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Essays on information, incentives and operational strategies
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Essays on information, incentives and operational strategies
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ESSAYS ON INFORMATION, INCENTIVES AND OPERATIONAL STRATEGIES by GUANGWEN KONG DISSERTATION Presented to the Faculty of The University of Southern California in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN BUSINESS ADMINISTRATION THE UNIVERSITY OF SOUTHERN CALIFORNIA August 2013 This thesis is dedicated to my family for their constant support and unconditional love. i ACKNOWLEDGMENTS Lookingback tothepastfive years ofmy lifeatUSC, Iamindebted tomany peoplefortheir support during my doctoral studies. It has been a real pleasure being here and working on various research problems. My today’s achievement would not have been possible without them. My first thanks belong to my advisor and dissertation chair Prof. Raj (Sampath) Ra- jagopalan. Iappreciate allyour years ofguidance in my academic research. Youhave guided me to the wonderland of academic research. You have inspired me to continue learning with an open and positive mind. You have always been there when I needed your advice, on research or otherwise. You have been a wonderful mentor and my role-model. Your unwavering support meant so much to me and you have generously paved the way for my development as a researcher. Iamdeeplygratefultomyadvisoranddissertationco-chair,Prof. HaoZhang. Ithasbeenan honor to be your first Ph.D. student. Thank you so much for your time, expertise, patience and support over the years of my doctoral study. I do appreciate the time and effort you have spent providing valuable suggestions, proofreading the manuscripts. You have taught me, both consciously and unconsciously, how perfect and rigorous analysis is done. It would not have been possible to write this doctoral thesis without the help and support from you. IamalsogreatlyindebtedtoProf. RamandeepRandhawafortheguidanceonthethesis. You have provided insightful discussions about the research. Your sincere interests in research, your vision and acumen have been a great inspiration to me. I would also like to thank committee members: Prof. Gad Allon, Prof. Sosic Greys and Prof. Isabelle Brocas for their extremely helpful comments and suggestions of this thesis. I am also grateful to Prof. Leon ii Chu, ProfessorSriramDasu,Prof. AmyWardandanumber ofotherfacultymembers atthe Marshall School of Business who have generously shared their time, experience knowledge with me throughout the past five years. I would like to thank Prof Nicholas Hall, who provide precious advice and encourage me to pursuit my degree in the area of operations management in the beginning. I would like to thank all my friends and my colleagues, with whom I enjoy a great life in the past five years. Finally and most importantly, I wish to thank my family for their unconditional love and enduring support throughout my life. June 2013 iii ABSTRACT The thesis consists of three projects on information asymmetry and strategic interactions within operations management. The first project explores the potential of revenue sharing contracts to facilitate information sharing in a supply chain and mitigate the negative effects of information leakage. We consider a supplier who offers a revenue sharing contract to two competing retailers, one of whom has private information about uncertain market potential and orders first. This order informationmaybeleakedtotheuninformedretailerbythesuppliertorealizehigherprofits. We show that the incentives of the supplier and retailers are better aligned under a revenue- sharingcontractunlikeinawholesalepricecontract,reducingthesupplier’sincentivetoleak. This is true for a wide range of wholesale prices and revenue share percentages and is more likely when the revenue share percentage is higher and when variation in demand is greater. Preventing informationleakagemayresult inhigherprofitsnotonly fortheinformed retailer and supplier, but surprisingly even the uninformed retailer. The results are robust when the model is generalized along various dimensions. Service outsourcing has become more prevalent online, but significant information asym- metry arises between a service provider and a customer. Both sides have information that benefits the other one but one may distort that information to take advantage of the other. The second project considers three contracts: fixed fee, time based payment (hourly rate times service time) and two-part tariff contract and investigate how a service provider’s iv optimal effort level and profit are impacted by the nature of the contracts. It identifies con- ditions under which (i) the firm will provide the same or different quality levels to different customer types and (ii) the pricing contracts produce the same ordifferent outcomes. It also explores the impact of the SP’s hidden effort: how it affects the SP’s ability to discriminate among buyers and whether it reduces or increases his profits. It turns out hidden effort does not change the SP’s profits under fixed fee and two part tariff. Hidden effort only has an impact under time based payment. Hidden effort reduces the profit of a high cost service provider because he cannot charge a high price rate. On the other hand, hidden effort im- proves the profit of a low cost service provider because it gains the flexibility to strategically misreport its actual effort. In addition, it finds that time-based payment can be as effective as two part tariff in certain parameter ranges. A service firm always faces a trade-off between exploring a larger market size and reducing the congestion in the business process. Especially nowadays a firm could easily use Social advertising tools such as Groupon to introduce a lot of new customers to their business in a short time. But When you introduce new customers by a Groupon promotion, you may attractdealhunterswhobringcongestiontoyourbusiness. Sonotonlydoyoulosemoneyon these deal hunters who may not return but you also increase congestion thatmay drive away your regular customers. The third project formally characterize these effects that promotion strategies have on a firm’s profit and come up with recommendations for effective promotion strategies. v TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi CHAPTER 1 REVENUE SHARING AND INFORMATION LEAKAGE IN A SUP- PLY CHAIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Review of Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Model Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Benchmark Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 Benchmark: Supplier Never Leaks . . . . . . . . . . . . . . . . . . . 11 1.4.2 Benchmark: Supplier Always Leaks . . . . . . . . . . . . . . . . . . . 12 1.4.3 Benchmark: Supplier’s First-Best Scenario . . . . . . . . . . . . . . . 15 1.5 Nonleakage Equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Existence of the Nonleakage Equilibrium . . . . . . . . . . . . . . . . 16 1.5.2 The Nonleakage Region . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Optimal Wholesale Price for the Supplier . . . . . . . . . . . . . . . . . . . . 26 1.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.7.1 Free Disposal by the Incumbent . . . . . . . . . . . . . . . . . . . . . 31 1.7.2 Active Entrant Who Can Reject Information . . . . . . . . . . . . . . 32 1.7.3 Simultaneous Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.7.4 Entrant Ordering First . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 CHAPTER 2 SERVING HETEROGENEOUS BUYERS THROUGH PRICE AND QUALITY DIFFERENTIATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 vi 2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 The Basic Model with Public Effort (Deterministic Quality) . . . . . . . . . 43 2.3.1 Buyer’s Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.2 Service Provider’s Utility . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.3 Sequence of the Game . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4 Optimal Contracts in the Public Effort (Deterministic Quality) Case . . . . . 46 2.4.1 Fixed Fee Contract-Deterministic Quality . . . . . . . . . . . . . . . 46 2.4.2 Time Based Contract-Deterministic Quality . . . . . . . . . . . . . . 50 2.4.3 Two part tariff-deterministic quality . . . . . . . . . . . . . . . . . . 59 2.5 Optimal Contracts in the Hidden Effort (Stochastic Quality) Case . . . . . . 62 2.5.1 Effort Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.5.2 Time-Based Contract: Stochastic Quality . . . . . . . . . . . . . . . . 66 2.5.3 Two Part Tariff-Stochastic Quality . . . . . . . . . . . . . . . . . . . 75 2.5.4 When Time Based Contract is Equivalent to Two Part Tariff . . . . . 76 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.6.1 Comparison Across Contracts . . . . . . . . . . . . . . . . . . . . . . 78 2.6.2 Comparison Across Deterministic Case and Stochastic Quality Cases. 79 CHAPTER 3 OPTIMAL PROMOTIONSTRATEGY FORASERVICE FIRMWITH DELAY SENSITIVE CUSTOMERS . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3.1 The Customer’s Decision . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.3.2 The Service Firm’s Revenue . . . . . . . . . . . . . . . . . . . . . . . 88 3.3.3 Properties of Equilibrium Waiting Time . . . . . . . . . . . . . . . . 89 3.3.4 When are Promotions Profitable? . . . . . . . . . . . . . . . . . . . . 90 3.4 Approximate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.4.1 A Large Market Approximation . . . . . . . . . . . . . . . . . . . . . 92 3.4.2 Refined Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 vii APPENDIX A PROOFS OF CHAPTER 1 . . . . . . . . . . . . . . . . . . . . . . . 101 A.1 Proofs of Propositions and Theorems . . . . . . . . . . . . . . . . . . . . . . 101 A.2 Pooling Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.3 Impacts of r and p on the Nonleakage Region . . . . . . . . . . . . . . . . . 122 A.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 APPENDIX B PROOFS OF CHAPTER 2 . . . . . . . . . . . . . . . . . . . . . . . 140 APPENDIX C PROOFS OF CHAPTER 3 . . . . . . . . . . . . . . . . . . . . . . . 166 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 viii LIST OF FIGURES 1.1 Supplier’s relative profit gain from the nonleakage equilibrium, in either demand state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 The nonleakage region for: (a) r ≥ 3, (b) 1 < r < 23+p 11+p , (c) 23+p 11+p ≤ r < 3, p≥ 0.25, and (d) 23+p 11+p ≤r <3, p< 0.25. . . . . . . . . . . . . . . . . . . . . . . 25 1.3 Optimal wholesale prices in the leakage andnonleakage regions givenα, forr =4 and p =0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 Firm profits under optimal wholesale prices in the leakage and nonleakage cases, for r =4 and p =0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Nonleakageregionwithfreedisposal,forr =3.5,p =0.2. The darkly shaded area represents the original nonleakage region without free disposal. The lightly shaded area represents the region in which nonleakage is enabled by the incumbent’s free disposal option in the high demand case. The bound UB FD is given by w μ = ( 3r pr−p+1 +1) α(1−α) 12+(p−5)α .) . . . . . . . . . . . . . . . 32 2.1 Buyer’s valuation against with time . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 Value and cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 Serving only high type-fixed fee . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 Serving both type-fixed fee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5 Optimal effort regime under fixed fee when θ 2 L θ 2 H ≤ v L v H . . . . . . . . . . . . . . . 50 2.6 Optimal effort regime under fixed fee when θ 2 L θ 2 H ≥ v L v H . . . . . . . . . . . . . . . 50 2.7 Optimal effort for type x buyer as p increases. . . . . . . . . . . . . . . . . . . . 52 2.8 Optimal solution in parameter regime v L θ H −θ L < θ L 2c . . . . . . . . . . . . . . . . . . 55 2.9 Optimal solution in parameter regime v L −v H θ H −θ L ≤ θ L 2c ≤ v L θ H −θ L ≤ p d L 2c . . . . . . . . . . 58 2.10 Optimal solution in parameter regime v L θ H −θ L > p d L 2c , when θ H 2c < v L −v H θ H −θ L . . . . . . . 59 2.11 Serving only low type-two part tariff . . . . . . . . . . . . . . . . . . . . . . . . 60 2.12 Serving both types-two part tariff when θ L −λθ H 2c(1−λ) ≥ ˜ t,v L <v H . . . . . . . . . . . 61 2.13 Serving both types- two part tariff when θ L −λθ H 2c(1−λ) ≥ ˜ t,v L >v H . . . . . . . . . . . 61 ix 2.14 Serving both types- two part tariff when θ L 2c < ˜ t< θ H 2c . . . . . . . . . . . . . . . 61 2.15 Optimal effort regime under two part tariff when θ L −λθ H 2c(1−λ) ≥ ˜ t . . . . . . . . . . . 63 2.16 Optimal effort regime under two part tariff when ˜ t≥ θ H −(1−λ)θ L 2cλ . . . . . . . . . 63 2.17 Relationshipamongtrueeffortt,randomqualityq,reportedeffort b t,report-based payment p b t, and quality-effective payment s(q), when p>(1−Δ)θ x . . . . . . 66 2.18 Time based payment-compare profit when θ L 2c > v L θ H −θ L and p ∗ <θ H . . . . . . . 72 2.19 Time based payment-compare profit when θ L 2c ≤ v L θ H −θ L . . . . . . . . . . . . . . . 72 2.20 Equivalence Region of Two Part Tariff and Time Based Payment . . . . . . . . 77 3.1 Threshold δ and the operational cases as p changes. . . . . . . . . . . . . . . . 95 3.2 Threshold δ and the operational cases as λ μ changes. . . . . . . . . . . . . . . . 96 3.3 Threshold δ and the operational cases as α changes. . . . . . . . . . . . . . . . 96 A.1 Solution to inequality (2) in case (b.ii). . . . . . . . . . . . . . . . . . . . . . . . 113 A.2 Supplier’s profit functions given wholesale prices w and w ′ . . . . . . . . . . . . . 117 A.3 Supplier’s profit functions given wholesale prices w, ˆ w, and w ∗ . . . . . . . . . . . 117 A.4 The range of θ and p in which pooling cannot be ignored. . . . . . . . . . . . . . 122 A.5 Nonleakage region with free disposal, for r =3.5, p= 0.2. . . . . . . . . . . . . . 128 A.6 Nonleakage region given an active entrant, for r =3.5, p= 0.2. . . . . . . . . . . 130 A.7 Supplier’s leakage decision when entrant orders first, for r = 4 and p =0.2. . . . 137 B.1 Optimalobjective values fromdifferent solutions asfunctions ofλ, when v L −v H θ H −θ L ≤ θ L 2c ≤ v L θ H −θ L ≤ p d L 2c . (a) λ mh ≤λ h H ; (b) λ mh >λ h H . . . . . . . . . . . . . . . . . . . 148 B.2 Optimal objective values fromdifferent solutions when θ L 2c ≤ v L −v H θ H −θ L ≤ v L θ H −θ L ≤ p d L 2c and 0<λ h L <λ mh <λ m H < 1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 x LIST OF TABLES 1.1 A Leakage-Proof Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2 Nonleakage Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3 Definition of the Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4 Quantities and Profits Under Various (α, w μ ) for r =4, p= 0.2 . . . . . . . . . . 29 2.1 The SP’s profit under fixed fee contract . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 The SP’s profit and strategy in parameter regimes I and II . . . . . . . . . . . . 57 2.3 The SP’s profit and strategy in regime III . . . . . . . . . . . . . . . . . . . . . 58 2.6 Change of the SP’s profits and effort levels when v L θ H −θ L ≥ θ L 2c . . . . . . . . . . 73 2.7 Change of the SP’s profits and effort levels when v L θ H −θ L ≤ θ L 2c and p ∗ <θ H . . . 73 2.8 Change of the SP’s profits and effort levels when v L θ H −θ L ≤ θ L 2c , p ∗ = θ H and λ =0.15, 0.2, or 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.4 The SP’s profit and optimal solution under two part tariff . . . . . . . . . . . . 80 2.5 Dominant Strategy in Different Parameter Regions . . . . . . . . . . . . . . . . 81 2.9 SP’s optimal profit and contract . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.1 Channel Quantity Distortion from Supplier’s First-Best . . . . . . . . . . . . . . 108 A.2 Conditions for Supplier to Prefer Nonleakage . . . . . . . . . . . . . . . . . . . . 108 A.3 Channel Quantity Distortion from Supplier’s First-Best . . . . . . . . . . . . . . 131 A.4 Conditions for Supplier to Prefer Nonleakage . . . . . . . . . . . . . . . . . . . . 132 xi CHAPTER 1 REVENUE SHARING AND INFORMATION LEAKAGE IN A SUPPLY CHAIN 1.1 Introduction Advances in information technology have had a dramatic impact on the ability of firms in a supply chain to share information and numerous firms have taken advantage of these advances. Greater collaboration between firms in a supply chain has resulted in initiatives such as Collaborative Planning, Forecasting and Replenishment (CPFR), in which well- knownmanufacturerssuchasProctorandGambleandBlackDeckeraswellasmajorretailers such as Home Depot and Wal-Mart have participated. But a major challenge has been the reluctance of some firms to share information vertically with suppliers due to the fear of leakage of this information to their competitors. In this case, the benefits of information sharing are not realized and all the parties in the supply chain may be worse off. Wisner et al. (2008)pointoutthat“retailers are reluctant toshare thetype ofproprietary information requiredbyCPFR.”Adewole(2005)pointsoutthatretailersintheUKclothingindustryare “reluctant to share information with suppliers, recognizing that those suppliers might also be supplying competitors and could wittingly or unwittingly divulge sensitive information.” Anand and Goyal (2009) have provided several other examples of information leakage and evidence of firms’ reluctance to share information. Several recent academic works have explored the issues of information sharing, distortion and leakage in a strategic context. Many of these works such as Li and Zhang (2008) show that firms competing with each other will be reluctant to share information vertically with a supplier for fear of leakage. 1 Anand and Goyal (2009)show that asupplier will always leak information froman informed retailer to an uninformed one under a wholesale price regime. Theissue ofinformationsharingandleakageismostsalient whentheinformationisvalu- able and demand information is often particularly valuable in a supply chain. One retailer may have a considerable advantage over others in access to or in acquiring demand infor- mation and the retailer can exploit this informational advantage, for instance by producing and selling more if demand is likely to be higher and thus achieving higher revenues. A toy retailer may be popular with young mothers and may be able to identify market trends earlier than, say, discount or mass market retailers selling toys. The same may be true for a retailer selling computers, video games or other products exclusively. A retailer may be able to exploit this informational advantage only with the cooperation of the supplier who manufactures the product. But the advantage may dissipate if the common supplier leaks the demand information or an equivalent signal to a competitor. The supplier also derives several benefits from advance information such as the ability to better plan capacity or inventory levels and achieve better service levels. While we have framed our discussion in terms of retailers sharing information with a supplier, the same issues are salient in the context of a manufacturer sharing information with the supplier of, say, a critical part. Our goal in this work is to explore the potential of revenue sharing contracts to facilitate information sharing in a supply chain without the negative effects of information leakage. Thus, the paper bridges and makes a contribution to two streams of research in operations management that have been fairly disparate: (1) contracting mechanisms, (2) information leakage in a supply chain. There have been numerous papers on contracting mechanisms, includingrevenuesharingcontractsthatcanhelpcoordinateasupplychain. Therehavebeen fewer works on information leakage in a supply chain. While the role of revenue sharing in coordinatingasupplychainiswell-known, theirpotentialinmitigatingtheharmfuleffectsof informationleakage(orthethreatthereof)onfirmandchannelprofitshasnotbeenexplored. 2 Inparticular, weconsider astylized supply chainwithonesupplier sellingtotworetailers who compete for customers. One of these retailers (the incumbent) is privy to private infor- mation about market potential and places an order based on this information. The supplier may leak this information to the second retailer (the entrant) if it benefits her and in turn this would impact the order quantity placed by the first retailer. The two retailers engage in Cournot competition, i.e. they sell in a market where the price is a decreasing function of their cumulative order quantities. Anand and Goyal (2009) analyzed an identical scenario and showed that the supplier will always leak information under a wholesale price contract. The information leakage may result in a less efficient supply chain with lower profits po- tentially for all parties. If the supplier can make a credible commitment not to leak, the supplier might benefit from it. So, we consider what would happen if the supplier uses a revenue sharingcontractwherein shesells theproducttothetworetailersatapossibly lower wholesale price but also receives a share of the retail revenue. Revenue sharing contracts are common in several industries including entertainment (see rentrak.com), sports leagues, software (drkarlpopp.com), laser-based devices used in plastic surgery (Frentzen 2010) and consumer goods in India (livemint.com 2009). The site www.techagreements.com provides specific examples of revenue sharing agreements in several industries. We explore the po- tential of a revenue sharing contract to reduce the supplier’s incentive to leak information and the impact of such a contract on the decisions and profits of the supplier and retailers in the supply chain. Our analysis reveals that a revenue sharing contract leads to many interesting effects on the actions and profits of the supplier and retailers. First, we show that a revenue sharing contract reduces a supplier’s incentive to leak demand information and there exist many combinations of wholesale price and revenue share percentage under which the supplier will not leak. Second, preventing information leakage may result in higher profits for both the incumbent and supplier relative to a scenario where the supplier leaks information and, 3 interestingly, even the entrant may be better offsometimes. Third, we show that nonleakage is more likely when the supplier’s revenue share percentage is relatively high and when the demand variation is greater. Finally, we show that our conclusions are robust when some of the model assumptions are relaxed or altered. Overall, the incentives of the supplier and retailers are better aligned under revenue-sharing: the supplier is no longer trying to push product under all circumstances as in a wholesale price contract and a non-cooperative “partnership” is achieved between the supplier and retailers. While revenue sharing contracts represent one approach to enabling information sharing in a supply chain, there are other enablers—for instance, the level of trust between parties can play acrucial role. Infact, ¨ Ozer et al. (2011)show thattrust andtrustworthiness play a crucial role in sharing forecast information in the face of uncertainty and asymmetric infor- mation. They consider a supply chain in which a manufacturer has private information and hasanincentive toexaggerate herdemandforecast toasupplier andexplore theroleoftrust and trustworthiness between supply chain partners using controlled laboratory experiments. Interestingly, they find that supply chain partners may share information truthfully without distortion even under a simple wholesale price contract depending on their willingness to trust and the trustworthiness of their partner, contradicting the results of game-theoretic models and provide an innovative model that incorporates trust. While we do not model trust and trustworthiness which are important elements, we provide mechanisms to faciltate information sharing even when parties do not necessarily trust each other. 1.2 Review of Literature There are several streams of research that have explored the role of information sharing in supply chains. Early work on information sharing, primarily in the economics literature, studied the incentives for sharing information in oligopoly markets, some examples being Gal-Or(1985), Vives (1984), Li (1985), and Raith (1993). Inthe Operations area, the initial 4 stream of research focused on the value of information sharing in improving supply chain decisions and supply chain efficiency. Lee and Whang (2000) provide numerous examples of firms in a supply chain that benefit from sharing demand and forecast information but they point out, based on an empirical study, that the benefits of sharing information may vary greatly among the different parties in a supply chain. They suggest that vertical informa- tion sharing usually leads to horizontal information leakage which in turn can be a serious deterrent to sharing information. Chen (2003) surveys various models in the literature that quantify the value of information sharing arising from reduction of inventories, alleviation of the bull-whip effect, etc. but these models do not consider strategic issues related to information sharing and leakage, which is our focus. Our work does not explicitly consider the direct benefits of information sharing but focuses on the potential of revenue sharing contracts to mitigate the possible negative effects of information sharing. So, the supply chain would be even better off if the benefits of information sharing were incorporated. A more recent stream of research has focused on vertical information sharing between two supply chain partners, one of whom may be better informed about some key element such as demand or capacity and has an incentive to be strategic in sharing information. ¨ Ozer and Wei (2006) show that a wholesale price contract can result in distortion of demand forecastinformationandanappropriatelydesignedcontractcanachievecredibleinformation sharing and channel coordination between a manufacturer and its supplier. Mishra et al. (2007) show that a profit margin sharing agreement can facilitate information sharing and make both parties better off in a one-retailer one-manufacturer supply chain. Oh and ¨ Ozer (2012)showhowasuppliercanelicitcredibledemandinformationfromitscustomerandhow forecasts evolve in a multi-period setting with asymmetric information. Even though these papers analyze a supply chain in an asymmetric information setting, they do not consider either information leakage or horizontal competition, which is our focus. Ha and Tong (2008) consider the problem of vertical information sharing between a manufacturer and its retailer when there are two competing supply chains and emphasize 5 the importance of contract type in inducing information sharing. In follow-up work, Ha et al. (2011) consider two competing supply chains and show when information sharing can benefit a supply chain under both Bertrand and Cournot competition. Shin and Tunca (2010) show that retailers may overinvest in improving their forecasts when there is more intense retail competition and greater demand uncertainty under wholesale price or two- part tariff contracts and suggest alternative contracts that can mitigate these problems and improve supply chain efficiency. But these works do not consider information leakage issues. Anotherstreamofresearchconsidersverticalinformationsharingwhenthereishorizontal competition along with information leakage. Li (2002) and Zhang (2002) consider a model with one supplier supplying retailers who compete by offering substitute products. Both works suggest that retailers will not share private demand information voluntarily with the supplier and Zhang (2002) suggests that they will need special incentives to do so. Li and Zhang (2008) also consider a similar supply chain and explore the effects of information leakage. They assume thatinformationleakageis throughthe supplier’s observable behavior as a response to the information shared by a retailer. Jain et al. (2011)consider an identical setup and develop pricing mechansims that will induce truthful information sharing by all parties. However, in these works, firms decide their information revelation behavior before receiving the private information and moreover, the supplier has the information, if shared, before making its pricing decision. We adopt a different approach, similar to that in Anand andGoyal(2009)intermsofwheninformationbecomesavailabletothesupplier. Whilethey show thatinformationleakagebythe supplier always occursunder wholesale price contracts, we demonstrate that information can be protected by a properly designed revenue sharing contract. The final stream of research related to our work is on revenue sharing contracts, which have been shown to be an effective tool in aligning firms’ incentives by inducing appropriate behavior in a supply chain. Mortimer (2008) shows that revenue sharing contracts have 6 improved the profits of firms in the supply chain and social welfare in the video-rental industry. Cachon (2003) shows that revenue sharing contracts achieve coordination in a one supplier one retailer supply chain. Dana and Spier (2001), Cachon and Lariviere (2005) and Yaoetal. (2008)considerdownstreamcompetitionandshowthatarevenuesharingcontract is a powerful tool in coordinating a supply chain with competing retailers. Although these papers study downstream competition in the face of uncertain demand, they assume that the demand information is “symmetric” among downstream retailers—either every retailer only knows the common distribution of demand or they know their own demand but their demands are independent. As a result, the revenue sharing contract literature has not considered the effects of information leakage in a supply chain, unlike in our work. 1.3 Model Framework The basic framework of the model is similar to that in Anand and Goyal (2009). We discuss some importantfeatures ofthemodelbelow forthereader’s convenience. Weconsider asup- ply chain with one upstream supplier serving two competing retailers (or two manufacturers selling directly to consumers), an incumbent retailer and an entrant retailer. The retailers compete in a market characterized by demand uncertainty. The supplier, the incumbent, and the entrant are indexed by s, i, and e respectively. All firms are risk neutral and aim to maximize their own expected profits. We study a dynamic game among the three players under incomplete information (Gibbons 1992). Demand Information. The incumbent and the entrant engage in a Cournot compe- tition, i.e. compete on quantity. The market clearing price is given by P = A−Q, where Q =q i +q e is the total supply in the market, and q i and q e are the incumbent and entrant’s order quantities, respectively. We assume that the intercept A of the (linear) inverse de- mand function is uncertain, which takes the value A H with probability p ∈ (0,1) and A L with probability 1−p, for some A H > A L > 0. We denote the mean demand potential by 7 μ=pA H +(1−p)A L . The distribution ofA iscommon knowledge toallparties, yet only the incumbentknowstheexactvalueofA,whichwillbereferredtoasthe“demandinformation” or “demand state” throughout this paper. This assumption is motivated by the observation that a retailer who has been in a market long enough often has an advantage over a retailer that has entered recently or an upstream supplier in acquiring market information. This is why we refer to the retailer with private information on A as the incumbent. Revenue Sharing Contract. We assume that the transactions between the supplier and the retailers are governed by a revenue sharing contract. This represents a crucial difference between ourmodel andthatofAnandand Goyal(2009),who consider awholesale price contract. Under the latter contract, a retailer pays a wholesale price w for each unit purchased from the supplier. Under a revenue sharing contract, a retailer not only pays a wholesale price for each unit purchased but also shares part of the revenue with the supplier at a certain rate α. In this work, we explore whether a revenue sharing contract affects the supplier’s incentive to leak the information obtained from the incumbent retailer. Since a wholesale price contract is a special case of a revenue sharing contract with the revenue sharing rate α = 0, this study also generalizes the results of Anand and Goyal (2009) and provides broader managerial insights. Under a revenue sharing contract, denoted by (α,w), if the marginal supply cost is normalized to zero (without loss of generality), the supplier’s profit is given by π s =w(q i +q e )+α(q i +q e )P, (1.3.1) where the first term is the supplier’s payment from selling the product at the wholesale price w, and the second term is her share of the retail revenue. The incumbent and the entrant’s profits are π i =(1−α)q i P−wq i =(1−α)(P− w 1−α )q i , and π e =(1−α)q e P−wq e = (1−α)(P− w 1−α )q e , 8 respectively. We assume that the retailers bring to market their entire order quantities. No free disposal is assumed in other work in the literature, e.g., Anand and Goyal (2009), Ha and Tong (2008), and Li (2002), and one does not observe significant deliberate disposal of products such as toys, computers, etc. in practice. We will relax this assumption in Section 1.7.1. Sequence of Events. The following events take place in sequence. (1) The supplier offers a revenue sharing contract, consisting of a wholesale price w and a revenue sharing rateα. (2) The incumbent observes the actual demand state A, which is either A H (high) or A L (low). (3) The incumbent places an order q i to the supplier. (4) The supplier may leak the incumbent’s order quantity to the entrant. (5) The entrant places an order q e to the supplier. (6)ThedemandstateAisrevealedtoallparties, themarketpriceisdeterminedby P =A−q i −q e , andtheprofitforeachplayer isrealized. In thispaper, we willinitially focus on the situation where the wholesale price w and revenue sharing rate α are exogenously given (Sections 1.4 and 1.5) and later allow the supplier to optimally choose w for any given α (Section 1.6). We do not consider the situation where the supplier selects both α and w, because her optimal choice would be α = 1 and w =0, which is not particularly interesting or realistic (see Footnote 7 for more details). Players’ Incentives and Decisions. Next, we discuss the players’ decisions in the game and, in particular, how the supplier’s decision on information leakage would impact the decisions and profits of the two retailers. Supplier: Amajordecisionfacedbythesupplieriswhetherornottoleaktheincumbent’s orderquantitytotheentrant. AnandandGoyal(2009)showthatthesupplieralwaysbenefits from leaking under a wholesale price contract. However, as will be discussed in the next section, information leakage not only reduces the incumbent’s profit but also may hurt the supplier, and if the supplier can make a credible commitment not to leak, both parties may benefit. Thus, we assume that the supplier can communicate to the incumbent whether or 9 notshe intends toleakthe incumbent’s order quantity totheentrant, which iscredible if the supplier can make a higher profit with the intended action. Our question is whether there exists a revenue sharing contract under which the supplier will not leak the incumbent’s order information. To focus on the main issue, we assume that the supplier has unlimited capacity and that she may not lie about the incumbent’s order quantity when she leaks it. 1 Incumbent: The incumbent’s decision in the game is his order quantity and he plays a signaling game with the supplier and the entrant. Since the incumbent has access to the private demand information, the order placed by the incumbent would naturally reflect this information. However, since the incumbent realizes that the supplier may leak this infor- mation to the entrant, the incumbent will be strategic when placing the order. Specifically, the incumbent may distort his order quantity, which may negatively impact all the parties in the supply chain. Entrant: The entrant’s only decision in the game is alsohis order quantity. Nevertheless, the entrant plays a passive role compared to the incumbent and makes his decision based on the supplier’s leakage decision and the incumbent’s order decision. The type of game played between the entrant and the incumbent is determined by the supplier’s leakage decision: if the supplier does not leak information, the two retailers play a simultaneous game; while if the supplier leaks, the retailers play a Stackelberg game, in which the incumbent is the leader andtheentrant isthe follower. Weconsider gameswith alternative sequence ofmoves in Section 1.7. 1.4 Benchmark Analysis In this section, we consider benchmark scenarios in which the order quantities and profits of the players are determined given the supplier’s leakage decision. Section 1.4.1 discusses the 1 In our model, the incumbent’s order quantity is fully revealed to the entrant through their competition and the market price. Thus, the supplier’s misrepresentation of the incumbent’s order can be verified by the entrant. Whereas information leakage is difficult to verify (by the incumbent) and arguably poses a more severe incentive problem in the situation we study. 10 case in which the supplier never leaks information, while Section 1.4.2 considers the scenario in which the supplier always leaks. We also study the supplier’s first-best scenario in Section 1.4.3, assuming that the supplier can control the total order quantity in the channel. The benchmarkanalysiswillbehelpfulinderivingandunderstandingthenonleakageequilibrium in subsequent sections. Our analysis in Sections 1.4.1 and 1.4.2 is similar to that in Anand and Goyal (2009) except that we are considering a revenue sharing contract instead of a wholesale price one. Propositions 1.4.1 and 1.4.2 in these sections can be proved in the same way as in Anand and Goyal (2009), by replacing A H and A L with A H − w 1−α and A L − w 1−α , respectively, andthustheproofsareomittedforbrevity. Theproofsforallsubsequent results in this chapter are available in Appendix A. The notational scheme used for order quantities and profits is as follows. In general, q E ∗ ad and π E ∗ ad denote the order quantity and profit, respectively, of player a in equilibrium (or game) E given demand state d: a can be i (incumbent), e (entrant), or s (supplier); E can be N (nonleakage), S (separating leakage), or P (pooling leakage); and d can be H (high) or L (low). For example, q N ∗ iH denotes the incumbent’s order quantity in a nonleakage equilibrium when the demand is high. The subscript d may be absent when a nonleakage or pooling (leakage) equilibrium is considered, as in q N ∗ e and π P ∗ s . Additional notation will be introduced later. 