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Delivery service assortment and product pricing in online retailing : the impact of pricing flexibility… Yu, Junhao 2015

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DELIVERY SERVICE ASSORTMENT AND PRODUCT PRICING IN ONLINE RETAILING: THE IMPACT OF PRICING FLEXIBILITY AND CUSTOMER RATING by  Junhao Yu  B.Mgt., Southwestern University of Finance and Economics, 2012  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Business Administration)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  August 2015  ©  Junhao Yu, 2015 AbstractThis research studies the delivery service assortment and product pricing problemin the context of online retailing where the seller selects a series of delivery optionsfrom a set of available alternatives for the customers to choose from and decides theprice for the product and the listed surcharge for each delivery service. We aim toexamine the impact of seller’s pricing flexibility and customer rating on the optimaldecisions and the optimal expected profit of the seller.By solving and comparing the results of four related problems, we find that usu-ally it would be optimal for the seller to include all the available delivery optionsand charge a constant mark-up for all the options. But when the customer rating isaggregated, the seller would have to solve a combinatorial optimization problem tofind out the optimal assortment when pricing is restricted to the product only and heshould differentiate the mark-up for each option when he enjoys the pricing flexibilityto re-price the quoted surcharges. We also show that two simple heuristic algorithmsprovide very good performance for the mentioned combinatorial optimization prob-lem.We explain why the aggregated rating would only hurt the seller and how pric-ing flexibility could remove its negative effect while assortment adjustment can onlyweaken its impact. In addition, numerical studies present the comparison betweenthe two main problems with aggregated customer rating and provide some observa-tions of the impact on delivery service providers and the customers. The findings inthis thesis yield useful managerial insights for the delivery service providers as wellas the seller for making their strategic decisions.iiPrefaceThis dissertation is original, unpublished, independent work by the author, JunhaoYu.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Main Results and Contributions . . . . . . . . . . . . . . . . . . . . . . . 41.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Interaction between Price and Lead Time . . . . . . . . . . . . . . . . . . 92.2 Assortment and Pricing in MNL Model . . . . . . . . . . . . . . . . . . . 142.2.1 MNL Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14iv2.2.2 Assortment in Logit Models . . . . . . . . . . . . . . . . . . . . . 162.2.3 Pricing in Logit Models . . . . . . . . . . . . . . . . . . . . . . . . 172.2.4 Joint Assortment and Pricing Problem . . . . . . . . . . . . . . . 192.3 Service Quality and Customer Rating . . . . . . . . . . . . . . . . . . . . 222.3.1 Delivery Service Quality and Quality Perception . . . . . . . . . 222.3.2 Rating Behavior and Rating Dynamics . . . . . . . . . . . . . . . 242.3.3 Impact of Rating and Firm Intervention . . . . . . . . . . . . . . . 253 A: Pricing Product without Customer Rating . . . . . . . . . . . . . . . . . . 293.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.1 Customers’ Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.2 Seller’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Assortment Decision . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Pricing Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 B: Pricing Product Bundle without Customer Rating . . . . . . . . . . . . . . 424.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1.1 Customers’ Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.1.2 Seller’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.1 Assortment Decision . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.2 Pricing Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 505 C: Pricing Product with Customer Rating . . . . . . . . . . . . . . . . . . . . 535.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.1 Customers’ Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.1.2 Customers’ Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.1.3 Seller’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59v5.2.1 Pricing Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2.2 Assortment Decision . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2.3 Structure of an Optimal Assortment . . . . . . . . . . . . . . . . . 665.2.4 Two Heuristic Algorithms . . . . . . . . . . . . . . . . . . . . . . . 705.3 Further Remarks on Example 5.1 . . . . . . . . . . . . . . . . . . . . . . . 775.3.1 Possible Collusion among the DSPs . . . . . . . . . . . . . . . . . 785.3.2 Possible Actions of the Seller . . . . . . . . . . . . . . . . . . . . . 836 D: Pricing Product Bundle with Customer Rating . . . . . . . . . . . . . . . 866.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.1.1 Customers’ Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.1.2 Customers’ Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.1.3 Seller’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.1 Pricing Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.2 Assortment Decision . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.3 Summary of Optimal Assortment and Pricing Decisions . . . . . . . . . 1076.4 Value of Pricing Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.1 Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.2 Impact on Seller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.3 Impact on DSPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.4 Impact on Customer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.1 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1308.2.1 DSPs’ Strategic Decisions and Interactions . . . . . . . . . . . . . 1318.2.2 Multiple Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 132vi8.2.3 Competing Sellers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338.2.4 Flexible Lead Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.2.5 Other Possibilities in Customer Rating Framework . . . . . . . . 134Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Appendix A: R Code for All Functions . . . . . . . . . . . . . . . . . . . . . . . 143Appendix B: R Code for Numerical Examples and Experiments . . . . . . . . 155viiList of TablesTable 1.1 Structure of the four problems in this thesis . . . . . . . . . . . . . 7Table 2.1 Comparison of literatures in section 2.1 . . . . . . . . . . . . . . . 13Table 4.1 Optimal results in different assortments in Example 4.1 . . . . . . 50Table 5.1 Optimal results in different assortments in Example 5.1 . . . . . . 61Table 5.2 Options in A in Example 5.2 . . . . . . . . . . . . . . . . . . . . . . 63Table 5.3 Options in A in Example 5.3 . . . . . . . . . . . . . . . . . . . . . . 64Table 5.4 Options in A in Example 5.4 . . . . . . . . . . . . . . . . . . . . . . 65Table 5.5 Options in A in Example 5.5 . . . . . . . . . . . . . . . . . . . . . . 66Table 5.6 Treatment on numerical experiments on proposed algorithms . . 73Table 5.7 Summary of numerical experiments on greedy policy . . . . . . . 74Table 5.8 Summary of numerical experiments on proposed algorithms . . . 76Table 5.9 Optimal results in different assortments in Example 5.6 . . . . . . 79Table 5.10 Optimal results in different assortments in Example 5.7 . . . . . . 81Table 5.11 Optimal results in different assortments in Example 5.8 . . . . . . 82Table 6.1 Optimal results in different assortments in Example 6.1 . . . . . . 101Table 6.2 Optimal results in different assortments in Example 6.2 . . . . . . 106Table 6.3 Comparison of optimal assortment decisions . . . . . . . . . . . . 107Table 6.4 Comparison of optimal pricing decisions . . . . . . . . . . . . . . 107Table 6.5 Comparison of optimal profits in six scenarios . . . . . . . . . . . 109viiiList of FiguresFigure 3.1 Plot of g(x) in Example 3.1 . . . . . . . . . . . . . . . . . . . . . . 36Figure 3.2 Plot of h(x) in Example 3.1 . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.3 Plot of pi(p) in Example 3.1 . . . . . . . . . . . . . . . . . . . . . . 37Figure 5.1 Time-line of events . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 5.2 Plot of yˆ in Example 5.2 when adding options sequentially . . . 63Figure 5.3 Plot of yˆ in Example 5.3 when adding options sequentially . . . 64Figure 5.4 Plot of yˆ in Example 5.4 when adding options sequentially . . . 65Figure 5.5 Plot of yˆ in Example 5.5 when adding options sequentially . . . 66Figure 7.1 Graphs for piD − piC when γ = 0.5 . . . . . . . . . . . . . . . . . 114Figure 7.2 Graphs for r when γ = 0.2 . . . . . . . . . . . . . . . . . . . . . . 115Figure 7.3 Graphs for ΛD −ΛC when γ = 0.5 . . . . . . . . . . . . . . . . . 117Figure 7.4 Graphs for λ1 when γ = 0.5 . . . . . . . . . . . . . . . . . . . . . 118Figure 7.5 Graphs for λ2 when γ = 0.5 . . . . . . . . . . . . . . . . . . . . . 119Figure 7.6 Graphs for λ3 when γ = 0.5 . . . . . . . . . . . . . . . . . . . . . 120Figure 7.7 Graphs for total service revenues when γ = 0.5 . . . . . . . . . . 123Figure 7.8 Graphs for AvgPayD −AvgPayC when γ = 0.5 . . . . . . . . . . 124Figure 7.9 Graphs for u¯ when γ = 0.5 . . . . . . . . . . . . . . . . . . . . . . 125Figure 7.10 Graphs for u¯D − u¯C when γ = 0.5 . . . . . . . . . . . . . . . . . . 127ixList of SymbolsNotation DescriptionA The set of available delivery options for the seller to select from.S The assortment selected by the seller for the customers to choose from.ui Customers’ mean utility associated with delivery option i.ε i The random Gumble variable in customer’s utility for delivery option i.Ui Customers’ utility associated with delivery option i. (Ui = ui + ε i)λi The probability of customers to choose delivery option i in an assortment.α The product’s base value.β The customers’ price sensitivity parameter.