2.1: The information structure signals (s j i )\states (w j ) GOOD BAD s j g t j (transparent) 1-f j (bluffing) s j b 1-t j (humble) f j (frank) 6 It is also related to the LIE (Law of Iterated Expectations) 17 Figure 2.1: The information structure In particular, the tactics beyond the optimal fit revelation activity by the firm j is summarized as in Table 2.1 and Figure 2.1. Where t j , 1-t j , f j and 1-f j represent the conditional probability, it is defined as follows: π{s j g |Good}=t (transparent), π{s j b |Good}=1-t (humble), π{s j g |Bad}=1- f (bluffing 7 ), and π{s j b |Bad}=f (frank). All the representation of signaling structure can be described as this two variable, t j and f j . Accordingly the problem of optimal informativeness of fit revelation activity becomes the choice of optimal level of the level of t j and f j , publicly announced to the public. That is, the strategy profile of each firm φ j consists of μ j i , q j and corresponding t j , and f j where j=A,B and i=g,b. φ j = μ j g ≡ prob{Good|s j g } = μ0t μ0t+(1−μ0)(1−f) with probability q j μ j b ≡ prob{Good|s j b } = μ0(1−t) μ0(1−t)+(1−μ0)f with probability 1− q j . 7 Deception can be defined as intentionally causing someone to have false beliefs. Bluffing in negotiations involves attempting to deceive others about one’s intentions or negotiating position (Carson, 2015). The United States FTC (Federal Trade Commission) defined puffery advertisement as a term frequently used to denote the exaggerations reasonably to be expected of a seller as to the degree of quality of his product, the truth or falsity of which cannot be precisely determined. (Wikipedia) 18 Note that the diagonal term in [Table 1], t j and f j increases the informativeness. In other words, there are two ways to increase the informativeness of signals: increasing t and f. For ex- ample, several factors affect the informativeness of fit revelation activities, such as the number of attributes contained in the advertising, the length of free trial, the level of generosity of return policy, and the number of experienced staffs in the mall. Whereas the off-diagonal term, 1- t j and 1-f j decreases the informativeness of signals. In particu- lar, 1-t j (humble signals) and 1-f j (frank signals) can be interpreted as Type-I error and Type-II er- ror, respectively: Type-I error=probability{s j b |Good}=1-t j , Type-II error=probability{s j g |Bad}=1- f j . For example, when the FDA considers approving for new medicine, it closely monitors the experiments conducted by the pharmaceutical firm. The quality information such as chemical com- ponents, is easy to observe. However, if we define the fit information as the safety, it is hard to observe. The FDA has a chance to update its beliefs about the level safety, after watching the experiment results: passing the test (having a good impression (s g )) or failing the test (hav- ing a bad-impression (s b )). In this situation, the firm carefully designs the informativeness of the experiment by strategically determining the level of 1-t and 1-f. If she designs a very tough test (lower t < 1, Type-I error=probability{s b |Good} >0), such as with thousands of clinical tests during several decades, it is less likely for the firm to generate a good signal (passing the test). But if the FDA knows that this test was difficult to pass, it might infer that this medicine is safe even when the test fails . Meanwhile, if she implements a very easy test (lower f < 1, Type-II error=probability{s g |Bad} >0), such as with a few clinical tests over only a period of months, it is more likely for the FDA to observe a good signal. But, in this case, the results−‘passing the test’−may not credible enough, hence the FDA might doubt safety of the drug. Thus, Bayesian persuasion can be explained as the problem of optimal choice of Type-I error and Type-II error. 19 Generally speaking, the sender designs her signaling structure to minimize Type-I (t=1) errors and incur Type-II errors (1-f>0) 8 . In many cases, the firm would like to generate the good signals as much as possible rather than to build the credibility of the good signals in order to attract the consumer, as a result, bluffing or exaggeration becomes the optimal advertising strategy. However, an interesting result arises when two firms compete in fit revelation activities and the consumer shops online. Compared to the high quality firm who offers bluffing signals (Type-II error), the low quality firm starts to offer humble signals (Type-I error) to survive in the market. There are two extreme strategies in fit revelation activities −fully revealing the information and not providing any information. This is exactly the same as the case in which the firm only considers whether to participate in fit revelation activities or not (Gu and Xie, 2013). When the firm uses a full revelation strategy, it resolves all uncertainty about the fit information. Thus, the consumer can perfectly find his match values between his taste and the product once he observes the advertising. On the other hand, when the firm provides a totally uninformative signal, it conveys no additional information to consumer. In this case, the consumer cannot learn anything from the advertising, he should decide whether to buy only based on the prior belief he had. These two cases are the subset of the question, how much information to provide. This Bayesian-way-signaling model contrasts with classic models where the advertising does signal the quality of products (Milgram and Robert, 1986). If there is uncertainty in quality and quality-heterogenous firms implement signaling, then it is always optimal for the high quality firm to reveal her type as much as possible. And the low-quality firm tends to consider whether or not to mimic high quality firm. However, in this case, when there is fit uncertainty and quality het- erogeneous firms implement the signals regarding the fit, it is not always good for the high quality 8 Since, in order to make a consumer buy, the sender always gives just favorable beliefs, less than 1, unless the consumer extremely shirks the Bad state (when the utility of the Bad state is very low), the posterior belief after observing a good signal does not have to be totally informative, μ g 6= 1. 20 firm to help consumers reduce their uncertainties, since such an action can harm the profit if the fits are not good. Thus, each firm should consider the optimal degree of fit revelation activities by carefully determining the relative levels of f and t. Figure 2.2: The timing of the game Figure 2.2 shows the timing of the game. In the first stage, firms choose their optimal advertising strategies, the optimal distributions over the posteriors, φ j (μ j i=g,b , q j ) 9 . Once firm(s) choose the signals, both the signals and their realizations (si=s g , s b ) are observed by a consumer. Based on these impressions, the consumer updates his beliefs and decides whether or not to buy the goods. 10 In particular, there are two ways of observing the signal realizations in the second stage, depend- ing on the shopping environment: offline shopping and online shopping. This is important because the timing of the game is different. The main difference lies in, whether or not a consumer can observe all the signal realizations before he enters the store. When the consumer shops offline stores such as Apple store and Samsung mini shop, the consumer can update his belief only after he en- ters each store and physically inspects one product at a time. On the contrary, when the consumer shops online, the consumer can observe and compare all the signal realizations simultaneously, which derives a harsh competition between two firms. 9 The price or the quality is determined ahead of the signalling stage, if the price competition or the investment in the quality is considered. See section 5 for the results. 10 Each consumer buys the goods iff, EU(.)= 0. 21 2.3 Results and Analysis In this section, I solve the model for the competition between two firms in optimal information revelation. There are two things affect the optimal information revelation of the firm: first, the quality advantage one firm (firm A) has and second, the timing of the game. Before analyzing the latter one, I consider the former one which directly derives an intuitive result. Regardless of the market channel−either offline or online store, there exists a trivial equilibrium in which firm A takes all the market without participating in any fit revelation activities, when the quality advantage of a firm A is high enough ( αμ 0 ≥ 1). The condition whether αμ 0 ≥ 1 or αμ 0 < 1, which determines the trivial equilibrium implies the feasibility of the best scenario for the high quality firm A −to win the low quality firm B with probability 1. This scenario is available if she is affordable to send the high enough signals to win firm B, no matter what happens. That is, whether or not it is possible for firm A to send a single strong enough signal to attract all demand. The condition, αμ 0 ≥ 1, exactly reflects this scenario. The analysis is consistent with the feature of price cutting in Bertrand competition. When there are two firms and one firm has an advantage in non-price attribute over the other, finding the equilibrium of Bertrand model is closely related to the corner solution. Similar logic applies in the signaling competition between the high quality firm and low quality firm. The price cutting incentive in Bertrand model is exactly the same as the belief building (provoking) incentive in this model. That is, μ B i =1 is the most aggressive signal firm B can send. Against this firm B’s attack, firm A’s minimum response to win firm B is, μ A g = 1 α . The problem is, whether firm A can always choose this signal or not. Due to the rationality condition, the maximum probability mass she can choose is q A =αμ 0 when μ A g = 1 α . Thus, the αμ o ≥ 1 means that firm A is affordable enough to do this, even with a over-bidding in μ A g , such that 1 α ≤ μ A g ≤ μ 0 without sacrificing the probability 22 mass, q A =1. As a result, firm A always wins in the belief building competition, firm B cannot gain any positive profits 11 . In this situation, the firm B’s optimal strategy is full revelation (2.1). φ B = μ B g =1, with probability q B =μ 0 μ b =0, with probability 1-q B =1-μ 0 (2.1) Proposition 1 When firm A’s quality advantage is high enough ( αμ 0 ≥ 1), there is a unique equilibrium in which, firm A captures all the market ( Eπ A =1, Eπ B =0) even though firm A gives no information and firm B chooses full-revelation (t B =f B =1). The following section analyzes the optimal fit revelation activities of firms in two market chan- nels, the brick-and-mortar store and online store, when firm A’s quality advantage is not high enough (αμ 0 < 1). Then even the high quality firm should participate in the fit revelation activities, and the low quality firm has a chance to earn the positive market share. 2.3.1 Brick-and-Mortar Store Shopping: Sequential Game I consider the brick-and-mortar store shopping case where a consumer can observe only one signal at a time. 12 This is consistent with offline store shopping where the consumer can physically observe the signals only after he enters the store, as in the experience shops of Alpple and Samsung. There are two main assumptions which describe the offline shopping. First, similar to the Pandora box problem, a consumer sequentially searches the market, one firm at a time. After observing the 11 In an off-the equilibrium path, firm B always gains nothing, the full revelation strategy ( μ B g , μ B b )={1,0} is indifferent to any other strategies, such that Bayes-plausible. However, firm B chooses the most aggressive signal which induces the maximum belief of consumers, full revelation, in the equilibrium. (See the Appendix for the proof.) 12 Brick-and-mortar store refers to businesses that have physical (rather than virtual or online) presences. In other words, those are stores (built of physical material such as bricks and mortar) that you can drive to and enter physically to see, touch, and purchase merchandise. (Whatls.com) 23 signals from one firm, the consumer has a chance to leave the firm without purchasing the product and move to the other firm. Second, each consumer faces the limited time and physical energy. In other words, a consumer can freely leave the first store without purchasing the product, but after visiting the second store he should buy the product from there. The window shopping at the second period is not available, due to the physical limits. The consumer cannot go back to the first store, no matter how bad the signal is, meaning there is no free recall. 13 If we assume the perfect free recall with no search cost in off-line shopping setup, then such an assumption will prove to be equivalent to the online shopping setup. Consider the offline store shopping situation of the consumer who wishes to buy a new released cell-phone, either iPhone7 or Galaxy7. It is widely believed that Apple’s product is superior to Samsung, but he is still uncertain about the fit values of iphone7 and Galaxy7, which draws him into their physical stores. At the beginning, the consumer considers where to stop first. The visits either Apple or Samsung store, which gives higher flows of expected utility. Suppose that he visits Apple store first. After physically touching and seeing iPhone7, he will purchase it if he receives the favorable enough impression compared to Galaxy7. If not, the consumer leaves Apple store and goes to Samsung store to inspect Galaxy7. Since he cannot enter two stores at the same time, he can update only one signal at a time. In this case, when the consumer decides whether to leave Apple store or not after finishing the inspection, he compares his updated beliefs of iPhone7 and prior belief of Galaxy7−while leaving his beliefs for Galaxy7 unchanged during this stage. As a result, the consumers who receive the unfavorable impression from Apple store become the market share of Samsung. 13 Shopping behaviors for certain categories of product also explain the situation of no-free recalls. For instance, the practical shopping skill for fast fashion brand such as H&M, or FOREVER21, is to not drag on your inspection of items: buy it at once if you like it. Usually, the complexity in the mall or high out-of-stock regret refrain consumers from window shopping (Ozer and Zheng, 2016). 24 Note that the store choice at the beginning is the key of the problem, as the first product sampled by a consumer (the first-stop store) is more likely to make a higher market demand. Therefore, the equilibrium lies on this question: who has a higher incentive to be the first-stop store? A consumer will first visit the store that gives a higher expected utility flows. EU j (j=A, B) denotes the expected utility when a consumer visits the store j first. EU A ={q A ∗ μ A g ∗ α + (1− q A )∗ μ 0 }V, EU B ={q B ∗ μ B g + (1− q B )∗ μ 0 ∗ α}V. If a consumer visits the store A first, then first-stop store firm A makes sales when the consumer observes the good signal, the second-stop store firm B takes the rest of them. The expected profit of firm A and B is, Eπ A1 =q A ,Eπ B2 =1-q A . If a consumer visits the store B first, then the high quality firm A takes the remaining demand of the low quality firm B. The expected profit of each firm Eπ A2 =1-q B , Eπ B1 =q B . The index such as j1 and j2, represents the order of store choice by the consumer. For example, Eπ A1 and Eπ A2 denotes the expected profit of firm A when she is the first-stop store and the second-stop store, respectively. The following lemma supports this ordering dependent expected profits. It comes from Bayes-plausibility. Since firms cannot make their distribution over the posterior beliefs be higher than the prior belief, no firms can make the high enough posterior beliefs to persuade the consumer to buy her product, even after observing the bad signals (μ j b , j=A,B). Formally, αμ A b μ o , such that{q A ∗ μ A g ∗ α + (1-q A )∗ μ 0 }V ≥{q B ∗ μ B g + (1-q B )∗ μ 0 ∗ α}V μ B b αμ o , such that{q A ∗ μ A g ∗ α + (1-q A )∗ μ 0 }V <{q B ∗ μ B g + (1-q B )∗ μ 0 ∗ α}V . 