1.4.1 Benchmark: Supplier Never Leaks We first consider the situation in which the supplier can make a credible commitment not to leak the incumbent’s order quantity to the entrant. Since neither retailer knows the other retailer’s orderquantity, thegamebetween theincumbent andtheentrant isasimultaneous- move game, and we have the following result. Proposition 1.4.1. Suppose that the supplier can credibly commit in advance not to leak the incumbent’s order information. If w μ ≤ 1 2 ( 3A L μ −1)(1−α), the high-type incumbent will 11 order q N ∗ iH = 1 2 A H − 1 6 μ− 1 3 w 1−α , the low-type incumbent will order q N ∗ iL = 1 2 A L − 1 6 μ− 1 3 w 1−α , and the entrant will order q N ∗ e = 1 3 (μ− w 1−α ) in both demand states. Undernonleakage,theincumbent’sorderquantitydependsonthedemandstatewhilethe entrant’s order quantity only depends on the mean demand. Intuitively, if the supplier can make a credible commitment not to leak, the incumbent will place an order that truthfully reveals the demand state to the supplier. In other words, vertical information sharing is enabled while horizontal information leakage is prevented (Li 2002). Note that nonleakage is assumed in this benchmark, while in Section 1.5, when we analyze the sustainability of the nonleakage equilibrium under various revenue sharing contracts, nonleakage will be the supplier’s equilibrium decision. Theparticipation constraint w μ ≤ 1 2 ( 3A L μ −1)(1−α)isequivalenttoq N∗ iL ≥ 0,whichimplies q N∗ iH ≥ 0 and q N∗ e ≥ 0. Because the entrant places the same order in both demand states under nonleakage, it is harder to guarantee the incumbent’s participation and maintain a positive market price when the demand is low. 1.4.2 Benchmark: Supplier Always Leaks Next we consider the case in which the supplier always leaks the incumbent’s order quantity totheentrant. Insuchacase,whenplacingtheorder,theincumbentwouldtakeintoaccount the entrant’s updated belief of the demand state and corresponding action after learning the incumbent’s order. As is common in the literature, we study two types of pure strategy Perfect Bayesian Equilibria (PBE)—the separating equilibrium and the poolingequilibrium. Separating Equilibrium In a separating equilibrium, the incumbent orders different quantities in the two demand states, and the entrant can infer the demand information from the incumbent’s order. The entrantbelievesthatthedemandislowiftheincumbent’sorderq i islowenoughandthatthe 12 demandishighotherwise. Denotethelow-typeincumbent’s orderquantityinequilibrium by q S ∗ iL and that of the high type by q S ∗ iH (> q S ∗ iL ). Suppose that, as in Anand and Goyal (2009), the entrant’s belief is: Pr e (A =A H ) = 0, if the supplier leaks and q i ≤q S ∗ iL , 1, if the supplier leaks and q i >q S ∗ iL . (1.4.1) The main incentive issue here is that the high-type incumbent is tempted to signal a low demand (i.e., pretend to be a low type) in hoping that the entrant orders less. Thus, the separating equilibrium requires that (i) it is valuable for the low-type incumbent to separate and (ii) it is too costly for the high-type incumbent to pool with the low type. We have the following result. Proposition 1.4.2. Define θ = A H − w 1−α A L − w 1−α . Suppose the supplier always leaks the incumbent’s order quantity to the entrant and the entrant’s belief about the demand is given by (1.4.1). Then, the following separating equilibrium exists: (i) The incumbent orders q S ∗ iH = 1 2 (A H − w 1−α ) if the demand is high, and q S ∗ iL = q S1 ∗ iL , 1 2 (A L − w 1−α ), if θ≥ 3, q S2 ∗ iL ,A H − A L 2 − 1 2 w 1−α − q (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 , if θ <3, (1.4.2) if the demand is low; (ii) The entrant orders q S ∗ eH = 1 4 (A H − w 1−α ) if Pr e (A =A H ) = 1, and q S ∗ eL = 1 4 (A L − w 1−α ), if θ≥ 3, 3 4 A L − A H 2 − 1 4 w 1−α + 1 2 q (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 , if θ < 3, if Pr e (A =A H ) = 0. It can be shown that in a sequential-move (Stackelberg) game under perfect information (i.e., the demand is public information and the incumbent is the Stackelberg leader), the incumbent’s orderquantityunderarevenue sharingcontract(α,w)is 1 2 (A H − w 1−α )or 1 2 (A L − 13 w 1−α ), depending on the demand state. We observe that: when the demand is high, the incumbent’s order quantity in the separating equilibrium q S ∗ iH is the same as the one in the perfect information game; however, when the demand is low, his order quantity q S ∗ iL depends on θ. Note that θ measures the relative variation in demand, from the retailers’ point of view and under a given contract (α,w). If θ ≥ 3, q S ∗ iL is still equal to the order quantity under perfect information; in this case, the high and low demand states are far apart so that the low-type incumbent separates out at no cost. Whereas if θ < 3, q S ∗ iL is strictly less than 1 2 (A L − w 1−α ); in this case, since the two demand states are relatively close, the low-type incumbent has to order less to prevent the high-type from mimicking. Thus, one consequence of the supplier’s leakage behavior is the downward distortion inflicted on the low-type incumbent. The supplier’s leakage behavior may have a negative impact on herself as well, because she sells less to the incumbent when the demand is low. Therefore, the total supply chain profit may fall due to information leakage. We note that the incumbent would participate in the separating leakage game in both demand states if the contract parameters satisfy w μ ≤ A L μ (1−α), which is implied by the participation constraint in the nonleakage case. The entrant would participate as long as the incumbent does. Pooling Equilibrium In a pooling equilibrium, the incumbent’s order quantity is independent of the demand, and hence the entrant cannot infer the demand state from it and would not update his belief. A pooling equilibrium differs from a nonleakage equilibrium even though the entrant retains the same belief in both cases. In a nonleakage equilibrium, the entrant has to guess the incumbent’s order quantity and they effectively play a simultaneous-move game, while in a pooling equilibrium, the entrant makes his best response to the incumbent’s order and they play a sequential-move game. In the rest of this paper, we ignore pooling because a 14 pooling equilibrium does not exist in a fairly general situation and even when it exists, the incumbent may be better off playing a nonleakage game. Intuitively, pooling does not exist when demand variation is high and is likely to be dominated by nonleakage when demand variation is smaller. A detailed discussion is provided in Appendix A.2. 1.4.3 Benchmark: Supplier’s First-Best Scenario The supplier’s profit under a given revenue sharing contract is determined by the total order quantity in the channel, Q =q i +q e , whether in a leakage or nonleakage game. It would be revealing to compare the channel quantities in those benchmark cases with the one in the supplier’s first-best scenario in which the supplier has perfect information and can control the channel quantity. We call the optimal Q in the latter setting the supplier’s first-best channel quantity, denoted by Q FB H = argmax Q {wQ+αQ(A H −Q)} when the demand is high and Q FB L = argmax Q {wQ+αQ(A L −Q)} when the demand is low. We identify the quantity distortion from the first-best values in both leakage and nonleakage equilibria. Proposition 1.4.3. (i) The supplier’s first-best channel quantity is Q FB H = 1 2 A H + 1 2 w α when the demand is high or Q FB L = 1 2 A L + 1 2 w α when the demand is low. (ii) Under a nonleakage Cournot competition, in both demand states, the retailers as a whole underorder (i.e., Q N∗ H < Q FB H and Q N∗ L < Q FB L ) if w μ > α(1−α) 3+α , overorder (i.e., Q N∗ H >Q FB H and Q N∗ L >Q FB L ) if w μ < α(1−α) 3+α , and perfectly align with the supplier’s objective if w μ = α(1−α) 3+α . (iii) Let UB N = α(1−α) 2+α A H μ and LB N = α(1−α) 2+α A L μ , if θ≥ 3 w μ such that Q S2∗ L =Q FB L , if θ < 3 ,where Q S2∗ L =q S2∗ iL + 1 2 (A L −q S2∗ iL − w 1−α ). Under a leakage Cournot competition with the incumbent being the Stackelberg leader, the retailers as a whole underorder if w μ > UB N , overorder if w μ <LB N , and underorder (overorder) when the demand is low (high) if LB N ≤ w μ ≤UB N . 15 Inthenextsection, weallowthesuppliertodecidewhetherornottoleaktheincumbent’s order information and show that the nonleakage equilibrium exists under certain revenue sharing contracts. 1.5 Nonleakage Equilibria Under a wholesale price contract, as shown in Anand and Goyal (2009), the supplier always benefits from leaking the incumbent’s order quantity. Because a larger order translates into higher profit for the supplier, the supplier would always like to inform the entrant when the demand is high. This is no longer true under a revenue sharing contract, where a larger quantityisnotalwaysbetter. AsshowninProposition1.4.3,intheseparating(leakage)case, the downstream retailers together may underorder when the demand is low and overorder when the demand is high, compared with the supplier’s first-best quantity. This type of quantity distortion may be mitigated in both demand states simultaneously if the supplier does not pass the demand information to the entrant so that the entrant has to order an intermediate quantity, aimed at the average demand. Thus, the supplier may benefit from nonleakage in both demand states. In this section, we first show the existence of the nonleakage equilibrium under a given revenue sharing contract (α,w) and then characterize the set of α and w that sustain such an equilibrium. We assume that in the leakage case the demand state can be inferred from the incumbent’s order quantity. Ignoring the pooling equilibria is justified in Appendix A. 1.5.1 Existence of the Nonleakage Equilibrium We first consider the supplier’s incentive compatibility with nonleakage and then the in- cumbent’s, from which we establish the existence of the nonleakage equilibrium under a given (α,w). Finally, we present a numerical example to demonstrate how a revenue sharing contract can prevent information leakage. 16 Supplier’s Incentive Compatibility with Nonleakage In contrast to the benchmark cases considered in Section 1.4, the supplier’s leakage choice is now her endogenous decision. For the nonleakage option to prevail, the supplier must make a higher profit in both demand states under nonleakage. If, say, the supplier only benefits from nonleakage when the demand is low and hence she only leaks the information when the demand is high, the entrant can still infer the demand state (low) when the information is concealed by the supplier. Thus, the supplier would like (and is able) to commit not to leak the incumbent’s order information if the following conditions hold, given the incumbent’s order quantities q iH and q iL : π S sH (q iH )≤π N ∗ sH and π S sL (q iL )≤π N ∗ sL , (1.5.1) where π N ∗ sH and π N ∗ sL are the supplier’s profits in the nonleakage equilibrium in the high and low demand states, respectively, and π S sH (q iH ) and π S sL (q iL ) are the supplier’s profits under leakage. 2 More specifically, π N ∗ sH andπ N ∗ sL can be derived from Proposition 1.4.1and equation (1.3.1), and π S sH (q iH ) =w(q iH +q ∗ e (q iH ))+α(q iH +q ∗ e (q iH ))(A H −q iH −q ∗ e (q iH )), π S sL (q iL ) =w(q iL +q ∗ e (q iL ))+α(q iL +q ∗ e (q iL ))(A L −q iL −q ∗ e (q iL )), for q ∗ e (q i ) =argmax qe (1−α)q e (A L −q i −q e )−wq e , if q i ≤q S∗ iL , (1−α)q e (A H −q i −q e )−wq e , if q i >q S∗ iL . The constraints in (1.5.1) ensure that the supplier makes more profits under nonleakage in both demand scenarios, depending on the contract (α,w) and the incumbent’s order 2 Ifthesupplierdoesnotleaktheincumbent’sorderinformation,shecaninducethenonleakageequilibrium and garner the profit π N ∗ sL or π N ∗ sH . The incumbent’s order quantity q iL (or q iH ) other than q N ∗ iL (or q N ∗ iH ) will be off the Nash-equilibrium path. Therefore, we compare the supplier’s profit from leaking q iL (or q iH ) with π N ∗ sL (or π N ∗ sH ). There are multiple ways for the supplier to make her nonleakage commitment credible. For example, if low demand is inferred, she can cap the incumbent’s order quantity at q N∗ iL and make the profit π N ∗ sL . If the supplier cannot make a credible commitment, we may have to compare the supplier’s profits from leaking and not leaking q i directly. It can be shown that the nonleakage region characterized in Theorem 1.5.4 later would shrink but the main insights of the paper would be unchanged. 17 Figure 1.1. Supplier’s relative profit gain from the nonleakage equilibrium, in either demand state. Incumbent’s order quantity q i Difference in supplier’s profit π N∗ s −π S s (q i ) q i ¯ q i Nonleakage Interval Leakage Interval Nonleakage Interval 0 quantity. The next result provides conditions for the supplier not to leak in terms of these variables. Proposition 1.5.1. Suppose the incumbent orders no more than q S∗ iL when the demand is low and more than q S∗ iL when the demand is high. In either demand state, the supplier prefers leaking the incumbent’s order quantity q i to playing the nonleakage equilibrium if and only if q i ∈ (q i ,¯ q i ), where q i = w α + w 1−α − 1 3 μ−3 w α −4 w 1−α , and ¯ q i = w α + w 1−α + 1 3 μ−3 w α −4 w 1−α . The first assumption in the proposition guarantees that the entrant’s belief is consistent with the incumbent’s strategy should the incumbent’s order quantity be leaked. We call (q i ,¯ q i ) the supplier’s leakage interval and [0,q i ] and [¯ q i ,∞) the nonleakage intervals. The supplier’s profit gains from nonleakage, π N ∗ sH −π S sH (q iH ) when the demand is high and π N ∗ sL − π S sL (q iL ) when it is low, happen to have the same functional form, illustrated by π N ∗ s −π S s (q i ) in Figure 1.1. Thus, the leakage and nonleakage intervals depend solely on the contract (α,w) and the mean demand μ, not on the actual demand state. The function π N ∗ s −π S s (q i ) always intersects the horizonal axis and its quadratic form leads to the thresholds q i and ¯ q i . Notice that q i and ¯ q i can be rewritten as q i = min{ 1 3 (μ− w 1−α ),2 w α + 7 3 w 1−α − 1 3 μ} and ¯ q i = max{ 1 3 (μ− w 1−α ),2 w α + 7 3 w 1−α − 1 3 μ}, and that 2 w α + 7 3 w 1−α − 1 3 μ≤ 1 3 (μ− w 1−α ) if and only 18 if w μ ≤ α(1−α) 3+α . In the special case w μ = α(1−α) 3+α , we have q i = ¯ q i and the leakage interval (q i ,¯ q i ) is empty, which confirms the result of Proposition 1.4.3(ii) that the channel quantity under nonleakage is first-best to the supplier in this situation. In all other cases, q i < ¯ q i . Incumbent’s Incentive Compatibility with Nonleakage To support the nonleakage equilibrium, the incumbent’s order quantities in the equilibrium in both demand states must lie in the supplier’s nonleakage intervals, i.e., q N ∗ iH ,q N ∗ iL ∈ [0,q i ]∪ [¯ q i ,∞). Therearefourcasesregardingtheintervalsinwhichq N ∗ iH andq N ∗ iL mayfall. Toreduce the number of cases and identify conditions that deter the incumbent’s deviation into the supplier’s leakage interval, we compare the incumbent’s order quantities and profits under leakage and nonleakage. Proposition 1.5.2. The incumbent’s order quantities and profits in the nonleakage and leakage (separating) equilibria satisfy: (i) q N ∗ iH ≤ q S ∗ iH and q N ∗ iL ≤ q S ∗ iL , (ii) q N ∗ iH ≥ q i and q N ∗ iL ≤ ¯ q i , and (iii) π N ∗ iH ≥ π S ∗ iH if and only if θ≥ 1−p 3(1− √ 2 2 )−p ≥ 0. According to part (i), the incumbent’s order quantity is larger in the separating equilib- rium in both demand scenarios, because the incumbent has the first-mover advantage in a Stackelberg game. Part (ii) asserts that the incumbent’s order quantity in the nonleakage equilibrium cannot be too low if the demand is high or too high if the demand is low, com- pared withthe supplier’s leakagethresholds. Itleaves only onepossible nonleakagecase, i.e., q N ∗ iH ≥ ¯ q i and q N ∗ iL ≤ q i . Part (iii) shows that under a mild condition, when the demand is high, the incumbent’s profit is higher in the nonleakage case, even though his order quantity isactuallyloweraccordingtopart(i). Thisisbecause whenthedemandishightheentrant’s order quantity is much lower under nonleakage. Existence of the Nonleakage Equilibrium The following theorem presents a set of sufficient conditions and necessary ones for the existence of the nonleakage equilibrium. 19 Theorem 1.5.3. Assume θ ≥ 1−p 3(1− √ 2 2 )−p ≥ 0 and w μ ≤ 1 2 ( 3A L μ − 1)(1−α). A nonleakage equilibrium exists if q N ∗ iH ≥ ¯ q i and q S ∗ iL ≤q i , and only if w μ ≤ ( 3A H μ +2) α(1−α) 12+5α and q S ∗ iL ≤q i . 3 The assumption θ≥ 1−p 3(1− √ 2 2 )−p ensures that the high-type incumbent prefers nonleakage to leakage and w μ ≤ 1 2 ( 3A L μ − 1)(1− α) ensures q N ∗ iL ≥ 0 and q N ∗ iH ≥ 0. 4 The conditions q N ∗ iH ≥ ¯ q i andq S ∗ iL ≤q i aresufficient toensure thatthe incumbent actually choosesq N ∗ iH orq N ∗ iL in the corresponding demand state and does not deviate into the supplier’s leakage interval (q i ,¯ q i ). The low-type incumbent would not deviate if (q N ∗ iL ≤) q S ∗ iL ≤ q i because the entrant would then believe that the demand is high and place an order too large to the incumbent’s liking. A necessary condition for nonleakage is that the channel quantity distortion (from the supplier’s point of view) is milder under nonleakage than under leakage, which amounts to q S ∗ iL ≤ q i in the low demand scenario and w μ ≤ ( 3A H μ +2) α(1−α) 12+5α in the high demand one. The above sufficient conditions and necessary ones differ by only one condition, q N ∗ iH ≥ ¯ q i versus w μ ≤ ( 3A H μ +2) α(1−α) 12+5α . The former condition is more restrictive than the latter, but the differenceissmall,asevidentfromtheexampleillustratedinFigure1.2(a)later. Norevenue sharing contract violating the necessary conditions can support a nonleakage equilibrium: when w μ > ( 3A H μ +2) α(1−α) 12+5α ,thesupplierhastheincentivetoleakthehighdemandinformation; when q S ∗ iL >q i , the supplier is tempted to leak the low demand information. For robustness of the results, we will consider the sufficient conditions in the rest of this paper. 5 Next, 3 By Proposition 1.4.2, q S ∗ iL takes the value q S1 ∗ iL when θ≥ 3 and q S2 ∗ iL when θ < 3. In addition, it can be shown that q S2 ∗ iL < (=, or >) q S1 ∗ iL when θ < (=, or >)3. 4 The range of θ and p satisfying the first assumption is illustrated in Figure A.4 in Appendix A. The assumption is equivalent to π N ∗ iH ≥ π S ∗ iH . It is not entirely necessary when q S ∗ iH lies in the supplier’s (upper) nonleakageinterval,i.e.,q S ∗ iH ≥ ¯ q i ,becausetheincumbentcannotrealizetheprofitπ S ∗ iH byorderingq S ∗ iH ,which will not be leaked by the supplier. Nevertheless, because this condition is not restrictive and is satisfied by all examples in this paper, we take it as given for convenience. 5 The nonleakage equilibrium may exist if w μ ≤ ( 3AH μ + 2) α(1−α) 12+5α and q N ∗ iH < ¯ q i , because the incumbent may benefit from ordering some q iH ≥ ¯ q i (> q N ∗ iH ) and persuading the supplier not to leak. Unfortunately, such a deviation (from q N ∗ iH ) would alter the supplier’s nonleakage interval and complicate the analysis, as evident from a related analysis in Section 7.1. Thus, for the ease of exposition, we do not derive necessary and sufficient conditions for nonleakage in the paper. 20 Table 1.1. A Leakage-Proof Contract High Demand Low Demand Expected Value NonleakageSeparating NonleakageSeparating NonleakageSeparating q i 216.8 272.7 66.8 99.7 - - q e 111.8 136.4 111.8 72.8 - - Q =q i +q e 328.6 409.1 178.6 172.5 - - π i 23,501 18,591 2,231 3,632 8,612 8,120 π e 12,119 9,296 3,734 2,652 6,250 4,645 π s 53,562 50,221 15,717 15,706 27,070 26,061 π Total =π i +π e +π s 89,182 78,108 21,682 21,991 41,932 38,826 we illustrate with an example the nonleakage equilibrium under a given revenue sharing contract. Example 1.5.1. Assume A H = 600, A L = 300, and p = 0.3. Thus, the inverse demand function is P(Q) = 600−Q with probability 0.3 or 300−Q with probability 0.7, and the mean demand is μ =pA H +(1−p)A L = 390. Consider a revenue sharing contract (α,w)= (0.5,27.3), under which the relative demand variation θ = A H − w 1−α A L − w 1−α =2.22<3. The players’ order quantities and profits in the nonleakage and separating (leakage) equilibria are provided in Table 1.1. According to the table, if the supplier protects the incumbent’s order information, her profits will be 53,562 and 15,717 in the two demand scenarios, respectively; and if she leaks the information,herprofitswilldropto50,221and15,706. Thus,thesupplierprefersnonleakage in both demand states. The first-best channelquantities for the supplier areQ FB H = 327.3 and Q FB L = 177.3. Thus, in the separating equilibrium, the two retailers overorder (underorder) when the demandis high (low); under nonleakage, the entrant orders an intermediateamount 111.8∈ (72.8,136.4),bringingthetotal orderquantitiesclosertothesupplier’sfirst-best. The supplier’sleakageintervalisgivenby(q i ,¯ q i )= (106.6,111.8).Hence, theincumbentwouldnot deviate from the nonleakage equilibrium, because (i) when the demand is high, he makes more profit under nonleakage; (ii) when the demand is low, if he deviates to q i (>q S ∗ iL = 99.7), the 21 entrant will believethat the demandis high and make the incumbent worse off. Therefore, the nonleakage equilibrium is sustained under the revenue sharing contract (α,w)=(0.5,27.3). Interestingly, Table 1.1 shows that the entrant’s profits in the two demand states increase from 9,296 and 2,652 to 12,119 and 3,734, respectively, and the expected profits for all players are higher under nonleakage. This is not alwaysthe casehowever. Fromthe entrant’s point of view, there is a trade-off: on the one hand, he benefits from nonleakage because he has more leverage in a simultaneous-move game than being the follower in a sequential- move game; on the other hand, he benefits from leakage because he can adjust his order quantity according to the demand information. Interested readers can refer to Section 7.2 for a comparison of the entrant’s expected profits under leakage and nonleakage. 1.5.2 The Nonleakage Region We now characterize the set of revenue sharing contracts under which nonleakage can be sustained,forgivenmodelparametersA H ,A L ,andp. Forconvenienceinexposition,wescale the wholesale price by the mean demand and refer to w μ as the wholesale price henceforth. Our goal is to identify the set of (α, w μ ) pairs supporting nonleakage, namely the nonleakage region. (This section is somewhat technical and the reader may skim it to grasp the main results.) Define r = A H A L , which measures the relative variation in demand, regardless of the con- tract. Recall that the other relative demand variation measure θ = (A H − w 1−α )/(A L − w 1−α ) depends on the contract. The latter can be expressed more precisely as θ(α, w μ ) = ( A H μ − 1 1−α w μ )/( A L μ − 1 1−α w μ ), but for brevity we will keep using the simple notation θ. Notice that A H μ = A H pA H +(1−p)A L = r pr+1−p , A L μ = A L pA H +(1−p)A L = 1 pr+1−p , and the horizontal axis of the (α, w μ ) plane satisfies θ =r. The next theorem describes the nonleakage region in the cases r≥ 3, 23+p 11+p ≤r < 3, and 1<r < 23+p 11+p , corresponding to high, medium, and low demand variations, respectively. 22 Theorem 1.5.4. Given the demand variation r, the nonleakage equilibrium exists if w μ lies in one of the subregions defined in Table 1.2. Each subregion is determined by a set of lower and upper bounds on w μ defined in Table 1.3. Table 1.2. Nonleakage Region r Subregions Lower Bounds Upper Bounds Illustration (a) r≥ 3 max(LB S1 L− , LB S1 L+ ) min(UB H , UB P ) Figure 1.2 (a) (b) 1<r < 23+p 11+p i LB S2 L− min(UB S2 L− , B FB ) Figure 1.2 (b) ii max(LB S2 L+ , B FB ) min(UB S2 L+ , UB H ) i max(LB S1 L− , LB S1 L+ , B θ=3 )min(UB H , UB P ) (c) 23+p 11+p ≤r < 3 ii LB S2 L− min(UB S2 L− , B FB , B θ=3 ) Figure 1.2 (c)&(d) iii max(LB S2 L+ , B FB ) min(UB S2 L+ , UB H , B θ=3 ) Table 1.3. Definition of the Bounds LB S1 L− :( 3 pr+1−p +2) α(1−α) 12+5α LB S1 L+ :( 3 pr+1−p −2)(1−α) LB S2 L− : (72r−24p+4α+24pr−10pα+21rα+10prα−12)−6 √ δ pr+1−p α(1−α) (12+5α) 2 LB S2 L+ : θ−r θ−1 1−α pr+1−p UB S2 L− : (72r−24p+4α+24pr−10pα+21rα+10prα−12)+6 √ δ pr+1−p α(1−α) (12+5α) 2 UB S2 L+ : θ−r θ−1 1−α pr+1−p UB H : ( 3r pr+1−p +1) α(1−α) 12+4α UB P : 1 2 ( 3 pr+1−p −1)(1−α) B FB : α(1−α) 3+α B θ=3 : 3−r 2 1−α pr+1−p δ : (24α−108r−12pα−54rα+12prα+6α 2 −5pα 2 −6rα 2 +5prα 2 +36)(1−r) θ : 2(6−3 √ p−11p+2p 2 ) 9−24p+4p 2 , if p∈ [0,0.4019]; 1, if p∈ [0.4019,1]. θ : 2(6+3 √ p−11p+2p 2 ) 9−24p+4p 2 , if p∈ [0,0.4019]; 2(6−3 √ p−11p+2p 2 ) 9−24p+4p 2 , if p∈ [0.4019,1]. The bound B FB specifies the (α, w μ ) pairs under which the total quantity Q ordered by the retailers is first-best for the supplier, as discussed in Proposition 1.4.3, and the bound B θ=3 specifies those (α, w μ ) that satisfy θ = 3. The upper bound UB P corresponds to the participation constraint w μ ≤ 1 2 ( 3A L μ − 1)(1− α), and UB H corresponds to the condition q N ∗ iH ≥ ¯ q i in Theorem 1.5.3. 6 The lower bound LB S1 L− is derived from the condition q S1 ∗ iL ≤q i in Theorem 1.5.3 when θ≥ 3, and the lower bound LB S2 L− and upper bound UB S2 L− are both obtained from the condition q S2 ∗ iL ≤ q i when θ < 3; the “-” sign in the subscript of a bound 6 Theupperbound(line)UB P hasnoimpactonthenonleakageregionifitissteeperthantheupperbound (curve) UB H at α = 1, i.e.,− 1 2 ( 3 pr+1−p −1)≤ [( 3r pr+1−p +1) (1−α)α 12+4α ] ′ α=1 , or r≤ 5+3p 1+3p , which is implied by r≤ 23+p 11+p in case (b). Also, UB P has no impact in cases (c.ii) and (c.iii), because (r−1)(1−p)≥ 0 implies UB P ≥B θ=3 . 23 indicates that the bound is only applicable when it lies below B FB . The bounds LB S1 L+ , LB S2 L+ , and UB S2 L+ have similar interpretations, except that they are only applicable above B FB . As discussed after Proposition 1.5.1, q i has different expressions for w μ ≤ B FB and w μ >B FB . In case (a), where r is relatively large, we have θ≥ 3 in the entire region of (α, w μ ) and hence the condition q S1 ∗ iL ≤q i applies. Notice that the bound LB S1 L+ lies below the horizontal axis (and hence the bound B FB ) if p > 1 2(r−1) , as in Figure 1.2 (a). The upper bound UB Nec corresponds to the necessary condition w μ ≤ ( 3A H μ +2) α(1−α) 12+5α in Theorem 1.5.3. In case (b), when r is small, we have θ < 3 and hence the condition q S2 ∗ iL ≤ q i is in effect. Subregions (i) and (ii) correspond to the scenarios w μ ≤ B FB and w μ ≥ B FB , respectively. In the example illustrated in Figure 1.2 (b), UB S2 L− > B FB > LB S2 L+ for all α ∈ [0,1] and hence the nonleakage region is given by LB S2 L− ≤ w μ ≤ min{UB S2 L+ ,UB H }. In case (c), for medium r, the region is obtained from those defined in cases (a) and (b) by imposing the conditions θ ≥ 3 and θ < 3, respectively. Note that in subregion (i), when p < 0.25 (as in Figure 1.2(d)), LB S1 L+ dominates B θ=3 and the nonleakage region may consist of two disjoint pieces: LB S1 L+ ≤ w μ ≤ UB H and LB S2 L− ≤ w μ ≤ B FB (the subregion (iii) does not exist in this example because the upper bound UB S2 L+ lies below the horizontal axis). It is clear from the figures that the nonleakage region (i.e. the range of wholesale prices that support nonleakage) is relatively wide when α lies in the middle of the interval [0,1] and it shrinks as α moves toward 0 or 1. Intuitively, when α is close to 0, the supplier’s profit comes mainly from selling to the retailers, and hence a larger order quantity is better for the supplier. As a result, the supplier is tempted to leak the high demand information. In the extreme case α = 0, there is no nontrivial revenue sharing contract (with w > 0) that can prevent the supplier from leaking the high demand information. This is consistent with the result in Anand and Goyal (2009) that the supplier always leaks under a wholesale price contract (with α = 0). When α increases, the supplier’s profit is more in line with the 24 Figure 1.2. The nonleakage region for: (a) r ≥ 3, (b) 1 < r < 23+p 11+p , (c) 23+p 11+p ≤ r < 3, p≥ 0.25, and (d) 23+p 11+p ≤r < 3, p<0.25. 0 . 2 0 . 4 0 . 6 0 . 8 1 0 . 1 0 0 . 1 0 . 2 B F B B F B L B L ! S 1 U B H U B P U B N e c B F B L B S 1 L + 0 . 2 0 . 4 0 . 6 0 . 8 1 E 0 . 1 0 0 . 1 0 . 2 L B S 2 L U U B S 2 L [ L B S 2 L + U B H U B S 2 L + B F B 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 1 0 . 2 0 . 3 L B S 1 L U B H L B S 2 L U B S 2 L B θ = 3 U B S 2 L + B F B 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 1 0 . 2 0 . 3 L B S 1 L ´ L B S 1 L + U B H L B S 2 L U B S 2 L B θ = 3 B F B 25 supply chain profit and she is more willing to control the total quantity in the channel by hiding the demand information from the entrant (recall the discussion at the beginning of Section 1.5). However, as α approaches 1, the feasible range of w that induces the retailers’ participation diminishes, resulting in a narrow nonleakage band near α = 1. The nonleakage region depends on two parameters, r and p. Their impacts are investi- gated in detail in Appendix A.3 and a brief discussion is given here. According to Theorem 1.5.3, the nonleakage region is determined by the relationships q N ∗ iH ≥ ¯ q i and q S ∗ iL ≤ q i . As r increases, A H μ increases and A L μ decreases. Thus, the incumbent’s order quantity q N ∗ iH in- creases while q S ∗ iL decreases, and both are more likely to fall into the supplier’s nonleakage intervals, causing the nonleakage region to expand. Intuitively, when the demand variation increases, the incumbent’s information advantage exacerbates the quantity distortion from the supplier’s perspective (Proposition 1.4.3 (iii)) and prompts the supplier to prevent in- formation leakage. As p increases, both A H μ and A L μ decrease, which relaxes the condition q S ∗ iL ≤q i for the lower boundary yet tightens the condition q N ∗ iH ≥ ¯ q i for the upper boundary. Thus, both boundaries shift downward and the net impact is not transparent, depending on whether p≥ 1 2(r−1) or p< 1 2(r−1) . 1.6 Optimal Wholesale Price for the Supplier In the previous section, we characterized the range of revenue sharing contracts that sustain nonleakage. Since contractparameters areprovided by thesupplier, wenow explore whether the supplier has an incentive to offer a leakage-proof contract. We allow the supplier to optimally choose the wholesale price w given a revenue sharing rate α, i.e., w becomes an endogenousdecision variableforthesupplier. 7 Wedemonstratethroughexamples thatwhen 7 If the contract parameters α and w are both endogenized, our results show that α = 1 and w = 0 is the unique optimal revenue sharing contract for the supplier (see Figures 1.3 and 1.4). This specific result does not provide any substantial managerialinsights as it suggests that the supplier should effectively control the entire channel. 26 αislargeenoughthewholesalepricethatmaximizesthesupplier’sprofitinducesnonleakage. Thissuggeststhatnonleakageisarobustoutcomeundercertainrevenue sharingcontractsas thesuppliercannotdobetterbyalteringthecontract(thewholesaleprice,morespecifically). First, we identify the supplier’s optimal wholesale prices in the leakage and nonleakage regions for a given α. Theorem 1.6.1. Assume r ≥ 7+2p 3+2p . Given the revenue sharing rate α, (i) if the supplier does not leak information, her optimal wholesale price is ( w μ ) N ∗ = (1−α)(3−2α) 6−2α , and (ii) if the supplier leaks information and θ ≥ 3, her optimal wholesale price is ( w μ ) S ∗ = (α−2)(α−1) 4−α . If the w μ defined in case (i) or (ii) above does not belong to the interior of the corresponding nonleakage or leakage region, the optimal wholesale price in that region, given α, lies on the boundary of the region. According to Lemma A.1.1 (in the proof of Theorem 1.6.1), under leakage, the supplier’s profit is larger when θ ≥ 3 than when θ < 3. Thus, to show that the supplier’s profit is larger under nonleakage than under leakage, it is sufficient to focus on the simpler leakage case θ≥ 3. It can be shown that, if unrestricted, ( w μ ) N ∗ and ( w μ ) S ∗ decrease with α. The exam- ple below demonstrates that there exists a threshold α above which the supplier’s optimal wholesale price w μ lies in the nonleakage region. Example 1.6.1. Assume r = 4 and p = 0.2. Let α N ∗ and α S ∗ be the revenue sharing rates at which the (locally) optimal wholesale prices ( w μ ) N ∗ and ( w μ ) S ∗ intersect the upper boundary of the nonleakage region. Then, (1−α)(3−2α) 6−2α = ( 3r pr+1−p +1) α(1−α) 12+4α implies α N ∗ = 0.623, and (α−2)(α−1) 4−α = ( 3r pr+1−p +1) α(1−α) 12+4α implies α S ∗ = 0.688. It can be shown that ( w μ ) N ∗ and ( w μ ) S ∗ do not intersect the lower boundary of the nonleakage region. The (locally) optimal wholesale prices ( w μ ) N ∗ and ( w μ ) S ∗ are illustrated in Figure 1.3. The shaded area in the figure represents the nonleakage region depicted in Figure 1.2 (a). 27 Figure1.3. Optimalwholesalepricesintheleakageandnonleakageregionsgivenα,forr =4 and p= 0.2. 0.2 0.4 0.6 0.8 1.0 0 0.1 0.2 LB S1 L− UB H UB P revenue sharing rate α scaled wholesale price ω μ Optimal ω μ in the Non-leakage Region Optimal ω μ in the Leakage Region α N* α S* Example 1.6.2. The monotonicity of ( w μ ) N ∗ and ( w μ ) S ∗ imply that ( w μ ) N ∗ lies within the nonleakage region when α ≥ 0.623 and ( w μ ) S ∗ lies within the leakage region when 0 ≤ α ≤ 0.688; for other ranges of α, the optimal prices lie on the upper boundary of the nonleakage region ( 3r pr+1−p + 1) α(1−α) 12+4α . Given the optimal prices, the supplier’s corresponding expected profits π N ∗ s and π S ∗ s can be determined, as in the proof of Theorem 1.6.1. The supplier’s maximum expected profits ˜ π N ∗ s =π N ∗ s /μ 2 and ˜ π S ∗ s =π S ∗ s /μ 2 (normalized by μ 2 ) in the leakage and nonleakage regions are depicted in Figure 1.4 (a), as functions of α. Because ˜ π N ∗ s ≥ ˜ π S ∗ s when α ≥ 0.51, the optimal revenue sharing contract (α, w ∗ μ ), given α, resides in the nonleakage region when α≥ 0.51. Figure 1.4 (b) shows that the incumbent is always better off under a leakage-proof revenue sharing contract with the wholesale price optimally chosen, while the entrant may or may not be better off, depending on the revenue sharing rate. To compare the revenue sharing contracts with the wholesale price contract studied by Anand and Goyal (2009), we list the order quantities and all parties’ profits under various α in Table 1.4. To reveal the impact of the wholesale price w, we consider two w μ under each α, i.e., ( w μ ) N ∗ and ( w μ ) S ∗ , the optimal w μ in the “N”onleakage and “S”eparating (leakage) regions, respectively. 28 Figure 1.4. Firm profits under optimal wholesale prices in the leakage and nonleakage cases, for r = 4 and p =0.2. 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 1 0 . 2 0 . 3 0 . 4 ( a ) S u p p l i e r ' s p r o f i t 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 1 0 . 2 ( b ) I n c u m b e n t a n d e n t r a n t ' s p r o f i t s Table 1.4. Quantities and Profits Under Various (α, w μ ) for r =4, p= 0.2 α 0 (benchmark) 0.3 0.6 0.9 w/μ 0 0.500 0.1350.322 0.142 0.165 0.029 0.049 Equilibrium N S N S N S N S q iH /μ 1.083 1.000 1.0191.020 0.965 1.044 0.988 1.005 q eH /μ 0.333 0.500 0.2690.510 0.215 0.522 0.238 0.502 Q H /μ 1.417 1.500 1.2881.530 1.181 1.566 1.226 1.507 Q FB H /μ ∞ ∞ 1.4751.786 1.368 1.387 1.266 1.277 π iH /μ 2 1.174 0.500 0.7270.364 0.373 0.218 0.098 0.050 π eH /μ 2 0.361 0.250 0.1920.182 0.083 0.109 0.024 0.025 π sH /μ 2 0 0.750 0.6420.937 1.102 1.135 1.441 1.421 π Total H /μ 2 1.535 1.500 1.5611.484 1.558 1.463 1.562 1.496 q iL /μ 0.146 0.063 0.0810.083 0.028 0.107 0.051 0.067 q eL /μ 0.333 0.031 0.2690.041 0.215 0.053 0.238 0.034 Q L /μ 0.479 0.094 0.3500.124 0.243 0.160 0.289 0.101 Q FB L /μ ∞ ∞ 0.5380.849 0.431 0.450 0.328 0.340 π iL /μ 2 0.021 0.002 0.0050.0020.00030.0020.00030.0002 π eL /μ 2 0.049 0.001 0.0150.001 0.002 0.001 0.001 0.0001 π sL /μ 2 0 0.047 0.0760.059 0.090 0.071 0.096 0.053 π Total L /μ 2 0.070 0.050 0.0960.062 0.093 0.074 0.097 0.053 29 As can be seen, the first-best channel quantity decreases with α, under both ( w μ ) N ∗ and ( w μ ) S ∗ , in both demand states. The retailers always underorder under ( w μ ) N ∗ , and they underorder under ( w μ ) S ∗ except in the high demand case with α = 0.6 or 0.9. Under nonleakage, the quantity distortion (weakly) decreases with α, in both demand states. Under leakage, this is not the case in the high demand case. The quantity distortion is smaller under nonleakage than under leakage except in the high demand case with α = 0 or 0.6. The supplier’s benefit from nonleakage steadily increases with the revenue sharing rate α, as the difference between her profits under nonleakage and leakage varies from−0.750 to −0.295, then−0.034, and finally 0.020, when the demand is high, and from−0.047 to 0.017, 0.019, and 0.043, when the demand is low. The supply chain always benefits from nonleakage in both demand scenarios, and the incumbent and the entrant are better off under nonleakage in most scenarios. As another example, when r = 2.5 and p = 0.3, we find that α N ∗ = 0.789, α S ∗ = 0.898, ˜ π N ∗ s ≥ ˜ π S ∗ s if α ≥ 0.72, and interestingly, both the incumbent and the entrant are better off under a leakage-proof revenue sharing contract. Comparing the two examples, we see that the threshold α at which the supplier starts to prefer nonleakage moves to the left as r increases. That is, as r increases, the demand states are more distinguishable, and a smaller shareofrevenueisneededtopersuadethesuppliernottoleak(especiallyinthehighdemand case). This is consistent with the sensitivity analysis of the nonleakage region with respect to r (Appendix A.3). 1.7 Extensions Inthissection,weextendourresultsbyalteringsomeassumptionsinthemodel. Wefindthat these extensions do not change the main results of the paper but may alter the nonleakage region in different ways. We summarize the main findings here; the details of the analyses and examples are provided in Appendix A.4. 30 1.7.1 Free Disposal by the Incumbent In previous analysis, we implicitly assumed that the incumbent sells all units ordered from thesupplieranddoesnotwithholdanyunitsfromthemarket. Nowwerelaxthisassumption and explore how it impacts the nonleakage region. Free disposal cannot benefit the incumbent if the type of equilibrium is unchanged. If a leakage game is played, the supplier and entrant know the incumbent’s order quantity and can calculate his selling quantity in equilibrium if it differs from his order. Thus the incumbent has no reason to purchase more than his actual selling amount. Similarly, if a nonleakagegameisplayed, theincumbent’sorderquantityisunknowntotheentrantandthe tworetailers’sellingquantitiesarefullydeterminedbytheequilibrium; again,theincumbent cannot gain from free disposal. The free disposal option can affect the type of equilibrium in two ways. (1) If q i < q N∗ iH < ¯ q i , theincumbent may persuade (or“bribe”)thesupplier nottoleak thehighdemand informationbyordering ¯ q i (butsellinglesslater),whichmayenableanonleakageequilibrium that is nonsustainable without free disposal. (2) If q S∗ iL < q i , the incumbent may stimulate the supplier to leak the low demand information by ordering q i (but selling less), which would topple an original nonleakage equilibrium; when free disposal is possible, a deviation to leakage may become more attractive to the incumbent. The first effect enlarges the nonleakage region by pushing its upper boundary upward (as q N∗ iH ≥ ¯ q i is no longer required), while the second effect reduces the region by raising its lower boundary (as q S∗ iL ≤ q i may not be sufficient any more). In Appendix A, we conduct an analysis similar to the case without free disposal analyzed in previous sections. 8 The main finding is that the first effect is more prominent and, as a result, the nonleakage region 8 The nonleakageand leakageequilibria under free disposaldiffer from those without free disposalbecause the incumbent’s order cost is sunk after ordering when free disposal is possible while it is linked to the incumbent’s selling quantity (which is also his order quantity) when free disposal is disregarded. 31 Figure 1.5. Nonleakage region with free disposal, for r = 3.5, p = 0.2. (Notes The darkly shaded area represents the original nonleakage region without free dis- posal. The lightly shaded area represents the region in which nonleakage is enabled by the incumbent’s free disposal option in the high demand case. The bound UB FD is given by w μ = ( 3r pr−p+1 +1) α(1−α) 12+(p−5)α .) 0.2 0.4 0.6 0.8 1.0 0 0.1 0.2 0.3 0.4 LB S2 L− UB H UB P UB FD revenue sharing rate α scaled wholesale price ω μ expands under typical model parameters; the second effect has no impact when θ < 3 and limited impact when θ≥ 3. An example is shown in Figure 1.5. 1.7.2 Active Entrant Who Can Reject Information Inthebasic model, theentrant playsapassive roleandalways accepts theincumbent’s order informationleakedbythesupplier. Becausethenonleakagegamemaybenefittheentrant, he may want to reject the information and drag the incumbent into a nonleakage game (given that the entrant will ignore the incumbent’s order information, the best order quantity for the incumbent is the one in the nonleakage equilibrium). Thus, an active entrant can turn a leakage game into a nonleakage one, whereas an original nonleakage game is unaffected because there is no information in the first place. As a result, allowing an active entrant would strictly enlarge the nonleakage region. More details can be found in Appendix A.3. 32 1.7.3 Simultaneous Ordering In our original model, the incumbent enjoys two potential advantages that are intertwined together: an information advantage due to his private demand information, and a first- mover advantage in the leakage game. In this extension, we disentangle these two effects by studying the following game: first, the incumbent decides whether to share the demand information (high or low) with the supplier, then the supplier decides whether to leak it to the entrant, and finally the incumbent and entrant place orders simultaneously. This new game takes away any potential first-mover advantage enjoyed by the incumbent and allows us to focus on his informational advantage. We assume that the incumbent cannot lie if he should decide to disclose the demand information. Otherwise the incumbent would always report low demand and there would notbeaninterestinggame. Becausehewouldalwaysliketheentranttolearntheinformation when the demand is low to restrain the entrant’s order quantity, the demand information is effectively always revealed to the supplier, and it is up to the supplier whether to leak it to the entrant or not. If the supplier does not leak, the game will be the same as the originalnonleakagegameandthe equilibrium orderquantities aregiven inProposition1.4.1. If the supplier leaks the information, the incumbent and entrant will play a simultaneous Cournot gameunder perfect information. Foranonleakage equilibrium toexist, thesupplier should prefer nonleakage in both demand scenarios. As shown in Appendix A, nonleakage is sustainable in the region α(1−α) 3+α A L +μ 2μ ≤ w μ ≤ α(1−α) 3+α A H +μ 2μ , in which the channel quantity distortionfromthesupplier’spointofviewislesssevereundernonleakagethanunderleakage in both demand states. This new nonleakage region is qualitatively similar to the nonleakage region found under thebasicmodel. Bothcontainthefirst-bestcurveB FB : w μ = α(1−α) 3+α andthusoverlap around that curve. We conclude that asking the incumbent to share information first and the two retailers to order simultaneously does not alter the key findings from the basic model. 33 1.7.4 Entrant Ordering First Wehaveshowninthepaperthataproperlydesignedrevenuesharingcontractcandiscourage thesupplierfromleakingtheincumbent’sorderinformationtotheentrant. Alternatively,the supplier can commit not to leak the incumbent’s private information by taking the entrant’s order first. In this extension, we consider a new game in which steps 3 to 5 in the original sequence ofevents (Section1.3)arereplaced by thefollowing: the entrant places anorderq e ; the supplier decides whether toleak ittothe incumbent ornot; andthe incumbent places an order q i . Because the entrant has no private information, his order quantity can be inferred by the incumbent in any equilibrium. Thus, leaking the entrant’s order quantity provides no extra information to the incumbent but it makes the entrant the Stackelberg leader in the retailers’ game. If the supplier does not leak the entrant’s order quantity, the two retailers play the same nonleakage game as under the basic model when the incumbent orders first. A thorough analysis of the new game is provided in Appendix A.3. The equilibrium of the game, under a given revenue sharing contract (α,w), can be summarized as follows: if w μ < α(1−α) 3+α , the supplier would not leak the entrant’s order information and the two retailers effectively play a simultaneous game; if α(1−α) 3+α ≤ w μ ≤ 5α(1−α) 12+5α , the supplier would be indifferent between leaking and not leaking the entrant’s order quantity; and if w μ > 5α(1−α) 12+5α , the supplier would leak the entrant’s order information and appoint him as the Stackelberg leader. Comparing the new game with the original one, we find that the supplier weakly prefers the entrant to be the first mover in all the above regions. In the intersection of the region w μ ≤ 5α(1−α) 12+5α and the original nonleakage region, the supplier would be indifferent to who orders first. The above result seems to suggest that the supplier would be better off taking the order from the uninformed retailer first. However, there are potential issues with this sequence of events. First,byholdingbacktheinformedretailer,thesupplierforfeitsallpotentialbenefits from advance information about the market discussed in the introduction and literature 34 review (which are not included in our model). Recall that the motivation of this paper was toenable informationsharingbetween the incumbent andthesupplier by preventing leakage to the entrant. Second, whether the supplier can actually dictate the order sequence and force the incumbent to order second is an open question. The incumbent can voluntarily share the demand information with the supplier or even go ahead and place an order and it may be difficult in practice for the supplier to ignore this action and force the incumbent to react to the entrant’s order. 