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Essays on bounded rationality and revenue management
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Essays on bounded rationality and revenue management
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ESSAYS ON BOUNDED RATIONALITY AND REVENUE MANAGEMENT by Seungbeom Kim A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) May 2014 Copyright 2014 Seungbeom Kim Dedication To my family, for their unconditional love and support. ii Acknowledgments First, I would like to thank my advisor, Professor Sriram Dasu. His great guidance and support allowed me to clear the hurdles throughout my PhD study. I deeply appreciate the time and eort he spent advising me and proofreading my thesis. He kept me motivated to nd valuable research topics and challenged me to become the best researcher I could be. He was not only my academic advisor but also my mentor in life. Due to his guidance, I have been able to make my rst steps as a researcher. I also want to express my deep gratitude for Professor Tim Huh. I truly appreciate his time and support during my doctoral and master's studies. He oered me valuable guidance both inside and outside of the academic realm. I am also thankful to Professor Ramandeep Randhawa and Professor Isabelle Brocas, for being committee members and providing invaluable comments on this thesis. I also thank Prof. Greys Sosic for her help as a PhD advisor, and Prof. Ravi Kumar for his support and advice. I'd like to thank my colleagues during my PhD study: Youngki Park, Guangwen Kong, Jeunghyun Kim, Fang Tian, Dongyuan Zhan, and Dongkyun Yim. I owe much appreciation to Bohyun Choi for helping me with programming, as well as Daniel Lee and Jacob Lee for helping me design and manage experiments. iii Finally, I would like to thank my parents, Jaeyoung Kim and Moungsook Hong, and sister, Bomi Kim, for their unconditional love and support. Feb 2014 iv Table of Contents Dedication ii Acknowledgments iii List of Tables viii List of Figures x Abstract 1 Chapter 1: Introduction 3 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Experiment and Modeling of Strategic Customers . . . . . . . . . . . . . . 6 1.3 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 2: An Experimental Study of Posted Prices 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Analytical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Design of the Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Experiments LOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.1 Analysis of the LOT Experiments . . . . . . . . . . . . . . . . . . 24 2.6 SIM: Simulation of the Two-Period Posted Pricing Problem . . . . . . . . 29 2.6.1 Analysis of the SIM Experiments . . . . . . . . . . . . . . . . . . . 31 2.6.2 Comparison of Behaviors In LOT and SIM Experiments . . . . . . 37 2.6.3 Changes in Decision Making over Time . . . . . . . . . . . . . . . 39 2.7 Implications for the Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7.1 Errors in Estimated Revenues under the Dierent Models . . . . . 41 2.7.2 Impact on Optimal Pricing and Revenues . . . . . . . . . . . . . . 42 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 3: Dynamic Learning in Strategic Customer Behavior 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Dynamic Learning of In-Stock Probability . . . . . . . . . . . . . . . . . . 48 3.2.1 Experimental Results: Dynamic Decision Making . . . . . . . . . . 48 3.2.2 Estimation with Simple Learning Model with Exponential Smoothing 51 v 3.2.2.1 Model Description and Estimation . . . . . . . . . . . . . 51 3.2.2.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . 56 3.2.3 Anchoring Bias and Asymmetry . . . . . . . . . . . . . . . . . . . 57 3.2.4 Implications for the Firm . . . . . . . . . . . . . . . . . . . . . . . 59 3.3 EWA Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.1 Experimental Design: CONV Experiment . . . . . . . . . . . . . . 60 3.3.1.1 Experimental Design: CONV Experiment . . . . . . . . . 60 3.3.2 Results of the CONV Experiments . . . . . . . . . . . . . . . . . . 61 3.3.3 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.3.1 EWA Learning Model and Estimation Result . . . . . . . 63 3.3.3.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . 67 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Chapter 4: Dynamic Posted Pricing Scheme: Existence and Uniqueness of Equilib- rium Bidding Strategy 72 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.1 Model Description and Threshold Policy . . . . . . . . . . . . . . . 74 4.2.2 Multiplicity of Equilibrium . . . . . . . . . . . . . . . . . . . . . . 77 4.2.2.1 Deterministic Distribution Case . . . . . . . . . . . . . . 78 4.2.2.2 Continuous Distribution Case . . . . . . . . . . . . . . . . 80 4.2.2.3 Normal Distribution Case . . . . . . . . . . . . . . . . . . 85 4.2.3 Existence of the Unique Equilibrium: Uniform Distribution Case . 87 4.2.3.1 Existence of the Unique Equilibrium . . . . . . . . . . . . 87 4.2.3.2 Revenue Function . . . . . . . . . . . . . . . . . . . . . . 92 4.3 Behavioral Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3.1 Two Symmetric Buyers . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3.1.1 Modeling Bounded Rationality . . . . . . . . . . . . . . . 94 4.3.1.2 Equilibrium of the buyer's bidding strategy . . . . . . . . 97 4.3.1.3 Revenue of the rm . . . . . . . . . . . . . . . . . . . . . 100 4.3.2 Multiple Symmetric Buyers . . . . . . . . . . . . . . . . . . . . . . 102 4.4 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 References 110 Appendix A Tables in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Appendix B SIM Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Appendix C LOT Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 vi Appendix D Sample Screenshots of SIM experiments . . . . . . . . . . . . . . . . . . . . . . 130 Appendix E Sample LOT Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 vii List of Tables 2.1 Description of the experiments . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Parameters for the LOT experiments . . . . . . . . . . . . . . . . . . . . . 24 2.3 Percentage of subjects employing a threshold policy in the LOT treatments (total: 142) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Goodness of Fit Test for the QRM to LOT Treatments . . . . . . . . . . . 28 2.5 Log likelihood and likelihood ratio tests for RUMM and QRM . . . . . . . 29 2.6 Parameters for SIM(SIM-I) Treatments . . . . . . . . . . . . . . . . . . . 31 2.7 Percentage of subjects employing a threshold policy . . . . . . . . . . . . 32 2.8 Percentage of subjects acting as bargain hunters or myopic customers . . 32 2.9 Goodness of Fit Test for the QRM to SIM(SIM-I) . . . . . . . . . . . . . 34 2.10 Log likelihood and likelihood ratio tests for RUMM and QRM . . . . . . . 35 2.11 Comparison of estimated parameters in rst 30 trials and last 30 trials . . 40 2.12 Predictions of the percentage who buy in the rst period (Number in paren- thesis are error %) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.13 Revenues predicted by RUMM and QRM . . . . . . . . . . . . . . . . . . 41 2.14 Changes in Optimal Pricing and Additional Revenue: RUMM vs. QRM . 43 2.15 Changes in Optimal Pricing and Additional Revenue: RA-LOT vs. QRM 43 3.1 Trend in the proportion of the decision \Buy in the rst period" at each trial ( t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 viii 3.2 Estimation of the Parameters by the Exponential Smoothing, RUMM, and Random Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Estimation of the parameters by Exponential Smoothing Model . . . . . . 54 3.4 Average Number of Trials to Find the True In-Stock Probability . . . . . 57 3.5 Goodness of Fit Test for the QRM-A to SIM-R . . . . . . . . . . . . . . . 59 3.6 Expected Revenue in each SIM experiments . . . . . . . . . . . . . . . . . 59 3.7 Parameters for CONV experiments . . . . . . . . . . . . . . . . . . . . . . 61 3.8 Trend in the percentage of subjects who purchase in the rst period . . . 61 3.9 Percentage of subject that employed the same decision in the last few trials of CONV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.10 Estimation of the Parameters by the Learning Models . . . . . . . . . . . 67 4.1 Comparison of resulted and actual in-stock probabilities in SIM experiment in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Parameters and results for experiment set SIM1-E and SIM2-E . . . . . . 107 A.1 Out of Sample Test: Goodness of Fit Test of SIM-I Treatments . . . . . . 119 A.2 Out of Sample Test: Goodness of Fit Test for SIM Treatments . . . . . . 120 A.3 Model Comparison for SIM2-I: QRM, RUMM and LOT . . . . . . . . . . 120 A.4 Model Comparison for SIM2: QRM, RUMM and LOT . . . . . . . . . . . 121 ix List of Figures 2.1 Aggregate results for LOT treatments. . . . . . . . . . . . . . . . . . . . . 25 2.2 Estimations by QRM Model. . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Aggregate results for SIM(SIM-I) treatments . . . . . . . . . . . . . . . . 31 2.4 Estimations by QRM Model to SIM(SIM-I) treatments . . . . . . . . . . . 35 2.5 Estimations by QRM Model to SIM(SIM-I) experiments . . . . . . . . . . 38 3.1 Proportion of the decision, \Buy in the rst period" at each trial ( t ) . . 49 3.2 perceived in-stock probability at trial, t . . . . . . . . . . . . . . . . . . . 55 3.3 Percentage of customers who purchase in the rst period . . . . . . . . . . 62 3.4 Probability of Buying in the First Period and EWA Estimates . . . . . . . 66 3.5 Samples of Asymptotic Analysis from CONV1 . . . . . . . . . . . . . . . . 68 3.6 Samples of Asymptotic Analysis from CONV2 . . . . . . . . . . . . . . . . 69 3.7 Samples of Asymptotic Analysis from CONV3 . . . . . . . . . . . . . . . . 69 3.8 histogram of P 1 (t = 200) in CONV1, CONV2, and CONV3 . . . . . . . . 70 4.1 Uniform Distribution Examples. In (a), the equilibrium 1 = 2 value is 0.68. In (b), the the equilibrium 1 = 2 value is 1, i.e., not bidding in the rst period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2 Revenue as a Function of Rationality( ). . . . . . . . . . . . . . . . . . . 101 D.1 Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 x D.2 Result - In Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 D.3 Result - Out of Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 xi Abstract This thesis aims to shed light on the bounded rationality of customers and its impact on revenue management policies. Understanding the decision process employed by customers is vital for analyzing dynamic pricing policies. This thesis consists of three research projects. The rst chapter is concerned with a laboratory experiment to study how customers make purchase decisions when they have the option of buying at a higher price or waiting for a lower price but incur the risk of the product going out of stock. We tested some of the core assumptions about consumer decision making processes that are commonly employed in normative theories. Most subjects are not perfect optimizers, bargain hunters, or myopic. We nd that they vacillate between the two options and hence act as if their assessment of the two options includes a random term. Surprisingly, the apparent randomness in evaluating the utility of options dominates the impact of the errors in the estimations of the availability of the product. We nd that the quantal response model (QRM), a quasi-rational model, provides a more accurate description of customers' decisions. We show that ignoring bounded rationality can result in substantial errors in estimation of revenues. 1 In the second chapter, we study the learning process of strategic customers. We adopted the exponential learning model to study how customers discover in-stock prob- abilities. We nd that there is an anchoring eect in learning and losses loom larger than gains. We also implemented an experience-weighed attraction learning model to study the impact of actions and outcomes on decisions. Results show that subjects place a high value on past history and move gradually from the initial anchor. In addition, subjects pay more attention to outcomes they experienced and less from ctitious play. Furthermore, revealing information about product availability is benecial only when the in-stock probability is no greater than 50%. In the third chapter, we conducted a multi-methodological analysis to study equilib- rium behavior in strategic customers. First, the uniqueness issues in theoretical models are investigated. We show that multiple equilibria can exist even in the simple setting where two identical customers compete for one unit of an item. We prove the existence of a unique equilibrium when their valuations follow uniform distribution. Second, bounded rational models are analyzed. We show that a unique equilibrium exists when there are two identical buyers who are bounded rational. We also nd that interactions among multiple bounded rational customers can yield an equilibrium. Finally, our experimental results suggest the existence of equilibrium behavior when subjects interact with each other. 2 Chapter 1 Introduction 1.1 Motivation Ever since dynamic pricing became one of the essential tools of the revenue management, rms' pricing scheme and customers' strategic behavior have evolved hand in hand. Dy- namic pricing schemes have developed dramatically since the deregulation of the US air- line industry in the late 1970s and the practice is now common in many other industries including hospitality, rental car, cruise line, railway, energy, air cargo, and broadcasting industries (Talluri and Van Ryzin, 2005). 2-8% increase in revenue is attributed to these pricing technologies (Smith et al., 1992). Analyzing 1,000 companies, McKinsey & Co. reported that improvement in a rm's revenue management skill could generate 7.4% of increase in prots (Anthes, 1998). With wide implementation of dynamic pricing schemes, customers also began to react strategically. Nowadays, price volatilities of airfares and huge discounts during Black Friday are so established that they loom large in the minds of consumers. Aware of such price changes, consumers have begun to research price changes and time their purchases. 3 Such strategic behavior is based on their personal experiences and help from information from third parties such as travel agents (Anderson and Wilson, 2003) and websites like Microsoft's Bing Travel. Some travel websites such as Kayak.com also advice customers about likely direction of the prices in the future. Changes in buying behavior have begun to in uence the value of dynamic pricing to the rm and are causing them to re-examine their pricing policies. According to the Wall Street Journal, the Chief Executive Ocer of Best Buy, Brand Anderson, even called some of these customers, devils (McWilliams, 2004). Firms and researchers have recognized the importance of considering how cus- tomers have adopted to the new pricing regimes. In this thesis, we dene \Strategic Customer" as one who times his or her purchase so as to increase value (Shen and Su, 2007). In the last few decades numerous papers have been written on dynamic pricing. This rich literature has progressed through several phases. In the rst phase, pricing schemes such as the EMSR policy (Belobaba and Weatherford, 1996) and the GVR policy (Gallego and Van Ryzin, 1994) were introduced. These models were based on the assumption that customers are myopic and will bid for a product as long as the price is lower than their valuations. Subsequent research has explored the possibility that at least some of the customers may be strategic. In the next phase a new body of research emerged that focused on the consequences of strategic behavior on optimal pricing paths and stocking decisions. Anderson and Wilson (2003) showed that customers may postpone purchase until the last minute hoping for a discount, regardless of aordability. Asvanunt and Kachani (2006) investigated EMSR and GVR policies and showed that ignoring strategic customer behavior could result in a substantial revenue loss. 4 Studies during the third phase proposed new optimal pricing policies that explicitly accounted for strategic behavior. Game theoretic approaches for pricing and stocking strategies were developed (Elmaghraby et al., 2008; Yin et al., 2009; Dasu and Tong, 2010). These researchers assumed that strategic customers are completely rational. Cus- tomers either computed equilibrium pricing paths and stocking decisions perfectly or were able to infer them accurately through experience (Liu and van Ryzin, 2008). All these rational models predicted that buyers purchase according to a threshold policy (Zhou et al., 2005; Cachon and Swinney, 2009; Guo et al., 2009). In other words, a consumer makes a purchase when he or she perceives the value of the product to be greater than a threshold. Otherwise, the customer delays purchasing even if the price is lower than his or her valuation. Although the third phase of studies aimed to incorporate a more realistic model of customer behavior, only few attempted to take consumer's psychology into consideration. Nasiry and Popescu (2012) investigated the behavioral bias of strate- gic customers when the product is oered in advance of the realization of its valuation. They incorporated considerations of regret into their analysis. Liu and Shum (2013) studied the impact of customers' loss aversion. Nasiry and Popescu (2012) and Liu and Shum (2013), nevertheless, assume that customers have the computational capacity to solve game theoretical decision problems with a complete understanding of their own behavioral biases. In all of the studies, the common assumption is that the strategic customers are perfect optimizers (Su, 2008). This assumption enables models to be tractable because they perfectly predict customers' decisions. The common steps are to derive the equilibrium buying behavior for a given set of pricing decisions, and then based on this equilibrium 5 nd the threshold policy that will be employed by the customers. Based on the threshold policy, rms optimize their pricing strategy (Elmaghraby et al., 2008; Dasu and Tong, 2010; Osadchiy and Vulcano, 2010; Correa et al., 2011b). To the best of our knowledge there is no study that makes a more realistic assumption that customers are boundedly rational and do not perfectly optimize. To shed light on this unexplored area, in this thesis, we focus on bounded rationality of customers. In order to optimize their utility customers must acquire all relevant information and process it without making any mistake. However, due to bounded rationality customers make errors, do not acquire all the needed information, selectively acquire information, and display systemic biases. There is now abundant evidence that decision making is frequently subject to systematic biases and is based on heuristic and not elaborate opti- mization routines (Simon, 1955; Kahneman and Tversky, 1972, 1979; Tversky and Kah- neman, 1973; Engelbrecht-Wiggans et al., 2007; Engelbrecht-Wiggans and Katok, 2008; Diecidue et al., 2012; Nasiry and Popescu, 2012). 1.2 Experiment and Modeling of Strategic Customers To gain insights into the buying behavior of strategic customers, we conducted a series of laboratory experiments. Laboratory experiments are one of the major methodological ap- proaches in behavioral operations management (Loch, 2007) that complement theoretical approaches (Katok, 2011). Katok (2011) reinterpreted the three major purposes of the laboratory experiments proposed by Roth and Kagel (1995). The rst purpose is to test and rene exiting theory, the second purpose is to characterize new phenomena leading 6 to new theory, and the last purpose is to test the new institutional designs. She con- tends that most of the studies in behavioral operations have focused heavily on the rst purpose, (Schweitzer and Cachon, 2000; Osadchiy and Bendoly, 2010) and few studies on the second purpose (Loch and Wu, 2008; Cui et al., 2007). She encourages researchers to put more eort into testing more sophisticated operations management models. This thesis covers all three components. Our examinations of the normative theories is con- sistent with the rst purpose. Findings from the explorative experimental study led us to propose new approaches for modeling behavior, thereby meeting the second purpose by providing evidence in support of a new framework for modeling strategic customers. Finally, we develop new analytic methods that incorporate bounded rationality. To model bounded rationality of strategic customers, we utilize the quantal response framework. Quantal response models capture the cognitive or computational limitations of decision makers. In the quantal response framework (McFadden, 1976; McKelvey and Palfrey, 1995), the decision makers often { but not always { choose the strategy that provides bigger expected payo. The quantal response model is based on the assumption that we estimate the utility of an option with an error. As a result we are not able to distinguish between options that are close in value. Another attractive feature of the quantal response framework is that it permits a range of \rationality". In absence of rationality decisions would be purely random and under perfect rationality expected utility is maximized. Quantal response framework, in addition to accounting for the lack of capability to estimate the accurate expected utility (Huang et al., 2013; Wu and Chen, 2013), can also incorporate other behavioral traits such as risk aversion (Holt and Laury, 2002) and loss aversion (Ho and Zhang, 2008). 7 In the realm of operations management, quantal response models have been used to study newsvendor problems (Lim and Ho, 2007; Ho and Zhang, 2008; Su, 2008; Wu and Chen, 2013; Chen et al., 2012), and queuing models (Kremer and Debo, 2012). Huang et al. (2013) investigate the impact of bounded rationality on revenue and social welfare in queuing systems. Their nding suggests that when customers are rationally bounded, losses in revenue and welfare from the visible queues will be incurred even with optimal pricing. The study also shows that when the level of bounded rationality is suciently high, it benets the rm, but a low level will result in signicant losses in revenue and welfare. To the best of our knowledge, there are no analogous research in the context of dynamic pricing. 