25 Lemma 1 At the first-stop store, a consumer buys the product if he observes the good signals, otherwise leaves the store and moves to the other store. Accordingly, we can ask the same question in a different way. Who has more willingness to yield the first visited store? The answer is, it is optimal for firm A to be the first-stop store and serve all the full market (the consumers whose mass of 1) to inspect. And it is optimal for firm B to be the second-stop store and take the remaining demand who does not observe the good enough signal from the first-stop store A. The intuition behind lies in the fact that firm B should sacrifice greater probability mass (q B ) in order to attract the consumer at the beginning than firm A ( q A ). If the consumer visits firm B first but observes the bad signal, then he can move to firm A which guarantees α times higher expected utility (αμ 0 ) than the reverse case (visits firm A first, then moves to firm B ( μ 0 )). Thus, firm B has more incentive to decrease the probability of exposing her good signals (q B ), in order to attract the consumers at the beginning. However, the consumer is persuaded to buy only if he observes the good signals, thus sacrificing the probability mass ( q j ) hurts the expected profit of the firm who wished to be a winner. In the equilibrium, firm A attracts all the consumers at the beginning and firm B satisfies to be the second-stop-store. Instead, by op- timally choosing the signaling strategy, the firm B indirectly maximizes her expected profit (1- q A ). The quality advantage of firm A, α, plays an important role in determining the optimal fit revelation strategies of two firms. The basic tactics of firm B is as follows. The consumer who buys from firm B gains Eu B =μ B i V whereas Eu A =μ A i αV where{i=g, b} and α > 1. Since the store B has the inferiority in quality, she should give α times favorable beliefs to consumers than A. This disadvantage forces firm B to sacrifice its more probability mass ( q B ) than the high quality firm. When the consumer visits firm B first, firm B’s tactic is to maximize the probability mass on a 26 good signal to be just favorable enough not to lose consumers to firm A: μ B g ≥ αμ 0 , such that μ B g q M +μ B b (1-q M )=μ 0 . φ B = μ B g =αμ 0 , with probability q B =1/α μ B b =0, with probability 1-q B =1- 1 α . Due to the rationality condition, both firms prefer to deliver totally transparent signals ( t A =t B =1). In the relation to lemma 1, since the consumer who observes the bad signal from one firm will not buy the its product anyway, each firm does not have to build any credibilities via bad signals. It is a strictly dominated strategy for firm A, and a weakly dominated strategy for firm B that leaving positive posterior beliefs to the consumer when he observes the bad signals, μ j b . In addition, the low quality firm B sets her good signal ( μ B g ) to be greater than αμ o , the expected utility when a consumer moves to the store A. It it is a necessary and sufficient condition to persuade a consumer not to leave her store. This condition, μ B g > αμ o always holds in the equilibrium even though all the consumers first visit firm A at the beginning. The firm never participate in the fit revelation activities such as μ B g αμ o in the equilibrium, since it derives EU B < αμ o , which enables her rival, firm A to capture all the market without providing any information. The following lemma summarizes theses preliminary properties of the optimal information structure. Lemma 2 i) Any positive bad signals are dominated strategy for both firms, μ A b = μ B b = 0. ii) μ B g > αμ o in the equilibrium. Lemma 2-(i) simplifies the expected utility of consumer at the beginning. EU A ={μ 0 ∗ α + (1− q A )∗ μ 0 }V, and EU B ={μ 0 + (1− q B )∗ μ 0 ∗ α}V . 27 The optimal strategy profile of each firm is derived in this way. First of all, firm A who wishes to be the first-stop store, would like to send the good signals as much as possible, as long as she can sustain the enough credibility to attract consumers at the beginning. Thus, firm A maximizes her expected profit ( Eπ A =q A ), such that{μ 0 ∗ α + (1-q A )∗ μ 0 }V ≥ μ 0 + (1-q B )∗ μ 0 ∗ α}V . Thus, q A =α∗ q B . At the same time, firm B who becomes the second-stop store, indirectly maximizes her expected profit (1- q A ) by carefully designing her probability mass (q B ). That is, firm B maximizes her expected profit by inducing firm A to minimized q A . Tactically, firm B pressures her q B as low as possible, but not too low, such that firm A earns high enough expected profits as the first-stop store not to yield this position to firm B. Thus, firm B minimizes q B , such that Eπ A1 =q A ≥ Eπ A2 =1- q B . The following proposition summarizes the optimal fit revelation strategies of brick-and-mortar stores. Proposition 2 When brick-and-mortar stores compete in fit revelation activities, there is a unique equilibrium unless the quality advantage of firm A is high enough ( αμ 0 < 1). The optimal strategy profile is {φ A∗ , φ B∗ }. Firm B provides α time informative signals than firm A, but firm A earns α times higher expected profits than firm B (Eπ A =q A = α α+1 , Eπ B =1-q A = 1 α+1 ). The consumer first visits firm A and buys from A if he observes the good signal, otherwise the consumer moves to firm B and buys from B. The optimal strategy profile is {φ A∗ , φ B∗ } is as follows: φ A∗ = μ A g = α+1 α ∗ μ 0 with probability q A = α α+1, μ A b =0 with probability 1-q A = 1 α+1 , 28 φ B∗ = μ B g =(α + 1)μ 0 with probability q B = 1 α+1, μ A b =0 with probability 1-q B = α α+1 . Proposition 2 shows that when two firms compete in offline, the optimal informativeness of signals depends on the level of quality. The firm who has the quality dis-advantage should offer more informative fit revelation activities to survive in the market, than the firm who has the quality advantage. This is the opposite finding of Gu and Xie (2013) that the firm offering the high quality product implements fit revealing activities in a greater intensity than the firm offering low quality product, if the quality gap between two products is small enough. Corollary 1 Both firm (j=A,B) design the bluffing signals (Type-I error=1-f j >0), but firm B reveals more information than firm A (f B >f A ). However, there is no Type-II error (t j =1). Corollary 1 suggests that the consumer perfectly discerns the fit information when he observes a bad signal, since both firms use totally transparent signals ( t j =1). However, it is hard to discern the state when he observes a good signal, since both firms bluff their products ( f j 6= 1). These result hold only when the quality level of both firms is comparable ( αμ 0 < 1). In this case, the firm who has the inferior quality can survive in the market via intensive fit revelation activities. However, when the quality advantage of one firm is high enough ( αμ 0 ≥ 1), as stated in proposition 1, the low quality firm cannot gain the positive market via information revealing activities regardless of the market channels. Recall that the first generation iphones television adverting, calledHello, was totally uninformative. It ?delivered no specific information about the phone to help consumers uncover their valuation, but Apple dominated the smart phone market at that time. The success 29 of Apple’s advertising strategy comes from its overwhelming quality advantage. When the first generation iphone was introduced, it was the very innovative and there were no comparable smart phones in the market. However, as time goes by, the quality gap between smart phone has been decreased, Galaxy series of Samsung has been believed to offer comparable to iphone series of Apple, Apple opens Apple stores to help consumer learning. Also, as Samsung starts to build Samsung mini stores, there exist a harsh competition in fit revelation activities. The interesting feature is, Samsung has aggressively invested in experience shops these days, and their boutique shops will be around 460 square feet, even larger than Apple store. It is also consistent with the widely held belief that the lower quality firm is more eagerly participating in fit revelation activities such as hiring more experienced staff in the mall, or distributing more free trials, which supports this result. Note that a full revelation does not arise as an equilibrium outcome in offline, unless the quality advantage of firm A is high enough. It implies that there is less severe signaling competition between two firms, compared to simultaneous game in online. The sequential search setup in which a consumer can observe a single signal from one firm at a time, sets apart one firm from the other spatially or timely, it mitigates the signaling competition. The simultaneous game in the following section may drive a harsher signaling competition, and a full revelation may appear as an equilibrium outcome. 2.3.2 Online Shopping: Simultaneous Game I consider the simultaneous game of signaling competition between two firms. In contrast to the offline shopping case where a consumer can observe one signal realization at a time, he can observe all the signal realizations at the same time when the consumer shops online. Since each consumer directly compares all the signals from both firms simultaneously, there will be a more harsher 30 signaling competition in online firms than offline firms. Hence there might be no pure equilibrium in certain range of values (α,μ 0 ). 14 Similar logic of a Bertrand competition applies to the simultaneous signaling competition. It is called the competition in belief building, meaning each firm builds up a slightly higher beliefs such as , than the rival’s, to win the market. The price cutting incentive in Bertrand model is exactly the same as the belief building (provoking) incentive in this model. 15 The quality advantage of one firm plays a crucial role in features of the belief building. Since firm A has the inherent quality advantage (α) over firm B, she can always induce α times higher utility than firm B. Thus, when firm A competes with firm B, firm A responses in a less aggressive way by 1 α times to the given firm B’s signal. In turns, firm A can allocates greater probability mass on a her favorable signal ( s A b ), which increases her profit. In other words, firm A is affordable to send the good signals α times more often, without hurting her credibility. When there are two firms and one firm has an advantage in non-price attribute over the other in Bertrand model, the main logic of finding the equilibrium is closely related to the corner solution. Similarly, the main logic of finding the equilibrium in the signaling game lies in how to find the maximum (or minimum) point that one firm cannot follow the signal strategy of the other. Until one of their beliefs reached at certain level, each firm always has an incentive to increase his belief by to win. The firm A responses in less aggressive way by 1 α times and firm B does in more aggressive way by α times. It goes up to the maximum point, 1, a full revelation may arise. {φ j0 } 14 Boleslavsky and Cotton (2014) verified the property of mixed strategy. In particular, they specify the optimal number of signals as a choice variable of each firm, which increases the richness of analysis. In this model, I implicitly assume that there is binary signal. When two firms compete offline, the binary signal is enough to support the equilibrium outcome of each firm. Whereas, when two firms compete online, then the firm (especially, the high quality firm) may wishes to increase the number of signals she can use. 15 This model is similar to Bertrand competition in Hotelling model with consumers who are not evenly distributed between two firms. For example, when two firms are located at extreme to points, 0 and 1, consumers are located closer to high quality firm. 31 represents the full revelation strategy. However, there always exist the deviation incentives once the firm reaches a full revelation strategy. Lemma 3 A full-revelation,{φ j0 } is always not optimal for both firms, A and B. φ j0 = μ j g =1, with probability q j =μ 0 , μ j b =0, with probability 1-q j =1-μ 0 . (2.2) It is easy to find that a full-revelation is always not optimal for the high quality firm A. The firm A can induce the same expected value as firm B does by imposing 1 α times less favorable posterior beliefs−indicating that firm A is affordable to send the good signals α times more often. The maximum belief firm B can induce cannot exceed 1 ( μ B i =1), thus firm A needs only μ A i = 1 α as the maximum response to keep the consumers from the her rival. Any μ A i > 1 α are strictly dominated as it only decreases the ability to allocate greater mass on q A . Thus, the maximum belief firm A wants to induce is, μ A i = 1 α ; μ A i =1 cannot be happened in the equilibrium, a full revelation never happens 16 . In addition, the full revelation is not optimal for firm B, neither. Consider firm B chooses a full revelation,{φ B0 }. Once firm A expects {φ B0 }, then firm A chooses {φ A0 }. Subsequently, given {φ A0 }, firm B always has an incentive to deviate to {φ B1 }, and so on. This subsequent deviation reminds of the properties of Perfect Bayesian Equilibrium: the distribution over posteriors made by each firm {φ j } is the equilibrium, when no firms have incentives to deviates, given {φ −j }. In this part, I solve for the pure strategy equilibrium of the optimal fit revelation when two firms compete in online and obtain the following proposition. 16 This is one of the major differences from the studies of Bolenslavsky and Cotton (2014. 2015). 32 Proposition 3 Unless the quality advantage of firm A is not high enough ( αμ 0 <1), two online firms use the optimal strategy profile {φ A1 , φ B1 } where b=b*. Both firm A and B earn positive profits; A takes the market as long as her good signal is realized, and B takes the market whenever A’s bad signals is realized. The optimal strategy profile, {φ A1 , φ B1 } is described as below: φ A1 = μ A g = 1 α , with probability q A =αμ 0 μ A b =0, with probability 1-q A =1-αμ 0 (2.3) φ B1 = μ B g =1, with probability q B = μ 0 −b 1−b μ B b =b > 0, with probability 1-q B = 1−μ 0 1−b (2.4) The following corollary 2 and corollary 3 supports the results in proposition 3. (See in the appendix A.1) Corollary 2 The firm A designs the bluffing signals (positive Type-II errors, 1-f A >0), with zero Type=I error (t A =1). However, firm B designs the humble signals (positive Type-I error, 1-t B >0), with zero Type-II error (f B =1). The corresponding optimal t j and f j is as follow: t A =1, f A = 1−αμ0 1−μ0 < 1, and t B = μ0−b∗ μ0(1−b∗) < 1, f B =1. Corollary 3 There exists b=b*, as long as μ 0 ≥ 3 4 . There exists b = b* such that Eπ A {φ A1 }≥ Eπ A {φ A2 }, 33 where φ A2 = μ A g = 1 α , with probability q A = αμ0−b 1−b μ A b = b α , with probability 1-q A = 1−αμ0 1−b. Note that providing no-information is not optimal for firm A anymore. The firm A starts to send some level of informative signals as she responds to firm B. Whereas, firm B sends out some level of humble signals (t B < 1). That is, π{s B b |Good B } = 1− t B > 0, which means that firm B intentionally sends out some level of bad signal even in Good state with some probability, in order to prevent firm A from deviating. This enables A to keep μ A b =0. This is better news for B than μ A b = b α (A response to B’s provoke, by imposing μ A b = 1 α μ B b ) since B can take a market as long as (μ A b =0, μ B b =b) is realized. By giving out some humble signals−a risky more that can resulted in losing the market−firm B can make firm A stay with the profile φ A and not follow B. Thus, the key point of equilibrium is whether or not there is certain level of ‘b = b∗’, so as to not stimulate firm A to deviate from 0 to b > 0: b = b* exists, if and only if μ 0 ≥ 3 4 . Due to the advantage in quality, A always responds to B in a 1 α times in a less aggressive way. Thus, we can find a certain value of b, in which firm A does not have incentive to follow up while firm B does. Corollary 2 and Corollary 3 describe these incentives. Intuitively, as μ 0 increases, the profit from ( μ A b ,μ B b ) market becomes less attractive and two firms are less likely to send bad signals ( ∂(1−q j ) ∂μ0 <0), the gain from deviating decreases. ( ∂G ∂μ0 < 0). The results shown in proposition 3 is also supported by following features. First, firm A has no deviation incentive at{φ B1 }, μ A g < 1 α is not optimal, given{φ B1 } for μ o ∈ [ 3 4 , 1). φ A2 = μ A g = 1−A α , with probability q A2 = αμ0 1−A μ A b =0, with probability 1-q A2 = 1−A−αμ0 1−A (2.5) 34 As long as{φ A1 } is played in equilibrium, firm B has no deviation incentives. Decreasing μ B b =b ∈ [0,b∗) is strictly