1.8 Conclusion Information sharing between parties in a supply chain has many benefits and thus we have witnessed many efforts to do so in the past decade. But many firms are reluctant to share information due to the negative effects on their revenues and profits from potential leakage of sensitive information. Inthiswork,weexplorethepotentialofrevenuesharingcontractsinpreventingleakageof demand information in a supply chain and making the supply chain better off. We modeled the interactions between the supplier and its two clients as a dynamic game with incomplete information. Several key insights were obtained from our analysis. The incumbent reveals the demand information to the supplier through his order quantity. To prevent the supplier from choosing to leak the information, the incumbent should order a quantity that would actually lead to an under-order or over-order situation if the information gets leaked. As a result, information sharing between vertical partners is enabled yet information leakage to horizontal competitors is prevented. Our examples show that preventing leakage may make the incumbent, supplier and sometimes even the entrant better off. Our results also suggest that a revenue share percentage that is relatively high is more capable of preventing leakage. This is consistent with studios’ revenue share percentages (40-60%) in the video rental industry as well as their share of revenues with movie theaters (Danaand Spier 2001). 35 In this scenario, the supplier may even be able to choose a wholesale price that maximizes her profits while preferring not to leak information. Further, we have established that a revenue sharing contract can be robust in its ability to prevent information leakage under various extensions. There are many interesting avenues forfuture research on this issue. A natural extension of our paper is to study leakage–proof contracts under Bertrand competition, or when the products provided by the downstream retailers are imperfect substitutes or complements. Second,weassumeinthispaperthatthesuppliercanchoosewhethertoleaktheincumbent’s order quantity or not but cannot lie about it. It would be interesting to study the situation in which the supplier may distort such information and design contracts that still prevent information leakage. 36 CHAPTER 2 SERVING HETEROGENEOUS BUYERS THROUGH PRICE AND QUALITY DIFFERENTIATION 2.1 Introduction Professional services involve a high degree of information asymmetry: a service provider has little information about a buyer’s valuation ex ante. Conversely, the buyer has limited ability to verify a service provider’s effort ex post. The two sided information asymmetry induces strategic behavior by both service providers and service buyers and can result in inefficiencies and less desirable outcomes. For example, the service provider may be unable topricetheserviceinamannerthatadequatelycaptureswhatsomecustomersmaybewilling to pay for the service. Conversely, the outcome of professional service is often characterized by uncertainty and the effort is not directly observable. So, service providers may tend to strategically report their effort in order to maximize their own profit. One example is professional website design. A website designer potentially may serve different customers with different valuations for the same quality of output. Large corpora- tions such as Walmart may expect and be willing to pay more for a higher quality website than a small retailer. For instance, user friendliness and various features such as customer reviews, ability toorder onlineforin-store pickup and in-storestock availability will be more important to Walmart. Walmart may incur a huge loss if the website is not attractive or malfunctions unlike a small business owner. A website designer may have to devote a lot of time to make it more attractive and to avoid malfunctions. For these reasons, Walmart may value a high quality website much more than a small business owner. But a web designer cannot explicitly distinguish between buyer’s types prior to contracting. A web designer has 37 to price the service appropriately in order to satisfy different buyers and extract as much revenue as possible. When the outcome of a service is observable and a service provider’s effort determines service quality deterministically, we have an adverse selection problem in which a service provider differentiates among buyer’s valuations by providing different quality of service. Services such as home cleaning and SPA massage represent such types of service. However, in many professional services such as website design and programming, service quality is stochastically increasing with service effort and the quality may not be fully controlled by a service provider (SP). Another example is taxconsulting, where aSP cannot commit tosave a certain amount of taxes or provide a guarantee that the client will not be audited. These depend on the specific characteristics of the client’s tax situation and IRS policies which are not fully within the control of the tax consultant. We assume that in such situations quality is observable but not contractible. This assumption is common in many papers in the literature (Hermalin and Katz. (1991) and in practice. For example, while a web design firm and its clients can distinguish between high and low quality websites, they would not contract on the quality. This is partly because quality is in the eye of the beholder – two customers may perceive the same website to be of different quality. These aspects bring about a double information asymmetry in pricing of a service. A buyer’s valuation type is not known to a service provider and a service provider’s effort is not directly observable. We provide an analytical model to characterize the issues and tradeoffs and explore a service provider’s optimal pricing and operational strategy under different contracts: fixed fee, time based fee and two part tariff. Such contracts are common for professional services and are the most frequently used pricing schemes at online platforms for professional services such as eLance and oDesk. Our analysis of the contracts is for two scenarios: in one, service quality is a deterministic function of the service provider’s effort and in the second, service quality is a stochastic function of the service provider’s effort. We refer to the first case 38 as the public effort (or deterministic quality) case and the second case as hidden effort (or stochastic quality) case. In the case of public effort (deterministic quality), we find that fixed fee cannot differ- entiate between customers based on their valuation. A service provider either serves both types with the same effort or only serves one type. Time based payment is effective in differentiating customers to a certain degree. While two part tariff dominates both fixed fee and time based payment schemes, it does not always achieve the social optimal solution as it gives rent to one type of buyer. This is similar to the adverse selection problem in product design except that we also consider the case when high type customer may value a low quality service even lower than the low type customer. Therefore, we show that there exists a scenario in which two part tariff achieves a social optimal solution. In the case of hidden effort (stochastic quality), a service provider could strategically misreport the effort as a buyer cannot identify the true effort to a certain degree. The effects of one sided information asymmetry (adverse selection) on a service provider are well known. We explore how the other side of information asymmetry impacts the service provider, viz. how it affects the SP’s ability to discriminate among buyers and whether it reduces or increases his profits. It turns out hidden effort does not change the SP’s profits under fixed fee and two part tariff. Under a fixed fee contract, a service provider still cannot differentiate buyers based on their valuation and hidden effort does not have any impact because the service provider treats buyers as the same type. Hidden effort only has an effect when a service provider serves different types of buyers with varying effort levels. However, eventhoughatwo-parttariffprovidesdifferenteffortlevelstodifferentbuyertypes,theprofit doesn’t change under hidden effort even though the contract parameter changes, because a two-part tariff already extracts the surplus as much as possible. Hidden effort only has an impact under time based payment. Our finding is hidden effort reduces the profit of a high cost service provider because he cannot charge a high price rate. On the other hand, hidden 39 effort improves the profit of a low cost service provider because it gains the flexibility to strategically misreport its actual effort. In addition, we identify the region in which time based payment achieves the same profit as two part tariff. The paper is organized as follows: Section 2.2 provides a literature review. Section 2.3 develops the basic model characterizing the service provider’s problem and the buyer’s util- ity and constraints. Section 2.4 studies the service provider’s problem when service effort deterministically determines service quality under three different contracts (pubilic effort). Section 2.4 studies the service provider’s problem when service effort stochastically deter- mines service quality under three different contracts (hidden effort) and compares service provider’s profit under hidden with versus public effort. 2.2 Literature Review Our model studies professional service outsourcing under adverse selection (buyers vary in theirvaluationoftheservicequality)andhiddeneffort(serviceprovider’seffortisnotdirectly observable). The literature most closely related to our work considers service contracting with information asymmetry. Sundararajan (2004) analyzes the optimal pricing strategy for an information good with heterogeneous customers under fixed fee, usage-based pricing and two part tariff. In this work, the consumer decides the consumption level based on his valuation and the payment scheme, unlike in our work where it is the service provider who induces customer’s consumption level by offering different service levels at different prices. Sundararajan (2004) is appropriate for packaged information goods while our work is ap- propriate for professional services. Akan et al. (2011) consider information asymmetry in service outsourcing where the outsourcing firm has private information about its demand volume. The authors prove that a menu of two-part tariffs achieves the full-information solution, as we show in one of our scenarios. Ren and Zhang (2009) examine service out- sourcing contracts when the service provider’s capacity cost is private information. Cachon 40 and Zhang (2006)also consider a contract design problem where a buyer sources a good or service fromasupplier whose capacity costisprivateinformation. Inalloftheabovepapers, the service effort is considered tobe observable and contractible. Incorporatingboth buyer’s heterogeneous utility and service provider’s hidden effort is the main feature that distin- guishes our paper from these and other works in service outsourcing. Roels et al. (2010) analyzes the contracting issues that arise in collaborative services and investigate how the choiceofcontracttypeamongfixedfee,timeandmaterials,andperformancebasedcontracts is driven by service characteristics. Similar to our paper, this paper assumes that quality is observable but effort is unverifiable and output is uncertain. Unlike our paper, the paper does not consider buyer’s heterogeneous utility. Duetotheheterogeneityinbuyer’svaluation,ourpaperisalsorelatedtotheliteraturein product design and segmentation with adverse selection. Moorthy (1984) develops a theory of market segmentation based on consumer self-selection when consumers are of two types withdifferentvaluationsofquality, asinourwork. Heshowsthatconsumerself-selection has significantimplicationsforhowproductquality(andprices)arechosenbyamonopolist. Our work extends this framework to a service setting with unique features that are not discussed in the product design literature. In service outsourcing, the production of the service only occurs after the contract is signed and service provider’s effort is not necessarily observable. So it is possible that a service provider may misreport his effort to a buyer and the service provider’s strategic behavior may influence a buyer’s choice ex-ante. There is also a literature, in economics and operations management (Liu, T., (2011), Fong, Y. (2005), Emons, W. (1997), Debo et al. (2008) that considers expert services as credence goodsandstudy service provider’s fraudulent behavior. These works assume thata serviceprovider(andnotthecustomer)determinestheservicetimeoreffort,asinourpaper. However two critical assumptions in this literature are that service outcome (or quality) has only two states: a service problem has been solved or not, and service effort is observable. 41 The only way that a service provider cheats is over-treatment. In our paper, we assume that quality is increasing (perhaps stochastically) with a service provider’s effort and the buyer has knowledge of the service quality after the service is provided. Unlike Debo et al. (2008),service provider’s effort in our setting is not directly observable. As a result, we proposes another possible way that a service provider cheats: overcharge by over-reporting the effort. In addition, since we assume that a buyer can value service quality after the service, the incentive compatibility constraints should not only be satisfied ex ante but also ex post. There is also a literature in Operations that considers the relationship between service time and quality (Chase (1981), Karmarkar and Pitbladdo (1995), and Hopp et al. (2007). These papers consider service effort as a “discretionary task” and the quality of a service increases with service provider’s effort. Anand et. al (2011) consider the tradeoff between quality and speed in a queuing framework. and Kostami and Rajagopalan (2009) explore service quality and speed tradeoff in a dynamic model. Tong and Rajagopalan (2012) con- sider pricing and operational decisions when the service value is a function of service time and customers have heterogeneous valuations. Our research is different from these works in several aspects: first, we not only consider service quality as a deterministic function of service effort as in these works, but we also consider the scenario where service quality is stochastically increasing in the service provider’s effort. The stochastic relation between service effort and service quality may induce the service provider strategically report service effort. Second, these papers assume that buyer’s valuation are homogeneous or and are known to the service provider. We assume thatvaluations areheterogeneous and are private information. Third, these papers model high volume service systems with congestion and wait time unlike in our paper. There is a vast literature on service systems in operations management that model them as queuing systems and consider measures of quality such as wait time or percentage of 42 services (e.g. calls in a call center) resolved. Cachon and Zhang (2006) consider fast lead time as a measure of high quality when a supplier exerts the effort to invest in capacity. Bitran etc. (2008) ,Cachon and Feldman (2011), and Akan et al. (2011)examine the impact of a service provider’s policies on pricing and service level on a firm’s profitability. Waiting cost is considered as a measure of service provide r’s service quality in these papers. In addition, Ren and Zhou (2008) models the call center as a multiserver queue with customer abandonment and study contract design issues in order to induce the system-optimal effort. The common feature in the above literature is that quality of service is not only observable but also contractible. In our paper, we assume that service quality is observable but not necessarily contractible. Contractible means both observable and verifiable, however in the examples of website design, a court would find it hard to determine whether the service provider designs a good website. In addition, none of the above papers consider information asymmetry which is the main focus of our work. Moreover, unlike most of the literature using a queuing framework in which lower service times imply lower wait time and therefore high quality, we consider a scenario where service quality is increasing with service providers effort (often equivalent to time spent) on the service. 2.3 The Basic Model with Public Effort (Deterministic Quality) We consider a Service Provider (SP) who provides a professional service whose service qual- ity increases with the SP’s effort. Potentially the SP is facing buyers with heterogeneous valuations of the same quality of service. We assume that there are two types of buyers, either “high” or “low”. The proportion of high type buyers is λ and the proportion of low type buyers is 1−λ. Buyers and the SP’s utility function is defined in Sections 3.1 and 3.2. 2.3.1 Buyer’s Utility In the public effort (deterministic quality) case, the SP’s effort is equivalent to the quality of the service. We assume that buyer’s valuation of the service v(t) linearly increases with 43 the SP’s effort (or quality) t, i.e. v(t) = θ x t + v x . A buyer’s type is either high or low, i.e. x = H or L. The information of a buyer’s type is not known by the service provider. If a service provider spends time (effort) t and charges price p, the high type buyer’s ex- pected utility is tθ H +v H −P if t≥t 0 0 if t<t 0 , and the low type buyer’s expected utility is tθ L +v L −P if t≥t 0 0 if t<t 0 , where θ H ≥ θ L . Hence, t 0 is the basic or initial service required to deliver any value to the buyer. Let ˜ t satisfy ˜ tθ H +v H = ˜ tθ L +v L , i.e ˜ t = v L −v H θ H −θ L , which is the effort at which a high type buyer and a low type share the same valuation. Figure 2.1 shows a high type and a low type buyer’s valuation against the SP’s time (effort). We make the following assumptions so as to focus on the most interesting and essential trade-offs in the problem: • Assumption (nonnegative values): v L ≥ 0, v H ≥ 0; • Assumption (negligible prior service): t 0 is sufficiently small. The last assumption allows us to disregard the boundary constraint on t in subsequent analysis. Also note that we do not require v H ≥ v L , so it is possible for the two value lines to cross. 2.3.2 Service Provider’s Utility The SP’s profit equals the payment received from buyers minus the cost of serving those buyers. If the SP serves only high type buyers, π s = λ(P −c(t)); similarly, if the SP serves only low type buyers, π s = (1−λ)(P −c(t)); and if the service provider serves both types of buyers, π s =λ(P H −c(t H ))+(1−λ)(P L −c(t L )), where c(t) =ct 2 is the cost incurred by devotingefforttforthebuyer. Dependingonthecontract,thepaymentP hasthreedifferent forms, fixed fee w, time based fee pt and two part tariff pt+a, which will be discussed in 44 Figure 2.1. Buyer’s valuation against with time Figure 2.2. Value and cost S e r v i c e T i m e V a l u e S e r v i c e T i m e V a l u e the next section. When the SP serves both types of buyers, P H represents the payment of high type buyers and P L represents the payment of low type buyers. The SP determines the service effort for buyers and the contract parameters. Under each contract, the SP has to ensure that a high type buyer does not want to pretend to be a low type (ICH) and a low type buyer does not want to pretend to be a high type (ICL), i.e. t H θ H +v H −P H ≥ t L θ H +v H −P L (2.3.1) t L θ L +v L −P L ≥ t H θ L +v L −P H (2.3.2) 2.3.3 Sequence of the Game The sequence of events is as follows: the SP first provides the contract (fixed fee, time based payment or two part tariff), a buyer decides whether to use, and if the contract consists of a menu of options, which option to choose; Then the SP exerts effort according to the chosen option and the quality is realized. Finally, the buyer makes the payment according to the prescribed contract. 45 2.4 Optimal Contracts in the Public Effort (Deterministic Quality) Case In this section, we focus on the case when the SP’s effort determines the service quality deterministically. It applies to a variety of services such as home cleaning, SPA message ets. Since the quality is observable and determined directly by the SP’s effort, a buyer can infer the SP’s effort precisely. It is not necessarily true when the SP’s effort determines service quality stochastically, which will be discussed in details in the next section. 2.4.1 Fixed Fee Contract-Deterministic Quality Under a fixed fee contract, a service provider charges a fixed payment P =w from a buyer. The payment depends neither on the service provider’s effort nor the buyer’s type. The following discussion shows that the SP either serves only one type of buyer or both types of buyers with the same effort, but cannot serve both types of buyers with different efforts. One Type Problem When the service provider prefers to serve only one type of buyer, she solves the following problem, for x =H or L, and−x ={H,L}\x: max w,tx λ x (w−ct 2 x ) (2.4.1) s.t. v x +θ x t x −w≥ 0, (2.4.2) v −x +θ −x t x −w < 0 (2.4.3) whereλ x istheproportionoftypex. TheIRconstraint(2.4.2)ensures thatthetypexbuyer would join and (2.4.3) ensures that the type−x buyer is excluded. Lemma 2.4.1. If θx(θ −x −θx) 2c ≤ v x −v −x , the optimal solution to problem (2.4.1)-(2.4.3) is t ∗ x = θx 2c , p ∗ =θ x + 2cvx θx , and the optimal objective value is π fb x =λ x (v x + θ 2 x 4c ). 46 Lemma 2.4.1 suggests that the service provider’s profit of serving only the high type is π H f =λ( θ 2 H 4c +v H ) if θ H 2c ≥ v L −v H θ H −θ L and the service provider’s profit of serving only the low type isπ L f =(1−λ)( θ 2 L 4c +v L )if θ L 2c ≤ v L −v H θ H −θ L . Figure2.3shows the scenario ofserving only the high type. Define t FB H = θ H 2c and t FB L = θ L 2c . They represent the efficient efforts for the high type buyers and low type buyers respectively since they optimize social benefit λ x (θ x t x +v x −ct 2 x ) for type x. When serving only one buyer type, the efficient effort for that type of buyers is achieved and it leaves zero surplus to that type but the SP incurs profit loss by driving away the other type of buyers. Figure 2.3. Serving only high type-fixed fee Figure 2.4. Serving both type-fixed fee S e r v i c e T i m e V a l u e S e r v i c e T i m e V a l u e Two Type Problem Analternativestrategyistochoose fixedpaymentw toserve bothtypes ofbuyers. However, SP cannotdifferentiate buyers based ontheir valuationand cannotserve them with different efforts. Lemma 2.4.2. Under any given fixed payment w, to serve both types of buyers, the SP must exert the same effort, i.e. t H =t L . Lemma 2.4.2 suggests that the SP needs to solve the following problem in order to serve both types of buyers: max w,t w−ct 2 (2.4.4) 47 s.t. v H +θ H t−w≥ 0, (2.4.5) v −H +θ −H t−w≥ 0, (2.4.6) Lemma 2.4.3. The service provider’s profit from serving both types of buyers is: (1) π f = θ 2 L 4c +v L with effort t H =t L = θ L 2c and payment w = θ 2 L 2c +v L , if θ L 2c ≥ v L −v H θ H −θ L ; (2) π f = θ 2 H 4c +v H with t H = t L = θ H 2c and payment w = θ 2 H 2c +v H , if θ H 2c ≤ v L −v H θ H −θ L ; or (3) π f = θ L (v L −v H ) θ H −θ L +v L − c( v L −v H θ H −θ L ) 2 with t H =t L = v L −v H θ H −θ L and w = θ L (v L −v H ) θ H −θ L +v L , if θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c . Figure 2.4 shows that the SP serves both types of buyers with the same effort when θ L 2c ≥ v L −v H θ H −θ L . The effort is efficient for a low type buyer (t FB L = θ L 2c ) but downward distorted for a high type buyer. It leaves zero surplus for a low type buyer but positive surplus for a high type buyer. Overall Optimal Solution CombiningtheresultsinLemma2.4.1andLemma2.4.3,theSPeitherservesonlyonebuyers type or serves both types with the same effort. In the the first case, the SP provides efficient effort to the type of buyer he serves but ignores the other type whereas in the second case, the SP provides efficient effort to one type of buyers (at most) but inefficient effort to the other type. Which cases above occurs depends on the proportion of high type and low type buyers in the market. Table 2.1 presents the SP’s profit in three scenarios: serve only the high type, serve only the low type and serve both types. The following proposition provides the SP’s optimal strategy in different parameter regions. Proposition 2.4.4. (i) If θ L 2c ≥ v L −v H θ H −θ L , the service provider serves only high type if λ ≥ θ 2 L +4cv L θ 2 H +4cv H , and both types other wise; (ii) If θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c , the service provider serves only high type if λ≥ max( θ 2 L +4cv L θ 2 H +4cv H +θ 2 L +4cv L , θ L (v L −v H ) θ H −θ L +v L −c( v L −v H θ H −θ L ) 2 θ 2 H 4c +v H ), 48 Table 2.1. The SP’s profit under fixed fee contract Equilibrium Regime θ L 2c ≥ v L −v H θ H −θ L θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c v L −v H θ H −θ L ≥ θ H 2c serve only high type (H) π =λ( θ 2 H 4c +v H ) t H = θ H 2c w = θ 2 H 2c +v H π =λ( θ 2 H 4c +v H ) t H = θ H 2c w = θ 2 H 2c +v H N/A serve only low type (L) N/A π = (1−λ)( θ 2 L 4c +v L ) t L = θ L 2c w = θ 2 L 2c +v L π = (1−λ)( θ 2 L 4c +v L ) t L = θ L 2c w = θ 2 L 2c +v L server both types (B) π = θ 2 L 4c +v L t L =t H = θ L 2c w = θ 2 L 2c +v L π = θ L (v L −v H ) θ H −θ L +v L −c( v L −v H θ H −θ L ) 2 t L =t H = v L −v H θ H −θ L w = θ L (v L −v H ) θ H −θ L +v L π = θ 2 H 4c +v H t L =t H = θ H 2c w = θ 2 H 2c +v H only low type if λ≤ min( θ 2 L +4cv L θ 2 H +4cv H +θ 2 L +4cv L ,1− θ L (v L −v H ) θ H −θ L +v L −c( v L −v H θ H −θ L ) 2 θ 2 L 4c +v L ), and both types otherwise; (iii) If v L −v H θ H −θ L ≥ θ H 2c , the service provider serves only low type if λ≤ 1− θ 2 H +4cv H θ 2 L +4cv L , and both types otherwise. If v L ≤ v H , only case (i) of Proposition 2.4.4 is valid. In that case, the shadow area in Figure 2.5 represents the region of (λ,c) in which serving only high type is a dominant strategy. Theshapeoftheareaisdifferentinthescenarios θ 2 L θ 2 H ≥ v L v H and θ 2 L θ 2 H ≤ v L v H . Intuitively, if λ is large enough, the SP’s profit is mainly from serving high type, as a result serving only high type is a dominant strategy since serving both types involves large inefficiency for high 49 type. Notice that when θ L 2c ≥ v L −v H θ H −θ L the SP charges a higher price and devotes more effort when it serves only high type buyer than it serves both types of buyers. Similarly, when v L −v H θ H −θ L ≥ θ H 2c , the SP charges a higher price but devotes less effort when it serves only low type buyer than it serves both types of buyers. When θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c , the SP charges a medium price and devotes medium effort compared with serving only one type of buyers. Figure 2.5. Optimal effort regime under fixed fee when θ 2 L θ 2 H ≤ v L v H Figure 2.6. Optimal effort regime under fixed fee when θ 2 L θ 2 H ≥ v L v H ܿ ߣ ߠ ଶ ߠ ு ଶ ܿ ߣ ߠ ଶ ߠ ு ଶ ݒ ݒ ு ݒ ݒ ு Ͳ Ͳ ߠ ଶ െ ߣ ߠ ு ଶ Ͷ ሺ ߣ ݒ ு െ ݒ ሻ S e r v e o n l y h i g h t y p e S e r v e b o t h t y p e s ߠ ଶ െ ߣ ߠ ு ଶ Ͷ ሺ ߣ ݒ ு െ ݒ ሻ ܿ ߣ ߠ ଶ ߠ ு ଶ ܿ ߣ ߠ ଶ ߠ ு ଶ ݒ ݒ ு ݒ ݒ ு Ͳ Ͳ ߠ ଶ െ ߣ ߠ ு ଶ Ͷ ሺ ߣ ݒ ு െ ݒ ሻ S e r v e o n l y h i g h t y p e ߠ ଶ െ ߣ ߠ ு ଶ Ͷ ሺ ߣ ݒ ு െ ݒ ሻ S e r v e b o t h t y p e s 2.4.2 Time Based Contract-Deterministic Quality Under a time based contract, a service provider charges a buyer based on an hourly rate p and the time spent on the service. If t H is the time for a high type buyer and t L is the time for a low type buyer, the payments from the buyers should be P H = pt H and P L = pt L , respectively. Facing two types of buyers, the SP has three choices: serving one of the types only, serving both types with different rates, and serving both types with the same rate (but different efforts). We discuss these problems one by one. 50 One-Type Problem When the service provider prefers to serve only one type of buyer, she solves the following problem, for x =H or L: max p,tx λ x (pt x −ct 2 x ) (2.4.7) s.t. v x +θ x t x −pt x ≥ 0, (2.4.8) v −x +θ −x t −x −pt −x ≤ 0 (2.4.9) where λ x is the proportion oftype x. (2.4.8) is the participation constraint for typex buyers and (2.4.9) is the non-participation constraint for type−x buyers. Proposition 2.4.5. If θx(θ −x −θx) 2c ≤v x −v −x , the optimal solution to problem (2.4.7)-(2.4.9) is t ∗ x = θx 2c and p ∗ =θ x + 2cvx θx , and the optimal objective value is π fb x =λ x (v x + θ 2 x 4c ). Proof. The IR constraint (2.4.8) implies p ≤ vx tx + θ x . Because the objective function is increasinginp,(2.4.8)mustbebindingatoptimality. Bysubstitution, theobjectivefunction becomes λ x (v x +θ x t x − ct 2 x ), which is maximized at t ∗ x = θx 2c . The corresponding price is p ∗ = vx t ∗ x +θ x =θ x + 2cvx θx . And the optimal solution satisfies constraint (2.4.9) if θx(θ −x −θx) 2c ≤ v x −v −x . The optimal solution to the one-type problem targeting the high (or low) type achieves thefirstbestprofitπ fb H (orπ fb L )facingthattype. Thefollowingresult, asillustratedinFigure 2.7, will be referred to repeatedly in subsequent analysis. Lemma 2.4.6. Define p d x = 1 2 θ x + p θ 2 x +8cv x , (2.4.10) 51 Figure 2.7. Optimal effort for type x buyer as p increases. ! ! !! !! ! ! !!! ! !!! ! ! !! ! ! ! ! ! ! !! ! !! ! ! !! ! ! ! ! !" ! ! !!!! ! ! ! ! ! ! ! ! !! ! "#$%&!'! (#)*&+,! !! ! ! !!! ! ! !!! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! !! ! ! ! !! ! !! !! ! !!! ! !!!! ! !! ! ! !! !! ! ! ! !! !! ! ! ! !! ! ! ! ! and assume v x ≥ 0. 1 Given price p, the service provider’s optimal effort under problem (2.4.7)-(2.4.8) is t ∗ x (p) = p 2c , if p∈ [0,p d x ], vx p−θx , if p∈ (p d x ,∞). Given the optimal effort t ∗ x (p), the objective value in (2.4.7) increases with p in [0,θ x + 2cvx θx ] and decreases with p in (θ x + 2cvx θx ,∞). Furthermore, θ x ≤ p d x ≤θ x + 2cvx θx . Intuitively, p d x is the threshold price above which type x would get zero surplus under the optimal effort level. Two-Type Two-Rate Problem To serve both types of buyers with different price rates, the service provider should solve the following problem: max t H ,t L ,p H ,p L λ(p H t H −ct 2 H )+(1−λ)(p L t L −ct 2 L ) s.t. v H +θ H t H −p H t H ≥ 0, v L +θ L t L −p L t L ≥ 0, 1 The superscript “d” in p d x represents “deterministic” quality. 52 v H +θ H t H −p H t H ≥v H +θ H t L −p L t L , v L +θ L t L −p L t L ≥v L +θ L t H −p H t H , Notice that if we replace the total prices p H t H andp L t L by variables s H and s L , the problem becomes exactly the two-part tariff problem. Because we will study that problem in a later section (Section 4.3), we will not discuss it here and will instead assume that the SP only charges a single price rate over the entire range of effort levels. Two-Type One-Rate Problem To charge both types of buyers the same price rate, the service provider needs to solve the following problem: max p,t H ,t L λ(pt H −ct 2 H )+(1−λ)(pt L −ct 2 L ) (2.4.11) s.t. v H +θ H t H −pt H ≥ 0, (2.4.12) v L +θ L t L −pt L ≥ 0, (2.4.13) v H +θ H t H −pt H ≥v H +θ H t L −pt L , (2.4.14) v L +θ L t L −pt L ≥v L +θ L t H −pt H , (2.4.15) The constraints are (IRH), (IRL), (ICH), and (ICL), respectively. We will focus on this problem in subsequent analysis and will simply refer to it as “the two-type problem.” We have the following result: Lemma 2.4.7. The (ICH) and (ICL) constraints are equivalent to (1) t H > t L and θ H ≥ p≥θ L , or (2) t H =t L and p is unrestricted. This lemma suggests that the IC constraints permit a single effort level or different effort levels for the two types, but the latter is only possible when the price falls in the interval [θ L ,θ H ]. 53 Solving the Two-Type Problem Inthissubsection,wederivetheoptimalsolutiontothetwo-type(one-rate)problem(2.4.11)- (2.4.15),indifferentparameterregimes. First, we findheoptimaleffortlevels given theprice rate p. Lemma 2.4.8. Given price p, the service provider’s optimal effort levels under problem (2.4.11)-(2.4.15) are t ∗ L (p)= p 2c , if p∈ [0,p d L ], v L p−θ L , if p∈ (p d L ,p 0 ), v H p−θ H , if p∈ (p 0 ,∞). t ∗ H (p) = p 2c , if p∈ [0,θ H ], v L p−θ L , if p∈ (θ H ,p 0 ), v H p−θ H if p∈ (p 0 ,∞) wherep 0 = v L θ H −v H θ L θ H −θ L istheratesuchthatthepricelinep 0 tpassesthroughtheintersection of the two value lines, i.e. p 0 satisfies θ H t+v H =θ L t+v L =p 0 t, for t = v L −v H θ H −θ L . The combination of t ∗ L (p) and t ∗ H (p) depends on the relative value of θ H , p d L , p FB L (defined as p FB L =θ L + 2cv L θ L ) and p 0 . The assumption v H ≥ 0 implies that v L −v H θ H −θ L ≤ v L θ H −θ L . According to Figure 2.8, it implies θ H < p 0 . In addition Lemma 2.4.6 suggests that p d L < p FB L . The following discussion is based on three possible parameter regimes: (i) p d L < p FB L < θ H < p 0 ; (ii) p d L <θ H <p FB L and it has two sub cases, p d L <θ H <p FB L <p 0 and p d L <θ H <p 0 <p FB L ; (iii) θ H < p d L < p FB L and it has three sub cases, θ H < p d L < p FB L < p 0 , θ H < p d L < p 0 < p FB L and θ H < p 0 < p d L < p FB L . The following lemma simplifies the condition of the above parameter regimes. Lemma 2.4.9. (i) If v L θ H −θ L < θ L 2c , then p d L <p FB L <θ H <p 0 ; (ii) If θ L 2c ≤ v L θ H −θ L ≤ p d L 2c , then p d L <θ H < min(p FB L ,p 0 ); (iii) If v L θ H −θ L > p d L 2c , then θ H <p d L <p FB L . 54 Figure 2.8. Optimal solution in parameter regime v L θ H −θ L < θ L 2c . "#$%&!'! (#)*&+,! ! ! !! ! ! ! ! ! !! ! ! ! ! !!! ! ! ! ! !!! ! !!! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! !! ! ! !" !!! ! ! !!! ! !!!! ! !! ! ! !! !! !! !!! ! ! !! ! !! ! !! !! Proposition 2.4.10. (i) In Parameter Regime I ( v L θ H −θ L < θ L 2c ): Problem (2.4.11)-(2.4.15) reduces to max p∈[θ L + 2cv L θ L ,θ H ] π(p), for π(p)=λ p 2 4c +(1−λ) " pv L p−θ L −c v L p−θ L 2 # . (2.4.16) The optimal price is p ∗ = θ H if π(θ H )≥π(ˆ p) ˆ p if π(θ H )<π(ˆ p) , where b p ∈ (θ L + 2cv L θ L ,θ H ) solves v L p−θ L = θ L 2c − λp(p−θ L ) 2 4c 2 (1−λ)v L , and the optimal efforts are t ∗ L = v L p ∗ −θ L and t ∗ H = p ∗ 2c . 2 (ii) In Parameter Regime II ( θ L 2c ≤ v L θ H −θ L ≤ p d L 2c ): The optimal solution to problem (2.4.11)-(2.4.15) is p ∗ =θ H , t ∗ L = v L θ H −θ L , and t ∗ H = θ H 2c , if λ≥ ( v L θ H −θ L −t) t+ v L θ H −θ L − θ L c θ H 2c − v L θ H −θ L −2 p ∗ =θ L + v L t and t ∗ L =t ∗ H =t, otherwise, where t = max{ v L −v H θ H −θ L , θ L 2c }. 2 A necessary condition for π(θ H ) ≥ π(ˆ p) is vL θH−θL ≥ θL 2c − λθH(θH−θL) 2 4c 2 (1−λ)vL and a sufficient condition for π(θ H )<π(ˆ p) is vL θH−θL ≤ θL 2c − λθH(θH−θL) 2 4c 2 (1−λ)vL . 55 (iii) In Parameter Regime III ( v L θ H −θ L > p d L 2c ): The optimal solution to problem (2.4.11)- (2.4.15) is: p ∗ =θ H + 2cv H θ H and t ∗ L =t ∗ H = θ H 2c , if v L −v H θ H −θ L > θ H 2c , p ∗ = θ H v L −θ L v H v L −v H and t ∗ L =t ∗ H = v L −v H θ H −θ L , if θ L 2c < v L −v H θ H −θ L ≤ θ H 2c , p ∗ =θ L + 2cv L θ L and t ∗ L =t ∗ H = θ L 2c , if v L −v H θ H −θ L ≤ θ L 2c . The solution in Parameter Regime I is illustrated in Figure 2.8, where the optimal price line is constrained in the grey area (cone) bounded by rays p =θ H and p =p FB L =θ L + 2cv L θ L . The characteristics of this parameter regime are: the SP should provide different service levels for the two types; the low type’s IR constraint is binding; The low type gets no surplus and the high type gets positive surplus.The main trade-off in this regime is that raising the price can garner more profit from the high type but less profit from the low type. The effort for high type is efficient if the high type’s IC constraint is binding (p = θ H ), but has a downward distortion if the high type’s IC constraint is not binding (p<θ H ). In Parameter Regime II, two strategies compete against each other: providing different service levels for the two types or a single (relative low) service level. When the probability of the high type (λ) is relatively large, the first strategy prevails. In the case t = v L −v H θ H −θ L , the condition on λ in the solution becomes λ≥ v H θ H −θ L 2v L −v H θ H −θ L − θ L c θ H 2c − v L θ H −θ L −2 . The case t = θ L 2c is illustrated in Figure 2.9 where the condition reduces to λ≥ v L θ H −θ L − θ L 2c 2 θ H 2c − v L θ H −θ L −2 independent ofv H . Weobserve acommon trendthatwhenλislargeenough, the twoservice level solution prevails, consistent with our intuition ( The SP can charge the price p = θ H and provides the first best service for high type). In both cases, the SP leaves zero surplus to the low type and positive surplus to the high type. 56 In Parameter Regime III, as illustrated in Figure 2.10, the main characteristic of the solution is: providing a single service level for both types. The optimal solution is trying to gain the first best profit from one buyer type (unless it is prevented by the IR constraint of the other type). Table 2.2 and Table 2.3 summarizes the results analyzed in this section. Table 2.2. The SP’s profit and strategy in parameter regimes I and II Equilibrium Regime Serve only high type Serve only low type Both types same effort Both types different effort Regime I v L θ H −θ L < θ L 2c π = λ( θ 2 H 4c +v H ), t H = θ H 2c , p = θ H + 2cv H θ H N/A N/A π = λp ∗2 4c +(1−λ) ( p ∗ v L p ∗ −θ L −c( v L p ∗ −θ L ) 2 ), t L = v L p ∗ −θ L , t H = p ∗ 2c , p ∗ =θ H or ˆ p, v L ˆ p−θ L = θ L 2c − λˆ p(ˆ p−θ L ) 2 4c 2 (1−λ)v L Regime II θ L 2c ≤ v L θ H −θ L ≤ p d L 2c , θ L 2c ≤ ˜ t π = λ( θ 2 H 4c +v H ), t H = θ H 2c , p = θ H + 2cv H θ H π = (1−λ)( θ 2 L 4c +v L ), t L = θ L 2c , p =θ L + 2cv L θ L π = θ L (v L −v H ) θ H −θ L +v L −c ¯ t 2 , t H =t L = ˜ t, p =θ L + 2cv L ˜ t π = λθ 2 H 4c +(1−λ) ( θ H v L θ H −θ L −c( v L θ H −θ L ) 2 ) t L = v L θ H −θ L t H = θ H 2c p =θ H Regime II θ L 2c ≤ v L θ H −θ L ≤ p d L 2c , θ L 2c ≥ ˜ t π = λ( θ 2 H 4c +v H ), t H = θ H 2c p = θ H + 2cv H θ H N/A π = θ 2 L 4c +v L , t L =t H = θ L 2c p =θ L + 2cv L θ L , π = λθ 2 H 4c +(1−λ) ( θ H v L θ H −θ L −c( v L θ H −θ L ) 2 ), t L = v L θ H −θ L , t H = θ H 2c , p =θ H 57 Table 2.3. The SP’s profit and strategy in regime III Equilibrium Regime Serve only high type Serve only low type High & Low same effort Regime III v L θ H −θ L > p d L 2c , ˜ t> θ H 2c N/A π = (1−λ)( θ 2 L 4c +v L ), t L = θ L 2c , p=θ L + 2cv L θ L π = θ 2 H 4c +v H , t L =t H = θ H 2c , p =θ H + 2cv H θ H Regime III v L θ H −θ L > p d L 2c , θ L 2c ≤ ˜ t≤ θ H 2c π =λ( θ 2 H 4c +v H ), t H = θ H 2c , p =θ H + 2cv H θ H π = (1−λ)( θ 2 L 4c +v L ), t L = θ L 2c , p=θ L + 2cv L θ L π =θ L ˜ t+v L −c ˜ t 2 , t H =t L = ˜ t, p=θ L + v L ) ˜ t Regime III v L θ H −θ L > p d L 2c , ˜ t≤ θ L 2c π =λ( θ 2 H 4c +v H ), t H = θ H 2c , p =θ H + 2cv H θ H N/A π = θ 2 L 4c +v L , t L =t H = θ L 2c , p =θ L + 2cv L θ L Figure 2.9. Optimal solution in parameter regime v L −v H θ H −θ L ≤ θ L 2c ≤ v L θ H −θ L ≤ p d L 2c . !! !! ! ! !! ! ! ! ! ! !! ! ! ! ! !!! ! ! ! ! !!! ! ! !! ! ! "#$%&!'! (#)*&+,! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! !! ! ! !" !!! ! ! !!! ! !!!! ! !! ! ! ! !! ! !! ! ! ! !! ! !! ! !! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! !! ! !! ! ! !! ! ! !! ! ! !! ! !! ! ! ! ! ! !! !! ! When is Serving A Single Type Optimal? If the optimal solution to the two-type problem found in subsection 2.4.2 that includes a non-first-bestoutcomeforthehigh(orlow)typewouldbedominatedbytheoptimalsolution 58 Figure 2.10. Optimal solution in parameter regime v L θ H −θ L > p d L 2c , when θ H 2c < v L −v H θ H −θ L . !! !! ! ! !! !! !! ! "#$%&!'! (#)*&+,! ! ! !! ! ! ! ! ! !! ! ! ! ! !!! ! ! ! ! !!! ! ! !!! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! !! ! ! ! !! ! ! ! !! ! ! ! !!! ! ! !!! ! !!!! ! !! ! ! ! !! ! !! ! !! ! !! !! !!! ! !!!! ! !! ! ! !! to the high (or low) type-only problem when λ is large (or small) enough. More specifically, we can show that: Proposition 2.4.11. (1) The service provider should serve the high type only when λ is large enough in: parameter regime I ( v L θ H −θ L < θ L 2c ), if p ∗ = b p; parameter regime II ( θ L 2c ≤ v L θ H −θ L ≤ p d L 2c ); and parameter regime III ( v L θ H −θ L > p d L 2c ), if v L −v H θ H −θ L < θ H 2c . (2) The service provider should serve the low type only when λ is small enough in: parameter regimes II and III, if θ L 2c < v L −v H θ H −θ L . 