1.3 Dissertation Overview The rest of the dissertation consists of the following three chapters. In Chapter 2, we conduct an experimental study of strategic customers. In the experiment, subjects make repeated purchasing decisions in a computerized simulation game. The setting of the game is taken from the two-period posted pricing model of Dasu and Tong (2010). First, we check the validity of the assumptions of the normative theories. Our observation demon- strates that threshold policy is rarely employed by the decision makers and that most of the subjects cannot be categorized into any of the customer types frequently assumed by extant theories. We nd that surveys typically employed to measure risk aversion and the simulation games are theoretically identical, subjects' responses in the simulation games are not consistent with their answers to survey questions. We then demonstrate how the 8 quantal response model more accurately characterizes buying behavior. We also study the implications of these ndings to rms and show that loss in revenue resulting from ignoring bounded rationality can be substantial. The results described in this chapter serve as the fundamental basis of the following chapters. In Chapter 3, we take a deeper look at the dynamic decision making and the learning process of the strategic customers. First, we investigate how the customers learn the in- stock probability in the second period. The experiments in Chapter 2 are revisited. We adopt a well-known exponential smoothing model (Brown and Meyer, 1961) to describe the learning process of decision makers. We observe that the decision makers have an anchoring bias and the degree of anchoring depends on the true in-stock probability. This nding explains why it is benecial for rms to reveal the in-stock probability when it is equal or less than 50%. We have also utilized experience-weighted attraction (EWA) learning model (Camerer and Ho, 1999) to study how actions and outcomes aect sub- jects' decision making processes. For this we designed a new set of experiments where the valuation is xed throughout the simulation. EWA model suggests that subjects converge very slowly to a specic decision and learn more from outcomes they experienced and less from ctitious play (Camerer and Ho, 1999). In Chapter 4, we conduct theoretical analysis to shed light on the equilibrium behavior of strategic customers. We investigate both full rationality and bounded rationality of the customers. First we scrutinize the uniqueness of the equilibrium behavior of strategic customers who make inter-temporal purchase decisions. We demonstrate that multiple equilibria can exist even in a simple setting where two identical customers compete for one unit of item. For this we present the unique equilibrium when the valuations follow 9 uniform distribution. Then, by employing the quantal response model, we study the equilibrium behavior of the bounded rational customers whose valuations follow uniform distribution. We conclude that when two customers are presented, a unique equilibrium exists. Also, even in the case of multiple customers, there still exists equilibrium. Finally, we set up a simple experiment to see the equilibrium behavior in a more realistic setting and show that it is possible for the human subjects to form a unique equilibrium bidding strategy. 10 Chapter 2 An Experimental Study of Posted Prices 2.1 Introduction Marcy nds a dress priced at $150 that she likes. From her past experience she knows that two weeks later she could buy the dress for a 20% discount provided it is still available. Should Marcy buy the dress or should she wait? Assumptions about how customers make these decisions lie at the core of optimal pricing and quantity decisions for rms selling perishable goods. A number of authors have developed pricing strategies and stocking decisions employing game theoretic approaches (Aviv and Pazgal, 2008; Elmaghraby et al., 2008; Yin et al., 2009; Shen and Su, 2007). In all these papers the seller is a Stackelberg leader and customers are followers. Customers have private valuations and try to maximize their expected surplus by anticipating future prices, timing of price changes, and stock-out probabilities. Models dier by types of pricing policies that are considered by rms and factors that in uence customers' decision. However, there are some core assumptions. Customers discover the equilibrium strategies of other players, which enable them to estimate stock out probabilities and price paths. They purchase if 11 the expected utility of buying now is greater than that obtained by waiting. All these models predict that customers will employ a threshold strategy - each customer will purchase if her valuation exceeds a threshold. There is now growing experimental evidence, in a range of decision problems, that cus- tomers do not employ strategies predicted by game theoretic models. While the rational utility maximization models (RUMM) are elegant, they require each customer to either compute or discover through careful observation of the strategies of other customers and the rm and to act optimally. In the context of Newsvendor problems researchers (Bolton and Katok, 2008; Lim and Ho, 2007; Ho et al., 2010; Wu and Chen, 2013; Chen et al., 2012) have found that bounded rationality models provide a more accurate description of the actual buying behavior. In this chapter we employ experiments to investigate buying behavior in a two-period posted pricing scheme (Aviv and Pazgal, 2008; Dasu and Tong, 2010). Subjects are aware of their private valuation, the distribution of the valuations of all other customers, the number of other customers, and the number of units for sale. They can buy at a higher price or wait for a discount but face the risk of a stock-out. This model was analyzed by Dasu and Tong (2010) assuming that customers are fully rational. We are interested in the following questions: 1. Are there dierent types of customers - myopic, bargain hunters, and strategic? 2. How do customers evaluate the option of a certain current payo against an uncer- tain future payo? 12 (a) Do they conceptualize the problem as a utility maximization problem with known (or accurately computed) probabilities? (b) Is their decision making consistent with boundedly rational models that assume that individual customers' utility estimates include a random error term. 3. Through repetition will customers discover the optimal strategy predicted by the- oretical models? 4. What models best describe customer's strategies? 5. What are the implications for the rm? In the following section we review related literature. In section 2.3, we summarize the theoretical predictions for our game. Design of the experiments is presented in section 2.4. In sections 2.5 and 2.6, we present our experimental ndings and suggest an alternative behavioral model. We show that quantal response models (QRM) t the experimental data well (McKelvey and Palfrey, 1995). In section 2.7 we explore the implications for the rm. We conclude in section 2.8. 2.2 Literature Review Dynamic pricing models either assume that customers are myopic and purchase whenever their valuation exceeds the current price of the product (Gallego and Van Ryzin, 1994) or customers are strategic and anticipate future opportunities to purchase the product at a lower price (Aviv and Pazgal, 2008) and time their purchases so as to maximize expected utility. Cachon and Swinney (2009) identify a third type of customer that 13 they refer to as the bargain hunter, who always waits for the lowest price. Talluri and Van Ryzin (2005), Elmaghraby and Keskinocak (2003), and Shen and Su (2007) provide comprehensive reviews of dynamic pricing models. Researchers have considered dierent types of pricing schemes. In the posted pricing scheme the rm announces the prices at the beginning of the season and in the con- tingent pricing scheme (Cachon and Swinney, 2009) the rm's pricing strategy evolves dynamically based on observed sales. Aviv and Pazgal (2008) nd that neither approach dominates the other. Researchers (Liu and van Ryzin, 2008; Dasu and Tong, 2010) have also found that bulk of the benets of dynamic pricing is derived by a single price change. Based on these ndings, in this chapter we restrict ourselves to a two-period posted pric- ing scheme. We further assume that the rst period price (P 1 ) is higher than the second period price (P 2 ). Although, a declining price path is not always optimal (Su, 2007), it is optimal in our setting (Dasu and Tong, 2010). This is also a common practice in retailing. Each customer has a private valuationv, that corresponds to his or her willingness to pay. If the current price isP 1 , then a myopic customer purchases if her valuationvP 1 . A bargain hunter always waits for the lower price P 2 . A strategic customer purchases in period 1 only if the utility of buying in the rst period exceeds the expected utility of buying in the second period: U 1 (vP 1 )E[U 2 (vP 2 )] 14 The expected utility of buying in second period clearly depends on the stock out probability. Customer utility may also depend on several factors such as the time value of consumption and temporal changes in valuation (Besanko and Winston, 1990; Su, 2007). While these factors are likely to in uence customer's decisions, they also introduce complexity making it dicult analyze how customers evaluate the following choice: Buy at high price and derive a value of vP 1 , or wait and incur the risk of earning zero if there is a stock out or earning a value of vP 2 otherwise. This type of decision lies at the core of all dynamic pricing models. To make this decision subjects have to determine (i) the probability of getting the product in the second period () and (ii) compare the option of getting (vP 1 ) for sure or getting with probability ^ (vP 2 ) and 0 with probability 1 ^ where ^ is their estimated in-stock probability. The stock out probability depends on the stock levels, the number of customers in the market, arrival dynamics of the customers, whether customer's valuations change with time, heterogeneity in customer valuation, information available to the customer, and how customers make purchasing decisions (Yin et al., 2009; Aviv and Pazgal, 2008; Allon et al., 2011). Here we assume that valuations do not change from one period to the next, all customers are present in both periods, and there is a single class of customers. The customers are aware of the number of other customers in the market, the number of goods available for sale, and the distribution of other customers' valuations. In our experiments, the time gap between the rst and second period is too small for discounting to matter. 15 Although theoretical models dier in terms of the factors that in uence customer's choice (Nasiry and Popescu, 2012), they all assume that customers are rational, anticipate equilibrium behavior, and maximize their expected utility. We will use the abbreviation RUMM to denote rational utility maximization models. Liu and van Ryzin (2008) provide a very nice review of how dierent researchers have modeled decision making by strategic customers. They identify two broad approaches. In the rst approach, strategic customers explicitly compute the equilibrium strategies of other players in the game, compute the stock-out probabilities, and determine their optimal strategies. In the second approach, customers do not compute the equilibrium strategies. Instead, it is assumed that the properties of the equilibrium such as stock out probabilities and price paths are observable characteristics of the market. Liu and van Ryzin (2008) argue that the rst approach is more applicable in small markets and the second in large markets where the impact of a single customer on the overall market is innitesimal. Anderson and Wilson (2003) suggested that customers may be able to learn about the probabilities of future outcomes through third parties. However, even under their assumption, customers have to compute expected value of waiting, hence employ RUMM. In all of the RUMMs, independent of the underlying assumptions that factor into the customer's utility, and how customers discover the equilibrium, customers purchase a product if their valuation exceeds a threshold. In our setting this would mean the following: In all of the RUMMs, independent of the underlying assumptions that factor into the customer's utility, and how customers discover the equilibrium, customers pur- chase a product if their valuation exceeds a threshold. In our setting this would mean the following: 16 a. Each strategic customer has a threshold that depends on her utility function. She will purchase in the rst period if her private valuation (v) exceeds this threshold. b. A myopic customer will purchase in the rst period if her valuation exceeds P 1 ; i.e =P 1 . c. A bargain hunter will always wait for P 2 ; i.e =1. Researchers have long recognized the limitations of RUMM (McKelvey and Palfrey, 1995; Kahn and Baron, 1995). These models do not accurately capture either the decision rules employed by the customers or the factors that in uence these decisions. Kahn and Baron (1995) nd that even when the stakes are high subjects fail to evaluate trade-os in a manner proposed by utility theory. As a result experimental methods have been used to gain some insights into how customers make these complex trade-os (G uth et al., 1995, 2004; Bearden et al., 2008; Bendoly, 2012). Much of the early experimental work on dynamic pricing was concerned with durable goods and focused on the discount rate (G uth et al., 1995, 2004). Bearden et al. (2008) use experiments to study the behavior of sellers. In their exper- iments nancially motivated subjects have to sell a small number of objects over a nite time horizon. At various points in time a customer arrives and oers to buy one unit at some price. The subjects have to decide whether or not to accept the bid. The objective is to maximize the sales revenue over the time horizon. They nd that subjects use poli- cies that are sub-optimal but are qualitatively similar in structure to optimal policies. In our experiments subjects are customers and have to decide whether to purchase or wait for a lower price. 17 Osadchiy and Bendoly (2010) study how customers respond to a two-period posted pricing mechanism that is similar to the problem we study. In their experiment customers can either buy at a higher price or wait for a lower price. If they do wait they run the risk of a stock-out. We assume that all customers are present from the start, while they inform their subjects that customers arrive according to a Poisson process. In each trial in their experiment, subjects are informed about their time of arrival and their private valuation. In half the trials subjects are also informed about the likelihood of getting the product. Based on subject's purchasing decisions the authors try to determine whether or not each subject is myopic, strategic, or a bargain hunter. They nd that in aggregate, information about the stock-out probability in uences customer's behavior. A greater percentage of the subjects employ a strategy that appears to maximize expected earnings. Although our context is similar to that of Osadchiy and Bendoly (2010), the objectives of our experiments and how they are carried out are vastly dierent from theirs. In our experiment the stock-out probability remains the same but the private valuations change from one trial to the next. The same valuations, however, are repeated a few times. As a result we are able to gain very dierent insights. Because subjects have to make repeated decisions under the same parameter settings we have more information about whether or not they are myopic, bargain hunters, or strategic. We are also able to detect biases in how they estimate stock-out probabilities. We are able to approximately characterize the aggregate demand curve using the quantal response framework (McFadden, 1976; McKelvey and Palfrey, 1995). Quantal response models (QRM) are considered quasi-rational models. As stated earlier, a rational expected utility maximizing customer will have, regardless of ambiguity 18 or risk aversion, a threshold policy. For a rational customer there is a threshold , such that if the private value is below they will wait for a lower price and if the private valuation is above they will purchase in the rst period. Under a quantal response model, the likelihood of buying is monotone non-decreasing in the private value and the probability will increase rapidly around the threshold. Su (2008) suggests that QRM would be the alternative modeling method in OM area. In recent years there have been a number of experimental studies examining how decisions are made by subjects faced with Newsvendor type problems. Here too, not surprisingly, experimental data provides little support for rational decision models (Ho and Zhang, 2008; Ho et al., 2010; Wu and Chen, 2013). Wu and Chen nd that the quantal response frame-work provides a more accurate description of the decisions. 2.3 Analytical Background We implemented the two-period posted pricing problem analyzed by Dasu and Tong (2010) and Aviv and Pazgal (2008). The seller who is a monopolist announces two prices, P 1 and P 2 for the rst period and the second period, respectively, with P 1 >P 2 . There are two types of customers: high type and low type. Low type customer's valuation lies between P 2 and P 1 , while the high type customer's valuation exceeds P 1 . The number of high type customers (N 1 ), the number of low type customers (N 2 ), and the number of units for sale (K) are common information. The number of high type customers is less than the number of units for sale (N 1 < K); however, the total number of customers is greater than the number of units for sale (N 1 +N 2 >K); therefore, there is a possibility 19 of stock out in the second period. High type customers and low type customers valuations are drawn from Uniform(P 1 ; 1) and Uniform(P 2 ;P 1 ), respectively. Customers' valuations are private but the distribution of the valuations is common information. High type subject can buy in the rst period or buy in the second period provided there is supply. The expected payo for a risk neutral customer with valuation v, in the rst period and the second period is (vP 1 ) and 2 (vP 2 ), respectively. Value of 2 , the probability of getting the product in the second period, depends on how consumers make their decisions. If consumers are risk neutral rational utility maximizers, then theory predicts that there is a unique Bayesian Nash equilibrium for equation (1) which is given by the smallest solution in the range [P 1 ; 1]. All high type customers with valuations v2 [; 1] will bid in period 1 and the customers with valuationv2 [P 2 ;] will bid in period 2. We call this a threshold policy with a threshold at . (P 1 ) = 2 ()(P 2 ) (2.1) where 2 () : the probability of getting the product in period i when all the customers use the threshold policy with a threshold at . 2.4 Design of the Experiments 179 nancially motivated subjects participated in three sets of experiments termed: SIM, SIM-I, and LOT. The set called SIM, was a simulation of the two-period posted pricing decision problem that was in accordance with the model analyzed above. SIM consisted of 3 treatments and each treatment in turn consisted of 60 trials. In each trial in an 20 experiment private valuations changed. Each experiment in SIM diered only in terms of the size of the market (the number of low type customers). The number of goods for sale, and the purchase prices (P 1 andP 2 ) were not altered. As a result, the in-stock probability in the second period was dierent in each experiment. The in-stock probabilities were 50%, 83%, and 16% in treatment 1, 2, and 3. We will refer to these as SIM1, SIM2, and SIM3. In SIM-I subjects were informed of the in-stock probability in the second period. SIM and SIM-I were identical otherwise. We refer to the 3 treatments in SIM-I as SIM1-I, SIM2-I and SIM3-I. The focus of our study is on SIM. We want to identify whether or not subjects employ RUMM or whether their decisions are better described by quasi-rational models such as quantal response models. By comparing SIM and SIM-I, we sought to understand the impact of in-stock probability information. SIM and SIM-I also allow us to determine if some subjects are consistently myopic or bargain hunters. Each subject participated in only one of the experiments to eliminate any learning eect that carry over from one experiment to the next. LOT consisted of 5 surveys. We sent an email to the participants of the SIM and SIM-I experiment two or more days after they participated in the study asking them to ll out a survey. During the SIM and SIM-I experiments we informed subjects that they will be asked to participate in a dierent type of experiment and collected their email address. 142 out of 179 subjects who participated in SIM or SIM-I (79% of the subjects) participated in the LOT experiment. In these surveys the decision problems faced by the subjects in SIM(SIM-I) were stated in their most direct form by providing probabilities of success in the second period. For example subjects were asked if they would prefer $10 for 21 sure or $50 with 49% chance. The probability corresponded to probability of the product being in-stock in the second period. Out of 5 surveys, one survey had the same in-stock probability as the one in the SIM(SIM-I) experiment that the subject had participated. In theoretical papers, researchers always assume that the subjects reformulate the two- period dynamic pricing problem in terms of the form given in LOT. By comparing the decisions made by subjects in LOT with those made in SIM(SIM-I), we get insights as to whether or not subjects employ the same decision process in these dierent types of problems. Table 2.1 below summarizes the experiments. Experiment Type Number of Treatments Description Objective SIM-I 3 Simulation of a two-period posted pricing problem. Valu- ations varied from one trial to the next. In-stock probability is given. To identify the decision rules em- ployed by subjects when exact in-stock probability is oered as well as information about market size, valuation distribution. SIM 3 Simulation of a two-period posted pricing problem. Valua- tions varied from one trial to the next. In-stock probability is not given. To identify the decision rules em- ployed by subjects when they have to learn the in-stock proba- bility through information about market size and valuation distri- bution. LOT 3 Surveys in which the deci- sion problem encountered in SIM(SIM-I) is presented as a lot- tery To explore whether the decisions in SIM(SIM-I) are structurally similar to decisions made while evaluating lotteries Table 2.1: Description of the experiments Subjects were recruited from Amazon Mechanical Turk's (AMT) online pool. Paolacci et al. (2010) and Buhrmester et al. (2011) provide an excellent overview of this tool and its reliability. Readers are also referred to work by Archak et al. (2011), Kaufmann et al. 22 (2013) and Toubia et al. (2013) for additional validation of AMT for experiments such as ours 1 . Subjects were given a brief overview of the experiment and were asked to answer a few questions to ensure that they understood the objective of the experiment. Screen shots of the simulation tool used in SIM and the survey used in LOT, and the detailed instructions given to the subjects are available in the online appendix. The average time the subjects spent in the SIM(SIM-I) experiment and in the LOT experiment is approximately 21 minutes and 7 minutes, respectively. 