γ The customers’ delivery time sensitivity parameter.δ The customers’ rating sensitivity parameter.p The base price for the product.pi The price for the product and delivery option bundle i.mi The mark-up for the product and delivery option bundle i.ci The quoted surcharge for delivery option i.ti The quoted lead time for delivery option i.θi The reliability of delivery option i.r The aggregated average rating for the seller.xAcknowledgementsFirst of all, I would like to extend my heartfelt gratitude to my thesis supervisorProfessor Tim Huh for his guidance and help during my thesis preparation. Withouthis support, I wouldn’t have had the chance to choose a suitable topic and carry outthe research. He is a knowledgeable and solid scholar and an encouraging role modelfor his students like me.I owe thanks to my committee member Professor Robin Lindsey for his detailedand constructive comments to improve this thesis.I would also like to thank Professor Hao Zhang for sparing his time to be myExternal Examiner and his insightful suggestions.Most importantly, I want to give special thanks to my parents for their generousfinancial and psychological support throughout my studies at UBC.xiDedicationTo my parentsxiiChapter 1Introduction1.1 BackgroundOnline retailing is a very important business area that creates much profit for thecompanies and many interesting topics for the researcher. The speed of developmentof online retailing, especially in emerging markets, shows the vitality of this specialbusiness segment. It not only requires the seller to provide products that have valuesto the customers, but also involves the delivery service to physically transport theproducts directly to the buyers, which is not needed in traditional retailing channel.Without either part, this business cannot exist.Online retailing also changes the dynamics of retailing industry and forces the tra-ditional channels to compete with it directly. The competition in price becomes fiercersince long lead time drives online retailers to decrease price to attract customers andtheir lower unit costs allow them to do so. Some firms develop dual channel system(Hua et al., 2010) to embrace chances in new businesses and address to the threatsmentioned above.The life of consumers has also been changed by this business. Customers can en-joy more choice variety and potentially lower price by shopping online, but a long1and unpredictable lead time is still a concern. Delivery service providers (DSPs) offerdifferent delivery options with various combinations of price, lead time, and reliabil-ity to satisfy different needs and preferences of the customers so that the sellers maychoose a few from available ones to display in their online shops and let the customersto choose from.Although online retailing has been developing for many years, it still has greatpotential that cannot be ignored in emerging markets as well as niche markets to beexplored in developed economies. The historical IPO (initial public offering) of Al-ibaba in the year of 2014 attracts the international spotlight as well as global staketo this rising e-commerce giant in China and the great Chinese online shopper pop-ulation that supports its success. Its accomplishment not only makes the shoppingtask of customers easier, but also creates many business opportunities for millions ofindependent sellers and big corporations to open their online shops with very lowcost. But those prospects also bring challenges to both the seller as well as the DSPssince the transportation becomes an important issue, especially for some productswith very high price (electronic devices) or very low price (second-hand items).This research studies two typical problems that would be faced by small indepen-dent sellers as well as big firms (all referred to as “sellers” hereafter) in such onlineretailing context: how to select appropriate DSPs for their prospective customers tochoose from, and how to price the product jointly with the assortment consideration.Since it is the seller, not the end customers, who is going to deal with those DSPsdirectly and pass the payment from customers to pay for the surcharge of deliveryservice, we would naturally ask the following question: can the seller make a profitfrom this transaction by listing a surcharge that is higher than the quoted one fromthe DSP? While it should be safe for us to assume that the seller normally enjoys suchfreedom, it is possible that there are certain constraints that prohibit him from doingso, such as contract with the DSPs or the regulation for the online platform on which2he is operating. We would refer to such ability to re-price the quoted surcharge asflexible pricing and examine the impact of pricing flexibility on the optimal decisionsof the seller.As we mentioned, one important distinction of online retailing is its service com-ponent. The intangible nature of service decides that it may not always be launchedas planned. Online rating or review serves as a very good information channel tohelp customers communicate and disseminate the messages. This platform enablesthem to share their experience and perspectives that are not restricted to service it-self. For example, on Taobao.com, customers have the chance to provide feedback onthe performance of the seller in four categories: Description (whether description isconsistent with actual product), Service (service attitude and manners of the seller),Logistics (how long it takes to ship the product after the order is confirmed), andDelivery Service (overall experience with the DSPs).1 When the rating is aggregatedto seller’s level, we might wonder whether such aggregation helps customers to makeinformed decisions. On the other hand, we can also ask whether it would help theseller to earn a higher profit since some unfavorable experiences reflected in bad rat-ings might be disguised by good ratings provided by those customers who actuallyenjoy or happen to be served with high quality service, and a higher average ratingin turn would encourage more customers to buy (Chevalier and Mayzlin, 2006). Sowe would like to inspect the impact of rating aggregation, too.To summarize, we study the optimal assortment and pricing decisions of the sell-er and the impact of pricing flexibility and customer rating in the online retailingcontext. Our research provides new insights to this kind of joint assortment andpricing problems to the research community and offers managerial implications forpractitioners who have to face and solve those problems in their businesses and fordelivery service providers who need to design their service plans. The setting and1Delivery service rating is not available for the customer to view and is currently only for “internaluses” for the platform.3results can also be applicable to other scenarios in which customers make purchasingdecisions regarding core good and side good, such as for the flight and services at theairport.1.2 Research QuestionsIn this research, we intend to provide answers to the following questions thatcan be categorized into two groups. The first three problems are concerned with thedecision maker — the seller:• Whether the seller shall include all the available options in his assortment? Ifnot, how should the decision be made?• What is the impact of seller’s pricing flexibility on his profitability?• How would the customer rating of delivery service quality impact the results?The second group of two questions aims at checking the other parties in the mar-ket:• Is the constant industry-level reliability standard a valid assumption? Is thereany incentive for the DSPs to deviate from the standard?• How does the involvement of the seller affect the DSPs and customers?1.3 Main Results and ContributionsThe main results of this thesis are:1. Generally speaking, “the more, the better” policy is good with and without thecardinality constraint on the assortment size and the seller should usually set aconstant mark-up for all the options selected in the assortment (if he can pricethe product only, then the base product price should equal to that mark-up).42. Aggregated customer rating would make the solutions mentioned above sub-optimal but if the rating is option-specific (equivalent to let the customers knowthe actual reliabilities), they become optimal again.3. When rating is aggregated, the seller needs to solve a combinatorial problemto get the optimal assortment, but the product price determination structureis still the same. We show that although ordering-based algorithm does notguarantee optimal assortment, two very simple heuristic algorithms that built onthat philosophy can provide very good performance. When rating is aggregated,the seller should differentiate the mark-ups of the options in his assortmentaccording to the reliability of each option, but the seller should still include allthe available options in his assortment.4. Pricing flexibility doesn’t necessarily translate into higher profitability and itwould only help when the information distortion is present, such as when therating is aggregated. It can help to improve from a suboptimal state to anoptimal state.5. When rating is aggregated and the seller can only price the product, the DSPs’choice of reliabilities could be tricky: if it’s too low then this option might be e-liminated, but if it’s too high then this option might get hurt in the competition.Then it’s expected that an industry-level reliability standard could be sustainedand no one is hurt or benefiting from that. When rating is not aggregated orwhen the seller has pricing flexibility, the DSPs should make joint decision onquote surcharge, quoted lead time, and reliability directly without consideringsuch coordination problem. If a DSP has necessary capabilities to deviate fromhis competitors under such circumstance, his deviation would bring him com-petitive advantage without hurting himself.6. There exists the possibility of collusion among the DSPs when certain DSPs facethe risk of being eliminated from the seller’s assortment. The collusion wouldlead to