dominated. Due to the quality inferiority, firm B cannot take the ( μ A g , μ B g ) market. Regardless of whether firm B chooses a high signal, her profit is independent of its level of μ B g . Second, firm B has no incentive to decrease its μ B g < 1 in equilibrium given{φ A1 }. Even though profit of firm B seems to be independent of q B , decreasing μ B g < 1 does not happen in equilibrium. Since ∂Eπ B ∂qμ B g > 0, the decrease in μ B g < 1 induces firm A to reduce its good signals (μ A g < 1 α ), which in turn increases the q A . Here, firm B is more likely to face the good signal of firm A. Thus, firm B has no incentive to decrease her μ B g < 1. (See the appendix A.2.) Lemma 4 (Posterior beliefs-followings) As long as μ A b =0 i) the best response of firm A for a good signal is ∂μ A g ∂μ B g = 1 α , given any values of μ B g . ii) the best response of firm B for a good signal is ∂μ B g ∂μ A g =α, given any values of μ A g d (μ B g is always greater than αμ 0 ) As long as μ A b =0, the optimal response of firm A’s good signal against firm B’s good signal is to follow B. Whether or not to follow the rival’s beliefs (μ j i , i ={g, b}) is the main concern in this model. Firm A has two options: (i) following: for whatever values of μ B g follows B, and takes the market with (μ A g ,μ B g ) and (μ A g ,μ B b ), or (ii) escaping: just to give up the market with (μ A g , μ B g ), only targeting the market with (μ A g ,μ B b ) by allocating the highest as possible q A instead. Since the maximum gains from (ii) escaping is greater than loss from it, A always chooses to respond by following B by 1 α times. Putting together Lemma 4 and **, we can deduce that the strategy profile φ A0 and φ B1 is the unique equilibrium, as long as μ A l =0 and as long as the existence of equilibrium depends on whether or not μ A l =0. In the relation to Lemma 4, the existence of the equilibrium depends firm A’s incentive to build up her bad signals in order to capture the market where (μ A b , μ B b ) is realize. The question of whether 35 μ A b =0 or μ A b 6=0, determines the uniqueness of equilibrium. The following proposition completes the distinct features of the optimal fit revelation strategies made by online firms. Proposition 4 The result shown in proposition 3 is a unique pure strategy equilib- rium outcome. Corollary 4 There is no equilibrium with μ A b = A > 0. (See the appendix for the proof.) The results above indicates several marketing implications when two firms compete in online. First, unless her quality advantage is overwhelming, firm A always has an incentive to participate in fit revelation activities against to firm B who always participates in fit revelation activities 17 . Even though a consumer always earns non-negative utility from the purchasing the product, and even it is strictly higher if he buys from firm A, she should participate in fit revelation activities. Second, the optimal signaling strategies of online stores is different from the strategy of offline stores. Recall the results that the brick-and-mortar stores always send bluffing signals and the lower quality firm provides more informative signals than the high quality firm, which is consistent with a widely held belief. Meanwhile, when two firms compete in online, the low quality firm uses a something new signaling strategy−sending a humble signal (Type-I error). Take the advertising as an example. Sometimes companies intentionally reveal heir bad aspects by admitting their flaws or faults in order to survive in the market. This strategy of sending a hum- ble signal is called bluntvertising, which has been highlighted as a brilliant advertising strategy these days. In contrast to the traditional advertising approach of exaggeration, which praises the goods with fancy words and attracts consumers attention with dazzling images, bluntvertisng offers the information in an opposite way with brutally honest messages, which builds new type of credibility. 17 Giving no information about the fit such as using image advertising, is always dominated strategy for firm B. 36 As in Dominos 2009 television commercial http://www.youtube.com/watch?v=AH5R56jILagMae Culpa ads (Turn-around campaign) 18 which frankly reveals the harsh critics of consumers, some- times the firm intentionally reveals her bad aspects in order to build the credibility, even bearing the risk that consumers are more likely to be exposed to bad impressions. While this strategy seemed like a huge risk initially, advertisements stressing transparency and candor ultimately built a lot of equity with consumers, Mark Smith, an analyst at Feltl&co. said. In other words, Domi- nos turnaround campaign to reveal truthfully how bad their pizza tastes increased the credibility of their promise to develop new recipes and improve the tastes. This humble advertising strategy even increased the expectation of customers who imagined the poor tastes after watching the harsh critics of advertising might wonder, does the sauce really taste like ketchup? Actually, Domino’s strategy was successful. After two-and-a-half years of sales declines, same-store sales for 2010 increased 14.3% year-over-year, after Dominos launched the campaign. Figure 2.3: The Partition of Equilibrium 18 In Dominos 2009 Television commercial http://www.youtube.com/watch?v=AH5R56jILagMae Culpa ads (Turnaround campaign), it admits something startling−namely, that its pizza is pretty terrible: Totally devoid of flavor, Dominos pizza crust to me is like cardboard, says a woman in a clip taken from a focus-group panel. Another employee reads another review: The sauce tastes like ketchup. (Paul Farhi, Washington post, ?2010) 37 However, this candid advertising includes a huge risk, hence it does not always guarantee success. Following the proposition 3, we can specify three conditions to succeed; first, it should be a lower- quality firm when there exists a competition (more than two firms in the market) second, consumers can simultaneously observe all the advertising and the last, the quality advantage of one firm should not be high enough. Figure 2.3 summarizes the results so far. Proposition 1 presents the region (A). When firm A’s quality advantage is high enough (αμ 0 ≥ 1), there is a unique equilibrium in which firm A captures all the market (Eπ A =1) without providing. And proposition 3 and 4 is represented by the region (B). When the quality of two firms is similar enough ( αμ 0 < 1), then firm A should participate in the fit revelation activities. There is a severe fit revealing competition between two firms in online, the low quality firm B sends the humble signals (positive Type-I error). That is, in order to survive in the online market, the low quality firm sometimes intentionally reveals her bad states to build the credibility of her signals. This is the opposite way of delivering the information from the strategy of firm A, who designs the bluffing signals (positive Type-II error) in order to send (only) the good signals as much as possible. This equilibrium exists if only if, the prior probability is high enough (μ 0 ≥ 3 4 ) to prevent firm A from deviating (stay at μ A b =0). As in Lemma 4, the existence of the equilibrium depends firm A’s incentive to capture the market where ( μ A b , μ B b ) is realize. As μ 0 increases and the market in with μ A b becomes less important, firm A is less likely to cling to this market. Thus, the pure equilibrium exists. On the contrary, there is no pure equilibrium when the prior probability is low (μ 0 < 3 4 ). In this case, we may extend the equilibrium concept to consider the mixed strategy. (See the appendix) 38 2.4 Application In this section, I extend the main analysis in two ways. First, I incorporate the firms fit revelation activities into a simple model of price competition between two firms. Second, I analyze the quality choice problem as a pre-stage game. Figure 2.4 shows the new timing of the game. Figure 2.4: Timing of the game 2.4.1 Price Competition The firm’s problem consists of two stage. At the first stage, each firm determines the price P A and P B . At the second stage, the firm chooses her optimal distribution over the beliefs. Once the signals are realized, then a consumer buys from the firm either A or B, and the profit of each firms is realized. Note that there is a no-buy-option when the price is considered in the model. When the consumer receives the bad signals from the firm, he may not purchase the product. This feature leads to something new results: both firms always bluff. Tactically, this no-buy option refrains each firm from deviating from μ j b = 0, and mitigates the harsh belief-building (provoking) competition between two firms. The intuition behind is, in order to link any positive μ j b = b(> 0) to the own profit, the firm needs to lower the price just enough to guarantee non-negative expected utility of 39 a consumer. However, the lowered price always hurts the profit of the firm, thus each firm has less willingness to build up the belief of bad signals. As a result, both two firms ( j=A,B) stick their μ j b = 0. I analyze the equilibrium of each subgame by backwards induction. At the second stage, each firm simultaneously chooses the optimal signal structure, taking the prices ( P A , P B ) as given. Each firm adjusts the optimal level of informativeness of signaling to maximize the market share (expected demand) by responding to the prices which have already been set. At the first stage, they know how their choices influence the subsequent choice of signaling structures and the market share. Thus, both firms choose the optimal prices which maximize the profit. When signals respond to prices, the equilibrium in two market channels is characterized in following proposition. The detailed analysis about the two stage game is established in appendix (A.3). Proposition 5 When two brick-and-mortar stores compete in prices, the optimal price and fit revelation activities are described as follows: • Firm A chooses the price P A = αμ 0 V , and firm B sets P B = μ 0 V . • Price competition does not affect the optimal fit revealing strategies of brick- and-mortar stores. The optimal strategy profile is still {φ A∗ , φ B∗ } and firm B provides α time informative signals than firm A. • All the consumers first visit firm A. Consumers buy from firm A if they receives good impressions (μ A g ), and leave the store without buying the goods otherwise. Among these consumers, those with good impressions of product B (μ B g ) buy 40 from firm B. The first-stop store firm A earns the higher profit (Eπ A =q A ∗ P A = α α+1 *αμ 0 V ) than firm B (Eπ B =(1-q A )*q B ∗ P B = 1 (α+1) 2 ∗ μ 0 V ). • The ex-ante consumer surplus is zero, but the ex-post consumer surplus is posi- tive. Proposition 6 When two firms compete in online, the optimal price and fit revelation activities are described as follows: • Firm A chooses any price inside an interval P A ∈ (P L , P H ]⊂ (αμ 0 V, αV ]. and firm B sets P B = V . • Firm A reveals a certain level of information with exaggeration (t A =1 and f A ∈ (0, 1]), chooses any level of informativeness of signals which induce μ A g ∈ (μ 0 , 1] and μ A b =0. Firm B fully-reveals (t B =1 and f B =1). The low quality firm sends more informative signals (Blackwell informative) than the high quality firm. • Those consumers with good impressions of product A (μ A g ) buy from firm A, the consumers with good impressions of product B (μ B g ) and bad impressions of prod- uct A (μ A b ) buy from firm B. Firm A earns the monopolistic profit ( Eπ A =αV μ 0 ) and firm B earns positive profit ( Eπ B ∈ (0, (1− μ 0 )μ 0 V ]). • Consumer surplus is always zero. 41 Proposition 5 and 6 shows several interesting results. First, price competition does not affect the fit revelation activities of offline stores, but it does affect those of online stores. When two firms compete in prices, the optimal fit revelation strategy of the offline stores remains the same as the Section 3.1, but the result of online stores is different from the Section 3.2. The role of the signaling in this part is designed not to dampen competition by segmenting the market demand, but to persuade consumers to buy her product (form a non-negative expected utility). Thus, the price competition at the previous stage mitigates the harsh belief-building competition between two firms. As a result, regardless of the market channel −offline or online, the optimal advertising strategy of both firms is exaggeration. Meanwhile, the low quality firm offers more informative advertising than the high quality firm. Second, both firms’ prices of online stores are higher than the prices of offline stores. In other words, consumers face higher prices when they shop online than offline. This result is opposite with a widely held belief that the online shopping increases the price competition and lowers the prices. The intuition is as follows. Compared to the prices which are common knowledge, the fit values are uncertain. Thus, in order to obtain this information, the consumer needs the ‘search process’which is different in market channels. When the consumer shops offline, he compares the values sequentially. When the consumer shops online, whereas, he compares those things at the same time. The different timing of the search process brings the different results. That is, there is more severe signaling competition in online than offline, which allows online firms to set the higher prices than offline firms. Also, when they compte in offline, the entry decision at the beginning should be taken into account when they choose prices. In order to attract consumers to the offline store at the beginning, each firm should set the prices low enough to make consumers’ ex-ante utility be greater than zero. 42 Third, if they have a chance to choose the channels of market, the high quality firm prefers online to offline store, whereas the low quality firm prefers offline to online store. As we can see above, high (low) quality firm earns higher profit when they compete in online (offline). The similar logic applies in understanding this result. The less direct competition in offline brings a higher profits for the low quality firm. Last, the channels of market determines the consumer surplus. When the consumer shops online, he gains the zero surplus. Meanwhile, when the consumer shops offline, he gains positive ex-post surplus regardless of his store choice. Their ex-ante consumer surplus is zero, since both firms design their strategy to make the consumer be indifferent between buying and not. However, once the consumer enters the store, either A or B, and purchases the goods then he gains positive surplus. Because when the consumer shops offline, he has a chance to switch the store if he receives a bad signal without purchasing the goods (window shopping is available offline). Similar to the stock option, the consumer purchases the goods only if the good signal is realized, otherwise he has one more chance to draw a signal from the second firm. Thus the positive ex-post surplus is realized in offline stores. 2.4.2 Quality choice problem In this part, I consider the quality choice problem. Before the signaling competition stage, each firm has a chance to increase the quality level with some costs. The high quality firm A incurs less cost of improving the quality than the low quality firm B. That is, when they produce the level of quality (V j ), firm A incurs the cost − V A2 α 2 (α > 1), whereas firm B incurs the cost −V B . Similar to the previous study, I analyze the equilibrium of each subgame by backwards induction. At the second stage, each firm simultaneously chooses the optimal information revelation, taking 43 the prices (V A ,V B ) as given. At the first stage, both firms determines the level of optimal quality development by considering the subsequent effect on the signaling competition in following stage. Note that the optimal information revealing strategy is the same as the case with exogeneous quality level. As long as we consider the quadratic cost function, the quality choice problem does not affect the fit revealing strategy, regardless of the market channel −either offline or online. Regarding the quality choice, there is no big difference in a general analysis between two market channels, except those numerical values. The equilibrium in two market channels is characterized in the following proposition. The detailed analysis about the two stage game is established in appendix (A.4). Proposition 7 When two brick-and-mortar firms consider the quality choice problem, the optimal quality level and fit revelation activities are as follows: • There always exist the positive quality investments by both firms. Firm A invests √ α times more in the quality than firm B. • The quality choice problem does not affect on the fit revelation activities. The optimal fit revealing strategy is the same as proposition 2. Proposition 8 When two firms compete in online, the optimal quality level and fit revelation activities are as follows: • The trivial equilibrium exists when firm A’s cost advantage is high enough (αμ 0 ≥1). There is no quality investment (V A =0=V B ), all buyers buy from firm A ( Eπ A =1,Eπ B =0). 