2.5 Experiments LOT To gain insights into how subjects evaluate the trade-o arising in the two-period posted pricing problem we simplied the decision problem by informing the subjects about the in-stock probabilities and explicitly presented the trade-o. As stated earlier, RUMMs assume that subjects transform the two-period posting problem to an equivalent lottery in which the in-stock probability is known. The experiment set LOT consisted of 5 surveys. The surveys are modeled after the lotteries proposed by Holt and Laury (2002). Subjects were asked to choose between Option B a certain outcome or Option A a risky outcome with a higher payo. Decisions in the survey corresponded to the decisions arising in the SIM experiments. In SIM we xP 1 = $120,P 2 = $80, therefore; when the valuation was $125 then the payo obtained from buying in period 1 was $125-$120 = $5. If the subject decided to wait then there 1 We have previously conducted the same experiments that are only dierent in parameters and model descriptions. 138 nancially motivated subjects attended in the laboratory and received $10 for partici- pation and up to $10 of performance fee. The subjects from AMT outperformed the ones from the lab experiment. 23 was a chance of earning $125 $80 = $45. In SIM1 and SIM1-I there was 49% chance of success in the second period, in SIM2 and SIM2-I it was 83%, and in SIM3 and SIM3-I it was 16%. The 5 surveys have the same set of payos but with dierent chance of winning: 16%, 33%, 49%, 66%, and 83% (Table 2.2). Because subjects participate in only one SIM(SIM-I) experiment, only one of the 5 had the same in-stock probability as their SIM(SIM-I) experiment. To reduce ordering eect we randomized the order of the 5 surveys. In each survey subjects had to make 16 decisions. While making decision j (j2f1; 2; 3; ; 16g) subjects had to choose between Option B that guaranteed $5j or a risky Option A that paid $40 + $5j. Thus decision j corresponded to a resale price of $120 + $5j in SIM. The full surveys are in the appendix. SURVEY Probability of success in the risky option A Corresponding Treat- ments in SIM(SIM-I) LOT16% 16% SIM3, SIM3-I LOT33% 33% N/A LOT49% 49% SIM1, SIM1-I LOT66% 66% N/A LOT83% 83% SIM2, SIM2-I Table 2.2: Parameters for the LOT experiments 2.5.1 Analysis of the LOT Experiments Figure 2.1 shows the aggregate data for LOT16%, LOT49% and LOT83% treatments. For dierent decision numbers j, this gure shows the percentage choosing the certain option, Option B. The solid line is the percentage that would be observed under RUMM. We tested the t of RUMM using chi-square goodness of t test. The result rules out the possibility that all subjects are rational, risk neutral, and employ the threshold policy 24 Figure 2.1: Aggregate results for LOT treatments. predicted by theory. (All p-values = 0.00). We tested for all possible threshold policies but once again none of them has a signicant explanation power. (All p-values = 0.00). Observation 1.1: In the aggregate data we fail to observe the threshold policy predicted by theoretic models that assume that all customers are risk neutral rational utility maximizers. Figure 2.1 does not, however, rule out the possibility that subjects may have dierent thresholds due to individual dierences in risk or ambiguity aversion. To address this concern we studied individual decisions. Any subject who always chooses Option B if the decision number j exceeds some number, and always selects Option A below this number is considered to have employed a threshold policy. Subjects who always chose Option A or always selected Option B were also included in this set. Subjects who always chose Option B may be considered to be myopic customers, and those who always pursued the risky. Option A may be bargain hunters. Table 2.3 contains the percentage of subjects that employed a threshold policy in each survey. The percentage is similar to what is reported by Holt and Laury (2002). Nevertheless, it is surprising that 26.76% do not employ a threshold policy when you consider all the surveys together. The decision problem is 25 one trivial step away from how the expected utility trade-o equation is evaluated. Yet RUMM is not consistently employed by at least 25% of the subjects. Treatment Percentage employing Percentage always Percentage always a threshold policy choosing option A choosing option B LOT16% 129 (90.85%) 0 (0.00%) 61 (42.96%) LOT33% 126 (88.73%) 2 (1.41%) 29 (20.42%) LOT49% 116 (81.69%) 6 (4.23%) 10 (7.04%) LOT66% 122 (85.92%) 22 (15.49%) 6 (4.23%) LOT83% 130 (91.55%) 89 (62.68%) 0 (0.00%) In ALL 104 (73.24%) 0 (0.00%) 0 (0.00%) Table 2.3: Percentage of subjects employing a threshold policy in the LOT treatments (total: 142) Given the limited experimental support for RUMM, we turn to behavioral models to explain the aggregate curves observed in Figure 2.1. We employ the quantal response frame-work that is based on the assumption that the likelihood of adopting the best option increases as the gap between the optimal option and the next best increases. Under the rational model, it is assumed that each subject accurately estimates the utility of each of the decisions; i.e. U(V B;j ) if they choose Option B and U(V A;j ) if they choose Option A, where is the likelihood of success in Option A, and V A;j and V B;j are the payos of Option A and B at decision number j, respectively. In the quantal response framework, it is assumed that each of these utilities is estimated with some error. ^ U B =U(V B;j ) +" 1;k (2.2) ^ U A =U(V A;j ) +" 2;k (2.3) 26 Due to the error terms " i;k , the likelihood of picking the dominant decision depends on the dierence between the two options. Subjects are more likely to vacillate when the decision j is such that the two options are almost equal in value, and subjects are more likely to pick the optimal option as the decision j is away from the break-even point. If the error terms (" i;k ) in equations (2.2) and (2.3) are unbiased and have extreme value distributions, then the probability of purchasing in the rst period is given by a Logit model (McKelvey and Palfrey, 1995). We employ the exponential utility function to capture the risk attitude. Readers are referred to the work by Holt and Laury (2002) for a justication of this utility function. We let r denote the risk aversion parameter. U(V ) = (1 exp(r(V ))) r (2.4) where, r > 0, r = 0 and r < 0 would imply risk aversion, risk neutrality, and risk preference, respectively and V is the payo. Let : Rationality parameter. LetR B;j =U(V B;j ),R A;j =U(V A;j ), andR(V A;j ;V B;j ;) = R B;j R A;j =U(V B;j )U(V A;j ). The probability of choosing Option B at decision j is given by: Pr j (V A;j ;V B;j ;; ;r) = exp( R(V A;j ;V B;j ;))) (1 +exp( R(V A;j ;V B;j ;)) (2.5) 27 The rationality parameter is estimated by maximizing the log likelihood function: max ;r X j2J [O B (j) log(Pr j (V A;j ;V B;j ;; ;r)) +O A (j) log(1Pr j (V A;j ;V B;j ;; ;r))] (2.6) whereO a (j) is the number of observation in which subjects chose Option a,a =A, B, when the decision number is j. We call this model QRM. When the rationality parameter equals1 and the risk aversion parameter equals 0, the quantal response model converges to a threshold policy, consistent with RUMM. Figure 2.2 shows the t of QRM model. Table 2.4 below shows Goodness of Fit Test for QRM to LOT experiment. The QRM model ts our experimental data very well. The log-likelihood and the likelihood ratios for the QRM model are in Table 2.5 This observation is consistent with that of Holt and Laury (2002). Figure 2.2: Estimations by QRM Model. Treatment Rationality( ) Risk Aversion(r) Chi-Square Test (p) RMSD LOT16% 0.1898 0.0416 1 0.0274 LOT49% 0.1051 0.0039 0.9598 0.0425 LOT83% 0.1489 0.0086 1 0.0173 Table 2.4: Goodness of Fit Test for the QRM to LOT Treatments Observation 1.2: The aggregate data in LOT can be modeled by a quantal response model with risk parameter. 28 Treatment Model RUMM QRM LOT16% Number of parameters 0 2 Log-likelihood -1446 -266.29 Likelihood-ratio test against QRM 2 =2359.43 (p = 0.00) - LOT49% Number of parameters 0 2 Log-likelihood -3534 -550.43 Likelihood-ratio test against QRM 2 =5967.14 (p = 0.00) - LOT83% Number of parameters 0 2 Log-likelihood -1998 -370.51 Likelihood-ratio test against QRM 2 =3254.98 (p = 0.00) - Table 2.5: Log likelihood and likelihood ratio tests for RUMM and QRM 2.6 SIM: Simulation of the Two-Period Posted Pricing Problem As stated in the analytical background in section 2.3, we implemented the Dasu and Tong (2010) model with a minor change. In the rst period, all subjects are assured that they will get the product. By doing this, we allow the subject to focus only on estimating the in-stock probability in the second period. Subjects were informed that there were N 1 1 other high type customers in the market in the rst period interested in buying the product and that there would be an additional N 2 low type customers in the second period. Subjects always played the role of the high type customers. The characteristics of the product were not specied to avoid any bias. They were told that K objects were available for sale. They could resell the product at a resale price (v) and thereby make a prot. The resale price corresponds to the private valuation. Hence forth we will refer to the private valuation as resale price. Their objective was to maximize prot in each period by deciding whether to buy in period 1 or wait for a lower price in period 2. They were guaranteed to get the product 29 if they purchased in the rst period. However, there was no such assurance of product availability in the second period. Whether or not a product was available depended on resale prices observed by and purchasing decisions of other customers. Subjects were also informed that resale prices were dierent for each high type cus- tomer and varied from one trial to next. For each high type customer the resale prices were distributed uniformly in the range [P 1 ; 200]. Subjects were also informed that they were not competing with other subjects but with a \computer". The decisions for other customers were made by the computer. Although the decisions made by the computer were based on equilibrium threshold policies (Dasu and Tong, 2010), subjects were not informed about the decision rule being used by the computer. By having subjects play against other \players" who are employing the equilibrium strategy, we can determine whether or not each subject gravitates to the predicted threshold policy. Each subject may either compute the equilibrium strategy, or they can learn through repeated play the probability of obtaining the product in the second period and therefore converge to a threshold strategy. Thus both approaches that have been employed by theoretical models, as described by Liu and van Ryzin (2008) are amenable to our subjects. The parameters for the six treatments are given in Table 2.6. The threshold values (), and the in-stock probabilities () are based on the equilibrium for a game among risk neutral rational customers. In each trial using this probability the simulation randomly determined whether or not the product was in stock. Resale price (v) is also randomly generated from uniform distribution U(P 1 ; 200] and assign to each subject at each trial. 30 Parameters SIM1, SIM1-I SIM2, SIM2-I SIM3, SIM3-I Number of High Type Customers(N 1 ) 10 10 10 Number of Low Type Customers(N 2 ) 26 14 72 Number of units for sale(K) 20 20 20 First period price(P 1 ) 120 120 120 Second period price(P 2 ) 80 80 80 In-stock probability in the second period() 0.49 0.83 0.16 Threshold() 160 > 200 128 Number of trials 60 60 60 Table 2.6: Parameters for SIM(SIM-I) Treatments 2.6.1 Analysis of the SIM Experiments Figure 2.3 shows the aggregate data for each of the 6 treatments. For dierent resale prices, this gure shows the percentage of decisions that resulted in the subjects buying in the rst period at a price P 1 . The solid line is the percentage that will be observed under RUMM. Figure 2.3: Aggregate results for SIM(SIM-I) treatments Similar to the result of LOT, in SIM(SIM-I) we also reject the possibility that subjects employ any threshold policy predicted by RUMM at the aggregate level. (All p-values = 0:00) We looked for threshold policies at the individual level. Table 2.7 contains the 31 percentage of subjects that employed a threshold policy. We also employed a more lenient criterion for determining whether or not a subject was employing a threshold policy. Subjects who deviated from a threshold policy at most two times were also counted as having employed a threshold policy. One clear conclusion we can draw from this data is that the percentages are signicantly lower than that we observed in LOT. Treatment Threshold policy For all resale prices (%) Except for two resale prices (%) SIM1 0 (0.0%) 5 (16.7%) SIM2 2 (6.9%) 7 (24.1%) SIM3 2 (6.7%) 11 (36.7%) SIM1-I 2 (6.7%) 6 (20.0%) SIM2-I 2 (6.7%) 8 (27.7%) SIM3-I 4 (13.3%) 13 (43.3%) Table 2.7: Percentage of subjects employing a threshold policy Treatment Bargain Hunters Myopic Customers For all resale prices (%) Except for two resale prices (%) For all resale prices (%) Except for two resale prices (%) SIM1 0 (0.0%) 0 (0.0%) 0 (0.0%) 3 (10.0%) SIM2 2 (6.9%) 3 (10.3%) 0 (0.0%) 0 (0.0%) SIM3 1 (3.3%) 6 (20%) 0 (0.0%) 0 (0.0%) SIM1-I 0 (0.0%) 0 (0.0%) 2 (6.7%) 3 (10.0%) SIM2-I 1 (3.3%) 5 (16.7%) 1 (3.3%) 2 (6.7%) SIM3-I 0 (0.0%) 0 (0.0%) 1 (3.3%) 10 (33.3%) Table 2.8: Percentage of subjects acting as bargain hunters or myopic customers In SIM2 and SIM2-I, for a risk neutral customer it is optimal to always wait for the second period act; i.e. act as a bargain hunter. As a result in these two experiments, bargain hunting is confounded with optimal decision making. In our experiment we draw attention to stock out risks which may cause subjects to occasionally purchase in the rst period. Would this be a limitation that prevents us from asserting that the experiment 32 fails to support the claim that there are bargain hunters? By denition bargain hunters keep track of price reductions. This would also mean that they are very likely to be aware of stock outs. In that case our experiments imply that customers who are classied as bargain hunters because they always buy discounted products, are doing so not because it is a trait but because they have a lower willingness or ability to pay. Out of 179 subjects only 4 appeared to be bargain hunters. Observation 1.3: Less than 3% of the subject whose valuation is greater than the rst period price displayed the trait of being a bargain hunter. In our study the percentage of buyers who acted myopically is also small. Our study, however, does not rule out the possibility that subjects outside the experimental setting may behave in a myopic manner. The experiment explicitly draws subject's attention to the possibility of buying at a lower price. In absence of such framing, subjects may not have considered the option of waiting. Payos of buying in the rst period and waiting are (vP 1 ) and (vP 2 ), respectively, where v is resale price, P 1 and P 2 are the prices. Let : Rationality parameter. Let R 1 =U(vP 1 ),R 2 =U(vP 2 ), andR(v;P 1 ;P 2 ;) =R 1 R 2 =U(vP 1 )U(vP 2 ). The probability of purchasing in the rst period when the resale price is v, is given by: Pr(v;P 1 ;P 2 ;; ;r) = exp( R(v;P 1 ;P 2 ;; ;r)) 1 +exp( R(v;P 1 ;P 2 ;; ;r) (2.7) 33 The rationality parameter is estimated by maximizing the log likelihood function: max ;r X v2V [O 1 (v) log(Pr(v;P 1 ;P 2 ;; ;r)) +O 2 (v) log(1Pr(v;P 1 ;P 2 ;; ;r))] (2.8) whereO i (v) is the number of observation in which subjects buy in periodi,i = 1; 2; when the resale price is v. Although this model is static and does not incorporate dynamic eects such as learn- ing 2 or regret, we found that this model has a strong explanation power. Table 2.9 below shows the result of goodness of t test for the QRM to SIM and SIM-I experiment. Treatment Rationality Risk Aversion Chi-square Root Mean Parameter ( ) Parameter (r) Test p value Square Deviation SIM1 0.1030 0.0089 0.7603 0.0560 SIM2 0.1825 0.0351 0.7245 0.0429 SIM3 0.0675 0.0068 0.2333 0.0760 SIM1-I 0.1092 0.0183 0.9916 0.0392 SIM2-I 0.1372 0.028 0.7507 0.0570 SIM3-I 0.0958 0.0262 0.7994 0.0590 Table 2.9: Goodness of Fit Test for the QRM to SIM(SIM-I) Table 2.10 below compare dierent models. The rst is the RUMM. Here the ratio- nality parameter is set equal to1. The next model is the QRM in which the in-stock probabilities are xed at the true values but we estimate the rationality parameter and risk aversion parameter. The Log-likelihood of each model and Likelihood-ratio test for each treatment for each of these models is as follows. The QRM model outperformed RUMM in every case. (P-values = 0:00 for all treat- ments). Figure 2.4 illustrates how well the QRM model conforms to the experimental data. 2 The learning eect will be re-visited in Chapter 3 34 Treatment Model RUMM QRM SIM1 Number of parameters 0 2 Log-likelihood -4032.00 -433.35 Likelihood-ratio test against QRM 2 = 7197.31 (p = 0.00) - SIM2 Number of parameters 0 2 Log-likelihood -6056.00 -454.18 Likelihood-ratio test against QRM 2 = 11205.65 (p = 0.00) - SIM3 Number of parameters 0 2 Log-likelihood -2528.00 -314.12 Likelihood-ratio test against QRM 2 = 4447.59 (p = 0.00) - SIM1-I Number of parameters 0 2 Log-likelihood -4224.00 -463.95 Likelihood-ratio test against QRM 2 = 7611.22 (p = 0.00) - SIM2-I Number of parameters 0 2 Log-likelihood -5712.00 -477.14 Likelihood-ratio test against QRM 2 = 10469.73 (p = 0.00) - SIM3-I Number of parameters 0 2 Log-likelihood -2344.00 -352.01 Likelihood-ratio test against QRM 2 = 4050.89 (p = 0.00) - Table 2.10: Log likelihood and likelihood ratio tests for RUMM and QRM Figure 2.4: Estimations by QRM Model to SIM(SIM-I) treatments 35 The QRM model has a two parameters and the added exibility should enable us to nd a good t. To control for this we use out of sample data to measure the quality of the t. We randomly selected 50% of the subjects to t the QRM model and tested the t on the remaining 50% of the subjects. Table A.1 and Table A.2 in the appendix A summarize the results for 5 dierent random partitions. QRM model still has a very strong estimation power in the treatments having the in-stock probability of 16% and 49%. Chi-square tests have p values well above 0.05 except for one sampling, and RMSDs are also below 0.15 and 0.10 for the most of the samplings in SIM1 and SIM2, respectively. In SIM2, Chi-square test p values are greater than 0.01 in the two samplings out of ve samplings. Maximum RMSD is 0.1682. When the in-stock probability is 83%, Chi-square test p values are greater than 0.01 in the two samplings out of ve samplings in both SIM2-I and SIM2. Maximum RMSD is 0.1701 and 0.1682 in SIM2-I and SIM2-I, respectively. To take a deeper look at SIM2 and SIM2-I, we compare QRM to RUMM and LOT by using Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for model comparison. AIC is evaluated as AIC= (2) LL + 2k (Akaike, 1974) and BIC is evaluated as BIC= (2) LL + k log(n) (Schwarz, 1978; Liddle, 2007), where LL denotes the maximum log likelihood, k is the number of parameters estimated and n is the number of datapoints used in the t. By checking AIC and BIC, we nd that QRM model dominates RUMM and LOT estimations in all partitions. 36 2.6.2 Comparison of Behaviors In LOT and SIM Experiments Researchers routinely assume that strategic customers transform the decision problem in the two-period posted pricing problem into one in which a deterministic payo is compared with an uncertain payo (Elmaghraby et al., 2008; Cachon and Swinney, 2009). If this assumption is correct, then decisions in LOT and SIM-I experiments should display some consistency. Similarly, although the in-stock probability is not presented to the subjects, if the subjects compute or learn the in-stock probability correctly, then decisions in LOTand in SIM should also be consistent. To verify whether or not this is true, for each subject we compare the average probability the subjects bought in the rst period in SIM(SIM-I) and the percentage of times they choose Option B in LOT (Holt and Laury, 2002). We dene the average probability of times the subjects bought in the rst period for SIM(SIM-I) and LOT as follows: 1) The average probability buying in the rst period in SIM(SIM-I): X v2V [O 1 (v)=O(v)]=16 (2.9) ,where O 1 (v) is the number of choosing \buy now" at v and O(v) is the total number of trials at v. 2) The percentage of choosing option B in LOT (Holt and Laury, 2002) : X j2J [I 1 (j)]=16 (2.10) 37 ,whereO 1 (j) = 1, if Option B is chosen at decision number j, and,O 1 (j) = 0, otherwise. The results of these comparisons are displayed in Figure 2.5. Here each point corre- sponds to one subject. Figure 2.5: Estimations by QRM Model to SIM(SIM-I) experiments The coecient of determinations(AdjustedR 2 ) of the linear regression are 0.0%, 0.0%, 0.0%, 0.0%, 21.2%,and 48.9%, for SIM1,2,3,1-I,2-I,and 3-I, respectively. Therefore, we fail to nd evidence that individuals employ the same decisions in LOT and SIM(SIM-I). There is a possibility that in the early trials the subjects focus on learning the in stock probability so we also conducted the same analysis with the last 30 trials. Here too we did not nd strong linear relationships between the decisions in SIM(SIM-I) and LOT. All of the coecients of determinations were less than 15%. Then, we compare SIM, SIM-I and LOT together. We utilize Bonferroni method which is frequently used non-parametric method for multiple comparison 3 . This multiple 3 We also perform parametric method such as Sid ak method and Dunnett method, and both generate the same results as Bonferroni method. 