decreased option quality and decreased expected profit for the seller.5The seller may want to include all options in order to deter such collusion. Butwe also find that even if the seller decides to include all the options, such crisisis still not solved completely.7. The intervention of the seller generally decreases the business of the DSPs be-cause of double marginalization effect, but options with low reliabilities mayhave the chance to benefit from aggregated rating when the seller has no pric-ing flexibility. On the other hand, if the rating is aggregated, the customers’average mean utility is not always lower when the seller has pricing flexibilitiesand it would depend on the market characteristics.We have the following contributions:1. We propose a general framework to include customer rating, an important busi-ness factor, into the consideration of assortment and pricing decisions.2. We analyze the function of delivery services in determining product price inonline retailing.3. We study the assortment and pricing problems with the presence of customerrating serving as a mechanism to communicate credible information among thecustomers and examine the impact of seller’s pricing flexibility. We give theclosed-form pricing solutions to these two problems and find the way to selectthe optimal assortment.4. We show that when the rating is aggregated and the seller can only price theproduct, the ordering-based policy would lead to sub-optimal solutions evenwhen there is no cardinality constraint but the average performance of algo-rithms built on such philosophy may not be bad.5. We prove in our setting that information aggregation cannot make the sellerbetter off if such information is user-generated and the seller cannot manipulateit directly, and we also show that this tool can backfire and hurt the seller if hedoesn’t have the flexibility to re-price the quoted surcharges of delivery options.66. We identify the role of pricing flexibility in helping to remove information dis-tortion and reveal the true information to the customers.7. We examine the industry-level reliability standard assumption from another per-spective.8. We discover an interesting area that should receive more attention and attractmore research efforts: the interactions among the DSPs as well as the interac-tions between DSPs and the seller considering endogenous decision structure.1.4 Organization of the ThesisIn Chapter 2, we review a few important literatures related to this research topic.The succeeding four chapters constitute the main part of this thesis and wouldcover four related problems:Table 1.1: Structure of the four problems in this thesisPricing flexibilityRating availabilityNo rating Rating aboutdelivery service qualityPricing the product only Problem A: Chapter 3 Problem C: Chapter 5Pricing the product and delivery options Problem B: Chapter 4 Problem D: Chapter 6The first two problems mainly serve as benchmarks to compare against. In Chap-ter 3, we consider the situation when the seller can only price the product and thereis no customer rating. We prove that it’s optimal to include all the available deliveryoptions in the assortment and present the closed-form optimal pricing function. InChapter 4, we assume the seller is granted with the flexibility to re-price the deliv-ery options, and we show that after decision variable transformation, this problem isan example of previously solved problem. We provide a new and simple method toprove that “the more, the better” assortment policy applies before obtaining structural7property of pricing decision. The next two problems are the main problems in thisthesis. In Chapter 5, we take into account the influence of aggregated customer ratingon the basis of Chapter 3 and show that the pricing function would have the samestructure but we need to solve a combinatorial optimization problem in order to findthe optimal assortment. Two heuristic algorithms are proposed and their performanceis tested and compared. In Chapter 6, we combine the two features and look at theproblem in which the seller has the flexibility in pricing and the aggregated customerrating is available. We show that to include all the options in the assortment is stilloptimal but the optimal mark-ups should be adjusted by option reliabilities now andare generally not constant.In Chapter 7, we present numerical studies to compare the results in Problem Cand D together with a hypothetical scenario in which there is no intervention of theseller, and check the impact of parameters on the DSPs and the customers as wellas the seller. In Chapter 8, firstly we present the conclusions and elaborate on theimplications, and then we discuss some possible approaches for future research.8Chapter 2Literature ReviewThis study is directly related with three areas of research: the first part helps usunderstand how firms make pricing and lead time decisions according to the charac-teristics of the firm and the market conditions, the second part introduces the choicemodel we are going to use and reviews a few related literatures in assortment andpricing, and the last part reviews some studies on service quality and customer rating.2.1 Interaction between Price and Lead TimeThere are plenty of studies in price-sensitive market and we are interested in thestream of literatures that investigate the firms’ decision when facing with price andtime sensitive markets.Among one of the first studies in the interaction between lead time and pricing, Soand Song (1998) examine how a firm should jointly consider pricing, guaranteed leadtime and capacity expansion decisions to achieve maximal average profit. They findthat increased capacity should lead to shorter quoted lead time but the decision onpricing could vary, when higher service level is expected the firm should quote longerlead time while reduce price, and guaranteed lead time decision is more delicate forhigh-cost firms than for low-cost firms since if the lead time decision is suboptimal,9the profit loss for such firms are higher. Another important discovery is that the ser-vice level constraint should always be binding at optimality, which is confirmed inother later studies.Looking at a similar problem, Palaka et al. (1998) take into account the effect ofcongestion on firm cost. They find that there exist conditions that would allow a firmto choose a service level that is higher than industry level, which is triggered by thefirm’s profit maximization interests, instead of goodwill towards consumers.Unlike other studies in which price and lead time are treated as separate and in-dependent decision variables, Ray and Jewkes (2004) provide a new perspective byexplicitly enforcing the market price to be determined by the lead time decision di-rectly (higher lead time leads to lower price: p = d− eL, where p is price, L is leadtime, and d and e are constants) and examine the impact of such changes comparedwith independent decisions case. They identify some conditions under which the ig-norance of such explicit relationship would lead to negative impact on the firm andshow their implications are consistent with actual situations.In previous three papers, capacity is a joint decision with lead time, but So (2000)argues that the capacity is usually a long-term strategic decision and cannot be changedeasily so that it should be treated as a characteristic, rather than a decision variable inthe short run, but it still allows for sensitivity analysis.It’s noted that the capacity expansion cost and the unit operation cost are twoimportant cost factors that have been taken into consideration to reflect two specificcharacteristics of the firms and help us understand the impact of different capabilitiesof the firms on their decisions. The former would influence the capacity decision di-rectly while the latter influences the financial performance. As an important factor inthe industry, the economies of scale is also considered in some papers to acknowledgethe benefits of large volume of demand (e.g., Cachon and Harker 2002 and Ray and10Jewkes 2004).To reflect the impact of demand on the service time which then impacts the reli-ability of the guaranteed lead time, queueing model is often utilized. However, wenote that there are also other possible ways to model this effect. For example, Wuet al. (2012) propose an alternative model using multiplicative variables. They look ata newsvendor problem in which the firm decides quoted lead time, price, as well asstocking level before the selling season.While the majority of the literatures consider the monopoly case, a few othersshed lights on alternative market structures and interactions. A duopoly time-basedcompetition situation is studied by Li and Lee (1994). They derive the demand func-tion from a queuing system with competing servers and show that when firms chooseprices to compete, the firm with higher capacity can charge a higher price and wouldeven enjoy a higher market share when both firms have adequate capacity.Liu et al. (2007) consider a Stackelberg game framework for a two-level decentral-ized supply chain in which the supplier leads and the retailer follows. Comparingwith a centralized benchmark case, this paper highlights the importance of improvinginternal operational efficiency before pursuing options such as coordinating to weak-en the impact of double marginalization effect.More recently, Zhu (2015) adopts a similar structure in which the supplier decidesthe capacity and the wholesale price, while the retailer chooses the retail price andthe quoted lead time, which is different from the setting in Liu et al. (2007) where thesupplier decides the quote lead time. The author finds that the supplier’s insufficientcapacity would lead to increased retail price and longer lead time on the retailer’sside, and shows that inclusion of capacity decision would reduce profit loss that re-sults from double marginalization, compared with a benchmark case without suchdecision. He then proposes a franchise contract to coordinate the supply chain since11revenue-sharing and two-part