2.4.3 Two part tariff-deterministic quality Under a two part tariff contract, the payment of a buyer contains two parts, a time based payment pt and a fixed fee a. Depending on the time effort t H or t L , the payments from the high type buyer is P H = pt H + a, and that the payment from the low type buyer is P L = pt L +a. Since the fixed fee and time based contracts are special cases of two part tariff, the latter contract dominates the former ones. In this section, we discuss the SP’s optimal strategy under two part tariff contracts. 59 One-Type Problem If the SP prefers to serve only one type ofbuyer, she solves the following problem, forx =H (or L) and−x =L (or H): max p,a,tx λ x (pt x +a−ct 2 x ) (2.4.17) s.t. v x +θ x t x −pt x −a≥ 0, (2.4.18) v −x +θ −x t x −pt x −a≤ 0 (2.4.19) where λ x is the proportion of type x. Lemma 2.4.12. If θx(θx−θ −x ) 2c ≥ v −x −v x , the optimal solution to problem (2.4.17)-(2.4.19) is t x = θx 2c and p θx 2c +a =v x + θ 2 x 2c , and the optimal objective value is π fb x =λ x (v x + θ 2 x 4c ). Lemma 2.4.12 suggests that the SP’s total profit from serving only the high type buyer is π H t = λ θ 2 H 4c +v H if θ H 2c ≥ v L −v H θ H −θ L , and that from serving only the low type buyer is π H t = (1−λ) θ 2 L 4c +v L if θ L 2c ≤ v L −v H θ H −θ L . Figure 2.11 is the scenario when the SP serves only the low type. It provides efficient service level to the type and leaves zero surplus to the low type. Figure 2.11. Serving only low type-two part tariff Service Time Value ! "+# ! " $ % "+# % # ! # % "& ' ("+' )" * " $ " % +, 60 Figure 2.12. Serving both types-two part tariff when θ L −λθ H 2c(1−λ) ≥ ˜ t,v L <v H Figure2.13. Serving bothtypes- twopart tariff when θ L −λθ H 2c(1−λ) ≥ ˜ t,v L >v H S e r v i c e T i m e V a l u e S e r v i c e T i m e V a l u e Figure 2.14. Serving both types- two part tariff when θ L 2c < ˜ t< θ H 2c S e r v i c e T i m e V a l u e ߠ ு ݐ ݒ ு ݐ ߠ ݐ ݒ ݒ ு ݒ ݐ ǁ ܽ ݐ ܽ ܿ ݐ ଶ ߠ ு ʹ ܿ ߠ ʹ ܿ Two-Type Problem To serve both types of buyers with a two part tariff contract, the SP needs to solve the following problem: max p,a,t H ,t L λ(pt H +a−ct 2 H )+(1−λ)(pt L +a−ct 2 L ) (2.4.20) s.t.v H +θ H t H −pt H −a≥ 0, (2.4.21) v L +θ L t L −pt L −a≥ 0, (2.4.22) v H +θ H t H −pt H −a≥v H +θ H t L −pt L −a, (2.4.23) v L +θ L t L −pt L −a≥v L +θ L t H −pt H −a. (2.4.24) The constraints are IRH, IRL, ICH, and ICL respectively. We have the following result: 61 Lemma 2.4.13. The ICH (2.4.23) and ICL (2.4.24) constraints are equivalent to θ H ≥p≥ θ L and t H 6=t L in an optimal solution of problem (2.4.20)-(2.4.24). Unlike the time based payment, Lemma 2.4.13 suggests that the two part tariff induces the optimal strategy in which the SP serves the two types of buyers with different effort. That is, two part tariff is more effective in terms of differentiating buyers based on their valuation. Proposition 2.4.14. The optimal solution to problem (2.4.20)-(2.4.24) is given by the last column of Table 2.4. When is Serving One Type Optimal? If an optimal solution to a one-type problem is also feasible to the two-type problem, it is dominated by the latter. On the other hand, any two-type solution found in subsection 2.4.3 that includes a non-first-best outcome for the high (or low) type would be dominated by the high (or low) type-only solution when λ is large (or small) enough. Table 2.5 provides the optimality condition for serving one type or two types in three mutually exclusive and exhaustive regions: θ L −λθ H 2c(1−λ) ≥ ˜ t, θ L −λθ H 2c(1−λ) ≤ ˜ t ≤ θ H −(1−λ)θ L 2c(1−λ) and ˜ t ≥ θ H −(1−λ)θ L 2c(1−λ) . Figure 2.15 shows that serving only high type is optimal if λ is sufficiently large. Figure 2.16 illustrates that serving only low type is optimal if λ is in the middle. Intuitively, ifλ is large enough, it isoptimaltoleavepositivesurplustolowtypeinordertoservebothtypes. Ifλissufficiently small, θ L −λθ H 2c(1−λ) ≤ ˜ t≤ θ H −(1−λ)θ L 2cλ is more likely to be true. In that case, it is possible for the SP to serve both types and leaves zero surplus to them. 2.5 Optimal Contracts in the Hidden Effort (Stochastic Quality) Case In this section, we consider the situation where the service provider’s effort (or time) de- termines the service quality only stochastically. A type x (H or L) buyer derives utility 62 Figure 2.15. Optimal effort regime under two part tariff when θ L −λθ H 2c(1−λ) ≥ ˜ t Figure 2.16. Optimal effort regime under two part tariff when ˜ t≥ θ H −(1−λ)θ L 2cλ ݒ ݒ ு ߠ ߠ ு ߣ Ͳ ߣ ͳ െ ݒ ு ݒ Ͳ ሺ ߠ ு െ ሺ ͳ െ ߣ ሻ ߠ ு ሻ ଶ Ͷ ߣ ሺ ሺ ͳ െ ߣ ሻ ݒ െ ݒ ு ሻ ݐ ǁ ൌ ߠ ு െ ሺ ͳ െ ߣ ሻ ߠ ʹ ܿ ߣ S e r v e o n l y h i g h t y p e S e r v e b o t h t y p e s ܿ ܿ ሺ ߠ െ ߣ ߠ ு ሻ ଶ Ͷ ሺ ͳ െ ߣ ሻ ሺ ߣ ݒ ு െ ݒ ሻ ݒ ݒ ு ߠ ߠ ு ߣ Ͳ ߣ ͳ െ ݒ ு ݒ Ͳ ሺ ߠ ு െ ሺ ͳ െ ߣ ሻ ߠ ு ሻ ଶ Ͷ ߣ ሺ ሺ ͳ െ ߣ ሻ ݒ െ ݒ ு ሻ ݐ ǁ ൌ ߠ ு െ ሺ ͳ െ ߣ ሻ ߠ ʹ ܿ ߣ S e r v e o n l y l o w t y p e S e r v e b o t h t y p e s ܿ ܿ ሺ ߠ െ ߣ ߠ ு ሻ ଶ Ͷ ሺ ͳ െ ߣ ሻ ሺ ߣ ݒ ு െ ݒ ሻ v x + θ x q from quality q, not from the service provider’s effort t directly. For tractability of the analysis, we assume q = (1+ǫ)t, where ǫ follows a zero-mean uniform distribution U[−Δ,Δ] and hence E(q|t)=t. We assume that the degree of quality uncertainty is common knowledge, i.e., Δ is known to both parties. However, due to this uncertainty, although the quality q can be observed by the buyer, the effort t cannot. In contrast to the deterministic quality case, the service provider’s effort could not be inferred exactly in the stochastic quality case. As service quality is not usually contractible, we could only contract on the service provider’s reported time effort. However, the service provider may misreport her true effort as long as it is undetectable by the buyer. More specifically, if quality q is observed (or realized), the service provider may report an effort b t∈ [(1+Δ) −1 q,(1−Δ) −1 q] (assuming, as commonly seen in reality, that there aren’t enough observations for the buyer to make a statistical inference). 2.5.1 Effort Reporting In a stochastic quality model, the reported time effort can be different from the actual time effort. This is amajordeparture fromthe deterministic quality setting. The service provider 63 prefers to report the effort as much as possible but within the participation constraint of the buyer. In the following discussion, we include this assumption into the analysis of both time based payment and two parttariff. Notice thatwe don’tdiscuss fixed fee payment with stochastic quality because it could be easily verified that it achieves the same profit as in the deterministic case since the payment does not depend on the reported service time. We make the following assumption about the SP’s report: • Assumption (Extreme Reporting): The service provider reports her effort to be as large as possible. That is, if the service provider could report her effort from an interval [ b t 1 , b t 2 ], she would choose the upper bound b t 2 to report. This conforms with the notion of subgame perfection. In this buyer-provider sequential game, after the buyer has revealed his type x, the SP has exerted effort t, and the quality q has been observed, the unique subgame perfect move for theservice provider istoreportthemaximum allowableefforttocollectthelargestpayment. Ideally, the SP would like to over-report her effort to the maximum extent, i.e., b t = (1−Δ) −1 q, which would yield the expected report E( b t|t) = E((1−Δ) −1 q|t) = (1−Δ) −1 t. However, this is not always practically feasible. We assume: • Assumption (Ex-Post IR): The buyer’s participation constraint must be satisfied ex post. The reason is that buyers are not happy about paying more than their valuation of the observed quality and may inflict damage to the SP’s reputation. Under this assumption, the device provider may even have to under-report in some circumstances to ensure that p b t≤v x +θ x q (under time based payment) or p b t+a≤ v x +θ x q (under two part tariff). The SP’s (expected) report under time based payment is characterized in the following lemma more precisely. The lemma illustrates how the service provider would report her efforttothe buyer given price p and true effort t under a time based payment contract. 64 Lemma 2.5.1. Assume Ex-Post IR and Extreme Reporting. Given type x (H or L) and price p: (1) If p≤ (1−Δ)θ x , the expected report given the true effort t is E( b t|t)= (1−Δ) −1 t; (2) If p>(1−Δ)θ x , the expected report given the true effort t is E( b t|t)= (1−Δ) −1 t, if t≤ q x (p) 1+Δ , (1−Δ) −1 t− (1−Δ) −1 p−θ x 4Δpt (1+Δ)t−q x (p) 2 , if q x (p) 1+Δ) <t< q x (p) (1−Δ) , v x +θ x t p , if q x (p) (1−Δ) ≤ q x (p) 1+Δ , where q x (p) = vx (1−Δ) −1 p−θx and q x (p) = vx (1+Δ) −1 p−θx , if p>(1+Δ)θ x , ∞, if p≤ (1+Δ)θ x . When t > (1+ Δ) −1 q x (p), the ex-post IR constraint for type x will be violated with positive probability. In all cases above, the expected payment to the service provider given true effort t is pE( b t|t). Intuitively, part (1) of Lemma 2.5.1 shows that the service provider can over report her serviceefforttothemaximumextentifthepricerateissolowthattheactualpaymentlinep ˆ t anddoesnotcrossthetypexbuyer’s valueline(frombelow). Ifthepricerateishigher, asin part(2)ofLemma 2.5.1suggests theexpected effortreportdepends onthetrue effortt. Itis useful toobserve thatwhenp> (1−Δ)θ x and(1−Δ) −1 q x (p)≤t≤ (1+Δ) −1 q x (p), theSP’s expected compensation for effort t is pE( b t|t) =v x +θ x t. In this case, the expected payment tracesthetypexbuyer’svaluelinebasedontheSP’strueeffortt,independentofthenominal pricep(aslongasitisinthegivenrange),andasaresult,thereissomeflexibilityinchoosing theprice tosupport acertaineffortandpayment pair. Part(2)ofthelemma isillustrated in Figures2.17(a)and(b). (a)isforGeneraltand(b)isfort∈ ((1+Δ) −1 q x (p),(1−Δ) −1 q x (p)). Note that t 1 = (1+Δ) −1 q x (p), t 2 = (1−Δ) −1 q x (p), and t 3 = (1+Δ) −1 q x (p). The red lines in the figures represent the payment as a function of realized quality q. Notice that when 65 p > (1− Δ)θ x , we have (1− Δ) −1 q x (p) = vx p−(1−Δ)θx < vx p−(1+Δ)θx = (1 + Δ) −1 q x (p) if p>(1+Δ)θ x , or (1−Δ) −1 q x (p)<∞= (1+Δ) −1 q x (p) if p≤ (1+Δ)θ x . Thus, in any case, all sub cases in part (2) of the lemma are non-degenerate. Figure 2.17. Relationship among true effort t, random quality q, reported effort b t, report- based payment p b t, and quality-effective payment s(q), when p>(1−Δ)θ x ݐ ଵ ݐ ଶ ݐ ଷ ݒ ௫ ߠ ௫ ݍ ݐ ݍ ݐ Ƹ ݐ Ƹ ݍ ௫ ݍ ௫ Ͳ ݏ ሺ ݍ ሻ ȟ ݐ ଶ ȟ ݐ ଶ ȟ ݐ ଵ ȟ ݐ ଵ ȟ ݐ ଷ ȟ ݐ ଷ ݐ Ƹ Ȁ ሺ ͳ െ ȟ ሻ ݐ Ƹ Ȁ ሺ ͳ ȟ ሻ ݐ ଵ ݐ ଶ ݐ ଷ ݒ ௫ ߠ ௫ ݍ ݐ ݍ ݐ Ƹ ݐ Ƹ ݍ ௫ ݍ ௫ Ͳ ݏ ሺ ݍ ሻ ȟ ݐ ଶ ȟ ݐ ଶ ȟ ݐ ଵ ȟ ݐ ଵ ȟ ݐ ଷ ȟ ݐ ଷ ݐ Ƹ Ȁ ሺ ͳ െ ȟ ሻ ݐ Ƹ Ȁ ሺ ͳ ȟ ሻ ݐݐݐ ଵ ଶଶ ݐݐݐ ଶ ݐݐ ଷଷ ݐ ଷ ݒݒ ௫௫ ݒݒݒݒݒݒݒ ߠߠ ௫௫ ݍݍ ݒ ௫ ߠ ௫ ݍ ݐݐ ݍݍ ݐݐ ƸƸ ݐ ݍ ݐ Ƹ ݐݐ ƸƸ ݐ Ƹ ݍݍ ௫௫ ݍݍ ଵଵଵଵ ௫௫ ݐݐݐݐ ݍ ௫ ݍݍ ௫௫ ݍ ௫ ͲͲͲ ݏݏ ሺሺ ݍݍ ሻሻ ݏ ሺ ݍ ሻ ȟȟ ݐݐ ȟ ݐ ଶ ȟȟ ݐݐ ଶଶ ȟȟ ݐݐݐݐ ଶଶଶଶ ȟ ݐ ଶ ȟȟ ݐݐ ȟ ݐ ଵ ȟȟ ݐݐݐݐ ଵଵ ȟȟȟȟ ȟȟ ݐݐݐݐ ଵଵଵଵ ȟ ݐ ଵ ȟȟ ݐݐ ଷଷ ȟ ݐ ଷ ȟȟ ݐݐ ଷଷ ȟ ݐ ଷ ݐݐ ƸƸ ȀȀ ሺሺ ͳͳ െെ ȟȟ ሻሻ ݐ Ƹ Ȁ ሺ ͳ െ ȟ ሻ ݐݐ ƸƸ ȀȀ ሺሺ ͳͳ ȟȟ ሻሻ ݐ Ƹ Ȁ ሺ ͳ ȟ ሻ 2.5.2 Time-Based Contract: Stochastic Quality In this section, we consider the SP’s optimization problem facing stochastic quality under a time-based contract. Section 5.2.1 discusses the one type pricing problem, 5.2.2 discusses the two-type pricing problem. And the comparison between stochastic and deterministic quality under a time-based contract is discussed in section 5.2.3. In section 5.2.4, we provide numerical examples to illustrate the effects of stochastic quality on the service provider’s profitability and service effort in equilibrium. The One-Type Pricing Problem As in the deterministic quality setting, we first study the following one-type problem (the proportion factor λ x is dropped from the objective function for convenience without loss of generality): max p,t pE( b t|t)−ct 2 (2.5.1) 66 s.t. E( b t|t) is given in Lemma 2.5.1 for type x. (2.5.2) We first characterize how the optimal effort and profit vary with the price. Proposition2.5.2. Define r s x = 1 2 θ x + p θ 2 x +8c(1+Δ) −1 v x . Given price p, the optimal effort to problem (2.5.1-2.5.2) is given by: t ∗ x (p) = p 2c(1−Δ) , if p∈ [0,(1−Δ)r s x ], the positive root of: −2ct 3 + (1+Δ) 2 θx−(1−Δ) 2 r 4Δ t 2 + v 2 x 4Δ(r−θx) , if p∈ ((1−Δ)r s x , 2cvx θx +(1−Δ)θ x ), θx 2c , if p∈ [ 2cvx θx +(1−Δ)θ x , 2cvx θx +(1+Δ)θ x ], vx p−(1+Δ)θx , if p∈ ( 2cvx θx +(1+Δ)θ x ,∞). Under this optimal effort plan, the objective value in (2.5.1) increases with p in [0, 2cvx θx + (1−Δ)θ x ), stays constant when p∈ [ 2cvx θx +(1−Δ)θ x , 2cvx θx +(1+Δ)θ x ], and decreases with p in ( 2cvx θx +(1+Δ)θ x ,∞). Notice that when the price p is in the range [ 2cvx θx + (1− Δ)θ x , 2cvx θx + (1 + Δ)θ x ] the objective is constant and the efficient effort θx 2c is achieved. Recall from Lemma 2.4.6, the efficient effort θx 2c could be achieved in the deterministic case only when p= 2cvx θx +θ x . Thus, we have some flexibility in choosing p to achieve the first best one-type solution. The Two-Type Pricing Problem Now, we focus on the service provider’s problem to serve both types of buyers (with a single rate). Without loss of generality, the problem can be formulated as the following: max t H ,t L ,p λ[pE( b t H |t H )−ct 2 H ]+(1−λ)[pE( b t L |t L )−ct 2 L ] (2.5.3) s.t. pE( b t L |t L )≤θ L t L +v L , (2.5.4) pE( b t H |t H )≤θ H t H +v H , (2.5.5) 67 v H +θ H t H −pE( b t H |t H )≥v H +θ H t L −pE( b t L |t L ), (2.5.6) v L +θ L t L −pE( b t L |t L )≥v L +θ L t H −pE( b t H |t H ). (2.5.7) The constraints are IRH (2.5.5), IRL (2.5.4), ICH (2.5.6), ICL (2.5.7), respectively. In the ICH constraint (2.5.6), the left hand side is simplified from E q, b t (v H +θ H q−p b t x = H,t = t H ) and the right hand side from E q, b t (v H +θ H q−p b t x =L,t =t L ). Notice that the type x in the above expressions is the type perceived by the service provider, not the buyer’s actual type. Individual Rationality Constraints and Incentive Compatibility Constraints In ordertosimplifytheoptimizationproblem(2.5.3)-(2.5.7),wefirstdiscusstheIndividualRa- tionality Constraints - IRH (2.5.5) and IRL (2.5.4) and Incentive Compatibility Constraints - ICH (2.5.6) and ICL (2.5.7) below. Individual Rationality Constraints The following lemma will help us reduce the Individual Rationality Constraints if t L or t H is in a certain range. Lemma 2.5.3. (i) IRH (2.5.5) is implied by IRL (2.5.4) and ICH (2.5.6) if t L ≥ v L −v H θ H −θ L . (ii) IRL (2.5.4) is implied by IRH (2.5.5) and ICL (2.5.7) if t H ≤ v L −v H θ H −θ L In order to simplify the analysis, we assume that v L ≤v H in the following discussion. It follows that the condition in Lemma 2.5.3 (i) always holds and thus we can get rid of IRH in the SP’s optimization problem. Incentive Compatibility Constraints The ICH (2.5.6) and ICL (2.5.7) constraints can be simplified as follows: θ L (t H −t L )≤pE( ˆ t H |t H )−pE( ˆ t L |t L )≤ θ H (t H −t L ) (2.5.8) 68 The first inequality is ICL and the second inequality is ICH. The optimization problem is simplified to max t H ,t L ,p λ(pE( ˆ t H |t H )−ct 2 H )+(1−λ)(pE( ˆ t L |t L )−ct 2 L ) (2.5.9) s.t.pE( ˆ t L |t L )≤t L θ L +v L (2.5.10) θ H (t H −t L )≥p(E( ˆ t H |t H )−E( ˆ t L |t L ))≥θ L (t H −t L ) (2.5.11) The next lemma investigates properties of the incentive constraints. Lemma 2.5.4. (i) The SP serves the high type with weakly higher time effort, i.e. t H ≥t L ; (ii) If (1−Δ)r S L ≤p≤θ H (1−Δ), ICL is implied by ICH and IRL; (iii) If p>θ H (1−Δ), ICH and ICL are satisfied if and only if t H =t L . This lemma suggests that the IC constraints permit a single service level or different service levels for the two types depending on the range of p. Notice that if p < (1−Δ)r S L , Proposition 2.5.2 suggests that t H (p) = t L (p) = p 2c(1−Δ) . ICL is also satisfied in this case, yet the objective is increasing with p. So the optimal price rate p must lies above (1−Δ)r S L . When p≥ θ H (1−Δ), the problem is reduced to a problem in which t H = t L , i.e. serving both types with the same effort. Serving both types with different effort exists only when (1−Δ)r S L ≤ p≤ θ H (1−Δ) in which ICL could be ignored as suggested by Lemma 2.5.4. In the following lemma, we identify a crucial property of the optimization problem (2.5.9)- (2.5.11). Lemma 2.5.5. If (1−Δ)r S L ≤p≤θ H (1−Δ), the IRL constraint (2.5.10) is binding in the optimal solution to problem (2.5.9) - (2.5.11). Or equivalently, any optimal solution to the problem satisfies v L p−(1−Δ)θ L ≤t L ≤ v L p−(1+Δ)θ L . If t H 6= t L , Lemma 2.5.4 (iii) implies that p≤ θ H (1−Δ) < (1−Δ)r S H and by Lemma 2.5.1 (i), E( ˆ t H |t H ) = t H 1−Δ . In addition, Lemma 2.5.5 and Lemma 2.5.1 (ii) suggest that 69 pE( ˆ t L |t L ) = θ L t L +v L , and thus the incentive compatibility for the high type (second in- equality of (2.5.8)) is reduced to (θ H − p 1−Δ )t H ≥ (θ H −θ L )t L −v L . By Lemma 2.5.4 (ii), ICL is implied by IRL and ICH, so the problem can be simplified to: max t H ,t L ,p λ( pt H 1−Δ −ct 2 H )+(1−λ)(θ L t L +v L −ct 2 L ) (2.5.12) s.t. v L p−(1−Δ)θ L ≤t L ≤ v L p−(1+Δ)θ L (2.5.13) (θ H − p 1−Δ )t H ≥ (θ H −θ L )t L −v L . (2.5.14) If t H =t L , the problem can be simplified to max t,p pE( ˆ t|t)−ct 2 (2.5.15) s.t. pE( ˆ t|t)≤tθ L +v L . (2.5.16) The analysis of problem (2.5.12)-(2.5.14) is provided in the Appendix. Problem (2.5.15)- (2.5.16)is the same as in the deterministic case under time based payment and hence can be ignored. In the following discussion, we explore if the SP makes more profit under stochastic or deterministic quality. Comparison of Optimal Solutions in Deterministic and Stochastic quality Cases (v L ≤v H ) In this subsection, we restrict our attention to the scenario v L ≤ v H . In the deterministic case, since θ L t L +v L =pt L , the optimization problem faced by the SP is max t H ,t L ,p λ(pt H −ct 2 H )+(1−λ)(θ L t L +v L −ct 2 L ) (2.5.17) s.t.θ H (t H −t L )≥pt H −(θ L t L +v L ), (2.5.18) t L = v L p−θ L . (2.5.19) 70 In the stochastic case, since θ L t L +v L = pE ˆ t L , , the optimization problem that the SP faces with is, max t H ,t L ,p λ( pt H 1−Δ −ct 2 H )+(1−λ)(θ L t L +v L −ct 2 L ) (2.5.20) s.t. v L p−(1−Δ)θ L ≤t L ≤ v L p−(1+Δ)θ L (2.5.21) (θ H − p 1−Δ )t H ≥ (θ H −θ L )t L −v L (2.5.22) Notice that the two problems are consistent when Δ =0. Since the SP cannot efficiently serve both types simultaneouly even in the deterministic quality case (Δ = 0), the rest of this section investigates if the stochastic quality reduces or increase the inefficiency and if the SP gains or loses profit. Instead of going though all possible values of Δ, we focus on the scenario that Δ→0, and check the direction of the SP’s profit change. Lemma 2.5.6. Let ξ = v L (θ H −p ∗ ) p ∗ −θ L , and p ∗ solves v L b p−θ L = θ L 2c − λb p(b p−θ L ) 2 4c 2 (1−λ)v L . If p ∗ < θ H and v L p ∗ −θ L ≤ θ L 2c , then (i) 2p ∗ θ H − √ θ 2 H −8cξ ≥ 1; (ii) 1− 2p ∗ θ H + √ θ 2 H −8cξ ≥ 0; (iii) If Δ≤ 1− 2p ∗ θ H + √ θ 2 H −8cξ , then (θ H − p ∗ 1−Δ ) p ∗ 2c(1−Δ) ≥ v L (θ H −p ∗ ) p ∗ −θ L . The following theorem compares the SP’s profit under stochastic quality with that under deterministic quality. Let π h denotes the former and π nh the latter. Theorem 2.5.7. If Δ> 0 is sufficiently small, (i) If θ L 2c ≤ v L θ H −θ L , the SP makes more profit under deterministic quality, i.e. π h ≤π nh ; (ii) If θ L 2c > v L θ H −θ L the SP makes more profit with stochastic quality, i.e. π h > π nh if 0≤ λ≤ ˆ λ; and π h (λ)≤ π nh (λ) if ˆ λ≤λ≤ 1, for a ˆ λ∈ ( 2cv L (2cv L +θ L (θ H −θ L )) (2cv L +(θ H −θ L ) 2 )(θ 2 H −2cv L −θ H θ L) ,1). Theorem 2.5.7 suggests that the SP may gain or lose profit from quality certainty. For example, if θ L 2c > v L θ H −θ L and p ∗ < θ H (p ∗ is defined in Proposition 2.4.10), Figure 2.18 shows that the SP may gain from the flexibility of reporting time effort because it enables the 71 SP to serve the low type buyer more efficiently (it is closer to θ L 2c , the first best time effort for low type) without sacrificing profit by decreasing price p. Intuitively, it applies to the scenario when the SP’s cost parameter c is small. On the other hand, if θ L 2c ≤ v L θ H −θ L (when c is large), even though the SP has the flexibility of reporting time effort, Figure 2.19 shows that it cannot improve the efficiency as θ L 2c ≤ v L θ H −θ L ≤t nh L . Even worse is that the SP cannot charge a higher price rate since a high type buyer may not be willing to pay if he expects that the SP will over report the time effort. When c is in the middle, there is a trade off between earning more profit from serving high type buyer or that from low type buyer: on the one hand, stochastic quality may benefit the SP by serving low type with more efficient effort; but onthe other hand, itdrives down the price rateandtherefore the profit fromhigh type is compromised. the SP has to balance these two effects in optimality and which effect dominates depends on the proportions of high and low type buyers. Figure 2.18. Time based payment- compare profit when θ L 2c > v L θ H −θ L and p ∗ <θ H . Figure 2.19. Time based payment- compare profit when θ L 2c ≤ v L θ H −θ L . ఏ ಽ ଶ ௩ ಽ ఏ ಹ ି ఏ ಽ ߠ ு ݐ ݒ ு ߠ ݐ ݒ ܿ ݐ ଶ V a l u e & P a y m e n t ݒ ு ݒ Ͳ ఏ ಹ ଶ t ൌ ߠ ு ௩ ಽ ሺ ଵ ି ௱ ሻ ሺ ఏ ಹ ି ఏ ಽ ሻ ఏ ಹ ଶ ሺ ଵ ି ௱ ሻ ൌ ሺ ͳ െ ߂ ሻ ߠ ு ఏ ಽ ଶ ௩ ಽ כ ି ఏ ಽ ߠ ு ݐ ݒ ு ߠ ݐ ݒ ܿ ݐ ଶ V a l u e & P a y m e n t ݒ ு ݒ Ͳ t כ ଶ ൌ כ כ ଶ ሺ ͳ െ ߂ ሻ ൌ כ ͳ െ ߂ Numerical Example We provide three numerical examples to illustrate the results found in section 5.2.3. Example 2.5.1. Let λ= 1/2; v L = 3 2 ; v H =2, θ H = 3, θ L = 1, c = 1. 72 This example falls into the case of v L θ H −θ L ≥ θ L 2c . Table 2.6 shows the SP’s total profit, profit from the high type π H s , profit from the low type π L s , optimal price p and service levels (t H and t L ), and consumer surplus for the high type CS H and low type CS L . Notice that “FB” represents the social optimal case and Δ = 0 represents the deterministic quality case. Table 2.6. Change of the SP’s profits and effort levels when v L θ H −θ L ≥ θ L 2c Δ SP’s Profit π H s π L s p t H t L CS H CS L ICH FB 3 2.1250.875N/A 1.5 0.5 0 0 N/A 0 1.969 1.1250.844 3 1.5 0.75 2 0 Binding 0.1 1.949 1.1100.8402.951.5130.7652.031 0 Binding 0.2 1.928 1.0930.8352.901.5280.7812.063 0 Binding 0.3 1.907 1.0760.8312.851.5450.7982.096 0 Binding 0.4 1.883 1.0580.8252.801.5650.8151.130 0 Binding Comparing Δ = 0 and Δ> 0 in Table 2.6, we observe that Δ> 0 reduces the SP’s profit by reducing the price rate p and meanwhile pushing both t H and t L away from the efficient effort θ H 2c and θ L 2c . In this scenario, both high type and low type buyers receive a higher quality service from the SP under stochastic quality. In addition, with the increase of Δ, low type buyers’ surplus remains 0. Example 2.5.2. Let λ= 1 5 ; v L = 1 2 ; v H = 2; θ H =2; θ L = 1; c = 1 6 (λ≤ θ L θ H ). This example falls in the case v L θ H −θ L ≤ θ L 2c and p ∗ < θ H . The resutls are shown in Table 2.7. Again “FB” represents the social optimal case and Δ = 0 represents the deterministic quality case. Table 2.7. Change of the SP’s profits and effort levels when v L θ H −θ L ≤ θ L 2c and p ∗ <θ H Δ SP’s Profit π H s π L s p t H t L CS H CS L ICH FB 3.2 1.6 1.6 N/A 6 3 0 0 N/A 0 2.011 0.4151.59641.17633.52902.8354 4.9 0 Not binding 0.1 2.027 0.4241.59621.17663.56542.83164.893 0 Not Binding 0.2 2.043 0.4481.59581.19703.66442.82424.853 0 Not Binding 0.3 2.064 0.4651.59551.20743.73482.81824.820 0 Not Binding 0.4 2.078 0.4821.59531.21783.8056 2.812 4.784 0 Not Binding 73 Comparing Δ = 0 and Δ > 0 in Table 2.7, we observe that Δ > 0 improves the SP’s profit by allowing the SP to charger a higher price rate p. In this scenario, high type buyers receive higher service effort from the SP, but low type buyers receive lower service effort from the SP under stochastic quality. With the increase of Δ, low type buyer’s surplus remains 0 but the SP is able to extract more surplus from high type buyers because it can charge a higher price rate p. Table 2.8. Change of the SP’s profits and effort levels when v L θ H −θ L ≤ θ L 2c , p ∗ = θ H and λ =0.15, 0.2, or 0.3. λ =0.15SP’s Profit π H s π L s p t H t L CS H CS L ICH FB 1.329 0.55 0.779 N/A 1.667 0.5 0 0 N/A Δ = 0 0.973 0.25 0.723 2 1.667 0.5 3.122 0 Binding Δ = 0.1 0.990 0.2170.772 1.70 1.6670.6862.217 0 Binding Δ = 0.2 0.989 0.2170.772 1.50 1.6880.7172.217 0 Binding Δ =0.25 0.982 0.2030.779 1.37 1.7930.8012.301 0 Binding λ =0.2 SP’s Profit π H s π L s p t H t L CS H CS L ICH FB 1.467 0.7340.734 N/A 1.667 0.5 0 0 N/A Δ = 0 1.013 0.333 0.68 2 1.667 0.5 2 0 Binding Δ = 0.1 1.021 0.3080.713 1.73 1.6670.6252.125 0 Binding Δ = 0.2 1.017 0.2900.726 1.50 1.7080.7152.215 0 Binding Δ =0.25 1.004 0.2710.732 1.38 1.8000.8002.300 0 Binding λ =0.3 SP’s Profit π H s π L s p t H t L CS H CS L ICH FB 1.741 1.1 0.641 N/A 1.667 0.5 0 0 N/A Δ = 0 1.095 0.5 0.595 2 1.667 0.5 2 0 Binding Δ = 0.1 1.090 0.4740.6161.7541.6640.5852.085 0 Binding Δ = 0.2 1.071 0.4350.6361.5011.7270.7122.213 0 Binding Δ =0.25 1.048 0.4070.6411.3761.8060.8002.300 0 Binding Example 2.5.3. Let v L = 1 2 ; v H = 2, θ H =2, θ L = 1, c = 3 5 . This example falls into the case v L θ H −θ L ≤ θ L 2c and p ∗ =θ H The results are shown in Table 2.8. Whether the stochastic quality decreases or increases the SP’s profit depends on the value of λ. If λ is small, e.g. λ = 0.15 or λ = 0.2, comparing Δ = 0 and Δ > 0 in Table 2.8, we observe that Δ > 0 may improve the SP’s profit. In this scenario, both high type buyers 74 and low type buyers receive higher service effort from the SP under stochastic quality (over treatment to both types). With the increase of Δ, the SP’s profit from the high type is decreasing and that from the low type is increasing, but the second effect can be dominant because λ is small. If λ is relatively large, e.g. λ = 0.3, Δ > 0 reduces the SP’s profit. In this scenario, as Δ increases, the SP’s profit from the high type is decreasing, that from the low type is increasing, and the first effect is dominant because λ is relatively large. 2.5.3 Two Part Tariff-Stochastic Quality Under a two part tariff contract, the payment from a buyer contains two parts, a time based payment p ˆ t and a fixed fee a. Depending on the reported time effort, the payment is P H =p ˆ t H +a or P L =p ˆ t L +a. One-Type Problem If the service provider prefers to serve only one type of buyer, she solves the following problem, for x =H or L: max p,a,tx λ x (pE( ˆ t x |t x )+a−ct 2 x ) (2.5.23) s.t. v x +θ x t x −pE( ˆ t x |t x )−a≥ 0, (2.5.24) v −x +θ −x t x −pE( ˆ t x |t x )−a≤ 0 (2.5.25) where λ x is the proportion of type x. Lemma 2.5.8. If θx(θx−θ −x ) 2c ≥v −x −v x , the optimal solution to problem (2.5.23)-(2.5.25) is t x = θx 2c and v x + θ 2 x 2c (1−Δ)≤ p θx 2c +a≤ v x + θ 2 x 2c (1+Δ) and the optimal objective value is π fb x =λ x (v x + θ 2 x 4c ). 75 Two-Type Pricing Problem Now we focus on the service provider’s problem to serve both types of buyers. The problem can be formulated as the following: max t H ,t L ,p λ[pE( b t H |t H )+a−ct 2 H ]+(1−λ)[pE( b t L |t L )+a−ct 2 L ] (2.5.26) s.t. pE( b t L |t L )+a≤ θ L t L +v L , (2.5.27) pE( b t H |t H )+a≤θ H t H +v H , (2.5.28) v H +θ H t H −pE( b t H |t H )≥v H +θ H t L −pE( b t L |t L ), (2.5.29) v L +θ L t L −pE( b t L |t L )≥v L +θ L t H −pE( b t H |t H ). (2.5.30) One way to deal with the above optimization problem is to first derive the expression for E( b t H |t H ) and E( b t L |t L ), as we did under the time based contract. However, the following proposition provide a shortcut to address the problem (2.5.26)-(2.5.30) . Proposition 2.5.9. Given the same parameter c, v H , v L , θ H and θ L , the optimization problem (2.5.26)-(2.5.30) and (2.4.20)-(2.4.24) have the same optimal solution of t H and t L , and achieve the same optimal profit. Proposition 2.5.9 suggests that the SP makes the same profit under stochastic quality anddeterministic quality. Eventhoughtheexactoptimalcontractsaredifferent, theoptimal service effort levels do not change with the stochastic quality. Intuitively, the flexibility of two part tariff itself has already enabled the SP extract the surplus as much as possible, and more flexibility gained from stochastic quality cannot improve the SP’s profit further. Table 2.9 provides the optimal contract (p,a) with stochastic quality. 2.5.4 When Time Based Contract is Equivalent to Two Part Tariff As we know, two part tariff dominates time based contract in general because the SP can leverageatoextractmoresurplusfrombuyersineitherdeterministicqualitycaseorstochas- tic quality case. We are interested in when a time based contract can generate the same 76 profit as a two part tariff. One obvious scenario when that is true is when the SP serves a single type under a two part tariff contract. The next proposition shows that the SP may achieve the same profit under time based contract as that under two part tariff even in the case of serving both types with different efforts. Proposition 2.5.10. If c ∈ (c 1 ,c 2 ),where c 1 = (θ L −θ H λ)((1−Δ)θ 2 H +(1−Δ)θ 2 L −2θ H θ L (1−Δλ)) 2v L (1−λ)(θ H (1−Δλ)−(1−Δ)θ L ) and c 2 = (1−Δ)(θ H −θ L ) 2 (θ L −θ H λ) 2v L (1−λ)(θ H (1−Δλ)−(1−Δ)θ L ) ) and ˜ t≤ θ L −λθ H 2c(1−λ) , the time based contract generates the same profit as the two part tariff with t H = θ H 2c and t L = θ L −λθ H 2c(1−λ) . Intuitively, the flexibility of reporting effort in the stochastic quality case sometimes can play a similar role as the flexibility gained by the fixed fee a. The flexibility is limited but not trivial. The SP gains the same flexibility from reporting service effort when the optimal effort for low type θ L −λθ H 2c(1−λ) is in the range[ v L p ∗ −(1−Δ)θ L , v L p ∗ −(1+Δ)θ L ]. The SP cannot gain the flexibility in the deterministic quality case because Δ = 0 and the range degenerates to one point. Figure 2.20. Equivalence Region of Two Part Tariff and Time Based Payment ܿ ߣ ߠ ߠ ு Ͳ ܿ ଵ ܿ ଶ I I I I I I The proof of this proposition follows from Table 2.9. Region II in Figure 2.20 is the region in which the SP can achieve the same profit under time based payment as that under two part tariff. Region I and Region III are the regions in which the SP makes more profit under time two two part tariff than under time based payment. 77 2.6 Conclusion We have explored the SP’s optimal strategy under three different contracts: fixed fee, time based contract, and two part tariff in both deterministic and stochastic quality cases. We conclude our paper by a comparison across different contracts and a comparison across deterministic and stochastic quality cases. 2.6.1 Comparison Across Contracts It is clear that time based contract dominates fixed fee contract in both deterministic and in stochastic case because the time based payment can differentiate buyers based on their valuation. It is also clear that two part tariff dominates time based contract because the extra leverage of the fixed fee a enables the SP to choose the optimal service effort that is closer to the efficient effort level. However, one interesting question is whether and when they generate the same profit for the SP: Inthedeterministic quality case, timebased payment equivalents tofixed feewheneither the SP serves only one type or serves both types with the same effort. Essentially, this is the scenario even time based payment fails to differentiate buyers. In addition, two part tariff may be equivalent to time based payment only when v L θ H −θ L = θ L 2c . But it is a very special case. In general, two part tariff strictly dominates time based payment when v L θ H −θ L > θ L 2c or v L θ H −θ L < θ L 2c . In the stochastic quality case, time based contract may still be equivalent to fixed fee contractwhentheSPservesonlyonetypeorbothtypeswiththesameeffortinthestochastic quality case. In contrast to the deterministic quality case, we can find a range in which the two part tariff is equivalent to the time based contract. As mentioned in the previous section, it is when the flexibility gained from reporting service effort plays the same role as the flexibility gained from choosing the fixed fee under the two part tariff. 78 2.6.2 Comparison Across Deterministic Case and Stochastic Quality Cases. Another comparison is between deterministic quality case andstochastic quality case. As we know, even in the deterministic quality case, the SP can not necessarily achieve the efficient effort level because of the information asymmetry in an adverse selection problem. What is the impact if the SP can misreport service effort that are unknown by buyers? Our analysis shows that the SP makes the same profit under fixed fee contract because the strategic behavior of misreporting doesn’t affect the SP’s profit under fixed fee contract; In addition, the SP also makes the same profit under two part tariff contract because the SP have gained enough flexibility even in the deterministic quality case by choosing a. However, the answer for time based contract is that it depends: a low cost the SP may benefit from stochastic quality case because of the flexibility it gains from misreporting. But it also occurs that the SP can be worse off from stochastic quality because it can not charge a higher price rate from high type. It is especially true for a high cost SP. 79 Table 2.4. The SP’s profit and optimal solution under two part tariff Parameter Regime H only L only H&L (a) ˜ t≤ θL−λθH 2c(1−λ) π = λ( θ 2 H 4c +v H ), t H = θH 2c , p θH 2c +a = v H + θ 2 H 2c N/A π =v L + θ 2 L 4c + λ(θ H −θ L ) 2 4c(1−λ) , t H = θ H 2c ,t L = θL−θHλ 2c(1−λ) , p =θ H a =v L − (θ H −θ L )(θ L −θ H λ) 2c(1−λ) Figure 2.12 , Figure 2.13 (b) θH−(1−λ)θL 2cλ ≤ ˜ t N/A π = (1−λ)( θ 2 L 4c +v L ), t L = θ L 2c , p θL 2c +a =v L + θ 2 L 2c Figure 2.11 π =v H + θ 2 H 4c + (1−λ)(θ L −θ H ) 2 4cλ , t H = θ H −(1−λ)θ L 2cλ ,t L = θ L 2c p=θ L , a = (θ H −θ L )(θ H −(1−λ)θ L ) 2cλ +v H (c1) θL 2c ≤ ˜ t ≤ θH 2c N/A N/A π =λ( θ 2 H 4c +v H )+ (1−λ)( θ 2 L 4c +v L ), t H = θ H 2c ,t L = θL 2c , p =θ H +θ L − 2c(v L −v H ) θ H −θ L a = v L θ H −v H θ L θ H −θ L − θ L θ H 2c Figure 2.14 (c2) θL−λθH 2c(1−λ) < ˜ t< θL 2c N/A N/A π = λθ 2 H 4c +v H −c ˜ t 2 (1−λ) +(1−λ) ˜ tθ H , t H = θ H 2c ,t L = ˜ t, p =θ H ,a=v H (c3) θ H 2c ≤ ˜ t≤ θ H −(1−λ)θ L 2cλ N/A N/A π = (1−λ)θ 2 L 4c +v L −c ˜ t 2 λ+ ˜ tθ L λ, t H = ˜ t,t L = θL 2c , p =θ L ,a=v L 80 Table 2.5. Dominant Strategy in Different Parameter Regions Parameter Region Dominant Strategy θL−λθH 2c(1−λ) ≥ ˜ t Serving only high type if v L −λv H + (θ L −λθ H ) 2 4c(1−λ) ≤ 0 Serving both types with different effort if v L −λv H + (θ L −λθ H ) 2 4c(1−λ) ≥ 0 Figure 2.15 θL−λθH 2c(1−λ) ≤ ˜ t≤ θH−(1−λ)θL 2cλ Serving both types with different effort ˜ t≥ θH−(1−λ)θL 2cλ Serving only low type if v H −(1−λ)v L + (θ H −(1−λ)θ L ) 2 4cλ ≤ 0 Serving both types with different effort if v H −(1−λ)v L + (θ H −(1−λ)θ L ) 2 4cλ ≥ 0 Figure 2.16 81 Table 2.9. SP’s optimal profit and contract Parameter Regime H L H&L ˜ t≤ θ L −λθ H 2c(1−λ) π =λ( θ 2 H 4c +v H ) t H = θ H 2c v H + θ 2 H 2c (1−Δ) ≤ p θ H 2c +a≤ v H + θ 2 H 2c (1+Δ) N/A π =v L + θ 2 L 4c + λ(θ H −θ L ) 2 4c(1−λ) t H = θ H 2c ,t L = θ L −θ H λ 2c(1−λ) ,p = (1−Δ)(θ H −θ L ) θ H (1−Δλ)−θ L (1−Δ) a = (θ L −θ H λ) 2c(1−λ)(θ H (1−Δλ)−θ L (1−Δ)) ((1−Δ)(1−θ L )θ L +v L + θ H ((1−Δ)−θ L (1−Δλ) θ L −λθ H 2c(1−λ) < ˜ t ˜ t< θ L 2c N/A N/A π =λ( θ 2 H 4c +v H )−c ˜ t 2 (1−λ) +(1−λ) ˜ tθ H t H = θ H 2c ,t L = ˜ t p = (1−Δ)θ H (2c(v H −v L )+θ H (θ H −θ L )) 2c(v H −v L )(1−Δ)+θ H (θ H −θ L ) a =v H + (v L −v H )Δθ 2 H 2c(v H −v L )(1−Δ)+θ H (θ H −θ L ) θ L 2c ≤ ˜ t≤ θ H 2c N/A N/A π =λ( θ 2 H 4c +v H )+ (1−λ)( θ 2 L 4c +v L ) t H = θ H 2c ,t L = θ L 2c p =θ H +θ L − 2c(v L −v H ) θ H −θ L a = v L θ H −v H θ L θ H −θ L − θ L θ H 2c θ H 2c ≤ ˜ t≤ θ H −(1−λ)θ L 2c(1−λ) N/A N/A π = (1−λ)( θ 2 L 4c +v L )−c ˜ t 2 λ + ˜ tθ L λ t H = ˜ t,t L = θ L 2c p = (1−Δ)θ L (2c(v H −v L )+(θ H −θ L )θ L ) 2c(v H −v L )(1−Δ)+θ L (θ H −θ L ) a = 2c(v L −v H )v L (1−Δ) 2c(v L −v H )(1−Δ)−θ L (θ H −θ L ) + θ L (v H Δθ L −v L (θ H −(1−Δ)θ L )) 2c(v L −v H )(1−Δ)−θ L (θ H −θ L ) θ H −(1−λ)θ L 2c(1−λ) ≤ ˜ t N/A π = (1−λ)( θ 2 L 4c +v L ) t L = θ L 2c , v L + θ 2 L 2c (1−Δ) ≤ p θ L 2c +a≤ v L + θ 2 L 2c (1+Δ) π =v H + θ 2 H 4c + (1−λ)(θ L −θ H ) 2 4cλ t H = θ H −(1−λ)θ L 2cλ ,t L = θ L 2c , p = (1−Δ)(θ H −θ L )θ L (1−Δ)θ H −θ L +Δθ L (1−λ) a = v H + θ H θ L 2c + (θ H −θ L ) 2 2cλ − (θ H −θ L )θ 2 L (1−Δ)θ H −θ L +Δθ L (1−λ) 82 CHAPTER 3 OPTIMAL PROMOTION STRATEGY FOR A SERVICE FIRM WITH DELAY SENSITIVE CUSTOMERS 3.1 Introduction Daily deals websites such as Groupon and Living Social have grown rapidly in recent years. Many firms utilize such daily deals as way for price discrimination and as an effective tool to attract new customers and explore a larger market size. However, introducing new cus- tomers comes along with a challenge to the firm’s profitability: according to a survey pro- vided in Dholakia (2006), while 66% of businesses reported that they have benefited from the Groupon’s promotion because it brings them many new customers, 32% of businesses reported it as being unprofitable, because it brings too many customers with respect to the firm’s capacity to process them. The problem is especially prominent for service businesses: one cafe owner described a Groupon promotion as “the single worst decision he has ever made.” Some small business owners said “it is a great marketing tool, it is just not great at making profit”. The main challenge that a promotion brings to a service firm is the negative externality imposed by its customers: while a promotion may generate more potential customers and enlarge the market size of the service provider, discount-seeking customers may only come during the promotion period and may further impose an externalize on the system that could drive away regular customers who pay full price for the service. In this paper, we explore how the congestion of a service system impacts a service firms’ promotion strategy. When should a firm have a promotion to draw new customers? How should firms decide how many customers they would like to handle by issuing coupons? What is the impact of 83 such a promotion strategy on customer flows in the short and long run. We answer these questionsbyusingatwo-stageM/M/1queuemodelandcharacterizingthetradeoffsbetween attracting new customers and creating too much congestion to current customers: When should a firm have a promotion to draw new customers? Unlike a promotion for products in which customers do not impose significantly negative externatility on each other, a promotion in service does not necessarily improve firms’ revenue even though it does attracts new customers in the service system. In Section 3.3, we provide a threshold on current customer’s waiting time as a criteria for firms to decide whether do promotion or not. How should firms decide how many customers they would like to handle by issuing coupons and what are the impact of promotion strategy on customer flows in the short run and long run? We answer this question in Section 3.4. The exact analysis involves equi- librium waiting time turns out to be intractable. We introduce an approximation analysis that are widely used in large service systems. We find two operating cases: (i) overloaded case and (ii) underloaded case depending on the future discounting factor. It decomposes the effect of the firm’s promotion decision. If the future discounting factor is large, i.e. the present revenues are more valuable compared with the future, the the firm is more likely to bring in only enough new customers in the promotion period so that it is critically loaded (with demand and supply nearly equal). Hence it would remain underloaded in the post promotion period as some discount seeking customers do not return. If the future discount- ing factor is small, i.e., future revenues are more valuable compared with the present, then the firm is more likely to bring in a sufficient number of new customers so that it is critically loaded post promotion and then overloaded in the promotion period. We call this the over- loaded case. Based on those analysis, we devise recommendations as to how a promotion is likely to be effective and beneficial. 84 3.2 Literature Review The work we present here is inspired by three streams of research. The first is the economic of the effect of queueing in operations management, the second is large-scale approximation methods in service systems, and the third is price discrimination and advertising literature in marketing. The first is the economic analysis of effects of queue delays, Mendelson (1985) establishesamicro-economicsframeworktomodelbehaviorofcustomerswhoaresensitiveto both price and delay and to study optimal capacity and pricing decisions incorporating such queuing effects. Dewan and Mendelson (1990) and Chen and Frank (2004) also consider the optimal capacity and pricing decision but generalizes linear delay structure to various delay structures. Mendelson andWhang(1990)extendsMendelson (1985)’sframeworktoderiving an incentive-compatible pricing scheme in an M/M/1 queueing system with multiple user classes. All the papers mentioned above focus on static decision making. Ata and Shneorson (2006) explores a dynamic control problem with delay sensitive customers by choosing price or service rate. The common feature of the above literature is that the arrival rate of a service system is endogenously influenced by the firms’ price decision. In other words, price is the only leverage that a firm could use to influence arrival rate of a service system. In contrast to these papers, we build an economic framework in which a firm could control arrival rate by promotion decisions. Our work is also inspired by the economics and marketing literature in advertising and price discrimination. Varian (1980), Jeuland and Narasimhan (1984), Narasimhan (1988)il- lustrate that promotional pricing help firms attract larger market segments and introduce new customers. Dholakia (2006) provides empirical evidence that customers attracted by promotionmaybecomerelationalcustomers. ThepaperclosesttoourworkisEdelmanetal. (2010). They model price discrimination and advertising effect of voucher promotion. How- ever, none of the models in this literature has ever included congestion effect in promotion as we considering in our work even though it is observed that promotions bring challenges 85 to firm’s operations. To the best of our knowledge, this is the first paper that combines the effect of congestion in operations management and the value of promotion in marketing and economics. The third is the heavy traffic approximation on large service system. A substantial literature is devoted to optimal staffing policies in a service system: Halfin and Whitt (1981)presents an asymptotic analysis of queueing systems with many servers. Garnett et al. (2002) apply the method to approximate design rules for large call centers. Borst et al. (2004) propose optimal staffing level for call centers based on an asymptotic framework. However, most papers that study large-capacity service system by utilizing heavy traffic approximations typically aim to achieve certain service quality by staffing decisions with- out economic considerations. Recent papers such as Maglaras and Zeevi (2003), Maglaras and Zeevi (2005), Randhawa and Kumar (2008), Allon and Gurvich (2010) and Kumar and Randhawa (2010) connect the economic approach in the first research stream with heavy traffic approximation analysis in the third research stream. The main framework of these papers is to first solve a deterministic analysis for the relaxed problem and then refine it to incorporate stochastic fluctuations. Our work utilizes the mechanism developed in these papers. Interestingly, asymptotically we obtain two different operating regimes in the same system. For instance, the promotion period maybe overloaded (critically loaded) and then the post promotion period would be critically loaded (underloaded). 