38 comparison method requires to set a control level which is one of the experiments. Because the setting of SIM-I is between SIM and LOT, it is chosen for the control level. The method results that when in-stock probability is 49% the decisions in SIM, SIM-I and LOT are similar. However when in-stock probabilities are 83% and 16%, the decisions in LOT treatments is signicantly dierent from both SIM and SIM-I. SIM treatments are not signicantly dierent from SIM-I. The condence level for the tests is 95%. This is evidence showing that there is a structural dierence between the decisions in SIM(SIM-I) and LOT, which even dominates the availability of the real in-stock probability. Observation 1.4: Subjects do not employ the same decision strategy in LOT and SIM(SIM-I) experiments 2.6.3 Changes in Decision Making over Time It is conceivable that subjects learn and progressively gravitate towards RUMM. There- fore we tested if the rationality and risk parameters change over the course of the SIM experiment. We computed these parameters for the rst 30 trials and compared them to those of the last 30 trials. We did this at an individual level. Table 2.11 summarizes the results. Means and p-values given by paired-t tests are presented. Rationality level signicantly increases from the rst 30 trials to the last 30 trials. (p value = 0.023). We, however, failed to observe statistically signicant changes in the risk parameter. These nding suggest that rationality is aected by the level of experience or learning. Although, there is some evidence of learning, the rationality parameter continues to be very small. In the last 30 trials it is signicantly smaller than 1.00 (p value = 0.00). This 39 leads us to conclude that even after 60 trials, subjects continue to demonstrate bounded rationality. Rationality Risk Attitude Median in First 30 0.4132 0.1071 Median in Last 30 0.4961 0.1151 Hypothesis Test First 30 = Last 30 First 30 = Last 30 vs. First 30 < Last 30 vs. First 30 <>Last 30 T value -2.01 -0.33 P value 0.023 0.744 Table 2.11: Comparison of estimated parameters in rst 30 trials and last 30 trials In summary we nd that decision making in the two-period posted pricing problem is best modeled by quasi-rational models. There is little evidence that repeated play results in decision making that is consistent with RUMM. Further, the decision heuristics employed in LOT and SIM are dierent. 2.7 Implications for the Firm The goal of theoretical models is to nd prices that maximize expected revenues. For a given set of prices these models, either implicitly or explicitly, compute the equilibrium threshold policy and the expected revenues. In the next sub-section, we evaluate the error in the revenues predicted by theoretical models based on RUMM. For a given set of prices we compute the dierence between the revenues the rm would have realized if customers employed the policy predicted by RUMM against the actual revenues. This analysis is based on the data from the SIM experiments. In the following sub-section we study how rms should modify their pricing policies if customer decision making is 40 consistent with a quantal response model. Finally we explore the impact of disclosing in-stock probabilities on the rm's prot. 2.7.1 Errors in Estimated Revenues under the Dierent Models In Table 2.12 we compare the percentage who bought in the rst period in SIM1, SIM2 experiments and with those estimated by the RUMM, QRM model, and LOT respectively. Table 2.13 compares the estimated expected revenues. In-stock probability 49% 83% 16% Experiment SIM1 SIM1-I SIM2 SIM2-I SIM3 SIM3-I Observed (experiment) 60.53% 65.48% 43.91% 39.66% 79.81% 80.96% RUMM 56.25% 56.25% 0.00% 0.00% 93.75% 93.75% (-7.07%) (-14.10%) (-100.00%) (-100.00%) (-17.47%) (-15.80%) LOT 57.76% 57.76% 14.35% 14.35% 89.74% 89.74% (-4.58%) (-11.79%) (-67.32%) (-63.82%) (-12.44%) (-10.84%) QRM 65.63% 61.61% 40.01% 43.12% 82.13% 82.00% (-1.78%) (-0.23%) (-1.80%) (-0.88%) (-2.74%) (-1.45%) Table 2.12: Predictions of the percentage who buy in the rst period (Number in paren- thesis are error %) Experiment Experimental RUMM QRM Error in the Error in Result Estimation Estimation RUMM estimate QRM estimate SIM1 1842.11 1825 1846.44 -0.93% 0.24% SIM1-I 1861.91 1825 1862.52 -1.98% 0.03% SIM2 1775.65 1600 1772.48 -9.89% 0.00% SIM2-I 1758.64 1600 1760.04 -9.02% 0.08% SIM3 1919.24 1975 1928 2.91% 0.01% SIM3-I 1923.83 1975 1928.52 2.66% 0.24% Table 2.13: Revenues predicted by RUMM and QRM The error in expected revenues based on the RUMM varies between 0.93% and -9.89%. The QRM based estimates have a considerably smaller error; less than 0.24%. Although the absolute errors in estimation based on the RUMM are small, they are not insignicant. The gains in revenues from employing optimal posted pricing schemes that account for 41 strategic behavior relative to pricing schemes that are based on the assumption that customers are myopic is under 8% (Dasu and Tong, 2010). 2.7.2 Impact on Optimal Pricing and Revenues Therefore, we estimate the consequences of QRM on pricing and revenues in equilibrium. For a given number of customers (N 1 ) and (N 2 ), stocking level (K), prices (P 1 and P 2 ), rationality parameter ( ), and risk aversion parameter (r), we have to nd the equilibrium in-stock probabilities 4 . We do so through an iterative process, we begin by assuming that the in-stock probability is 0 . Based on this assumption we compute the in-stock probability ( 1 ) if all customers act according to a QRM with parameters 0 , ,and r. We repeat the exercise but replace 0 with 1 . We continue until we nd the equilibrium in-stock probability. This approach allows us to determine the revenues for a given set of prices. With P 2 xed, we nd the optimal P 1 . We evaluated the consequence of assuming RUMM when customers' decision making is consistent with QRM. In our experiments, the number of customers is greater than the number of items and the rm is assured a revenue of KP 2 . The value of a two-price scheme should be measured in terms of the additional revenue it generates in excess of this guaranteed minimum. Table 2.14 contains the optimal rst period price and the expected additional revenues. We use a rationality parameter of 0.05 and risk parameter of 0.086 estimated from LOT83%. We xedP 2 at 80 and found the optimal prices under these two models. The optimal rst period under QRM is higher than that under RUMM. The additional revenues under QRM are higher by 13.4%. 4 Deeper analysis in equilibrium follows in Chapter 4 42 RA-LOT Prediction QRM Prediction Additional Revenue comparison Optimal P 1 Predicted Additional Additional Optimal P 1 Additional % Loss due to Revenue QRM Revenue QRM Revenue RUMM Assumption 94 250.9 157.6 112 178.7 11.80% Table 2.14: Changes in Optimal Pricing and Additional Revenue: RUMM vs. QRM RUMM model above assumes that all subjects are risk neutral. An alternative is to assume that subjects are risk averse with varying risk aversion parameters. We call this the RA-LOT model. We infer the risk parameters for the subjects based on the LOT experiment. We evaluated the consequence of employing a RA-LOT model based on the distribution of the risk parameters obtained in LOT83% when customer's decision making is consistent with QRM. Table 2.15 summarizes the result. RA-LOT Prediction QRM Prediction Additional Revenue comparison Optimal P 1 Predicted Additional Additional Optimal P 1 Additional % Loss due to Revenue QRM Revenue QRM Revenue RA-LOT assumption 97 270 168.7 112 178.7 5.6% Table 2.15: Changes in Optimal Pricing and Additional Revenue: RA-LOT vs. QRM Once again we nd that ignoring the level of rationality of the customers can result in signicant decline in revenues. 2.8 Conclusions In this chapter, we employed experiments to gain insights into how customers make purchase decisions for products that are dynamically priced. In particular we conducted experiments for a two-period posted pricing scheme. We investigated some of the common assumptions in theory and one of our main ndings is that a threshold policy, central to all rational utility maximizing models, is employed by very few subjects. Subjects are more 43 likely to randomize their decisions when the dierence between the two alternatives being evaluated is small. As a result the aggregate data can be modeled by quantal response models. Even when the stock out probabilities and the pay-o are explicitly presented in the form of a survey, over 20% of the subjects did not employ a threshold policy. It is surprising that, even for this relatively simple decision problem, many subjects employed a mixed strategy. Our experiment employed a partial equilibrium model. Subjects were gaming against other fully rational customers simulated by the computer. In our initial experiments we found that the simulated in-stock probabilities deviated from those that would have been observed if all subjects were quasi-rational. We performed another set of experiments where we carefully selected the simulated in-stock probabilities. In these experiments the simulated in-stock probabilities coincided with the in-stock probabilities we would have observed if all subjects in the experiment employed the same quasi-rational decision heuristics. This suggests the existence of an equilibrium in these settings. Next we used numerical experiments to compute the optimal prices when all subjects employ quasi-rational models. In our examples we nd that optimal prices based on quantal response models have a higher rst period price compared to that prescribed by pricing models based on the assumption that all customers are rational. Further, assuming that subjects rational utility maximizers can reduce a rm's revenues. Next step would be to determine the equilibrium policies when all players in the exercise are subjects. It would also be useful to develop analytic pricing models in which customer decision making is based on quantal response models. 44 Chapter 3 Dynamic Learning in Strategic Customer Behavior 3.1 Introduction In this chapter, we study how strategic customers learn when they make purchasing decisions repeatedly. Many products and services inducing strategic customer behav- iors incorporate repetitive interactions with customers. Many business travelers tend to choose the same airline and hotels that they have used previously. Air cargo forwarders purchase cargo spaces of aircrafts on a regular basis. Fast fashion brands such as Zara, H&M, and Forever 21 adapt new fashion trends every year so as to induce demand for the new seasonal items. One of the fundamental factors for a strategic customer making choice is the timing of purchase (Shen and Su, 2007). Hence, when making purchasing decisions, customers repeatedly contemplate about the optimal timing of purchase. Al- though researchers have had many studies on this issue, they have mostly focused on a single purchase decision (Aviv and Pazgal, 2008; Elmaghraby et al., 2008; Yin et al., 2009). Such limitation can be explained by the fact that they have all assumed the 45 Rational Utility Maximization Model (RUMM). Since the RUMM is based on the tradi- tional assumption that customers are fully rational, it is applicable only when a customer repeats the same strategy in making purchasing decision. However, our experiments discussed in Chapter 2, disprove the assumption of per- fect rationality. Even in the SIM-I experiment where the in-stock probability is given, customer's decision uctuates for the same valuation. In the SIM experiment where the in-stock probability is learned through experience, the magnitude of the uctuation is greater. Although the quantal response model implemented in Chapter 2 oered numer- ous ndings about the behaviors of the strategic customers, it is limited by the fact that it is inherently a static model. Hence, the need for an analysis of the dynamics of the decision making emerged. In Chapter 3, we focus on the learning process of subjects in a two period dynamic posted pricing scheme by utilizing two well-known dynamic learn- ing models: exponential smoothing model and experience-weighted attraction learning model. First, we focus on the learning process of the in-stock probability. In-stock probability plays a central role in the analysis (Liu and van Ryzin, 2008; Elmaghraby et al., 2008; Cachon and Swinney, 2009; Dasu and Tong, 2010). By implementing the exponential smoothing model (Brown and Meyer, 1961), we investigate how perception of in-stock probability changes over time. The exponential smoothing model is the most popular modeling method employed in literature to analyze price learning (Winer, 1986; Kalya- naram and Little, 1994; Popescu and Wu, 2007). In the price learning model, the reference price is updated based on past experiences. By the nature of the exponential smoothing model, the recent price receives more weight. There is a memory parameter that captures 46 the relative emphasis placed between the latest price and the past experience. Instead of studying price learning, we study how subjects learn about the availability of the prod- uct. In our model, the experience of whether the product was available or went out of stock aects the perception of the in-stock probability. Similar to the case of price, the recent experience may have a greater impact on the perception. We use a parameter, , to weigh the outcome in the previous trial. To the best of our knowledge, this is the rst work to study learning of in-stock probability in dynamic pricing schemes. The exponen- tial smoothing model has strong estimation power. One of our main ndings is that the estimated depends on the level of in-stock probability. When the in-stock probability is 16%, is smaller. This causes the perceived in-stock probability to deviate from 0.5 and to reach the true in-stock probability faster. However, when the in-stock probability is 83%, becomes larger and the perceived in-stock probability fails to reach the true level by the end of the experiment. This result is consistent with the nding that the proportion that buy in the rst period in the SIM2 experiment never reaches the optimal level for risk neutral agents, while the proportion in SIM3 does. The exponential learning model only focuses on product availability but not the im- pact of the past actions and outcomes on the subject. To investigate the relationship among past actions, outcomes, and new decisions, we designed an additional set of ex- periment, CONV. Dierent from the SIM experiment, here each experiment consists of 20 trials and the valuation (v) remains the same in all 20 trials in a treatment. In other words, the valuation of the product,v; the rst and second period prices,p 1 andp 2 respec- tively; the number of items, K, and the number of customers, N, are all xed throughout a treatment. CONV consisted of 3 treatments. We continue to assume that there is 47 only one type of customers in this experiment, identical to that of the original model of Dasu and Tong (2010). We employed the experience-weighted attraction (EWA) learn- ing model (Camerer and Ho, 1999) to analyze the data from CONV. The EWA leaning model is one of the methods most frequently used to investigate dynamic games. One of the areas in the behavioral economics which eectively implement this model is the market for durable goods. EWA model incorporates two types of learning: belief learning and reinforcement learning. In belief learning, decision makers choose the strategy that gives the highest payo based on the belief that they have developed about other players' strategies. Reinforcement learning assumes that decision makers only take into account their previous payo but not the actions of others. In CONV data, EWA model fails to detect belief learning. There is greater support for reinforcement learning. The EWA model also has a strong estimation power. The experimental study on pricing and inventory behavior is still an unexplored area (Amaldoss et al., 2008). To the best of our knowledge, this is the rst experimental study on the learning of strategic customers. 3.2 Dynamic Learning of In-Stock Probability 3.2.1 Experimental Results: Dynamic Decision Making First, we re-investigate the SIM experiment from the point of view of dynamic deci- sion making. In this analysis, instead of aggregating all the trials, we take a look at each trial. In the SIM experiments, the outcome of whether or not the product is avail- able in the second period is identical for all subjects. Hence, all subjects receive the 48 Figure 3.1: Proportion of the decision, \Buy in the rst period" at each trial ( t ) same information about the product availability. Valuation is randomly drawn from f125; 130; 135; ; 190; 195; 200g. As a result, by aggregating the subjects' decisions, we can see the change in the proportion that buy in the rst period in each trial. The proportion that buy in the rst period at trial t is given by, t = P v=200 v=125 (O 1 t (v t )=O t (v t )) 16 where O 1 t (v t ) is the number that buy in the rst period when valuation is v t at t and O 1 t (v) = total number of subjects whose valuation is v t att. 16 is the number of possible valuations. Then, we estimated the model, t = +t Figure 3.1 shows the t at each t for SIM2, SIM2-I, SIM3, and SIM3-I and Table 3.1 shows the result of the regression. 49 Treatment In-Stock Probability Constant Slope R 2 (p-value) (p-value) SIM2 83% 0.4379 (0.000) -0.0001 (0.892) 0 SIM2-I 0.4811 (0.000) -0.0021 (0.000) 0.261 SIM3 16% 0.7373 (0.000) 0.0021 (0.000) 0.302 SIM3-I 0.7083 (0.000) 0.0037 (0.000) 0.519 Table 3.1: Trend in the proportion of the decision \Buy in the rst period" at each trial ( t ) The RUMM predicts that t will be 0 and 0.9375 when in-stock probability is 83% and 16%, respectively. The results show that the slopes are higher in the SIM-I experiments. Therefore, information about in-stock probability aects the speed of convergence. In early trials, t is far below the RUMM prediction even in SIM-I. When the in-stock probability is 83%, the speed of learning becomes slower. Moreover, t does not reach to RUMM prediction even after 60 trials. When the in-stock probability is 16%, t nearly reaches the RUMM prediction. It is interesting to see that, in SIM-I, t is far from the decision that maximizes utility in early trials. Theoretically, information about the true in-stock probability should enable subjects to maximize their utility. It appears that in the beginning the subjects underestimate the in-stock probability when it is 83%, and overestimate when it is 16%. This may be due an anchoring bias (Schweitzer and Cachon, 2000). In addition, the speed of learning depends on the in-stock probability. 50 3.2.2 Estimation with Simple Learning Model with Exponential Smoothing 3.2.2.1 Model Description and Estimation To take a closer look at how subjects' learn in-stock probabilities, we utilize a simple dynamic learning model, and reinterpret the result of the SIM experiments. Here we introduce the concept of perceived probability (Kahneman and Tversky, 1979). We as- sume that perceived probability weighs the risk in our decision problem and see how this perceived probability deviates from the initial value, 0.5. We use a single exponential smoothing model (Brown and Meyer, 1961) to describe the change in subjects' perceived in-stock probability. In this model, the outcome from the previous trial has a weight of and the pervious perceived in-stock probability has weight of (1). Therefore, the impact of the past experience depreciates with time while the most recent outcome has a constant weight (P otzelberger and S ogner, 2003). Let result t denote the outcome in trial t as, result t = 8 > > < > > : 0 if product goes out of stock at t 1 if f product is in stock at t Then, the perceived in-stock probability at trial t, t , is updated as, t =result t1 + (1) t1 where 0 1 is the weight to the most recent outcome. In the SIM experiments, the outcome at each trial t, result t , is the same for all subjects. Because the perceived in-stock probability at trial t, t , only depends on the 51 outcome, we can assume that t is identical for all subjects. Note that the valuation observed by subject i at t, v i ;t, is randomly generated. Thus, each subject may have a dierent valuation. Then, subject i's probability of choosing \buy in the rst period" at t is, Pr i;t (v i;t ;p 1 ;p 2 ; t ; ) = exp( R(v i;t ;p 1 ;p 2 ; t )) 1 +exp( R(v i;t ;p 1 ;p 2 ; t )) wherev i;t ,p 1 andp 2 are valuations to the subjecti at trialt, the rst period price and the second period price, respectively, and is the aggregate level of rationality parameter. Note that, although the perceived in-stock probability is assumed to be consistent for all subjects, the choice probability diers by subjects, because each subject may have a dierent valuation even at the same trial 1 . Then, we set the log-likelihood function, LL(; ), as follows, LL(; ) = T X t=1 N X i=1 ln(I(Action(i;t); 0)Pr i;t (v i;t ;p 1 ;p 2 ; t ; )) where Action(i;t) = 8 > > < > > : 1 if the decision of the subject i is \wait for the second period" 0 if the decision of the subject i is \buy in the rst period" 1 Exponential smoothing model does not consider the action of the decision maker. It only depends on the result of the availability of the product in the second period. The model which takes both the action of the customers and the feedback into consideration will be introduced and investigated in the following chapter. 52 and I(x;y) = 8 > > < > > : 1 if x=y 0 if x6= y We also check the error function dened as, SSD = X i;t [Pr i;t (v i;t ;p 1 ;p 2 ; t ; )Action(i;t)] 2 where Action(i;t) = 8 > > < > > : 0 if the decision of the subject i is \wait for the second period" 1 if the decision of the subject i is \buy in the rst period" We estimate the parameters with maximum likelihood method. In addition to the exponential smoothing model, we also check the RUMM model and the random model. The RUMM model assumes that Pr i;t (v i;t ;p 1 ;p 2 ; t ; ) is 1 if v i ;t is greater than the threshold, otherwise 0. Random model assumes that Pr i;t (v i;t ;p 1 ;p 2 ; t ; ) is always 0.5. The comparisons of the exponential smoothing model to other models are in Table 3.2. The estimated parameters are shown in Table 3.3, and the perceived in-stock proba- bility at t, t , is in Figure 3.2. The result shows that alpha is smaller when the in-stock probability is 83%, than when the in-stock probability is 16%. This implies that the speed with which perceived in- stock probability deviating from the initially perceived in-stock probability 0.5 is slower, when the in-stock probability is 83%. Therefore, we may conclude that there is a bigger anchoring bias when the in-stock probability is 83%. 