tariff contracts fail to do so.In a similar two-level supply chain setting, Hua et al. (2010) compare the optimaldecision of a supplier (manufacturer) and a retailer in a centralized and a decentral-ized dual-channel retailing system, and provide new insights on the role that deliverylead time plays in influencing pricing decision of the manufacturer and the retailer inboth cases.So (2000) looks at oligopoly competition in the market that has identical servicereliability (industry standard) for all the firms, derives the Nash equilibrium when theparticipating firms are identical, and provides some insights in the non-identical firmscase through numerical experiments on two competing companies with different char-acteristics. He finds that: a firm with higher capacities offers better time guaranteewhile a firm with lower operation cost charges lower price; when the market is moreprice-sensitive, the price gap narrows and the gap in time guarantee would increase;when the market is more time-sensitive, both firms would increase their prices andthe gap in prices increases, but the gap in time decreases. In addition, a low-cost firmbenefits more and more from increasing price-attraction factor while a high-capacityfirm benefits from increasing time-attraction factor in the same manner.While the major research efforts have been devoted to the study of one product,Boyaci and Ray (2003) analyze a problem with two substitutive products. While thelead time of the regular product is fixed by industry standard, the firm can decide thelead time for the express product as well as the prices for both products. The impactof capacity constraint and cost on the product differentiation strategy is examined.As we can see, linear demand model is still a popular choice in this area of stud-ies, but we note that So (2000) and Ho and Zheng (2004) distinguish themselves byassuming alternative probabilistic choice and demand models. The former researchemploys a multiplicative competitive interaction (MCI) model to represent the mar-12ket demand, and the latter one utilizes a multinomial logit (MNL) model to studythe interaction between quoted lead time and delivery service. However, both papersassume the firms can capture all the possible demands in the market and thus theno-purchase behavior is not considered. We will spend more time talking about logitmodels in the second part of the literature review.We would like to note that the majority of the studies assume there is an industry-level standard or internal performance target for delivery reliability that is consideredas fixed even in competition settings and relatively less effort has been exerted intounderstanding the competitive role of it. To try to provide some insights regardingthis problem is also one of the goals of this thesis.The table below summarizes some of the important comparable characteristics ofthe literatures mentioned in this section. “Price” and “Time” indicate whether suchdecision is considered explicitly. If there is more than one decision maker, then thetable cells indicate who make that decision. 2Table 2.1: Comparison of literatures in section 2.1Paper Competition/Game Price Time Joint decision ReliabilityDemand impacton service Demand Model(Boyaci and Ray, 2003) No Yes Yes1 Capacity Internal2 Queue Linear(Hua et al., 2010) Stackelberg Both3 Supplier N/A N/A No Linear(Li and Lee, 1994) Duopoly Yes No N/A N/A Queue Derived4(Liu et al., 2007) Stackelberg Both Supplier Capacity N/A Combinational5 Linear(Palaka et al., 1998) No Yes Yes Capacity Industry6 Queue Linear(Ray and Jewkes, 2004) No No Yes Capacity Internal Queue Linear(So, 2000) Oligopoly Yes Yes N/A Industry Queue MCI(So and Song, 1998) No Yes Yes Capacity Internal Queue Cobb-Douglas(Wu et al., 2012) No Yes Yes Inventory N/A Multiplicative Random7(Zhu, 2015) Stackelberg Both Retailer Capacity N/A Queue Linear2Notes for the table: 1. For express product only, the other (regular product) is set by industry; 2.Identical for two products; 3. For decentralized case only, in the centralized case all decisions are madeby the supplier; 4. Indirect utility is linear; 5. See (15) on Page 718 for details; 6. Minimal level is set byindustry but the firm can decide one that is higher than that; 7. Expected demand is linear.132.2 Assortment and Pricing in MNL Model2.2.1 MNL ModelCustomers’ demand and firm’s business strategy interact with each other, so un-derstanding the mechanism behind customers’ choice is one of the fundamental tasksfor the firm to make optimal decisions. Many models have been proposed and devel-oped to reflect customers’ decision.MNL model is one of such important approaches. After its introduction, re-searchers contributed a lot of theoretical as well as empirical work using it. Andersonet al. (1992) provide a detailed background of MNL model and Train (2003) also con-tributes some discussion and comparison with other related choice models.MNL model and its variants are widely utilized in research for transportation (e.g.,McFadden 1974, Williams and Ortu´zar 1982, and Ben-Akiva and Bierlaire 1999), mar-keting (e.g., Cachon et al. 2008), and operations management (e.g., Dong et al. 2009).The usage of the MNL model is not new in the study of delivery service area andHo and Zheng (2004) is just an example. The Multiplicative competitive interaction(MCI) model adopted in So (2000), which we’ve talked about in the first part of theliterature review, also shares some similar characteristics. In fact, in a recent compre-hensive review on those important demand functions, Huang et al. (2013) point outthat MNL model is one of few popular functional forms that can incorporate all thesix categories of factors that are commonly considered: price, rebate, lead time, space,quality, and advertising.One often mentioned disadvantage of MNL model is the independence of irrele-vant alternatives (IIA). It means that the ratio of probabilities for any two alternativesis independent from alternatives other than themselves. A famous illustration is thered-bus-blue-bus example, and we refer to Train (2003) for more discussion on this.However, in our setting this problem might be less important because the delivery op-14tions are essentially the same “product” that only varies on some specific attributes,so we are not letting the customers to compare “bus” against “car”, and the compar-ison is restricted to one between “bus” and another “bus”.To overcome such imperfection that leads to problems in some scenarios, a nestedlogit model (McFadden, 1980) is proposed to group all similar items in one nest whileallowing multiple nests that are somewhat different from each other. The standardMNL model is just a special case of the nested model in which there is only one nestand its dissimilarity parameter is 1.On the other hand, MNL model is suitable in the online retailing environment be-cause the total market size is considered fixed and not impacted by the price decision.We consider the fact that online shopping for a specific product is only appealing fora certain group of customers who constitute our prospective customer base and payspecial attention to the market condition. In contrast, those who don’t have interestsin the product or online shopping at all are generally not aware of changes in priceand sometimes may even not be able to have access to such information. That’s to say,the market size may not expand just because the price of the product is reduced.Throughout the thesis, we assume that the model parameters can be estimatedfrom the market information and we would not deal with the parameter estimationin this research. For the estimation methods, we would recommend Train (2003) andHess and Train (2011).The assortment and pricing problems are usually two distinctive domains and wewould first review them separately in the following two parts and then look at thestudies that examine joint decision problems.152.2.2 Assortment in Logit ModelsIn assortment problems, we often assume that the revenue of each available item isfixed and the problem for the decision maker is to select a subset of items from the setof all alternatives in order to achieve the maximal expected revenue. Ko¨k et al. (2015)provide a comprehensive review on assortment planning in research and practice.Talluri and van Ryzin (2004) and Liu and van Ryzin (2008) show that under thestandard MNL model, the optimal assortment can be found by revenue-ordered pol-icy: order all available items by their revenues from highest to lowest and there is acritical item i∗ such that all items before it should be selected in the assortment.Rusmevichientong et al. (2010) consider a static capacitated assortment (assort-ment under cardinality constraint) optimization problem in which the customer pref-erence is known and a dynamic problem where such preference is unknown and hasto be estimated from past data. In the former problem they show that a greedy pol-icy may lead to suboptimal solution and derive some structural properties that helpthem to propose an efficient search algorithm. In the latter problem they develop anadaptive method to optimize the problem while estimating the customer preferencefrom past sales and assortment decisions.In Rusmevichientong and Topaloglu (2012), the authors consider the robust op-timization problem with the objective of maximizing the worst-case revenue over anuncertainty set of customer preference. They show that the revenue-ordered assort-ments are still robust. A dynamic problem to allocate the capacity over time is con-sidered in addition to the static problem, but the uncertainty set is not refined whenobserving customers’ decision. They also discuss how to construct the uncertainty setto balance between the worst-case and average revenue.More recently, Rusmevichientong et al. (2014) study the optimization problem16with random parameters under mixed logit model, show an example in which therevenue-ordered assortment method leads to non-optimal choice, and prove that eventhe two-class logit scenario is NP hard. They first identify two cases in which