44 • Unless firm A’s quality advantage is high enough ( V A V B μ 0 < 1), the optimal strat- egy of each firm is the same as proposition 3, and firm A invests α times more in the quality than firm B. Proposition 7 and 8 implies several features. First of all, compared to the price competition in Section 4.1, the market channel does not bring significant difference in results. Regardless of the market channel, the quality choice problem does not affect the choice of the optimal fit revelation problems. The same results hold as in Section 3, so the main analysis is unchanged. By contrast, there is a slight difference between two channel. First, the numerical values of the quality is different. Firm A who has a cost advantage always produces a higher level of the quality. Especially, firm A invests √ α times, and α times more in the quality production than firm B, in the offline model, and online model respectively. The equilibrium outcome of the quality in offline and online, V j * and V j **, is derived in appendix (A.5). This result also implies the role of the market channel in deciding the quality level. The level of quality made by online firms is higher than the quality made by offline firms, which implies each firm has more willingness to invest in the quality in online than offline. The intuition is, in order to prepare a severe competition in fit revelation activities in online, each firm is more eagerly developing her physical quality level. The interesting thing is, firm B’s willingness to invest for the quality is even increasing with respect to the cost advantage of firm A. From the strategic perspective, the quality is the variable which has the strategic complimentary property. Second, the trivial equilibrium appears only in the online shopping model. That is, when the cost advantage one firm has is dominant then neither of online firms invest in the quality. Even firm A who has the cost advantage in producing the quality, does not invest at all. Even though this superior production structure is not utilized, the fact that firm A has a superior production function enables firm A to take all the market. This equilibrium does 45 not arise in offline store, both brick-and-mortar stores always produce the positive quality. Finally, the quality choice problem highlights several policy implications. Note that each con- sumer’s surplus largely depends on the level of quality in this situation. And we can easily find that there is no quality investment in a monopoly market. The results in proposition implies that the government should encourage the competitive market in order to increase the investment in a quality and the consumer surplus. At same time, the government should foster the low quality firm in order to decrease the quality gap between firms and prevent the trivial equilibrium. Without the proper research and development of the low quality firm, the desired outcome may not achieved in the equilibrium. Otherwise, improving the environment of online shopping might be one way, since each firm has more willingness to invest in quality in online than offline, in order to prepare the harsher competition in signaling. One extreme case of signaling, if the government forces the firm to fully reveal the fit information, then the model with the quality choice becomes the Bertrand model. Due to the strategic compliment features of the quality, two firms competitively invests in quality until the low quality gives up. From the consumer surplus point of view, regulating each firm’s blurring of signals and making them fully revel is desirable. 2.5 Discussion In this section, I discuss some of the assumptions of the model in more detail. I briefly analyze the results when the assumption is relaxed. 2.5.1 Tie-breaking rule What happens if I change the tie-breaking rule? In my model, for a simplification issue, I assumed firm A preferred tie-rule : if EU A (.)=EU B (.), then a consumer buys from firm A, who gives weakly 46 better values. The unique pure equilibrium was found in which each firm uses the strategy profile {φ A1 ,φ B1 } where μ o ≥ 3 4 =0.75 and, b∈ B∗. Now I assume the share the market tie-rule: if two firms ties, then the consumer randomizes fairly between two firms (the expected profit of each firm is 1/2). Then, the results are as follows. In general, the same results holds. Except the fact that the value of is included in firm As strategy: firm A imposes μ A g = 1 α + , and takes all the market whenever μ A g is realized. All the same logic of analysis applies into the model of the new tie-breaking rule, since tie never happens in equilibrium (always, each firm deviates by + and takes a whole market), the share the market tie-rule does not hurts the general results. When it comes to simultaneous game, the change in tie-breaking rule slightly shifts the values of the critical value of μ 0 and b, which support the pure equilibrium; μ o ≥ .6568 = μ o **, and b = b**. (See the appendix B.3.) First of all, note that, for both values of b* and b** exist if,f μ o is high enough. As μ 0 increases, the market in with μ A b becomes less important, it becomes easier to prevent firm A from deviating (stay at μ A b =0). Thus, the range of b* and b** becomes greater, as μ 0 increases. However, the values which support the equilibrium varies. The main intuition comes from the fact that firm A has more incentive to deviate from φ A1 to φ A2 under the new tie-breaking rule. This incentive is represented by the result that b** is less dispersed compared to b*. Since it is harder for firm A to refrain from deviating under share the market-tie rule, the range of b** which sustains the equilibrium, shrinks. The tricky thing is, the critical value μ 0 ** which defines the real value of b**, becomes less compared to the crictical value μ o *(b*). That is, in order to sustain the equilibrium under share the market-tie rule, it is harder to define the value of b**, meaning that the available range of b** shrinks. Whereas, under firm A preferred-tie rule , there are more candidates for value of b* to support equilibrium. Inversely,? in this set-up, it might be harder to define b* with this 47 broader range. This is the reason the critical value of μ 0 ∗ which guarantees the real value of b*, falls behind those of μ 0 **. 2.5.2 Consider heterogenous consumer (θ i ) If we consider the heterogeneous quality preferences of consumers, then we observe interesting features in two stage game: the consumer with low preference for quality chooses the low-qualify firm at the first stage, but may leave the store and go to high-quality firm in the second stage, and vice versa. If both firms can choose their optimal levels of advertising at each period (similar to Coase conjecture), the high- quality firm will increase the beliefs μ H g (t=1)≤ μ H g (t=2) in the 2nd stage to serve low-quality lovers who visited the high-quality firm in the 1st period (as they may have observed bad signals μ H b (t=1) at the previous stage, consequently experiencing very unfavorable impression for high-quality firm). Meanwhile, the low-quality firm will decrease its signal to be μ L g (t=1)≥ μ L g (t=2) and the other way around. That is, the market in 2nd period will be differentiated, which consist of the consumers with heterogeneous prior beliefs. This market division effect may relax the signal-competition between two firms, while exacerbate belief updating and Bayes-plausible problems. Also, if consumers with lower preferences may choose high quality firm in the 2nd period and vice versa, the reverse store choice logic may arise. 2.5.3 Competitive market structure (N firms) What happens when the number of firms increases in the online market? Suppose that there are three firms who offer High, Medium, and Low quality (H ,M, and L) in the market which gives α H V,α M V,α L V , where α H > α M > α L =1. Then the following equilibrium exists when 48 α M μ o < α H μ 0 < 1. μ H g = 1 α H μ H g =0 μ M g = 1 α M μ M g =b1 μ H g =1 μ H g =b2 where, b1<b2 There exists b1 (b2) where H is indifferent between following M, L and not. The value of b1 is determined where Eπ H ( μ H b μ M b = α M α H ) = Eπ H ( μ H b μ M b =0). Similarly, the value of b2 is determined where Eπ M ( μ H b μ L b = 1 α M ) = Eπ H ( μ H b μ L b =0). The level of b2 is expected to be greater than b1, since it is harder to prevent H from following M thanH from following L: firm H will not easily give up the market where (μ H b ,μ M b ,μ L b ) is realized by imposing μ H b =0. Then, firm H takes the market as long as μ H g is realized: Eπ H =q H . If the relative quality advantage of firm M is high enough ( α M ∗b1 > b2), then Eπ M =(1-q H )(q M + (1− q M )(1− q L )) and Eπ L =(1-q H )(1-q M )q L . If α M ∗ b1 < b2, firm M takes the market as long as μ H b and μ M g is realized: Eπ M = (1− q H )q M , and Firm L takes the market only if bad signals of two firms, μ H b and μ M b , are realized: Eπ L =(1-q H )(1-q M ). μ H g = 1 α H μ H g =0 μ M g = 1 α M μ M g =0 μ H g =1 μ H g =b2 if b1 > α M ∗ b2 (0< b2) When the relative level of α M is not high enough, firm L can make firm M as well as H not to follow her, in such a case, there may exist an equilibrium where μ M b =0. When there are N firms that have heterogenous quality advantage- α i U[1, A], then there will be continuous good signals of 1 α i ∈ [ 1 A , 1]. And firm with the highest quality (denoted by A*) can take the market as long as μ A g ∗ is realized, thus (Eπ A∗ =q A∗ ) is always guaranteed−meaning that the firm can adequately enjoy its quality advantage in a highly competitive market. On the contrary, 49 faced with discrete bad signals of bi∈ [0, b], the firm with lowest quality (denoted by A) can take the market only if all the bad signals of others (μ −i b , i=A −i ) are realized at some value of b>0. Also, there will be continuous level of t i ≤ 1, and the market will be very competitive. Recall that the monopolist provides no information. We can conclude that the number the firms increases, the market becomes more informative. This is consistent with the study of Kamenica and Genztkow (2015). 2.6 Conclusion I used the Bayesian persuasion model (Kamenica and Genztkow, 2011) of signaling competition to study how much information the firm provides to consumers, who are uncertain about of the products fit. Fit revelation activities such as advertising, free trials, consumer reviews, or return policies help the consumer learn about the value of the product before the purchase, while the levels of informativeness of these activities vary. I found that the optimal signaling strategy of each firm depends on two factors: first, the degree of the quality advantage one firm has and second, the timing of the game-sequential search (offline shopping) or simultaneous search (online shopping). In general, the low quality firm reveals more information than the high quality firm, regardless of the timing of the game. In some situations, the low quality firm has a chance to earn positive profits by using optimal strategy such as bluffing signals (Type-II error) or even humble signals (Type-I error). The results of this research have both managerial implications and policy implications. From a marketing strategy perspective, this paper provides a starting point to understand the role of the quality and market channels in designing firms advertising strategy. In particular, the use of bluntvertising, which has been highlighted as a brilliant advertising strategy, does not always 50 guarantee success. It requires that three conditions be satisfied: the strategy must be played by a lower-quality firm; the quality advantage of one firm should not be too high; and the consumer can observe all advertisements simultaneously. On the other hand, from a public policy point of view, this study implies that it should encourage the competition between firms in order to increase the consumer surplus and the investment in quality improvement. At same time, government needs to support the firms who produce lower quality products in order to decrease the quality gap between firms. Without the proper research and development (R&D) on the low quality firm, the desired outcome may not be achieved in the equilibrium. This model can be extended to a more general framework. First, the binary signals of the signaling strategy simplifies the analysis as it is enough for each brick-and-mortar store, but the model which allows the firm to choose the number of signals will enrich the analysis. Second, I assumed the complimentary between two attributes−fit and quality with specifying the utility two levels−positive or zero. As long as the product gives a nonnegative utility and the high quality product gives a higher utility than the low quality firm, the general result is not altered, but a more general setup of utility function deliberates the equilibrium. There are two ways for future studies. One direction is to consider the dynamics of signaling competition. When a consumer is exposed to the signals over N periods, he has a chance to switch the brand at every period. As the preferences of a consumer evolve over time, I conjecture that the dynamic approach will generate a probability distribution of a Markov kernel, and so it will enrich the information structure in the context of dynamic persuasion. Another direction is to explore the collusive game, such as the production line design problem of one firm. When a firm produces two products which have different quality- luxury line and distribution line, such as iPhone and iPhone SE (Galaxy and Galaxy Mini), the optimal fit revealing strategy of each line will vary. The results of this inter-brand competition are 51 also different from those of intra-brand competition. It would be interesting to examine how to coordinate the cannibalization effect of one product on the other, which involves many practical issues. I leave these issues to future explorations. 52 Chapter 3 Consumer Loyalty and Spatial Competition between shopping centers (Co-worked with JongJae Lee 1 ) 3.1 Overview O’Kelly (1981) reported that 63% of grocery shopping are multi-stop shopping, and 74% of non- grocery shopping are multi-stop shopping. Multi-stop shopping behavior is well-accepted and com- monly observed phenomenon. For example, some consumers stop into Starbucks for coffee while also stop by Dunkin’ Donuts for donuts, even though both store sell coffee and donuts. Similarly, some consumers purchase the skincare products at SK-II whereas purchase the makeup products at Bobbi Brown, even though both brands have the cosmetics of skincare and makeup line. In this paper, we aim to explain these multi-stop shopping behaviors of consumers in light of the customer loyalty, which is one of the well-known factors to affect these behaviors. We define it broadly as a psychological benefit from purchasing a product in addition to its intrinsic value. 