53 Treatment Model LL Number of Parameters AIC BIC Rank SIM2-I Exponential -481.14 2 966.27 968.78 1 Smoothing RUMM -541.85 0 1083.71 1083.71 2 Random -2142.47 0 4284.95 4284.95 3 SIM3-I Exponential -343.81 2 691.62 694.13 1 Smoothing RUMM -542.16 0 1084.31 1084.31 2 Random -879.66 0 1759.31 1759.31 3 SIM2 Exponential -459.56 2 923.12 925.6 1 Smoothing RUMM -523.49 0 1046.98 1046.98 2 Random -2271.43 0 4542.85 4542.85 3 SIM3 Exponential -296.29 2 596.58 599.09 1 Smoothing RUMM -541.55 0 1083.1 1083.11 2 Random -873.63 0 1747.26 1747.26 3 Table 3.2: Estimation of the Parameters by the Exponential Smoothing, RUMM, and Random Models Treatment In-Stock Probability Rationality ( ) Weight parameter ( ) SIM2 83% 0.0809 0.0048 SIM2-I 0.0701 0.0102 SIM3 16% 0.0871 0.0925 SIM3-I 0.1013 0.0945 Table 3.3: Estimation of the parameters by Exponential Smoothing Model 54 Figure 3.2: perceived in-stock probability at trial, t . 55 By observing the perceived in-stock probability at each trial, we can see that the perceived in-stock probability in the case of 83% does not reach the actual in-stock prob- ability within 60 trials. On the other hand, when the in-stock probability is 16%, the perceived in-stock probability becomes the same as the true in-stock probability in 30 trials. After 30 trials, the perceived in-stock probability uctuates. But the range is still within10%, approximately, from the true probability, 16%. After the 30th trial, t is between 0.0542 and 0.2320 in SIM3-I, and t is between 0.0621 and 0.2305 in the SIM3. Therefore, the gap between t and the true in-stock probability is always smaller when the in-stock probability is 16%. Additionally we can see that the perceived in-stock probability in the SIM2 is always below that in SIM2-I. Observation 3.1: The experimental data shows that losses loom larger than gains. Subjects learn faster when the in-stock probability is lower. 3.2.2.2 Asymptotic Analysis To understand how long it takes for customers to fully learn the true in-stock probability, we conducted a numerical simulation. By assuming the simple exponential smoothing, we simulate the perceived in-stock probability. For this we use the parameter estimated from the experiment SIM2-I, SIM2, SIM3-I, and SIM3 given in Table 3.1. For each trial, we randomly generate the results of whether or not the product is available based on the true in-stock probability. Then, the perceived in-stock probability is computed based on the estimated parameters, and . The initial perceived in-stock probability is set to 0.5. Then we repeat the trials until the perceived in-stock probability and the true in-stock probability match. We iterate 10,000 times and observe the average number of 56 trials at which the perceived in-stock probability and the true in-stock probability become identical. The result is summarized in Table 3.4. Treatment In-Stock Probability Average Number of Trials to the True In-Stock Probability (Std.) SIM2 83% 621.84 (148.44) SIM2-I 274.66 (84.54) SIM3 16% 22.99 (10.35) SIM3-I 21.85 (9.67) Table 3.4: Average Number of Trials to Find the True In-Stock Probability 3.2.3 Anchoring Bias and Asymmetry We designed an additional experiment to measure the anchoring eect. We label this additional set of experiment, SIM-R. In this experiment, we control risk aversion by informing subjects that their performance evaluation would be based on the expected prot. In other words, when a subject waits, instead of receiving either (vp 2 ) or 0, they receive expected prot of (vp 2 ) (Osadchiy and Bendoly, 2010). This design makes it easier to nd the perceived in-stock probability. Except for the compensation received by subjects, SIM2-R and SIM3-R are identical to SIM2 and SIM3, respectively. We estimate the in-stock probability subject uses when making decisions by using the QRM with anchoring bias. The QRM with Anchoring is modeled as follows (We call this QRM-A): Payos of buying in the rst period and waiting are (vp 1 ) and (vp 2 ), respectively, where v is the resale price, and P 1 and P 2 are the prices. Let : Rationality parameter. 57 LetR 1 = (vP 1 ),R 2 =(vP 2 ), andR(v;P 1 ;P 2 ;) =R 1 R 2 = (vP 1 )(vP 2 ). In this model, the perceived in-stock probability (b ) is described as follows: b = (1) + anchor , where anchor = 0:5 and 0 1 is anchoring parameter. This model corresponds to the adoption of a uniform error over the choice domain of size two. This approach is widely adopted in the experimental economics. Readers are referred to the work by Harless and Camerer (1994) and Chen et al. (2012). The probability of purchasing in the rst period when the resale price is v, is given by: Pr(v;P 1 ;P 2 ;b ; ;r) = exp( R(v;P 1 ;P 2 ;b )) 1 + (exp( R(v;P 1 ;P 2 ;b )) The rationality parameter is estimated by maximizing the log likelihood function: max ; X v2V [O 1 (v) log(Pr(v;P 1 ;P 2 ;b ; ;r)) +O 2 (v) log(1Pr(v;P 1 ;P 2 ;b ; ;r))] ,where O i (v) is the number of observation in which subjects buy in period i, i = 1; 2; when the resale price is v. The results of the estimation with the QRM-A are in Table 3.5. As above, the anchoring parameters are 0.81 and 0.60 in SIM2-R and SIM3-R, respec- tively. Since anchoring parameters range from 0 to 1, those estimations show that the 58 Experiment In-Stock Prob. Rationality Anchoring Bias Chi-Square RMSD ( ) ( ) () Test (p) SIM2-R 0.83 0.0925 0.81 0.8339 0.0458 SIM3-R 0.16 0.0669 0.6 0.9774 0.048 Table 3.5: Goodness of Fit Test for the QRM-A to SIM-R in-stock probability perceived by the subjects is closer to 0.5 than its original in-stock probabilities (0.83 and 0.16). 3.2.4 Implications for the Firm We explore the value of disclosing the in-stock probability. This analysis is based entirely on experimental data. In Table 3.6 we compare the revenues in the SIM-I and the SIM, in order to determine whether presenting explicit probability information benets the rm. The expected revenues were computed based on the percentage that were bought in the rst period in the experiments. In-stock probability 49% 83% 16% Experiment SIM1-I SIM1 SIM2-I SIM2 SIM3-I SIM3 Expected Revenue 1861.91 1842.11 1758.64 1775.65 1923.83 1919.24 Table 3.6: Expected Revenue in each SIM experiments The SIM generates higher revenue when the in-stock probability is 83%, and the SIM-I generates more when the in-stock probability is 16%. This result can be explained by the anchoring eect. When the probability is 83%, without explicit information, the subjects tend to underestimate the in-stock probability (anchoring parameter = 0.28); therefore, they are more likely to buy in the rst period. This results in higher expected revenues. Similarly, when the probability is 16%, without explicit information, the subjects tend to overestimate (anchoring parameter = 0.81); this results in lower expected revenue in the 59 SIM relative to the SIM-I. These results suggest that rms should inform subjects about stock-out probability only when the stock out probability is greater than 50%. 3.3 EWA Model Estimation The study that uses exponential smoothing model in the previous chapter focuses only on the product availablity. To capture the impact of the decisions made in previous trials on the current trial, we analyze customer decisions using the EWA model (Camerer and Ho, 1999). This model allows us to understand the impact of the action and the feedback. It measures the impact of both the realized payos from their actions and the ctitious payo from actions they did not take. EWA model was employed on CONV data. 3.3.1 Experimental Design: CONV Experiment 3.3.1.1 Experimental Design: CONV Experiment The two-period posted pricing problem is a complicated problem. Even if subjects at- tempt to derive equilibrium probabilities through observation, they are faced with a fairly complex estimation problem. In CONV, we simplify the decision problem by xing the valuation. There were 3 treatments in the CONV set. Each treatment had 20 trials. The rst objective of the CONV experiment was to discover if subjects converge to a specic decision or continue to employ randomized strategies that are consistent with the quantal response model. The second objective was to implement the EWA learning model to gain a deeper understanding of the learning mechanisms. The details of the 3 experiments are given below in Table 3.7. 60 Experiment N K P 1 P 2 2 v 1 CONV1 10 4 60 30 0.49 80 88.82 CONV2 10 5 50 20 0.53 60 83.83 CONV3 10 3 70 50 0.48 89 88.46 Table 3.7: Parameters for CONV experiments In CONV2 the valuation is well below the threshold level for rational risk neutral customers. Here we expect subjects to converge to a decision by consistently deferring purchase to the second period after a few trials. In CONV1 and CONV3 the valuation is closer to the threshold. Convergence to a specic decision in these experiments would provide a compelling evidence that subjects can discover the utility of waiting. 3.3.2 Results of the CONV Experiments 86 subjects participated in the CONV experiment. In gures 3.3 we plot the percentage of times subjects choose to buy in the rst period in CONV1, CONV2, and CONV3. Assuming that the subjects are learning during the experiment, we can expect either an increasing or a decreasing trend. We use a regression model to detect such tendency. Table 3.8 contains the results of the regression analysis. Treatment Slope R-Sq p CONV1 0.00934 37.50% 0.004 CONV2 0.00558 11.80% 0.137 CONV3 0.00829 44.60% 0.001 Table 3.8: Trend in the percentage of subjects who purchase in the rst period We nd a slight trend in CONV1. In the other two experiments at an aggregate level there is no signicant trend. However, the trend in CONV1 is not very signicant either that it leads to the following observation. 61 Figure 3.3: Percentage of customers who purchase in the rst period Observation 3.2: The aggregated purchase decisions shows that subjects do not converge to a single decision but continue to employ a randomized strategy consistent with QRM. We also examined individual decisions to see if some subjects converged to a single decision. Table 3.9 below shows the percentage of subjects who employed the same decision in the last 5 and 7 decisions. % who made the same decisions % who made the same decisions In the last 5 trials In the last 7 trials Buy in 1st Period Buy in 2nd Period Total Buy in 1st Period Buy in 2nd Period Total CONV1 16.28% 11.63% 27.91% 13.95% 9.30% 23.26% CONV2 4.65% 34.88% 39.53% 0.00% 32.56% 32.56% CONV3 20.93% 13.95% 34.88% 9.30% 9.30% 18.60% Table 3.9: Percentage of subject that employed the same decision in the last few trials of CONV 62 Observation 3.3: More than 60% of the subjects do not converge to a deci- sion and their decision making process continues to be consistent with quasi- rational models such as the quantal-response model. The result shows that the subjects do not converge to either one of the decisions: buying in the rst period and waiting to the second period. 3.3.3 Model Estimation 3.3.3.1 EWA Learning Model and Estimation Result EWA model is a behavioral learning model incorporating both belief basis model and reinforcement model (Camerer, 2003). There are two core variables in this model: A j i (t) andN(t). A j i (t) is the attraction for subjecti at trialt, wherej = 1 and 2 correspond to buying in the rst period and the second period, respectively. N(t) measures the weight of past trials. N(t) is updated as follows: N(t) =N(t 1) + 1;t 1 where (0 1) indicates the depreciation rate of previous trial. Then the updating rule for A j i (t) can be described by the following equation: A j i (t) = (N(t 1)A j i (t 1) + [ + (1)I(s j ;s i (t))](s j )) N(t) The parameter(0 1) is the decay rate of the previous attraction. Lower decay rate means that subjects tend to forget about the old observations and put more weight 63 on the most recent payos. s j is the strategy chosen: s 1 ands 2 denote \buying in the rst period" and \wait for the second period", respectively. I s j ;s i (t) ! becomes 1 if s(t) , which is the strategy at trial t, is s j , otherwise, 0:(s j ) is the payo when strategy s j is chosen: (s 1 ) is always (vp 1 ); however,(s 2 ) = (vp 2 ) if the product is available and (s 2 ) = 0 if the product goes out of stock. is a parameter weighing the payo actually received. We setA 1 i (0) equal to (vp 1 ) and setA 2 i (0) is constrained to be between 0 and (vp 2 ). We also x N(0) at one. Attraction weights determine the choice probability. The logit form for probability of choosing strategy s j is given as follows: P j i (t + 1) = e A j i (t) e A 1 i (t) +e A j 2 (t) where denotes the rationality in the aggregated level. Then, we set the log-likelihood function, LL(A 2 (0);;;;), as follows, LL(A 2 (0);;;;) = T X t=1 N X i=1 ln 2 X j=1 I(s j i ;s i (t))P j i (t) ! We also computed the mean squared deviation dened as, MSD = ( P T t=1 P N i=1 P 1 i (t)I(s 1 ;s i (t)) 2 ) NT where T and N are the total number of trials and subjects, respectively. We estimate the parameters with the maximum likelihood method. In addition to the EWA model, we also tested the reinforcement learning model, belief based learning model, and random choice model. In the random choice model each decision is chosen 64 with equal probability. Belief based model focuses on the strategies adopted by others. If = 1 and = , EWA model becomes equivalent to the belief based model. On the other hand, reinforcement model focuses on payos from chosen strategies. The EWA model reduces to the reinforcement model if = 0 and = 0 (Camerer and Ho, 1999). Note that the parameters estimated are at the aggregated level. In other words, the estimated choice probabilities of all of subjects are averaged by trial. Figure 3.4 shows the result of the EWA model estimation. The estimation results of EWA and other learning models are in Table 3.10. EWA dominates the belief based model and random model in all criteria in all treat- ments. In all treatments, EWA dominates the reinforcement model based on LL and MSD. However, when we use AIC and BIC that considers the number of parameters, EWA does not always dominate the reinforcement model. Rather, the performances of both models are similar. This may indicate that the impact of to the model is not very signicant. However, there are some clear implications from the parameter estimation. First, is either zero or close to zero, indicating that the belief based model has the smallest explanation power. Thus, we can infer that the signicance of the expected pay- o from the unchosen strategy is very small. Instead, subjects tend to rely on the result from the chosen strategy. Second, is close to 1, meaning that subjects put more weight on previous attraction. Lastly,A 2 (0) values for CONV1, CONV2, and COMV3 are 29.97, 39.80 and 22.38, respectively. Because we set the range of A 2 (0) to be between zero and the maximum payo, the value measures the initial attraction of the risky option. All three are greater than the risk neutral expected payo. 65 (a) (b) (c) Figure 3.4: Probability of Buying in the First Period and EWA Estimates 66 Experiment Learning Model A 2 (0) LL MSD AIC BIC CONV1 EWA 0.84 0.7733 0.1056 0.0661 29.97 -443.5 0.2039 897.03 903.21 Belief 0.772 0.772 1 0.0367 39.48 -506.37 0.2424 1018.74 1022.44 Reinforcement 0.8102 0 0 0.0164 36.93 -445.93 0.2055 897.85 901.56 Random - - - - - -517.77 0.25 1035.54 1035.54 CONV2 EWA 0.7921 0.6602 0.0005 0.0664 39.8 -359.61 0.1581 729.23 735.41 Belief 0.9531 0.9531 1 0.0721 40 -415.35 0.1854 836.71 840.41 Reinforcement 0.7742 0 0 0.0258 40 -363.92 0.1609 733.84 737.54 Random - - - - - -517.77 0.25 1035.54 1035.54 CONV3 EWA 0.8695 0.5166 0 0.0341 22.38 -445.38 0.2063 900.77 906.95 Belief 0.918 0.918 1 0.0481 23.05 -511 0.2455 1028 1031.71 Reinforcement 0.8623 0 0 0.0172 21.77 -445.56 0.2064 897.11 900.82 Random - - - - - -517.77 0.25 1035.54 1035.54 Table 3.10: Estimation of the Parameters by the Learning Models 3.3.3.2 Asymptotic Analysis We extended the number of trials by conducting a simulation to check whether or not the decisions converge. Assuming the decision of customers can be described by EWA learning model, we conducted the asymptotic analysis employing estimated parameters. Since the valuation is xed in each treatment, we know that the decisions must converge to either \buying in the rst period" or \waiting until the second period" depending on the risk attitude of a subject. The procedure of the simulation is as follows: We use the parameters, ;;;;A 1 (0), and A 2 (0), estimated by EWA model to determine purchase probabilities. The results of whether or not the product is available are simulated based on the in-stock probability, and the decisions of customers at t are simulated based on the choice probability, P 1 (t). Then, A 1 (t), and A 2 (t) are updated. We repeat this until the 200th trial. Finally, when t = 200, we record the probability of choosing \buy now", P 1 (t = 200). 67 Figure 3.5: Samples of Asymptotic Analysis from CONV1 We iterated this procedure 1,000,000 times. The result shows that the probability of buying in the rst period never converges. Figure 3.5, Figure 3.6 and Figure 3.7, display the rst 200 trials of some of the samples in CONV1, CONV2, and CONV3, respectively. If decisions converge to the optimal decision in 200 trials, P 1 (t = 200) should be either 0 or 1. We simulated P 1 (t = 200) 1,000,000 times and graph the distribution of P 1 (t = 200). The distribution is in the following Figure 3.8. The results in the Figure 3.8 indicate that there is no convergence to the optimal decision after 200 trials because P 1 (t = 200) is not concentrated at any point. 3.4 Conclusions In this Chapter, we investigated the learning process of strategic customers. First, we studied how the perception of the in-stock probabilities evolves. We re-interpreted the 68 Figure 3.6: Samples of Asymptotic Analysis from CONV2 Figure 3.7: Samples of Asymptotic Analysis from CONV3 69 (a) (b) (c) Figure 3.8: histogram of P 1 (t = 200) in CONV1, CONV2, and CONV3 70 experimental results of the previous chapter. Even with information about the true in- stock probability, the decisions in the beginning deviate much from the optimal decision which is predicted by the RUMM. However, the decisions in the latter trials approached the decision predicted by RUMM. Hence, we introduced the concept of the perceived probability and anchoring, and studied the strategic customers' learning of the in-stock probability by using the exponential smoothing model. The results showed that the amount of anchoring on 0.5 was depended on the true in-stock probability: when it is 83% the anchoring eect is bigger. Therefore, it takes more time to converge to the true in-stock probability of 83%. We then extended the range of our analysis by considering actions and outcomes. We designed a new set of experiment called CONV. In the CONV experiment, subjects repeatedly encountered the same decision making problem. In this simpler setting we observed whether the decisions eventually converged. The result showed a slight but insignicant convergence tendency. Then, we introduced the EWA model to study the interactions between actions and outcomes. EWA model had a very strong estimation power. We found that subjects updated the attraction of the chosen strategy, not the unchosen one, consistent with a reinforcement learning model. To the best of our knowledge, this is the rst work analyzing the dynamic decisions making of strategic customers. Our future work includes a deeper investigation on the impact of customers' learning process on the nancial performance of a rm. We believe that the exponential smoothing and EWA models would be useful tools for analyzing the behavior of strategic customers in the setting of dynamic decisions making. 71 Chapter 4 Dynamic Posted Pricing Scheme: Existence and Uniqueness of Equilibrium Bidding Strategy 4.1 Introduction In the posted dynamic pricing scheme, a path of prices is announced to the customers at the beginning of the planning horizon. Since the posted pricing scheme eliminates the uncertainty of future prices, it simplies the decision making for strategic customers and, therefore, oers a simpler theoretical ground for analyzing dynamic pricing schemes. This pricing scheme has been studied by a number of researchers such as Elmaghraby et al. (2008), Aviv and Pazgal (2008) and Dasu and Tong (2010). They investigate the best response for each customer, nd the Nash equilibrium of the customers' purchasing strategies, and based on this equilibrium, extended the analysis to the impact of this strategic behavior on the seller's decision and the expected revenue. In this chapter, we focus on the equilibrium bidding behavior. We investigate the uniqueness of equilibrium under RUMM and the existence of an equilibrium under QRM. To start, we consider a simple setting where two strategic customers compete for one 72 unit of item oered by a monopolist in two periods. The two customers are identical in a sense that their valuations are drawn from the same distribution. Similar to Dasu and Tong (2010), the customers arrive at the start of selling season. Prices are preannounced in the beginning, and the second-period price represents markdown. Even in this simple setting, we nd some examples showing multiple equilibria. Furthermore, we show the uniqueness of the equilibrium when the buyers' valuations are uniformly distributed. Our second assumption on the decision making of strategic customers is that they are bounded rational. We model the bounded rationality with the quantal response framework (McFadden, 1976; McKelvey and Palfrey, 1995). In the quantal response framework, the option yielding greater prot possesses a higher choice probability but not always equal to one. This is one of the most popular approaches to describe the bounded rationality of the decision makers (Su, 2008). We nd that when the distribution of the valuations of the two identical buyers is drawn from uniform distribution, there is also a unique equilibrium. In addition, when there are multiple customers, there exists equilibrium. Then, we design an experimental study to validate the ndings from our analysis. Our experimental results show the possibility of an equilibrium with multiple buyers. Subjects play a simulation game based on the theoretical model. To control the dynamics, we employ a partial equilibrium model. In other words, each subject plays not with other human subjects, but with computerized buyers employing RUMM. The experimental results point to the possibility of an equilibrium. While the unique equilibrium behavior is often essential to meaningfully predicting the outcomes of a game with strategic customers, this issue has received limited attention in the literature. For example, Dasu and Tong (2010) do not acknowledge that the buyer's 73 equilibrium bidding strategy may not be unique. Osadchiy and Bendoly (2010) and Correa et al. (2011a) have investigated uniqueness in their extensions to the two-period posted pricing scheme of Aviv and Pazgal (2008). Osadchiy and Bendoly (2010) oer a sucient condition for the uniqueness of an equilibrium, and Correa et al. (2011a) show the existence of unique equilibrium in a more general setting with one item. These papers assume that the buyers arrive dynamically according to a Poisson process, which makes it easier to establish the uniqueness of the equilibrium since the buyers bid sequentially (one at a time) instead of biding simultaneously. In this chapter, we consider a setting where all of the buyers present from the beginning of the planning horizon, and show that multiple equilibria can exist even if the rm oers only one unit of the item. The model by Elmaghraby et al. (2008) adopts an assumption that \valuations are drawn from non- overlapping intervals", which sidesteps the diculty associated with proving uniqueness by assuming that only one of the two buyers faces a nontrivial bidding decision. To our knowledge, this is the rst work in the area of studying strategic customers, which investigates the equilibrium behavior of bounded rational buyers and also extends the range to the experimental study. 