therevenue-ordered assortment method is still valid for optimal solution and then derivetight approximation guarantee for that method in the more general case. The mod-el then is extended to multi-period capacity allocation setting. By using numericalanalysis on two sets of test problems, they also find that the revenue-ordered methodtends to work well. Feldman and Topaloglu (2015) develop an approach that canquickly obtain quite tight upper bounds on optimal expected revenue. Kunnumkal(2015) compares different approximation methods utilized to get the upper bound ofexpected revenue under mixture of logit model and proposes an alternative method.Under a variant of the Nested Logit (NL) model, Davis et al. (2014) show thatthe competitive products and fully captured nests (customers would not leave a nestafter choosing it, so no-purchase behavior is not considered within a nest) is a specialcase in which the solution can be obtained in polynomial time while the relaxationon either of the two conditions makes this problem NP hard. Motivated by thosemore general cases, they develop collections of assortments with various worst-caseperformance guarantees and formulate a tractable convex problem that gives an up-per bound on the optimal expected revenue and provides a benchmark to compareagainst.2.2.3 Pricing in Logit ModelsIn pricing problems, we often assume that we know the cost of items in a set ofalternative items and the decision marker shall price all of items in order to attain thehighest expected profit.In their seminal paper, Hanson and Martin (1996) find that under MNL model,the profit function is not concave in the price vector. After that, there are mainly twoapproaches to handle this problem.17One approach is to transform decision variables. Dong et al. (2009) and Song andXue (2007) show that profit function under standard MNL model is actually concavein the market share (or the probability of choosing each item) vector. However, bothworks consider identical price-sensitivity parameter among all the products whileSong and Xue (2007) restrict such parameter to be in the unit interval.Motivated by these facts, Li and Huh (2011) take into account the variant price-sensitivity parameters and extend the concavity property to the NL model. Afterderiving the optimal solutions in the monopoly condition, they use the results toexplore the oligopoly market in the Bertrand competition and in the Cournot com-petition. Closed-form analytical solutions are obtained by using Lambert W functionand a modified Lambert W function. They find competition leads to lower price whilethe price under price competition is the lowest case.The other approach is to check unimodality property directly. Akc¸ay et al. (2010)show that in the MNL model the profit function is unimodal in prices and thus theFirst-Order-Condition is sufficient to obtain the global optima.Looking at the pricing problem in a NL model (same as in Li and Huh 2011),Gallego and Wang (2014) show that the market share transformation cannot guaranteethe concavity property of the profit function when the price sensitivity parameters forthe products within a nest are differentiated. Also relaxing the unit interval constrainton the dissimilarity parameter among the nests, the authors establish an adjustedmarkup and prove that it’s constant for all products within a nest while an adjustednest-level markup is constant for all nests at optimality so that the complexity of theproblem is greatly reduced in the nested logit model as well as its two variants. Ifcertain conditions are met, the profit function is shown to be unimodal in the nestinvariant adjusted nest-level mark-up. The optimal adjusted nest-level mark-up canstill be bounded in an interval even if such conditions are violated and the profit18function is not unimodal. The results are used to consider the oligopolistic pricecompetition and study the Nash equilibrium.2.2.4 Joint Assortment and Pricing ProblemIn this part, we will review the stream of research that studies the joint assortmentand pricing problem.It is noted that sometimes the assortment decision can be incorporated into thepricing decision when modelling the problem and analyzing the model, because onecan always price one item to infinity if that item should be excluded from the assort-ment. However, the decision maker may not always enjoy such flexibility because ofsome constraints that prohibit him from doing so. Therefore, the assortment problemin the joint decision still needs careful consideration under such circumstances.Aydin and Ryan (2000) is one of the first studies that find out that in MNL model,the optimal mark-ups of the items in one assortment should be equal when makingpricing decision, and that the decision maker should include as many items in theassortment as allowed and the first K items with highest quality-cost gap should beselected if only K items are allowed when choosing appropriate assortment.Hopp and Xu (2005) extend the constant positive mark-up and “including mostitems” property to the Bayesian logit model (Go¨nu¨l and Srinivasan, 1993) with risk-sensitive decision maker. Comparative statics are also provided to show how themark-up and optimal assortment size are influenced.Maddah and Bish (2007) study the joint assortment, pricing, and inventory deci-sion problem in which consumer choice is modeled by MNL and inventory setting isnewsvendor. They first derive some structure for the optimal assortment and find thatwhen the total market size is large enough the optimal mark-up for all items shouldbe equal. After the discussion about the properties of optimal prices, they propose an19effective heuristic solution.A recent work that’s very similar to our research is Wang (2012) who considers acapacitated problem in the MNL model in which the decision maker can only offerup to K items in the assortment. By establishing the property that connects the op-timal mark-ups and the optimal profit, the author shows that the optimal size of theassortment should always equal to K and an algorithm is developed to find such Kitems.To understand the role of assortment and pricing in a competitive setting, Besbesand Saure´ (2015) look at the equilibrium in assortment-only and joint assortment-and-pricing competition in a duopoly market in which the consumer choice is modeled byMNL model and highlight the impact of set of available products to the two retailers.It has been shown that when they provide exclusive products there always exists apareto-optimal equilibrium while when they have access to common products as wellas exclusive products, a pure-strategy Nash equilibrium is guaranteed only if there isno cardinality constraint on the assortment.Rayfield et al. (2015) impose price bounds to each products in the pricing problemin the NL model and study two pricing tasks: the first fixes the assortment while thesecond determines the joint assortment and pricing plan. The authors show that theunimodality property found in Gallego and Wang (2014) is lost with the presence ofprice boundaries but provide approximating methods that can produce an expectedprofit that deviates from the optimal profit by a factor of specified precision in theworst case.Also as a side note to this part, Li et al. (2015) look at the assortment and pricingproblems in a NL model with d levels but the two problems are considered as sepa-rate instead of a joint decision. The authors propose efficient algorithms to find theoptimal assortment and prices.20To summarize, the research mainly focuses on two kinds of logit models: MN-L and NL, while a group of researchers are also exploring diverse variants of logitmodels. After some main results have been established, people start to look at moregeneral cases by adding constraints and investigating more into details to completeprevious research. In assortment problem a usual restriction is the cardinality con-straint that requires the size of the assortment doesn’t exceed a certain number andin pricing problem attention has been drawn to the price boundaries for the product-s. On the other hand, researchers also highlight the impact of product-differentiatedprice sensitivities, nest dissimilarity parameter, and within-nest no-purchase behaviorin NL model.These works are only similar to Problem B in this thesis and we note it may notbe always true that Problem B can be transformed into a problem that can be solvedby the methods introduced in those literatures. We differ with them in the followingways:1. We use a very simple alternative method to prove that even before any structuralproperty of the optimal pricing policy is obtained, the solution to our assortmentproblem is ”the more the better” with or without cardinality constraint in Prob-lem A and Problem B.2. Closed-form solutions are obtained in Problem A and Problem B to help us tohave a better understanding of the optimal assortment policy in the capacitatedcase: a simple ranking policy would suffice to solve the problem and the methodis very easy to implement.3. Most importantly, according to our knowledge, nobody has studied the impactof customer rating and pricing flexibility in this structure.4. We obtain closed-form solution of optimal pricing strategies when the rating isaggregated and the seller has pricing flexibility. We show that the mark-upsshould be differentiated and they might be negative for some options.215. We further prove the unimodality property of the profit function with respect tothe reliability-adjusted mark-up in Problem D.2.3 Service Quality and Customer RatingThirdly, we will briefly review some literatures related to service quality and cus-tomer rating.2.3.1 Delivery Service Quality and Quality PerceptionService quality is an antecedent of consumer satisfaction, which then has a sig-nificant effect on purchase intentions (Cronin Jr and Taylor, 1992). So to understandhow service quality is delivered to and perceived by customers is very important tothe maintenance and growth of business.Many studies show that the delivery service is one of the most important criteriafor