3.3 The Model Weconsideraservicefirmservingpriceanddelaysensitivecustomers. Thefirmisconsidering a promotion strategy such as Groupon, in which for a short time, customers can prepay for services atahugediscounts. Suchapromotionmaybringtwoeffectstoaservicefirm: onthe one hand, it brings new customers to the firm, who may potentially repeat to purchase the service from the firm even after the promotion. On the other hand, it may bring congestion 86 to the service system and as a result, increase the delay cost for both regular customers and discount customers. In other words, it may drive away regular customer by introducing deal hunters who may not purchase again after the promotion period. In this section, we formulate the model that will be used in our analysis. 3.3.1 The Customer’s Decision We model the service firm using an M/M/1 queuing regime. Our model has two periods. These periods are of long enough time length that the system may be considered to be in steady-state in each period. The firm ex ante commits to a price p in both periods. The firm and customers share a common discounting factor δ, i.e. $1 in period 2 is worth $δ in period 1. In the promotion period, the firm decides a promotion size, i.e. the number of discount coupon to be sold with each coupon entitling its recipient to purchase the service at a price αp, where 0 < α < 1. The promotion size G could equal zero in which the firm does not have any promotion. In the post promotion period, the firm offers the service at the fixed price p. We assume customers arrive to the firm as a Poisson process and customers are drawn from two populations, one with proportion size λ. These are the regular customers of the firm and the other populations is that of those attracted by the discount promotion. We assume that the size of the customers attracted by promotion is determined by the firm’s promotion strategy, denoted by G. Each customer (regular and discount customer) has a valuationontheservice thatisrandomlydistributed with acumulative distributionfunction denoted byF, withsupporton[v,¯ v]. Weassume there isstrictly positive density denoted by f. Fortractabilityoftheanalysiswe assumethatthedistributionF hasanincreasingfailure rate (IFR). We assume there is a single queue so both customer types have identical waiting time. Each customer incurs a waiting cost of h per unit of time spent in the queue. For customers from either regular customer pool or discounted customer pool, the equilibrium 87 steady-statetimeinqueueisw 1 inthepromotionperiodandw 2 inthepostpromotionperiod. A customer with valuationv in the regular customer poolpurchases in the promotion period ifv−p−hw 1 ≥ 0,andacustomerinthepromotioncustomerpoolpurchase inthepromotion period if v−αp−hw 1 ≥ 0. Thus, the total demand in the promotion period that the firm faces is λ ¯ F(p+hw 1 )+G ¯ F(αp+hw 1 ); a customer with valuation v in the post promotion period will continue to purchase the service from the firm if v−p−hw 2 ≥ 0, and the total demand that the firm faces in this period is (λ+G) ¯ F(p+hw 2 ), where ¯ F(·) = 1−F(·) is tail distribution of F. The firm’s service rate is μ. Consequently, the average time spent in the queue for each customer in the promotion period satisfies w 1 = λ ¯ F(p+hw 1 )+G ¯ F(αp+hw 1 ) μ(μ−(λ ¯ F(p+hw 1 )+G ¯ F(αp+hw 1 )) , (3.3.1) In the post promotion period, the size G customers join in the regular customer pool and a customer with valuation v in the regular customer pool purchases in the post promotion period if v− p− hw 2 ≥ 0, and the average time in queue for each customer in the post promotion period satisfies w 2 = (λ+G) ¯ F(p+hw 2 ) μ(μ−(λ+G) ¯ F(p+hw 2 )) . (3.3.2) 3.3.2 The Service Firm’s Revenue The firm’s total discounted revenue over the two periods is π =p(λ ¯ F(p+hw 1 )+Gα ¯ F(αp+hw 1 )+δ(λ+G) ¯ F(p+hw 2 )) (3.3.3) The first term pλ ¯ F(p+hw 1 ) represents the firm’s revenue from current regular customer pool in the promotion period, the second term αpG ¯ F(αp+hw 2 ) represents the discounted revenue from discount coupon buyers in the promotion period, and δ(λ + G) ¯ F(p + hw 2 ) represents the future revenue from both customer pools who repeat purchase from the firm. To make the dependence ofthe waiting times and revenue on the promotion size explicit, we denote the solution to (3.3.1) and (3.3.2) by w 1 (G) and w 2 (G), respectively, and the revenue function in (3.3.3) by π(G) =p( αμ 2 w 1 (G) 1+μw 1 (G) + δμ 2 w 2 (G) 1+μw 2 (G) +(1−α)λ ¯ F(p+hw 1 (G))). 88 3.3.3 Properties of Equilibrium Waiting Time A promotion for a service firm may bring delay to both regular customers and discount customers. Now we derive relations between the steady state time spent in queue (in the promotion period and the post promotion period) and the promotion size. Proposition 3.3.1 below proves that both the promotion period time spent in queue w 1 (G) and the post promotion period time spent in queue w 2 (G) are increasing with promotion size G. Intuitively, with the increase of promotion size G, the service firm brings more customers to the system and the system is more congested. As a result, the expected delays are longer in both the promotion period and the post promotion period. Proposition 3.3.1. Customer’s expected time in queue both in the promotion period and in the post promotion period increases with promotion size G, i.e. dw 1 (G) dG ≥ 0 and dw 2 (G) dG ≥ 0. The next proposition identifies how a customer’s expected steady-state time spent in queue change with other metrics of the firm such as price p, current market size λ when a firm does not have promotions. We would like to know what firms tend to have longer customer’s time in queue even without promotion. Proposition3.3.2. If the service firm does not have promotion, i.e. (G= 0), its customer’s expected time in queue (i) decreases with price p, i.e. dw 1 dp | G=0 ≤ 0; (ii) increases with market size λ, i.e. dw 1 dλ | G=0 ≥ 0; (iii) does not depend on α. Proposition 3.3.2 suggests that a high price firm with low market size and high unit waiting cost tends to have a shorter average system time. The firm’s profit is a function of the promotion period’s customer’s expected time in queue. The next section explores whether the firm’s decision of whether promotion could be reduced to a simple threshold of the firm’s current customer’s expected time in queue and how such threshold changes with price p, unit waiting cost h and market size λ 89 3.3.4 When are Promotions Profitable? One of the central question for a business firm in coupon promotion is that whether it is profitable for the business. Specifically, if the firm’s revenue function π(G) is increasing at G = 0, then a promotion is profitable. To answer this question, it will be convenient to write the revenue as a function of post promotion period waiting time. As will be shown in the proof of Proposition 3.3.3, w 2 is increasing with w 1 and we represent the service firm’s profit as a function of the time spent in queue in promotion period, i.e. π w (w 1 ) = αμ 2 w 1 1+μw 1 + δμ 2 w 2 (w 1 ) 1+μw 2 (w 1 ) +(1−α)λ ¯ F(p+hw 1 ). Taking the derivative with respect to w 1 , we have dπ w (w 1 ) dw 1 | G=0 = αμ 2 (1+μw 1 ) 2 + δμ 2 (1+μw 2 ) 2 dw 2 (w 1 ) dw 1 | G=0 +(1−α)hλ ¯ F ′ (p+hw 1 ) (3.3.4) So promotion is profitable if dπw(w 1 ) dw 1 | G=0 > 0. The analysis of Theorem 3.3.3 is based on this idea and it shows that there exists a threshold with respect to current customer’s expected time in queue so that promotion is profitable if current customer’s expected time in queue is below the threshold. Theorem 3.3.3. There exists a unique ¯ w such that α+δ ¯ F(h¯ w+p) ¯ F(h¯ w+pα) = (1−α)h¯ wf(p+h¯ w) ¯ F(p+h¯ w) . Promo- tion is profitable if the expected time in queue in the promotion period at G= 0 ( i.e. w 1 (0)) satisfies w 1 (0)≤ ¯ w. Theorem 3.3.3 provides a simple criteria for a service firm to decide whether promotion is profitable. Intuitively, if waiting cost h and market size λ are large enough, promotion is not profitable to the firm. The term ¯ F(p) ¯ F(pα) characterizes the ratio of repeat customers to the discount customer pool after promotion. If ¯ F(p) ¯ F(pα) is large, then the repeat ratio is large and the promotion is more likely to be a profitable strategy. The next proposition investigates how the threshold policy ¯ w changes with price p, dis- count rate α and market size λ 90 Proposition 3.3.4. The threshold ¯ w (i) decreases with price p; (ii) increases with discount rate α ; (iii) does not change with market size λ. Comparing Proposition 3.3.2 and Proposition 3.3.4, it follows that promotion is more likely to be a profitable strategy for firms with small market size λ and small discounts, i.e. largeα. However, the trend with respect top is not obvious: intuitively, a high price implies that the repeated customer ratio in the discount customer pool ¯ F(p) ¯ F(pα) is small because F is IFR, or in other words, it is more likely to induce more deal hunters to the service. On the other hand, a high price p also reduces the actual waiting time w 1 . The dominating effects depends on parameters and the properties of distribution F. For example, when F follows an uniform distribution we found cases in which promotion profitability increases with price and cases in which it decreases with price. However, when f(v) F(v) is bounded, the following proposition characterizes a nice property. Proposition 3.3.5. If the hazard rate f(v) F(v) is bounded, then there always exists p such that promotion is profitable if p≥ ¯ p. As a summary, we have explored when the promotion is profitable in this section and characterized how the threshold changes with parameters. A more general question is what is the optimal promotion size G that maximize the firm’s revenue. A firm essentially face the following optimization problem: max G π(G) =λpF(p+hw 1 )+GαpF(αp+hw 1 )+δ(λ+G)p ¯ F(p+hw 2 ) (3.3.5) s.t.w 1 = λF(p+hw 1 )+G ¯ F(αp+hw 1 ) μ(μ−λ ¯ F(p+hw 1 )−G ¯ F(αp+hw 1 )) (3.3.6) w 2 = (λ+G)(1−p−hw 1 ) μ(μ−(λ+G)(1−p−hw 2 )) (3.3.7) 91 λ ¯ F(p+hw 1 )+G ¯ F(αp+hw 1 )≤ μ (3.3.8) (λ+G) ¯ F(p+hw 2 )≤μ (3.3.9) The complexity in solving the model explicitly is due the equilibrium time in queue in promotion and post promotion period. In order to yield insights that does not come out of exact analysis, we next explain an asymptotic analysis is more tractable and thus provided in the next section. We study a large market problem based on the original problem. 3.4 Approximate Analysis We consider a sequence of systems indexed by n, in which the (regular) market size and service rate in system ‘n’ are λ n = nλ and μ n = nμ. Denote the total revenue as π n (G), the optimal solution as G ∗ n and the average time in queue in the promotion period and post promotion period as w n 1 and w n 2 respectively. By applying a law of large numbers argument, Proposition 3.4.2 shows that on the O(n) scale 1 , the system operates in a deterministic regime, this is called a fluid approximation. We will use this to compute an approximation to the optimal promotion size. 3.4.1 A Large Market Approximation We consider the following limiting problem, max G,¯ w 1 ,¯ w 2 ¯ π(G) :=λp ¯ F(p+h¯ w 1 )+Gαp ¯ F(αp+h¯ w 1 )+δ(λ+G)p ¯ F(p+h¯ w 2 ) (3.4.1) s.t.λ ¯ F(p+h¯ w 1 )+G ¯ F(αp+h¯ w 1 )≤μ (3.4.2) (λ+G) ¯ F(p+h¯ w 2 )≤μ (3.4.3) Proposition 3.4.1. We have G ∗ n n →G ∗ where ¯ G ∗ solves (3.4.1)-(3.4.3). 1 For a sequence of non-negative real-valued numbers{a n ,b n },{a n } is said to be O(b n ) if limsup n→∞ an bn <∞ 92 We next characterize the solution to the optimization problem (3.4.1)-(3.4.3). First consider the case λF(p) > μ, then ¯ w 1 > 0 and ¯ w 2 > 0. The system is overloaded in both promotion periods and hence no promotion is an optimal strategy. Suppose instead that μ≥ λF(p). The constraints in the optimization problem (3.4.1)-(3.4.3) could be divided in the following two scenarios: (i) ¯ w 1 > 0, ¯ w 2 = 0 and (ii) ¯ w 1 = 0, ¯ w 2 = 0. (Notice that ¯ w 1 =0, ¯ w 2 >0wouldnever occurbecauseλ ¯ F(p)+G ¯ F(αp)≥ (λ+G) ¯ F(p)>(λ+G) ¯ F(p+h¯ w 2 ) =μ, which leads to a contradiction.) As a result, the feasible region of a service provider’s fluid limit optimization problem is decomposed into two parts. (i) ¯ w 1 >0, ¯ w 2 = 0 (Overloaded case): the service system is overloaded in the promotion period and critical loaded in the post promotion period. The expected time spent in queue in the post promotion period is 0. It satisfies λ ¯ F(p)+G ¯ F(αp) > μ and (λ+G) ¯ F(p)≤ μ. The optimization problem in this scenario is: max G,¯ w 1 ¯ π(G) =λp ¯ F(p+h¯ w 1 )+Gαp ¯ F(αp+h¯ w 1 )+δ(λ+G)p ¯ F(p) (3.4.4) s.t. λ ¯ F(p+h¯ w 1 )+G ¯ F(αp+h¯ w 1 )=μ (3.4.5) (λ+G) ¯ F(p)≤μ (3.4.6) λ ¯ F(p)+G ¯ F(αp)>μ. (3.4.7) (ii) ¯ w 1 = 0, ¯ w 2 = 0 (Underloaded case): the service system is critical loaded in the promotion period and under loaded in the post promotion period. The expected time spent in queue in both periods are 0, or equivalently, λ ¯ F(p)+G ¯ F(αp)≤μ and (λ+G) ¯ F(p)≤μ. The optimization problem in this scenario is max G ¯ π(G)=λp ¯ F(p)+Gαp ¯ F(αp)+δ(λ+G)p ¯ F(p) (3.4.8) s.t. λ ¯ F(p)+G ¯ F(αp)≤μ. (3.4.9) Proposition 3.4.2 identifies the optimal promotion size in each of the cases discussed above. The logic is as follows: the firm’s approximate revenue ¯ π(G) increases with G if 93 G< μ−λ ¯ F(p) ¯ F(αp) and decreases with GifG> μ−λ ¯ F(p) ¯ F(p) , so the optimal promotionsize ¯ G ∗ is within the range [ μ−λ ¯ F(p) ¯ F(αp) , μ−λ ¯ F(p) ¯ F(p) ]. Further, the firm’s approximate revenue ¯ π(G) is convex with respect to G in this range, and then the optimal solution has to be either one of the corner solution: ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) or ¯ G ∗ = μ−λ ¯ F(p) ¯ F(p) . Proposition 3.4.2. If μ>λ ¯ F(p) and F is a concave function, then defining δ = λ(1−α)( ¯ F(p)− ¯ F(p+h¯ w 1 )) (1− ¯ F(p) ¯ F(αp) )(μ−λ ¯ F(p)) , we have: (i) If δ ≥ δ, then ¯ G ∗ = μ ¯ F(p) −λ, ¯ π ∗ ( ¯ G ∗ ) = (α+δ)pμ+(1−α)λp ¯ F(p+h¯ w 1 ), ¯ w ∗ 1 > 0, ¯ w ∗ 2 =0 where ¯ w ∗ 1 satisfies λ ¯ F(p+h¯ w ∗ 1 )+( μ ¯ F(p) −λ) ¯ F(αp+h¯ w ∗ 1 ) =μ; (ii) If δ ≤ δ, ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) , ¯ π ∗ ( ¯ G ∗ ) = αpμ + λp(1− α) ¯ F(p) + δ(λ( ¯ F(αp)− ¯ F(p))+μ)p ¯ F(p) ¯ F(αp) , ¯ w ∗ 1 = ¯ w ∗ 2 =0. Intuitively, if the service provider does not have enough capacity (μ < λ ¯ F(p)), the promotion is not necessary, ¯ G = 0. If a service firm has enough capacity (μ > λ ¯ F(p)), Proposition 3.4.2 suggest that a service firm operates either in the overloaded case, or in the underloaded case depending on the future discounting factor δ: If δ is large, the service firm wouldlike tousethepromotiontointroducemorecustomers. Thesystem iscritically loaded in the promotion period such that the supply and demand is balanced after the promotion. If δ is small, a service firm may find it is crucial to use the promotion to balance promotion period’s supply and demand so that the expected delay in both periods is zero. The next proposition provides the comparative analysis of ¯ w 1 and δ in the overloaded case. Proposition 3.4.3. If δ > ¯ δ, then ¯ w ∗ 1 >0 and we have: (i) dw ∗ 1 dλ ≥ 0, dw ∗ 1 dα ≤ 0, dw ∗ 1 dp ≤ 0 (ii) dδ dλ ≥ 0, dδ dα ≤ 0; If F(x) is concave and has an increasing failure rate, then dδ dp ≤ 0 94 Figure 3.1. Threshold δ and the operational cases as p changes. ߜ ҧ ߜ ҧ ൌ ሺ ͳ െ ߙ ሻ ሺ ͳ െ ߙ ሻ ߣ ߤ ͳ ͳ ߙ ߣ ሺ ͳ െ ߙ ሻ ߤ O v e r l o a d e d C a s e U n d e r l o a d e d C a s e ߜ ҧ ߜ ҧ ൌ ሺ ͳ െ ߙ ሻ ሺ ͳ െ ߙ ሻ ߣ ߤ ߣ ߤ ͳ ͳ െ Ͳ Ͳ Proposition 3.4.3 implies that a service firm is more likely to operate in the underloaded case and focus on balancing the supply and demand in the system in the promotion period if the market size λ is large. In contrast, a service firm is more likely to operate in the overloaded case and focus on balancing the supply and demand in the system in the post promotionperiodifpriceishighordiscountislarge,i.e. αissmall. Foruniformlydistributed valuation, we have hw ∗ 1 =p(1−α)(1− λ(1−p) μ ), δ = (1−α)(1−pα)λ μ . Numerical Illustration Figure 3.1-3.3 show the overloaded case, underloaded case and the threshold ¯ δ as a function of price p, ratio λ μ and price discount rate α. Figures 3.1 shows that if the discounting factor δ is large enough, the service provider should operate in the overloaded case because it helps to balance the future supply and demand in the system. However, if δ is relatively small, the service provider should operate in the underloaded case when p is small and in the overloaded case when p is large because the service provider may have an incentive to introduce more populations when p is large, should operate in the overloaded case when λ μ is small and in the underloaded case when λ μ is large because there is more capacity in the system when λ μ is large, and should operate in the underloaded case when α is small and overloaded case when α is large because the promotion profit margin is high when α is large. 95 Figure 3.2. Threshold δ and the operational cases as λ μ changes. ߜ ҧ ߜ ҧ ൌ ሺ ͳ െ ߙ ሻ ሺ ͳ െ ߙ ሻ ߣ ߤ ͳ ͳ ߙ ߣ ሺ ͳ െ ߙ ሻ ߤ ߜ ҧ ߜ ҧ ൌ ሺ ͳ െ ߙ ሻ ሺ ͳ െ ߙ ሻ ߣ ߤ ߣ ߤ O v e r l o a d e d C a s e U n d e r l o a d e d C a s e ͳ ͳ െ Ͳ Ͳ Figure 3.3. Threshold δ and the operational cases as α changes. ߜ ҧ ߜ ҧ ൌ ሺ ͳ െ ߙ ሻ ሺ ͳ െ ߙ ሻ ߣ ߤ ߙ O v e r l o a d e d C a s e U n d e r l o a d e d C a s e ߣ ߤ ͳ Ͳ 96 3.4.2 Refined Approximation In the fluid approximation, we find that if ¯ G ∗ = μ ¯ F(p) −λ, then the service system is critically loaded in the second period and if ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) the service system is critically loaded in the first period. Now we refine the optimal promotion size in each scenario at the O( √ n) level and investigate the diffusion approximation of the promotion size. The next proposition demonstrates that in the overloaded case, the system operates near 100% utilization in post promotion period according to 1−ρ n 2 ≈ γ 2 √ n ; In the underloaded case, the system operates near to full utilization in promotion period according to 1−ρ n 1 ≈ γ 1 √ n . Proposition 3.4.4. If the distribution F has an increasing failure rate, the optimal promo- tion size G ∗ n must be of the form G ∗ n =n ¯ G ∗ − √ ng+o( √ n), for some g > 0. Proposition3.4.4specifiestheappropriateregimeforasymptoticanalysis. Thepromotion sizemustadmitthedecompositionG ∗ n ≈n ¯ G ∗ − √ ng+o( √ n),where ¯ G ∗ hasbeencomputedin Proposition3.4.2. Sotheoptimalpromotionproblemisnowreducedtoselectingtheoptimal value g that will maximize revenues. Let G n = n ¯ G ∗ − √ ng+o( √ n), ˆ π = lim n→∞ √ n(¯ π( ¯ G ∗ )− π n (Gn) n ). The next lemma illustrates that in order to maximize the service provider’s profit, we should find g ∗ such that g ∗ = argmin g ˆ π. Lemma 3.4.5. If g ∗ =argminˆ π, then G ∗ n =n ¯ G ∗ − √ ng ∗ +o( √ n). Intuitively, when we take into account the stochastic of the system, we should introduce less customers than the fluid limit in order to buffer the uncertainty of the arrival process so g≥ 0. In addition, the service firm will make less revenue than the fluid revenue and our goal is to find g such that the difference is as small as possible. The key of the analysis is to estimate the average system time when the system is critical loaded. Let e w n 2 := w n 2 √ n if ¯ G ∗ = μ−λF(p) F(p) and e w n 1 :=w n 1 √ n if ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) . The following proposition computes e w n 2 and e w n 1 in each scenario respectively 97 Proposition 3.4.6. (i) If μ≥λ ¯ F(p) and δ > ¯ δ, then e w n 2 = 1 g ¯ F(p) +O( 1 √ n ); (ii) If μ≥ λ ¯ F(p)) and δ≤ ¯ δ, then e w n 1 = 1 g ¯ F(αp) +O( 1 √ n ). The next step is to estimate ˆ π = lim n→∞ √ n(¯ π( ¯ G ∗ )− π n (G ∗ n ) n ). We use Taylor’s expansion to estimate ˆ π in Lemma 3.4.7. Lemma 3.4.7. (i) If μ ≥ λ ¯ F(p) and δ > ¯ δ, then ˆ π = gαp ¯ F(αp) + δ(λ + ¯ G ∗ ) phf(p) g ¯ F(p) + δgp( ¯ F(p)− f(p) g ¯ F(p) ); (ii) If μ≥λ ¯ F(p)) and δ≤ ¯ δ, then ˆ π = (λ+α ¯ G ∗ ) pf(p)h g ¯ F(αp) +gαp( ¯ F(αp)− f(αp)h g ¯ F(αp) )+δgp ¯ F(p). The next result optimize ˆ π to obtain the optimal refinement. Proposition 3.4.8. (i) If μ ≥ λ ¯ F(p) and δ > ¯ δ, we have g ∗ = q δμhf(p) F 2 (p)(αF(αp)+δF(p)) . And further, G ∗ n =n( μ ¯ F(p) −λ)−g ∗ √ n+o( √ n); (ii) Similarly, μ≥λ ¯ F(p) andδ≤ ¯ δ, we haveg ∗ = q (λ( ¯ F(αp)−α ¯ F(p))+αμ)f(p)h (α ¯ F(αp)+δ ¯ F(p)) ¯ F 2 (αp) . And further, G ∗ n =n( μ−λ ¯ F(p) ¯ F(αp) )−g ∗ √ n+o( √ n) The following proposition investigate how the refined optimal promotion size g ∗ changes with parameters μ, h, and δ. g ∗ is non-decreasing in p is from the fact that both f(p) ¯ F(p) and − ¯ F(p) ¯ F(αp) is increasing with p if F has an increasing failure rate. Proposition 3.4.9. If for the case μ≥λ ¯ F(p): (i) If δ > ¯ δ, then dg ∗ dμ ≥ 0, dg ∗ dh ≥ 0, dg ∗ dδ ≥ 0; if F has an increasing failure rate, dg ∗ dp ≥ 0 ; (ii) If δ≤ ¯ δ, then dg ∗ dμ ≥ 0, dg ∗ dh ≥ 0, dg ∗ dδ ≤ 0; if F has an increasing failure rate dg ∗ dp ≥ 0. The relation between g ∗ and μ, h or δ can be directly observed from the representation of g ∗ . The relation between g ∗ and δ is in the opposite direction when δ ≥ ¯ δ and δ ≤ ¯ δ. It means that it hits the minimum point at δ = ¯ δ. In addition, if F has an increasing failure rate, then g ∗ is increasing with p independent of δ since F has an increasing failure rate implies both f(p) ¯ F(p) is increasing and ¯ F(p) ¯ F(αp) is increasing. 98 3.5 Conclusions A service firm always faces a tradeoff between exploring a larger market size and reducing congestion in the system. This paper considers a two period M/M/1 queue model to model this trade-off when the firm has the option of using a promotion strategy to inject more customers into the system. We find a threshold waiting time such that if the firm’s current wait time exceeds this threshold then such promotions would not be profitable strategy. In order to explore the effective promotion size to maximize a firm’s profit, we develop an asymptotic optimization problem and characterize the fluid approximation solution. Herein, we find that a dual-asymptotic regime in which the firm either operates in an overloaded regime in the promotion period and a critically loaded regime in the post-promotion period, or critically loaded in the promotion period and underloaded in the post-promotion period. The former occurs if the firm’s discount factor is small so that the firm balances the system supply and demand in the post-promotion period at the cost of additional congestion in the promotion period. We refine this fluid approximation further using standard diffusion approximations. In this work, we have taken price and capacity of the service firm as a given. One direction of future research is to incorporate capacity or price as a decision variable in the model. For example, one may suspect that price and capacity by themselves could be effective tools to control customer flow. So, the combination of these along with the promotion decision would make for an interesting study. Another extension that we are interested in is to consider introducing the promotions in a controlled manner, perhaps by some form of scheduling, to minimize externalities. The current work assumes that both regular customers and discount customers decide whether to use the service based on the steady state waiting time and the price and the firm treats customersinbothpoolsequallybasedonafirstcomefirstserverule. Inthenextstep,wewill develop priority rules to serve regular customers and discount customers and differentiate them through scheduling. While a regular customer’s decision to use the service is still 99 dependent on the steady waiting time, a discount customer’s decision to use the service may depend on the system state and firm’s priority rule. 100 APPENDIX A PROOFS OF CHAPTER 1 A.1 Proofs of Propositions and Theorems Proof of Proposition 1.4.3: (i) The first-best channel quantity for the supplier is Q FB H = argmax Q {α(A H −Q)Q+wQ}= argmax Q αQ(A H −Q+ w α )= 1 2 A H + 1 2 w α when the demand is high, or Q FB L = argmax Q {α(A L −Q)Q+wQ} = argmax Q αQ(A L −Q+ w α ) = 1 2 A L + 1 2 w α when the demand is low. (ii)Inthe nonleakageequilibrium, by Proposition1.4.1,we have Q N∗ H = 1 2 A H + 1 6 μ− 2 3 w 1−α and Q N∗ L = 1 2 A L + 1 6 μ− 2 3 w 1−α . Thus, Q N∗ H < (>, or =)Q FB H and Q N∗ L < (>, or =)Q FB L if w μ > (<, or =) α(1−α) 3+α . (iii) In the leakage equilibrium, by Proposition 1.4.2, we have Q S∗ H = 3 4 (A H − w 1−α ) and Q S∗ L = 3 4 (A L − w 1−α ), if θ≥ 3, q S2∗ iL + 1 2 (A L −q S2∗ iL − w 1−α ), if θ < 3. Thus, Q S∗ H ≥ Q FB H (i.e., the retail- ers overorder when the demand is high) if and only if A H ≥ 2 w α +3 w 1−α , or, w μ ≤ α(1−α) 2+α A H μ = UB N . Similarly, when θ ≥ 3, Q S∗ L ≤ Q FB L (underorder when the demand is low) if and only if w μ ≥ α(1−α) 2+α A L μ . When θ < 3, Q S∗ L ≤ Q FB L if and only if w μ is greater than the threshold w μ such that Q S2∗ L = Q FB L (Q S∗ L becomes Q S2∗ L in the case θ < 3). Let LB N = α(1−α) 2+α A L μ , if θ≥ 3, the w μ such that Q FB L =Q S2∗ L , if θ < 3. Then, the retailers underorder in both demandstatesif w μ >UB N ,overorderinbothstatesif w μ <LB N ,andunderorder(overorder) when the demand is low (high) if LB N ≤ w μ ≤UB N . 101 Proof of Proposition 1.5.1: Suppose the incumbent orderse q iH > q S∗ iL when the demand is high and e q iL ≤ q S∗ iL when thedemand is low. Ifthesupplier leaks the incumbent’s orderquantity q i , the entrant places an order q e , as the best response to q i . This is a standard Stackelberg game. However, if the supplier does not leak, the entrant has no information about q i (and hence the demand state) when placing the order q e . The game is equivalent to a simultaneous game and the Cournot equilibrium is achieved. We analyze these two situations below. (a) If supplier leaks the incumbent’s order quantity e q iH and the demand is believed to be high (which requirese q iH >q S∗ iL ), the entrant places the ordere q eH as the best response to e q iH , i.e., e q eH = argmax qe {(1−α)(A H −e q iH −q e )q e −wq e } = 1 2 (A H −e q iH − w 1−α ). Thus, when the demand is high, the price is P =A H −e q iH −e q eH = 1 2 (A H −e q iH + w 1−α ), and the supplier’s profit is π S sH =w(e q iH +e q eH )+αP(e q iH +e q eH ) = 1 2 [w+ α 2 (A H −e q iH + w 1−α )](A H +e q iH − w 1−α ). (A.1.1) Similarly, if the supplier leaks the incumbent’s order quantity e q iL and the demand is believed to be low (which requires e q iL ≤ q S∗ iL ), the entrant will place the order e q eL , as the best response toe q iL , i.e., e q eL = argmax qe {(1−α)(A L −e q iL −q e )q e −wq e }= 1 2 (A L −e q iL − w 1−α ). Thus, when the demand is low, the price is P =A L −e q iL −e q eL = 1 2 (A L −e q iL + w 1−α ), and the supplier’s profit is π S sL =w(e q iL +e q eL )+αP(e q iL +e q eL ) = 1 2 [w+ α 2 (A L −e q iL + w 1−α )](A L +e q iL − w 1−α ). (A.1.2) (b) If the supplier does not leak, the result of Proposition 1.4.1 applies. That is, the incumbent orders q N ∗ iH = 1 2 A H − 1 6 μ− 1 3 w 1−α if the demand is high andq N ∗ iL = 1 2 A L − 1 6 μ− 1 3 w 1−α 102 if the demand is low. The entrant orders q N ∗ e = 1 3 (μ− w 1−α ). When the demand is high, the price is P =A H −q N ∗ iH −q N ∗ e = 1 2 A H − 1 6 μ+ 2 3 w 1−α , and the supplier’s profit is: π N ∗ sH =w(q N ∗ iH +q N ∗ e )+αP(q N ∗ iH +q N ∗ e ) = (w+α( 1 2 A H − 1 6 μ+ 2 3 w 1−α ))( 1 2 A H + 1 6 μ− 2 3 w 1−α ) = α 4 ( 2w α +A H − 1 3 μ+ 4 3 w 1−α )(A H + 1 3 μ− 4 3 w 1−α ). (A.1.3) When the demand is low, the price is P = A L −q N ∗ iL −q N ∗ e = 1 2 A L − 1 6 μ+ 2 3 w 1−α , and the supplier’s profit is: π N ∗ sL =w(q N ∗ iL +q N ∗ e )+αP(q N ∗ iL +q N ∗ e ) =(w+α( 1 2 A L − 1 6 μ+ 2 3 w 1−α ))( 1 2 A L + 1 6 μ− 2 3 w 1−α ) = α 4 ( 2w α +A L − 1 3 μ+ 4 3 w 1−α )(A L + 1 3 μ− 4 3 w 1−α ). (A.1.4) Thesupplierprefersthenonleakageequilibrium ifthesupplier’s profitisweakly higherunder nonleakage no matter whether the demand is high or low. That is, the following inequalities are satisfied simultaneously: π S sH ≤π N ∗ sH , and π S sL ≤π N ∗ sL . Thus, from (A.1.1)–(A.1.4), we have (e q iH ) 2 −2( w α + w 1−α )e q iH +( 2w α +A H − 1 3 μ+ 4 3 w 1−α )(A H + 1 3 μ− 4 3 w 1−α ) −( 2w α +A H + w 1−α )(A H − w 1−α )≥ 0, (e q iL ) 2 −2( w α + w 1−α )e q iL +( 2w α +A L − 1 3 μ+ 4 3 w 1−α )(A L + 1 3 μ− 4 3 w 1−α ) −( 2w α +A L + w 1−α )(A L − w 1−α )≥ 0. These two inequalities turn out to be the same after simplification, i.e., both e q iH and e q iL satisfy e q 2 i −2( w α + w 1−α )e q i + 1 3 (μ− w 1−α )( 2w α + 7 3 w 1−α − 1 3 μ)≥ 0. (A.1.5) 103 Notice that ( w α + w 1−α ) 2 − 1 3 (μ− w 1−α )( 2w α + 7 3 w 1−α − 1 3 μ) = 1 9 (μ−3 w α −4 w 1−α ) 2 ≥ 0. Thus (A.1.5) is satisfied and the supplier prefers the nonleakage equilibrium if q i ≥ w α + w 1−α + 1 3 μ−3 w α −4 w 1−α , or 0≤q i ≤ w α + w 1−α − 1 3 μ−3 w α −4 w 1−α . Proof of Proposition 1.5.2: (i)ByPropositions1.4.1and1.4.2,wehaveq N ∗ iH −q S ∗ iH = ( 1 2 A H − 1 6 μ− 1 3 w 1−α )− 1 2 (A H − w 1−α )= − 1 6 (μ− w 1−α )≤ 0 (as w 1−α ≤ A H ). Similarly, we have q S1 ∗ iL −q N ∗ iL = 1 2 (A L − w 1−α )−( 1 2 A L − 1 6 μ− 1 3 w 1−α ) = 1 6 (μ− w 1−α )≥ 0 for θ≥ 3, and q S2 ∗ iL −q N∗ iL =A H − A L 2 − 1 2 w 1−α − s (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 −( 1 2 A L − 1 6 μ− 1 3 w 1−α ), for θ≤ 3. The inequality q S2 ∗ iL −q N ∗ iL ≥ 0 is equivalent to the following: A H −A L − 1 6 w 1−α + 1 6 μ− s (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 ≥ 0, (A.1.6) i.e., θ−1+ 1 6 (pθ+(1−θ))− r (θ− 1 2 ) 2 − 1 4 θ 2 ≥ 0, (A.1.7) where θ = A H − w 1−α A L − w 1−α . Inequality (A.1.7) can be transformed to: 1 6 p+1 2 − 3 4 ! θ 2 + 1−2( 1 6 p+1)( 5 6 + 1 6 p) θ+ 5 6 + 1 6 p 2 − 1 4 ≥ 0. (A.1.8) This is a quadratic inequality in θ and because Δ = 1−2( 1 6 p+1)( 5 6 + 1 6 p) 2 −4 1 6 p+1 2 − 3 4 ! 5 6 + 1 6 p 2 − 1 4 ! =− 1 18 p<0, the inequality (A.1.8) holds and therefore q S2 ∗ iL ≥ q N ∗ iL . (ii) Since ¯ q i = w α + w 1−α + 1 3 μ−3 w α −4 w 1−α , q i = w α + w 1−α − 1 3 μ−3 w α −4 w 1−α and q N ∗ iH = 1 2 A H − 1 6 μ− 1 3 w 1−α , we have q N ∗ iH − ¯ q i = 1 2 A H − 1 6 μ− 1 3 w 1−α − w α + w 1−α + 1 3 μ−3 w α −4 w 1−α 104 = 1 2 A H − 1 6 μ− 4 3 w 1−α − w α − 1 3 μ−3 w α −4 w 1−α , q N ∗ iH −q i = 1 2 A H − 1 6 μ− 1 3 w 1−α − w α + w 1−α − 1 3 μ−3 w α −4 w 1−α = 1 2 A H − 1 6 μ− 4 3 w 1−α − w α + 1 3 μ−3 w α −4 w 1−α . If μ≥ 3 w α +4 w 1−α , then q N ∗ iH −¯ q i = 1 2 A H − 1 2 μ≥ 0 and hence q N ∗ iH ≥ ¯ q i ≥q i ; if μ≤ 3 w α +4 w 1−α , then q N∗ iH −q i = 1 2 A H − 1 2 μ≥ 0 and hence q N∗ iH ≥q i . In both cases, we have q N ∗ iH ≥q i . Similarly, replacing q N ∗ iH by q N ∗ iL = 1 2 A L − 1 6 μ− 1 3 w 1−α , we have: if μ ≥ 3 w α +4 w 1−α , then q N ∗ iL − ¯ q i = 1 2 A L − 1 2 μ ≤ 0; if μ ≤ 3 w α +4 w 1−α , then q N ∗ iL −q i = 1 2 A L − 1 2 μ ≤ 0 and hence q N ∗ iL ≤q i ≤ ¯ q i . So, in both cases, we obtain q N ∗ iL ≤ ¯ q i . (iii) It is equivalent to showing that π N ∗ iH π S ∗ iH ≥ 1. From Propositions 1.4.1 and 1.4.2, we can show that π N ∗ iH = 1 4 (A H − 1 3 μ− 2 3 w 1−α ) 2 (1−α), π S ∗ iH = 1−α 8 (A H − w 1−α ) 2 . Thus, the above inequality is equivalent to 1 4 (A H − 1 3 μ− 2 3 w 1−α ) 2 (1−α) 1 8 (A H − w 1−α ) 2 (1−α) ≥ 1 ⇔(1− 1 3 μ− w 1−α A H − w 1−α ) 2 ≥ 1 2 ⇔(1− 1 3 (p+ 1−p θ )) 2 ≥ 1 2 . Sinceθ≥ 1,andhence1− 1 3 (p+ 1−p θ )≥ 0,thelastinequality isequivalenttoθ≥ 1−p 3(1− √ 2 2 )−p ≥ 0. Thus, π N ∗ iH ≥π S ∗ iH if and only if θ≥ 1−p 3(1− √ 2 2 )−p ≥ 0. Proof of Theorem 1.5.3: I. Sufficient Conditions. Thereexistsanonleakageequilibriumiftheincumbent’sorderquantityintheequilibrium lies in the supplier’s nonleakage intervals. 105 (a) When the demand is low, for the nonleakage equilibrium to exist, we require that either q N ∗ iL ≤ q i or q N ∗ iL ≥ ¯ q i . From Proposition 1.5.2 (ii), it must be q N ∗ iL ≤ q i . Moreover, we need to check the condition that no player wants to deviate from the equilibrium. In other words, the incumbent should notplace anorder other thanq N ∗ iL and the supplier should stick to not leaking. If q S ∗ iL ≤ q i , notice that even if the incumbent deviates to the leakage interval and moti- vates the supplier to leak the order information, the entrant will believe that the demand is high, given the entrant’s belief structure (Pr e (A = A H ) = 1 if q i ≥ q S ∗ iL ). Define the profit for the low-type incumbent when he deviates to the interval (q i ,¯ q i ) as: π D ∗ iL =max q i ≥q i (1−α)(A L −q i −q eH (q i )− w 1−α )q i where q eH (q i ) =argmax qe (1−α)(A H −q i −q e − w 1−α )q e = 1 2 (A H −q i − w 1−α ). By substitution, we obtain π D ∗ iL = max q i ≥q i (1−α)(A L − 1 2 A H − 1 2 q i − 1 2 w 1−α )q i . (A.1.9) Notice that the optimal solution of the unconstrained version of (A.1.9) is q i = A L − A H 2 − 1 2 w 1−α ≤ 1 2 (A L − w 1−α ) ≤ q i . So the constraint q i ≥ q i is binding, and π D ∗ iL = (1−α)(A L − 1 2 A H − 1 2 q i − 1 2 w 1−α )q i . If μ−3 w α −4 w 1−α ≤ 0, then q i = 1 3 (μ− w 1−α ) and π D ∗ iL = 1 3 (1−α)(A L − 1 2 A H − 1 6 μ− 1 3 w 1−α )(μ− w 1−α ). If μ−3 w α −4 w 1−α ≥ 0, then q i = 2 w α + 7 3 w 1−α − 1 3 μ and π D ∗ iL =(1−α)(A L − 1 2 A H − w α + 1 6 μ− 5 3 w 1−α )(2 w α + 7 3 w 1−α − 1 3 μ). Now we compare π D ∗ iL with π N∗ iL = 1 4 (A L − 1 3 μ− 2 3 w 1−α ) 2 (1−α): 106 If μ−3 w α −4 w 1−α ≤ 0, π N ∗ iL π D ∗ iL = 1 4 (A L − 1 3 μ− 2 3 w 1−α ) 2 (1−α) 1 3 (1−α)(A L − 1 2 A H − 1 6 μ− 1 3 w 1−α )(μ− w 1−α ) = 3(1− 1 3 (pθ+(1−p)) 2 4(1− 1 2 θ− 1 6 (pθ+1−p))(pθ+1−p) = 3( 1 pθ+(1−p) − 1 3 ) 2 4( 1− 1 2 θ pθ+(1−p) − 1 6 ) . If μ−3 w α −4 w 1−α ≥ 0, i.e., pθ+(1−p)≥ 3Φ, for Φ = w α + w 1−α A L − w 1−α , π N ∗ iL π D ∗ iL = 1 4 (A L − 1 3 μ− 2 3 w 1−α ) 2 (A L − 1 2 A H − w α + 1 6 μ− 5 3 w 1−α )(2 w α + 7 3 w 1−α − 1 3 μ) = (1− 1 3 (pθ+(1−p)) 2 4(1− 1 2 θ+ 1 6 (pθ+1−p)−Φ)(2Φ− 1 3 (pθ+1−p)) = ( 1 pθ+(1−p) − 1 3 ) 2 4( 1− 1 2 θ−Φ pθ+(1−p) − 1 6 )( 2Φ pθ+(1−p) − 1 3 ) ≥ 3( 1 pθ+(1−p) − 1 3 ) 2 4( 1− 1 2 θ−Φ pθ+(1−p) − 1 6 ) ≥ 3( 1 pθ+(1−p) − 1 3 ) 2 4( 1− 1 2 θ pθ+(1−p) − 1 6 ) . Thus, to show π D ∗ iL ≤ π N ∗ iL , we show that 3( 1 pθ+(1−p) − 1 3 ) 2 4( 1− 1 2 θ pθ+(1−p) − 1 6 ) ≥ 1. (A.1.10) Let y = 1 pθ+(1−p) . Since p∈ [0,1] and θ≥ 1, we have y∈ [ 1 θ ,1]. Inequality (A.1.10) can be transformed to 3(y− 1 3 ) 2 ≥ 4((1− 1 2 θ)y− 1 6 ), which is equivalent to 3y 2 − (4+ 2 p )y + 2 p +1 ≥ 0. Solving this inequality, we have either y ≤ 1 or y ≥ 2 p +1 3 . Because y ∈ [ 1 θ ,1], 3( 1 pθ+(1−p) − 1 3 ) 2 . 4( 1− 1 2 θ pθ+(1−p) − 1 6 ) ≥ 1 holds. Therefore, π D ∗ iL ≤ π N ∗ iL as long as q S ∗ iL ≤q i . (b)When the demand is high, by Proposition 1.5.2(ii), the incumbent’s order quantity that will not be leaked by the supplier must satisfy q N ∗ iH ≥ ¯ q i . From Proposition 1.5.2(iii), π N ∗ iH ≥ π S ∗ iH and hence the incumbent would not deviate if θ ≥ 1−p 3(1− √ 2 2 )−p ≥ 0. Thus, when the demand is high, a nonleakage equilibrium exists if q N ∗ iH ≥ ¯ q i and θ≥ 1−p 3(1− √ 2 2 )−p ≥ 0. Therefore, a nonleakage equilibrium exists under the assumptions of the theorem. II. Necessary Conditions. 107 Table A.1. Channel Quantity Distortion from Supplier’s First-Best Nonleakage, H Leakage, H Nonleakage, LLeakage, L w μ Q N∗ H −Q FB H Q S∗ H −Q FB H Q N∗ L −Q FB L Q S∗ L −Q FB L (i) w μ ≥UB N − − − − (ii) α(1−α) 3+α ≤ w μ ≤UB N − + − − (iii) LB N ≤ w μ ≤ α(1−α) 3+α + + + − (iv) w μ ≤LB N + + + + Table A.2. Conditions for Supplier to Prefer Nonleakage w μ High Demand Low Demand (i) w μ ≥ UB N Q S∗ H ≤Q N∗ H (≤Q FB H ) Q S∗ L ≤Q N∗ L (≤Q FB L ) (ii) α(1−α) 3+α ≤ w μ ≤UB N Q FB H −Q N∗ H ≤Q S∗ H −Q FB H (same as above) (iii) LB N ≤ w μ ≤ α(1−α) 3+α (Q FB H ≤)Q N∗ H ≤Q S∗ H Q N∗ L −Q FB L ≤ Q FB L −Q S∗ L (iv) w μ ≤LB N (same as above) (Q FB L ≤)Q N∗ L ≤Q S∗ L The participationconstraint w μ ≤ 1 2 ( 3A L μ −1)(1−α)isnecessary toensure thatq N ∗ iL ,q N ∗ iH ≥ 0. A necessary condition for nonleakage is that the supplier prefers nonleakage to leakage in both demand states, i.e., the channel distortion (from the supplier’s first-best quantity) is less severe under nonleakage than under leakage in both demand scenarios. Table A.1 compares the downstream retailers’ total order quantity with the supplier’s first-best in the two equilibria and two demand states, given different ranges of w μ (and α). The results are directly derived from Proposition 1.4.3. Table A.2 gives the conditions that ensure less channel distortionundernonleakage, fordifferent rangesof w μ andα. These conditions follow from the fact that the supplier’s profit function is quadratic in the total channel quantity. We examine the conditions in Table A.2 in more detail below. (i) If w μ ≥ UB N , the conditions are Q S∗ H ≤ Q N∗ H (≤ Q FB H ) and Q S∗ L ≤ Q N∗ L (≤ Q FB L ). From thefirstinequality, wehave 3 4 (A H − w 1−α )≤ 1 2 A H + 1 6 μ− 2 3 w 1−α andhence w μ ≥ (3 A H μ −2)(1−α). This inequality cannot be satisfied because the region lies outside of the region determined by the participation constraint, i.e., (3 A H μ −2)(1−α)> 1 2 ( 3A L μ −1)(1−α). In other words, w μ ≤UB N is necessary for the nonleakage equilibrium. 108 (ii)If α(1−α) 3+α ≤ w μ ≤UB N , the conditionsareQ FB H −Q N∗ H ≤Q S∗ H −Q FB H andQ S∗ L ≤Q N∗ L (≤ Q FB L ). Because Q S∗ L −Q N∗ L = 1 2 (A L +q S∗ iH − w 1−α )−( 1 2 A L + 1 6 μ− 2 3 w 1−α )= 1 2 (q S∗ iL −q i ), the second condition Q S∗ L ≤ Q N∗ L is equivalent to q S∗ iL ≤ q i . The first condition can be rewritten as 1 2 (A H + w α )−( 1 2 A H + 1 6 μ− 2 3 w 1−α )≤ 3 4 (A H − w 1−α )− 1 2 (A H + w α ), which reduces to w α + 17 12 w 1−α ≤ 1 4 A H + 1 6 μ, or w μ ≤ (3 A H μ +2) α(1−α) 12+5α . The new upper bound, denoted UB Nec : (3 A H μ +2) α(1−α) 12+5α , lies below the bound UB N : α(1−α) 2+α A H μ forallα,because(3 A H μ +2) α(1−α) 12+5α < α(1−α) 2+α A H μ isimpliedby(3 A H μ +2)/( A H μ )< 12+5α 2+α , 3+2 μ A H <5< 12+5α 2+α , or μ A H < 1. (iii) If LB N ≤ w μ ≤ α(1−α) 3+α , the necessary conditions in Table A.2 are (Q FB H ≤)Q N∗ H ≤Q S∗ H and Q N∗ L −Q FB L ≤ Q FB L −Q S∗ L . The first condition is equivalent to 1 2 A H + 1 6 μ− 2 3 w 1−α ≤ 3 4 (A H − w 1−α ), and w μ ≤ (3 A H μ − 2)(1− α). According to the argument in part (i), the last inequality is implied by the participation constraint w μ ≤ 1 2 ( 3A L μ −1)(1−α). Because (Q N∗ L −Q FB L )−(Q FB L −Q S∗ L ) =[( 1 2 A L + 1 6 μ− 2 3 w 1−α )−( 1 2 A L + 1 2 w α )]−[( 1 2 A L + 1 2 w α )− 1 2 (A L + q S∗ iL − w 1−α )] = 1 6 μ− 7 6 w 1−α − w α + 1 2 q S∗ iL = 1 2 (q S∗ iL −q i ), the second condition is equivalent to q S∗ iL ≤ q i . In addition, it can be shown that q S∗ iL ≤ q i implies LB N ≤ w μ . (iv) If w μ ≤ LB N , the necessary conditions given in Table A.2 are (Q FB H ≤)Q N∗ H ≤ Q S∗ H and (Q FB L ≤)Q N∗ L ≤ Q S∗ L . As in part (iii), the first condition is implied by the participation constraint. BecauseQ N∗ L −Q S∗ L = 1 2 A L + 1 6 μ− 2 3 w 1−α − 1 2 (A L +q S∗ iL − w 1−α ) = 1 2 [ 1 3 (μ− w 1−α )−q S∗ iL ] = 1 2 (¯ q i −q S∗ iL ), the second condition Q N∗ L ≤ Q S∗ L is equivalent to q S∗ iL ≥ ¯ q i . However, as shown below, this inequality cannot be satisfied. By Proposition 1.5.2 (ii), we have q N ∗ iL ≤ ¯ q i . If q i ≤ q N ∗ iL ≤ ¯ q i , the nonleakage equilibrium (q N ∗ iL ,q N ∗ iH ,q N ∗ e ) cannot be sustained by the supplier in the low demand state. Therefore, we assume q N ∗ iL < q i . Then, if the incumbent deviates to q iL ∈ [q i ,¯ q i ], the entrant will believe 109 that the demand is low (because q iL ≤ ¯ q i ≤q S∗ iL ) and the incumbent’s profit will be π D iL = max q i ≤q iL ≤¯ q i (1−α)(A L −q i −q eL (q i )− w 1−α )q i where q eL (q i )= argmax qe (1−α)(A L −q i −q e − w 1−α )q e = 1 2 (A L −q i − w 1−α ). By substitution, we obtain π D iL = 1 2 max q i ≤q iL ≤¯ q i (1−α)(A L −q i − w 1−α )q i . Because the solution to the unconstrained version of the problem is q S1∗ iL (≥ q S∗ iL )≥ ¯ q i , the constraint q iL ≤ ¯ q i must bind and π D iL = 1 2 (1−α)(A L − ¯ q i − w 1−α )¯ q i . Because w μ ≤ α(1−α) 3+α , we have ¯ q i = 1 3 (μ− w 1−α ) and π D iL = 1 6 (1− α)(A L − 1 3 μ− 2 3 w 1−α )(μ− w 1−α ). Because q N ∗ iL = 1 2 A L − 1 6 μ− 1 3 w 1−α and q N ∗ e = 1 3 (μ− w 1−α ),wehaveπ N∗ iL = (1−α)(A L −q N ∗ iL −q N ∗ e − w 1−α )q N ∗ iL = 1 4 (A L − 1 3 μ− 2 3 w 1−α ) 2 (1−α). Thus, π D iL −π N∗ iL = 1 12 (1−α)(A L − 1 3 μ− 2 3 w 1−α ) 2(μ− w 1−α )−3(A L − 1 3 μ− 2 3 w 1−α ) = 1 4 (1−α)(A L − 1 3 μ− 2 3 w 1−α )(μ−A L )≥ 0, which implies that the incumbent would deviate to ¯ q i in the low demand case and the nonleakage equilibrium (q N ∗ iL ,q N ∗ iH ,q N ∗ e ) is nonsustainable. Therefore, from parts (i)-(iv) above, the necessary conditions from the supplier’s per- spective, as given in Table A.2, (plus the incumbent’s IC constraint in case (iv)) reduce to q S ∗ iL ≤q i and w μ ≤ (3 A H μ +2) α(1−α) 12+5α . Proof of Theorem 1.5.4: Based on Theorem 1.5.3, we derive the region of (α, w μ ) for a nonleakage equilibrium to exist. Consider the three cases of r below. Because the participation constraint w μ ≤ 1 2 ( 3A L μ −1)(1−α) can be treated independently, it is omitted in the proof for simplicity. (a) Suppose r≥ 3. Because θ = A H − w 1−α A L − w 1−α =3 is equivalent to w μ = 3A L /μ−A H /μ 3−1 (1−α)= 3−r 2 1−α pr+1−p , every nonnegative (α, w μ ) satisfies w μ ≥ 3−r 2 1−α pr+1−p when r ≥ 3. Thus, θ ≥ 3 and q S ∗ iL =q S1 ∗ iL . 