4.2 Theoretical Model 4.2.1 Model Description and Threshold Policy Our model is adapted from Elmaghraby et al. (2008) and Dasu and Tong (2010). Suppose that there exist N buyers, indexed by i = 1;:::;N, and each buyers is interested in ob- taining one unit of a particular product. LetV i denote the random variable representing 74 buyeri's valuation of the product, and we usev i to denote its realization. We assume that fV 1 ;:::;V N g are independent and identically distributed with the common distribution V , and we refer to its cumulative density function and probability density function as G() and g(), respectively. Let K be the number of units for sale. The price changes from the rst period to the second period, and let p 1 and p 2 denote the price per unit in the rst period and the second period, respectively. Both p 1 and p 2 are exogenously given, and we assume p 1 p 2 . We describe the sequence of events. The value of v i is realized for each i, and any customer i with valuation v i p 2 leaves the system immediately. Then, the remaining buyers decides, independently and simultaneously, whether or not to place a bid in the rst period,N. If the seller (monopolist) receivesK or more bids, then the she randomly selects the K bidders to whom the unit will be sold at the price of p 1 , and there is no more unit remaining for the second period. If the number of bids is less than K, then each buyer who bid will buy the product atp 1 , and any remaining unit will be sold in the second period atp 2 , when buyers who did not buy in the rst period have an opportunity to bid. This model oers a simple structure to analyze the buyer's problem in the second period. If he does not place a bid, then his payo will be 0. If he places a bid, then there is some chance of being able to buy a unit, in which case his payo is vp 2 , where v is the buyer's valuation of the product. Thus, it is optimal that the buyer places his bid if and only if his valuation exceeds p 2 . The rst period problem involves a more delicate tradeo between the price and the probability of obtaining the product. 75 Below, we characterize the buyers' equilibrium bidding strategies and show that they follow a threshold policy. Lemma 4.2.1. Fix p 1 and p 2 , where p 1 p 2 . Let i2f1;:::;Ng. (a) The dominant strategy of bidder in the second period is to bid if and only if v i p 2 . (b) Any equilibrium of among the buyers can be characterized by i p 1 , for each i2f1;:::;Ng such that bidder i's strategy is given by 8 > > > > > > > > > < > > > > > > > > > : do not bid if v i <p 2 bid in period 2 if p 2 v i < i bid in period 1 if v i i . (4.1) Proof. If the second period bids are accepted, then the buyer's expected prot from bidding in the second period is (v i p 2 ) multiplied by the probability that the buyer will obtain a unit in the second period conditioned on bids being accepted in the second period. Since this depends only on the other bidders' strategies, not on v i , the optimal decision for buyer i is to bid if and only if v i p 2 . This proves (a). Now, it suces to consider the rst period bid for the case v i p 2 . Suppose that we x the strategies of buyers other than buyer i. Let 1 and 2 denote the probability that buyeri will obtain a unit if he bids in the rst period and if he bids in the second period, 76 respectively. Clearly 1 2 . Suppose that it is optimal to bid in the rst period with v i . Then, 1 (v i p 1 ) 2 (v i p 2 ): Then, for any ^ v>v i , we have 0 1 (v i p 1 ) 2 (v i p 2 ) 1 (v i p 1 ) 2 (v i p 2 ) + ( 1 2 ) (^ vv i ) = 1 (^ vp 1 ) 2 (^ vp 2 ) ; where the second inequality follows from 1 2 and ^ v >v i . Thus, we obtain 1 (^ v p 1 ) 2 (^ vp 2 ), implying that it is also optimal for buyer 1 to bid in the rst period when his value is ^ v. We remark that Lemma 4.2.1 is applicable for a general model with arbitrary N, K and any distribution for V . 4.2.2 Multiplicity of Equilibrium We rst show that the equilibrium strategy for buyers may not be unique. 77 4.2.2.1 Deterministic Distribution Case Proposition 4.2.2. Suppose N = 2 and K = 1. Suppose that the buyers' valuation is deterministic and identical, i.e., V 1 = V 2 = ^ v. If p 1 ^ v 2p 1 p 2 , then the buyers' bidding strategy forms multiple equilibria. Proof. If both players bid in the rst period, then two buyers have an equal probability of obtaining the object. In this case, the payo to a buyer i is V i p 1 = ^ vp 1 if the buyer obtains the object; otherwise, it is 0. Thus, the expected payo is (^ vp 1 )=2. If only one buyer submits the bid in the rst period, this buyer will have the payo of ^ vp 1 , whereas the other buyer has the payo of 0. If neither buyers submits the bid in the rst period, each buyer receives the object in the second period with probability 0.5, yielding the expected payo of (V i p 2 )=2 = (^ vp 2 )=2. We summarize this in the following table. (Payo to 1, Payo to 2) Buyer 2 bids in period 1 Buyer 2 bids in period 2 Buyer 1 bids in period 1 ^ vp 1 2 ; ^ vp 1 2 (^ vp 1 ; 0) Buyer 1 bids in period 2 (0; ^ vp 1 ) ^ vp 2 2 ; ^ vp 2 2 Note that there are two equilibria above since p 1 ^ v 2p 1 p 2 : (i) both buyers bid in the rst period, and (ii) both buyers do not bid in the rst period and bid in the second period. (This is essentially the well-known matching pennies game.) The result of Proposition 4.2.2 can be extended to non-identical valuations between the buyers. 78 Lemma 4.2.3. In the case of asymmetric buyers with deterministic valuation, the buyers' strategy may not form a unique equilibrium. And, equilibrium of buyer's mixed strategy can exist. Proof. Same results are from Proposition 4.2.2 and Lemma 4.2.4 with V 1 6=V 2 . Under the setting of Proposition 4.2.2, it can be shown that an equilibrium exists in mixed strategies, where each of the buyers bid in the rst period with probability 2p 1 p 2 ^ v p 1 p 2 . Lemma 4.2.4. If the value distribution is deterministic, then the equilibrium of buyers' mixed strategy can exist. Proof. In the setting of Proposition 4.2.2, suppose both buyers employ a mixed strat- egy such that buyer 1 and buyer 2 bid in the rst period with probability of 1;1 and 1;2 , respectively. The buyer 1's expected payo from buyer 2's mixied strategy is (Vp 1 ) (1 1;2 ) + (Vp 1 ) 2 1;2 = (Vp 1 ) 1 1;2 2 if buyer 1 bids in the rst period and is 1 2 (Vp 2 ) (1 1;2 ) if buyer 1 bids in the second period. Similarly, the buyer 2's ex- pected payo from buyer 1's mixed strategy is (Vp 1 ) 1 1;1 2 and 1 2 (Vp 2 ) (1 1;1 ) if bids in the rst period and second period, respectively. Therefore the equilibrium mixed strategy 0 1;1 1 is such that 79 Vp 1 (1 1;1 2 = 1 2 (Vp 2 )(1 1;1 ) , (Vp 1 )(1 1;1 2 ) = 1 2 (Vp 2 )(1 1;1 ) , 2(Vp 1 ) (Vp 1 ) 1;1 = (Vp 2 ) (Vp 2 ) 1;1 , (Vp 2 ) 1;1 (Vp 1 ) 1;1 = (Vp 2 ) 2 (Vp 1 ) , (p 1 p 2 ) 1;1 =Vp 2 2V + 2p 1 , (p 1 p 2 ) 1;1 = 2p 1 p 2 V , 1;1 = 2p 1 p 2 V p 1 p 2 And the equilibrium payo is (Vp 1 ) 2(p 1 p 2 )2p 1 +p 2 +V 2(p 1 p 2 ) = (Vp 1 )(Vp 2 ) 2(p 1 p 2 ) . Likewise, 1;2 = 2p 1 p 2 V p 1 p 2 and the equilibrium payo isrev 2 = (Vp 1 )(Vp 2 ) 2(p 1 p 2 ) . The equilibrium exists only when p 1 V 2p 1 p 2 4.2.2.2 Continuous Distribution Case Multiple equilibria can exist even when the valuations are uncertain. Proposition 4.2.5. If the value distribution is continuous, then the buyers' strategy may not form a unique equilibrium. 80 Proof. Suppose now that eachV i has the following distribution. It has support on [0; 1]. Let > 0. Let f denote the density of V i such that it has the support of [0; 1], and satises f(x) = 8 > > > > < > > > > : + 1 2 if x2 [0:9; 0:9 +] otherwise Additionally, suppose that 0 <p 2 <p 1 < 0:9 and p 2 < 2p 1 1 . (Because p 1 > 0:5, there exists p 2 > 0 such that p 2 < 2p 1 1) Then, for suciently small , we still have multiple equilibra. It is shown that if V i <p 2 buyer i does not bid, and if p 2 <V i <p 1 buyer i bids in the second period by Lemma 4.2.1. Suppose buyer 2 follows a Strategy A: Bid in period 2 if V 2 p 2 , Do not bid otherwise. Now let's consider the decision of buyer 1. IfV 1 <p 2 buyer 1 does not bid, and ifp 2 <V 1 <p 1 buyer 1 bids in the second period. When V 1 > p 1 , buyer 1 can bid in either period 1 or period 2. If buyer 1 bids in the rst period, the payo to the buyer is (V 1 p 1 ) because buyer 2's does not bid in period 1. If buyer 1 bids in the second period, the payo to the buyer is as follows: p 2 (V 1 p 2 ) + (1p 2 ) 2 (V 1 p 2 ) = (1 +p 2 ) 2 (V 1 p 2 ) Therefore, buyer 1 bids in the rst period if (V 1 p 1 ) (1+p 2 ) 2 (V 1 p 2 ). Otherwise, buyer 1 bids in the second period. (V 1 p 1 ) (1 +p 2 ) 2 (V 1 p 2 ), (V 1 p 1 ) (V 1 p 2 ) (1 +p 2 ) 2 81 Meanwhile, (V 1 p 1) (V 1 p 2 ) (1p 1 ) (1p 2 ) because xa xb is a non-decreasing function of x when a > b and V i 2 [0; 1] . In addition, (1p 1 ) (1p 2 ) < 1 2 because we suppose that p 2 < 2p 1 1. (p 2 < 2p 1 1, (1p 1 ) (1p 2 ) < 1 2 ) Therefore, it is always (V 1 p 1) (V 1 p 2 ) < (1+p 2 ) 2 and buyer always bids in the second period when V 1 >p 1 . Thus, buyer 1 also followes the same strategy, Strategy A, as buyer 2. By symmetricity, buyer 2 follows Strategy A if buyer 1 follows Strategy A. Dene > 0 s.t. p 1 +< 0:9 or < 0:9p 1 . Since, p 1 < 0:9, such exists if is suciently small. Suppose buyer 2 has a Strategy B as follows: Bid in period 1 if V 2 p 1 +, bid in period 2 if p 2 V 2 < p 1 +, and do not bid otherwise. When V 1 >p 1 , if buyer 1 bids in the rst period, the payo to the buyer is as follows: (1 (p 1 +)) 2 V 1 p 1 + (p 1 +) V 1 p 1 = (1 + (p 1 +)) 2 V 1 p 1 If buyer 1 bids in the second period, the payo to the buyer is as follows: ((p 1 +)p 2 ) 2 V 1 p 2 +p 2 V 1 p 2 = ((p 1 +) +p 2 ) 2 V 1 p 2 82 Therefore, whenV 1 >p 1 ,the buyer 2 bids in the rst period if (1+(p 1 +)) 2 V 1 p 1 ((p 1 +)+p 2 ) 2 V 1 p 2 and the buyer 2 bids in the second period otherwise. (1 + (p 1 +)) V 1 p 1 ((p 1 +) +p 2 ) V 1 p 2 ,f(1 + (p 1 +)) ((p 1 +) +p 2 )gV 1 (1 + (p 1 +))p 1 ((p 1 +) +p 2 )p 2 , (1p 2 )V 1 p 1 + (p 1 +)p 1 (p 1 +)p 2 p 2 2 , (1p 2 )V 1 p 1 + (p 1 p 2 ) (p 1 +)p 2 2 ,V 1 p 1 + (p 1 p 2 ) (p 1 +)p 2 2 (1p 2 ) We can say that the buyer 2 follows the same strategy, Strategy B, if p 1 +(p 1 p 2 )(p 1 +)p 2 2 (1p 2 ) = p 1 +. Equivalently, the buyer 2 follows Strategy B if, = p 1 + (p 1 p 2 ) (p 1 +)p 2 2 (1p 2 ) p 1 , (1p 2 ) =p 1 + (p 1 p 2 ) (p 1 +)p 2 2 p 1 (1p 2 ) , (1p 2 ) =p 1 + (p 1 p 2 )p 1 + (p 1 p 2 )p 2 2 p 1 (1p 2 ) , (1p 2 ) (p 1 p 2 ) =p 1 + (p 1 p 2 )p 1 p 2 2 p 1 (1p 2 ) , (1p 1 ) =p 1 +p 2 1 p 1 p 2 p 2 2 p 1 +p 1 p 2 , (1p 1 ) = p 2 1 p 2 2 , = p 2 1 p 2 2 (1p 1 ) 83 Let's check the existence of = (p 2 1 p 2 2 ) (1p 1 ) s.t. 0 < < 0:9p 1 . Since p 1 > p 2 and > 0, p 2 1 p 2 2 > 0 , and therefore (p 2 1 p 2 2 ) (1p 1 ) > 0 for suciently small . And, p 2 1 p 2 2 (1p 1 ) < 0:9p 1 , p 2 1 p 2 2 < (0:9p 1 ) (1p 1 ) , p 2 1 p 2 2 < 0:9p 1 0:9p 1 +p 1 2 +p 2 1 , 0:9p 1 0:9p 1 +p 1 2 +p 2 1 p 2 1 p 2 2 > 0 ,p 1 2 1 + 0:9p 1 +p 2 2 + 0:9p 1 > 0 If we let r () = p 1 2 1 + 0:9p 1 +p 2 2 + 0:9p 1 , dr() d = 2p 1 1 + 0:9p 1 +p 2 2 and d 2 r() d 2 = 2p 1 . When = 0, r () = 0:9p 1 and dr() d = 1 + 0:9p 1 +p 2 2 . Let s () = 1 + 0:9p 1 +p 2 2 + 0:9p 1 is a support of r (). Since 0:9p 1 > 0, s ()> 0 iif 0< 0:9p 1 1+0:9p 1 +p 2 2 . And therefore, r ()>s ()> 0 if 0< 0:9p 1 1+0:9p 1 +p 2 2 . ( 0:9p 1 1+0:9p 1 +p 2 2 > 0 because p 1 < 0:9.) Therefore, = (p 2 1 p 2 2 ) (1p 1 ) < 0:9p 1 if < 0:9p 1 1+0:9p 1 +p 2 2 . Hence, there is a equilibrium strategy, Strategy B with = (p 2 1 p 2 2 ) (1p 1 ) for suciently small . Suppose buyer 2 has a Strategy C as follows: Bid in period 1 if V 2 p 1 , bid in period 2 if p 2 V 2 < p 1 , do not bid otherwise. When V 1 > p 1 , if buyer 1 bids in the rst period, the payo to the buyer is as follows: (1p 1 ) 2 V 1 p 1 +p 1 V 1 p 1 = (1 +p 1 ) 2 V 1 p 1 84 If buyer 1 bids in the second period, the payo to the buyer is as follows: (p 1 p 2 ) 2 V 1 p 2 +p 2 V 1 p 2 = (p 1 +p 2 ) 2 V 1 p 2 Therefore, when V 1 > p 1 ,the buyer 2 bids in the rst period if (1+p 1 ) 2 V 1 p 1 (p 1 +p 2 ) 2 V 1 p 2 and the buyer 2 bids in the second period otherwise. (1 +p 1 ) V 1 p 1 (p 1 +p 2 ) V 1 p 2 ,f(1 +p 1 ) (p 1 +p 2 )gV 1 (1 +p 1 )p 1 (p 1 +p 2 )p 2 , (1p 2 )V 1 p 1 +p 2 1 p 1 p 2 p 2 2 ,V 1 p 1 + p 2 1 p 1 p 2 p 2 2 (1p 2 ) p 1 +(p 2 1 p 1 p 2 p 2 2 ) (1p 2 ) !p 1 as! 0. Therefore, the buyer 2 also follows Strategy C as ! 0. Note that Strategy C is a special case of Strategy B. In conclusion, there exist multiple equilibria for suciently small . 4.2.2.3 Normal Distribution Case Proposition 4.2.6. If the value distribution is Normal distribution, then the buyers' strategy may not form a unique equilibrium. Proof. We now consider the case where the value distribution is no longer a uniform distribution but is given by a general distribution. Let f and F denote the probability density function and cumulative density function of the value distribution. With this 85 generalization, we see that Lemma 4.2.1 remains valid. Fix the value of j such that j p 1 lies within the support of the value distribution. Consider the optimal response of buyer i. Suppose v i p 1 . Bidding in the rst period. The expected prot is given by u i 1 ( j ) = F ( j ) + (1F ( j ))=2 v i p 1 = F ( j ) 2 + 1 2 v i p 1 : Bidding in the second period. The expected prot in this case is given by u i 2 ( j ) = F (p 2 ) + (F ( j )F (p 2 ))=2 v i p 2 = F ( j ) 2 + F (p 2 ) 2 v i p 2 : Note that it is better to bid in the rst period if the following condition is met: F ( j ) 2 + 1 2 v i p 1 F ( j ) 2 + F (p 2 ) 2 v i p 2 1F (p 2 ) 2 v i p 1 p 2 F (p 2 ) 2 + p 1 p 2 2 F ( j ) v i p 1 p 2 F (p 2 ) 1F (p 2 ) + p 1 p 2 1F (p 2 ) F ( j ) : This characterizes the optimal response of buyer i. We denote i ( j ) = p 1 p 2 F (p 2 ) 1F (p 2 ) + p 1 p 2 1F (p 2 ) F ( j ) : Now, let's look for a symmetric equilibrium, i.e., i ( j ) = j . Then, j = p 1 p 2 F (p 2 ) 1F (p 2 ) + p 1 p 2 1F (p 2 ) F ( j ) (4.2) 86 Consider, for example, p 2 = 0. Then, the equation (4.2) becomes j = p 1 +p 1 F ( j ) Assume that p 1 = 0:2, F (x) is a c.d.f of Normal distribution whose mean is 0.3 and standard deviation is 0.05. Then, i (0) = 0:2 0 = j i (0:25) = 0:2 + 0:2F (0:25) = 0:2 + 0:2 0:1587 = 0:2317 0:25 = j i (0:35) = 0:2 + 0:2F (0:35) = 0:2 + 0:2 0:8413 = 0:3683 0:35 = j 4.2.3 Existence of the Unique Equilibrium: Uniform Distribution Case 4.2.3.1 Existence of the Unique Equilibrium Now, we focus our attention to the case with N = 2 and K = 1 where each V i is uniformly distributed between 0 and 1, for i2f1; 2g. In this section, we show that a unique equilibrium exists for buyers' strategies, and provide a close-form expression for the equilibrium bidding strategy. Suppose i;j2f1; 2g where i6=j. Based on Lemma 4.2.1(b), the threshold values i and j play an important role in describing the strategy, where i ; j 2 [p 1 ; 2]. Suppose we x the value of j , and study the optimal response of buyer i. Suppose v i 2 [p 1 ; 1]. The buyer needs to decide between bidding in the rst period and bidding in the second period. Let j 2 [p 1 ; 1]. 87 Suppose buyer i bids in the rst period. Then, if buyer j's valuation v j < j , then buyeri obtain the unit. Ifv j 2 [ j ; 1] then buyeri obtains the unit with probability 1=2. Thus, the expected prot is given by u i 1 ( j ) = j + (1 j )=2 v i p 1 = j 2 + 1 2 v i p 1 : Suppose buyer i does not bid in the rst period, and bids in the second period. Then, buyer i obtains the unit only if buyer j did not bid in the rst period. In this case, if buyer j does not bid in the second period (which occurs if v j < p 2 ), then buyer i will get the unit. Otherwise, if buyer j bids in the second period (i.e., v j 2 [p 2 ; j )), then buyer i will get the unit with probability 1=2. Thus, the expected prot in this case is given by u i 2 ( j ) = p 2 + ( j p 2 )=2 v i p 2 = j 2 + p 2 2 v i p 2 : Then, the buyer i's decision is based on comparing u i 1 ( j ) and u i 2 ( j ). Based on this approach, we obtain the following result. Proposition 4.2.7. Suppose N = 2, K = 1, and V i U[0; 1] for i2f1; 2g. Fix p 1 and p 2 , where p 2 < 1. Then, the unique equilibrium bidding strategy is symmetric and characterized by Lemma 4.2.1(b) and the following: (i) if p 1 p 2 2 1p 1 2 [p 1 ; 1], then 1 = 2 = p 1 p 2 2 1p 1 ; and (ii) if p 1 p 2 2 1p 1 > 1, do not bid in the rst period. Furthermore, p 1 p 2 2 1p 1 p 1 . 88 The proof of Proposition 4.2.7 follows after Proposition 4.2.8. Proposition 4.2.8. Under the conditions of Proposition 4.2.7, if bidder j's strategy follows by a threshold policy of (4.1) with xed j 2 [p 1 ; 1], then bidder i's best response strategy also follows (4.1) where the threshold value i satises i ( j ) = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 j : Proof. We compare u i 1 ( j ) and u i 2 ( j ). Both of these functions are linear functions of v i with an intersection at i ( j ). To ses this, we equate u i 1 ( j ) and u i 2 ( j ) and solve for v i : j 2 + 1 2 v i p 1 = j 2 + p 2 2 v i p 2 1p 2 2 v i = p 1 p 2 2 2 + p 1 p 2 2 j v i = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 j : Note that u i 1 ( j ) has a higher slope than u i 2 ( j ). Thus, if v i < i ( j ), then u i 1 ( j ) > u i 2 ( j ), i.e., it is better to bid in the rst period; similarly, if v i > i ( j ), then u i 1 ( j )< u i 2 ( j ), i.e., it is better to bid in the second period. This completes the proof. Proof of Proposition 4.2.7. We rst note that p 1 p 2 2 1p 1 p 1 p 2 1 1p 1 = p 1 : We now argue that (i) and (ii) form an equilibrium bidding strategy. 89 Suppose p 1 p 2 2 1p 1 2 [p 1 ; 1]. If j = p 1 p 2 2 1p 1 , then from Proposition 4.2.8, buyer i's best response satises i ( j ) = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 j = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 p 1 p 2 2 1p 1 = 1p 1 1p 1 p 1 p 2 2 1p 2 + p 1 p 2 1p 2 p 1 p 2 2 1p 1 = (1p 2 )(p 1 p 2 2 ) (1p 1 )(1p 2 ) = p 1 p 2 2 1p 1 = i : Thus, ( 1 ; 2 ) forms an equilibrium strategy. Now, suppose p 1 p 2 2 1p 1 > 1. If buyer j's strategy is not to bid in the rst period, and it is equivalent to having j = 1. Thus, bidder i's best response satises i ( j ) = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 j = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 > 1p 2 1p 2 = 1 ; where the above inequality follows from (p 1 p 2 2 ) + (p 1 p 2 ) > (1p 1 ) + (p 1 p 2 ) = 1p 2 : Thus, bidder i's strategy is not to bid in the rst period. We now argue that any equilibrium bidding strategy should be symmetric. Let (^ 1 ; ^ 2 ) be any equilibrium bidding strategy, where ^ 1 ; ^ 2 2 [p 1 ; 1]. Assume, by way of contradic- tion, ^ 1 6= ^ 2 . Without loss of generality, assume ^ 1 < ^ 2 ; thus ^ 1 < 1. 90 Then, from Proposition 4.2.8, we must have ^ 1 < 2 (^ 1 ) = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 ^ 1 : (Otherwise, we would not have ^ 1 < ^ 2 .) Then, 1 (^ 2 ) = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 ^ 2 > p 1 p 2 2 1p 2 + p 1 p 2 1p 2 ^ 1 > ^ 1 : This implies that ^ 1 is not the best response strategy to ^ 2 . This contradiction shows ^ 1 = ^ 2 . Finally, we argue for the uniqueness of the equilibrium. Suppose (~ ; ~ ) and (^ ; ^ ) represent two distinct equilibrium strategies, where ~ ; ^ 2 [p 1 ; 1]. Suppose, by way of contradiction, that ~ 1 6= ^ 1 . Without loss of generality, there exists > 0 such that ^ = ~ +. Note that this implies p 1 < 1. From Proposition 4.2.8, i (~ ) = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 ~ j and i (^ ) = p 1 p 2 2 1p 2 + p 1 p 2 1p 2 ^ j : Case ^ < 1. Then, i (~ ) = ~ and i (^ ) = ^ . It follows = i (^ ) i (~ ) = p 1 p 2 1p 2 [^ ~ ] = p 1 p 2 1p 2 < : Case ^ = 1. Then, i (~ ) = ~ and i (^ ) 1 = ^ . We have = ^ ~ i (^ ) i (~ ) = p 1 p 2 1p 2 [^ ~ ] = p 1 p 2 1p 2 < : 91 In both cases, we obtain a required contradiction. Thus, we conclude that ~ = ~ . Proposition 4.2.7 provides not only a closed-form expression for an equilibrium strat- egy, but also shows that this equilibrium is unique and symmetric. See Figure 4.1 for illustration. (a) (p 1 ;p 2 ) = (0:5; 0:4) (b) (p 1 ;p 2 ) = (0:6; 0:4) Figure 4.1: Uniform Distribution Examples. In (a), the equilibrium 1 = 2 value is 0.68. In (b), the the equilibrium 1 = 2 value is 1, i.e., not bidding in the rst period. 4.2.3.2 Revenue Function Now, we turn our attention to the seller's pricing problem, where she determine the value of p 1 and p 2 to maximize her revenue. First of all, we argue that it suces to consider the value of p 1 where the threshold value in the statement of Proposition 4.2.7 satises p 1 p 2 2 1p 1 1 : 92 To see this, suppose that this fraction exceeds 1, implying that neither buyer submits any bid in the rst period. Then, consider decreasing p 1 until this ratio becomes 1. It would not aect the probabilities that a buyer submits a bid in the rst period and the second period under the equilibrium solution. Let (p 1 ;p 2 ) = p 1 p 2 2 1p 1 : Consider the buyers' strategy (;), where 2 [p 1 ; 1]. The seller will sell the unit at the price of p 1 if at least one buyer i satises v i . She will sell at the price of p 2 if both buyers' values are less than , but at least one of them exceeds p 2 . Proposition 4.2.9. R (p 1 ;p 2 ) is a concave function of p 1 Proof. R (p 1 ;p 2 ) =p 1 p 1 p 2 2 1p 1 2 (p 1 p 2 )p 3 2 The rst derivative of the function is as follows: dR (p 1 ;p 2 ) dp 1 = 1 ( 2 p 1 p 2 2 1p 1 ( (1p 1 ) + p 1 p 2 2 (1p 1 ) 2 ) (p 1 p 2 ) + p 1 p 2 2 1p 1 2 ) = 1 ( 2 p 1 p 2 2 1p 2 2 (p 1 p 2 ) (1p 1 ) 3 + p 1 p 2 2 1p 1 2 ) And the second derivative of the function is as follows: d 2 R(p 1 ;p 2 ) dp 2 1 = 2 f(p 1 p 2 2 )(1p 2 2 )+(1p 2 2 )(p 1 p 2 )g(1p 1 ) 3 +3(p 1 p 2 2 )(1p 2 2 )(p 1 p 2 )(1p 1 ) 2 (1p 1 ) 6 p 1 p 2 2 1p 1 2 93 Then, the concavity can be shown as follows: d 2 R (p 1 ;p 2 ) dp 2 1 < 0*p 1 >p 2 2 ; 1>p 2 2 ;p 1 >p 2 4.3 Behavioral Model In this chapter, we investigate the equilibrium of a buyer's purchasing behavior in a more realistic setting supported by the behavioral model. We have previously shown the uniqueness of the equilibrium when the valuation is drawn from uniform distribution in the existence of two symmetric buyers. Subsequently, we now question if a unique equilibrium still exists when buyers are bounded rational. Thus, we develop a model in which buyers make errors due to their computational or cognitive limitations (Su, 2008). To our knowlege, the study on the equilibrium of not-fully-rational buyers is virtually unprecedented in the literature of strategic customers. 