customers’ assessment of overall service quality in the online retailing context.Yang and Jun (2008) find that for the “Internet Purchasers”, the most importantquality dimension is reliability, in which the on-time delivery is a criterion. This en-sures us that on-time delivery is indeed a very important aspect for the overall qualityfor the group of prospective customers we are interested in.Heim and Field (2007) empirically examine the relationships between e-serviceprocess and quality dimensions. They find that the breadth of delivery modes does-n’t seem to affect the customer ratings while individual carriers present significanteffect — some exhibit negative effect while others exhibit positive effect.Focusing specifically on logistics-related operational factors in the e-commercecontext, Ramanathan (2010) shows that there is strong moderating effect of efficiency,which is defined as the “ability to achieve good ratings in terms of operational fac-22tors and also in terms of customer loyalty” and assessed by DEA (Data EnvelopmentAnalysis), on the relationship between on-time delivery and intention of shoppingwith the merchant again, while on-time delivery alone is not significant in explainingcustomer loyalty.Rao et al. (2014) discover that the consistency between promised delivery time,especially promises of expedited delivery, and actual delivery performance would in-fluence the product return chance, which reflects the impact of service on perceivedquality and experience from another perspective.There are also researches who analyze the relationship between different stagesin customers’ online shopping experience. Cho (2014) looks at the relationship be-tween customer price perception and customer perception of Internet retailer’s ser-vice quality in three different phases in online shopping experience and finds out thatcustomers’ positive price perception helps to achieve positive perception of order-fulfilment quality. This confirms Cao and Zhao (2004)’s finding that when the cus-tomers perceive the price to be low, they would be more tolerant to longer deliverytime.Ho and Zheng (2004) propose a model to incorporate delivery service qualityconsideration into the process of determining quoted lead time and the quality hereis defined as the conformance of the perceived delivery time to the expected deliverytime (the ratio of deliveries that take no more than the quoted time). They investigatethe tradeoff between a more alluring commitment (short quoted lead time) that wouldattract more customers and a secured commitment (long quoted lead time) that wouldensure excellent service. However, we note there is a possible missing logic link in themodel: there lacks a concrete mechanism to facilitate the disseminating of such qualityinformation to the customers. The customers cannot know the actual performance inadvance (at the time of choice) because they have no way to know the overall marketcondition and thus we may say they have no information for the actual service quality.23We suggest that online rating is a natural choice that would serve this role and fill thisgap.2.3.2 Rating Behavior and Rating DynamicsThe second stream of literatures investigate the behavioral aspects and chronolog-ical dynamics of customer/user rating.Moe and Schweidel (2012) look at how previous ratings would influence subse-quent customers’ decision at the individual level on whether they should post theirrating and what rating they should provide. The important influence of the rating en-vironment (previous ratings) and composition of customer base (customer type andunderlying distribution of evaluation) is identified in their empirical research andsubsequent simulation study.The temporal dynamics of rating also receives attention from the researchers.Godes and Silva (2012) study the evolution of online rating for books and observethat on average the ratings decline in both time (the book’s available time on the mar-ket) and order (the sequence of the ratings itself). They recognize two simultaneousdynamics processes regarding time and order. In the former one they find that theaverage pattern is actually increasing after controlling for calendar trend, rather thandecreasing suggested in existing findings, and in the later one they find that a ratingprovided later is on average lower than one provided earlier.Zhao et al. (2013) propose a structural model using Bayesian learning frameworkto study how the subsequent purchase of experiential products is influenced by prod-uct reviews whose credibility is indicated by how precise the product reviews reflectcustomers’ own evaluation. Their empirical study finds that the customers learn morefrom customer reviews than from their own past experience with other products ofthe same type.24To understand the impact of aggregated information on individual decision mak-ing, Muchnik et al. (2013) design a randomized experiment on a social news aggregatewebsite and show that manipulated initial rating result in topic-dependent asymmet-ric herding effects: negative influence would inspire users to correct the manipulatedrating while positive influence increases the likelihood of subsequent positive ratingas well as final rating on average.In our research, we would not focus on the influence of sequential effect and otherinternal dynamics or interactions among the ratings. While adding those elementsmay help to make the setting more realistic, it may also mask the effects of individualfactors and make it harder to isolate the impact of different strengths. We want tostart with the most basic mechanism that emphasizes on rating’s fundamental role ofinformation dissemination and later we may add other components to complete theresearch.2.3.3 Impact of Rating and Firm InterventionChevalier and Mayzlin (2006) find that there is empirical evidence to support apositive relationship between average rating and sales, the magnitude of impact forone-star reviews is greater than that for five-star reviews on the customers, and cus-tomers do not rely merely on the average star ranking.While usually the average of the rating is the focus, Sun (2012) looks at how vari-ance of rating would influence sales and concludes from her empirical studies ofAmazon.com and BN.com that when average rating is low, a high-variance ratingwould improve the sales of a product.Through empirical study on transaction level data before and after a policy changein rating filtering in a large online retailer, Awad and Zhang (2007) compare the im-pact of two filtering strategies: one aims at removing negative reviews (before) whilethe other aims at noise reduction (after) and find that before the change, only posi-25tive reviews are positively associated with transaction amount but after the change,both positive and negative reviews are positively associated with transaction amoun-t. They also confirm the finding in Chevalier and Mayzlin (2006) regarding positivecorrelation between average rating and online purchase.Kuksov and Xie (2010) study how a firm should properly utilize different combi-nations of pricing and frill (“extras that are valued by consumers but not consideredby consumers or promised to them at the time of purchase”) tools in order to maxi-mize firm profit under different time horizon and different price history observabil-ity when the consumers can exchange information through a binary rating system.Market growth rate is identified as one important factor that determines the optimalcombination under different circumstances. The authors also find out that when con-sumers are heterogeneous, larger idiosyncratic part of consumer uncertainty wouldinduce higher firm’s interests in affecting the rating through the combination of priceand frill.On the other hand, some papers study firms’ involvement in online forum as partof their marketing campaign in a competitive game model.Mayzlin (2006) models the “promotional chats” (sending out messages to promotetheir products regardless of the true quality) of two quality-differentiated firms andlooks at a scenario in which the uninformed customer has to learn the quality informa-tion from online messages provided by the two firms as well as the informed customer(expert). She finds that there exists a unique equilibrium in which the high-qualityfirm always enjoys higher profit but under three different conditions the manipula-tion combinations could be varied: both firms manipulate while the low-quality firmexerts more effort; only low-quality firm manipulates; neither firm manipulates.Looking at both monopoly and oligopoly settings, Dellarocas (2006) models firms’manipulation (such as posting anonymous reviews that praise their product) in on-26line forums and checks the impact on informativeness from an information systemperspective. He shows that in some settings such manipulation is beneficial to theconsumers. He also characterizes a “rat race” in which the firms have to invest in ma-nipulations only to avoid bias against them since the customers would expect firms’manipulations, even if such actions are profit-reducing. It is also shown that firmshave the most to gain from developing technologies to make manipulations more d-ifficult for firms when the volume and quality of user-generated content increases.To sum up, we may see there are roughly four main subareas in the rating litera-tures and they study: why and how customers provide rating, whether and how therating influence rating (dynamics and evolution), what and how customers learn fromprevious rating, and how should the firms’ strategies interact with customer ratingwith and without competition considerations.In our thesis, we would not look at the seller’s direct rating manipulation action,as we want to see first the explicit role of rating’s information dissemination mecha-nism and then we may check later regarding how the seller’s involvement can impacthis own profit as well as customer welfare.However, we would compare the differences between two distinct rating struc-tures: option-specific rating and aggregated rating. We rarely see