1 Department of Economics, The Johns Hopkins University, Baltimore, MD 21218, USA. Email: jlee427@jhu.edu 53 As in the study of Zhang at el. (2013), we view this store loyalty as a category-specific trait of consumers who are loyal to different stores in different categories. The goal in this paper is to explore the effect of customer loyalty on the existence of multi-stop behaviors, and the patterns of such a shopping behaviors, when two shopping centers compete each other. The shopping centers located at the extrems of the linear city, and can be either a department store or shopping malls depending on the retail mode. A department store is a multi- product firm that sells two goods at the same location, while a shopping mall is a shopping center of single-product firms that sell each of two goods at the same location. Consumers are uniformly distributed across the city and have unit demand of each good. They may travel to either shopping centers and buy both goods at there, or travel to both shopping centers and buy each good where the perceived price is cheaper. Our study represents following two main results. First of all, we show the existence of two-stop shopping equilibrium when the relative benefits from two-stop shopping is great enough. Secondly, we confirm that if a two-stop shopping equilibrium exists, such an equilibrium exhibits the loyalty- seeking behaviors of consumers−eonsumers always purchase the product with loyalty at each store and take advantage of the extra benefit from this loyalty, which is intuitive. One of our contributions to multi-stop shopping literature lies in the fact that our model yields to describe the determination of the features of multi-stop shopping in addition to the mere existence of such behaviors by applying the customer loyalty. Related Literature The present study makes several contributions to the existing Hotelling lit- erature. Our model is a variant of Hotelling’s celebrated model, which account for the model of multi-products and multi-stop shopping. Of course, we are not the only one who study this multi- stop shopping behaviors. As multi-stop shopping behavior becomes well-accepted and commonly 54 observed phenomenon, several studies of store choice have started to focus on not only the problem of where to stop, but also the problem of how many stops (Bell, Ho, and Tang (1998), Hanson (1980), and Popkowski et al. (2004)). These studies reported that more than half of consumers in general exhibits multi-stop shopping behaviors and pointed out that the assumption of a single-stop single-purpose shopper is highly unrealistic. Particularly, our model is closely related to Brand˜ ao et al.(2014). They relate multi-stop shop- ping behaviors to the number of goods and the retail mode of shops−the department store or the shopping mall. One of main implications of Brand˜ ao et al.(2014) is the number of goods should be not too low nor too high (between 7 and 11), although they caution that these numbers are not meant to be taken literally. They point out that the failure to explain multi-stop shopping behaviors in spatial competition model may be due to the usual restriction that the number of goods in those models are two, and thus they argue that when the number of goods are not restricted, their vari- ant of Hotelling model can present the multi-stopping shopping equilibrium. As we argue earlier, nevertheless, they do not provide a prediction regarding the features of these shopping behaviors (the product choice problem and store choice problem). Hence, we extend the study of Brand˜ ao et al.(2014) in two ways. First of all, we show that the multi-stop shopping equilibrium exists even when we hold the number of goods to be two, as long as the level of loyalty is high enough. Secondly, we figure out the pattern of multi-stop shopping, such as loyalty seeking shopping behaviors in addition to the existence of multi-stop shopping equi- librium. These are our main contributions to multi-stop shopping literature. Our result also contributes to the vast literature on customer loyalty. The customer loyalty was traditionally defined in terms of the relationship between an individual’s relative attitude and repeat patronage, which is seen as mediated by social norms and situational factors (Dick, A. and Basu, 55 K. (1994)). The recent studies of customer loyalty distinguish the store-specific loyalty from the category-specific loyalty (Fazlul K. Rabbane et al. (2012), Zhang et al. (2013)). Our model broadly defines the customer loyalty as a psychological benefit from purchasing a product in addition to its intrinsic value, and considers a general customer loyalty as a category-specific trait of consumers, following the study of Zhang et al. (2013). Compared to the studies on loyalty which mainly focused on the empirical study of purchase decision problem, our study incorporates a concept of loyalty into theoretical study of store choice problem. To the best of our knowledge, our model is the first model that introduces the customer loyalty into the spatial competition model of Hotelling-type. We take a different approach to accommodate multi-stop shopping behavior in the Hotelling-type model. Rather than relating multi-stop shopping behaviors to the number of goods (Brand˜ ao et al. (2014)), we relate it to the customer loyalty as a main factor which causes multi-stop shopping. The rest of paper proceeds consists of as follows. Section 2 describes the model setup and preliminary analysis of the profit maximization problem and the optimal pricing by each shopping centers who differ in retail mode. Section 3 presents the main results and section 4 concludes with a brief discussion of possible future works. 3.2 Model 3.2.1 Environment We consider a variant of the model of Brand˜ ao et al.(2014) which is a multi-product version of Hotelling (1929). There are two firms and the spatial distance between them is normalized to unity. As in Hotelling (1929), we consider a unit interval [0, 1] and each firm is located at the end of this interval. Let L(Left) and R(Right) denote the firms located at 0 and 1, respectively. Each firm 56 consists of two shops selling two different goods, good 1 and good 2, thus in what follows, we shall refer to a firm as a shopping center or a store for short. There is a continuum of customers uniformly distributed on this unit interval. Each customer has a unit demand for each good, thus she always purchase both goods. All customers share the same preference. Specifically, the purchase of both goods yields the intrinsic value of V > 0. In addition to the intrinsic value V , every customer gains some additional values by choosing to use a particular store or to buy one particular product. This so-called customer loyalty is captured by a matrix A = a 1L a 2L a 1R a 2R where a ij is the value derived from buying good i = 1, 2 at store j = L, R. For simplicity, we assume that a 1L = a 2R = a > 0 and a 2L = a 1R = 0. Customers perceive good 1 to have a higher value (better quality) when it is on sale in store L, while good 2 is expected to have a higher value when it is on sale in store R. In order to buy a good from store j, a customer whose location is x∈ [0, 1] must take a trip to the store by incurring a traveling cost t > 0 per distance. To be specific, the traveling cost is tx if she buys from store L, and it is t(1− x) if she buys from store R. In what follows, we shall refer to a customer located at x simply as customer x. Let p ij be the price of good i charged by each store j for i = 1, 2 and j = L, R. As each customer’s demand for both goods is perfectly inelastic, she chooses among four possibilities: (i) buying both goods from store L, (ii) buying both goods from store R, (iii) buying good 1 from L and good 2 from R, or (iv) buying good 1 from R and good 2 from L. Let L, R, LR, and RL denote each possibility. Hence, the payoff of customer x∈ [0, 1] in each case can be expressed as follows: u L (x) = V− tx + a− p 1L − p 2L , u R (x) = V− t(1− x) + a− p 1R − p 2R u LR (x) = V− t + a− p 1L + a− p 2R , u RL (x) = V− t− p 1R − p 2L 57 For customer x drops by two stores in the latter two cases, we shall refer to these cases as two-stop shopping, while the rest are called one-stop shopping. The payoffs belonging to the case of two-stop shopping may be summarized into one single expression: u TS (x) = max{u LR (x),u RL (x)} = V− t− [min{p 1L − a, p 1R } + min{p 2R − a, p 2L }] . Note that the presence of the customer loyalty a > 0 reduces the cost, psychologically though, paid by a customer. The customer may find the effective price for good i at store j is not the price p ij on the price tag (set by the store), but p ij − a ij . Based on this observation, we shall work with the notion of the effective price of good i at store j, μ ij = p ij −a ij , instead of p ij . Then, the payoff of customer X in each case is expressed as follows: u L (x) = V− tx− μ 1L − μ 2L , u R (x) = V− t(1− x)− μ 1R − μ 2R u LR (x) = V− t− μ 1L − μ 2R , u RL (x) = V− t− μ 1R − μ 2L and u TS (x) = V− t− [min{μ 1L ,μ 1R } + min{μ 2R , μ 2L }]. For simplicity, we define μ L = μ 1L + μ 2L , μ R = μ 1R + μ 2R , and μ TS = min{μ 1L , μ 1R } + min{μ 2R , μ 2L }. Similarly to Brand˜ ao et al.(2014), the demand for store j = L, R selling good i = 1, 2 depends upon whether there exists a customer who would travel to both stores in search for a cheaper price or a loyalty value. If there is no such a customer, which we refer to as the one-stop shopping scenario, a customer buys both goods either from store L or from store R. These possible scenarios are illustrated in Figure 3.1. 58 Figure 3.1: The partition of consumers (one-stop shopping) Figure 3.2: The partition of consumers (two-stop shopping) 59 Otherwise if there is such a customer, which we refer to as the two-stop shopping scenario, the customers are divided into three groups according to their demand behaviors: customers buying both goods from L, those buying both goods from R, and those buying both goods from differ- ent stores. The two-stop shopping can be either loyalty seeking shopping or non-loyalty seeking shopping. These two scenarios of two-stop shopping was depicted in Figure 3.2. dfd In these scenarios, the demand for each good can be determined by identifying the location of a customer who is indifferent between any two adjacent choices. Specifically, in the one-stop shopping scenario, a customer chooses where to buy both goods altogether, L or R. The indifferent customer ˜ x = 1 2 + μ R−μ L 2t can be found by u L (˜ x) = u R (˜ x), and all the customers lie on the left side of ˜ x would buy both goods at L, while those on the right side of ˜ x would buy at R. Similarly, in the two-stop shopping scenario, we can identify the customer ˜ x L = 1− μ L −μ T S t , who is indifferent from purchasing both goods at L and purchasing only each good from each different shopping center, and the customer ˜ x R = μ R −μ T S t (indifferent between buying all at R and buying each good from a different shopping center. Note that ˜x L < ˜ x R must hold in the two-stop shopping scenario. In other words, the necessary and sufficient condition for the existence of two-stop shopping is determined by ˜ x L < ˜ x R , which is X i=1,2 |μ iL − μ iR | > t ⇐⇒ |p 1L + a− p 1R | +|p 2L − p 2R − a| > t. (3.1) The effective price vector μ = (μ 1L ,μ 2L , μ 1R , μ 2R )∈ 4 + belongs either of the following two sets: P 1 = n μ∈ 4 + : P i=1,2 |μ iL − μ iR |≤ t o P 2 = n μ∈ 4 + : P i=1,2 |μ iL − μ iR | > t o 60 Denote byI L ={i∈{1, 2}|μ iL < μ iR } andI R the sets of goods that are strictly cheaper at L and R, respectively. Then, we may rewrite the expressions for the indifferent consumers as follows: ˜ x L = 1− μ L − μ TS t = 1− 1 t X i∈I R (μ iL − μ iR ), ˜ x R = 1 t X i∈I L (μ iR − μ iL ) When μ iL = μ iR , the half of the consumers buys good i at l and the other half buys it at R (tie-breaking rule). The demand for good i at L is thus q iL = ˜ x if μ∈P 1 min(˜ x R , 1) if μ∈P 2 and i∈I L max(0, ˜ x L ) if μ∈P 2 and i∈I R max(0,˜ x L )+min(˜ x R ,1) 2 if μ∈P 2 and μ iL = μ iR and q iR = 1− q iL . 3.2.2 Profit Maximization Problem and the Optimal Pricing Rules Both shopping centers L and R consist of two different shops, but they may differ from each other in their organization. Each shopping center may take either of the following two forms, a department store and a shopping mall. In the department store, the two shops are under the control of the headquarter and thus they behave like a single firm. On the other hand, however, when the two shops are located in the same shopping mall, the two shops may choose the price of their product independently of each other. dfd We assume that store L is a department store, while store R is a shopping mall. In this section, we shall investigate how these two different organizations of a shopping center set their prices of 61 two goods. Firstly, we consider the department store L’s optimal pricing rules given the prices p 1R and p 2R of the shopping mall R. Under the scenario of one-stop shopping, the profit maximization problem of L is max p 1L ,p 2L (p 1L + p 2L )˜ x In terms of μ, the above problem can be rewritten as max μ 1L (μ L + a) 1 2 + μ R − μ L 2t and the optimal pricing rule is computed as μ L = 1 2 [t + μ R − a] , or equivalently, p L = 1 2 [t + p R ] The optimal pricing is the same as the one without loyalty. 2 This is not surprising. With one-stop shopping, the department may coordinate the prices its shops are selling, thus choosing the price of a bundle that consists of the two goods. Moreover, the demand for the bundle depends on the bundling price of its opponent. That is, both goods as a bundle lie in the market for the bundle, and the department store L is competing with its opponent R in one integrated market. As we assume that the size of the customer loyalty is symmetric across two shopping centers, the effect of the loyalty to the store L exactly cancels out that of the loyalty to the store R. dfd Under the scenario of two-stop shopping, however, this is no longer true. The objective function depends on whether good i belongs toI L orI R . If i∈I L , good i is perceived to be cheaper at store L (μ iL < μ iR ). Otherwise, it is perceived to be cheaper at store R (μ iL > μ iR ). As there are 2 See (4) in Brand˜ ao et al.(2014) 62 two goods, good 1 and good 2, we may write the expression explicitly contingent on whether the two-stop shoppers exhibit behaviors of LR-type or RL-type. dfd Consider first the case where the two-stop shoppers exhibit LR-type behaviors, i.e. buying good 1 at store L, and good 2 at store R. The profit maximization problem of store L is max p 1L ,p 2L p 1L ˜ x R + p 2L ˜ x L or, equivalently, max μ 1L ,μ 2L (μ 1L + a) μ 1R − μ 1L t + μ 2L 1− 1 t (μ 2L − μ 2R ) and the resulting pricing rule for each product is μ 1L = μ 1R − a 2 , μ 2L = t + μ 2R 2 which can be expressed in terms of the prices charged by each store: p 1L = 1 2 p 1R + a 2 and p 2L = 1 2 p 2R + t 2 − a 2 . dfd Similarly for the case where the two-stop shoppers exhibit RL-type behaviors, the profit max- imization problem of store L and its pricing rules are obtained as follows: max p 1L ,p 2L p 1L ˜ x L + p 2L ˜ x R , and μ 1L = μ 1R + t− a 2 , μ 2L = μ 2R 2 or, equivalently, p 1L = 1 2 (p 1R + t + a), p 2L = 1 2 (p 2R − a). 63 dfd To compare, we state that Equation (6) and (7) in Brand˜ ao et al.(2014) (restricted to the case of two goods and LR-type two-stop shopping) as follows: p 1L = 1 2 p 1R , p 2L = 1 2 p 2R + t 2 . dfd Notice that the department store L’s price of good 1 is increasing in the loyalty value a while the price of good 2 is decreasing in a. Due to the presence of two-stop shoppers, the department store L is now competing with its opponent R in two different markets. In the market for good 1, the higher loyalty value attracts more customers to L from R, hence L can raise its price of good 1. On the other hand, in the market for good 2, L does not have the advantage of loyalty in good 2, and thus it needs to lower its price to attract customers who is loyal to good 2 being sold in its opponent R. dfd Now, we turn to the optimal pricing rules of the shopping mall R. As the two shops in the shopping mall set their prices independently, we need to analyze the pricing rule of each shop i = 1, 2 given the prices (p 1L ,p 2L ) of the department L, and the price p jL , j6= i of the other shop in the shopping mall R. By modifying Brand˜ ao et al.(2014) to take account for the customer loyalty a, we study the pricing rule of an individual shop i by partitioning the domain of μ iR into the five different cells, (D1) all customers buy good i at R; (D2) there is two-stop shopping with i∈I R ; (D3) there is one-stop shopping; (D4) there is two-stop shopping with i∈I L ; (D5) no customer buys good i at R: 64 D 1 = [0,−t + μ iL + s Ri ] D 2 = (−t + μ iL + s Ri ,−t + μ iL + s Li + s Ri ) D 3 = [−t + μ iL + s Li + s Ri ,t + μ iL − s Li − s Ri ] D 4 = (t + μ iL − s Li − s Ri ,t + μ iL − s Li ) D 5 = [t + μ iL − s Li , +∞) and the corresponding demand for good i is q iR = 1, μ iR ∈ D 1 = [0,−t + μ iL + s Ri ] 1− ˜ x L = 1 t (μ iL − μ iR ), μ iR ∈ D 2 = (−t + μ iL + s Ri ,−t + μ iL + s Li + s Ri ) 1− ˜ x = 1 2 + 1 2t (μ L − μ R ), μ iR ∈ D 3 = [−t + μ iL + s Li + s Ri , t + μ iL − s Li − s Ri ] 1− ˜ x R = 1− 1 t (μ iR − μ iL ), μ iR ∈ D 4 = (t + μ iL − s Li − s Ri , t + μ iL − s Li ) 0, μ iR ∈ D 5 = [t + μ iL − s Li , +∞) where s Li = P j∈I L \{i} (p jR − p jL ) and s Ri = P j∈I R \{i} (p jL − p jR ). dfd For each shop i = 1, 2 located at R, we may derive its optimal pricing rule in the relevant ranges (D 2 ,D 3 ,D 4 ). For shop i, its profit maximization problems and the corresponding pricing rules can be computed as follows: D 2 : max μ iR (μ iR + a(i− 1))(1− ˜ x L ) and μ iR = μ iL −a(i−1) 2 , i.e. p iR = p iL −a 2 + a(i− 1). D 3 : max μ iR (μ iR + a(i− 1))(1− ˜ x) and μ iR = t + μ L − μ R , i.e. p iR = t + p L − p R . D 4 : max μ iR (μ iR + a(i− 1))(1− ˜ x R ) and μ iR = t+μ iL −a(i−1) 2 , i.e. p iR = t+p iL−a 2 + a(i− 1). 