110 (i)Ifμ−3 w α −4 w 1−α ≥ 0,i.e. w u ≤ α(1−α) 3+α ,wehaveq i =2 w α + 7 3 w 1−α − 1 3 μand ¯ q i = 1 3 (μ− w 1−α ). Then, by Theorem 1.5.3, q N ∗ iH ≥ ¯ q i ⇒ 1 2 A H − 1 6 μ− 1 3 w 1−α ≥ 1 3 (μ− w 1−α )⇒A H ≥μ, which always holds; q S1 ∗ iL ≤q i ⇒ 1 2 (A L − w 1−α )≤ 2 w α + 7 3 w 1−α − 1 3 μ⇒ w μ ≥ 1 3 + 1 2 A L μ 2 α + 17 6(1−α) = (3 A L μ +2) (1−α)α 12+5α . Thus, we obtain a nonleakage region (3 A L μ +2) α(1−α) 12+5α ≤ w μ ≤ α(1−α) 3+α . (ii)Ifμ−3 w α −4 w 1−α ≤ 0,i.e. w u ≥ α(1−α) 3+α ,wehaveq i = 1 3 (μ− w 1−α )and ¯ q i =2 w α + 7 3 w 1−α − 1 3 μ. Then, by Theorem 1.5.3, q N ∗ iH ≥ ¯ q i ⇒ 1 2 A H − 1 6 μ− 1 3 w 1−α ≥ 2 w α + 7 3 w 1−α − 1 3 μ⇒ w μ ≤ ( 3A H μ +1) α(1−α) 12+4α ; q S1 ∗ iL ≤q i ⇒ 1 2 (A L − w 1−α )≤ 1 3 (μ− w 1−α )⇒ w μ ≥ (1−α)(3 A L μ −2). Notice that 3A H μ +1≥ 4, or(3 A H μ +1) 1 12+4α ≥ 1 3+α , andhence the curve ( 3A H μ +1) α(1−α) 12+4α is above α(1−α) 3+α for all α∈ [0,1]. Thus, we obtain another nonleakage region max{ α(1−α) 3+α ,(1− α)( A L μ −2)}≤ w μ ≤ ( 3A H μ +1) α(1−α) 12+4α . Now, we combine the above two regions. If α ≥ α 0 = 3 A L μ −2 1− A L μ , we have α(1−α) 3+α ≥ (3 A L μ −2)(1−α) and region (ii) becomes α(1−α) 3+α ≤ w μ ≤ ( 3A H μ +1) α(1−α) 12+4α . Combining region (i), we obtain the total region (3 A L μ +2) α(1−α) 12+5α ≤ w μ ≤ ( 3A H μ +1) α(1−α) 12+4α . If α≤ α 0 , we have α(1−α) 3+α ≤ (3 A L μ −2)(1−α) and region (ii) becomes (3 A L μ −2)(1−α) ≤ w μ ≤ ( 3A H μ + 1) α(1−α) 12+4α . It can be shown that (3 A L μ + 2) α(1−α) 12+5α ≥ α(1−α) 3+α in this case. Hence the nonleakage region (i) is empty and the total region is given by (3 A L μ −2)(1−α) ≤ w μ ≤ ( 3A H μ +1) α(1−α) 12+4α . It turns out that (3 A L μ +2) α(1−α) 12+5α ≥ (3 A L μ −2)(1−α) if and only if α≥α 0 . Therefore, we can summarize the total nonleakage region as max n (3 A L μ +2) α(1−α) 12+5α ,(3 A L μ −2)(1−α) o ≤ w μ ≤ ( 3A H μ +1) α(1−α) 12+4α . Because A L μ = 1 pr+1−p and A H μ = r pr+1−p , we have α 0 = 3 pr+1−p −2 1− 1 pr+1−p = 1 p(r−1) −2. Case (a) of the theorem is proved. (b) Suppose 1 < r < 23+p 11+p . The curve w μ = (3 A H μ + 1) α(1−α) 12+4α is an upper boundary of the nonleakage region and is concave in α. We compare the line θ = 3 with the curve w μ = (3 A H μ +1) α(1−α) 12+4α at α = 1 (notice that the two intersect at α = 1): 111 ( 3−r 2 1−α pr+1−p ) ′ | α=1 =− 3−r 2 1 pr+1−p , ((3 A H μ +1) α(1−α) 12+4α ) ′ | α=1 = (3 A H μ +1) (1−2α)(12+4α)−4α(1−α) (12+4α) 2 | α=1 =− 1 16 ( 3r pr+1−p +1)≥− 1 4 . Thus the line θ = 3 is above the nonleakage region if 3−r 2 1 pr+1−p > 1 16 ( 3r pr+1−p + 1), or r < 23+p 11+p . So, we have θ < 3 and q S ∗ iL =q S2∗ iL . (i)Ifμ−3 w α −4 w 1−α ≥ 0,i.e., w μ ≤ α(1−α) 3+α ,wehaveq i =2 w α + 7 3 w 1−α − 1 3 μand ¯ q i = 1 3 (μ− w 1−α ). Then, according to Theorem 1.5.3, q N ∗ iH ≥ ¯ q i ⇒ 1 2 A H − 1 6 μ− 1 3 w 1−α ≥ 1 3 (μ− w 1−α )⇒A H ≥μ, which always holds; q S2 ∗ iL ≤q i ⇒ A H − A L 2 − 1 2 w 1−α − s (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 ≤ 2 w α + 7 3 w 1−α − 1 3 μ, i.e., r pr+1−p − 1 2(pr+1−p) − 1 2 w/μ 1−α − s ( r pr+1−p − 1 2(pr+1−p) − 1 2 w/μ 1−α ) 2 − 1 4 ( r pr+1−p − w/μ 1−α ) 2 ≤ 2 w/μ α + 7 3 w/μ 1−α − 1 3 . The inequality holds if either (1) r pr+1−p − 1 2(pr+1−p) − 1 2 w/μ 1−α − (2 w/μ α + 7 3 w/μ 1−α − 1 3 ) ≤ 0 or (2) q ( r pr+1−p − 1 2(pr+1−p) − 1 2 w/μ 1−α ) 2 − 1 4 ( r pr+1−p − w/μ 1−α ) 2 ≥ r pr+1−p − 1 2(pr+1−p) − 1 2 w/μ 1−α −(2 w/μ α + 7 3 w/μ 1−α − 1 3 )≥ 0. Thefirstinequalityaboveisequivalentto w μ ≥ 1 2 α + 17 6(1−α) r pr+1−p − 1 2(pr+1−p) + 1 3 , which is violated because 1 2 α + 17 6(1−α) r pr+1−p − 1 2(pr+1−p) + 1 3 ≥ α(1−α) 3+α ≥ w μ . The second in- equality above is equivalent to LB S2 L− (α)≤ w μ ≤UB S2 L− (α), where LB S2 L− (α)= (72r−24p+4α+24pr−10pα+21rα+10prα−12)−6 √ δ (pr+1−p) · α(1−α) (12+5α) 2 and UB S2 L− (α)= (72r−24p+4α+24pr−10pα+21rα+10prα−12)+6 √ δ (pr+1−p) α(1−α) (12+5α) 2 , 112 Figure A.1. Solution to inequality (2) in case (b.ii). for δ =(24α−108r−12pα−54rα+12prα+6α 2 −5pα 2 −6rα 2 +5prα 2 +36)(1−r). Therefore, we obtain a nonleakage region LB S2 L− (α)≤ w μ ≤ min n α(1−α) 3+α ,UB S2 L− (α) o . (ii)Ifμ−3 w α −4 w 1−α ≤ 0, i.e., w μ ≥ α(1−α) 3+α ,wehaveq i = 1 3 (μ− w 1−α )and ¯ q i =2 w α + 7 3 w 1−α − 1 3 μ. Then, according to Theorem 1.5.3, q N ∗ iH ≥ ¯ q i ⇒ 1 2 A H − 1 6 μ− 1 3 w 1−α ≥ 2 w α + 7 3 w 1−α − 1 3 μ ⇒ w μ ≤ (3 A H μ +1) α(1−α) 12+4α ⇒ w μ ≤ ( 3r pr+1−p +1) α(1−α) 12+4α ; q S2 ∗ iL ≤q i ⇒A H − A L 2 − 1 2 w 1−α − q (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 ≤ 1 3 (μ− w 1−α ) ⇒ r pr+1−p − 1 2(pr+1−p) − 1 2 w/μ 1−α − q ( r pr+1−p − 1 2(pr+1−p) − 1 2 w/μ 1−α ) 2 − 1 4 ( r pr+1−p − w/μ 1−α ) 2 ≤ 1 3 (1− w/μ 1−α ) ⇒ θ− 1 2 − q (θ− 1 2 ) 2 − 1 4 θ 2 ≤ 1 3 (pθ+1−p). Theinequality holdsifeither(1)θ− 1 2 − 1 3 (pθ+1−p)≤ 0or(2)0≤ θ− 1 2 − 1 3 (pθ+1−p)≤ q (θ− 1 2 ) 2 − 1 4 θ 2 . Thefirstinequalityreducestoθ≤ 5 6 − 1 3 p 1− 1 3 p ≤ 1,whichisalwaysviolated. The second inequality is equivalent to θ = 2(6−3 √ p−11p+2p 2 ) 9−24p+4p 2 ≤θ≤ 2(6+3 √ p−11p+2p 2 ) 9−24p+4p 2 =θ, if 0≤p≤ 0.40192, or θ = 1≤ θ ≤ 2(6−3 √ p−11p+2p 2 ) 9−24p+4p 2 = θ, if 0.40192≤ p≤ 1, as illustrated in Figure A.1. Because A H − w 1−α A L − w 1−α = θ, we have w μ = θA L /μ−A H /μ θ−1 (1−α) = θ−r θ−1 1−α pr+1−p . Therefore, we obtain another nonleakage region: max{ θ−r θ−1 1−α pr+1−p , α(1−α) 3+α }≤ w μ ≤ min{ θ−r θ−1 1−α pr+1−p ,( 3r pr+1−p + 1) α(1−α) 12+4α }. 113 Case (b) of the theorem is proved. Below, we show that for some model parameters, the region (ii) is empty. Notice that ( θ−r θ−1 1−α pr+1−p ) ′ | α=1 = − θ−r θ−1 1 pr+1−p and ((3 A H μ +1) α(1−α) 12+4α ) ′ = − 1 16 ( 3r pr+1−p +1). Thus the region (ii) is empty if (1) θ−r θ−1 1 pr+1−p ≥ 1 16 ( 3r pr+1−p +1) or (2) θ−r θ−1 1 pr+1−p ≤ 0. For 0≤ p≤ 0.40192, inequality (1) above is equivalent to 2(6−3 √ p−11p+2p 2 )/(9−24p+4p 2 )−r 2(6−3 √ p−11p+2p 2 )/(9−24p+4p 2 )−1 1 pr+1−p ≥ 1 16 ( 3r pr+1−p +1),i.e.,r≤ 21+32 √ p+11p 17+32 √ p+11p ;andinequality(2)aboveisequivalenttor≥ 2(6+3 √ p−11p+2p 2 ) 9−24p+4p 2 . For 0.40192 ≤ p ≤ 1, inequality (1) is violated, while inequality (2) reduces to r ≥ 2(6−3 √ p−11p+2p 2 ) 9−24p+4p 2 . Now, for any p ∈ [0,1], there exist some r such that the region (ii) is empty. (c) Suppose 23+p 11+p ≤ r < 3. Based on the analysis in cases (a) and (b), if 23+p 11+p ≤ r < 3, the line θ = 3 divides the nonleakage region into two parts, satisfying θ ≥ 3 and θ < 3 respectively. We consider the two cases below. (i) Assume w μ ≥ 3−r 2 1−α pr+1−p , i.e. θ ≥ 3. If this condition is disregarded, the nonleakage region would be given by max{( 3 pr+1−p +2) α(1−α) 12+5α ,( 3 pr+1−p − 2)(1− α)} ≤ w μ ≤ ( 3r pr+1−p + 1) α(1−α) 12+4α , from part (a) of the theorem. Thus, the nonleakage region for θ ≥ 3 is given by max{( 3 pr+1−p +2) α(1−α) 12+5α , 3−r 2 1−α pr+1−p ,( 3 pr+1−p −2)(1−α)}≤ w μ ≤ ( 3r pr+1−p +1) α(1−α) 12+4α . (ii)Assume w μ < 3−r 2 1−α pr+1−p , i.e. θ < 3. Imposing thiscondition onthenonleakageregions found in part (b) of the theorem, we obtain the following regions: LB S2 L− (α)≤ w μ ≤ min{UB S2 L− (α), α(1−α) 3+α , 3−r 2 1 pr+1−p } andmax{ θ−r θ−1 1−α pr+1−p , α(1−α) 3+α }≤ w μ ≤ min{ θ−r θ−1 1−α pr+1−p ,( 3r pr+1−p +1) α(1−α) 12+4α , 3−r 2 1−α pr+1−p }.Theproof is complete. Proof of Theorem 1.6.1: Beforeweprovethetheorem,weshowanintermediateresultthatsimplifiesthesupplier’s profitfunctionintheleakagecase. AscanbeseenfromProposition1.4.2,thesupplier’sprofit function in the case θ < 3 is rather complex due to the complex expressions of q S ∗ iL and q S ∗ eL . 114 Nevertheless, the next lemma suggests that we can focus on the case θ ≥ 3, because the supplier’s profit is larger when q S ∗ iL and q S ∗ eL assume the values intended for the case θ ≥ 3 than those for the case θ < 3 (i.e. the incumbent orders q S1 ∗ iL rather than q S2 ∗ iL when the demand is low). Thus, if the supplier’s profit is larger under nonleakage than under leakage assuming θ≥ 3, the conclusion would be even more robust if the actual case is θ < 3. LemmaA.1.1. Assumer≥ 7+2p 3+2p . Underthesupplier’soptimalwholesalepricew foragiven α, in a separating (leakage) equilibrium, the supplier’s profit is higher when the incumbent orders q S1 ∗ iL (w) rather than q S2 ∗ iL (w) in the low demand state. Proof of Lemma A.1.1: (i) First, we find the incumbent’s order quantity that is first-best for the supplier under leakage in each demand case. When the supplier leaks the information, the supplier’s profit is π s = w(q i +q e ) +α(A− q i − q e )(q i +q e ), and the entrant’s order quantity is q e (q i ) = 1 2 (A−q i − w 1−α ), where A = A H if the demand is high, or A = A L if the demand is low. After simplification, we have π sH (w,q i )=− α 4 (q i − w α − w 1−α ) 2 + (αA H +w) 2 4α , (A.1.11) π sL (w,q i )=− α 4 (q i − w α − w 1−α ) 2 + (αA L +w) 2 4α . Hence, the supplier’s profit is determined by the distance between q i and w α + w 1−α and the wholesale price w, given A H , A L and α. Clearly, q FB i (w)= w α + w 1−α is the incumbent’s order quantity that the supplier wants to induce in both demand states to maximize her profit. Notice that as w increases, π sH (w,q FB i (w)) = (αA H +w) 2 4α and π sL (w,q FB i (w)) = (αA L +w) 2 4α increase; q FB i (w) increases; q S1 ∗ iL (w), q S2 ∗ iL (w) and q S ∗ iH (w) decrease. (ii) Next, we show that the optimal wholesale price w ∗ for the supplier must satisfy w ∗ α + w ∗ 1−α ≥ q S ∗ iL (w ∗ ). Consider any wholesale price w such that w α + w 1−α <q S ∗ iL (w). We show that the supplier can do better by increasing w. 115 If θ ≥ 3, w α + w 1−α < 1 2 (A L − w 1−α ). Increase w to w ′ = w + 1 2 ( 1 2 A L 1 α + 3 2 1 1−α − w) > w. Because w ′ = 1 2 w+ 1 2 1 2 A L 1 α + 3 2 1 1−α , we have w ′ α + 3 2 w ′ 1−α = w 2 ( 1 α + 3 2 1 1−α )+ 1 4 A L ≤ 1 2 A L , and hence w ′ α + w ′ 1−α <q S ∗ iL (w ′ )= 1 2 (A L − w ′ 1−α ). If θ <3, w α + w 1−α <A H − A L 2 − 1 2 w 1−α − q (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 . Increase w to w ′ =w+ 1 2 ( A H − A L 2 − q (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 1 α + 3 2 1 1−α −w) = 1 2 A H − A L 2 − q (A H − A L 2 − w 2(1−α) ) 2 − 1 4 (A H − w 1−α ) 2 1 α + 3 2 1 1−α + 1 2 w. We have w ′ α + 3 2 w ′ 1−α = w 2 ( 1 α + 3 2 1 1−α )+ 1 4 (A H − A L 2 − s 1 2 (A H −A L )( 3 2 A H − A L 2 − w (1−α) )) ≤ 1 2 (A H − A L 2 − s 1 2 (A H −A L )( 3 2 A H − A L 2 − w (1−α) )) ≤ 1 4 (A H − A L 2 − s 1 2 (A H −A L )( 3 2 A H − A L 2 − w ′ (1−α) )) and hence w ′ α + w ′ 1−α <q S2 ∗ iL (w ′ ). When we increase w to w ′ (which satisfies w ′ α + w ′ 1−α < q S ∗ iL (w ′ )), w ′ 1−α + w ′ α moves closer to q S ∗ iL (w ′ ) and q S ∗ iH (w ′ ), as illustrated in Figure A.2. Therefore, the supplier’s profit strictly increases by offering w ′ and w cannot be optimal. (iii) Finally, we prove that the optimal wholesale price cannot satisfy q S2 ∗ iL (w)≤ w α + w 1−α <q S1 ∗ iL (w). (A.1.12) Letw 0 = 3A L −A H 2 (1−α) (sothatθ(w 0 )= 3). Iftheoptimalwholesale pricew≥w 0 , then θ(w)≥ 3. As shown in part (ii), the optimal wholesale price satisfies w α + w 1−α ≥ q S ∗ iL (w) = q S1 ∗ iL (w), so (A.1.12) does not hold. Now, suppose that the optimal wholesale price satisfies w ≤ w 0 and q S2 ∗ iL (w) ≤ w α + w 1−α < q S1 ∗ iL (w), as in Figure A.3. We show that there exists w ∗ > w such that the supplier’s profit is strictly better. 116 Figure A.2. Supplier’s profit functions given wholesale prices w and w ′ . Figure A.3. Supplier’s profit functions given wholesale prices w, ˆ w, and w ∗ . 117 First, increase w to ˆ w = A L α(1−α) 2+α , which satisfies 1 2 (A L − ˆ w 1−α ) = ˆ w α + ˆ w 1−α . Notice that q S1 ∗ iL (ˆ w) = 1 2 (A L − ˆ w 1−α ) = ˆ w α + ˆ w 1−α < q S ∗ iH (ˆ w) = 1 2 (A H − ˆ w 1−α ). Since |q S ∗ iH (ˆ w)− ( ˆ w α + ˆ w 1−α )| < |q S ∗ iH (w)− ( w α + w 1−α )| and (αA H +ˆ w) 2 4α ≥ (αA H +w) 2 4α , we have π sH (ˆ w,q S ∗ iH (ˆ w)) > π sH (w,q S ∗ iH (w)). In addition, π sL (ˆ w,q S1 ∗ iL (ˆ w)) = π sL (ˆ w, ˆ w α + ˆ w 1−α ) > π sL (w, w α + w 1−α ) ≥ π sL (w,q S2 ∗ iL (w)) (as illustrated in Figure A.3, point A is dominated by point B and B is dominated by C). Thus, we have π S1 s (ˆ w)>π S2 s (w), where π S1 s (w)=pπ iH (w,q S ∗ iH (w))+(1−p)π iL (w,q S1 ∗ iL (w)) (A.1.13) =p − α 4 ( 1 2 (A H − w 1−α )− w α − w 1−α ) 2 + (αA H +w) 2 4α +(1−p) − α 4 ( 1 2 (A L − w 1−α )− w α − w 1−α ) 2 + (αA L +w) 2 4α , π S2 s (w)=pπ iH (w,q S ∗ iH (w))+(1−p)π iL (w,q S2 ∗ iL (w)). (A.1.14) (Notethatπ S1 s (w)isthesupplier’sactualexpectedprofitwhenw≥w 0 ;whenw≤w 0 ,π S2 s (w) is her expected profit.) From q S1 ∗ iL (ˆ w) = 1 2 (A L − ˆ w 1−α ) = ˆ w α + ˆ w 1−α < 1 2 (A H − ˆ w 1−α ) = q S ∗ iH (ˆ w), we can easily show that (π S1 s ) ′ (ˆ w)≥ 0. Consider two cases below: If ˆ w ≥ w 0 , since π S1 s (w) is concave, there must exist w ∗ > ˆ w such that π S1 s (w ∗ ) > π S1 s (ˆ w)>π S2 s (w) and we are done. If ˆ w <w 0 , increase w further to w 0 . From (A.1.11), dπ sH (w,q S ∗ iH (w)) dw = 3[2α 2 A H +4(A H −w)−4w+α(−6A H +2w)] 16(1−α) 2 , dπ sL (w,q S1 ∗ iL (w)) dw = 3[2α 2 A L +4(A L −w)−4w+α(−6A L +2w)] 16(1−α) 2 , and hence (π S1 s ) ′ (w)=p dπ sH (w,q S ∗ iH (w)) dw +(1−p) dπ sH (w,q S ∗ iL (w)) dw = 3[2α 2 μ+4(μ−w)−4w+α(−6μ+2w)] 16(1−α) 2 , 118 where μ=pA H +(1−p)A L . Therefore, the optimal wholesale price is w ∗ μ = (2−α)(1−α) 4−α , (A.1.15) and (π S1 s ) ′ (w) ≥ 0 if w μ ≤ w ∗ μ . Since w 0 = 3A L −A H 2 (1− α), we have w 0 μ ≤ w ∗ μ (and hence (π S1 s ) ′ (w 0 )≥ 0)if 3−r 2(pr+1−p) ≤ 2−α 4−α . Becauser≥ 7+2p 3+2p implies 3−r 2(pr+1−p) ≤ 2−α 4−α forallα∈ [0,1], we have (π S1 s ) ′ (w 0 ) ≥ 0 if r ≥ 7+2p 3+2p . Since π S1 s (w) is concave and w ∗ ≥ w 0 , the supplier’s optimal wholesale price satisfies π S1 s (w ∗ )≥π S1 s (w 0 )>π S1 s (ˆ w)>π S2 s (w). Thus, we have shown that the optimal wholesale price must satisfy q S2 ∗ iL (w)≤q S1 ∗ iL (w)≤ w α + w 1−α =q FB i . So, the supplier prefers the incumbent to order q S1 ∗ iL (w) than q S2 ∗ iL (w) when the demand is low under the optimal wholesale price, provided that r≥ 7+2p 3+2p . Theaboveresultisbasedonthefollowingfacts: first,thesupplier’sprofitunderleakageis nondecreasing intheincumbent’s orderquantity givenanyw andα; second, theincumbent’s order quantity q S2 ∗ iL (w) is lower than q S1 ∗ iL (w). Now, we begin the proof of Theorem 1.6.1: (i) Suppose that the supplier does not leak. By Proposition 1.4.1, we have q N ∗ iH +q N ∗ e = 1 2 A H + 1 6 μ− 2 3 w 1−α , q N ∗ iL +q N ∗ e = 1 2 A L + 1 6 μ− 2 3 w 1−α , and hence π N ∗ sH =w(q N ∗ iH +q N ∗ e )+α(q N ∗ iH +q N ∗ e )(A H −q N ∗ iH −q N ∗ e ) =(− 2 3 1 1−α − 4 9 α (1−α) 2 )w 2 +( 1 2 A H + 1 6 μ+ 2 9 α 1−α μ)w+α( 1 2 A H + 1 6 μ)( 1 2 A H − 1 6 μ), π N ∗ sL =(− 2 3 1 1−α − 4 9 α (1−α) 2 )w 2 +( 1 2 A L + 1 6 μ+ 2 9 α 1−α μ)w+α( 1 2 A L + 1 6 μ)( 1 2 A L − 1 6 μ), π N ∗ s =pπ N ∗ sH +(1−p)π N ∗ sL =(− 2 3 1 1−α − 4 9 α (1−α) 2 )w 2 +( 2 3 μ+ 2 9 α 1−α μ)w+α( pA 2 H +(1−p)A 2 L 4 − 1 36 μ 2 ). Since π N ∗ s μ 2 is concave with respect to w μ for a given α, it is maximized by ( w μ ) N ∗ = 2 3 + 2 9 α 1−α 4 3 1 1−α + 8 9 α (1−α) 2 = 6(1−α) 2 +2α(1−α) 12(1−α)+8α = (1−α)(3−2α) 6−2α . (ii)Supposethatthesupplierleaks. ByProposition1.4.2andassumingθ≥ 3(byLemma A.1.1), we have q S ∗ iH +q S ∗ eH = 3 4 (A H − w 1−α ), q S ∗ iL +q S ∗ eL = 3 4 (A L − w 1−α ), and as we have shown 119 in equation (A.1.15) in the proof of Lemma A.1.1, the supplier’s profit is maximized by ( w μ ) S ∗ = (2−α)(1−α) 4−α . The final part of the theorem follows from the concavity of π N ∗ s μ 2 and π S ∗ s μ 2 . A.2 Pooling Equilibria In this Appendix, we characterize the pooling equilibria and show that pooling equilibria do not exist or are dominated by other equilibria in most scenarios and thus justify the omission of the pooling equilibria from our main analysis. We begin by identifying the pooling equilibria in the benchmark case where the supplier always leaks. Benchmark: Supplier Always Leaks—the Pooling Case A reasonable belief structure for a pooling equilibrium is the following: if the incumbent’s order quantity q i falls in a certain interval, the supplier and the entrant cannot infer the demand state from q i and hence retain their prior belief about the demand. As in Anand and Goyal (2009), we assume that the entrant holds the following belief: Pr e (A =A H )= 0, if the supplier leaks and q i <q P i , p, if the supplier leaks and q P i ≤q i ≤q P i , 1, if the supplier leaks and q i >q P i , (A.2.1) where q P i and q P i are the maximum and minimum quantities that enable a pooling equilib- rium. Thenextpropositioncharacterizesthepoolingboundsq P i andq P i andthePareto-dominant poolingequilibrium, inwhich theincumbent ordersthemaximum amount. Theproofissim- ilar to that in Anand and Goyal (2009) and is omitted. Proposition A.2.1. Suppose that the supplier always leaks and the entrant’s belief about the demand is given by (A.2.1). If θ ≤ 3+2p−p 2 1+4p−p 2 , there exists a pooling equilibrium in which 120 the incumbent orders q i ∈ [q P i ,q P i ], for q P i = (A H − 1 2 μ− 1 2 w 1−α − 1 2 q (A H −μ)(3A H −2 w 1−α )) and q P i = (A L − μ 2 − 1 2 w 1−α ) + . In a Pareto-dominant pooling equilibrium, the incumbent orders q P ∗ i = (A L − μ 2 − 1 2 w 1−α ) + , and the entrant orders q P ∗ e = 3 4 μ− 1 2 A L − 1 4 w 1−α . Note that a pooling equilibrium may not always exist, because if θ is large, it would be easy for the low-type incumbent to separate and costly for the high type to pool. Also note that the maximum pooling quantity q P ∗ i is less than the incumbent’s order quantity in the perfect information game under both demand states. Thus, it can be shown that, similar to the separating case, pooling demand information (when the supplier always leaks) reduces the profits of the incumbent, the supplier, as well as the supply chain. Justification for Ignoring Pooling WhenanalyzingtheexistenceofthenonleakageequilibriuminSection1.5.1,weassumedthat the demand state can be inferred from the incumbent’s order quantity, i.e., the possibility of the incumbent playing a pooling strategy was ignored. We can conduct a similar analysis when pooling is taken into account. However, that is unnecessary in most circumstances, for the following reasons: First, according to Proposition A.2.1, a poolingequilibrium does not exist ifθ > 3+2p−p 2 1+4p−p 2 . The condition θ > 3+2p−p 2 1+4p−p 2 includes the scenarios that are most interesting. For instance, if p≥ 0.2, the condition is equivalent to θ >1.9; if p≥ 0.4, it reduces to θ > 1.5. Second, if pooling is possible (when θ≤ 3+2p−p 2 1+4p−p 2 ) but θ≥ 0.1082 p +1, it can be shown that the incumbent is worse off by playing a pooling game than a nonleakage one. Third, if θ < 0.1082 p +1 but θ ≤ (p+1)(p+4) 3 , it can be shown that the condition q S ∗ iL ≤ q i (part of Theorem 1.5.3) implies q P ∗ i ≤ q p i , where q p i is the lower bound of a sustainable pooling quantity. That is, pooling is excluded by the condition q S ∗ iL ≤ q i in this case. Figure A.4 illustrates the range of p and θ that fall in the above cases so that either a pooling equilibrium does not exist or the incumbent does not want to deviate to such 121 an equilibrium. We believe that this range of parameters covers the majority of practical situations—only the shaded region B is uncovered. If necessary, pooling in region B can be prevented by adding a constraint q P ∗ i ≤q p i in Theorem 1.5.3. The proofs of the above claims aretedious anddonotprovide additionalinsights, andtherefore areomittedfromthe paper. Details are available from the authors. Figure A.4. The range of θ and p in which pooling cannot be ignored. 0.25 0.5 0.75 1 1 2 3 4 p θ B θ = 0.10819 p +1 θ = 3+2p−p 2 1+4p−p 2 θ = (p+1)(p+4) 3 θ = 1−p 3(1− √ 2 2 )−p A.3 Impacts of r and p on the Nonleakage Region We focus on the case r ≥ 3 to be concise. Similar analysis can be conducted when r is in another range. By Proposition 1.5.1, the supplier’s leakage interval (q i ,¯ q i ) (if normalized by μ) only depends on w μ and α, not on r or p. Thus, according to Theorem 1.5.3, the change in the nonleakage region will be driven by the incumbent’s order quantities q N ∗ iH and q S ∗ iL (normalized by μ) and the relationships q N ∗ iH ≥ ¯ q i and q S ∗ iL ≤q i . Impact of r on the Nonleakage Region As r increases, the distinction between the two demand states becomes more significant, with A H μ increasing and A L μ decreasing, by the equations A H μ = r pr+1−p and A L μ = 1 pr+1−p . As a result, the incumbent’s order quantity q N ∗ iH increases while q S ∗ iL decreases, and both are more 122 likely to fall in the supplier’s nonleakage intervals. More specifically, the gaps between the upper bound and the two lower bounds are: g 1 (r,p)=UB H −LB S1 L− =( 3r pr+1−p +1) α(1−α) 12+4α −( 3 pr+1−p +2) α(1−α) 12+5α , g 2 (r,p)=UB H −LB S1 L+ =( 3r pr+1−p +1) α(1−α) 12+4α −( 3 pr+1−p −2)(1−α). Notice that ∂ ∂r g 1 = 3α(1−α)(12+(5−p)α) 4(pr+1−p) 2 (36+27α+5α 2 ) ≥ 0, (A.3.1) ∂ ∂r g 2 = 3(1−α)(α+3p(4+α)) 4(pr+1−p) 2 (3+α) ≥ 0. (A.3.2) So, as r increases, the vertical distance between the upper bound ( 3r pr+1−p +1) α(1−α) 12+4α and the lower bound, ( 3 pr+1−p +2) α(1−α) 12+5α or ( 3 pr+1−p −2)(1−α), for a given α, increases. In addition, ∂ ∂r ( 3 pr+1−p −2 1− 1 pr+1−p ) = −1 p(1−r) 2 implies that with an increase in r, the threshold point α 0 moves to the left. Thus, asr increases, sodoesthegapbetween thelowerandupperboundsof w μ foragiven α, and the nonleakage region expands. Intuitively, when the demand variationincreases, the incumbent’s information advantage exacerbates the quantity distortion from the supplier’s perspective (Proposition1.4.3(iii))andpromptsthesuppliertopreventinformationleakage. Impact of p on the Nonleakage Region Recall that p is the probability that the demand is high. According to the equations A H μ = r pr+1−p and A L μ = 1 pr+1−p , both A H μ and A L μ decrease with p. While an increase in p relaxes the condition q S ∗ iL ≤ q i (which determines the lower boundary), it tightens the condition q N ∗ iH ≥ ¯ q i (which determines the upper boundary). Thus, both the lower and upper boundaries shift downward, but the total impact is less obvious. There are two cases: (1) If p≥ 1 2(r−1) , we have LB S1 L+ ≤B FB (i.e., LB S1 L+ is not applicable) for all α. Because ∂ ∂p g 1 = 3(r−1)α(1−α)(12(1−r)+4α−5αr) 4(pr+1−p) 2 (3+α)(12+5α) ≤ 0, (A.3.3) 123 the gap between the effective bounds UB H and LB S1 L− decreases with p for any α ∈ [0,1]. Hence the nonleakage region shrinks, which suggests that when the demand is likely to be high, the probability affects the upper boundary more than the lower boundary. (2) If p < 1 2(r−1) , there exists α 0 > 0 such that LB S1 L+ > B FB for any α ∈ [0,α 0 ). The above result still holds when α∈ [α 0 ,1]. For α∈ [0,α 0 ) however, because ∂ ∂p g 2 = 3(r−1)(1−α)(12+(4−r)α) 4(pr+1−p) 2 (3+α) ≥ 0 (for 3≤r≤ 4+ 12 α ), (A.3.4) the gap between the effective bounds UB H and LB S1 L+ increases with p and the nonleakage region expands. In addition, since ∂ ∂p ( 3 pr+1−p −2 1− 1 pr+1−p ) = −1 p 2 (r−1) < 0, the threshold point α 0 moves to the left as p increases. A.4 Extensions In this appendix, we extend our results by altering some assumptions in the model. We find that these extensions do not change the main results of the paper but may alter the nonleakage region in different ways. Free Disposal by the Incumbent As argued in the paper, the incumbent’s free disposal option may affect the type of equi- librium in two situations: (1) if q i < q N∗ iH < ¯ q i , when the demand is high, the incumbent may benefit from ordering ¯ q i (hence persuading the supplier notto leak) but selling less; and (2) if q S∗ iL < q i , when the demand is low, the incumbent may benefit from ordering q i (thus inducing the supplier to leak) but selling less. The first situation may enable a nonleakage equilibrium that is not sustainable without free disposal; while the second may topple an original nonleakage equilibrium because the deviation to a leakage equilibrium may become more attractive to the incumbent under free disposal. The first effect may push up the upper boundary of the nonleakage region; while the second may raise the lower boundary. We consider these effects below. 124 First, in part (a), we compute the new nonleakage equilibrium under free disposal when q i <q N∗ iH < ¯ q i . Then in part (b), we find a condition to ensure nonleakage when the demand is high. Finally, in part (c), we find a condition to ensure nonleakage when the demand is low. A nonleakage equilibrium exists when the conditions in (b) and (c) are both met. (a) We find the new nonleakage equilibrium under free disposal (which may be useful only in the high demand state). Suppose that when the demand is high, the incumbent orders q or iH from the supplier but sells q N iH ≤q or iH to the market. Then his profit is given by π N iH (q or iH ) = max q N iH ≤q or iH (1−α)(A H −q N iH −q N e )q N iH −wq or iH , with the first order condition q N iH = 1 2 (A H −q N e ) (FOC1) when q N iH ≤q or iH . When the demand is low, the incumbent does not order more than what is to be sold, so his profit is π N iL = max q N iL (1−α)(A L −q N iL −q N e )q N iL −wq N iL , with the first order condition q N iL = 1 2 (A L −q N e − w 1−α ) (FOC2). The entrant’s profit under nonleakage is π N e =max q N e (1−α) p(A H −q N iH −q N e − w 1−α )q N e +(1−p)(A L −q N iL −q N e − w 1−α )q N e , with the first order condition q N e = 1 2 (μ− w 1−α −(pq N iH +(1−p)q N iL )) (FOC3). Solving (FOC1), (FOC2), and (FOC3) together, we obtain the new nonleakage equilib- rium (q N ′ iH ,q N ′ iL ,q N ′ e ) (as long as the incumbent orders q or iH ≥q N ′ iH in the high demand state): q N ′ iH = 1 2 A H − 1 6 μ+ 1 6 (1+p) w 1−α ≥q N∗ iH q N ′ iL = 1 2 A L − 1 6 μ− 2−p 6 w 1−α ≥q N∗ iL q N ′ e = 1 3 (μ−(1+p) w 1−α )≤q N∗ e where (q N∗ iH ,q N∗ iL ,q N∗ e ) is the original nonleakage equilibrium without free disposal. The par- ticipation constraint for this new equilibrium is w μ ≤ (1−α) (2−p(r−1)) (2−p)(pr+1−p) . 125 Let π N ′ iH (q or iH ) and π N ′ iL denote the incumbent’s equilibrium profits in the high and low demand states and π N ′ e denote the entrant’s expected equilibrium profit. (b) Now we find a condition to ensure nonleakage when the demand is high, given that q i <q N∗ iH < ¯ q i . When the demand is high, if the incumbent orders q or iH , the supplier’s profit in the above nonleakage equilibrium is π N ′ sH (q or iH )=α(A H −q N ′ iH −q N ′ e )(q N ′ iH +q N ′ e )+w(q or iH +q N ′ e ), and her profit in the leakage (separating) equilibrium is π S sH (q or iH ) =α(A H −q or iH −q eH (q or iH )+ w α )(q or iH +q eH (q or iH )). The supplier prefers nonleakage if π N ′ sH (q or iH )≥π S sH (q or iH ). Solving this inequality, we obtain q or iH ≥ 1 3α(1−α) (6α−3)w+ v u u u t (α 2 −2α 3 +α 4 )μ 2 +2(1−α)α(9(1−α)A H +μ((5−p)α−6)w +(9+6α(2p−7)+α 2 (34−10p+p 2 )w 2 ,q ′ i . For the incumbent, a sufficient set of conditions for nonleakage is (i1) π N ′ iH (q ′ i )≥ π S∗ iH (q S∗ iH ), (i2) q N ′ iH ≤ q ′ i . We can show that condition (i2) is equivalent to w μ ≥ C 0 , r−1 pr−p+1 (1− α) or w μ ≤ C 1 , ( 3r pr−p+1 +1) α(1−α) 12+(p−5)α . Condition (i1) is non-tractable if we directly plug in q ′ i . So, instead we consider q ′′ i = w α(1−α) + w α(1−α) − μ 3 − 1−p 3 w 1−α , which isthe largersolutionofπ S sH (q or iH )−π N ′ sH (q N ′ iH )= 0. Morespecifically, q ′′ i =− μ 3 + 2w α(1−α) − 2−p 3 w 1−α if w μ ≥ α(1−α) 3−2α+αp , and q ′′ i = 1 3 (μ+(2−p) w 1−α ) if w μ ≤ α(1−α) 3−2α+αp . 126 Because π S sH (q ′ i ) =π N ′ sH (q ′ i )≥π N ′ sH (q N ′ iH ) =π S sH (q ′′ i ) (the inequality is implied by q ′ i ≥ q N ′ iH ), π S sH (·) is concave, and q ′′ i is the larger quantity that yields the profit π N ′ sH (q N ′ iH ), we must have q ′ i ≤q ′′ i . Thus, π N ′ iH (q ′ i )≥π N ′ iH (q ′′ i ), and condition (i1) follows from π N ′ iH (q ′′ i )≥π S∗ iH (q S∗ iH ), i.e., (1−α)(A H −q N ′ iH −q N ′ e )q N ′ iH −wq ′′ i ≥ (1−α)(A H −q S∗ iH −q eH (q S∗ iH ))q S∗ iH −wq S∗ iH . Solving this inequality, we obtain NB 1 ≤ w μ ≤NB 2 , where NB 1,2 = 1−α (55−28p−2p 2 )(1+p(−1+r)) 2 [−14−2p 3 (1−r) 3 +15r −p(−26+37r−15r 2 )−2p 2 (5−9r+4r 2 )± √ τ 1 ], if w μ ≤ α(1−α) 3−2α+αp , α(1−α) (−144+α(41−20p+2p 2 )(1+p(−1+r)) 2 [−10+22p−14p 2 +2p 3 −15r −11pr+30p 2 r−4p 3 r−15pr 2 −16p 2 r 2 +2p 3 r 2 ± √ τ 2 ], if w μ ≥ α(1−α) 3−2α+αp , for τ 1 =12(1−p)(1+p(−1+r)) 2 (17−60r−40r 2 +p(17−40r+24r 2 )), τ 2 =(8(2+2p 2 (r−1) 2 −12r+9r 2 −4p(1−4r+3r 2 )) +α(1+44r−8r 2 −p(2+52r−64r 2 )+p 2 (1+8r−8r 2 ))). Note that the curve w μ = α(1−α) 3−2α+αp lies between NB 1 and NB 2 in both cases. (c) When the demand is low, if q S∗ iL < q i , the incumbent may have an incentive to order more (to enter the supplier’s leakage interval) but sell less later. We find a constraint that prevents this type of deviation (from the nonleakage equilibrium). In a separating equilibrium, the high-type incumbent orders q S∗ iH = 1 2 (A H − w 1−α ). To prevent him from deviating to the low type’s order quantity q S iL , the following incentive compatibility (IC) constraint must be satisfied (A H −q S∗ iH −q S∗ eH )q S∗ iH (1−α)−q S∗ iH w≥ (A H −q S iL −q S eL (q S iL ))q S iL (1−α)−q S iL w, (A.4.1) where q S eL (q S iL ) = 1 2 (A L −q S iL − w 1−α ). The inequality can be simplified to π S∗ iH (q S∗ iH ) = 1 8 (A H − w 1−α ) 2 ≥ (A H − A L 2 − q S iL 2 − 1 2 w 1−α )q S iL Δ =π D iL|H (q S iL ) (A.4.2) 127 Figure A.5. Nonleakage region with free disposal, for r = 3.5, p =0.2. 0.2 0.4 0.6 0.8 1.0 −0.05 0.05 0.15 0.25 0.35 0.45 LB S2 L− UB H UB P UB FD C 0 NB 1 NB 2 revenue sharing rate α scaled wholesale price ω μ The incumbent’s order quantity in the low demand state in a separating equilibrium is q S∗ iL = q S1∗ iL = 1 2 (A L − w 1−α ), if θ≥ 3 q S2∗ iL =A H − A L 2 − 1 2 w 1−α − q (A H − A L 2 − 1 2 w 1−α ) 2 − 1 4 (A H − w 1−α ) 2 , if θ < 3. Inequality (A.4.2) is satisfied strictly at q S iL =q S1∗ iL for θ > 3 and is binding at q S iL = q S2∗ iL for all θ. We show below that q S2∗ iL ≤ q i is a sufficient condition to ensure nonleakage when the demand is low. (1) When θ < 3, to play a separating game, the incumbent cannot order more than q S2∗ iL when the demand is low, because such a quantity would violate the high-type incumbent’s IC constraint (A.4.2) and trigger the high type’s deviation. Thus, when the demand is low, q S2∗ iL ≤q i is sufficient to induce nonleakage. (2) When θ ≥ 3, the IC constraint for the high-type incumbent does not bind at q S1∗ iL , so the low-type incumbent might order some q iL > q S1∗ iL without triggering the high type’s deviation. The high-type incumbent is indifferent between the quantities q S∗ iH (revealing his true type) and q S2∗ iL (pretending to be a low type, if the entrant believes so). Thus, in a separating equilibrium, the incumbent cannot order more than q S2∗ iL in the low demand case. 128 Therefore, the condition q S2∗ iL ≤ q i is sufficient to prevent the incumbent from deviating to any q iL >q i in the low demand case. When θ < 3, because q S∗ iL = q S2∗ iL , the new condition q S2∗ iL ≤ q i is identical to the one in Theorem 1.5.3 without free disposal. Only when θ≥ 3, the new condition q S2∗ iL ≤q i is more restrictive than the original one, q S∗ iL ≤q i , but the difference is small and, after all, it is only a sufficient condition. Combining the conditions found in parts (b) and (c), we obtain the nonleakage region under free disposal. An example is illustrated in Figure A.5. The darkly shaded area represents the nonleakage region in which the incumbent does not make free disposal. The lightly shaded area represents the region in which nonleakage is enabled by the incumbent’s free disposal option in the high demand case (the bound UB FD is the curve C 1 ). Active Entrant Who Can Reject Information Inourbasicmodel, theentrantplaysapassive roleandalwaysaccepts theincumbent’s order information leaked by the supplier. Because the nonleakage game may benefit the entrant, he may gain by rejecting the leaked information. In this extension, we allow the entrant to decide whether or not to accept the information from the supplier. Weassumethattheentrantrejectstheinformationifhisexpectedprofitundernonleakage is greater than that under leakage. If the entrant rejects the information, he will make the same expected profit as that in the nonleakage equilibrium (the supplier and the incumbent can foresee the entrant’s reaction and hence play a nonleakage game), i.e., Eπ N∗ e = (1− α)[p(A H −q N∗ iH −q N∗ e )+(1−p)(A L −q N∗ iL −q N∗ e )]q N∗ e −wq N∗ e = 1−α 9 (μ− w 1−α ) 2 . If the entrant accepts the information leaked by the supplier, his expected profit is given by Eπ S∗ e = p(1−α)(A H −q S∗ iH −q S∗ eH − w 1−α )q S∗ eH +(1−p)(1−α)(A L −q S∗ iL −q S∗ eL − w 1−α )q S∗ eL , where q S∗ iH , q S∗ eH , q S∗ iL , and q S∗ eL are defined in Proposition 1.4.2. The entrant rejects the information if Eπ N∗ e ≥ Eπ S∗ e , which has a solution of the form w μ ≤ EB 1 or w μ ≥ EB 2 . In the case r ≥ 3, the solution is w μ ≤ (1−3 √ 7(pr +1−p)(r− 129 Figure A.6. Nonleakage region given an active entrant, for r = 3.5, p =0.2. 0.2 0.4 0.6 0.8 1.0 −0.05 0 0.05 0.10 0.15 0.20 0.25 LB S1 L− UB H EB1 EB2 revenue sharing rate α scaled wholesale price ω μ 1) p p(1−p))(1−α). ThedarklyshadedareainFigureA.6representstheoriginalnonleakage regionwhentheentrantispassive,whileinthelightlyshadedregion,anactiveentrantrejects the information leaked by the supplier and turns a leakage game into a nonleakage one. As a result, the nonleakage region is expanded. When r < 3, the analysis is similar but more tedious and is hence omitted from here. Simultaneous Ordering In this extension, the incumbent first shares the demand information with the supplier truthfully, then the supplier decides whether to leak it to the entrant or not, and finally the incumbent and the entrant place orders simultaneously. If the supplier leaks the demand information to the entrant, the two retailers engage in a simultaneous Cournot game under perfect information. Specifically, when the demand is high, they would order q PI iH = argmax q i {(1− α)(A H − q i − q PI eH )q i − wq i } and q PI eH = argmax qe {(1−α)(A H −q PI iH − q e )q e −wq e }, respectively (the superscript “PI” stands for “perfect information”). Solving the two first order conditions A H −2q PI iH −q PI eH − w 1−α = 0 and A H −q PI iH −2q PI eH − w 1−α =0 simultaneously, we obtain the total order quantity from the two retailers Q PI H =q PI iH +q PI eH = 2 3 (A H − w 1−α ). Similarly, when the demand is low, the total order quantity in the channel is Q PI L =q PI iL +q PI eL = 2 3 (A L − w 1−α ). 130 Table A.3. Channel Quantity Distortion from Supplier’s First-Best Nonleakage, H Leakage, H Nonleakage, L Leakage, L w μ Q N∗ H −Q FB H Q PI H −Q FB H Q N∗ L −Q FB L Q PI L −Q FB L (i) w μ ≥ α(1−α) 3+α A H μ − − − − (ii) α(1−α) 3+α ≤ w μ ≤ α(1−α) 3+α A H μ − + − − (iii) α(1−α) 3+α A L μ ≤ w μ ≤ α(1−α) 3+α + + + − (iv) w μ ≤ α(1−α) 3+α A L μ + + + + If the supplier does not leak the demand informtion to the entrant, the two retailers play the same nonleakage game as analyzed in the paper. By Proposition 1.4.1, the total order quantity in the channel is Q N∗ H = q N∗ iH +q N∗ e = 1 2 A H + 1 6 μ− 2 3 w 1−α (< Q PI H ) or Q N∗ L = q N∗ iL +q N∗ e = 1 2 A L + 1 6 μ− 2 3 w 1−α (>Q PI L ), depending on the demand state. Recall from Proposition 1.4.3 that the supplier’s first-best channel quantity is Q FB H = 1 2 A H + 1 2 w α when the demand is high or Q FB L = 1 2 A L + 1 2 w α when the demand is low. As a result, Q PI H ≥Q FB H ifandonlyif w μ ≤ A H μ α(1−α) 3+α , andQ PI L ≥Q FB L ifandonlyif w μ ≤ A L μ α(1−α) 3+α . From Proposition 1.4.3, we also have Q N∗ H ≥ Q FB H and Q N∗ L ≥Q FB L if and only if w μ ≤ α(1−α) 3+α . Nonleakage is preferred by the supplier if the channel distortion (from the supplier’s first- best) is less severe under nonleakage than under leakage in both demand scenarios. Table A.3 compares the total channel quantities in the two equilibria with the supplier’s first-best channel quantity, in the two demand states, given different ranges of w μ (and α). Table A.4 gives theconditionsthatensure less channeldistortionundernonleakage, fordifferentranges of w μ and α. Recall that Q N∗ H < Q PI H and Q N∗ L > Q PI L . As a result, at least one condition is violated in cases (i) and (iv) of Table A.4 and nonleakage cannot be sustained. In case (ii), i.e., when α(1−α) 3+α ≤ w μ ≤ α(1−α) 3+α A H μ ,theconditioninthehighdemandstate,Q FB H −Q N∗ H ≤Q PI H −Q FB H ,is equivalent to w μ ≤ α(1−α) 3+α A H +μ 2μ ; and the condition in the low demand state is always satisfied. 131 Table A.4. Conditions for Supplier to Prefer Nonleakage w μ High Demand Low Demand (i) w μ ≥ α(1−α) 3+α A H μ Q PI H ≤Q N∗ H (≤Q FB H ) Q PI L ≤Q N∗ L (≤Q FB L ) (ii) α(1−α) 3+α ≤ w μ ≤ α(1−α) 3+α A H μ Q FB H −Q N∗ H ≤Q PI H −Q FB H (same as above) (iii) α(1−α) 3+α A L μ ≤ w μ ≤ α(1−α) 3+α (Q FB H ≤)Q N∗ H ≤Q PI H Q N∗ L −Q FB L ≤Q FB L −Q PI L (iv) w μ ≤ α(1−α) 3+α A L μ (same as above) (Q FB L ≤)Q N∗ L ≤Q PI L In case (iii), i.e., when α(1−α) 3+α A L μ ≤ w μ ≤ α(1−α) 3+α , the condition in the high demand state is always satisfied; and the condition in the low demand state, Q N∗ L −Q FB L ≤ Q FB L −Q PI L , is equivalent to w μ ≥ α(1−α) 3+α A L +μ 2μ . In summary, the supplier prefers nonleakage in the region ( α(1−α) 3+α A L μ <) α(1−α) 3+α A L +μ 2μ ≤ w μ ≤ α(1−α) 3+α A H +μ 2μ (< α(1−α) 3+α A H μ ). If the supplier does not leak the demand information to the entrant, it does not matter to the incument whether or not to reveal the information to the supplier in the first place, so nonleakage can be sustained in the above region. The region is in fact the necessary region for nonleakage as well. The incumbent would always like the entrant to know when the demand is low so as to curb the entrant’s order quantity. Thus, the incumbent would always reveal the demand information to the supplier (if he does not reveal it, the supplier can correctly infer that the demand is high). The supplier’s preference is therefore the only factor in determining the nonleakage region. Entrant Ordering First In this extension, we study the implications if the supplier can let the entrant order first. The new sequence of events is the following, which differs from the original sequence by switching steps 3 and 5: (1) The supplier offers a revenue sharing contract (α,w). (2) The incumbent observes the demand state, A H or A L . (3) The entrant places an order q e to the supplier. (4) The supplier decides whether or not to leak q e to the incumbent. (5) The incumbent places an orderq i to the supplier. (6) The demand state is revealed to all parties, the market price is determined by P =A−q i −q e , and the profit for each player is realized. 132 We take several steps to find and compare the leakage and nonleakage equilibria in this extension. Because the entrant has no private information, his order quantity can be calculatedbytheincumbentinanyequilibrium. Thus,leakingtheentrant’sorderquantityq e totheincumbentprovidesnoadditionalinformationyetitmakestheentranttheStackelberg leaderbetweenthetworetailers. Ifthesupplierdoesnotleaktheentrant’sorderquantity,the two retailers play the same nonleakage game as in the basic model (in which the incumbent orders first). (i) Retailers’ order quantities in the leakage and nonleakage equilibria. Ifthesupplierdoesnotleaktheentrant’sorderquantity,thetworetailers’orderquantities in the equilibrium are given in Proposition 1.4.1, i.e., q N ∗ e = 1 3 (μ− w 1−α ) for the entrant, and q N ∗ iH = 1 2 A H − 1 6 μ− 1 3 w 1−α and q N ∗ iL = 1 2 A L − 1 6 μ− 1 3 w 1−α for the incumbent in the two demand states. If instread the supplier leaks the information to the incumbent, the entrant would max- imize his expected profit through the following order quantity q EF e (the superscript “EF” stands for “entrant first”): q EF e =argmax qe p[(1−α)(A H −q iH (q e )−q e )q e −wq e ]+(1−p)[(1−α)(A L −q iL (q e )−q e )q e −wq e ] where q iH (q e ) = argmax q i [(1−α)(A H −q i −q e )q i −wq i ] = 1 2 (A H −q e − w 1−α ), q iL (q e ) = argmax q i [(1−α)(A L −q i −q e )q i −wq i ] = 1 2 (A L −q e − w 1−α ). Solving the above problem, we obtain the entrant’s equilibrium order quantity q EF e = 1 2 (μ− w 1−α ) and the incubment’s equilibrium order quantities q EF iH = 1 2 (A H − 1 2 μ− 1 2 w 1−α ) and q EF iL = 1 2 (A L − 1 2 μ− 1 2 w 1−α ) in the two demand states. (ii) The supplier’s incentive compatibility with nonleakage. 