option-specific rat-ing for each delivery option and behind it there could be many reasons such as highimplementation cost or simplicity consideration for user interface design. We wouldwant to contrast these two structures to understand what would impact the seller’soptimal decision when such structure is a solid constraint, and whether and how aseller may benefit from it when such structure is an optional tool.The aggregated rating case is studied directly in Problem C and Problem D whilethe option-specific rating case can be seen as a variant of Problem A and Problem Band the results can be derived directly from the solution to those two problems. The27results are compared in Chapter 6.28Chapter 3A: Pricing Product withoutCustomer Rating3.1 Model FormulationWe consider a market in which a monopolistic seller sells a standard product to agroup of homogeneous prospective customers via the Internet. The seller possessesabundant supply of products but has to use the delivery service from the third partyin order to transport the product to the customers, who pay for such service via theseller. The DSPs offer a set of delivery options with various quoted lead times andsurcharges to the seller. Faced with these available options, the seller selects an as-sortment from them for the customers to choose from and then decides the price forthe product. We assume those options are exogenously given and there is no directinteraction among the DSPs.While shopping online, customers make the purchasing decision according toMNL model by taking into account the value of product, the base price of the productas well as the quoted surcharge and quoted lead time of each delivery option. Eachcustomer either decides to buy the product and chooses one delivery option from thepre-selected assortment, or leaves without buying.29While some studies discuss the late penalty that is implemented to compensatethe customers when the actual delivery takes longer than the promised in the deliv-ery service literatures (e.g., Palaka et al. 1998, Liu et al. 2007, and Zhu 2015), we willnot consider it explicitly in our research. The main rationale behind is that such prac-tice is different from real online retailing markets. On the other hand, the seller is nothandling the logistics and he is in a vulnerable position to be held responsible for latedeliveries, so even if the compensation is indeed necessary, it should be provided bythe DSPs, instead of the seller. So we will simply assume that the customers do nothave any expectation on such compensation when making purchase decision, and theseller is not involved in any potential negotiation between the customers and DSPsafter late deliveries.The decision problem for the seller is to select the optimal assortment of deliveryoptions and to decide the optimal base price for the product, in order to maximizethe expected profit from all the customers.3.1.1 Customers’ ChoiceFacing with an assortment S of delivery options selected by the seller, customerschoose among them according to random utility maximization. Specifically, eachcustomer has utility Ui for each option i ∈ S (including the product) and it can berepresented asUi = ui + ε i (3.1)and ε i is a random variable following a standard Gumbel distribution with locationparameter 0 and scale parameter 1. We refer to ui as the mean utility associated withoption i.We also consider the no-purchase situation in which the customer doesn’t buy theproduct and we normalize the no-purchase utility to zero.30Following the theory in standard MNL model, a customer chooses option i withthe probabilityλi =eui1 +∑j∈S euj . (3.2)We model customers’ mean utility associated with each delivery option i (includ-ing the product) asui = α− β(p + ci)− γti, (3.3)in which p is the base price of the product, ci is the quoted surcharge for deliveryoption i, and ti is the quoted lead time for delivery option i. For the model param-eters, α is the value of the product plus the delivery service needed to transport theitem from the seller to the buyer (for simplicity, we would refer to α as the “quality”or the “value” of the product), β is the customers’ sensitivity for price, and γ is thecustomers’ sensitivity for lead time. All the parameters are positive.Note that we are not differentiating the customer’s sensitivity for the product priceand the surcharge for delivery option here. The main reason is that on platforms suchas Taobao.com, the information for delivery options might be displayed together withproduct description and the customers can acquire necessary information to makedecision at the same time. There are studies that find the customers may responddifferently to item price and other price elements (e.g., Smith and Brynjolfsson 2001).However, our analysis for this problem doesn’t require identical price sensitivity andour method can easily be extended to the situation of differentiated price sensitivitiesas long as the linearity form is kept, and the structure of the solution still holds.3.1.2 Seller’s ProblemIn order to maximize the expected profit, the seller chooses an assortment of de-livery options S from the set of all available alternatives A for the customers to choosefrom, and then decides the base price for the product. The objective of the seller can31be written as3maxS,p ∑i∈S pλi. (3.4)In our problem, we don’t have constraint on “capacity”, or the number of optionswe can provide in the assortment, simply because there is no extra physical costfor providing more options. One might argue that only the first few options listedwould actually catch the attention of the customers and thus impact their decisions,but in practice the delivery options are usually displayed all together in a relativelysmall area to make it convenient for the customers to compare and we cannot ruleout the customers’ interests and willingness to make comparison against each option.On the other hand, the impact of display order is beyond the scope of this research.Therefore, it’s hard to impose an arbitrary restriction on the number of options in theassortment.3.2 AnalysisIn this section, we first look at the optimal assortment decision and then study thepricing decision.3.2.1 Assortment DecisionWhen the seller looks at A, he may find there are some “unfavorable” optionswith either high surcharge or long quoted lead time, or even “dominated” optionswith high surcharge as well as long quoted lead time. We are most concerned withthe question about whether he should include such options in the assortment S be-cause he may want to divert more demand flow to those favorable options with lowersurcharge or shorter lead time in order to achieve a higher expected profit.Counter-intuitively, we can show in the following theorem that the seller actuallyshould include all the available options in his assortment.3Without loss of generality, the cost of the product is normalized to be zero and the total market sizeis normalized to be one.32Theorem 3.1. The seller should always include all the available delivery options in the as-sortment, i.e. he should make S = A.Proof. We begin by assuming the seller can achieve some profit pi0 by finding an as-sortment S0 ⊂ A and corresponding price p0. Then we add another option k ∈ A \ S0to form a new assortment S = S0 ∪ {k}. We can prove that even if we keep the priceunchanged, we still can improve the expected profit.For the simplicity of notation, we first denote vi ≡ eα−β(p+ci)−γti . Then we canwrite the optimal profit aspi0 = ∑i∈S0p0λ0i = ∑i∈S0p0 vi1 +∑j∈S0 vj . (3.5)After we bring in the option k and form a new assortment S, if we keep the pricep0 unchanged, the new expected profit pi would bepi = ∑i∈Sp0λi = p0λk + ∑i∈S0p0λi= p0 vk1 +∑j∈S vj + ∑i∈S0 p0 vi1 +∑j∈S vj= p0 vk1 + vk +∑j∈S0 vj + ∑i∈S0 p0 vi1 + vk +∑j∈S0 vj= p0 vk1 + vk +∑j∈S0 vj +1 +∑j∈S0 vj1 + vk +∑j∈S0 vj ∑i∈S0 p0 vi1 +∑j∈S0 vj= p0 vk1 + vk +∑j∈S0 vj +1 +∑j∈S0 vj1 + vk +∑j∈S0 vj pi0.(3.6)in which the last equality follows (3.5).According to (3.5) and (3.6), we havepi − pi0 = p0 vk1 + vk +∑j∈S0 vj +1 +∑j∈S0 vj1 + vk +∑j∈S0 vj pi0 − pi0= p0 vk1 + vk +∑j∈S0 vj +−vk1 + vk +∑j∈S0 vj pi0= vk1 + vk +∑j∈S0 vj (p0 − pi0).33As it’s clear that vk1+vk+∑j∈S0 vj > 0 and the optimal expected profit cannot be higherthan the price (pi0 < p0) because some prospective customers would not buy, wewould have pi > pi0. That’s to say, by adding another option in the assortment, theseller can surely increase the expected profit. Therefore, he should include as manydelivery options as possible.This finding is constant with those in Aydin and Ryan (2000), Hopp and Xu (2005),and Wang (2012), in which the decision maker is assumed to have the authority toprice every item (product). We demonstrate in this theorem that even if the sellercan only price the product, he still should include as many options as possible inhis assortment. Furthermore, we show it before we obtain any structural property ofoptimal pricing decisions.It also confirms the famous “long-tail” effect in e-commerce. Since the seller does-n’t have cost for listing more options, providing varied choices would help to acquirecustomers with different preferences and capture niche markets. We show that it isalso true for deciding the assortment of delivery services to provide, in addition tothe selection of products to sell.3.2.2 Pricing DecisionAccording to Theorem 3.1, the seller’s problem now is to set the base price poptimally:maxp pi(p) = ∑i∈A pλi. (3.7)Non-concavity of profit functionFirst of all, we should check the property of function pi(p), then we have thefollowing theorem.Theorem 3.2. The profit function pi(p) is not concave in p ∈ (0,∞).34Proof. To begin with, we have∂pi(p)∂p =∑i∈A eα−β(p+ci)−γti[1− βp +∑i∈A eα−β(p+ci)−γti][1 +∑j∈A eα−β(p+ci)−γtj]2 . (3.8)Then we have∂2pi(p)∂p2 =β∑i∈A eα−β(p+ci)−γti[βp− 2− (βp + 2)∑i∈A eα−β(p+ci)−γti][1 +∑i∈A eα−β(p+ci)−γti]3 .It’s clear that except for the termβp− 2− (βp + 2) ∑i∈Aeα−β(p+ci)−γtiwhose sign we cannot determine, all the other terms in the product on the RHS of theequation above are positive.Let x ≡ βp + 2 (x > 2) and k ≡ ∑i∈A