65 dfd As in the case of the department store L, the optimal pricing rule of each shop in the shopping mall R is increasing in the customer loyalty a if the customers are loyal to the good the shop is selling, and decreasing in a otherwise. 3.3 Main Result In this section, we demonstrate that there exists a two-stop shopping equilibrium with a prediction about which type of two-stop shopping equilibria, LR-type or RL-type, arises. Moreover, we identify the range of the loyalty value (relative to the traveling cost t) in which such a two-stop shopping equilibrium is sustained. Although our focus is on the existence of a two-stop shopping equilibrium and its properties, we present the following result about the existence of a one-stop shopping equilibrium for completeness. Lemma 1. For a t ≤ 1 2 , there exists an one-stop shopping equilibrium, and the equilibrium prices are p 1L + p 2L = 5 4 t and p 1R = p 2R = 3 4 t. In addition, more than the half of customers buy both goods from the department store L ˜ x = 5 8 = 0.625 . The profits of the department store L and of each shop in the shopping mall R are π L = 25 32 t and π 1R = π 2R = 9 32 t. In order for the department store to induce one-stop shoppers, the loyalty level should not be high enough. If the level of loyalty is high enough to cover the additional transportation costs, consumers would travel to both stores L and R to seek the extra loyalty (two-stop shopping with LR) and one-stop shopping never arises. In this situation, the department store L should lower its bundle prices to compensate customers for the extra loyalty they would derive from the shopping mall in order to attract these customers to buy both goods from him. Specifically, the exact expression of 66 2a≤ t comes from the following logic. By deviating to induce two-stop Shopping behaviors, the department store L gains more demand in the market for good 1 by ˜ x R − ˜ x while losing ˜ x− ˜ x L in the market for good 2. Notice that ˜ x = ˜ x L+˜ x R 2 . This implies that the gains in the demand for good 1 cancels out the loss in the demand for good 2. That is, ˜ x− ˜ x L = ˜ x R − ˜ x = μ L +μ R −2μ 0 T S −t 2t , where μ L = μ L = μ 0 L . Hence, the profitability of this deviation that induces two-stop shopping depends entirely on the changes in the prices. dfd Now, we turn to the case of a two-stop shopping equilibrium, which is our main focus. Theorem 1. For a t ≥ 11+6 √ 2 7 (approximately, 2.78), there exists a two-stop shopping equilibrium. Then, the equilibrium prices of the products to which customers are loyal are p 1L = p 2R = t+a 3 and of the other products without loyalty are p 1R = p 2L = 2t−a 3 . The corresponding demands and the profits are q 1L = q 2R = 1 3 + a 3t , q 1R = q 2L = 2 3 − a 3t , π 1L = π 2R = (t+a) 2 9t , and π 1R = π 2L = (2t−a) 2 9t . Unlike the case of an one-stop shopping equilibrium, the department store (L) and the shopping mall (R) earn the same amount of profit, which is equal to π L = π R = (t+a) 2 +(2t−a) 2 9t . Now that there are two-stop shoppers, traveling to both shopping centers, the positive externalities one shop’s price has on the profit of the other shop in the same shopping center disappear. In the previous case of an one-stop shopping equilibrium, the two goods are tied as a bundle. Each customer buys both goods from the only one shopping center located either at the right or the left extreme. In other words, the market for good 1 and the one for good 2 are integrated as one market for the bundle of good 1 and 2. On the other hand, however, when it comes to a two-stop shopping equilibrium, the two markets are independent, i.e. the price of one good set by a shop does not affect the price of the other good. With the absence of the positive externalities, the department store cannot do better than the shopping mall by coordinating the prices of goods in the two different shops. 67 Corollary 1. The two-stop shopping equilibrium exhibits the loyalty-seeking behaviors of the two-stop shopping customers. Specifically, any two-stop shopping customers x∈ (˜ x L , ˜ x R ) would buy good 1 from the department store L, and good 2 from the shopping mall R to take advantage of the additional loyalty values. Corollary 1 indicates that consumers travel to both store to enjoy the extra loyalty values. The opposite pattern of two-stop shopping, traveling to both store and purchasing the product without loyalty (RL-type equilibrium), does not arise in the equilibrium. Notice that this non-loyalty seeking behavior corresponds to the non-existence result of Brand˜ ao et al.(2014) under the case in which the number of goods is low (n=2), for this pattern of shopping does not involve any loyalty. The intuition behind is also consistent with the case of low number of goods in Brand˜ ao et al.(2014). When there are two types of goods and the consumers choose not to gain any extra loyalty values during their shopping trips, there are very few shoppers who are willing to travel both stores and it is profitable for the shop to capture one-stop shoppers by setting a higher price to extract more surplus from one-stop shoppers, in spite of losing two-stop shoppers. Compared to the papers by Brand˜ ao et al.(2014), our model with customer loyalty yields to describe the features of two-stop shopping, which is our one of main contributions to multi-stop shopping model. dfdf As one most important keyword in this model is loyalty (a), there are several results to keep an eye on under the two-stop shopping equilibrium. First of all, the price, quantity of the shops without loyalty (shop 1R, or department store 1L) under two-stop shopping is decreasing with respect to the level of loyalty (a), which is intuitive. However, the profit of these inferior shops may also increase with respect to loyalty (even the shops who do not have the loyalty (such as shop 2R, or department store 2R), as long as the loyalty level is high enough (a > 2t). The intuition 68 behind is that the shop without loyalty directly suffers from the loyalty level since he reduces the price and loses the demand due to the absence of loyalty. Even the shop without loyalty, however, may indirectly benefits from the loyalty when the loyalty level is high enough. This is from the fact that the loyalty of the other shop, shop 2R for example, attracts the consumers who travels the shopping center at the same location and some of these consumers (whose location is inconvenient to stop by two stores) purchase both goods from this shopping center. Therefore, as the loyalty level of the other shop at the same shopping center is high enough to attract one-stop shoppers, this leads the shop 1R without the loyalty to enjoy the indirect reflected benefit from the loyalty of the other shop. ∂P 1R ∂a =− 1 3 < 0, ∂q 1R ∂a =− 1 3t < 0, but ∂π 1R ∂a =− 4 9 + 2a 9t > 0whena/t > 2 Secondly, the shop who sells the product with loyalty enjoys the higher price, demand, and profits are, as the level of loyalty increases, regardless of retails of mode. ∂P 2R ∂a = 1 3 > 0, ∂q 2R ∂a = 1 3t > 0, and ∂π 2R ∂a = 2 9 + 2a 9t > 0. Lastly, the retail modes−the department store or the shopping mall−play a different role under the two types of shopping behaviors, one-stop shopping and two-stop shopping. Under one-stop shopping, the retail modes yields the asymmetric results. The department store who internalizes the externality enjoys the higher profits than the independent decision maker, the shopping mall. However, under two-stop shopping equilibrium, the results of price, demand, and profit, are sym- metric across the retail modes. The only thing matter is the existence of loyalty: the shop with 69 loyalty enjoys the higher profits and these two shops’ profits are the same regardless of retail modes. The store without loyalty earns the less profits than those who have loyalty, but these two inferior shops’ profits are the same regardless of retail modes. 3.4 Conclusion As a variant of Hotelling’s celebrated model, the aim of this study is to explain the multi-stop shopping behaviors of consumers in light of the customer loyalty. We define this loyalty as a category-specific trait of consumers who are loyal to different stores in different categories and this loyalty is symmetric across the shopping center and consumers. According to our results, we find that there exists two-stop shopping equilibrium which exhibits the loyalty-seeking behaviors of the two-stop shoppers when there are two types of goods. Our main contribution to literature on Helling model and customer loyalty, maybe the fact that our model describes the the features of multi-stop shopping in addition to the mere existence of such behaviors. dfdf Our model can be extended to a more general framework. Firstly, we can explore the endoge- nous choice of retails mode by each shopping center. So far, we focus on the competition between a department store and shopping malls whose retail modes was given. The pre-stage game with the choice of retail mode would enrich this analysis. Secondly, we may consider asymmetric loyalty values. In our main model, we assume the symmetric loyalty values across the shopping centers. However, we can extend this to the case where each shop has asymmetric customer loyalty across the products or each shop has distinct category-specific and store-specific loyalty values. This general setup of loyalty could lead to further insights of the results. dfdf There are two clear directions for future study. One is to consider the problem of mixed bundling. In our model, each shopping center considers the prices of each product bundle, which 70 simplifies the model. 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[Figure 3] presents the values of b*, which supports the equilibrium. Figure A.1: The values of b* 76 Proof. The value of b∗ is derived in this way. Check the deviation incentives. φ A1 = μ A g = 1 α , with probability q A =αμ 0 , μ A b =0, with probability 1-q A =1-αμ 0 . φ B0 = μ B g =1, with probability q B =μ 0 μ B b =0, with probability 1-q B =1-μ 0 The outcome with both μ j b =0 is unstable; Since if the two beliefs are tied, then a consumer buys from A by assumption, B has always deviation incentive to increase his μ B b by to win the market where (μ A b ,μ B b ) is realized. Then, in responding to B’s provoke, A considers to follow her μ A b by 1 α times . When one of their beliefs reached at certain level, but if it is still dominated by the other’s, then she gains nothing from provoking while only wasting the probability mass for μ j b . Thus, it is optimal for her to go back to original point (minimum μ j b =0); similar to the price cutting in Bertrand competition, there is a beliefs building in competitive in Bayesian persuasion. Due to the advantage in quality, A always responses to B in a 1 α times less aggressive way. Thus, we can find a certain value of b, in which firm A does not have incentive to follow up, while firm B does. Compare the profit Eπ B {φ B0 } and Eπ B {φ B1 }; given φ A0 , and μ 0 . Eπ B {φ B0 } = μ 0 (1-αμ 0 ) < Eπ B1 {φ B1 }=(1-αμ 0 ) —————————————————–(A.0) The firm B’s profit always increase; the increased profit by taking the ( μ A b , μ B b ) market is strictly dominant than the loss of decreasing quality mass q B , actually the new profit is independent of q B , thus B always benefits from deviating. Next, consider the profit of firm A’s incentive to follow B. If A follows B’ provoke, by b α , her new strategy profile will be {φ A2 }, given{φ B1 }. 77 φ A2 = μ A g = 1 α , with probability q A = αμ0−b 1−b μ A b = b α , with probability 1-q A = 1−αμ0 1−b Compare the profit Eπ A {φ A1 }=αμ 0 , and Eπ A {φ A2 }= αμ0−b 1−b +{ 1−μ0 1−b 1−αμ0 1−b }, given{φ B1 }, and μ 0. G=the gains from deviation is recovering market of (μ A b , μ B b )= 1−μ0 1−b 1−αμ0 1−b L=the loss from deviation is sacrificing market with μ A g ( q A )=αμ 0 − αμ0−b 1−b In order to make no deviation incentives for A, Eπ A {φ A1 }≥ Eπ A {φ A2 }, there exists a certain level of b = b∗; it should be some point, where one of them (A) can not follow the rivla’s provocation (B). That is, there is a cut-off point in which one party’s gains of deviation equals to the loss of it, thus no incentive to increase or decrease her μ J i . Specifically, when (1-μ 0 )≤ b (1−b) ————————————————————————–(A.1) Thus, there is no incentive for A to build up her belief, as long as there is a certain value of b, which satisfies (A.1). That is, the existence of a pure strategy equilibrium depends whether or not the b = b∗ exists, such that b∗∈ [ 1 2 − √ 4μ 0 −3 2 , 1 2 + √ 4μ 0 −3 2 ], where b∗∈ R + , 0<b<μ 0 . Thus, as long as μ 0 ≥ 3 4 , we can define the value of b, which satisfies (A.1). Intuitively, as μ 0 increases, the profit from ( μ A b ,μ B b ) market becomes less attractive and two firms are less likely to send bad signals ( ∂q j ∂μ0 > 0), whereas the loss from deviating increases, as the market with μ j b becomes more attractive. ( ∂G ∂μ0 < 0, ∂L ∂μ0 < 0). Therefore, A does not follow B’s provoking by b α , 78 keep her μ A b = 0. As long as the{φ B1 }, is played by B, decreasing μ A g ∈ (μ 0 , 1) is not optimal for A, since she should keep the market where (μ A g , μ B g ) is realized, with μ A g ≥ 1 α to beat μ B h =1. A.1.1.2 Proofs for Proposition 3 Claim 1 Eπ{φ A2 } < Eπ{φ A1 }, given φ B1 for μ o ∈ [ 3 4 , 1). Proof. Suppose not, Eπ{φ A3 } gives firm A a better profit than Eπ A {φ A1 }, given φ B1 . φ A3 = μ A g = 1−A α , with probability q A2 = αμ0 1−A μ A b =0, with probability 1-q A2 = 1−A−αμ0 1−A Then, new profit of firm A by decreasing μ A b comes from the market where (μ A g , μ B b ) is real- ized. By decreasing μ A g by A α , A can allocate more probability mass on q A now, but A should give up the (μ A g ,μ B g ) market, instead. New {φ A2 } and old {φ A1 } profits are, Eπ A {φ A1 } = αμ 0 , Eπ A {φ A2 }=( 1−μ0 1−b )( αμ0 1−A ). When firm A deviates from φ A1 to φ A2 , the objective function of A is to maximize q A2 ; q A =1. Firm A prefers to send a single signal, μ 0 with probabil- ity 1. The maximum profit firm A can obtain when she considers the new profile φ A2 =μ 0 is, ˉ Eπ A {φ A2 =μ 0 }=( 1−μ0 1−b ). ˉ Eπ A {φ A2 =μ 0 } ≥ Eπ A {φ A1 }, iff 1−μ0 1−b ≥ αμ 0 where (1− b) ∈ (0, 1). However, (1−μ 0 ) αμ0 > 1, only if μ 0 ∈ (0, 1 2 ), contradiction. And, as long as {φ A1 } is played in equilibrium, B has no deviation incentives; decreasing μ B b =b∗∈ [0,b) is strictly dominated. Due to high enough μ A g = 1 α , the case in which B can not take the (μ A g ,μ B g ) market, no matter how she chooses a high signal, her profit is independent of her μ B g with a combination of μ B b > 0, thus B has no incentive to decrease her μ B g < 1 (Claim 2). 79 Claim 2. B has no incentive to decrease her μ B g < 1, given φ A0 . Proof. Eπ B (φ B1 )=(1-q A )=(1-αμ 0 ), independent of her q B . It seems that any combination of μ B g ,μ B b is indifferent to φ B1 , as long as she can win the market where μ A b is realized. However, the equilibrium is unique; when μ A l =0, the only equilibrium is φ A0 and φ B1 . Decreasing μ B g < 1 is not optimal for B, as long as μ A b =0. Suppose not, μ B g <1. Eπ B {φ B2 }=1-q A =(1-αμ 0 ), where φ B2 = μ B g < 1, with probability q B = μ0−b μ B g −b μ B b =b, with probability 1-q B = μ B g −μ0 μ B g −b Then, decreasing μ B g < 1 is not happened in equilibrium, since, ∂Eπ B ∂qμ B g = (1− ∂q A ∂μ A g )( ∂μ A g ∂μ B g ) > 0, where ∂q A ∂μ A g =- μ0 (μ A g ) 2 and ∂μ A g ∂μ B g = 1 α ———————– (A.2) The first term of (A.2) is from Bayes-plausible condition ( q A = μ o−μ A l μ A h −μ A l ) and the second part is guaranteed by the best response of firm A (Claim 3). Thus, φ B1 is the unique equilibrium, in the case where μ A l =0. 