133 Since the entrant’s order quantity carries no demand information, the supplier compares her expected profits under leakage and nonleakge. Given the entrant’s order quantity q e , under leakage (“entrant first”) the supplier’s profits in the two demand states are: π EF sH (q e )=α(A H −q iH (q e )−q e )(q iH (q e )+q e )+w(q iH (q e )+q e ) =α(A H + w α −q iH (q e )−q e )(q iH (q e )+q e ) = α 4 (A H + 2w α −q e + w 1−α )(A H +q e − w 1−α ) = α 4 [−q 2 e +2q e ( w α + w 1−α )+( 2w α +A H + w 1−α )(A H − w 1−α )], π EF sL (q e )=α(A L −q iL (q e )−q e )(q iL (q e )+q e )+w(q iL (q e )+q e ) = α 4 [−q 2 e +2q e ( w α + w 1−α )+( 2w α +A L + w 1−α )(A L − w 1−α )]. In the nonleakage equilibrium, the supplier’s profits are (by equations (A.1.3) and (A.1.4)): π N ∗ sH = α 4 ( 2w α +A H − 1 3 μ+ 4 3 w 1−α )(A H + 1 3 μ− 4 3 w 1−α ) = α 4 ( 2w α +A H + w 1−α − 1 3 μ+ 1 3 w 1−α )(A H − w 1−α + 1 3 μ− 1 3 w 1−α ) = α 4 [( 2w α +A H + w 1−α )(A H − w 1−α )+ 2 3 (μ− w 1−α )( w α + w 1−α )− 1 9 (μ− w 1−α ) 2 ], π N ∗ sL = α 4 ( 2w α +A L − 1 3 μ+ 4 3 w 1−α )(A L + 1 3 μ− 4 3 w 1−α ) = α 4 [( 2w α +A L + w 1−α )(A L − w 1−α )+ 2 3 (μ− w 1−α )( w α + w 1−α )− 1 9 (μ− w 1−α ) 2 ]. Thesupplier(strictly)prefersleakingq e tothenonleakageequilibriumifandonlyifπ EF s (q e ), pπ EF sH (q e )+(1−p)π EF sL (q e )>π N ∗ s , pπ N ∗ sH +(1−p)π N ∗ sL . Because π EF sH (q e )−π N ∗ sH = α 4 [−q 2 e +2q e ( w α + w 1−α )− 2 3 (μ− w 1−α )( w α + w 1−α )+ 1 9 (μ− w 1−α ) 2 ], =π EF sL (q e )−π N ∗ sL , (A.4.3) the inequality π EF s (q e )>π N ∗ s reduces to −q 2 e +2q e ( w α + w 1−α )− 2 3 (μ− w 1−α )( w α + w 1−α )+ 1 9 (μ− w 1−α ) 2 >0. (A.4.4) 134 Because Δ 4 =( w α + w 1−α ) 2 − 2 3 (μ− w 1−α )( w α + w 1−α )+ 1 9 (μ− w 1−α ) 2 =(( w α + w 1−α )− 1 3 (μ− w 1−α )) 2 = 1 9 (μ−3 w α −4 w 1−α ) 2 , inequality (A.4.4) is solved by q e ∈ (q e ,¯ q e ), where q e = w α + w 1−α − 1 3 μ−3 w α −4 w 1−α , and ¯ q e = w α + w 1−α + 1 3 μ−3 w α −4 w 1−α . Notice that this leakage interval (q e ,¯ q e ) is the same as (q i ,¯ q i ), the supplier’s leakage interval when the incumbent orders first. Recall that q N∗ e = 1 3 (μ− w 1−α ), which equals q e or ¯ q e depending on whether ( w α + w 1−α )− 1 3 (μ− w 1−α ) > 0 or not. Thus, we have q N∗ e / ∈ (q e ,¯ q e ) and the supplier has no incentive to leak the entrant’s order quantity if he orders q N∗ e . Next, we check the entrant’s incentive compatibility with the order quantity q N∗ e . (iii) The entrant’s incentive compatibility with nonleakage. Under leakage, given the entrant’s order quantity q e and the incumbent’s best responses q iH (q e ) = 1 2 (A H −q e − w 1−α ) and q iL (q e ) = 1 2 (A L −q e − w 1−α ) in the two demand states, the entrant’s expected profit is p[(1−α)(A H −q e −q iH (q e ))q e −wq e ]+(1−p)[(1−α)(A L −q e −q iL (q e ))q e −wq e ] =p 1−α 2 (A H −q e − w 1−α )q e +(1−p) 1−α 2 (A L −q e − w 1−α )q e = 1−α 2 (μ− w 1−α −q e )q e . Under the nonleakage equilibrium, the entrant’s expected profit is p[(1−α)(A H −q N∗ e −q N∗ iH )q N∗ e −wq N∗ e ]+(1−p)[(1−α)(A L −q N∗ e −q N∗ iL )q N∗ e −wq N∗ e ] =p 1−α 3 ( 1 2 A H − 1 6 μ− 1 3 w 1−α )(μ− w 1−α ) 135 +(1−p) 1−α 3 ( 1 2 A L − 1 6 μ− 1 3 w 1−α )(μ− w 1−α ) = 1−α 3 ( 1 2 μ− 1 6 μ− 1 3 w 1−α )(μ− w 1−α ) = 1−α 9 (μ− w 1−α ) 2 . The entrant would prefer leaking q e to the nonleakage equilibrium if and only if (μ− w 1−α −q e )q e > 2 9 (μ− w 1−α ) 2 , or 1 3 (μ− w 1−α )<q e < 2 3 (μ− w 1−α ). (iv) The supplier’s leakage and nonleakage regions. Thenonleakageequilibrium issustainableifandonlyifthesupplierandentrant’s leakage intervals (q e ,¯ q e ) and ( 1 3 (μ− w 1−α ), 2 3 (μ− w 1−α )) do not overlap, i.e., there is no order quantity q e which both the entrant and the supplier would like to leak to the incumbent. If w μ ≤ α(1−α) 3+α , we have q N∗ e = ¯ q e = 1 3 (μ− w 1−α ) and hence (q e ,¯ q e ) does not overlap with ( 1 3 (μ− w 1−α ), 2 3 (μ− w 1−α )). Inthiscase, theentrantordersthenonleakageequilibrium quantity q N∗ e and the supplier does not leak it to the incumbent. If w μ > α(1−α) 3+α , we have q N∗ e = q e = 1 3 (μ− w 1−α ) < ¯ q e = 2( w α + w 1−α )− 1 3 (μ− w 1−α ) and hence (q e ,¯ q e ) overlaps with ( 1 3 (μ− w 1−α ), 2 3 (μ− w 1−α )). Notice that the entrant’s best order quantity q EF e = 1 2 (μ− w 1−α ) in the leakage game lies at the center of ( 1 3 (μ− w 1−α ), 2 3 (μ− w 1−α )). If q EF e < ¯ q e , i.e., w μ > 5α(1−α) 12+5α , the entrant would order q EF e and the supplier’s expected profit would be π EF s (q EF e ) > π N ∗ s . If q EF e ≥ ¯ q e , i.e., w μ ≤ 5α(1−α) 12+5α , the entrant would order ¯ q e , and the supplier would be indifferent between leakage and nonleakage because π EF s (¯ q e )=π N ∗ s by the definition of ¯ q e . In summary, there are three regions: (1) If w μ ≤ α(1−α) 3+α = B FB , the entrant would order q N∗ e and the supplier would not leak the information to the incumbent; (2) If α(1−α) 3+α < w μ ≤ 5α(1−α) 12+5α = UB EF , the entrant would order ¯ q e and the supplier would be indifferent between leakage and nonleakage; (3) If w μ > UB EF , the entrant would order q EF e and the supplier would leak the information to the incumbent. In Figure A.7, the (darkly or lightly) shaded area represents the (weakly) nonleakage region when the entrant orders first. 136 Figure A.7. Supplier’s leakage decision when entrant orders first, for r =4 and p =0.2. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.10 0.15 revenue sharing rate α scaled wholesale price ω μ UB P B FB UB EF LB L− S1 UB H (v) The supplier’s preference between incumbent ordering first and entrant ordering first. In this part, we provide a sketch of a proof that the supplier weakly prefers the entrant to be the first mover in all regions (the rigorous proof is tedious and omitted). The darkly shaded area in Figure A.7 represents the intersection of the nonleakage regions under the original model (incumbent ordering first) and the new model (entrant first); in this area, the supplier is indifferent to who orders first. First, consider the region w μ ≤ α(1−α) 3+α . In this region, nonleakage is sustainable when the entrant orders first but unsustainable (in the most part of the region) when the incumbent orders first. To determine which scenario is better for the supplier, it amounts to compare the supplier’s expected profits under leakage and nonleakage when the incumbent orders first. FrompA 2 H +(1−p)A 2 L −μ 2 >0, we can show thatp(π N ∗ sH −π S∗ sH )+(1−p)(π N ∗ sL −π S∗ sL )> α 144 (μ− w 1−α )[5(μ− w 1−α )−12( w α + w 1−α )]. The right hand side of the inequality is nonnegative if w μ ≤ 5α(1−α) 12+5α , which is implied by w μ ≤ α(1−α) 3+α . Thus, in this region, the supplier prefers the nonleakage equilibrium which can be sustained by letting the entrant order first. Next, considertheregion α(1−α) 3+α < w μ ≤ 5α(1−α) 12+5α . Thelowerbound α(1−α) 3+α isjustB FB . The upper bound 5α(1−α) 12+5α is comparable with UB H = ( 3r pr+1−p +1) α(1−α) 12+4α . In fact, we can show that 3r pr+1−p +1≥ 5 and hence 5α(1−α) 12+5α ≤UB H , if p≤ (3r−4)/(4r−4). The last inequality 137 is satisfied by all examples discussed in this paper. Thus, the region α(1−α) 3+α < w μ ≤ 5α(1−α) 12+5α is typically contained in the nonleakage region when the incumbent orders first. The supplier is indifferent between leakage (with the entrant being the first mover) and nonleakage (with the incumbent ordering first). Finally, consider the region w μ > 5α(1−α) 12+5α . In this region, leakage is preferred by the supplier when the entrant orders first and is also sustained (in the most part of the region) when the incumbent orders first. To determine which scenario is better for the supplier, it amounts to compare the supplier’s expected profits (under leakage) when the entrant orders first and when the incumbent orders first. If the entrant orders first and the supplier leaks the information, the supplier’s profits in the two demand states are π EF sH (q EF e ) =w(q EF e +q iH (q EF e ))+α(A H −q EF e −q iH (q EF e ))(q EF e +q iH (q EF e )) =w( A H 2 + μ 4 − 3 4 w 1−α )+α( A H 2 − μ 4 + 3 4 w 1−α )( A H 2 + μ 4 − 3 4 w 1−α ), π EF sL (q EF e ) =w(q EF e +q iL (q EF e ))+α(A L −q EF e −q iL (q EF e ))(q EF e +q iL (q EF e )) =w( A L 2 + μ 4 − 3 4 w 1−α )+α( A L 2 − μ 4 + 3 4 w 1−α )( A L 2 + μ 4 − 3 4 w 1−α ). If the incumbent orders first and the supplier leaks the information, the supplier’s profits are (assuming θ≥ 3 for simplicity): π S∗ sH =w(q S∗ iH +q S∗ eH )+α(A H −q S∗ iH −q S∗ eH )(q S∗ iH +q S∗ eH ) = 3 4 w(A H − w 1−α )+α(A H − 3 4 (A H − w 1−α )) 3 4 (A H − w 1−α ), π S∗ sL =w(q S∗ iL +q S∗ eL )+α(A L −q S∗ iL −q S∗ eL )(q S∗ iL +q S∗ eL ) = 3 4 w(A L − w 1−α )+α(A L − 3 4 (A L − w 1−α )) 3 4 (A L − w 1−α ). As a result, π EF sH (q EF e )−π S∗ sH =w(− A H 4 + μ 4 )+α[( A H 4 ) 2 − 3 8 w 1−α A H + 3 8 w 1−α μ−( μ 4 ) 2 ], π EF sL (q EF e )−π S∗ sL =w(− A L 4 + μ 4 )+α[( A L 4 ) 2 − 3 8 w 1−α A L + 3 8 w 1−α μ−( μ 4 ) 2 ], 138 π EF s (q EF e )−π S∗ s =p[π EF sH (q EF e )−π S∗ sH ]+(1−p)[π EF sL (q EF e )−π S∗ sL ] = 1 16 α[pA 2 H +(1−p)A 2 L −μ 2 ] = 1 16 αp(1−p)(A H −A L ) 2 ≥ 0. Thus, in this region, the supplier would like to have the entrant order first and then leak his order quantity to the incumbent. In summary, the supplier weakly prefers the entrant to be the first mover in all regions. 139 APPENDIX B PROOFS OF CHAPTER 2 Proof of Lemma 2.4.1 The IR constraint (2.4.2) and (2.4.3) imply v −x + θ −x t −x ≤ w ≤ v x +θ x t x . Because the objective function is increasing in w, (2.4.2) must be binding at optimality. By substitution, the objective function becomes λ x (v x +θ x t x −ct 2 x ), which is maximized at t ∗ x = θx 2c . The corresponding price is w ∗ = v x + θ x t x = v x + θ 2 x 2c . Since θx(θ −x −θx) 2c ≤v x −v −x , it implies that v −x +θ −x t x ≤v x +θ x t x is satisfied when t ∗ x = θx 2c . As a result, (2.4.3) is also satisfied. Proof of Lemma 2.4.2 Suppose the SP charges w from the buyers, the ICH and ICL constraints are t L θ L +v L −w ≥ t H θ L +v L −w, (B.0.1) t H θ H +v H −w ≥ t L θ H +v H −w. (B.0.2) The above inequalities are equivalent to t H =t L . Proof of Lemma 2.4.3 IR constraints (2.4.5) and (2.4.6) imply that w ≤ min(v x + θ x t, v −x +θ −x t). The optimization problem (2.4.4) becomes max w,t v x +θ x t−ct 2 s.t. (θ x −θ −x )t≤v −x −v x . As a result, the optimal solution is t = min( θx 2c , v −x −vx θx−θ −x ) if θ x ≥ θ −x (i.e. x = H); and t =max( θx 2c , vx−v −x θx−θ −x )ifθ −x ≥θ x (i.e. x =L). Soif θ L 2c ≥ v L −v H θ H −θ L , theoptimalsolutionist = θ L 2c , w = θ 2 L 2c +v L , and the optimal profit is θ 2 L 4c +v L ; if θ H 2c ≤ v L −v H θ H −θ L , and the optimal solution is t = θx 2c ,w = θ 2 x 2c +v x andtheoptimalprofitis θ 2 x 4c +v x ; if θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c ,theoptimalsolution is t = v L −v H θ H −θ L , w = θ L (v L −v H ) θ H −θ L +v L and the optimal profit is π f = θ L (v L −v H ) θ H −θ L +v L −c( v L −v H θ H −θ L ) 2 . 140 ProofofLemma2.4.6 Givenanypricep,theeffortthatmaximizestheobjectivefunction (2.4.7) is t x = p 2c . By substitution, the objective function becomes λ x p 2 /(4c), increasing in p. The type x buyer’s surplus is f x (t x |p) = v x +θ x t x −pt x , which, under the relationship t x = p 2c , reduces to f x (p)= 1 2c (2cv x +θ x p−p 2 ). (B.0.3) We have f x (p) ≥ 0 if and only if p ∈ [p 1 ,p 2 ], where p 1 = 1 2 θ x − p θ 2 x +8cv x and p 2 = 1 2 θ x + p θ 2 x +8cv x . The assumption v x ≥ 0 implies that p 1 ≤ 0 and p 2 ≥θ x . Let p d x =p 2 . Thus, in the region p∈ [0, p d x ], the optimal effort is t ∗ x = p 2c , the objective value λ x p 2 /(4c) increases with p, and the IR constraint f x (p)≥ 0 is satisfied. When p > p d x , the IR constraint is violated under the relationship t x = p 2c . So, the optimal effort is determined by the binding IR constraint, i.e., t x = vx p−θx . By substitution, theobjective function(2.4.7)becomesλ x (pt x −ct 2 x ) =λ x (v x +θ x t x −ct 2 x ),which ismaximized by t x = θx 2c , corresponding to p =θ x + vx tx =θ x + 2cvx θx . Because p d x ≥θ x , we have p d x 2c ≥ θx 2c and, by Figure 2.7, p d x ≤θ x + 2cvx θx . Thus, as p increases from p d x to θ x + 2cvx θx , t x decreases from p d x 2c to θx 2c , and the objective value increases to its maximum. The objective value declines when p increases further. Proof of Lemma 2.4.7 The (ICH) and (ICL) constraints can be simplified as follows: (θ H −p)(t H −t L )≥ 0, (p−θ L )(t H −t L )≥ 0. Consider three cases: (1) If t H − t L > 0, the above inequalities imply θ H − p ≥ 0 and p−θ L ≥ 0, i.e., θ H ≥ p ≥ θ L ; (2) If t H −t L = 0, any p can satisfy these inequalities; (3) If t H −t L < 0, the above inequalities imply θ H −p≤ 0 and p−θ L ≤ 0, i.e., θ H ≤ p≤ θ L . Because θ H >θ L by our basic assumption, this case is infeasible. 141 Proof of Lemma 2.4.8 Notice that p d H >θ H , and if p> (<)p 0 , θ H t+v H > (<)θ L t+v L . The proof follows directly from Lemma 2.4.5 and Lemma 2.4.7. Proof of Lemma 2.4.9 (i) Since θ H < p 0 and p d L < p FB L , p d L < p FB L < θ H < p 0 only requires p FB L <θ H Or equivalently, v L θ H −θ L < θ L 2c . (ii) By Figure 2.9, the relationship θ L 2c ≤ v L θ H −θ L ≤ p d L 2c implies p d L < θ H < p FB L . The assumption v H ≥ 0 implies v L −v H θ H −θ L ≤ v L θ H −θ L or equivalently, θ H < p 0 . Combining all these inequalities, we obtain the following order of effort levels: max v L −v H θ H −θ L , θ L 2c ≤ v L θ H −θ L ≤ p d L 2c ≤ θ H 2c . (B.0.4) and p d L <θ H < min(p FB L ,p 0 ) (B.0.5) (iii) By Figure 2.10, the assumption v L θ H −θ L > p d L 2c (> θ L 2c ) is equivalent to θ H <p d L (<p FB L ). Combining the fact that p 0 > θ H , only θ H < p d L < min(p FB L ,p 0 ) and θ H < p 0 < p d L < p FB L are the possible scenarios. Proof of Proposition 2.4.10 (i) If v L θ H −θ L < θ L 2c , we obtain the following order of prices θ L <p d L <p FB L <θ H <p 0 (B.0.6) First, consider the price range [0,p FB L ]. The expected profits from both buyer types are increasing inpandthus the expected totalprofitis maximized atp † =p FB L =θ L + 2cv L θ L . Also by Lemma 2.4.6, the optimal efforts at price p FB L are t † L = v L p FB L −θ L and t † H = p FB L 2c . Because θ H >p FB L ≥θ L , by Lemma 2.4.7, the IC constraints reduce to t H ≥t L . By the definition of p d L , the inequality p FB L ≥ p d L implies that p FB L 2c ≥ v L p FB L −θ L , which is exactly t † H ≥ t † L . So, when the price is restricted in [0,p FB L ], the optimal solution to problem (2.4.11)e(2.4.15) is given by (p † ,t † H ,t † L ) =(p FB L , v L p FB L −θ L , p FB L 2c ). 142 Next, consider the price range[p FB L ,θ H ]. Whenp increases in this range, by Lemma 2.4.6 and inequalities (B.0.6), the expected profit from the low type decreases while that from the high type increases; the optimal efforts are still given by t L (p) = v L p−θ L and t H (p) = p 2c . By substitutingintheoptimalefforts,inthegivenpricerange,problem(2.4.11)-(2.4.15)reduces to (2.4.16). Let π(p) denote the objective function in (2.4.16). Its first order derivative is given by: π ′ (p)=λ p 2c +(1−λ) v L p−θ L −(1−λ) pv L (p−θ L ) 2 +(1−λ) 2c(v L ) 2 (p−θ L ) 3 =λ p 2c −(1−λ) θ L v L (p−θ L ) 2 +(1−λ) 2c(v L ) 2 (p−θ L ) 3 =λ p 2c +(1−λ) 2cv L (p−θ L ) 2 v L p−θ L − θ L 2c . Since π ′ (p FB L ) = λp FB L 2c ≥ 0, the optimal solution for (2.4.16) satisfies p > p FB L . The Lagrangian function is L(p,ξ)=λ p 2 4c +(1−λ) " pv L p−θ L −c v L p−θ L 2 # +ξ 1 (θ H −p) dL(p,ξ) dp =λ p 2c +(1−λ) 2cv L (p−θ L ) 2 v L p−θ L − θ L 2c −ξ 1 If ξ 1 = 0, the optimal solution p satisfies λ p 2c +(1−λ) 2cv L (p−θ L ) 2 v L p−θ L − θ L 2c =0 and p<θ H If ξ 1 6=0, the optimal solution p =θ H and λ θ H 2c +(1−λ) 2cv L (θ H −θ L ) 2 v L θ H −θ L − θ L 2c ≥ 0 The reduced problem may have an interior solution atb p such that π ′ (b p)= 0, i.e., v L b p−θ L = θ L 2c − λb p(b p−θ L ) 2 4c 2 (1−λ)v L if λ θ H 2c + (1− λ) 2cv L (θ H −θ L ) 2 v L θ H −θ L − θ L 2c ≤ 0 (sufficient condition). If the optimization has a boundary solution at p = θ H , λ θ H 2c +(1−λ) 2cv L (θ H −θ L ) 2 v L θ H −θ L − θ L 2c ≥ 0 (necessary condition). Because π ′ (p † ) = λ p † 2c > 0, the lower price bound p † cannot be optimal. 1 1 As a potentially useful note, the first order equation π ′ (p) = 0 can be transformed to (1 − λ) h pvL p−θL −c v 2 L (p−θL) 2 i = λ p 4c (p− θ L ) + (1− λ) h vL(p−θL/2) p−θL i , and thus the optimal objective function can be simplified to π(p) =λ p 2 2c −λ p 4c θ L +(1−λ) h vL(p−θL/2) p−θL i =λ p 2c (p− θL 2 )+(1−λ) v L + θL 2 t L . 143 Inaddition, considerthepricerange(θ H ,p 0 ). ByLemma2.4.7,theICconstraintsrequire that t H = t L . According to Figure 2.8, the IRL is more restrictive than IRH in this range, so IRL should bind. Problem (2.4.11)-(2.4.15)then reduces to the one-type problem (2.4.7)- (2.4.8), with x = L and λ x = 1. By Lemma 2.4.6, the objective value decreases with p in (p FB L ,p 0 )⊃ (θ H ,p 0 ), so the optimal p does not belong to this range. Finally, consider the price range (p 0 ,∞). By Lemma 2.4.7, the IC constraints require that t H = t L . According to Figure 2.8, the IRH is more restrictive than IRL in this range, so IRH should bind. Problem (2.4.11)-(2.4.15)then reduces tothe one-type problem (2.4.7)- (2.4.8), with x =H and λ x =1. Since θ L <p 0 implies v L −v H θ H −θ L < θ L 2c < θ H 2c , the objective value decreases with p in (p 0 ,∞), so the optimal p does not belong to this range. (ii) First, consider the price range [0,θ H ]. When p is restricted in this range, by Lemma 2.4.6, the expected profits from both buyer types are increasing in p and thus the expected total profit is maximized at p † =θ H . Also by Lemma 2.4.6, the optimal efforts at this price are t † L = v L p † −θ L and t † H = p † 2c . By a similar argument as in the proof of Proposition 2.4.10, we have t † H ≥ t † L and the IC constraints are met. The service provider’s expected profit is given by the objective function in (2.4.16) evaluated at p = θ H , i.e., π m = λ θ 2 H 4c +(1− λ) θ H v L θ H −θ L −c v L θ H −θ L 2 (the superscript “m” represents “medium” price). Next, consider the price range (θ H ,min(p FB L ,p 0 )). If p FB L ≤ p 0 , i.e. v L −v H θ H −θ L ≤ θ L 2c , by a similar argument as in the proof of Proposition 2.4.10, problem (2.4.11)-(2.4.15) reduces to the one-type problem (2.4.7)-(2.4.8).with x =L and λ x = 1. By Lemma 2.4.6, the objective value increases with p in (θ H ,p FB L ]. The objective value is maximized at p ∗ = p FB L and the IRH constraint is satisfied at that solution; if p FB L > p 0 , i.e. v L −v H θ H −θ L > θ L 2c , p can only be raised up to p 0 = θ L + v L ( θ H −θ L v L −v H ) such that IRH is binding. (Although it is possible to raise p further so that the point (p,t) would trace the high type’s value line, that would be suboptimal because θ H 2c ≥ v L −v H θ H −θ L by (B.0.4)). Thus, the best solution in the price range (θ H ,min(p FB L ,p 0 )) can be summarized as: p ∗ = min(p FB L ,p 0 )=θ L + v L t and t ∗ L =t ∗ H =t, (B.0.7) 144 for t = max{ v L −v H θ H −θ L , θ L 2c }. The service provider’s expected profit is π h = v L +θ L t−ct 2 (the superscript “h” represents “high” price). The optimalsolutiontoproblem (2.4.11)-(2.4.15)is between theabove twosolutions. So, we compare π m and π h below: π m −π h =λ θ 2 H 4c +(1−λ) " θ H v L θ H −θ L −c v L θ H −θ L 2 # −(v L +θ L t−ct 2 ) =λ " θ 2 H 4c − θ H v L θ H −θ L +c v L θ H −θ L 2 # + " θ H v L θ H −θ L −c v L θ H −θ L 2 −(v L +θ L t−ct 2 ) # =λc θ H 2c − v L θ H −θ L 2 + θ L ( v L θ H −θ L −t)+c t+ v L θ H −θ L t− v L θ H −θ L =λc θ H 2c − v L θ H −θ L 2 −c( v L θ H −θ L −t) t+ v L θ H −θ L − θ L c . Thus, π m −π h ≥ 0 if and only if λ≥ ( v L θ H −θ L −t) t+ v L θ H −θ L − θ L c θ H 2c − v L θ H −θ L −2 . Notice that in the current parameter regime with θ L 2c ≤ v L θ H −θ L , we have θ L 2c ≤ max{ v L −v H θ H −θ L , θ L 2c }≤ v L θ H −θ L . It implies that v L θ H −θ L −t ≥ 0, t + v L θ H −θ L − θ L c = (t− θ L 2c )+ ( v L θ H −θ L − θ L 2c ) ≥ 0, and hence ( v L θ H −θ L −t) t+ v L θ H −θ L − θ L c θ H 2c − v L θ H −θ L −2 ≥ 0. (iii) Sinceθ H < p d L < p FB L , the expected profit increases with p by Lemma 2.4.6 in the price range [0,θ H ],. So, the optimal price must lie in (θ H ,∞). By Lemma 2.4.7, the IC constraints require that t H = t L . There are three cases: (1) θ H < p d L < p FB L ≤ p 0 ; (2)θ H <p 0 <p d L <p FB L ; and (3)θ H <p d L <p 0 <p FB L . (1) θ H <p d L <p FB L ≤ p 0 Since p 0 ≥ p d L , i.e. v L −v H θ H −θ L ≤ p d L 2c . By Lemma 2.4.6, the expected profit is increased by raisingthepricepfromθ H top d L . Atp =p d L , theoptimal(p,t)pointhitsthelowtype’s value line and moves along that line when increasing p further. Since p FB L ≤ p 0 , i.e. v L −v H θ H −θ L ≤ θ L 2c , the optimal (p,t) stops at the first best solution for the low type, i.e., (θ L + 2cv L θ L , θ L 2c ). 145 (2)θ H <p 0 <p d L <p FB L If ( θ L 2c <) v L −v H θ H −θ L ≤ θ H 2c , the optimal (p,t) stops at the intersection of the two types’ value lines, i.e., ( θ H v L −θ L v H v L −v H , v L −v H θ H −θ L ); If ( θ L 2c <) θ H 2c < v L −v H θ H −θ L , as shown in Figure 2.10, the optimal (p,t) should switch to the high type’s value line and move toward the high type’s first best solution (θ H + 2cv H θ H , θ H 2c ). (3) θ H < p d L < p 0 < p FB L , i.e. v L −v H θ H −θ L > p d L 2c (> θ H 2c ). The optimal (p,t) hits the high type’s value line first, at price p = p d H , and move alone that line as p increases further. That is, problem (2.4.11)-(2.4.15) reduces to the one-type problem (2.4.7)-(2.4.8), with x = H and λ x = 1. By Lemma 2.4.5, the optimal solution is t ∗ = θ H 2c and p ∗ =θ H + 2cv H θ H . Notice that this is identical to the solution in the sub case (2) and (3) when θ H 2c < v L −v H θ H −θ L above, and they can be combined into one case, θ H 2c < v L −v H θ H −θ L , regardless of v L −v H θ H −θ L ≤ p d L 2c or v L −v H θ H −θ L > p d L 2c . Proof of Proposition 2.4.11 In this proof, we only consider the most intriguing case, parameter regime II ( θ L 2c ≤ v L θ H −θ L ≤ p d L 2c ). In the proof of Proposition 2.4.10, the optimal objective values under the medium and high prices are found to be π m = λ θ 2 H 4c + (1− λ) θ H v L θ H −θ L −c v L θ H −θ L 2 and π h = v L + θ L t− ct 2 , respectively, and the optimal solution shifts from the former to the latter at the threshold λ mh =( v L θ H −θ L −t) t+ v L θ H −θ L − θ L c θ H 2c − v L θ H −θ L −2 . ByLemma2.4.5,theoptimalobjectivevaluesfortheone-typeproblemsareπ fb H =λ(v H + θ 2 H 4c ) and π fb L =(1−λ)(v L + θ 2 L 4c ), respectively. First, we compare π fb H with π m and π h : π fb H −π m =λ(v H + θ 2 H 4c )−λ θ 2 H 4c −(1−λ) " θ H v L θ H −θ L −c v L θ H −θ L 2 # =λ " v H + θ H v L θ H −θ L −c v L θ H −θ L 2 # − " θ H v L θ H −θ L −c v L θ H −θ L 2 # , 146 π fb H −π h =λ(v H + θ 2 H 4c )−(v L +θ L t−ct 2 ). Thus,π fb H ≥π m ifandonlyifλ≥λ m H = θ H v L θ H −θ L −c v L θ H −θ L 2 v H + θ H v L θ H −θ L −c v L θ H −θ L 2 andπ fb H ≥π h ifandonlyifλ≥λ h H =(v L +θ L t−ct 2 )/(v H + θ 2 H 4c ). The high-type-onlysolution dominates if and only if λ≥ max{λ m H ,λ h H }. Next, we compare π fb L with π m and π h : π fb L −π m =(1−λ)(v L + θ 2 L 4c )−λ θ 2 H 4c −(1−λ) " θ H v L θ H −θ L −c v L θ H −θ L 2 # =λ " θ H v L θ H −θ L −c v L θ H −θ L 2 −(v L + θ 2 L 4c )− θ 2 H 4c # + " (v L + θ 2 L 4c )− θ H v L θ H −θ L +c v L θ H −θ L 2 # =λc " θ L v L c(θ H −θ L ) − v L θ H −θ L 2 − θ 2 L 4c 2 − θ 2 H 4c 2 # +c " θ 2 L 4c 2 − θ L v L c(θ H −θ L ) + v L θ H −θ L 2 # =−λc " v L θ H −θ L − θ L 2c 2 + θ 2 H 4c # +c v L θ H −θ L − θ L 2c 2 , π fb L −π h =(1−λ)(v L + θ 2 L 4c )−(v L +θ L t−ct 2 )=−λ(v L + θ 2 L 4c )+( θ 2 L 4c −θ L t+ct 2 ). Thus, π fb L ≥ π m if and only if λ ≤ λ m L = v L θ H −θ L − θ L 2c 2 v L θ H −θ L − θ L 2c 2 + θ 2 H 4c and π fb L ≥ π h if and only if λ ≤ λ h L = ( θ 2 L 4c −θ L t+ct 2 )/(v L + θ 2 L 4c ). The low-type-only solution dominates if and only if λ≤ min{λ m L ,λ h L }. In the case v L −v H θ H −θ L ≤ θ L 2c ≤ v L θ H −θ L ≤ p d L 2c , because t = θ L 2c , we have λ h L = 0 and hence the low-type-only solution cannot be optimal. Since λ h H = (v L + θ 2 L 4c )/(v H + θ 2 H 4c ) and λ mh = v L θ H −θ L − θ L 2c 2 θ H 2c − v L θ H −θ L −2 , we arrive at case (a) of Figure B.1 if λ mh ≤ λ h H , i.e., v H ≤ (v L + θ 2 L 4c ) θ H 2c − v L θ H −θ L 2 v L θ H −θ L − θ L 2c −2 − θ 2 H 4c , and at case (b) otherwise. In both cases, the high-type-only solution is optimal when λ is large enough. 147 FigureB.1. Optimalobjectivevaluesfromdifferentsolutionsasfunctionsofλ,when v L −v H θ H −θ L ≤ θ L 2c ≤ v L θ H −θ L ≤ p d L 2c . (a) λ mh ≤λ h H ; (b) λ mh >λ h H . ! ! !" ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! "! #! !! $%&! ! ! !" ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! "! #! !! $'&! Inthecase θ L 2c ≤ v L −v H θ H −θ L ≤ v L θ H −θ L ≤ p d L 2c , we havet = v L −v H θ H −θ L ,λ h L = ( θ 2 L 4c −θ L t+ct 2 )/(v L + θ 2 L 4c ), λ mh = v H θ H −θ L h 2v L −v H θ H −θ L − θ L c i θ H 2c − v L θ H −θ L −2 , andλ h H = (v H +θ H v L −v H θ H −θ L −c( v L −v H θ H −θ L ) 2 )/(v H + θ 2 H 4c ). The most complex situation, 0 < λ h L < λ mh < λ m H < 1, is illustrated in Figure B.2: as λ increases from 0 to 1, the optimal solution changes from low type only (π fb L ), to two types with single effort (π h ), then two types with different efforts (π m ), and finally high type only (π fb H ). FigureB.2. Optimalobjectivevaluesfromdifferentsolutionswhen θ L 2c ≤ v L −v H θ H −θ L ≤ v L θ H −θ L ≤ p d L 2c and 0<λ h L <λ mh <λ m H < 1. ! ! !" ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! "! #! !! ! ! ! ! ! ! !" ! Other thresholds where theoverall optimalsolution switches between aone-typesolution and a two-type solution can be determined by comparing the service provider’s expected profits directly. 148 Proof of Lemma 2.4.12 The IR constraint (2.4.18) implies pt x + a ≤ v x + θ x t x . Be- cause the objective function is increasing in p, (2.4.18) must be binding at optimality. By substitution, the objective function becomes λ x (v x + θ x t x − ct 2 x ), which is maximized at t ∗ x = θx 2c . The corresponding price satisfies p ∗ θ 2 x 2c +a ∗ = v x + θ 2 x 2c and (2.4.19) is satisfied since θx(θx−θ −x ) 2c ≥v −x −v x . Proof of Lemma 2.4.13 ICH and ICL are equivalent to θ L (t H − t L ) ≤ p(t H − t L ) ≤ θ H (t H −t L ). Now we show thatt H =t L never appears in anoptimalsolution ofoptimization problem (2.4.20)-(2.4.24). We show it in two steps: First we find the optimal solution if t H = t L = t . The objective function becomes maxpt+a−ct 2 . In order to satisfy both (IRH) and (IRL), one of them has to be binding. If t > v L −v H θ H −θ L , IRL is binding since θ H t+v H ≥ θ L t+v L ; If t≤ v L −v H θ H −θ L , IRH is binding since θ L t+v L ≥θ H t+v H . Substituting IRL or IRH into the objective function, we get If θ L 2c ≥ v L −v H θ H −θ L ,t = θ L 2c ,p θ L 2c +a = θ 2 L 2c +v L , IRLisbinding andtheoptimalprofitis θ 2 L 4c +v L ; If θ H 2c ≤ v L −v H θ H −θ L , t = θ H 2c , p θ H 2c +a = θ 2 H 2c +v L , IRH is binding and the optimal profit is θ 2 H 4c +v H ; If θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c , p v L −v H θ H −θ L +a = (v L −v H )θ L θ H −θ L +v L = (v L −v H )θ H θ H −θ L +v H , IRH and IRL are binding and the optimal profit is θ L (v L −v H ) θ H −θ L +v L −c( v L −v H θ H −θ L ) 2 Next, in each of the above scenario, we could construct a solution of serving both types with different effortand theprofit dominates the optimal solution ofserving bothtypes with the same effort. If θ L 2c ≥ v L −v H θ H −θ L , let p = θ H ,t L = θ L 2c , t H = θ L 2c +ǫ, IRL is binding and the total profit is greater than θ 2 L 4c +v L since the SP makes the same profit ((1−λ)( θ 2 L 4c +v L )) from a low type buyer but more profit from a high type buyer since p =θ H , t H = θ L 2c +ǫ is closer to the first best solution ( θ H 2c ) for the high type, i.e. θ L 2c < θ L 2c +ǫ≤ θ H 2c ; If θ H 2c ≤ v L −v H θ H −θ L , let p =θ L , t H = θ H 2c , t L = θ H 2c −ǫ, IRH is binding the total profit is greater than θ 2 H 4c +v H since the SP makes the same profit (λ( θ 2 H 4c +v H )) from a high type buyer but 149 less profit fromalow type buyer since ifp=θ H , t L = θ H 2c −ǫis closer tothe first best solution ( θ L 2c ) for the low type, i.e. θ L 2c ≤ θ H 2c −ǫ≤ θ H 2c ; If θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c , let t L = v L −v H θ H −θ L and t H = θ H 2c and p = θ H ,the total profit is greater than θ L (v L −v H ) θ H −θ L +v L −c( v L −v H θ H −θ L ) 2 since the SP makes the same profit ((1−λ) θ L (v L −v H ) θ H −θ L +v L − c( v L −v H θ H −θ L ) 2 ) from a low type buyer but more profit (λ( θ 2 H 4c +v H ))from a high type buyer since v L −v H θ H −θ L ≤ θ H 2c and θ H 2c is the first best effort for high type.. As a result, in each possible scenario when t H = t L , we could find a solution in which t H 6= t L that dominates t H = t L . So in the optimal solution, the SP will serve both types with different effort under a two part tariff contract. Proof of Proposition 2.4.14 The service provider offers t H to high type buyer and t L to low type buyer. The optimization problem (2.4.20) could be simplified as the following, max a,p,t H ,t L λ(pt H +a−ct 2 H )+(1−λ)(pt L +a−ct 2 L ) s.t.θ L t L +v L ≥pt L +a θ H t H +v H ≥pt H +a θ L (t H −t L )≤p(t H −t L )≤ θ H (t H −t L ) Lagrangian function of the above optimization problems is L(t H ,t L ,ξ,ν,η,ρ)=λ(pt H +a−ct 2 H )+(1−λ)(pt L +a−ct 2 L )+ξ(θ H t H +v H −pt H −a) +ν(θ L t L +v L −pt L −a)+η(θ H −p)(t H −t L )+ρ(p−θ L )(t H −t L ) where the Lagrangian parameters ξ, ν, η and ρ are nonnegative. Take the partial derivative of L(t H ,t L ,ξ,ν,η,ρ) with respect to t H , t L , ξ, ν, η, ρ respec- tively, we obtain: ∂L ∂t H =λ(p−2ct H )+(ξ+η)(θ H −p)+ρ(p−θ L ), 150 ∂L ∂t L = (1−λ)(p−2ct L )+(ν−ρ)(θ L −p)−η(θ H −p), ∂L ∂p =λt H +(1−λ)t L −ξt H −νt L +(ρ−η)(t H −t L ), ∂L ∂a = 1−ξ−ν. After simplification, the KKT conditions suggest, t H = p 2c + (ξ+η)(θ H −p)+ρ(p−θ L ) 2cλ (B.0.8) t L = p 2c + (ν−ρ)(θ L −p)−η(θ H −p) 2c(1−λ) (B.0.9) ν = 1−ξ (B.0.10) (λ−ξ−ρ+η)(t H −t L )= 0 (B.0.11) ξ(θ H t H +v H −pt H −a) = 0 (B.0.12) ν(θ L t L +v L −pt L −a) =0 (B.0.13) η(θ H −p)= 0 (B.0.14) ρ(p−θ L ) =0 (B.0.15) θ H t H +v H −pt H −a≥ 0, θ L t L +v L −pt L −a≥ 0, (θ H −p)(t H −t L )≥ 0, (p−θ L )(t H −t L )≥ 0. Equation (B.0.11) suggests that eitherλ−ξ−η+ρ = 0 or t H = t L in an optimal solution. Lemma 2.4.13 suggests that the second case cannot occur in optimum. So we have λ = ξ+η−ρ and t H 6=t L . (B.0.8) and (B.0.9) imply that t H = θ H 2c + ρ(θ H −θ L ) 2cλ and t L = (1−ξ+ρ)θ L −(λ+ρ−ξ)θ H 2c(1−λ) Inaddition,(B.0.10)suggeststhatonlyoneofthefollowingcaseshappen(a)ξ =0,ν =1; (b) ν =0, ξ =1; (c)ν6= 0 ξ6= 0. We discuss each case one by one in the following analysis. 151 (a) ξ = 0, ν = 1: Since ν > 0, (B.0.13) implies that θ L t L +v L = pt L +a. In addition, substitute ξ = 0 into λ=ξ+η−ρ, we obtainη =λ+ρ>0. And equation (B.0.14) suggest that p = θ H since t H 6= t L and η > 0. In addition, since p = θ H 6= θ L , (B.0.15) suggest that ρ = 0. Substitute the value of ξ, ν, η, ρand p into (B.0.8) and (B.0.9), we obtain t H = θ H 2c , t L = θ L −λθ H 2c(1−λ) are the optimal time effort on high type and low type buyer respectively. Since IRL is binding, a =v L +(θ L −p)t L =v L −(θ H −θ L ) θ L −λθ H 2c(1−λ) . Now we substitute t H , t L , a and p to θ H t H +v H −pt H −a≥ 0 and check if the optimal solution satisfies IRH constraint. IRH is satisfied if θ L −λθ H 2c(1−λ) ≥ v L −v H θ H −θ L = ˜ t. As a result, if θ L −λθ H 2c(1−λ) ≥ v L −v H θ H −θ L , the optimal contract parameter and time effort are: p = θ H , a = v L − (θ H −θ L )(θ L −θ H λ) 2c(1−λ) t H = θ H 2c , t L = θ L −θ H λ 2c(1−λ) . And the total profit is v L + θ 2 L 4c + λ(θ H −θ L ) 2 4c(1−λ) (b) ν = 0, ξ = 1: Substitute them into λ = ξ +η−ρ we get η = λ+ρ−1. Suppose ρ = 0, then η =λ−1 < 0 contradicts to η≥ 0. As a result ρ6= 0, and p = θ L as suggested by (B.0.15). In additionη = 0 from (B.0.14) and ρ = 1−λ. Substituting ρ, ν and ξ into (B.0.8) and (B.0.9), we obtain the optimal effort for high type and low type buyers are t H = θ H −(1−λ)θ L 2cλ , t L = θ L 2c respectively. Since ξ >0, (B.0.11) suggests that θ H t H +v H =pt H +a. Substituting the optimal value of t H , we get a = (θ H −θ L ) θ H −(1−λ)θ L 2cλ +v H . Now we check if the optimal solution satisfies IRL constraint, i.e. θ L t L +v L −pt L −a≥ 0. IRL is satisfied if θ H −(1−λ)θ L 2cλ ≤ v L −v H θ H −θ L = ˜ t. As a result, if θ H −(1−λ)θ L 2cλ ≤ v L −v H θ H −θ L , the optimal contract is p = θ L , a = (θ H −θ L ) θ H −(1−λ)θ L 2cλ +v H , andtheoptimaleffortforhightypeandlowtypecustomers aret L = θ L 2c andt H = θ H −(1−λ)θ L 2cλ . Total profit is v H + θ 2 H 4c + (1−λ)(θ H −θ L ) 2 4cλ . (c) ν > 0 and ξ > 0: since θ L ≤ p ≤ θ H , there are three scenarios to consider: (c1) θ L <p<θ H ; (c2) p=θ H ; (c3) p=θ L . (c1) If θ L < p < θ H , (B.0.14) , (B.0.15) and t H 6= t L implies that η = ρ = 0, and ξ = λ− η +ρ = λ. Since ν +ξ = 1, substitute them into (B.0.8) and (B.0.9), we have t H = p 2c + ξ(θ H −p) 2cλ = θ H 2c , t L = p 2c + (1−ξ)(θ L −p) 2c(1−λ) = θ L 2c . In addition, (B.0.11)and (B.0.13)suggest 152 thatθ H t H +v H −pt H −a =0andθ L t L +v L −pt L −a =0. As aresult, p=θ H +θ L − 2c(v L −v H ) θ H −θ L and a = v L θ H −v H θ L θ H −θ L − θ L θ H 2c Now we check that the constraint θ L ≤p≤ θ H is satisfied under the optimal solution. It requires that θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c . The total profit is λ( θ 2 H 4c +v H )+(1−λ)( θ 2 L 4c +v L ). Notice that this is the only scenario in which the SP achieves first best for both high type and low type buyers. (c2) If p = θ H , ρ = 0, t H = θ H 2c , t L = θ H 2c + (1−ξ)(θ L −θ H ) 2c(1−λ) . Since θ H t H +v H −pt H −a = 0 and θ L t L +v L −pt L −a =0, we obtain a=v H and t L = v L −v H θ H −θ L = ˜ t. Since ξ≥ 0 and ν≥ 0, ξ = λθ H −θ L +2c ˜ t(1−λ) θ H −θ L ≥ 0, i.e. ˜ t≥ θ L −λθ H 2c(1−λ) ν = 1−ξ = (θ H −2c ˜ t)(1−λ) θ H −θ L ≥ 0, i.e. ˜ t≤ θ H 2c . As a result, the service provider makes a profit of λθ 2 H 4c + (1− λ)(θ H ˜ t− c ˜ t 2 ) + v H if θ L −λθ H 2c(1−λ) ≤ ˜ t≤ θ H 2c . The result is only sustained for θ L −λθ H 2c(1−λ) ≤ ˜ t≤ θ L 2c because when θ L 2c ≤ ˜ t≤ θ H 2c , the result is dominated by (c1) since (c1) achieves first best. (c3) If p =θ L , η = 0, then (B.0.8) and (B.0.9) suggest that t H = θ L 2c + ξ(θ H −θ L ) 2cλ , t L = θ L 2c . Since θ H t H +v H − pt H −a = 0 and θ L t L +v L −pt L −a = 0, we obtain a = v L and t H = v L −v H θ H −θ L = ˜ t. Since ξ ≥ 0 and ν ≥ 0, substitute the optimal solution into ξ and ν respectively. If ˜ t = v L −v H θ H −θ L ≥ θ L 2c , then ξ = 2c θ H −θ L ( v L −v H θ H −θ L − θ L 2c ) ≥ 0. In addition, if ˜ t ≤ θ H 2c then ν = 1−ξ≥ 0. As a result, if θ L 2c ≤ v L −v H θ H −θ L ≤ θ H 2c , the optimal solution is p =θ L , a =v L , t L = θ L 2c , t H = v L −v H θ H −θ L and the total profit is (1−λ)θ 2 L 4c +v L −c ˜ t 2 λ+ ˜ tθ L λ. Proof of Lemma 2.5.1 (1) When p ≤ (1− Δ)θ x , we have v x + θ x q− (1− Δ) −1 qp = v x +[θ x −(1−Δ) −1 p]q≥v x ≥ 0 for all q≥ 0. Thus, the SP can always report the maximum effort (1−Δ) −1 q, which implies E( b t|t) =(1−Δ) −1 t. (2)Assumep> (1−Δ)θ x . Letq x bethemaximumqualityqatwhichthemaximumreport (1−Δ) −1 q will not violate typex buyer’s IR constraint. (We use simple notations q x and q x in the proofforconvenience.) Thus, q x satisfies v x +θ x q =(1−Δ) −1 qpandq x = vx (1−Δ) −1 p−θx . 153 Let q x be the maximum quality q at which the minimum report (1+Δ) −1 q will not violate typex buyer’s IR constraint. If p > (1+Δ)θ x , q x satisfies v x +θ x q = (1+Δ) −1 qp and thus q x = vx (1+Δ) −1 p−θx > 0; if p≤ (1+Δ)θ x , we have v x +θ x q ≥ (1+Δ) −1 qp for all q ≥ 0 and thus we set q x =∞. When the SP exerts an effort t≤ (1+Δ) −1 q x , the realized quality is q =(1+ǫ)t≤ (1+ Δ)t≤q x and, by the definition ofq x , the SP can report the maximum effort b t = (1−Δ) −1 q. As a result, E( b t|t) =(1−Δ) −1 t. Suppose,asillustratedinFigure2.17(b),thattheSPexertsaneffortt∈ ((1+Δ) −1 q x ,(1− Δ) −1 q x ), i.e., (1− Δ)t ≤ q x ≤ (1+Δ)t. By the definition of q x , if the realized quality q turns out to be in [(1−Δ)t,q x ], the largest feasible report is b t = (1−Δ) −1 q; if q turns out to be in [q x ,(1+Δ)t], the largest feasible report should make the IR constraint binding, i.e., p b t =v x +θ x q, or, b t =(v x +θ x q)/p. The expected report can be calculated as: E( b t|t) = 1 2Δt Z q x (1−Δ)t (1−Δ) −1 qdq+ 1 2Δt Z (1+Δ)t q x v x +θ x q p dq = 1 2Δt Z (1+Δ)t (1−Δ)t (1−Δ) −1 qdq+ 1 2Δt Z (1+Δ)t q x θ x −(1−Δ) −1 p p q+ v x p dq =(1−Δ) −1 t+ 1 4Δt θ x −(1−Δ) −1 p p (1+Δ) 2 t 2 −(q x ) 2 + 1 2Δt v x p (1+Δ)t−q x =(1−Δ) −1 t+ θ x −(1−Δ) −1 p 4Δpt (1+Δ)t−q x (1+Δ)t+q x +2 v x θ x −(1−Δ) −1 p =(1−Δ) −1 t− (1−Δ) −1 p−θ x 4Δpt (1+Δ)t−q x 2 . When the SP exerts an effort t ∈ [(1− Δ) −1 q x ,(1 + Δ) −1 q x ], the realized quality is q = (1+ǫ)t∈ [q x ,q x ] and, by the definition of q x , the largest feasible report should satisfy p b t =v x +θ x q, which implies E( b t|t)= (v x +θ x t)/p. Finally, when t > (1 + Δ) −1 q x , we have q x < (1 + Δ)t and q = (1 + ǫ)t > q x for 1 +ǫ ∈ (q x /t,1 + Δ]. By the definition of q x , given such a q, even the minimum report (1+Δ) −1 q will violate the ex-post IR constraint. 154 Proof of Lemma 2.5.2 (1) When p≤ (1−Δ)θ x , we have E( b t|t) =(1−Δ) −1 t by Lemma 2.5.1(1). Given p, problem (2.5.1)-(2.5.2) becomes max t {(1−Δ) −1 pt−ct 2 }, which is solved by t ∗ = p/(2c(1− Δ)). By substitution, the problem reduces to max p {p 2 /(4c(1− Δ) 2 )}. The objective value increases with p, so we should increase the price into the region p > (1−Δ)θ x . By Lemma 2.5.1(2), the expected report is still E( b t|t) = (1−Δ) −1 t, and hence the optimal effort is still t ∗ = p/(2c(1− Δ)), if and only if t ∗ ≤ (1 + Δ) −1 q x (p). This condition can be rewritten as (1−Δ) −1 p 2c ≤ (1+Δ) −1 vx (1−Δ) −1 p−θx , or f s x (r) ≤ 0, where f s x (r) = r 2 − θ x r−2c(1+Δ) −1 v x for r = (1−Δ) −1 p. Because the larger root of f s x (r) is given by r s x = 1 2 θ x + p θ 2 x +8c(1+Δ) −1 v x (≥θ x ), the condition reduces to r≤r s x , or p≤ (1−Δ)r s x . In this region, because problem (2.5.1)-(2.5.2) reduces to max p {p 2 /(4c(1−Δ) 2 )}, the objective value increases with p. (2) As p enters the region p > (1−Δ)r s x , we have t ∗ > (1+Δ) −1 q x (p) and the second sub case of Lemma 2.5.1(2) is activated. The SP will report according to E( b t|t) = (1− Δ) −1 t− (1−Δ) −1 p−θx 4Δpt (1+Δ)t−q x (p) 2 . Given price p, problem (2.5.1)-(2.5.2) becomes max t h x (t|p),forh x (t|p)= (1−Δ) −1 pt− (1−Δ) −1 p−θx 4Δt [(1+Δ)t−q x (p)] 2 −ct 2 ,ormax t h x (t|r), for h x (t|r) = rt− r−θx 4Δt [(1+Δ)t− vx r−θx ] 2 −ct 2 . The first and second order derivatives with respect to t are: d dt h x (t|r) = d dt rt− r−θ x 4Δ (1+Δ) 2 t−2(1+Δ) v x r−θ x + v 2 x (r−θ x ) 2 t −1 −ct 2 =−2ct+r−(1+Δ) 2 r−θ x 4Δ + r−θ x 4Δ v 2 x (r−θ x ) 2 t −2 =−2ct+ (1+Δ) 2 θ x −(1−Δ) 2 r 4Δ + v 2 x 4Δ(r−θ x ) t −2 , d 2 dt 2 h x (t|r) =−2c− v 2 x 2Δ(r−θ x ) t −3 <0 for t≥ 0. Thus, h x (t|r) is strictly concave in t ≥ 0 and, in that region, is maximized at the unique solutiontothefirstorderequation d dt h x (t|r)=0,or,−2ct 3 + (1+Δ) 2 θx−(1−Δ) 2 r 4Δ t 2 + v 2 x 4Δ(r−θx) =0. The closed-formexpression fortheoptimaltiscomplicated. However, such anexpression is unnecessary for our purpose. Suppose the optimal effort t ∗ for a given p satisfies (1 + 155 Δ) −1 q x (p) < t ∗ < (1−Δ) −1 q x (p) (i.e., the second sub case of Lemma 2.5.1(2)). According to Figure 2.17(b), the objective function (2.5.1) can be improved by increasing p a little bit while keeping the effort t ∗ unchanged. So, the optimal value increases with the price in this region. At the boundary, we must have t ∗ = (1−Δ) −1 q x (p). (3) As p increases further, the third sub case of Lemma 2.5.1(2) is activated. The SP will reportaccordingtoE( b t|t)= vx+θxt p ,andproblem(2.5.1)-(2.5.2)becomesmax t {v x +θ x t−ct 2 }. The optimal solution is t ∗ = θx 2c , which is first best when facing type x buyer. The third sub case of Lemma 2.5.1(2) requires (1−Δ) −1 q x (p)≤t ∗ ≤ (1+Δ) −1 q x (p), i.e., vx p−(1−Δ)θx ≤ θx 2c ≤ vx p−(1+Δ)θx if p > (1+Δ)θ x , or vx p−(1−Δ)θx ≤ θx 2c if p≤ (1+Δ)θ x . The condition can be rewritten as 2cvx θx +(1−Δ)θ x ≤p≤ 2cvx θx +(1+Δ)θ x if p> (1+Δ)θ x , or 2cvx θx +(1−Δ)θ x ≤p if p≤ (1+Δ)θ x . Notice that 2cvx θx +(1+Δ)θ x >(1+Δ)θ x , the two cases can be combined to 2cvx θx +(1−Δ)θ x ≤p≤ 2cvx θx +(1+Δ)θ x . Thefirst-besteffortt ∗ = θx 2c andprofitπ fb x =λ x (v x + θ 2 x 4c ) are achieved in this price range. (4)Finally, consider thecasep> 2cvx θx +(1+Δ)θ x . Fromthe abovediscussions, forafixed p> 2cvx θx +(1+Δ)θ x ,theobjectivefunctionincreaseswiththeeffortt(passingthroughthesub cases in Lemma 2.5.1(2)) until the maximum effort that satisfies the ex-post IR constraint, t =(1+Δ) −1 q x (p) = vx p−(1+Δ)θx . The assumption p≥ 2cvx θx +(1+Δ)θ x is equivalent to t≤ θx 2c (the equality is included to ensure that the subsequent problem has an optimal solution). The expected report is given by E b t = vx+θxt p , and thus problem (2.5.1)-(2.5.2) reduces to: max t {v x + θ x t− ct 2 } s.t. t ≤ θx 2c . The objective value increases with t when t ≤ θx 2c , or equivalently, decreases with p when p≥ 2cvx θx +(1+Δ)θ x . Proof of Lemma 2.5.3 (i) Suppose ICH constraint (2.5.6) and IRL constraint (2.5.7) are satisfied, pE( ˆ t H |t H ) ≤ pE( ˆ t L |t L )+θ H (t H −t L ) ≤ v L +θ L t L +θ H (t H −t L ) 156 = θ H t H +v H +(v L −v H )−(θ H −θ L )t L ≤ θ H t H +v H The first inequality is from ICH (2.5.6) and the second inequality is from IRL (2.5.4). The third inequality is true since (v L −v H )−(θ H −θ L )t L ≤ 0 by assumption. As a result, IRH (2.5.5) is satisfied. (ii) Similarly, suppose IRH and ICL are satisfied, pE( ˆ t L |t L ) ≤ pE( ˆ t H |t H )−θ L (t H −t L ) ≤ θ H t H +v H −θ L (t H −t L ) = θ L t L +v L −(v L −v H )+(θ H −θ L )t H ≤ θ L t L +v L The first inequality is from ICL (2.5.7) and the second inequality follows from IRH (2.5.5). The third inequality follows since t H ≤ v L −v H θ H −θ L . As a result, IRL (2.5.4) is satisfied. Proof of Lemma 2.5.4 (i) From ICH and ICL, pE( ˆ t H |t H )−pE( ˆ t L |t L ) ≤ θ H (t H −t L ) pE( ˆ t H |t H )−pE( ˆ t L |t L ) ≥ θ L (t H −t L ) From the first inequality minus the second inequality, we get (θ H −θ L )(t H −t L ) ≥ 0, i.e. t H ≥t L . (ii) From Lemma 2.5.1 (1) if p≤ θ H (1−Δ), then E( ˆ t H |t H ) = t H 1−Δ . From Lemma 2.5.1 (2) if p≥ (1−Δ)r S L ≥ (1−Δ)θ L , then E( ˆ t L |t L )≤ t L 1−Δ . Substituting them in to (2.5.8), we get pE( ˆ t H |t H )−pE( ˆ t L |t L ) ≥ pt H 1−Δ − pt L 1−Δ = p 1−Δ (t H −t L ) 157 ≥ r S L (t H −t L ) ≥ θ L (t H −t L ) Thesecondinequalityisfromp≥ (1−Δ)r S L . Since(1−Δ)r S L = 1−Δ 2 (θ L + p θ 2 L +8c(1+Δ) −1 v L > (1− Δ)θ L , i.e., r S L > θ L , the second inequality follows. As a result, ICL is satisfied if (1−Δ)r S L ≤p≤θ H (1−Δ). (iii)Ift H =t L , theincentive compatibility forthe hightypeandlow typearetrivial since both sides equal zero. On the other hand, we could show that for any p>θ H (1−Δ), there does not exist t H 6= t L such that pE( ˆ t H |t H )−pE( ˆ t L |t L )≤ θ H (t H −t L ). If IRL is binding, we get θ H t H −pE( ˆ t H |t H )≥ (θ H −θ L )t L −v L (B.0.16) Consider both the left side (LHS) and the right hand side (RHS) of (B.0.16). From Lemma 2.5.1, if p>θ H (1−Δ), then pE( ˆ t H |t H )≥ min( pt H 1−Δ ,θ H t H +v H ) and as a result, LHS =θ H t H −pE( ˆ t H |t H )≤θ H t H −min( pt H 1−Δ ,θ H t H +v H )≤ max((θ H − p 1−Δ )t H ,−v H )≤ 0. RHS = (θ H −θ L )t L −v L ≥ (θ H −θ L ) v L p−(1−Δ)θ L −v L ≥ v L 1−Δ −v L ≥ 0. In other words, ifp>θ H (1−Δ) and IRL is binding, (B.0.16) is violated for any t H 6=t L . In other words, pE( ˆ t H |t H )−(θ L t L +v L )≥θ H (t H −t L ) for any t H >t L when IRL is binding. Ifp>θ H (1−Δ)andIRLis notbinding, pE( ˆ t H |t H )−pE( ˆ t L |t L )≥pE( ˆ t H |t H )−(θ L t L +v L )≥ θ H (t H −t L ). ICH is gain violated for and t H >t L . As a result, ICH constraint is satisfied if and only if t H =t L when p>θ H (1−Δ). Proof of Lemma 2.5.5 Suppose, ξ,ν,η≥ 0 are the Lagrangian parameters of optimiza- tion problem (2.5.10). The Lagrangian function is, L(t H ,t L ,ξ,ν,η)=λ(pE( ˆ t H |t H )−ct 2 H )+(1−λ)(pE( ˆ t L |t L )−ct 2 L )+ξ(t L θ L +v L −pE( ˆ t L |t L ))+ ν(θ H (t H −t L )−p(E( ˆ t H |t H )−E( ˆ t L |t L )). Taking partial derivatives with respect to t H , t L , p , we get ∂L ∂t H =λ( dpE( ˆ t H |t H ) dt H −2ct H )+ν(θ H − dpE( ˆ t H |t H ) dt H ) 158 ∂L ∂t L = (1−λ)( dpE( ˆ t L |t L ) dt L −2ct L )+ξ(θ L − dpE( ˆ t L |t L ) dt L )+ν( dpE( ˆ t L |t L ) dt L −θ H ) ∂L ∂p =λ dpE( ˆ t H |t H ) dp +(1−λ) dpE( ˆ t L |t L ) dp −ξ dpE( ˆ t L |t L ) dp −ν( dpE( ˆ t H |t H ) dp − dpE( ˆ t L |t L ) dp ) Since dpE( ˆ t H |t H ) dp = p 1−Δ , KKT conditions suggest, t H = p 2c(1−Δ) + ν(θ H − p 1−Δ ) 2cλ (B.0.17) t L = (1−λ−ξ+ν) dpE( ˆ t L |t L ) dt L +ξθ L −νθ H 2c(1−λ) (B.0.18) λ dpE( ˆ t H |t H ) dp +(1−λ) dpE( ˆ t L |t L ) dp =ξ dpE( ˆ t L |t L ) dp +ν( dpE( ˆ t H |t H ) dp − dpE( ˆ t L |t L ) dp ) (B.0.19) ξ(t L θ L +v L −pE( ˆ t L |t L )) =0 (B.0.20) ν(θ H (t H −t L )−p(E( ˆ t H |t H )−E( ˆ t L |t L )) =0 (B.0.21) t L θ L +v L −pE( ˆ t L |t L )≥ 0 (B.0.22) θ H (t H −t L )−p(E( ˆ t H |t H )−E( ˆ t L |t L ))≥ 0 (B.0.23) ξ≥ 0, ν≥ 0 InordertoshowIRLisbindingatoptimality,wejustneedtoshowtheLagrangianparameter associated with IRL satisfies ξ 6= 0. Suppose ξ = 0, from (B.0.21), either (i) ν = 0 or (ii) θ H (t H −t L )−p(E( ˆ t H |t H )−E( ˆ t L |t L ))= 0. (i) If ξ =0 and ν =0, substitute ξ andν into (B.0.17) and (B.0.18), we get t H = p 2c(1−Δ) , t L = 1 2c dpE( ˆ t L |t L ) dt L . And (B.0.19) will be simplified to λ dpE( ˆ t H |t H ) dp +(1−λ) dpE( ˆ t L |t L ) dp =0 (B.0.24) It follows from Lemma 2.5.1 that dpE( ˆ t H |t H ) dp ≥ 0 and dpE( ˆ t L |t L ) dp > 0 . The left hand side of the above equation is strictly positive but the right hand side is zero. This scenario cannot occur. 