4.3.1 Two Symmetric Buyers 4.3.1.1 Modeling Bounded Rationality We extended the model in Section 4.2.3. where N = 2 , K = 1 and V i U [0; 1] for i2f1; 2g. We add an additional assumption that buyers' decisions can be decribed by the quantal response framework McKelvey and Palfrey (1995). In the area of behavioral operations, introducing the quantal response framwork is one of the most frequently used ways to extend the domain of modeling into behavioral decision making (Su, 2008). In 94 Chapter 2, we found that the choice behavior of a subject could be explained by the quantal response framework. In our model, we do not take the risk attitude of buyers into consideration, hence only the parameter that describes the bounded rationality of buyers is utilized. We call this parameter, . Quantal response framework is based on the condition that the better option is often likely to be chosen, although not always. In our setting, the two options will be bidding in both the rst and the second period. The larger becomes, the greater the likelihood of selecting the better option. It is clear that the buyer becomes fully rational as "1. Note that our interest in this chapter is the bounded rationality of customers. Hence, we assume that <1. To derive the model, we rst introduce the utility of each option. Letu i 1 v i ;p 1 ; 1 = 1 v i p 1 denote the buyer i's expected payo from bidding in the rst period when his valuation is v i . Similarly, the expected payo of the buyer i whose valuation is v i from bidding in the second period is u i 2 v i ;p 2 ; 2 = 2 v i p 2 . We use u i 1 and u i 2 for u i 1 v i ;p 1 ; 1 and u i 2 v i ;p 2 ; 2 , respectively, for the simplicity of the equations. Let i v i ;p 1 ;p 2 ; 1 ; 2 ; indicate the probability that buyer i bids in the rst period when his valuation is v i . We also use i v i and i v i ;p 1 ;p 2 ; 1 ; 2 ; interchangeably. Note that i v i ;p 1 ;p 2 ; 1 ; 2 ; plays an important role in the anlysis, and it has a logit choice form McFadden (1976) as follows: i v i ;p 1 ;p 2 ; 1 ; 2 ; = 8 > > > > < > > > > : exp( u i 1 ) exp( u i 1 )+exp( u i 2 ) ;v i p 1 0 ;v i <p 1 (4.3) where is a parameter associated with the degree of rationality. 95 Because each buyer does not know the valuation of another buyer, each buyer has to consider the expected realization of the valuation to strategically make the purchase decision. Therefore, we introduce one more term, ( 1 ; 2 ; ) = R v (v)g (v)dv, which denotes the probability that a buyer places a bid in the rst period whereg (v) is a pdf of v. Since both buyers are identical, we omit the superscripti in the equation. By assuming that V U[0; 1], ( 1 ; 2 ; ) can be described as the following closed form equation: ( 1 ; 2 ; ) = 8 > > > > < > > > > : ln h 1+exp( ((1p 1 ) 1 (1p 2 ) 2 )) 1+exp( (p 1 p 2 ) 2 ) i ( 1 2 ) ; ( 1 2 )6= 0 1 1+exp( (p 1 p 2 ) 1 ) ; ( 1 2 ) = 0 (4.4) Then, A buyer's probability of biding in the rst period is ( 1 ; 2 ; ). A buyer's probability of biding in the second period is 1 ( 1 ; 2 ; )p 2 . Let i t denote the probability that a buyeri will obtain an item when the buyer places a bid in period t where t2f1; 2g. Let i be the ( 1 ; 2 ; ) of buyer i. We assume that buyer 2's choice probability is realized by 2 . Note that 0 2 (1p 1 ) by (4.3). Then, the probability that the buyer 1 will obtain an item based on the bidding period is as follows: Bidding in the rst period. Then the probability that the buyer 1 will obtain an item: 1 1 = 2 2 + 1 2 = 1 2 2 : (4.5) 96 Bidding in the second period. Then the probability that the buyer 1 will obtain an item: 1 2 = 1 2 +p 2 2 : (4.6) Then, the dierence between two probabilities, 1 1 and 1 2 , only depends on p 2 ; there- fore, 1 1 1 2 = 1p 2 2 . As a result, by assuming that > 0, we can eliminate the possibility that ( 1 2 ) = 0 in the equation (4.4). 4.3.1.2 Equilibrium of the buyer's bidding strategy Here, we show that there is a unique equilibrium strategy. Before the analyis, we rule out the possibility that ( 1 ; 2 ; ) is either 0 or (1p 1 ). Lemma 4.3.1. Suppose 0 < p 2 < p 1 < 1 and 0 < <1. Then, 0 < ( 1 ; 2 ; ) < 1p 1 . Proof. ( 1 2 )> 0 from the assumption. Thus, the only condition for ( 1 ; 2 ; ) to be zero is 1+exp( ((1p 1 ) 1 (1p 2 ) 2 )) 1+exp( (p 1 p 2 ) 2 ) = 1. The equality is reduced to ( 1 2 ) (1p 1 ) = 0 and this cannot be satised by our assumption. Similarly, ln h 1+exp( ((1p 1 ) 1 (1p 2 ) 2 )) 1+exp( (p 1 p 2 ) 2 ) i ( 1 2 ) = (1p 1 ) contradicts our assumption that 0<p 2 <p 1 < 1 and 0< <1. The result of Lemma 4.3.1 shows that the bounded rationality of customers prevents them from adhering to a single strategy constantly. This is supported by the ndings in Chapter 2 that customers switch their decisions most of the time. Proposition 4.3.2. SupposeN = 2,K = 1,V i U [0; 1] fori2f1; 2g, 0<p 2 <p 1 < 1, and 0 < <1. Furthermore, the probability that buyer i bids in the rst period when 97 his valuation is v i can be described by the quantal response framework as in (4.3). Then, there exists the symmetric unique equilibrium bidding strategy. Proof. Let i t denote the probability that a buyer i will obtain an item when the buyer places a bid in period t where t2f1; 2g. Let i be the i ( 1 ; 2 ; ) of buyer i. To nd the unique equilibrium, we assume that buyer 2's choice probability is realized by 2 . Consider the response of buyer 1. Then, 1 1 1 ; 1 2 ; = ln " 1+exp ( ( (1p 1 ) 1 1 (1p 2 ) 1 2 )) 1+exp ( (p 1 p 2 ) 1 2 ) # ( 1 1 1 2 ) 1 1 and 1 2 are the functions of 2 as in the equation (4.5) and (4.6) respectively; therefore, 1 is also the function of 2 as follows: 1 2 = 1 ( 1 1 1 1 ) ln (1+exp( (( 1 1 1 2 )(p 1 1 1 p 2 1 2 )))) (1+exp( (p 1 p 2 ) 1 2 )) = 1 1p 2 2 ln 1+exp 1p 2 2 p 1 1 2 2 p 2 1 2 +p 2 2 1+exp (p 1 p 2 ) 1 2 +p 2 2 = dln (1+exp(b+a 2)) (1+exp(c+a 2)) ,where a = (p 1 p 2 ) 2 , b = 12p 1 p 2 2 2 , c = (p 1 p 2 )(1+p 2 ) 2 , and d = 1 1p 2 2 . First, we show that 1 2 is a strictly increasing function of 2 . We take the rst derivative of 1 2 w.r.t. 2 : d d 2 dln (1+exp(b+a 2)) (1+exp(c+a 2)) =d d d 2 ln 1 +exp b +a 2 ln 1 +exp c +a 2 =d aexp(b+a 2) 1+exp(b+a 2) aexp(c+a 2) 1+exp(c+a 2) =ad exp(b+a 2)exp(c+a 2) (1+exp(b+a 2))(1+exp(c+a 2)) =adexp a 2 exp(b)exp(c) (1+exp(b+a 2))(1+exp(c+a 2)) > 0 98 The last inequality comes froma> 0,d> 0 andexp (b)>exp (c). (exp (b)>exp (c) because, bc = n (1p 2 )(1p 1 ) 2 o > 0 ) Therefore, d d 2 1 2 > 0 and this shows that the function is strictly increasing. Then, we check the second derivative of 1 2 : d 2 d 2 2 dln (1+exp(b+a 2)) (1+exp(c+a 2)) =d n a 2 B(1+B)a 2 B 2 (1+B) 2 a 2 C(1+C)a 2 C 2 (1+C) 2 o =a 2 d n (BC)(1BC) (1+B) 2 (1+C) 2 o where B =exp b +a 2 and C =exp c +a 2 . Since a 2 > 0,d> 0, (1 +B) 2 > 0, (1 +C) 2 > 0, and (BC)> 0, 8 > > > > < > > > > : d 2 d 2 2 ln (1+exp(b+a 2)) (1+exp(c+a 2)) 0 ; (1BC) 0 d 2 d 2 2 ln (1+exp(b+a 2)) (1+exp(c+a 2)) > 0 ; (1BC)> 0 (4.7) By denision, 1BC = 1exp b +a 2 exp c +a 2 = 1exp 2a 2 +b +c . Hence the equation (4.7) is the same as: 8 > > > > < > > > > : d 2 d 2 2 ln (1+exp(b+a 2)) (1+exp(c+a 2)) 0 ; 2a 2 +b +c 0 d 2 d 2 2 ln (1+exp(b+a 2)) (1+exp(c+a 2)) > 0 ; 2a 2 +b +c< 0 (4.8) Note that b +c can be zero, positive, or negavitive according to the values of p 1 and p 2 . Suppose b +c 0 Then, 2a 2 +b +c 0, because a> 0 and 2 0. 99 Therefore, d 2 d 2 2 ln (1+exp(b+a 2)) (1+exp(c+a 2)) 0 and 1 2 is a concave function of 2 . The concavity of the function along with 0< 1 2 < 1p 1 by Lemma 4.3.1 supports that there is only one 2 such that 1 2 = 2 . Suppose b +c< 0 If 2 < b+c 2a , 2a 2 +b +c< 0. By equation (4.8), 1 2 is concave. If 2 b+c 2a , 2a 2 +b +c 0. Thus, 1 2 is convex. Therefore, in this case, the function changes its shape from concave to convex. The other way around is not possible. In other words, once the slope increases it never decreases again. The function intersects 1 2 = 2 once. Therefore, regardless of b +c, there is only one 2 such that 1 2 = 2 . This proves the exsitence of the unique symmetric equilibrium strategy 1 = 2 = , where denotes the equilibrium bidding probability that bidder i = 1; 2 will bid in the rst period. 4.3.1.3 Revenue of the rm Now, we turn our attention to the revenue of the rm. When both buyers have the valution less than p 2 the rm's revenue is zero. This event happens with the probability of p 2 2 . If at least one of the buyers bid for the rst period, the rm's revenue is p 1 . This event happens with the probability of 1 1 2 . 100 Otherwise, the rms revenue is p 1 with probability of 1p 2 2 n 1 1 2 o = 1 2 p 2 2 . Therefore, the rm's expected revenue will be n 1 1 2 o p 1 + n 1 2 p 2 2 o p 2 Then the revenue of the rm is: R p 1 ;p 2 ; = n 1 1 2 o p 1 + n 1 2 p 2 2 o p 2 = p 1 p 1 1 2 +p 2 1 2 p 3 2 = p 1 p 3 2 (p 1 p 2 ) 1 2 The revenue of the rm is a strictly increasing concave function of . Then, we show that, depending on the price levels, the revenue of the rm can both increase and decrease, as rationality ( ) increases. We conduct a numerical analysis. Based on the prices and rationality, we nd the equilibrium and evaluate the revenue. We iterate this with increasing the level of rationality ( ) while xing the rst and second period prices. The result shows that the revenue can either be an increasing or decreasing function of rationality ( ), depending on the prices. The result is shown in Figure 4.2. (a) (p 1 ;p 2 ) = (0:2; 0:1) (b) (p 1 ;p 2 ) = (0:7; 0:1) Figure 4.2: Revenue as a Function of Rationality( ). 101 The result is intuitively appealing. Suppose the case where a fully rational customer is better o by bidding in the second period at any valuations. Due to the bounded rationality, there is still a possibility for the boundedly rational customer to bid in the rst period. Thus, as rationality increases, the likelihood of the boundedly rational customer bidding in the rst period becomes lesser. As a result, the revenue of the rm decreases. In the case where it is better for the customers to bid in the second period, the result induced by increasing rationality is opposite. Observation 4.1: The revenue of the rm can both decrease and increase as the rationality of strategic customers increases. 4.3.2 Multiple Symmetric Buyers Here, we investigate the equilibrium of the buyer's purchasing decision when there are more than two buyers. Adding customers in the problem increases the complexity of the analysis. Hence we consider the case when the in-stock probability in the rst period is set to one. This is one of the most common scenarios that are frequently adapted in the real world, since rms often oer enough capacity in the beginning. Thus, we implement the model discussed in Section 2.3. The main premise of this model dierentiating it from previous ones is that there are two types of customers: high type and low type. The high type customers always have their valuations above the rst period price, while the low type customers have their valuations between the rst and second period price. By giving two types of customers and setting the number of the high type customer less than the product available, we can remove the uncertainty in the rst period. Here, we are only interested in the behavior of the high type customers, 102 because the decisions of low type customers is trivial: the low type customers always bid in the second period. Let N 1 and N 2 denote the number of high type and low type customers, respectively. The in-stock probability in the rst period, 1 is 1, and ( 1 ; 2 ; ) = ( 1 = 1; 2 ; ) . We use ( 2 ) instead of ( 1 = 1; 2 ; ) throughout this section. Similar to (4.4), we can get the closed form equation for ( 2 ) as follows: ( 1 ; 2 ; ) = 8 > > > > < > > > > : ln h 1+exp( ((1p 1 )(1p 2 ) 2 )) 1+exp( (p 1 p 2 ) 2 ) i (1 2 )(1p 1 ) ; (1 2 )6= 0 1 1+exp( (p 1 p 2 2 )) ; (1 2 ) = 0 (4.9) Lemma 4.3.3. ( 2 ) is a continuous function of 2 . Proof. Basically, when 2 6= 1, ( 2 ) is a continuous function of 2 because it is a composition of continuous functions of 2 . Therefore, all we need to show is that it is continuous at 2 = 1. Thus, we need to show that, lim 2 !1 ' ( 2 ) = lim 2 !1 n ln (1+exp( ((1p 1 )(1p 2 ) 2 ))) (1+exp( (p 1 p 2 ) 2 )) o (1 2 )(1p 1 ) = 1 1+expf (p 1 p 2 )g . By L'H^ opital's rule, lim 2 !1 n ln (1+exp( ((1p 1 )(1p 2 ) 2 ))) (1+exp( (p 1 p 2 ) 2 )) o (1 2 ) (1p 1 ) = lim 2 !1 @ @ 2 n ln (1+exp( ((1p 1 )(1p 2 ) 2 ))) (1+exp( (p 1 p 2 ) 2 )) o @ @ 2 (1 2 ) (1p 1 ) 103 The rst derivative in the numerator is, @ @ 2 ln (1 +exp ( ((1p 1 ) (1p 2 ) 2 ))) (1 +exp ( (p 1 p 2 ) 2 )) = @ @ 2 fln (1 +exp ( ((1p 1 ) (1p 2 ) 2 )))ln (1 +exp ( (p 1 p 2 ) 2 ))g = (1p 2 )exp ( ((1p 1 ) (1p 2 ) 2 )) 1 +exp ( ((1p 1 ) (1p 2 ) 2 )) + (p 1 p 2 )exp ( (p 1 p 2 ) 2 ) 1 +exp ( (p 1 p 2 ) 2 ) ; and, the rst derivative in the denominator is, @ @ 2 (1 2 ) (1p 1 ) = (1p 1 ) Hence, the limit function becomes: lim 2 !1 (1p 2 )exp( ((1p 1 )(1p 2 ) 2 )) 1+exp( ((1p 1 )(1p 2 ) 2 )) + (p 1 p 2 )exp( (p 1 p 2 ) 2 ) 1+exp( (p 1 p 2 ) 2 ) (1p 1 ) = (1p 2 )exp( (p 1 p 2 )) 1+exp( (p 1 p 2 )) + (p 1 p 2 )exp( (p 1 p 2 )) 1+exp( (p 1 p 2 )) (1p 1 ) = (1p 2 ) 1+exp( (p 1 p 2 )) + (p 1 p 2 ) 1+exp( (p 1 p 2 )) (1p 1 ) = 1 (1p 1 ) (1p 2 ) + (p 1 p 2 ) 1 +exp ( (p 1 p 2 )) = 1 1 +exp ( (p 1 p 2 )) Therefore, lim 2 !1 ' ( 2 ) = 1 1+expf (p 1 p 2 )g and ' ( 2 ) is continuous at 2 2 [0; 1]. 104 Lemma 4.3.4. Suppose, high type customers have a common belief that the in-stock probability is ^ 2 , then the expected in-stock probability in the second period is: 2 = N 1 X i=0 Ki N 1 +N 2 i 0 B B @ N 1 i 1 C C A (^ 2 ) i 1 (^ 2 ) Ni Proof. The probability that i out of N 1 buyers buy in the rst period is, Pr 1 (i) = 0 B B @ N 1 i 1 C C A (^ 2 ) i 1 (^ 2 ) Ni fori = 0;:::;N 1 Therefore, the expected in-stock probability in the second period is, 2 = N 1 X i=0 Ki N 1 +N 2 i 0 B B @ N 1 i 1 C C A (^ 2 ) i 1 (^ 2 ) Ni Proposition 4.3.5. There exists 2 such that 2 = N 1 X i=0 Ki N 1 +N 2 i 0 B B @ N 1 i 1 C C A ( 2 ) i 1 ( 2 ) Ni : Proof. We already showed that ( 2 )is continuous funtion of 2 2 [0; 1]. Since P N 1 i=0 Ki N 1 +N 2 i 0 B B @ N 1 i 1 C C A ( 2 ) i 1 ( 2 ) Ni is the sum, product and compo- sition of continuous functions of 2 , there exists 2 such that, 105 2 = P N 1 i=0 Ki N 1 +N 2 i 0 B B @ N 1 i 1 C C A ( 2 ) i 1 ( 2 ) Ni by xed-point theorem. 4.4 Experimental Study So far, we have analyzed the existence of equilibrium in the setting of posted pricing scheme. When there are two symmetric customers whose valuation is drawn from uniform distribution, there exists a unique symmetric equilibrium in both rational and behavioral models. However, when there are multiple boundly rational customers, the possibility of multiple equilibrium is not yet ruled out. Now, we extend our domain of study by conducting an experimental study to shed light on the case of multiple boundly rational customers. First, we revisit the ndings of the experiments of Chapter 2. In SIM experiments in Chapter 2, subjects did not interact with other subjects who employed quasi-rational models, but with subjects that employed RUMM. If subjects interact with each other the in-stock probability in the second period could change, and, as a result, the buying decisions could also change. Ideally speaking, we may have to conduct experiments in a setting where subjects actively interact with one another to understand the equilibrium behavior and the resulting in-stock probability. However, this design of having interacting subjects can increase the complexity of the dynamics and diculty of the analysis. Therefore, we evaluate the possibility of forming an unique equilibrium by studying the experiments in the partial equilibrium model. SIM experiments give us the probability that subjects will purchase at a given val- uation. By employing this data, we computed the in-stock probability. In other words, 106 we computed the in-stock probability if all buyers in the market employed the buying behavior of the experimental subjects. The computed in-stock probabilities are compared with actual in-stock probabilities in the following Table 4.1. SIM1 SIM2 SIM3 In-stock probability ( 2 ) 0.49 0.83 0.16 Re-calculated in-stock probability (~ 2 ) 0.46 0.80 0.16 Table 4.1: Comparison of resulted and actual in-stock probabilities in SIM experiment in Chapter 2 In SIM3, in-stock probability based on the actual decisions of the subjects is identical to that which was simulated by the computer. Therefore, it is possible that the observed buying behavior is the same as the equilibrium behavior that would have been observed if subjects interacted with each other. The recalculated in-stock probabilities in SIM1 and SIM2, however, are dierent from the ones we employed in the experiment. To determine the equilibrium in-stock probabil- ity for these parameter settings, we ran an additional set of treatments by only changing the in-stock probability to 46% and 80%, respectively. We call these new experiments SIM1-E and SIM2-E, respectively. The parameters and results of the new experiments are summarized in Table 4.2. Parameters SIM1-E SIM2-E Number of High Type Customers 10 10 Number of Low Type Customers 26 14 Number of units for sale 10 10 First period price 120 120 Second period price 80 80 In-stock probability in the second period 0.46 0.80 Number of trials 60 60 Re-Calculated In-stock Probability 0.4640 0.7983 Table 4.2: Parameters and results for experiment set SIM1-E and SIM2-E 107 Surprisingly, in these experiments the in-stock probabilities calculated using the ob- served buying behaviors are 46.40% and 79.83%. This result suggests the existence of the equilibrium behavior, when subjects interact with each other. Observation 4.2: There may exist equilibrium in-stock probabilities when subjects employ bounded rational purchasing strategies. 4.5 Conclusions In this chapter, we study the equilibrium behavior of strategic customers in the setting of two-period posted pricing scheme. The unique equilibrium behavior is a key to meaning- fully predicting the behavior of strategic customers. Hence, we make various assumptions about the decision making process of the customers and investigate the existence and the uniqueness of their equilibrium bidding behaviors. We observe the equilibrium behaviors by altering the level of rationality, the number of customers, and the valuation distribu- tion. First, we check the setting of two identical fully rational customers competing for one unit of item. In the simple case of deterministic distribution, the problem can be reduced to the well-known \matching pennies" game. Therefore, buyers can form two equilibria and employ mixed strategies. The results hold when the customers are non-identical. We extend the scope of the distribution to the continuous distribution and show that there can be multiple equilibria. In the case of normal distribution, the buyers' strategies may not form a unique equilibrium, either. The results in the case of fully rational customers show that there are many cases in which a unique equilibrium is not guaranteed. 108 Then, we show that when the valuation distribution is uniform, customers employ a unique symmetric bidding strategy. We move on to monitor the case where customers are bounded rational. We assume that the bounded rationality of customers can be described in the quantal response model. When there are two identical bounded rational customers, their strategies form a unique symmetric equilibrium. We extend our analysis to the case of multiple customers. We nd that in this case they can form an equilibrium. Lastly, we design an experiment to observe the possibility of the existence of equilib- rium. The result suggests the existence of equilibrium behavior when subjects interact with each other. 109 Bibliography Hirotugu Akaike. A new look at the statistical model identication. Automatic Control, IEEE Transactions on, 19(6):716{723, 1974. Gad Allon, Achal Bassamboo, and Itai Gurvich. "we will be right with you": Managing customer expectations with vague promises and cheap talk. Operations research, 59(6): 1382{1394, 2011. Wilfred Amaldoss, Teck-Hua Ho, Aradhna Krishna, Kay-Yut Chen, Preyas Desai, Ganesh Iyer, Sanjay Jain, Noah Lim, John Morgan, Ryan Oprea, et al. Experiments on strategic choices and markets. Marketing Letters, 19(3-4):417{429, 2008. CK Anderson and John G Wilson. Wait or buy? the strategic consumer: Pricing and prot implications. Journal of the Operational Research Society, 54(3):299{306, 2003. Gary H Anthes. The price had better be right. Computerworld, 32(51):65{66, 1998. Nikolay Archak, Anindya Ghose, and Panagiotis G Ipeirotis. Deriving the pricing power of product features by mining consumer reviews. Management Science, 57(8):1485{1509, 2011. 110 A Asvanunt and S Kachani. Optimal purchasing policy for strategic customers under dif- ferent dynamic pricing models. In Presentation at the 6th Annual INFORMS Revenue Management and Pricing Section Conference. Columbia University, New York, 2006. Yossi Aviv and Amit Pazgal. Optimal pricing of seasonal products in the presence of forward-looking consumers. Manufacturing & Service Operations Management, 10(3): 339{359, 2008. J Neil Bearden, Ryan O Murphy, and Amnon Rapoport. Decision biases in revenue management: Some behavioral evidence. Manufacturing & Service Operations Man- agement, 10(4):625{636, 2008. Peter P Belobaba and Lawrence R Weatherford. Comparing decision rules that incorpo- rate customer diversion in perishable asset revenue management situations. Decision Sciences, 27(2):343{363, 1996. Elliot Bendoly. Real-time feedback and booking behavior in the hospitality industry: Moderating the balance between imperfect judgment and imperfect prescription. Jour- nal of Operations Management, 2012. David Besanko and Wayne L Winston. Optimal price skimming by a monopolist facing rational consumers. Management Science, 36(5):555{567, 1990. Gary E Bolton and Elena Katok. Learning by doing in the newsvendor problem: A lab- oratory investigation of the role of experience and feedback. Manufacturing & Service Operations Management, 10(3):519{538, 2008. 