1 Claim 3. ∂μ A g ∂μ B g = 1 α , as long as μ A b =0. Proof. First of all, as long as μ A o =0, then μ B b > 0, by (A.0). Given any values of q A given μ A b =0 thus B has a chance to win against μ A b , Eπ B {μ B b =0}=(1-q A )(q B ) always dominates Eπ B {μ B b >0}=(1- q A ). Secondly, given that μ A b =0 and μ B b =b > 0, firm A has two choices; (i) for whatever values of μ B g , follows her, and takes the market with (μ A g , μ B g ) and (μ A g , μ B b ) are realized. (ii) just gives up the market with (μ A g ,μ B g ), only targets the market with (μ A g , μ B b ) by allocating the highest 1 Thus, the uniqueness of a pure equilibrium depends on whether or not firm A has an incentive to make her bad signals μ A b =0. 80 as possible q A , instead. The best response for (i) is, ∂μ A g ∂μ B g = 1 α , whereas, (ii) is, ∂μ A g ∂μ B g = 0. If A chooses (i), follow B by imposing 1 α times higher μ A g than μ B g , Eπ A {(i)}=q Ai , whereas, if A gives up the good signal market (ii), Eπ A {(ii)}=q Aii (1− q B ). Firm A always chooses to fight (i), if the loss from choosing escaping (ii) (losing the market where (μ A g , μ B g ) is realized) is greater than the maximum gains from escaping (ii) (increasing the winning rates). That is, the loss from choosing (ii); q Ai q B is greater than the gains from choosing (ii); (q Aii -q Ai )(1-q B ). The maximum gains from choosing (ii) is to set q Aii =1, as providing no information and no uncertainty at the same time, occupies the solid demand by targeting the market where (μ A b , μ B b ) is realized. q Ai = μ0 μ A g , q B = μ 0 −b μ B g −b by Bayes-plausible condition. The maximum loss from (ii) and the gains from (ii) is as follows, respectively. Then, the former is always dominated by the latter, the inequality always holds for (i) ∂μ A g ∂μ B g = 1 α , thus q Ai = μ o 1 α μ B g = αμ o μ B g , iff (A.3) holds. L=(1 −q Ai )(1− μ 0 −b μ B g −b ), and G= q Ai μ 0 −b μ B g −b , (1− q Ai )(1− μ0−b μ B g −b ) < q Ai μ 0−b μ B g −b ⇐⇒ (μ B g − μ 0 )(μ B g − αμ 0 ) > 0————————–(A.3) The first term is always positive due to Bayes-plausible condition, as μ B b =b > 0, the second term is positive iff μ B g > αμ 0 ; which depends on value of μ B g . By μ B g > αμ 0 holds for given q A . Therefore, it is always optimal for A to choose (i), ∂μ A g ∂μ B g = 1 α is the best response when μ A b =0. μ B g is always greater than αμ 0 , (αμ 0 < 1), given any values of μ A g . Suppose not, μ B g < αμ 0 ; μ B g < αμ 0 < 1. In this case, firm B also has two options (i) fight; imposing high μ B g ≥ αμ o + 2 and takes the markets where (μ A g ,μ B g ), (μ A b , μ B g ) and (μ A b , μ B b ) are realized. Or, (ii) escape; just focus on the winning rate, q B and takes the market where (μ A b , μ B g ) and (μ A b , μ B b ) are realized. Given any values of q A (μ A g ), the expected payoffs are Eπ B {(i)}=(1-q A )+q A q B , and Eπ B {(i)}=(1-q A ), which 2 The minimum beliefs firm A can has is, μ 0 , thus μ B g should higher enough than αμ 0 +, at least, in order to take the (μ A g , μ B g ) market. 81 is always optimal for B to choose (i). That is, μ B g ≥ αμ o + , at least, which is contradicted by assumption. Thus, μ B g > αμ 0 always holds for any given q A (μ A g ) as long as μ A b =0. Third, the last thing is, according to (A.2), ∂Eπ B ∂qμ B g = (1− ∂q A ∂μ A g )( ∂μ A g ∂μ B g ) > 0, given that ∂μ A g ∂μ B g = 1 α from previous part, thus μ B g =1. Thus, the strategy profile φ A0 and φ B1 is the unique equilibrium, in the case where μ A b =0. 3 A.1.2 Proposition 4 Proposition 4 states that there is no equilibrium with μ A b =A > 0. There are three claims to show that μ A b 6=0 in the equilibrium. That is, there is no equilibrium where μ A b =A < μ 0 α , μ A b =A = μ 0 α ,μ A b =A > μ 0 α . Claim 4-1. There is no equilibrium where μ A b = A < μ 0 α . Proof. Suppose that there is positive A, such that A< μ0 α in equilibrium; Eπ A (μ A b =A)≥ Eπ A (μ A b =A), the strategy profile of firm A which includes μ A b =A should be greater than any other values of A. First of all, between two options, (i) fight; ∂μ B b ∂μ A b =α + , or, (ii) escape; ∂μ B b ∂μ A b =0, taking firm A’s strategy profile as given, the best response of firm B in choosing her bad signal ( μ B b ) is (i); Eπ B (i)=(1-q A ) > Eπ B (ii)=(1-q A )q B . Secondly, given any values of μ B g , between two options, (i) fight; ∂μ A g ∂μ B g = 1 α , or, (ii) escape; ∂μ A g ∂μ B g =0, the best response of firm A in choosing her good signal (μ A g ) is also (i); Eπ A (i)=q A > Eπ A (ii)= (1− q B )q A . In this situation, where μ A b =A, μ B b =α∗ A, μ A g =α∗ μ B g , between two options, (i) fight; ∂μ B g ∂μ A g = α + , or, (ii) escape; ∂μ B g ∂μ A g = 0, the best response of B in choosing her good signal (μ B g ) is also (i); (by Calim 2). Given all the best responses above, the best response of firm A is to increase the value of A, such as A+ , until μ A b = μ0 α , the point 3 Thus, the uniqueness of a pure equilibrium depends on whether or not firm A has an incentive to make her bad signals μ A b =0. 82 where firm B can not follow her μ B b anymore, given that μ B g =1 (by Bayes-plausible condition), therefore firm A can take the market where ( μ A b , μ B b ) is realized. μ A b =A= μ0 α , contradiction. Claim 4-2. There is no equilibrium where μ A b = A = μ0 α . Proof. Suppose that there is positive A, such that A= μ0 α in equilibrium; Eπ A (μ A b =A)≥ Eπ A (μ A b =A’). When A= μ0 α , the strategy profile such as {μ B g =1, μ B b =μ 0 } is not available for firm B (by Bayes-plausible condition). Then, firm B has two options, (i) fight; ∂μb B ∂μ A b =α + , or, (ii) escape; ∂μ B g ∂μ A g =0. Taking firm A’s strategy profile as given, when she chooses (i) then μ B b =μ 0 + , and μ B g <1; Eπ B (i)=(1-q A ). Whereas when she chooses (ii), then she mainly targets the market where (μ A b ,μ B g ) is realized by maximizing q B ; μ B g =μ 0 + with probability 1-, and μ B b =μ 0 − with probability . Given this, the best response of firm A is to increase the value of A, such as A+, until two values of strategy profile becomes { 1 α , 0}; Eπ A (A)=q A +(1-q A ) < Eπ A (A + )=1. Therefore therefore firm A can take all the market by choosing μ A b =A+ > μ 0 α , contradiction. Claim 4-3. There is no equilibrium where μ A b = A > μ0 α . Proof. Suppose that there is positive A, such that A> μ 0 α in equilibrium; Eπ A (μ A b =A)≥ Eπ A (μ A b =A’). Then, firm B has two options, (i) fight; ∂μb B ∂μ A b =α + , or, (ii) escape; ∂μ B g ∂μ A g =0, tak- ing A’s strategy profile as given, the best response of firm B in choosing her bad signal ( μ B b ) is (i); Eπ B (i)=(1-q A ) > Eπ B (ii)=(1-q A )q B . Then, firm A is always better off, by increasing μ A b = 1 α (μ B b ) + , until two values of strategy profile becomes { 1 α , 0}, which means μ A b goes back to zero, contradiction. Thus, there exists the unique pure equilibrium in the simultaneous game. 83 A.1.3 Proposition 6 The problem of brick-and-mortar store consists of two parts: each firm determines the price levels at the first stage, and determines the fit revealing strategy at the second stage. I derive the equilibrium of each subgame 4 . A.1.3.1 At the 2nd stage: given (P A , P B ), optimal signaling structure Taking the prices (P A ,P B ) as given, each firm simultaneously chooses his optimal signal structure. Each firm adjusts the optimal level of informativeness of signaling to maximize the market share (expected demand) by responding to the prices which have already been set. The role of the signaling is designed to persuade consumers to buy her product, not to dampen competition by segmenting the market demands. Since the optimal signaling strategy responds to the prices which have been set at the previous stage, it varies the level of price gaps between two firms. i) When prices are sufficiently favorable for firm A (the price gap is small enough) ( αμ o V− P A ≥ V − P B ⇔ P A − P B ≤ αμ o V − V ), firm A always takes all the markets without giving any information. Meanwhile firm B cannot gain positive market share, regardless of her signaling strategy (even she fully reveals the information). ii) When prices are sufficiently favorable for firm B, (the price gap is large enough) ( αV− P A < V− P B ⇔ P A − P B > αV− V ) 5 , firm A cannot gain the market share regardless of her signaling 4 The main analysis of this part follows the study of Bleslavsky and Cotton (2016), which models competition between two firms: firm A offering an established product (no uncertainty), and firm B offering an innovative product for which consumers are uncertain about their valuations. They consider two cases: the committed prices with flexible demonstrations, and the committed demonstrations with flexible prices. My model refers to the first case. The price is determined first then, the fit revelation activity is adjusted in response to prices. 5 When the price gap can be extremely large, (αV−P A < μ oV−P B ⇔ αV−μ oV < P A −P B ) firm B always takes all the market without giving any information, meanwhile firm A cannot gain positive market share regardless of the signaling strategy(even firm A uses full revelation strategy). But (ii) condition is enough to ensure the equilibrium in which firm A has no incentive to deviate from μ A b = 0 and firm B uses full-revelation. 84 strategy (even she fully reveals the information), as long as the maximum belief μ B g = 1 is realized (firm B uses a full revelation strategy). iii) For the intermediate levels of price gap (αμ o V− V < P A − P B ≤ αV− V ), firm A reveals certain level of information (with exaggeration), firm B fully reveals the information. The optimal signalling{φ A4 ,φ B0 } is as follows. φ A4 = μ A g = 1 α + P A −P B αV μ A b = 0 , and φ B0 = μ B g = 1 μ B b = 0 (A.1) Firm A sells to consumers those with good impressions of product A (μ A g ), and firm B sells to consumers those with good impressions of product B (μ B g ) and bad impression of product A (μ A b ). Therefore, Eπ A = q A ∗ P A = αV μ 0 V +P A −P B ∗ P A , Eπ B = (1− q A )∗ q B ∗ P B = (1− αV μ0 V +P A −P B )∗ μ 0 ∗ P B . A.1.3.2 At the 1st stage: the optimal prices When choosing prices, the firms know how their choices influence the subsequent choice of signal- ing structures and the market share. Both firm know that they will use the optimal signalling profile {φ A4 ,φ B0 } in the following stage. When signals respond to prices, the optimal prices and corresponding profits are as follows. • Firm A chooses any price inside an interval P A ∈ (P L , P H ]⊂ (αμ 0 V, αV ] and firm B sets P B =V . 85 • Firm A achieves the monopolistic profit, Eπ A = αV μ 0 , and firm B achieves a positive profit, Eπ B ∈ (0, (1− μ 0 )μ 0 V ]. Note that Eπ A = αV μ0 V +P A −P B ∗ P A , ∂Eπ A ∂P A = 0 at P B = V , which means the profit of firm A is maximized with any P A in feasible range, as long as P B = V . Firm A achieves the monopolistic profits under the signaling strategy φ A0 , hence has no incentives to deviate. Thus, the equilibrium lies in the incentive for firm B to choose, P B = V. Firm B knows the fact that his profit depends on the level of P A . Given P A , firm B has no incentives to deviate from{φ B0 } to any other strategies. Claim 5:{φ B0 } is the optimal strategy profile for firm B. Proof. First, check that firm B prefers to stick his μ B b = 0. The deviation to{φ B1 }, any positive μ B b = b > 0 is not optimal. φ B1 = μ B g = 1 μ B b = b > 0 In order to make sales by deviating to any positive values (μ B b = b > 0), firm B needs to set the price level as low as μ B b ∗ V , which hurts the profit. Thus, the firm should consider the tradeoff between increasing the market demand and decreasing the prices. • At the 2nd stage, if the given P B is 0< P B ≤ bV , then firm B prefers {φ B1 } over{φ B0 } by setting P B = bV. Firm B takes the market as long as a bad signal of firm A is realized and achieves Eπ B1 = (1− q A )∗bV . • At the 2nd stage, if the given P B is bV < P B ≤ V : then firm B prefers {φ B0 } over{φ B1 } by setting P B = V , since Eπ B0 = (1−q A )∗μ 0 ∗V > Eπ B1 = (1−q A )∗q B1 ∗V (since, q B1 < μ 0 ). 86 By Bayes-plausibility, b < μ 0 , thus firm B prefers P B = V over P B = bV . Accordingly, firm B strictly prefers{φ B0 } over{φ B1 }. In equilibrium, μ B b = 0. Second, check that firm B prefers to stick his μ B g =1. The deviation to{φ B2 }, any values below 1, μ B g (=1-B) < 1, is not optimal. At the second stage, given P B , firm B may want to decrease his good signal μ B g (=1-B) < 1 to increase the probability of sending the good signal (q B2 ), which increases the market share. However, in order to link the increased market share to the profit, the price should be lower enough to make consumers have non-negative utility (i.g.P B =(1-B)V ). Eπ B2 =(1-q A )∗ q B2 ∗ P B2 =(1- q A )∗ μ0 1−B ∗ (1− B)V =Eπ B0 . The increased market share by deviation is exactly offset by the decreased price, no gains from deviation. Firm B weakly prefers{φ B0 } over{φ B1 }. In equilibrium, μ B g =1. Accordingly, the optimal price for firm B is, P B =V . This completes the proof for proposition 6. A.2 Remarks A.2.1 Remarks on Simultaneous Game I consider the case in which there are two independent states of the world, one state for each firm (A,B) −(Good A ,Bad A )∗(Good B ,Bad B )−and analyze the competition of two senders who have different vertical attribute. The most relevant model is the studies of Boleslavsky and Cotton (2014, 2015) (B&C). The high quality firm (A) has the inherent advantage over the low quality firm (B), which enables A to allocate more probability mass on her favorite option (μ A g ) than B. The main different setup from B&C is, how to define the high quality firm. In B&C paper, they assume that firm A has 87 a higher prior belief (μ A 0 > μ B 0 ), but both firms provide the same physical level of valuation from the product, V A =V B =V . However, in my model, I assume homogeneous priors (μ A =μ B =μ 0 ), but heterogenous quality valuations (V A =αV > V B =V ). The setup of heterogeneous prior belief in B&C model brings rich results and deliberate analysis, whereas the setup of the heterogeneous quality valuations in my model represents the quality difference between firms in a more visible way. When we consider the two-dimensional competition phrase, the key is on Bayes-plausible constraint of each firm; μ j g ∗q j +μ j b ∗(1−q j )=μ 0 . Apparently, the level of μ 0 plays the crucial role in determining the optimal signalling. Loosely speaking, the advantage in μ 0 affects more direct effects on increasing the profit ( q j ) than the advantage in quality valuation, thus more deliberate analysis is required to specify the equilibrium. The assumption of the homogeneous prior belief reduces one dimension of a degree of freedom, which simplifies the analysis. For example, compare the case where μ 0 has been increased by α times and the quality has been increased by α times. Bayes-plausible constraint indicates, q A = μ0−μ A b μ A g −μ A b . The increase in μ 0 directly increases the probability mass on a good signal by α times. However, the increase in the quality level enables the firm to decrease μ A g , then this saved proportion mass enables her to allocate more probability mass on a good signal. Thus, the advantage in the physical quality level affects the profits indirectly. other words, it is consistent with the setting in which the Bertrand competition with heterogeneous firms, taking μ j i as a price. B&C model is similar to Bertrand competition with different capacity constraints (related to cost functions, two firm have different marginal cost structures). And my model is similar to Bertrand competition in Hotelling model with consumers who are not evenly distributed between two firms (for example, when two firms are located at extreme to points, 0 and 1, consumers are located closer to high quality firm). The main logic remains the same, however, 88 there are subtle differences in analyzing the model. It brings the different results; for example, in B&C model, when both firms have high enough prior probability and the difference between two priors is not large enough, then both firms plays fully-revealing strategy, however, this result never arise in my model. Loosely speaking, there are harsher signaling competition when two firms have heterogeneous prior beliefs, compared to the setup in which two firms have heterogeneous quality levels. A.2.2 Remarks on Mixed Strategy Equilibrium The mixed strategy equilibrium always exists for any given prior probability level (μ 0 ). Similar to the first price auction with heterogeneous type of consumers model (Gadi, Gavious and Sela, 2002), each firm optimally designs the strategy profile of distribution as {M A , M B }. Taking the Bayes-plausible condition as a budget constraint, each firm chooses optimal distribution over the beliefs. M A μ A o = 0 with probability f A o . . . . μ A i with probability f A i . . . . μ A N = 1 α with probability f A N , M B μ B o = 0 with probability f B o . . . . μ B j with probability f B j . . . . μ B M = 1 with probability f B M where μ A i ∈ [0, 1/α], and μ B j ∈ [0, 1]. F A and F B is cdf. 89 The firm A’s problem is as follows. The firm A maximizes his expected profit such that it is Bayes-plausible. Note that firm A chooses the distribution over posterior beliefs, which is between μ A i ∈ [0, 1 α ]. 