159 (ii) If ξ = 0 and θ H (t H −t L )−p(E( ˆ t H |t H )−E( ˆ t L |t L )) = 0, take the derivative of p on both sides of the equation, dpE( ˆ t H |t H ) dp = dpE( ˆ t L |t L ) dp > 0 The left hand side of (B.0.19) is LHS =λ dpE( ˆ t H |t H ) dp +(1−λ) dpE( ˆ t L |t L ) dp > 0, but the right hand side of (B.0.19) is RHL = ξ dpE( ˆ t L |t L ) dp +ν( dpE( ˆ t H |t H ) dp − dpE( ˆ t L |t L ) dp ) = 0 since ξ = 0 and dpE( ˆ t H |t H ) dp = dpE( ˆ t L |t L ) dp . This scenario cannot occur either. As a result, ξ 6= 0 in optimal solution. It suggests that the optimal solution satisfies pE( ˆ t L |t L ) = v L +θ L t L . Recall from Lemma 2.5.1, the optimal solution satisfies t L ∈ [(1− Δ) −1 q L (p),(1+Δ) −1 ¯ q L (p)] i.e. v L p−(1−Δ)θ L ≤ t L ≤ v L p−(1+Δ)θ L . Proof of Lemma 2.5.6 (i) 2p ∗ θ H − √ θ 2 H −8cξ ≥ 1 if and only if p θ 2 H −8cξ≥ (θ H −2p ∗ ) 2 , and the latter is equivalent to p ∗2 −θ H p ∗ +2cξ≤ 0. Substituting ξ = v L (θ H −p ∗ ) p ∗ −θ L into p ∗2 −θ H p ∗ +2cξ≤ 0, we getp ∗ (p ∗ −θ H )+ 2cv L (θ H −p ∗ ) p ∗ −θ L ≤ 0. Since p ∗ <θ H , it is equivalent to v L p ∗ −θ L ≤ p ∗ 2c . The result above follows p ∗ 2c ≥ θ L 2c ≥ v L p ∗ −θ L where the first inequality comes from ICL, i.e. p ∗ ≥θ L . And the second inequality is based on the assumption. (ii) 1− 2p ∗ θ H + √ θ 2 H −8cξ ≥ 0 if (2p ∗ −θ H ) 2 ≤θ 2 H −8cξ or equivalently, p ∗ 2c ≥ v L p ∗ −θ L , which has been shown in (i). (iii)Letm = p ∗ 1−Δ ,ξ = v L (θ H −p ∗ ) p ∗ −θ L ,wefirstshowtheinequality(θ H −m) m 2c =ξhasasolution in terms of Δ. Let f(Δ) = (θ H −m) m 2c −ξ = (θ H − p ∗ 1−Δ ) p ∗ 2c(1−Δ) − v L (θ H −p ∗ ) p ∗ −θ L . Since f(Δ) is a continuous function on [0,1), f(0) = (θ H −p ∗ ) p ∗ 2c − v L (θ H −p ∗ ) p ∗ −θ L = (θ H −p ∗ )( p ∗ 2c − v L p ∗ −θ L ) > 0 and f(1−)<0, there exists a real solution for (θ H −m) m 2c ≥ξ in terms of Δ. Then we show that if Δ≤ 1− 2p ∗ θ H + √ θ 2 H −8cξ , then (θ H − p ∗ 1−Δ ) p ∗ 2c(1−Δ) ≥ v L (θ H −p ∗ ) p ∗ −θ L . Solving the function m 2 −mθ H +2cξ≤ 0, we obtain θ H − √ θ 2 H −8cξ 2 ≤m≤ θ H + √ θ 2 H −8cξ 2 , or equivalently, 1− 2p ∗ θ H − p θ 2 H −8cξ ≤ Δ≤ 1− 2p ∗ θ H + p θ 2 H −8cξ 160 From Lemma 2.5.6 (i) and (ii) , 1− 2p ∗ θ H − √ θ 2 H −8cξ < 0 and 1− 2p ∗ θ H + √ θ 2 H −8cξ ≥ 0 , so if 0≤ Δ≤ 1− 2p ∗ θ H + √ θ 2 H −8cξ , (θ H − p ∗ 1−Δ ) p ∗ 2c(1−Δ) ≥ v L (θ H −p ∗ ) p ∗ −θ L . Proof of Theorem 2.5.7 (i) If θ L 2c ≤ v L θ H −θ L , Proposition 2.4.10 suggest thatp nh =θ H and t nh L = v L θ H −θ L . Since t nh L = v L θ H −θ L ≤ v L p−(1−Δ)θ L ≤ t h L for any p h ≤ (1−Δ)θ H , the SP makes more profit from low type with deterministic quality since θ L 2c ≤ t nh L ≤ t h L , where θ L 2c is the first best time effort and is closer to t nh L than t h L . Now we look into the SP’s profit from high type: max t H λ( pt H 1−Δ −ct 2 H ) s.t. θ H (t H −t L )≥ pt H 1−Δ −(θ L t L +v L ) ≤max t H λ( pt H 1−Δ −ct 2 H ) = λp 2 4c(1−Δ) 2 ≤ λθ 2 H 4c = max t H λ(θ H t H −ct 2 H ) The firstinequality isbecause theoptimalprofitwithoutconstraintis greaterthanthatwith ICH constraint. The second inequality is because p ≤ θ H (1− Δ) from Lemma 2.5.4 (ii). As a result, the service provider also makes less profit from the high type with stochastic quality than that with deterministic quality. So the SP is worse off from the flexibility to report time effort. (ii) When v L θ H −θ L ≤ θ L 2c and 0 ≤ λ ≤ 2cv L (θ L (θ H −θ L )−2cv L ) (2cv L +(θ H −θ L ) 2 )(θ 2 H −2cv L −θ H θ L) , i.e. λ θ H 2c + (1− λ) 2cv L (θ H −θ L ) 2 v L θ H −θ L − θ L 2c ≤ 0 p nh = p ∗ < θ H , Proposition 2.4.10 suggests that t nh L = v L p ∗ −θ L and t nh H = p ∗ 2c . We will find a feasible solution with stochastic quality that achieves the same profit as deterministic quality. For any p h that satisfies p ∗ −θ L Δ ≤ p h ≤ p ∗ +θ L Δ, v L p h −(1−Δ)θ L ≤ v L p ∗ −θ L ≤ v L p h −(1+Δ)θ L . In other words, the optimal solution in the deter- ministic quality case for low type t L = v L p ∗ −θ L is also feasible in the stochastic case if p ∗ − Δθ L ≤ p h ≤ p ∗ + Δθ L . Let t h L = t nh L = v L p ∗ −θ L . the SP achieves the same profit from low type since the profit from low type in both cases equals (1−λ)(θ L t L +v L −ct 2 L ), which is only determined by t L . The profit from high type in the stochastic quality case is L maxλ( pt H 1−Δ −ct 2 H ) 161 s.t. pt H 1−Δ −(θ L t L +v L )≤ θ H (t H −t L ) (B.0.25) p ∗ −θ L Δ≤p≤ p ∗ +θ L Δ Let’s ignore the constraint (B.0.25) for now. Then a feasible solution of the above optimiza- tion is p = p ∗ and t H = p ∗ 2c(1−Δ) , and the objective is max t H ,p λ( pt H 1−Δ −ct 2 H ) ≥ λp ∗2 4c(1−Δ) 2 > λp ∗2 4c =π nh Now we check that the feasible solution t L = v L p ∗ −θ L , t H = p ∗ 2c(1−Δ) and p = p ∗ satisfies IC constraint(B.0.25), i.e. (θ H − p 1−Δ )t H ≥ (θ H −θ L )t L −v L orequivalently (θ H − p ∗ 1−Δ ) p ∗ 2c(1−Δ) ≥ v L (θ H −p ∗ ) p ∗ −θ L . From Lemma 2.5.6, if Δ ≤ 1− 2p ∗ θ H + √ θ 2 H −8cξ , t L = v L p ∗ −θ L , t H = p ∗ 2c(1−Δ) and p = p ∗ is a feasiblesolutionofthestochasticqualityoptimizationproblem. Anditachieves higherprofit than the optimal profit with deterministic quality. When v L θ H −θ L ≤ θ L 2c and 2cv L (θ L (θ H −θ L )−2cv L ) (2cv L +(θ H −θ L ) 2 )(θ 2 H −2cv L −θ H θ L) ≤ λ≤ 1 i.e. λ θ H 2c +(1−λ) 2cv L (θ H −θ L ) 2 v L θ H −θ L − θ L 2c ≥ 0. Since π h and π nh change as a function of λ, define π h (λ) and π nh (λ) as follows: π h (λ) = maxλ( pt H 1−Δ −ct 2 H )+(1−λ)(θ L t L +v L −ct 2 L ) s.t. pt H 1−Δ −(θ L t L +v L )≤θ H (t H −t L ) and v L p−(1−Δ)θ L ≤t L ≤ v L p−(1+Δ)θ L and π nh (λ) = maxλ(θ H t H −ct 2 H )+(1−λ)(θ L t L +v L −ct 2 L ) s.t. t L = v L θ H −θ L Let λ→ 2cv L (θ L (θ H −θ L )−2cv L ) (2cv L +(θ H −θ L ) 2 )(θ 2 H −2cv L −θ H θ L) , the previous analysis shows that 162 π h ( 2cv L (θ L (θ H −θ L )−2cv L ) (2cv L +(θ H −θ L ) 2 )(θ 2 H −2cv L −θ H θ L ) )≥π nh ( 2cv L (θ L (θ H −θ L )−2cv L ) (2cv L +(θ H −θ L ) 2 )(θ 2 H −2cv L −θ H θ L ) ) . In addition, π h (1−) = max( pt H 1−Δ −ct 2 H ) s.t. pt H 1−Δ −(θ L t L +v L )≤θ H (t H −t L ) and v L p−(1−Δ)θ L ≤t L ≤ v L p−(1+Δ)θ L < π nh (1−) =max(θ H t H −ct 2 H ) s.t. t L = v L θ H −θ L = π nh L (λ) The above inequality is satisfied because p h < θ H (1−Δ) for any t h > v L θ H −θ L . As a result π h (1+)−π nh (1−) < 0. Now we show that both π h (λ) and π nh (λ) are continuous functions with respect to λ. Suppose λ 1 and λ 2 are close and λ 1 ≥ λ 2 , i.e. 0≤ λ 1 −λ 2 ≤ δ . For any given (p,t H ,t L ), π h (λ 1 ,p,t H ,t L )−π h (λ 2 ,p,t H ,t L ) = (λ 1 −λ 2 )( pt H 1−Δ −ct 2 H −(θ L t L +v L −ct 2 L )) ≤ δ( pt H 1−Δ −ct 2 H −(θ L t L +v L −ct 2 L ))≤ δK where K is some constant. The last inequality is because t H , t L and p are bounded. π h (λ 1 ,p,t H ,t L )≤π h (λ 2 ,p,t H ,t L )+δK Since the constraints of the optimization problem do not depend on the specific value of λ, max p,t H ,t L π h (λ 1 ,p,t H ,t L )≤ max p,t H ,t L π h (λ 2 ,p,t H ,t L )+δK 163 As a result, π h (λ 1 )−π h (λ 2 )≤ δC = ǫ. As a result π h (λ) is continuous with respect to λ. Following the same logic we can show that π nh (λ) is also continuous. So there exists ˆ λ such that π h ( ˆ λ) =π nh ( ˆ λ). Next we show thatπ h (λ)−π nh (λ)is monotone with respect toλ, i.e. dπ h (λ) dλ − dπ nh (λ) dλ ≤ 0. Let L h = λ( pt H 1−Δ −ct 2 H )+(1−λ)(θ L t L +v L −ct 2 L )+ξ(θ H (t H −t L )− pt H 1−Δ +θ L t L +v L ) +γ 1 (pt L −(1−Δ)θ L t L −v L )+γ 2 (v L −pt L +t L (1+Δ)θ L ) The envelop theory with constraints suggests that dπ h (λ) dλ = ∂L h ∂λ = p h t h H 1−Δ −ct h∗2 H −(θ L t h∗ L +v L −ct h∗ 2 L )≤π h∗ (1−)−π h∗ L (λ) dπ nh (λ) dλ = ∂L nh ∂λ =pt nh∗ H −ct nh∗2 H −(θ L t nh∗ L +v L −ct nh∗ L )≥π nh∗ H (λ)−π nh∗ (0+) dπ h (λ) dλ − dπ nh (λ) dλ = ∂L h ∂λ − ∂L nh ∂λ ≤ (π h (1−)−π nh H (λ))−(π nh (0+)−π h L (λ))=(π h (1−)−π nh (1−))− (π h (0−)−π nh (0−))≤ 0 As a result π h (λ)−π nh (λ) is monotonously decreasing with λ. So there exists ˆ λ, π h (λ)≥π nh (λ) if λ≤ ˆ λ and π h (λ)≤π nh (λ) if λ≥ ˆ λ. ProofofProposition2.5.9 LetP H =pE( ˆ t H |t H )+a,P L =pE( b t L |t L )+a,theoptimization problem (2.5.26)-(2.5.30) could be transformed to max t H ,t L ,p,a λ(P H −ct 2 H )+(1−λ)(P L +a−ct 2 L ) (B.0.26) s.t. P L ≤θ L t L +v L , (B.0.27) P H ≤θ H t H +v H , (B.0.28) v H +θ H t H −P H ≥v H +θ H t L −P L , (B.0.29) v L +θ L t L −P L ≥v L +θ L t H −P H . (B.0.30) 164 In optimal solution of optimization problem (B.0.26)-(B.0.30), either IRL and ICH, or IRH and ICL, or IRL and IRH are binding. For example, if IRL and ICH are binding, the optimization problem is equivalent to max t H ,t L ,a,p λ(P H −ct 2 H )+(1−λ)(P L −ct 2 L ) s.t.P L =v L +θ L t L P H =θ H (t H −t L )+v L +θ L t L Since the optimal solution of (2.4.20)-(2.4.24) is t H = θ H 2c and t L = θ L −λθ H 2c(1−λ) , p =θ H , a = v L − (θ H −θ L )(θ L −θ H λ) 2c(1−λ) , let a ′ and p ′ satisfy a ′ +p ′ E( ˆ t H |t H ) = θ H (t H −t L )+v L +θ L t L (B.0.31) a ′ +p ′ E( ˆ t L |t L ) = v L +θ L t L (B.0.32) (B.0.32) suggests that v L +(1−Δ)θ L t L ≤p ′ t L +a ′ ≤v L +(1+Δ)θ L t L (B.0.33) Specifically, we letp ′ t L +a ′ =v L +θ L t L , whichobviously satisfies theaboveinequalities. And ifp ′ ≤ (1−Δ)θ H ,E( ˆ t H |t H ) = t H 1−Δ . (B.0.31)suggeststhata ′ + p ′ t H 1−Δ =θ H (t H −t L )+v L +θ L t L . Solving the equations, we obtain p ′ = (1−Δ)(θ H −θ L ) θ H (1−Δλ)−θ L (1−Δ) a ′ = v L + (θ L −θ H λ)((1−Δ)(1−θ L )θ L +θ H ((1−Δ)−θ L (1−Δλ) 2c(1−λ)(θ H (1−Δλ)−θ L (1−Δ)) (t H ,t L ,p ′ ,a ′ ) as a solution of (2.5.26)-(2.5.30) achieves the same profit as (2.4.20)-(2.4.24), where the optimal solution is (t H ,t L ,p,a). Since IRL and ICH are binding, the objective value ofoptimization problem (2.5.26)-(2.5.30)can notbeimproved further. As aresult, the optimization problem (2.5.26)-(2.5.30) and (2.4.20)-(2.4.24) have the same optimal solution of t H and t L and achieve the same optimal profit. A similar analysis can be done when IRH and ICL, or IRL and IRH are binding. The details of the optimal solution in all scenarios are presented in Table 2.9. 165 APPENDIX C PROOFS OF CHAPTER 3 ProofofProposition3.3.1 (i) The time spent in queue in the promotion periodsatisfies the equilibrium condition (3.3.1), i.e. w 1 = λ ¯ F(p+hw 1 )+G ¯ F(αp+hw 1 ) μ(μ−(λ ¯ F(p+hw 1 )+G ¯ F(αp+hw 1 )) . Applying implicit- function theorem to (3.3.1), we get dw 1 dG = 1 μ ¯ F(hw 1 +pα) (μ−λ ¯ F(p+hw 1 )−G ¯ F(hw+pα)) 2 −hλ ¯ F ′ (p+hw)−Gh ¯ F ′ (hw+pα) . (C.0.1) Since the tail distribution function ¯ F(·) = 1− F(·) is decreasing, ¯ F ′ (p + hw) ≤ 0 and ¯ F ′ (hw+pα)≤ 0. As a result, dw 1 (G) dG ≥ 0 . (ii) Similarly, the time spent in queue in the post promotion period satisfies the equi- librium condition (3.3.2), i.e. w 2 = (λ+G) ¯ F(p+hw 2 ) 1−(λ+G) ¯ F(p+hw 2 ) . Applying implicit-function theorem to (3.3.2), we get dw 2 dG = 1 μ ¯ F(hw 2 +p) (μ−(λ+G) ¯ F(p+hw 2 )) 2 −h(G+λ) ¯ F ′ (p+hw 2 ) . (C.0.2) As a result, dw 2 dG ≥ 0 follows from the fact that ¯ F ′ (p+hw 2 )≤ 0. Proof of Proposition 3.3.2 (i) Consider customers’ expected time in queue in the pro- motion period w 1 as a function of p. Taking the derivatives of both sides of (3.3.1) with respect to p, applying Implicit-function Theorem and substituting G = 0, we get dw 1 dp | G=0 = λ ¯ F ′ (p+hw 1 ) 1+λ 2¯ F(p+hw 1 ) 2 +2λ ¯ F(p+hw 1 )(−1)−hλ ¯ F ′ (p+hw 1 ) . Since the numerator is negative and the denominator is positive, dw 1 dp | G=0 ≤ 0 follows. (ii) Similarly, consider customers’ expected time in queue in the promotion period w 1 as a function of λ. Taking the derivatives of both sides of (3.3.1) with respect to λ, applying Implicit-function Theorem and substituting G = 0, we get 166 dw 1 dλ | G=0 = ¯ F(p+hw 1 ) 1+λ 2¯ F(p+hw 1 ) 2 +2λ ¯ F(p+hw 1 )(−1−hλ ¯ F ′ (p+hw 1 ) . As a result, dw 1 dp | G=0 ≤ 0 follows from the fact that both the numerator and the denominator are positive. (iii) Now consider customer’s expected time in queue in the promotion period w 1 as a function of α. Taking the derivative of both sides of (3.3.1) with respect to α, applying Implicit-function Theorem, and substituting G= 0, we get dw 1 dα | G=0 =0. ProofofTheorem3.3.3 The expected timeinqueue inanM/M/1queue satisfies (3.3.1) and (3.3.2), or equivalently λ ¯ F(p+hw 1 )+G ¯ F(αp+hw 1 ) = μ 2 w 1 1+μw 1 ; (C.0.3) (λ+G) ¯ F(p+hw 2 ) = μ 2 w 2 1+μw 2 . (C.0.4) As a result, the total revenue function of the firm could be represented as follows π(G)=p( αμ 2 w 1 (G) 1+μw 1 (G) + δμ 2 w 2 (G) 1+μw 2 (G) +(1−α)λ ¯ F(p+hw 1 (G))), where w 1 (G) and w 2 (G) satisfies (3.3.1) and (3.3.2). Since Proposition 3.3.1 implies that dw 2 dw 1 = dw 2 dG / dw 1 dG ≥ 0, the firm’s revenue can be represented by a function of w 1 , i.e., π w (w 1 ) = αμ 2 w 1 1+μw 1 + δμ 2 w 2 (w 1 ) 1+μw 2 (w 1 ) +(1−α)λ ¯ F(p+hw 1 ). Taking the derivative of π w (w 1 ) with respect to w 1 , we get dπ w (w 1 ) dw 1 = αμ 2 (1+μw 1 ) 2 + δμ 2 (1+μw 2 ) 2 dw 2 (w 1 ) dw 1 +(1−α)hλ ¯ F ′ (p+hw 1 ). 167 Substituting (C.0.1) and (C.0.2) into the above equation, we get dw 2 dw 1 = ¯ F(hw 2 +p) ¯ F(hw 1 +pα) (μ−λ ¯ F(p+hw 1 )−G ¯ F(hw 1 +pα)) 2 −hλ ¯ F ′ (p+hw 1 )−Gh ¯ F ′ (hw 1 +pα) (μ−(λ+G) ¯ F(p+hw 2 )) 2 −h(G+λ) ¯ F ′ (p+hw 2 ) . Substituting G =0 into the above equation, we have dw 2 dw 1 | G=0 = ¯ F(hw 2 +p) ¯ F(hw 1 +pα) (μ−λ ¯ F(p+hw 1 )) 2 −hλ ¯ F ′ (p+hw 1 ) (μ−λ ¯ F(p+hw 2 )) 2 −hλ ¯ F ′ (p+hw 2 ) = ¯ F(hw 1 +p) ¯ F(hw 1 +pα) (μ−λ ¯ F(p+hw 1 )) 2 −hλ ¯ F ′ (p+hw 1 ) (μ−λ ¯ F(p+hw 1 )) 2 −hλ ¯ F ′ (p+hw 1 ) = ¯ F(hw+p) ¯ F(hw+pα) . In addition, (3.3.1) and (3.3.2) implies that w 1 =w 2 if G =0. As a result, dπ(w 1 ) dw 1 | G=0 = αμ 2 (1+μw 1 ) 2 + δ (1+μw 1 ) 2 ¯ F(hw 1 +p) ¯ F(hw 1 +pα) +(1−α)hλ ¯ F ′ (p+hw 1 ). (C.0.5) And (C.0.3) implies that λ = μ 2 w 1 1+μw 1 1 ¯ F(p+hw 1 ) if G = 0. Substituting λ = μ 2 w 1 1+μw 1 1 ¯ F(p+hw 1 ) into (C.0.5), we get dπ(w 1 ) dw 1 | G=0 = αμ 2 (1+μw 1 ) 2 + δ (1+μw 1 ) 2 ¯ F(hw 1 +p) ¯ F(hw 1 +pα) − (1−α)hμ 2 w 1 f(p+hw 1 ) 1−F(p+hw 1 ) . Promotion is profitable if dπ(w 1 ) dw 1 | G=0 ≥ 0, i.e., αμ 2 1+μw 1 + δμ 2 ¯ F(hw 1 +p) ¯ F(hw 1 +pα) 1 1+μw 1 ≥ (1−α)hμ 2 w 1 f(p+hw 1 ) 1−F(p+hw 1 ) . (C.0.6) (i) Now we first show that the left hand side of(C.0.6), i.e. LHS = αμ 2 1+μw 1 + δμ 2 ¯ F(hw 1 +p) ¯ F(hw 1 +pα) 1 1+μw 1 decreases with w 1 . Since both αμ 2 1+μw 1 and 1 1+μw 1 decreases with w 1 . We only need to show that ¯ F(hw 1 +p) ¯ F(hw 1 +pα) also decreases with w 1 . Taking the derivative of ¯ F(hw 1 +p) ¯ F(hw 1 +pα) with respect to w 1 , we have d dw 1 ( ¯ F(hw 1 +p) ¯ F(hw 1 +pα) ) = −f(hw 1 +p) ¯ F(hw 1 +pα)+f(hw 1 +αp) ¯ F(hw 1 +p) ( ¯ F(hw 1 +pα)) 2 = ¯ F(hw 1 +pα) ¯ F(hw 1 +p) ( ¯ F(hw 1 +pα)) 2 ( f(hw 1 +αp) ¯ F(hw 1 +pα) − f(hw 1 +p) ¯ F(hw 1 +p) ). 168 Since F has an increasing failure rate, it implies that f(hw 1 +αp) ¯ F(hw 1 +pα) − f(hw 1 +p) ¯ F(hw 1 +p) ≤ 0. As a result, d dw 1 ( ¯ F(hw 1 +p) ¯ F(hw 1 +pα) )≤ 0, and LHS = αμ 2 1+μw 1 + δμ 2 ¯ F(hw 1 +p) ¯ F(hw 1 +pα) 1 1+μw 1 decreases with w 1 . (ii) Second, we show that the right hand side of (C.0.6), i.e RHS = (1−α)hμ 2 w 1 f(p+hw 1 ) 1−F(p+hw 1 ) increases with w 1 . Since F has an increasing failure rate, (RHS) ′ =(1−α)hμ 2 ( f(p+hw 1 ) 1−F(p+hw 1 ) +w 1 ( f(p+hw 1 ) 1−F(p+hw 1 ) ) ′ ≥ 0. As a result, RHS increases with w 1 . (iii) The results in (i) and (ii) suggest that RHS and LHS are at most single crossing. F(x) has a support on [v,¯ v], i.e. F(v) = 0 and F(¯ v) = 1. Without loss of generality, we assume that ¯ v ≤ p≤ ¯ v, then v < p+hw 1 < ¯ v implies that w 1 ∈ [ v−p h , ¯ v−p h ]. As a result, a promotion is always profitable (for any w 1 ∈ [ v−p h , ¯ v−p h ]) if LHS( ¯ v−p h )≥ RHS( ¯ v−p h ); And promotion is never profitable (for any w 1 ∈ [ v−p h , ¯ v−p h ]) if LHS( v−p h )≤ RHS( v−p h ). However since RHS( ¯ v−p h ) → ∞ if f(¯ v) > 0 and RHS( v−p h ) = 0, LHS( ¯ v−p h ) ≥ RHS( ¯ v−p h ) and LHS( v−p h )≤RHS( v−p h ) cannot occur. As a result, LHS( ¯ v−p h )−RHS( ¯ v−p h )≤ 0 and LHS( v−p h )−RHS( v−p h )≥ 0. So there exists ¯ w satisfies LHS(¯ w) =RHS(¯ w) and promotion is profitable if w 1 (0)≤ ¯ w. Proof of Proposition 3.3.4 f F is increasing implies that ¯ F ′′ ≤ ( ¯ F ′ ) 2 ¯ F and ¯ F ′ (pα+hw) ¯ F(pα+hw) − ¯ F ′ (p+hw) ¯ F(p+hw) = f(p+hw) ¯ F(p+hw) − f(pα+hw) ¯ F(pα+hw) ≥ 0. In addition, the threshold ¯ w satisfies the following equation αμ 2 +δμ 2 ¯ F(h¯ w+p) ¯ F(h¯ w+pα) = (1−α)hw f(p+h¯ w) ¯ F(h¯ w+p) (1+μ¯ w). (C.0.7) 169 (i) Consider ¯ w as a function of p. Take the derivative of both sides of C.0.7 with respect to p, we get d¯ w dp = − δ ¯ F ′ (p+h¯ w) ¯ F(pα+h¯ w) + αδ ¯ F(p+h¯ w) ¯ F ′ (pα+h¯ w) ¯ F 2 (pα+h¯ w) + h(−1+α)¯ w(1+¯ w)(− ¯ F ′2 (p+h¯ w)+ ¯ F(p+h¯ w) ¯ F ′′ (p+h¯ w)) ¯ F 2 (p+h¯ w) h δ ¯ F ′ (p+h¯ w) ¯ F(pα+h¯ w) + h(−1+α¯ w(1+w) ¯ F ′2 (p+h¯ w) ¯ F 2 (p+h¯ w) − δ ¯ F(p+h¯ w) ¯ F ′ (pα+h¯ w) ¯ F 2 (pα+h¯ w) + (1−α)((1+2¯ w) ¯ F ′ (p+h¯ w)+h¯ w(1+¯ w) ¯ F ′′ (p+h¯ w)) ¯ F(p+h¯ w) . The numerator is − δ ¯ F ′ (p+h¯ w) ¯ F(pα+h¯ w) + αδ ¯ F(p+h¯ w) ¯ F ′ (pα+h¯ w) ¯ F 2 (pα+h¯ w) + h(−1+α)¯ w(1+ ¯ w) − ¯ F ′2 (p+h¯ w)+ ¯ F(p+h¯ w) ¯ F ′′ (p+h¯ w) F 2 (p+h¯ w) = δ ¯ F(p+h¯ w) ¯ F(pα+h¯ w) ( α ¯ F ′ (pα+h¯ w) ¯ F(pα+h¯ w) − ¯ F ′ (p+h¯ w) ¯ F(p+h¯ w) ) + h(1−α)¯ w(1+ ¯ w)( ¯ F ′2 (p+h¯ w)− ¯ F(p+h¯ w) ¯ F ′′ (p+h¯ w)) ¯ F 2 (p+h¯ w) . Since ¯ F ′′ ≤ ( ¯ F ′ ) 2 ¯ F and α ¯ F ′ (pα+h¯ w) ¯ F(pα+h¯ w) − ¯ F ′ (p+h¯ w) ¯ F(p+h¯ w) =α( ¯ F ′ (pα+h¯ w) ¯ F(pα+h¯ w) − ¯ F ′ (p+h¯ w) ¯ F(p+h¯ w) )−(1−α) ¯ F ′ (p+h¯ w) ¯ F(p+h¯ w) ≥ 0, it follows that ¯ F ′2 (p+h¯ w)− ¯ F(p+h¯ w) ¯ F ′′ (p+h¯ w)≥ 0. As a result, the numerator is greater than 0. In addition, the denominator is h( δ ¯ F ′ (p+h¯ w) ¯ F(pα+h¯ w) + h(−1+α)¯ w(1+ ¯ w) ¯ F ′2 (p+h¯ w) ¯ F 2 (p+h¯ w) − δ ¯ F(p+h¯ w) ¯ F ′ (pα+h¯ w) ¯ F 2 (pα+h¯ w) + (1−α) (1+2¯ w) ¯ F ′ (p+h¯ w)+h¯ w(1+ ¯ w) ¯ F ′′ (p+h¯ w) F(p+h¯ w) ) =h( δF(p+h¯ w) ¯ F(pα+h¯ w) ( ¯ F ′ (p+h¯ w) ¯ F(p+h¯ w) − ¯ F ′ (pα+h¯ w) ¯ F(pα+h¯ w) ) − h(1−α)¯ w(1+ ¯ w) ¯ F 2 (p+h¯ w) ( ¯ F ′2 (p+h¯ w)− ¯ F ′′ (p+h¯ w) ¯ F(p+h¯ w)) + (1−α)(1+2¯ w) ¯ F ′ (p+h¯ w) ¯ F(p+h¯ w) ) 170 ≤0. As a result, d¯ w dp ≤ 0. (ii) Consider ¯ w as a function of α. Taking the derivative of both sides of (C.0.7) with respect to α, we get d¯ w dα = −1+ h¯ w(1+¯ w) ¯ F ′ (p+h¯ w) ¯ F(p+h¯ w) + pδ ¯ F(p+h¯ w) ¯ F ′ (pα+h¯ w) ¯ F 2 (pα+h¯ w) h δ ¯ F ′ (p+h¯ w) ¯ F(pα+h¯ w) + h(−1+α)¯ w(1+¯ w) ¯ F ′2 (p+h¯ w) ¯ F 2 (p+hw) − δ ¯ F(p+hw) ¯ F ′ (pα+hw) ¯ F 2 (pα+hw) + (1−α)((1+2w) ¯ F ′ (p+hw)+hw(1+w) ¯ F ′′ (p+hw)) ¯ F(p+hw) . The numerator −1+ hw(1+w) ¯ F ′ (p+hw) ¯ F(p+hw) + pδ ¯ F(p+hw) ¯ F ′ (pα+hw) ¯ F 2 (pα+hw) ≤ 0 and the proof in (i) implies the denominator is less than 0. As a result, dw dα ≥ 0. (iii) (C.0.7 ) implies that ¯ w does not depend on λ, and the conclusion follows. Proof of Proposition 3.3.5 (C.0.7) implies that the threshold waiting time¯ w satisfies α+δ ¯ F(h¯ w+p) ¯ F(h¯ w+pα) = (1−α)h¯ wf(p+h¯ w) ¯ F(p+h¯ w) . (C.0.8) On the other hand, the actual waiting time when no promotion w 1 satisfies w 1 = λ ¯ F(hw 1 +p) μ(μ−λ ¯ F(hw 1 +p)) . (C.0.9) If p+hw 1 → ¯ v, then (C.0.9) implies that w 1 → 0 ; if p +h¯ w → ¯ v, then (C.0.8) implies that ¯ w > 0 because ¯ w satisfies α = (1−α)h¯ wf(¯ v) ¯ F(¯ v) and f(¯ v) ¯ F(¯ v) is bounded. Since w 1 and ¯ w are continuous with respect to p ,there exists p such that promotion is profitable if p≥ ¯ p. Proof of Proposition 3.4.1 The service provider solves the following optimization prob- lem max G n π(G n ) n =λpF(p+hw n 1 )+ G n n αpF(αp+hw n 1 )+δ(λ+ G n n )p ¯ F(p+hw n 2 ) (C.0.10) s.t.w n 1 = λF(p+hw n 1 )+ G n n ¯ F(αp+hw n 1 ) nμ(μ−λ ¯ F(p+hw n 1 )− G n n ¯ F(αp+hw n 1 )) , (C.0.11) 171 w n 2 = (λ+ G n n )(1−p−hw n 2 ) nμ(μ−(λ+ G n n )(1−p−hw n 2 )) , (C.0.12) λ ¯ F(p+hw n 1 )+ G n n ¯ F(αp+hw n 1 )≤μ, (C.0.13) (λ+G) ¯ F(p+hw n 2 )≤μ. (C.0.14) Suppose G ∗ n is the optimal solution of problem (C.0.10)-(C.0.14). From equation (C.0.11) and (C.0.12), λ ¯ F(p+h¯ w 1 )+G ¯ F(αp+hw n 1 ) = nw n 1 μ 2 1+nw n 1 μ , (C.0.15) (λ+G) ¯ F(p+hw n 2 ) = nw n 2 μ 2 1+nw n 2 μ . (C.0.16) Next we show that lim n→∞ w n 2 exists. Taking the derivative of both sides of the equation C.0.16 with respect to n, we get−(λ+G)f(p+hw n 2 ) dw n 2 dn = μ (1+w n 2 nμ 2 ) ( dw n 2 dn n+w n 2 ) and as a result, dw n 2 dn < 0, i.e., w n 2 is a sequence that decreases with n and has a lower bound 0. So lim n→∞ w n 2 exists. Following the same logic, lim n→∞ w n 1 also exists. Let ¯ w 1 = lim n→∞ w n 1 , w 2 =lim n→∞ w n 2 and ¯ G ∗ =lim n→∞ G ∗ n n , then lim n→∞ π n (G ∗ n ) n =λp ¯ F(p+h¯ w 1 )+G ∗ n αp ¯ F(αp+h¯ w 1 )+δ(λ+G ∗ n )p ¯ F(p+h¯ w 2 ). Taking the limit n→∞, if ¯ w 1 > 0 and ¯ w 2 >0, we have λ ¯ F(p+h¯ w 1 )+ ¯ G ∗ ¯ F(αp+h¯ w 1 ) = μ, (C.0.17) (λ+ ¯ G ∗ ) ¯ F(p+h¯ w 2 ) = μ. (C.0.18) As a result, if n → ∞ and G n∗ n → ¯ G ∗ , the optimization (C.0.10)-(C.0.14) is reduced to (3.4.1)-(3.4.3) and ¯ G ∗ solves (3.4.1)-(3.4.3). Proof of Proposition 3.4.2 Since (λ + G) ¯ F(p) < λ ¯ F(p) + G ¯ F(αp), there are three possible scenarios to consider: μ≤ (λ+G) ¯ F(p), (λ+G) ¯ F(p)≤ μ≤ λ ¯ F(p)+G ¯ F(αp) and μ≥ λ ¯ F(p)+G ¯ F(αp). 172 (i)Ifμ≤ (λ+G) ¯ F(p),theconstraints(3.3.8)and(3.3.9)implythath¯ w 1 > 0andh¯ w 2 >0. According to (C.0.17) and (C.0.14), the total limiting revenue could be represented as ¯ π(G) =λp ¯ F(p+h¯ w 1 )+Gαp ¯ F(αp+h¯ w 1 )+δ(λ+G)p ¯ F(p+h¯ w 2 ) =(α+δ)pμ+(1−α)λp ¯ F(p+h¯ w 1 ), where ¯ w 1 satisfies λ ¯ F(p +h¯ w 1 ) +G ¯ F(αp +h¯ w 1 ) = μ. Since d ¯ F(p+h¯ w 1 ) d¯ w 1 ≤ 0 and d¯ w 1 dG ≥ 0, dπ dG = (1−α)λp d ¯ F(p+h¯ w 1 ) d¯ w 1 d¯ w 1 dG ≤ 0. Combining with the constraint μ < (λ+G) ¯ F(p), we have ¯ G ∗ = μ ¯ F(p) −λ if μ≥ λ ¯ F(p). (ii) If (λ+G) ¯ F(p)≤μ≤λ ¯ F(p)+G ¯ F(αp), the constraints (3.4.5) and (3.4.7) imply that hw 1 > 0 and hw 2 = 0. Since G≥ 0, it implies that λ ¯ F(p)≤ μ. And according to (3.4.5), the total limiting revenue could be represented as, ¯ π(G) =λp ¯ F(p+h¯ w 1 )+Gαp ¯ F(αp+h¯ w 1 )+δ(λ+G)p ¯ F(p+h¯ w 2 ) =λp ¯ F(p+h¯ w 1 )+Gαp ¯ F(αp+h¯ w 1 )+δ(λ+G)p ¯ F(p) =αpμ+(1−α)λp ¯ F(p+h¯ w 1 )+δ(λ+G)p ¯ F(p). where λ ¯ F(p+h¯ w 1 )+G ¯ F(αp+h¯ w 1 ) = μ. Taking the first and second derivative of ¯ π(G) with respect to G, we get d¯ π dG =δp(1−p)+(1−α)λp dF(p+h¯ w 1 ) d¯ w 1 d¯ w 1 dG , d 2 ¯ π dG 2 = (1−α)λp( dF(p+h¯ w 1 ) d¯ w 1 d 2 ¯ w 1 dG 2 + d 2 F(p+h¯ w 1 ) d¯ w 2 1 ( d¯ w 1 dG ) 2 ). Taking the first and second derivative of ¯ w 1 with respect to G, we get d¯ w 1 dG = ¯ F(pα+h¯ w 1 ) h(λf(p+h¯ w 1 )+Gf(pα+h¯ w 1 ) = ¯ F(pα+h¯ w 1 ) hμ ≥ 0, d 2 ¯ w 1 dG 2 = − f(pα+h¯ w 1 ) μ d ¯ w 1 dG ≤ 0. Substituting d¯ w 1 dG and d 2 ¯ w 1 dG 2 into d 2 ¯ π dG 2 , we get d 2 ¯ π dG 2 = (1−α)λp d¯ w 1 dG ( dF(p+h¯ w 1 ) d¯ w 1 ¯ w 1 d 2 ¯ w 1 dG 2 + d 2 F(p+h¯ w 1 ) d¯ w 2 1 ( d¯ w 1 dG ) 2 ) 173 = (1−α)λph μ d¯ w 1 dG (f(pα+h¯ w 1 )f(p+h¯ w 1 )−f ′ (p+h¯ w 1 ) ¯ F(pα+h¯ w 1 )). Since d 2 ¯ w 1 dG 2 ≤ 0 and f ′ (x) ≤ 0, it follows that d 2 ¯ π dG 2 ≥ 0 and ¯ π(G) is convex on the range of (λ+G) ¯ F(p)≤ μ≤ λ ¯ F(p)+G ¯ F(αp). As a result, the optimal solution is on the boundary of this range, i.e., either ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) or ¯ G ∗ = μ ¯ F(p) −λ. (iii) if μ≥λ ¯ F(p)+G ¯ F(αp), h¯ w 1 = 0 and h¯ w 2 =0. As a result ¯ π(G) =λp ¯ F(p)+Gαp ¯ F(αp)+δ(λ+G)p ¯ F(p)λp ¯ F(p)+Gαp ¯ F(αp)+δ(λ+G)p ¯ F(p). In addition, dπ dG =αp ¯ F(αp)+δ ¯ F(p)>0. Since the limiting revenue ¯ π(G) increases with G and G≥ μ−λ ¯ F(p) F(αp) , the optimal solution when μ≥λ ¯ F(p)+G ¯ F(αp) is ¯ G ∗ = μ−λ ¯ F(p) F(αp) . As a summary of scenario (i)-(iii), the potential optimal promotion size is either ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) or ¯ G ∗ = μ−λ ¯ F(p) ¯ F(p) . Next we derive the firm’s revenues given these two promotion size and compare which one is larger: If ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) , ¯ π 1 = λp ¯ F(p)+ ¯ G ∗ αp ¯ F(αp)+δ(λ+ ¯ G ∗ )p ¯ F(p) = λp ¯ F(p)+αp(μ−λ ¯ F(p))+δ(λ+ μ−λ ¯ F(p) ¯ F(αp) )p ¯ F(p) = αpμ+λp(1−α) ¯ F(p)+ δ(λ( ¯ F(αp)− ¯ F(p))+μ)p ¯ F(p) ¯ F(αp) . (C.0.19) If ¯ G ∗ = μ−λ ¯ F(p) ¯ F(p) , ¯ π 2 = (α+δ)pμ+(1−α)λp ¯ F(p+h¯ w 1 ). (C.0.20) where ¯ w 1 is the solution of λ ¯ F(p+h¯ w 1 )+( μ ¯ F(p) −λ) ¯ F(αp+h¯ w 1 ) = μ. Then, subtracting (C.0.20) from (C.0.20), we have ¯ π 1 − ¯ π 2 = λp(1−α)( ¯ F(p)− ¯ F(p+h¯ w 1 ))+ δ(λ( ¯ F(αp)− ¯ F(p))+μ)p ¯ F(p) ¯ F(αp) −δpμ 174 = λp(1−α)( ¯ F(p)− ¯ F(p+h¯ w 1 ))−δp(1− ¯ F(p) ¯ F(αp) )(μ−λ ¯ F(p)). Hence, ¯ π 1 ≥ ¯ π 2 if δ≤ λ(1−α)( ¯ F(p)− ¯ F(p+h¯ w 1 )) (1− ¯ F(p) ¯ F(αp) )(μ−λ ¯ F(p)) , where¯ w 1 satisfies λ ¯ F(p+h¯ w 1 )+( μ ¯ F(p) −λ) ¯ F(αp+ h¯ w 1 ) = μ. Let δ = λ(1−α)( ¯ F(p)− ¯ F(p+h¯ w 1 )) (1− ¯ F(p) ¯ F(αp) )(μ−λ ¯ F(p)) . As a result, the optimal solution for (3.4.1)-(3.4.3) is ¯ G ∗ = μ−λ ¯ F(p) ¯ F(p) if and only if δ≥δ, and ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) if and only if δ≤δ. Proof of Proposition 3.4.3 (i) Taking the partial derivative of λ ¯ F(p+h¯ w 1 )+( μ ¯ F(p) − λ) ¯ F(αp+h¯ w 1 )=μ with respect to λ, we have ¯ F(p+h¯ w 1 )+λ ∂ ¯ F(p+h¯ w 1 ) ∂¯ w 1 d¯ w 1 dλ +( μ ¯ F(p) −λ) ∂ ¯ F(αp+h¯ w 1 ) ∂¯ w 1 d¯ w 1 dλ = 0. Or equivalently, (λ ∂ ¯ F(p+h¯ w 1 ) ∂¯ w 1 +( μ ¯ F(p) −λ) ∂ ¯ F(αp+h¯ w 1 ) ∂¯ w 1 ) d¯ w 1 dλ =− ¯ F(p+h¯ w 1 ). Since λ ∂ ¯ F(p+h¯ w 1 ) ∂¯ w 1 +( μ ¯ F(p) −λ) ∂ ¯ F(αp+h¯ w 1 ) ∂¯ w 1 ≤ 0 and ¯ F(p+h¯ w 1 )≥ 0, it follows that d¯ w 1 dλ ≥ 0. Taking partial derivative of λ ¯ F(p+h¯ w 1 )+( μ ¯ F(p) −λ) ¯ F(αp+h¯ w 1 ) =μ with respect to α, we get ( μ ¯ F(p) −λ)( ∂ ¯ F(αp+h¯ w 1 ) ∂α + ∂ ¯ F(αp+h¯ w 1 ) ∂¯ w 1 d¯ w 1 dα ) =0. Since ∂ ¯ F(αp+h¯ w 1 ) ∂α ≤ 0 and ∂ ¯ F(αp+h¯ w 1 ) ∂ ¯ w 1 ≤ 0, it follows that d¯ w 1 dα ≤ 0. Taking partial derivative of λ ¯ F(p+h¯ w 1 )+( μ ¯ F(p) −λ) ¯ F(αp+h¯ w 1 )=μ with respect to p, we get λ ∂ ¯ F(p+h¯ w 1 ) ∂p +λ ∂ ¯ F(p+h¯ w 1 ) ∂¯ w 1 d¯ w 1 dp − μ ¯ F(αp+h¯ w 1 ) ( ¯ F(p)) 2 +( μ ¯ F(p) −λ)( ∂ ¯ F(αp+h¯ w 1 ) ∂p + ∂ ¯ F(αp+h¯ w 1 ) ∂¯ w d¯ w dp ) = 0, or equivalently, (λ ∂ ¯ F(p+h¯ w 1 ) ∂¯ w 1 +( μ ¯ F(p) −λ) ∂ ¯ F(αp+h¯ w 1 ) ∂¯ w ) d¯ w 1 dp 175 = μ ¯ F(αp+h¯ w 1 ) ( ¯ F(p)) 2 −λ ∂ ¯ F(p+h¯ w 1 ) ∂p −( μ ¯ F(p) −λ) ∂ ¯ F(αp+h¯ w 1 ) ∂p . Since ∂ ¯ F(p+h¯ w 1 ) ∂¯ w 1 ≤ 0 and ∂ ¯ F(αp+h¯ w 1 ) ∂ ¯ w ≤ 0, it follows that d¯ w 1 dp ≥ 0. (ii) First consider δ as a function of λ. We decompose δ = λ(1−α)( ¯ F(p)− ¯ F(p+h¯ w 1 )) (1− ¯ F(p) ¯ F(αp) )(μ−λ ¯ F(p)) into two parts: λ(1−α) (1− ¯ F(p) ¯ F(αp) )(μ−λ ¯ F(p)) and ¯ F(p)− ¯ F(p+h¯ w 1 ). The term λ(1−α) (1− ¯ F(p) ¯ F(αp) )(μ−λ ¯ F(p)) increases with λ. In addition, d¯ w 1 dλ ≥ 0 and ∂ ¯ F(p+h¯ w 1 ) ∂¯ w 1 ≤ 0 implied by (i). It follows that for the second term ¯ F(p)− ¯ F(p+h¯ w 1 ), d( ¯ F(p)− ¯ F(p+h¯ w 1 )) dλ =− ∂ ¯ F(p+h¯ w 1 ) ∂ ¯ w 1 d¯ w 1 dλ ≥ 0. As a result, dδ dλ ≥ 0. Next consider δ as a function of α. Taking the derivative of ¯ F(p) ¯ F(αp) with respect to α, we get d dα ( ¯ F(p) ¯ F(αp) ) = −α ¯ F ′ (αp) ¯ F(p) ( ¯ F(αp)) 2 ≥ 0. In addition d(1−α)( ¯ F(p)− ¯ F(p+h¯ w 1 )) dα =−( ¯ F(p)− ¯ F(p+h¯ w 1 ))− (1−α)( ∂ ¯ F(p+h¯ w 1 ) ∂ ¯ w 1 d¯ w 1 dα ). Since ¯ F(p)− ¯ F(p+h¯ w 1 )≥ 0, ∂ ¯ F(p+h¯ w 1 ) ∂¯ w 1 ≤ 0 and d¯ w 1 dα ≤ 0, it follows that d(1−α)( ¯ F(p)− ¯ F(p+h¯ w 1 )) dα ≤ 0. As a result, dδ dα ≤ 0. Finaly, consider δ as a function of p. Taking the derivative of ¯ F(p) ¯ F(αp) with respect to p, we get d dp ( ¯ F(p) ¯ F(αp) ) = ¯ F ′ (p) ¯ F(αp)− ¯ F ′ (αp) ¯ F(p) ( ¯ F(αp)) 2 = F ′ (αp) ¯ F(p)−F ′ (p) ¯ F(αp) ( ¯ F(αp)) 2 . Since F(x) has an increas- ing failure rate, F ′ (αp) ¯ F(αp) ≤ F ′ (p) ¯ F(p) and it results in d dp ( ¯ F(p) ¯ F(αp) ) ≤ 0. In addition d(μ−λ ¯ F(p)) dp ≥ 0. Since d¯ w 1 dp ≤ 0 and that F(x) is concave implies d( ¯ F(p)− ¯ F(p+h¯ w 1 )) dp ≤ 0, it follows that d( ¯ F(p)− ¯ F(p+h¯ w 1 )) dp = ¯ F ′ (p)− ¯ F ′ (p+h¯ w 1 )− ¯ F(p+h¯ w 1 ) d¯ w 1 dp ≤ 0. As a result, dδ dp ≤ 0. Proof of Proposition 3.4.4 In order to show the proposition, we first show that (i) If δ≥δ, ¯ G ∗ = μ ¯ F(p) −λ, w n 2 ∼ O( 1 √ n ), √ n(1−ρ n 2 )∼O(1); if δ≤δ, ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) , w n 1 ∼O( 1 √ n ), √ n(1−ρ n 1 )∼O(1); (ii) the optimal profit is given by π n (G ∗ n )=n(¯ π( ¯ G ∗ )− τ √ n +o( 1 √ n )), and (iii) if the distribution F has an increasing failure rate, the optimal promotion size G ∗ n must be in the form G ∗ n =n ¯ G ∗ − √ ng+o( √ n), for some g > 0. (i) Let ǫ n =n(μ−(λ+ ¯ G ∗ ) ¯ F(p+hw n 2 )). From Taylor’s expansion, ¯ F(p+hw n 2 ) = ¯ F(p+hw 2 )−f(p+hw 2 )(w n 2 −w 2 )+o(w n 2 −w 2 ). (C.0.21) . 176 Ifδ≥δ, ¯ G ∗ = μ ¯ F(p) −λandw ∗ 2 =0byProposition3.4.2. Substitutingw ∗ 2 = 0into(C.0.21), itfollowsthat ¯ F(p+hw n 2 ) = ¯ F(p)−f(p)w n 2 andǫ n =n(μ−(λ+ ¯ G ∗ ) ¯ F(p)+(λ+ ¯ G ∗ )f(p)w n 2 )= n(λ+ ¯ G ∗ )f(p)w n 2 . So ǫ n =O(nw n 2 ). (C.0.22) On the other hand, w n 2 = (λ+ ¯ G ∗ ) ¯ F(p+hw n 2 ) nμ(μ−(λ+ ¯ G ∗ ) ¯ F(p+hw n 2 )) = (λ+ ¯ G ∗ ) ¯ F(p+hw n 2 ) μǫn , so w n 2 ǫ n =O(1). (C.0.23) Asaresult, (C.0.22)and(C.0.23)implythatn(w n 2 ) 2 =O(1)anditfollowsthatw n 2 =O( 1 √ n ), ǫ n =O( √ n), √ n(1−ρ n 2 )=O(1) Similarly, let ǫ n =n(μ−(λ ¯ F(p+hw n 1 )+G ¯ F(αp+hw n 1 )). From Taylor’s expansion, we get ¯ F(p+hw n 1 ) = ¯ F(p+hw 1 )−f(p+hw 1 )(w n 1 −w 1 ) (C.0.24) and ¯ F(αp+hw n 1 ) = ¯ F(αp+hw 1 )−f(αp+hw 1 )(w n 1 −w 1 ). (C.0.25) If δ < δ, ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) and w ∗ 1 = 0 by Proposition 3.4.2. Substituting w ∗ 1 = 0 into (C.0.24) and (C.0.25), it follows that ¯ F(p+hw n 1 ) = ¯ F(p)−f(p)w n 1 , ¯ F(αp+hw n 1 )= ¯ F(αp)−f(αp)w n 1 and ǫ n =n(μ−(λ ¯ F(p)+ ¯ G ∗ ¯ F(αp)) =n(λf(p)+ ¯ G ∗ f(αp))w n 1 . So ǫ n =O(nw n 2 ). (C.0.26) On the other hand, w n 1 = λ ¯ F(p+hw n 1 )+ ¯ G ∗ ¯ F(αp+hw n 1 ) nμ(μ−(λ ¯ F(p+hw n 1 )+ ¯ G ∗ ¯ F(αp+hw n 1 )) = λ ¯ F(p+hw n 1 )+ ¯ G ∗ ¯ F(αp+hw n 1 ) μǫn , so w n 1 ǫ n =O(1). (C.0.27) As a result, (C.0.26) and (C.0.27) imply that n(w n 1 ) 2 = O(1) and it follows that w n 1 = O( 1 √ n ), ǫ n =O( √ n), √ n(1−ρ n 1 )=O(1) (ii) We now show that the optimal profit is given by π n (G ∗ n ) =n(¯ π( ¯ G ∗ )− τ √ n +o( 1 √ n )). 177 If δ≥ δ, ¯ G ∗ = μ ¯ F(p) −λ. It follows that ¯ w 2 =0 and w n 2 =O( 1 √ n ). On the one hand, π n (G ∗ n ) n ≥ π n ( ¯ G ∗ ) n =λp ¯ F(p+hw n 1 ( ¯ G ∗ ))+αp ¯ G ∗ ¯ F(αp+hw n 1 ( ¯ G ∗ ))+δ(λ+ ¯ G ∗ ) ¯ F(p+hw n 1 ( ¯ G ∗ )) =(λp ¯ F(p+h¯ w 1 )+αp ¯ G ∗ ¯ F(αp+h¯ w 1 )+δ(λ+ ¯ G ∗ ) ¯ F(p+h¯ w 2 )) −δ(λ+ ¯ G ∗ )f(p+h¯ w 2 )(w n 2 − ¯ w 2 )+o(w n 2 − ¯ w 2 ) =¯ π( ¯ G ∗ )−O( 1 √ n ) The first inequality is because G ∗ n is the optimal solution for π n (G ∗ n ) and the second equality is the Taylor’s expansion when ¯ w 2 =0 and w n 2 =O( 1 √ n ) by Proposition 3.4.2 and the results in (i). On the other hand, w n 2 is an decreasing function of n. The proof is as follows: (3.3.7) implies that (λ+G) ¯ F(p+hw n 2 ) = w n 2 nμ 1+w n 2 nμ 2 . Taking the derivative with respect to n on both sides of the above equation, we get−(λ+ G)f(p+hw n 2 ) dw n 2 dn = μ (1+w n 2 nμ 2 ) ( dw n 2 dn n+w n 2 ) and as a result, dw n 2 dn < 0. That is, for any given G, w n 1 ≥ ¯ w 1 . And similarly, w n 2 ≥ ¯ w 2 . So π n (G ∗ n ) n ≤ ¯ π( G ∗ n n )≤ ¯ π( ¯ G ∗ ). As a result, π n (G ∗ n ) n = ¯ π( ¯ G ∗ )− τ √ n +o( 1 √ n )) If δ ≤ δ, ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) , we get π n (G ∗ n ) n = π( ¯ G ∗ )− τ √ n + o( 1 √ n ) by following the same approach above, where τ may take different values. (iii)Nowweprovethepropositionbycontradiction. Suppose √ n( ¯ G ∗ − G ∗ n n )→−∞. From (C.0.17) and (C.0.18), we get λ ¯ F(p+h¯ w 1 ( G ∗ n n ))+ G ∗ n n ¯ F(αp+h¯ w 1 (G ∗ n )) =μ, (C.0.28) λ ¯ F(p+h¯ w 1 ( ¯ G ∗ ))+ ¯ G ∗ ¯ F(αp+h¯ w 1 ( ¯ G ∗ ))=μ, (C.0.29) 178 and (λ+ G ∗ n n ) ¯ F(p+h¯ w 2 ( G ∗ n n )) = (λ+ ¯ G ∗ ) ¯ F(p+h¯ w 2 ( ¯ G ∗ )) =μ. (C.0.30) Substracting (C.0.29) from (C.0.28), we get λ( ¯ F(p+h¯ w 1 ( G ∗ n n ))− ¯ F(p+h¯ w 1 ( ¯ G ∗ ))= ¯ G ∗ ¯ F(αp+h¯ w 1 ( ¯ G ∗ ))− G ∗ n n ¯ F(αp+h¯ w 1 ( G ∗ n n )) (C.0.31) As a result, (C.0.31) implies that λp ¯ F(p+h¯ w 1 ( G ∗ n n ))+αp G ∗ n n ¯ F(αp+h¯ w 1 ( G ∗ n n ))−(λp ¯ F(p+h¯ w 1 ( ¯ G ∗ ))+αp ¯ G ∗ ¯ F(αp+h¯ w 1 ( ¯ G ∗ )) (C.0.32) =λp(1−α)( ¯ F(p+h¯ w 1 ( G ∗ n n ))− ¯ F(p+h¯ w 1 ( ¯ G ∗ )). And (C.0.30) implies that δp(λ+ G ∗ n n ) ¯ F(p+h¯ w 2 ( G ∗ n n ))−δp(λ+ ¯ G ∗ ) ¯ F(p+h¯ w 2 ( ¯ G ∗ )) =0. (C.0.33) In addition, take the derivative of both sides of (C.0.17) with respect to G, we get d¯ w 1 dG = ¯ F(αp+h¯ w 1 ) λf(p+h¯ w 1 )+Gf(αp+h¯ w 1 ) >0. Since √ n( ¯ G ∗ − G ∗ n n )→−∞, G ∗ n ≥ ¯ G ∗ and it implies that ¯ w 1 ( ¯ G ∗ )≤ ¯ w 1 (G ∗ n ). √ n( π n (G ∗ n ) n − ¯ π(G)) ≤ √ n(¯ π( G ∗ n n )− ¯ π( ¯ G ∗ )) (C.0.34) = √ n(λp ¯ F(p+h¯ w 1 ( G ∗ n n ))+αpG ∗ n ¯ F(αp+h¯ w 1 ( G ∗ n n ))+δp(λ+G ∗ n ) ¯ F(p+h¯ w 2 ( G ∗ n n )) −(λp ¯ F(p+h¯ w 1 ( ¯ G ∗ ))+αp ¯ G ∗ ¯ F(αp+h¯ w 1 ( ¯ G ∗ ))+δp(λ+ ¯ G ∗ ) ¯ F(p+h¯ w 2 ( ¯ G ∗ ))) = √ np(1−α)( ¯ F(p+h¯ w 1 ( G ∗ n n ))− ¯ F(p+h¯ w 1 ( ¯ G ∗ )) (C.0.35) = √ np(1−α)f(p+h¯ w 1 ( ¯ G ∗ ))( ¯ G ∗ − G ∗ n n ) d¯ w 1 dG | G= ¯ G ∗ +o(( ¯ G ∗ − G ∗ n n ) d¯ w 1 dG | G= ¯ G ∗) (C.0.36) = √ np(1−α)( ¯ G ∗ − G ∗ n n ) ¯ F(αp+h¯ w 1 ( ¯ G ∗ ))f(p+h¯ w 1 ( ¯ G ∗ )) λf(p+h¯ w 1 ( ¯ G ∗ ))+ ¯ G ∗ f(αp+h¯ w 1 ( ¯ G ∗ )) +o(( ¯ G ∗ − G ∗ n n )) ≤−∞. (C.0.37) 179 Equality (C.0.35) is from (C.0.32) and (C.0.33). Equlity (C.0.36) is from Taylor’s expansion arround ¯ G ∗ . The inequality (C.0.37) is hold because λf(p+h¯ w 1 ( ¯ G ∗ ))+ ¯ G ∗ f(αp+h¯ w 1 ( ¯ G ∗ )) ¯ F(αp+hw 1 ( ¯ G ∗ )) < λf(p+h¯ w 1 ( ¯ G ∗ )) ¯ F(p+hw 1 ( ¯ G ∗ )) + ¯ G ∗ f(αp+h¯ w 1 ( ¯ G ∗ )) ¯ F(αp+hw 1 ( ¯ G ∗ )) ≤ (λ+ ¯ G ∗ )f(p+h¯ w 1 ) ¯ F(p+h¯ w 1 ) since ¯ F(αp+ hw 1 )> ¯ F(p+hw 1 ) and F has an increasing failure rate. As a result, ¯ F(αp+h¯ w 1 ( ¯ G ∗ ))f(p+h¯ w 1 ( ¯ G ∗ )) λf(p+h¯ w 1 ( ¯ G ∗ ))+ ¯ G ∗ f(αp+h¯ w 1 ( ¯ G ∗ )) > ¯ F(p+h¯ w 1 ( ¯ G ∗ )) λ+ ¯ G ∗ ≥ 0 and √ n( π n (G ∗ n ) n −¯ π( ¯ G ∗ )→−∞ Similarly, if √ n( ¯ G ∗ − G ∗ n n )→+∞. √ n(¯ π( ¯ G ∗ )− π n (G ∗ n ) n ) ≥ √ n(¯ π( ¯ G ∗ )− ¯ π( G ∗ n n )) = √ n(λp ¯ F(p+h¯ w 1 ( ¯ G ∗ ))+αp ¯ G ∗ ¯ F(αp+h¯ w 1 ( ¯ G ∗ ))+δp(λ+ ¯ G ∗ ) ¯ F(p+h¯ w 2 ( ¯ G ∗ )) −(λp ¯ F(p+h¯ w 1 ( G ∗ n n )+αp G ∗ n n ¯ F(αp+h¯ w 1 ( G ∗ n n ))+δp(λ+ G ∗ n n ) ¯ F(p+h¯ w 2 (G ∗ n )) = √ np(1−α)( ¯ F(p+h¯ w 1 ( ¯ G ∗ ))− ¯ F(p+h¯ w ∗ 1 (G ∗ n ))) = √ np(1−α)f(p+h¯ w 1 ( ¯ G ∗ ))( ¯ G ∗ − G ∗ n n ) d¯ w 1 dG | G= ¯ G ∗ +o(( ¯ G ∗ − G ∗ n n ) d¯ w 1 dG | G= ¯ G ∗) = √ np(1−α)( ¯ G ∗ − G ∗ n n ) ¯ F(αp+h¯ w 1 ( ¯ G ∗ ))f(p+h¯ w 1 ( ¯ G ∗ )) λf(p+h¯ w 1 ( ¯ G ∗ ))+ ¯ G ∗ f(αp+h¯ w 1 ( ¯ G ∗ )) +o(( ¯ G ∗ − G ∗ n n )) ≥+∞ As a result, (ii) is violated and G ∗ n = ¯ G ∗ − g √ n +o( 1 √ n ) is true. Proof of Lemma 3.4.5 By Proposition 3.4.4, the optimal promotion size G ∗ n must be in the form G ∗ n =n ¯ G ∗ − √ ng+o( √ n). Let G n =n ¯ G ∗ − √ ng+o( √ n), we have max Gn π n (G n ) = max g π n ( ¯ G ∗ − √ ng+o( √ n)) = max g n¯ π( ¯ G ∗ )− √ nˆ π(g)+o( √ n) = n¯ π( ¯ G ∗ )−min g √ nˆ π(g)+o( √ n). As a result, if g ∗ =argminˆ π, the optimal solution is G ∗ n =n ¯ G ∗ − √ ng ∗ +o( √ n). 180 Proof of Proposition 3.4.6 (i) Since w n 2 = e w n 2 √ n , e w n 2 √ n = (λ+G ∗ n ) ¯ F(p+h e w n 2 √ n ) nμ(μ−(λ+G ∗ n ) ¯ F(p+h e w n 2 √ n )) . SinceG ∗ n = ¯ G ∗ − g √ n +o( 1 √ n ), we have √ ne w n 2 = (λ+ ¯ G ∗ − g √ n +o( 1 √ n )) ¯ F(p+h e w n 2 √ n ) μ 2 −μ((λ+ ¯ G ∗ − g √ n +o( 1 √ n )) ¯ F(p+h e w n 2 √ n ) . Letn→∞, substituting G= μ−λ ¯ F(p) ¯ F(p) into the above equation, we get e w n 2 = 1 g ¯ F(p) +O( 1 √ n ). (ii) Since w n 1 = e w n 1 √ n , e w n 1 √ n = λF(p+h e w n 1 √ n )+GF(αp+h e w n 1 √ n ) nμ(μ−λF(p+h e w n 1 √ n )−GF(αp+h e w n 1 √ n )) . Since G ∗ n = ¯ G ∗ − g √ n +o( 1 √ n ), we have √ ne w n 1 = λ ¯ F(p+h e w n 1 √ n )+( ¯ G ∗ − g √ n +o( 1 √ n )) ¯ F(αp+h e w n 1 √ n ) μ 2 −μ(λ ¯ F(p+h e w n 1 √ n )+( ¯ G ∗ − g √ n ++o( 1 √ n )) ¯ F(αp+h e w n 1 √ n ) . Letn→∞, substituting ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) , we get e w n 1 = 1 g(1−F(αp)) +O( 1 √ n ). Proof of Lemma 3.4.7 (i) By Taylor’s expansion, we get ¯ F(p+hw n 1 )− ¯ F(p+h¯ w 1 ) = −(F(p+hw n 1 )−F(p+h¯ w 1 )) = −f(p+h¯ w 1 )h(w n 1 − ¯ w 1 )+o(w n 1 − ¯ w 1 ) = o( 1 √ n ), and ¯ F(p+hw n 2 )− ¯ F(p+h¯ w 2 ) = −(F(p+hw n 2 )−F(p+h¯ w 2 )) = −f(p+h¯ w 2 )h(w n 2 − ¯ w 2 ) = −f(p) he w n 2 √ n =−f(p) h √ n 1 g ¯ F(p) +o( 1 √ n ). 181 As ¯ G ∗ = μ−λ ¯ F(p) ¯ F(p) and G ∗ n =n ¯ G ∗ − √ ng+o( √ n), we have ˆ π = √ n(¯ π( ¯ G ∗ )− π n (G ∗ n ) n ) = √ n(λp ¯ F(p+h¯ w 1 )+ ¯ G ∗ αp ¯ F(αp+h¯ w 1 )+δ(λ+ ¯ G ∗ )p ¯ F(p+h¯ w 2 )) − (λpF(p+hw n 1 )+G ∗ n αpF(αp+hw n 1 )+δ(λ+G ∗ n )p ¯ F(p+hw n 2 )) = gαp ¯ F(αp)+δ(λ+ ¯ G ∗ ) phf(p) g ¯ F(p) +δgp( ¯ F(p)− f(p) g ¯ F(p) )+o( 1 n ). (ii)If n→∞, by Talor’s expansion, we have ¯ F(p+h e w n 1 √ n ) = ¯ F(p)−f(p) he w n 1 √ n +o( 1 √ n ), ¯ F(αp+h e w n 1 √ n ) = ¯ F(αp)−f(p) he w n 1 √ n +o( 1 √ n ), and F(p+hw n 1 )− ¯ F(p+h¯ w 1 ) = −f(p+h¯ w 1 )h(w n 1 − ¯ w 1 ) = −f(p) he w n 1 √ n = −f(p) h √ n 1 g ¯ F(αp) +O( 1 √ n ). As ¯ G ∗ = μ−λ ¯ F(p) ¯ F(αp) and G ∗ n =n ¯ G ∗ − √ ng+o( √ n), we have ˆ π = √ n(¯ π( ¯ G ∗ )− π n (G ∗ n ) n ) = √ n((λp ¯ F(p+h¯ w 1 )+ ¯ G ∗ αp ¯ F(αp+h¯ w 1 )+δ(λ+ ¯ G ∗ )p ¯ F(p+h¯ w 2 )) − (λp ¯ F(p+hw n 1 )+G ∗ n αp ¯ F(αp+hw n 1 )+δ(λ+G ∗ n )p ¯ F(p+hw n 2 )) = √ n((λp+ ¯ G ∗ αp)( ¯ F(p+h¯ w 1 )−F(p+hw n 1 ))+( ¯ G ∗ −G ∗ n )αp ¯ F(αp+hw n 1 ) + δ( ¯ G ∗ −G ∗ n )p ¯ F(p+hw n 2 ))+δ(λ+ ¯ G ∗ )p( ¯ F(p+h¯ w 2 )− ¯ F(p+hw n 2 )) = (λ+α ¯ G ∗ )pf(p) h g ¯ F(αp) +gαp( ¯ F(αp)− f(αp)h g ¯ F(αp) ) +δgp ¯ F(p)+O( 1 n ). 182 Proof of Proposition 3.4.8 By lemma 3.4.5, we have G ∗ n =n ¯ G ∗ − √ ng ∗ +o( √ n) where g ∗ = argmin g ˆ π. (i) If δ > ¯ δ, ˆ π = gαp ¯ F(αp)+δ(λ+ ¯ G ∗ ) phf(p) g ¯ F(p) +δgp( ¯ F(p)− f(p) g ¯ F(p) ). Taking the derivative of ˆ π with respect to g , we have dˆ π dg = αp ¯ F(αp)+δp( ¯ F(p)− f(p) g ¯ F(p) )− δ(λ+ ¯ G ∗ )hf(p) g 2¯ F(p) + δpf(p) g ¯ F(p) = αp ¯ F(αp)+δp ¯ F(p)− δ(λ+ ¯ G ∗ )phf(p) g 2¯ F(p) . The optimal g ∗ satisfifes dˆ π dg = 0 and d 2 ˆ π dg 2 ≥ 0. As a result, g ∗ = s δ(λ+ ¯ G ∗ )hf(p)p (αp ¯ F(αp)+δp ¯ F(p)) = s δμhf(p) F 2 (p)(αF(αp)+δF(p)) , where the second equality is by substituting ¯ G ∗ = μ−λ ¯ F(p) ¯ F(p) . (ii) Ifδ≤ ¯ δ, ˆ π =(λ+α ¯ G ∗ ) pf(p)h g ¯ F(αp) +gαp( ¯ F(αp)− f(αp)h g ¯ F(αp) )+δgp ¯ F(p). Taking the derivative of ˆ π with respect to g , we have dˆ π dg = αp( ¯ F(αp)− f(αp)h g ¯ F(αp) )+δp ¯ F(p)− (λ+α ¯ G ∗ )pF ′ (p)h g 2¯ F(αp) + αpf(αp)h g ¯ F(αp) = αp ¯ F(αp)+δp ¯ F(p)− (λ+α ¯ G ∗ )pf(p)h g 2 (1−F(αp)) The optimal g ∗ satisfifes dˆ π dg = 0 and d 2 ˆ π dg 2 ≥ 0. As a result, g ∗ = s (λ+α ¯ G ∗ )f(p)h (α ¯ F(αp)+δ ¯ F(p) ¯ F(αp) = v u u t (λ+α μ−λ ¯ F(p) ¯ F(αp) )f(p)h (α ¯ F(αp)+δ ¯ F(p)) ¯ F(αp) 183 Proof of Proposition 3.4.9 (i) Since g ∗ = q δμhf(p) F 2 (p)(αF(αp)+δF(p)) , taking the derivative of g ∗ with respect to μ,h respectively, we have dg ∗ dμ = 1 2 s δhf(p) F 2 (p)(αF(αp)+δF(p))μ ≥ 0, dg ∗ dh = 1 2 s δhf(p) F 2 (p)(αF(αp)+δF(p))h ≥ 0. Since g ∗ = r μhf(p) F 2 (p)( αF(αp) δ +F(p)) , as a result dg ∗ dδ ≥ 0. Since g ∗ = r δμhf(p)/F(p) F 2 (p)(αF(αp)/F(p)+δ) , where F has an increasing failure rate implies that f(p) ¯ F(p) is increasing with p and ¯ F(p) ¯ F(αp) and F 2 (p)is decreasing with p, we have dg ∗ dp ≥ 0. (ii) Since g ∗ = q (λ( ¯ F(αp)−α ¯ F(p))+αμ)f(p)h (α ¯ F(αp)+δ ¯ F(p)) ¯ F 2 (αp) ,taking the derivative of g ∗ with respect to μ,h, δ respectively, we have dg ∗ dμ = 1 2 s αhf(p) (α ¯ F(αp)+δ ¯ F(p)) ¯ F 2 (αp)μ ≥ 0, dg ∗ dh = 1 2 s (λ( ¯ F(αp)−α ¯ F(p))+αμ)f(p) (α ¯ F(αp)+δ ¯ F(p)) ¯ F 2 (αp)h ≥ 0, and dg ∗ dδ ≤ 0. 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Kong, Guangwen
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Core Title
Essays on information, incentives and operational strategies
School
Marshall School of Business
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Doctor of Philosophy
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Business Administration
Publication Date
08/07/2015
Defense Date
06/12/2013
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game theory,information asymmetry,OAI-PMH Harvest,service operations,supply chain management
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), Zhang, Hao (
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), Brocas, Isabelle (
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), Gad, Allon (
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), Randhawa, Ramandeep S. (
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), Sosic, Greys (
committee member
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Tags
game theory
information asymmetry
service operations
supply chain management