111 Robert G Brown and Richard F Meyer. The fundamental theorem of exponential smooth- ing. Operations Research, 9(5):673{685, 1961. Michael Buhrmester, Tracy Kwang, and Samuel D Gosling. Amazon's mechanical turk a new source of inexpensive, yet high-quality, data? Perspectives on Psychological Science, 6(1):3{5, 2011. G erard P Cachon and Robert Swinney. Purchasing, pricing, and quick response in the presence of strategic consumers. Management Science, 55(3):497{511, 2009. Colin Camerer and Teck-Hua Ho. Experience-weighted attraction learning in normal form games. Econometrica, 67(4):827{874, 1999. Colin F Camerer. Behavioural studies of strategic thinking in games. Trends in Cognitive Sciences, 7(5):225{231, 2003. Yefen Chen, Xuanming Su, and Xiaobo Zhao. Modeling bounded rationality in capacity allocation games with the quantal response equilibrium. Management Science, 58(10): 1952{1962, 2012. JOS E R Correa, RICARDO Montoya, and CHARLES Thraves. Contingent preannounced pricing policies with strategic consumers. Technical report, Working Paper, University of Chile, 2011a. JOS E R Correa, RICARDO Montoya, and CHARLES Thraves. Contingent preannounced pricing policies with strategic consumers. Technical report, Working Paper, University of Chile, 2011b. 112 Tony Haitao Cui, Jagmohan S Raju, and Z John Zhang. Fairness and channel coordina- tion. Management Science, 53(8):1303{1314, 2007. Sriram Dasu and Chunyang Tong. Dynamic pricing when consumers are strategic: Analy- sis of posted and contingent pricing schemes. European Journal of Operational Research, 204(3):662{671, 2010. Enrico Diecidue, Nils Rudi, and Wenjie Tang. Dynamic purchase decisions under regret: Price and availability. Decision Analysis, 9(1):22{30, 2012. Wedad Elmaghraby and Pnar Keskinocak. Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions. Manage- ment Science, 49(10):1287{1309, 2003. Wedad Elmaghraby, Altan G ulc u, and Pnar Keskinocak. Designing optimal prean- nounced markdowns in the presence of rational customers with multiunit demands. Manufacturing & Service Operations Management, 10(1):126{148, 2008. Richard Engelbrecht-Wiggans and Elena Katok. Regret and feedback information in rst-price sealed-bid auctions. Management Science, 54(4):808{819, 2008. Richard Engelbrecht-Wiggans, Ernan Haruvy, and Elena Katok. A comparison of buyer- determined and price-based multiattribute mechanisms. Marketing Science, 26(5):629{ 641, 2007. Guillermo Gallego and Garrett Van Ryzin. Optimal dynamic pricing of inventories with stochastic demand over nite horizons. Management science, 40(8):999{1020, 1994. 113 Pengfei Guo, John J Liu, and Yulan Wang. Intertemporal service pricing with strategic customers. Operations Research Letters, 37(6):420{424, 2009. Werner G uth, Peter Ockenfels, and Klaus Ritzberger. On durable goods monopolies an experimental study of intrapersonal price competition and price discrimination over time. Journal of Economic Psychology, 16(2):247{274, 1995. Werner G uth, Sabine Kr oger, and Hans-Theo Normann. Durable-goods monopoly with privately known impatience: A theoretical and experimental study. Economic Inquiry, 42(3):413{424, 2004. David W Harless and Colin F Camerer. The predictive utility of generalized expected utility theories. Econometrica: Journal of the Econometric Society, pages 1251{1289, 1994. Teck-Hua Ho and Juanjuan Zhang. Designing pricing contracts for boundedly rational customers: Does the framing of the xed fee matter? Management Science, 54(4): 686{700, 2008. Teck-Hua Ho, Noah Lim, and Tony Haitao Cui. Reference dependence in multilocation newsvendor models: A structural analysis. Management Science, 56(11):1891{1910, 2010. Charles A Holt and Susan K Laury. Risk aversion and incentive eects. American economic review, 92(5):1644{1655, 2002. Tingliang Huang, Gad Allon, and Achal Bassamboo. Bounded rationality in service systems. Manufacturing & Service Operations Management, 15(2):263{279, 2013. 114 Barbara E Kahn and Jonathan Baron. An exploratory study of choice rules favored for high-stakes decisions. Journal of Consumer Psychology, 4(4):305{328, 1995. Daniel Kahneman and Amos Tversky. Subjective probability: A judgment of represen- tativeness. Cognitive psychology, 3(3):430{454, 1972. Daniel Kahneman and Amos Tversky. Prospect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society, pages 263{291, 1979. Gurumurthy Kalyanaram and John DC Little. An empirical analysis of latitude of price acceptance in consumer package goods. Journal of Consumer Research, pages 408{418, 1994. Elena Katok. Laboratory experiments in operations management. TutORials in Opera- tions Research. INFORMS, Hanover, MD, pages 15{35, 2011. Christine Kaufmann, Martin Weber, and Emily Haisley. The role of experience sampling and graphical displays on one's investment risk appetite. Management Science, 59(2): 323{340, 2013. Mirko Kremer and Laurens G Debo. Herding in a queue: A laboratory experiment. Chicago Booth Research Paper, (12-28), 2012. Andrew R Liddle. Information criteria for astrophysical model selection. Monthly Notices of the Royal Astronomical Society: Letters, 377(1):L74{L78, 2007. Noah Lim and Teck-Hua Ho. Designing price contracts for boundedly rational customers: Does the number of blocks matter? Marketing Science, 26(3):312{326, 2007. 115 Qian Liu and Stephen Shum. Pricing and capacity rationing with customer disappoint- ment aversion. Production and Operations Management, 2013. Qian Liu and Garrett J van Ryzin. Strategic capacity rationing to induce early purchases. Management Science, 54(6):1115{1131, 2008. Christoph H Loch. Behavioral operations management. Now Pub, 2007. Christoph H Loch and Yaozhong Wu. Social preferences and supply chain performance: An experimental study. Management Science, 54(11):1835{1849, 2008. Daniel L McFadden. Quantal choice analaysis: A survey. In Annals of Economic and Social Measurement, Volume 5, number 4, pages 363{390. NBER, 1976. Richard D McKelvey and Thomas R Palfrey. Quantal response equilibria for normal form games. Games and economic behavior, 10(1):6{38, 1995. Gary McWilliams. Analyzing customers, best buy decides not all are welcome. The Wall Street Journal Online, 2004. Javad Nasiry and Ioana Popescu. Advance selling when consumers regret. Management Science, 58(6):1160{1177, 2012. Nikolay Osadchiy and Elliot Bendoly. Are consumers really strategic? implications from an experimental study. Technical report, Working paper, Emory University, Atlanta, GA, 2010. Nikolay Osadchiy and Gustavo Vulcano. Selling with binding reservations in the presence of strategic consumers. Management Science, 56(12):2173{2190, 2010. 116 Gabriele Paolacci, Jesse Chandler, and Panagiotis Ipeirotis. Running experiments on amazon mechanical turk. Judgment and Decision Making, 5(5):411{419, 2010. Ioana Popescu and Yaozhong Wu. Dynamic pricing strategies with reference eects. Operations Research, 55(3):413{429, 2007. Klaus P otzelberger and Leopold S ogner. Stochastic equilibrium: learning by exponential smoothing. Journal of Economic Dynamics and Control, 27(10):1743{1770, 2003. Alvin E Roth and John Henry Kagel. The handbook of experimental economics, volume 1. Princeton university press Princeton, 1995. Gideon Schwarz. Estimating the dimension of a model. The annals of statistics, 6(2): 461{464, 1978. Maurice E Schweitzer and G erard P Cachon. Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence. Management Science, 46 (3):404{420, 2000. Zuo-Jun Max Shen and Xuanming Su. Customer behavior modeling in revenue man- agement and auctions: A review and new research opportunities. Production and operations management, 16(6):713{728, 2007. Herbert A Simon. A behavioral model of rational choice. The quarterly journal of eco- nomics, 69(1):99{118, 1955. Barry C Smith, John F Leimkuhler, and Ross M Darrow. Yield management at american airlines. Interfaces, 22(1):8{31, 1992. 117 Xuanming Su. Intertemporal pricing with strategic customer behavior. Management Science, 53(5):726{741, 2007. Xuanming Su. Bounded rationality in newsvendor models. Manufacturing & Service Operations Management, 10(4):566{589, 2008. Kalyan T Talluri and Garrett J Van Ryzin. The theory and practice of revenue manage- ment, volume 68. Springer, 2005. Olivier Toubia, Eric Johnson, Theodoros Evgeniou, and Philippe Delqui e. Dynamic experiments for estimating preferences: An adaptive method of eliciting time and risk parameters. Management Science, 59(3):613{640, 2013. Amos Tversky and Daniel Kahneman. Availability: A heuristic for judging frequency and probability. Cognitive psychology, 5(2):207{232, 1973. Russell S Winer. A reference price model of brand choice for frequently purchased prod- ucts. Journal of consumer research, pages 250{256, 1986. Diana Yan Wu and Kay-Yut Chen. Supply chain contract design: Impact of bounded ra- tionality and individual heterogeneity. Production and Operations Management, 2013. Rui Yin, Yossi Aviv, Amit Pazgal, and Christopher S Tang. Optimal markdown pric- ing: Implications of inventory display formats in the presence of strategic customers. Management Science, 55(8):1391{1408, 2009. Yong-Pin Zhou, Ming Fan, and Minho Cho. On the threshold purchasing behavior of customers facing dynamically priced perishable products. Technical report, Working paper, 2005. 118 Appendix A Tables in Chapter 2 Table A.1: Out of Sample Test: Goodness of Fit Test of SIM-I Treatments Treatment Sampling No. Rationality Risk Aversion Log- Chi-square Test Root Mean Parameter( ) Parameter (r ) Likelihood p value Square Deviation SIM1-I 1 0.1494 0.0431 -211.16 0.0012 0.1483 2 0.0909 0.0219 -192.31 0.1623 0.1173 3 0.1246 0.0249 -208.16 0.9129 0.0608 4 0.1191 0.0267 -202.31 0.8852 0.0729 5 0.1143 0.0231 -200.96 0.8543 0.0805 SIM2-I 1 0.1093 0.0302 -230.37 0.0002 0.1187 2 0.1468 0.0327 -237.04 0.2931 0.0869 3 0.1588 0.0425 -236.99 0 0.1701 4 0.1141 0.0253 -237.04 0.1813 0.09 5 0.1158 0.0313 -230.86 0.0011 0.1207 SIM3-I 1 0.1206 0.0194 -209.44 0.6988 0.1037 2 0.1067 0.0229 -177.43 0.9967 0.065 3 0.1187 0.0297 -175.07 0.9931 0.0639 4 0.1263 0.053 -143.05 0.114 0.13 5 0.1023 0.0296 -156.61 0.9723 0.067 119 Table A.2: Out of Sample Test: Goodness of Fit Test for SIM Treatments Treatment Sampling No. Rationality Risk Aversion Log- Chi-square Test Root Mean Parameter( ) Parameter (r ) Likelihood p value Square Deviation SIM1 1 0.1029 0.011 -206.15 0.9851 0.0622 2 0.1113 0.0116 -212.36 0.8963 0.0765 3 0.1092 0.0183 -195.83 0.9946 0.0559 4 0.0913 0.0116 -195.82 0.5403 0.0872 5 0.0852 0.0133 -193.34 0.1647 0.1046 SIM2 1 0.1226 0.0294 -212.16 0.0349 0.1096 2 0.1784 0.0374 -215.05 0.5939 0.0842 3 0.1107 0.0228 -223.44 0 0.1612 4 0.3161 0.0437 -252.81 0 0.1682 5 0.2791 0.0368 -249.79 0 0.1346 SIM3 1 0.0594 0 -152.36 0.5875 0.0963 2 0.0652 0 -160.34 0.7238 0.0905 3 0.0697 0.0003 -163.81 0.9823 0.0651 4 0.0845 0.0192 -151.13 0.3454 0.1058 5 0.0783 0.0055 -170.98 0.8935 0.0739 Table A.3: Model Comparison for SIM2-I: QRM, RUMM and LOT Sampling Model RMSD Chi-square Log- Number of Parameters AIC BIC Rank No. Test p value Likelihood 1 QRM 0.1187 0.0002 -230.37 2 464.74 463.15 1 RUMM 0.4117 0 -1300.03 0 2600.05 2600.05 3 LOT 0.2564 0 -289.26 0 578.52 578.52 2 2 QRM 0.0869 0.2931 -237.04 2 478.09 476.5 1 RUMM 0.3991 0 -1304.02 0 2608.05 2608.05 3 LOT 0.2419 0 -307.15 0 614.3 614.3 2 3 QRM 0.1701 0 -236.99 2 477.97 476.38 1 RUMM 0.3804 0 -1164.03 0 2328.05 2328.05 3 LOT 0.2184 0 -261.8 0 523.61 523.61 2 4 QRM 0.09 0.1813 -237.04 2 478.07 476.48 1 RUMM 0.453 0 -1492.02 0 2984.05 2984.05 3 LOT 0.2906 0 -328.13 0 656.26 656.26 2 5 QRM 0.1207 0.0011 -230.86 2 465.72 464.13 1 RUMM 0.4179 0 -1320.02 0 2640.05 2640.05 3 LOT 0.2586 0 -292.02 0 584.05 584.05 2 120 Table A.4: Model Comparison for SIM2: QRM, RUMM and LOT Sampling Model RMSD Chi-square Log- Number of Parameters AIC BIC Rank No. Test p value Likelihood 1 QRM 0.109574813 0.034904 -212.16 2 428.33 426.73 1 RUMM 0.508978457 0 -1492.02 0 2984.04 2984.04 3 LOT 0.340481916 0 -317.09 0 634.18 634.18 2 2 QRM 0.084170186 0.593912 -215.05 2 434.11 432.52 1 RUMM 0.451081046 0 -1352.02 0 2704.04 2704.04 3 LOT 0.287227883 0 -298.18 0 596.37 596.37 2 3 QRM 0.161204824 0 -223.44 2 450.89 449.3 1 RUMM 0.547405102 0 -1672.02 0 3344.04 3344.04 3 LOT 0.381176692 0 -348.86 0 697.71 697.71 2 4 QRM 0.168215655 0 -252.81 2 509.62 508.03 1 RUMM 0.406035458 0 -1296.02 0 2592.05 2592.05 3 LOT 0.24782902 0 -304.6 0 609.21 609.21 2 5 QRM 0.13463468 0 -249.79 2 503.58 501.99 1 RUMM 0.440086431 0 -1404.02 0 2808.04 2808.04 3 LOT 0.2876622 0 -331.31 0 662.63 662.63 2 121 Appendix B SIM Instructions Welcome to an experiment in decision making. You will be participating in a simulation. Your earnings will consist of participation compensation and an additional compensation that is based on your performance in this simulation. The performance based compen- sation can double your earnings. Please read the instruction carefully. Your level of understanding can signicantly aect your performance. After you read the instruction, you will be asked to take a test. If your score in this test is below the qualication level, you will be disqualied and you will only earn a basic compensation of $0.50. THE SIMULATION GAME You will play the role of a retailer who buys one unit of a product and then resells it to a customer. You can purchase the product in one of two-periods. Price of the product in the rst period is $120. In the second period, there is a discount, and the price is $80.In the rst period, the product is always available. In the second period, however, you may not get the product because it may be sold out. If you can purchase the product, you will resell it to a customer at a given resale price. 122 And your prot = resale price - your purchase price If you chose to wait and the product is sold out, then your prot is zero. is guaranteed that you can resell it, so you are only supposed to focus on when to buy the product. MULTIPLE ROUNDS The game consists of 50 to 70 rounds. In each round you will be given information that includes your resale price (From $125 to $200), the rst period price ($120), and the second period price ($80). In each round, you have to decide if you want to (a) buy in the rst period or (b) wait and purchase in the second period. After you make your decision, you will learn whether or not the product is available in the second period. You will also be informed of your prot. Then the simulation will advance to the next round. Your objective is to maximize your prots. The additional performance based compensation you earn depends on your prot. OTHER RETAILERS In addition to you there are 35 other retailers. In the rst period, only 10 retailers including you are present. In the second period, 26 additional retailers join the market. In each round, the retailers who are present in the rst period have to decide whether to buy in the rst period or wait until the second period. The resale prices, however, are not the same for all of these retailers. For each retailer, a computer program randomly generates the resale price. The resale prices lie between $125 and $200 in $5 increments. As a result, dierent retailers will have dierent resale 123 prices. The 26 additional retailers who join the market in the second period have resale prices between $81 and $120; therefore, they only want to buy the product in the second period. The purchase prices (rst period price and second period price), however, will be the same for all the retailers. Total number of retailers is 36 but the total number of units available for sale is 20. This is why you may not get the product in the second period. Because there are only 10 retailers in the rst period, if you decide to buy the product in the rst period, you are guaranteed to get the product. In the second period all retailers who didn't buy in the rst period join the 26 other retailers. All these retailers try to buy the product in the second period. Because the number of retailers exceeds the number of units for sale in the second period, the computer randomly selects the retailers who will get the product. Total number of units for sale (20), total number of retailers present in the rst period (10), additional number of retailers joining the market in the second period (26), your resale price, rst period price ($120) and second period price ($80) will be displayed, and you don't have to remember them. You will not know the resale price of the other retailers. Each retailer knows only his or her own resale price. Remember you will always get the product if you decide to buy in the rst period. The computer also makes the purchasing decisions for all the other retailers. For the 9 other retailers who are present in the rst period, the computer makes a decision based on a rule that depends only on the resale price observed by that retailer and the other information displayed. Using this rule, the computer decides for each of the 9 retailers whether they should buy in the rst period or wait till the second period. 124 The purchase decisions of these 9 other retailers will not depend on your decision. The 26 additional retailers who join in the second period always want to buy the product in the second period. As stated earlier the game consists of 50 to 70 trials. In each trial your resale price and that of all the other retailers will change. The prices at which you can purchase, the number of retailers, and the number of units for sale will not change from one trial to the next. Example 1: 1st Pe- riod Price 2nd Period Price Your resale price Number of units for sale Number of retailers present in the rst period Number of addi- tional retailers join- ing in the second period $120 $80 $145 20 10 26 If you buy in the rst period, then your prot is: $145 - $120 = $25. If you wait and the product is available in the second period, then your prot is: $145 -$80 = $65. If you wait and the product is not available in the second period, then your prot is: $0 If 4 of the 10 retailers decide to buy in the rst period, then the number of units available in the second period: 20 - 4 = 16. And the number of retailers in the second period will be: 6 (from the rst period) + 26 = 31. Only 16 out of these 31 retailers will get the product in the second period. The computer randomly picks the 16 out of the remaining 31 and assigns them the product. 125 PERFORMANCE COMPENSATION Performance compensation can be $2 maximum. We will randomly pick 10 of your decisions. So out of 50 ~ 70 decisions, only 10 decisions will aect your performance. Because you don't know which decision will be selected, you should make each decision carefully. The prot you make in these 10 decisions will be summed up and the additional payment you receive will be proportional to this sum. 126 Example 2: 1st Pe- riod Price 2nd Period Price Your resale price Number of units for sale Number of retailers present in the rst period Number of addi- tional retailers join- ing in the second period $120 $80 $125 20 10 26 If you buy in the rst period your prot will be: Resale price = $125, Purchase price = $120, Prot = $125 - $120 = $5. If you wait for the second period: And if the product is out of stock, Prot = $0 And if the product is in stock: Resale price $125, Purchase price = $80 and Prot = $125 - $80 = $45. If 9 retailers decide to buy in the rst period, Number of units left for the second period = 20 - 9 =11 Number of retailers who decide to wait for the second period = 10 - 9 = 1 Number of additional retailers joining in the second period = 26 Total Number of retailers in the second period = 1 + 26 = 27 Only 11 out of these 27 retailers will get the product in the second period. 127 Appendix C LOT Instructions In this experiment, you are given an opportunity to choose between two dierent options and receive additional payments based on your decisions. One of the options (Option A) oers you a risky choice with a higher amount of points while the other option (Option B) oers a lower amount of points but there is no uncertainty. Two options will be displayed; one on the right and one on the left. Example 1 Option A: 10% chance at winning 200 points OR Option B: 10 points. In example 1, you can either choose option A with a 10% chance at 200 points, OR you can choose Option B that guarantees 10 points. If you select Option A, then a computer will generate a random number and determine whether or not you receive the 200 points. The chance of winning will be 10%. If you choose the guaranteed option, you will be given the 10 points. There are 5 decision sets. Each decision set has 20 decision questions. Please answer all of the decision questions in the 5 decision sets. A computer simulation will randomly select one of the four decision sets and then randomly choose one of the decision questions from the chosen decision set. If you selected 128 Option A then the computer will perform another simulation to determine whether or not you get the corresponding points. The likelihood of you winning the points will depend on the probability of success for Option A. On the other hand if you chose Option B, you will be given the corresponding points. The points achieved will be translated to monetary value between $0 to $1. Partici- pation fee for this experiment is $0.50 and additional payments based on the outcome in the decision you made can result in your total compensation to go up to $1. 129 Appendix D Sample Screenshots of SIM experiments Figure D.1: Decision Making 130 Figure D.2: Result - In Stock 131 Figure D.3: Result - Out of Stock 132 Appendix E Sample LOT Questions Option A Option B Your Choice A or B Decision 1 49% chance at winning $45 $5 Decision 2 49% chance at winning $50 $10 Decision 3 49% chance at winning $55 $15 Decision 4 49% chance at winning $60 $20 Decision 5 49% chance at winning $65 $25 Decision 6 49% chance at winning $70 $30 Decision 7 49% chance at winning $75 $35 Decision 8 49% chance at winning $80 $40 Decision 9 49% chance at winning $85 $45 Decision 10 49% chance at winning $90 $50 Decision 11 49% chance at winning $95 $55 Decision 12 49% chance at winning $100 $60 Decision 13 49% chance at winning $105 $65 Decision 14 49% chance at winning $110 $70 Decision 15 49% chance at winning $115 $75 Decision 16 49% chance at winning $120 $80 133
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Kim, Seungbeom
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Essays on bounded rationality and revenue management
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