6 , 1]isstrictlydominatedstrategy. EU A = P μ A i = 1 α μ A i =0 Prob{μ A i ∗ α≥ μ B j }= P μ A i = 1 α μ A i =0 F B (μ A i ∗ α). (s.t) E(M A ) = μ o . L= P μ A i = 1 α μ A i =0 F B (μ A i ∗ α)− λ(μ o − E(M A )) (i) ∂EU A ∂μ A i = α∗{ ∂ F B (μ A i ∗α) ∂μ A i }− λ(f A i ) = α∗ ∂F B (α∗μ A i ) ∂μ B j ∗ ∂μ B j ∂μ A i − λ(f A i )=0, where ∂μ B j ∂μ A i = α, α∗ f B (α∗ μ A i )∗ α = λ(f A i ) ——— holds for all μ A i ∈ [0, 1/α] Sum-up the optimality condition for all μ A i , then obtain α 2 ∗ P 1/α μ A i =0 f B (α∗ μ A i )=λ∗ 1, Then firm A’s problem is rewritten as below, (i) f A i ∗ = f B (α∗ μ A i )/ P 1/α μ A i =0 f B (α∗ μ A i ), such that, P 1/α μ A i =0 μ A i ∗ f A i = μ o . P 1/α μ A i =0 μ A i ∗ f B (α∗ μ A i )∗ 1/ P 1/α μ A i =0 f B (α∗ μ A i ) = μ 0 .————————————(B.1) Similarly, firm A’s problem is as follows. Firm B maximizes his expected profit such that it is Bayes-plausible. But firm B chooses the distribution over posterior beliefs, which is between μ B i ∈ [0, 1]. EU B = P μ B j =1 μ B j =0 Prob{μ A i < μ B j * 1 α }= P μB=1 μ B j =0 F A (μ B j * 1 α ), (s.t) E(M B ) = μ o . 6 The positive distribution over the μ A i = ( 1 α 90 L= P μ A i = 1 α μ B j =0 F A (μ B j * 1 α )− λ(μ o − E(M B )) (ii) ∂EU B ∂μ B j = 1 α ∗{ ∂F A (μ B j ∗ 1 α ) ∂μ B j }− λ(f B j ) = 1 α ∗ ∂F A (μ B j ∗ 1 α ) ∂μ A i ∗ ∂μ A i ∂μ B j − λ(f B j ) = 0, where ∂μ A i ∂μ B j = 1 α , 1 α ∗ f A (μ B j ∗ 1 α )∗ 1 α = λ(f B j ) ———————- holds for all μ B j ∈ [0, 1] Sum-up the optimality condition for all μ B i , then obtain 1 α 2 ∗ P μ B j =1 μ B j =0 f A (μ B j ∗ 1 α )=λ∗ 1. Then firm A’s problem is rewritten as below, (ii) f B j ∗ = f A (μ B j ∗ 1 α )/ P μ B j =1 μ B j =0 f A (μ B j ∗ 1 α ), such that, P μ B j =1 μ B j =0 μ B j ∗ f B j = μ o . P μ B j =1 μ B j =0 μ B j ∗ f A (μ B j ∗ 1 α )∗ 1/ P μ B j =1 μ B j =0 f A (μ B j ∗ 1 α ) = μ 0 ———————————–(B.2) Thus, the optimal f A i and f B j for μ A i and μ B j exist, such that (B.1)=(B.2). The mixed strategy equilibrium is derived, for any values of the prior probability, μ 0 . A.2.3 Remarks on Tie-Breaking Rule Under the share the market-tie rule*, the unique pure equilibrium exists iff, μ 0 ≥ 0.6568. The unique equilibrium is found when two firms use the strategy profile {φ A1∗∗ , φ B1 }. The new value of b** which supports the equilibrium is derived as follows. φ A1∗∗ = μ A g = 1 α + , with probability q A =αμ 0 μ A b =0, with probability 1-q A =1-αμ 0 ————————————–(B.3) 91 φ B1 = μ B g =1, with probability q B = μ0−b 1−b μ B b =b, with probability 1-q B = 1−μ0 1−b ————————————–(8) where μ o ≥ 0.6568 and, b = b**. A full revelation is always not optimal for firm B. Firm B always has the deviation incentive; Eπ B {φ B1 } < Eπ B {φ B1 } where, φ B1 = μ B g =1, with probability q B0 =μ 0 μ B b =0, with probability 1-q B0 =1-μ 0 Eπ B {φ B0 } = (1− q A )q B0 + (1− q A )(1− q B0 )∗ 1 2 Eπ B {φ B1 } = (1− q A )∗ 1 The firm A may stay under a certain condition ; Eπ A {φ A1∗ }≥ Eπ A {φ A1 } where, φ A1 = μ A g = 1 α + , with probability q A1 = αμ0−b 1−b μ A b = b α , with probability 1-q A1 = 1−αμ0 1−b Eπ A {φ A1∗ } = q A + (1− q A )(1− q B )∗ 1 2 Eπ A {φ A1 } = q A1 + (1− q A1 )(1− q B ) Loss from deviating=L= q A − q A1 = αμ 0 − αμ0−b 1−b Gains from deviating=G=(1− q B ){(1-q A1 ) - 1 2 (1-q A )} = (1−μ0)(1−αμ0) (1−b) 2 - (1−μ0)(1−αμ0) 2(1−b) Loss≥Gains if and only if, b**∈ [ (1+μ0)− √ μ 2 0 +10μ0−7 4 , (1+μ0)+ √ μ 2 0 +10μ0−7 4 ]. 92 Thus, we can define b** as long as, μ 0 ≥ 0.6568. Figure A.2: Compare: b*(firm A preferred-tie rule) vs b**(share the market-tie rule) b∗∈ [ 1− √ 4μ0−3 2 , 1+ √ 4μ0−3 2 ], b**∈ [ (1+μ 0 )− √ μ 2 0 +10μ 0 −7 4 , (1+μ 0 )+ √ μ 2 0 +10μ 0 −7 4 ] Any b* and b** in the range above, for given μ 0 ≥ 0.75 and μ 0 ≥ 0.6568, support the equilibrium. 93 Appendix B Appendix for Chapter 3 B.1 Proof of Lemma 1: One-stop Shopping Equilibrium First of all, we can compute the equilibrium prices by combining the optimal pricing rules of shops in L and R under one-stop shopping. μ R = μ 1R + μ 2R = 2t + 2μ L − 2μ R − a ⇐⇒ μ R = 2 3 (t + μ L )− 1 3 a ⇐⇒ p R = 2 3 (t + p L ) Then, combining this condition with the first-order condition for profit-maximization by the de- partment store (L), we obtain the candidate equilibrium prices. μ L = 1 2 [t + μ R − a] = 1 2 t + 2 3 (t + μ L )− 1 3 a − a ⇐⇒ μ L = 5t 4 − a, p L = 5t 4 ⇐⇒ μ R = 3t 2 − a with μ 1R = 3t 4 and μ 2R = 3t 4 − a, ⇐⇒ p R = 3t 2 with p 1R = p 2R = 3t 4 94 Under this candidate equilibrium price profile, ˜x = 1 2 + t 4 × 1 2t = 5 8 and the resulting profits are π L = 25 32 t and π 1R = π 2R = 9 32 t. To verify the candidate equilibrium profile to be an equilibrium that exhibits one-stop shopping behaviors, we need to show that both stores and the shops belonging to the store do not profit by deviating to induce either LR-type or RL-type two-stop shopping behaviors from the customers. Consider first the deviation incentive of the department store L. If it deviates to LR-type two- stop shopping, then the deviation prices are μ 0 1L = 3t−4a 8 and μ 0 2L = 4t+(3t−4a) 8 and the customers indifferent between the two stores are ˜x L = 1− μ 0 2L −μ 2R t = 7 8 − a 2t and ˜ x R = μ 1R −μ 0 1L t = 3 8 + a 2t . Store L would deviate if ˜ x L < ˜ x R and the deviation profit ( p 0 1L ˜ x R +p 0 2L ˜ x L = 3t−4a 8 + a 3 8 + a 2t + 4t+(3t−4a) 8 7 8 − a 2t = 1 t 3t+4a 8 2 + 1 t 7t−4a 8 2 is no less than π L = 25 32 t. It is not hard to see that the latter condition holds for any (a, t), and that the former condition is equivalent to 2a≤ t. Therefore, the department store L would not deviate to LR-type two-stop shopping if and only if 2a > t. On the other hand, if store L deviates to RL-type two-stop shopping, the deviation prices are μ 0 1L = 4t+(3t−4a) 8 and μ 0 2L = 3t−4a 8 . In this case, ˜ x L = 7 8 + a 2t < ˜ x R = 3 8 − a 2t , or equivalently, 2a + t < 0 must hold. As this condition cannot be met, the department store L would not deviate from the candidate one-stop shopping equilibrium if and only if 2a > t. As for the shopping mall R, each shop i = 1, 2 chooses the price of its product independently. We thus consider the deviation incentives of the shops in the shopping mall separately. Consider first the shop selling good 1 at the mall R, say shop 1R. To check for its deviation incentive, it suffices to show that the optimal pricing rule under one-stop shopping (in the region D 3 ), μ 1R = μ 1L 2 is actually a global maximizer, given μ L and μ 2R of an one-stop shopping equilibrium. Obviously, it is a local maximizer for the price range D 3 (one-stop shopping). The resulting profit is 9 32 t. Following the similar logic in [?], it is easy to see that there is no incentive for deviation when it comes to the 95 price range D 4 (LR). As the price increases, the demand for good 1 at the shopping mall R decreases by−1/2t in D 3 ∂1−˜ x ∂μ 1R =−1/2t , while the demand decreases by−1/t in D 4 ∂1−˜ x R ∂μ 1R =−1/t . That is, the demand for good 1 decreases more in D 4 . Therefore, it is not profitable to deviate by raising the price μ 1R to the range D 4 . Accordingly, it suffices to check whether μ 1R = μ1L 2 is still a local maximizer in D 2 . When deviating to D 2 (RL), the resulting demand is 1− ˜ x L = μ1L−μ1R t , and thus the profit is μ 1R μ1L−μ1R t . The deviation price is determined by the first-order condition, i.e. μ 0 1R = μ1L 2 . The profit is thus μ 2 1L 4t . For this deviation to be profitable, the following two conditions must be met: μ 2 1L 4t > 9 32 t, and μ 0 1R = μ 1L 2 ∈ D 2 = (−t + μ 1L ,−t + μ 1L + μ 2R − μ 2L ) Note that D 2 = (−t+μ 1L +s R1 ,−t+μ 1L +s L1 +s R1 ), but s R1 = P j∈I R ,j6=1 (μ jL −μ jR ) = 0 because j = 2 and the deviation to RL means that 26∈ I R (2∈ I L ), and s L1 = P j∈I L ,j6=1 (μ jR − μ jL ) = μ 2R − μ 2L because j = 26= 1 and j = 2∈ I L . Recall that in an one-stop shopping equilibrium, μ R − μ L = t 4 , thus implying μ R − μ L = μ 2R − μ 2L + 3 4 t− μ 1L = t 4 ⇐⇒ μ 2R − μ 2L = μ 1L − t 2 . By substituting μ 2R − μ 2L with the above expression, μ 1L ∈ (t, 2t). Moreover, recall that in order to guarantee the existence of D 3 , s R1 + s L1 ≤ t must hold. As s R1 = 0, the condition becomes μ 1L ≤ 3 2 t. The first no-deviation condition μ 2 1L 4t > 9 32 t becomes μ 1L > √ 18 4 t. In all, there is no deviation for shop 1R if and only if μ 1L ≤ √ 18 4 t, or equivalently, s L1 ≤ √ 18−2 4 t. For a more elegant expression, notice that in an one-stop shopping equilibrium, μ 1R + μ 2R − μ L = 96 3 2 t− a− 5 4 t + a = t 4 . For we cannot know that how μ L is decomposed into μ 1L and μ 2L in one-stop shopping equilibrium, the expression can be written as t 4 = μ 1R + μ 2R − μ L = μ 1R + μ 2R − μ LR − μ 1L − μ 2L + μ LR = |μ 2L − μ 2R |−|μ 1L − μ 1R | = (μ 2R − μ 2L )− (μ 1L − μ 1R ) ⇐⇒ μ 1L − μ 1R = (μ 2R − μ 2L )− t 4 . Therefore, we have X i=1,2 |μ iL − μ iR | = X i∈I L |μ iL − μ iR | + X i∈I R |μ iL − μ iR | (under RL, 2∈ I L , 1∈ I R ) = |μ 2L − μ 2R | +|μ 1L − μ 1R | = μ 2R − μ 2L + μ 1L − μ 1R = 2(μ 2R − μ 2L )− t 4 = 2s L1 − t 4 ≤ 6 √ 2− 5 4 t Now, we turn to the deviation incentive of shop 2R. Again, it suffices to show that the optimal pricing rule under an one-stop shopping equilibrium, μ 2R = 3 4 t− a is a global maximizer. By the same logic, it is never profitable to deviate to D 4 (RL), for the price raise only hurts the demand for good 2 more than the case of D 3 (One-stop). Hence, it suffices to show that shop 2 R has no incentive to deviate to D 2 (LR). The pricing rule under deviation (in D 2 ) is μ ∗∗ 2R = μ 2L −a 2 and the resulting profit is thus π ∗∗ 2R = μ 2L − a 2 + a μ 2L − μ2L−a 2 t ! = (μ 2L + a) 2 4t . 97 For this deviation to be profitable, the following two conditions must be met: (μ 2L + a) 2 4t > 9 32 t, and μ ∗∗ 2R = μ 2L − a 2 ∈ D 2 = (−t + μ 2L , t + μ 2L + μ 1R − μ 1L ) As in the previous case, s R2 = 0 and s L2 = μ 1R − μ 1L . The condition under OS equilibrium, μ R − μ L = t 4 implies μ R − μ L = 3 4 t− α− μ 2L + μ 1R − μ 1L = t 4 ⇐⇒ μ 1R − μ 1L = μ 2L − t 2 + α. Then, the latter deviation condition μ 2L −a 2 ∈ D 2 = (−t + μ 2L ,−t + μ 2L + μ 2L − t 2 + a) becomes μ 2L + a∈ (t, 2t). Moreover, recall that in order to guarantee the existence of D 3 , s R2 + s L2 ≤ t must hold. As s R2 = 0, the condition becomes μ 2L + a≤ 3 2 t. The first no-deviation condition (μ 2L +α) 2 4t > 9 32 t becomes μ 2L + a > √ 18 4 t. In all, there is no deviation for shop 2R if and only if μ 2L + a≤ √ 18 4 t, or equivalently, s L2 ≤ √ 18−2 4 t. Using the same condition of μ 1R + μ 2R − μ L = t 4 , we have μ 2L − μ 2R = (μ 1R − μ 1L )− t 4 . Therefore, we have the following: X i=1,2 |μ iL − μ iR | = X i∈I L |μ iL − μ iR | + X i∈I R |μ iL − μ iR | (under LR, 1∈ I L , 2∈ I R ) = |μ 1L − μ 1R | +|μ 2L − μ 2R | = μ 1R − μ 1L + μ 2L − μ 2R = 2(μ 1R − μ 1L )− t 4 = 2s L2 − t 4 ≤ 6 √ 2− 5 4 t 98 To sum up, any shop in the shopping mall has no incentive to deviate by inducing two-stop shopping behaviors if and only if X i=1,2 |μ iL − μ iR |≤ 6 √ 2− 5 4 t B.2 Proof of Theorem 1: Two-stop Shopping Equilibrium There are two cases of two-stop shopping, LR (loyalty seeking) two-stop shopping (μ 1L < μ 1R , and μ 2L > μ 2R ) and RL (non-loyalty seeking) two-stop shopping (μ 1L > μ 1R , and μ 2L < μ 2R ). We start by showing the existence of a LR-type two-stop shopping equilibrium, and then we shall show that a RL-type two-stop shopping equilibrium does not exist. To compute an equilibrium price profile of a LR-type two-stop shopping equilibrium, we use the optimal pricing rules derived in the main body (See Section ??). Recall that the optimal pricing rules of the two shops at the shopping mall 1R and 2R are μ 1R = t+μ 1L 2 and μ 2R = μ 2L −a 2 (Equivalently, p 1R = t+p 1L −a 2 and p 2R = p 2L +a 2 in terms of the price posted by both shopping centers). Together with the optimal pricing rule of the department store L, that is μ 1L = μ 1R −a 2 , μ 2L = t+μ 2R 2 , we have μ 1R = μ 2L = 2t− a 3 , μ 1L = μ 2R = t− 2a 3 . Notice that the bundle price (that sells good 1 and good 2 together) set by each shopping center is identical as p L = p R = t. Now, it remains to show that neither the department store (L) nor each shop in the shopping mall (R) has any incentive from this computed price profile. For the department store ( L), it is easy to show that it does not deviate to one-stop shopping if a/t > 1/2. Turn to check for the deviation incentives of the two shops in the shopping mall. Shop 1R would deviate to one-stop shopping 99 by setting its price to be μ 0 1R = t + μ L − μ R following the optimal pricing rule in D 3 given the effective prices of the other shop in the shopping mall μ 2R and of the department store μ L . By plugging the computed values μ 1L , μ 2L , and μ 2R , we obtain μ 0 1R = 5t 6 − a 6 = p 1R because shop 1R is selling good 1 without having customer’s loyalty. The resulting demand and the profit are q 0 1R = 1− ˜ x = 1 2t 5t 6 − a 6 and π 0 1R = 1 2t 5t 6 − a 6 2 . Comparing this with the profit under LR-type two-stop shopping equilibrium π 1R = (2t−a) 2 9t provides us with the condition a t < 11− 6 √ 2 7 ≈ 0.36 or a t > 11 + 6 √ 2 7 ≈ 2.78 that makes shop 1R stay in the LR-type two-stop shopping equilibrium. For shop 2R, selling good 2 with the benefit of customer loyalty, the deviating price to one-stop shopping is μ 0 2R = t + μ L − μ R − a = 4t 6 − 5a 6 and the resulting demand is q 0 2R = 1 2t 4t 6 + a 6 . The profit from this deviation is thus π 0 2R = 1 2t 4t 6 + a 6 2 . Again, similarly to the case for 1R, comparison of the profits leaves us with a t > −4+6 √ 2 7 ≈ 0.64 because a/t > 0. Notice that, given a loyalty-seeking two-stop shopping (LR-type), shop 1R has a stronger incentive for deviation to one-stop shopping. Unlike shop 2R which experiences a reduction in the demand for its product when deviating to one-stop shopping or the department store (L) which does not experience any changes in the demand as a whole, shop 1R would gain more under one-stop shopping by capturing more demand. Hence, the condition for the existence of a LR-type two-stop shopping equilibrium hinges on the no-deviation condition for shop 1R. In other words, LR-type two-stop shopping equilibrium exists if a t ≥ 11+6 √ 2 7 ≈ 2.78. To show the non-existence of a RL-type two-stop shopping equilibrium, suppose to the contrary that such an equilibrium exists. Using the optimal pricing rules derived in the main body, one can easily verify that the equilibrium (effective) price profile is μ 1R = μ 2L = t−a 3 and μ 1L = μ 2R = 100 2(t−a) 3 . This profile of effective prices must satisfy P i=1,2 |μ iR − μ iL | > t as a two-stop shopping equilibrium (effective) price profile. However, plugging the obtained expressions into the condition yields−2a > t. This is a contradiction because both a and t cannot take non-positive values. 101
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Asset Metadata
Creator
Kim, KyunHwa
(author)
Core Title
The essays on the optimal information revelation, and multi-stop shopping
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Economics
Publication Date
07/17/2017
Defense Date
05/01/2017
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Bayesian persuasion,multi-stop shopping,OAI-PMH Harvest,optimal information revelation
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Kocer, Yilmaz (
committee chair
), Dukes, Anthony (
committee member
), Tan, Guofu (
committee member
)
Creator Email
kyunhwak@usc.edu,tuture1107@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-401323
Unique identifier
UC11264407
Identifier
etd-KimKyunHwa-5525.pdf (filename),usctheses-c40-401323 (legacy record id)
Legacy Identifier
etd-KimKyunHwa-5525.pdf
Dmrecord
401323
Document Type
Dissertation
Rights
Kim, KyunHwa
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
Bayesian persuasion
multi-stop shopping
optimal information revelation