e2+α−βci−γti (k > 0), we can transform thementioned term intog(x) = x− kxe−x − 4. (3.9)It’s clear that when x is small, g(x) is negative. For example, g(3) = −3ke−3− 1 <0. On the other hand, we must also notice that g(x) could be positive. For example,we know that limx→∞ g(x) = ∞.Therefore, the sign of ∂2pi(p)∂p2 cannot be determined. That’s to say, the profit functionpi(p) is not concave in p ∈ (0,∞).Property of ∂pi(p)∂pAccording to (3.8), we know that the sign of ∂pi(p)∂p depends on1− βp + ∑i∈Aeα−β(p+ci)−γti .Let x ≡ βp (x > 0) and k ≡ ∑i∈A eα−βci−γti (k > 0), we can transform the men-tioned term intoh(x) = 1− x + ke−x. (3.10)35Then we have∂h(x)∂x = −1− ke−x.It’s clear that ∂h(x)∂x < 0. That’s to say, h(x) is decreasing in (0,∞). We alsoknow h(e) = 1− e + ke−e > 0 in which e is a arbitrarily small positive value, andlimx→∞ h(x) = −∞. So we can conclude that h(x) is decreasing on its domain (0,∞):it has positive value at first and then becomes negative, and that there is only one realroot x for h(x) = 0.We provide the following example to visualize the function g(x) and h(x) as de-fined above and the corresponding pi(p).Example 3.1. We set n = 3, α = 20, β = 2, γ = 1, (c1 = 5, t1 = 4), (c2 = 10, t2 = 3),and (c3 = 8, t3 = 4).0 5 10 15 20−800−600−400−2000xg(x)Figure 3.1: Plot of g(x) in Example 3.1As we can see, g(x) is negative when x is small and then becomes positive whenx increases. On the other hand, h(x) is positive when x is small and then becomesnegative when x increases. The plot for pi(p) clearly shows the “one-peak” property.360 5 10 15 200100200300400xh(x)Figure 3.2: Plot of h(x) in Example 3.10 2 4 6 8 1000.511.52ppi(p)Figure 3.3: Plot of pi(p) in Example 3.1Unimodal profit functionFrom previous part, we know that ∂pi(p)∂p is positive when p is small and becomenegative when p increases, and only has one real root for ∂pi(p)∂p = 0. Therefore,although the profit function is not concave in p, it’s unimodal, so we still can use37First-Order-Condition to find the global maximum. The unique solution to this opti-mization problem is presented in the following theorem.Theorem 3.3. There is a unique optimal price p∗ satisfyinge−β(p∗− 1β)= βeα−1 ∑i∈A e−βci−γti(p∗ − 1β).Proof. According to (3.7), by using First-Order-Condition, we have∂pi(p)∂p =A(1 +∑j∈A eα−β(p+ci)−γtj)2 ∑i∈A e−βci−γti = 0in whichA = (1− βp)eα−βp(1 + ∑i∈Aeα−β(p+ci)−γti)+ βpeα−βp ∑i∈Aeα−β(p+ci)−γti= eα−βp − βpeα−βp + eα−βp ∑i∈Aeα−β(p+ci)−γti= eα−βp(1− βp + ∑i∈Aeα−β(p+ci)−γti).To solve the function, we need to let A = 0. Then we have:1− βp + ∑i∈Aeα−β(p+ci)−γti = 0.Rearranging the equation above, we can gete−β(p− 1β)= βeα−1 ∑i∈A e−βci−γti(p− 1β).Let p˜ = p− 1β , we have:e−β p˜ = βeα−1 ∑i∈A e−βci−γtip˜. (3.11)As we can see in (3.11), there is only one unknown value p˜. On the left-hand-side,we know that e−β p˜ is decreasing on p˜ and its plot crosses (0, 1) and approximates 0when p˜ is large. On the right-hand-side, βeα−1 ∑i∈A e−βci−γti p˜ is a straight line crossing(0, 0) with positive slop. Therefore, there is one and only one intersection for theirplots and we can guarantee to find such p˜.Then it’s straightforward to get p∗ = p˜ + 1β .38In order to get the closed-form solution, we can use the Lambert W function(Corless et al., 1996), which is defined to be the function satisfyingW(z)eW(z) = z.When z ∈ R, this function is non-negative and monotonically increasing on [0,∞).Corollary 3.1. The optimal p∗ can be expressed asp∗ = 1β[1 + W(∑i∈Aeα−1−βci−γti)]in which W(·) is the Lambert W function.Proof. From (3.11), we can get∑i∈Aeα−1−βci−γti = β p˜eβ p˜. (3.12)Then we haveβ p˜ = W(∑i∈Aeα−1−βci−γti)which givesp˜ = 1βW(∑i∈Aeα−1−βci−γti).Then we should havep∗ = 1β + p˜ =1β[1 + W(∑i∈Aeα−1−βci−γti)].Then, we can also use Lambert W function to express the optimal profit pi∗.Corollary 3.2. The optimal pi∗ can be expressed aspi∗ = 1βW(∑i∈Aeα−1−βci−γti)in which W(·) is the Lambert W function.39Proof. We can rewrite pi(p) aspi(p) = ∑i∈Apλi = p ∑i∈A eα−βp−βci−γti1 +∑i∈A eα−βp−βci−γti= p e∑i∈A eα−1−βci−γtieβp + e∑i∈A eα−1−βci−γti.According to (3.12), we then should havepi∗ = p∗ eβ p˜eβ p˜eβp∗ + eβ p˜eβ p˜= p∗eβ(p∗ − 1β)eβ(p∗− 1β)eβp∗ + eβ(p∗ − 1β)eβ(p∗− 1β)= p∗ (βp∗ − 1)1 + (βp∗ − 1)= p∗ − 1β= 1β[1 + W(∑i∈Aeα−1−βci−γti)]− 1β= 1βW(∑i∈Aeα−1−βci−γti).in which the last but one equality uses Corollary 3.1.Note that Corollary 3.2 also confirms that the seller should include all the avail-able options in his assortment because W(·) is increasing in (0,∞). On the other hand,if the seller faces cardinality constraint for some reason, a simple sorting algorithmwould help to find the optimal assortment. We can sort all the items by ki = −βci−γtiand the first K options with highest kis should be selected in the seller’s assortment ifhe can only choose up to K options.According to Corollary 3.1 and Corollary 3.2, we know that pi∗ = p∗ − 1β . So thereis a positive relationship between product price and seller’s profit at optimality.Recall that Lambert W function is increasing in [0,∞). From Corollary 3.1 andCorollary 3.2 we can see that the increase of α would lead to higher p, which is con-sistent with our intuition that a product with higher value should have a higher price.40These equations also show that the decrease of γ would increase p and pi. This tellsus that when the customers are less sensitive to lead time, the seller can charge ahigher price and gain more profit. It corresponds to the reality that if the customersare more willing to wait, the seller is more profitable because if the customers are ex-tremely impatient, they may pursue other outside options such as buying the productin local market (if available), instead of shopping online. More importantly, these twocorollaries show the important role of the price sensitivity parameter β as it not onlylinks the optimal price and optimal profit, but also appears both within and outsidethe Lambert W function. This suggests that the seller’s pricing strategy and expectedprofit are very responsive to the customers’ price sensitivity and thus a small mistakeon the incorrect estimation of this parameter may lead to a huge loss.Although the quoted surcharges and lead times are exogenous, they still haveimportant influence on the seller’s profit. Corollary 3.2 shows that when ci or ti de-creases, the expected profit pi∗ would increase. This reveals the fact that the sellerwould directly benefit from lower service surcharge, even if he can only quote it di-rectly without adding any mark-up. On the other hand, shorter lead time would alsomake the seller become better off. The reason behind them is that customers’ betterperception about the delivery service because of decreased surcharge or lead timewould help to decrease the no-purchase probability and thus contribute to customeracquisition.41Chapter 4B: Pricing Product Bundle withoutCustomer Rating4.1 Model FormulationIn this problem, we consider a similar setting with only one difference: the selleris granted with the authority to list a different surcharge other than the quoted onefrom the DSP and he needs to give the DSP the quoted surcharge no matter howmuch the customers pay.The seller now enjoys more flexibility in pricing because he is able to control thelisted surcharge for each delivery option. Compared with previous problem in whichthe relationship between the seller and DSPs is more cooperation-oriented, this prob-lem features a more outsourcing-like interaction between the seller and DSPs.The decision problem for the seller is to select the appropriate assortment of de-livery options and to decide the corresponding listed surcharge for those options aswell as the base price for the product, in order to maximize the expected profit fromall the customers.424.1.1 Customers’ ChoiceFacing with an assortment S of delivery options selected by the seller, customerschoose among them according to random utility maximization. Following the theoryin standard MNL model, a customer chooses option i with the probability specifiedin (3.2).We model customers’ mean utility associated with each delivery option i asui = α− β(p0 + p˜i)− γti, (4.1)in which p0 is the base price of the product, p˜i is the listed surcharge for deliveryoption i, and ti is the quoted lead time for delivery option i. All parameters are thesame as in Chapter Seller’s ProblemThe seller chooses an assortment of delivery options S from the set of all availablealternatives A for the customers to choose from, and decides the listed surcharge foreach delivery option as well as the base price for the product. The objective of theseller is to maximize his expected profit from all the customers, and this problem canbe written asmaxS,p0,p˜i ∑i∈S(p0 + p˜i − ci)λi, (4.2)in which λi is customers’ probability of choosing option i (market share of option i)and is influenced by p0 and p˜i.We notice that because of their special structure, we can actually simplify theseller’s objective and the customers’ mean utility function by combining the p0 andp˜i terms together. Then the seller’s pricing decision is essentially to set the pricefor each product bundle consisting of one product and the corresponding delivery43service. Therefore, we may rewrite (4.2) asmaxS,pi ∑i∈S(pi − ci)λiin which pi is the price for the bundle of product plus delivery option i.Note that when implementing the pricing policy, there could be more than oneway for the seller to set the value of p0 and p˜i. One example would be p0 = min{pi}and p˜i = pi − p0. This example also illustrates one of the reasons that result in op-tions with surcharges lower than costs or even free delivery options — the paymentis simply shifted from the service surcharge to the item base price.Similarly, we can rewrite (4.1) asui = α− βpi − γti. (4.3)In this current model, the customers’ sensitivities for product price and deliveryservice surcharge are assumed to be the same, but we note that it’s also possible toincorporate differentiated price sensitivity parameters into this problem. The analysiscan be extended easily and the structure of the solution would remain the same aslong as linearity form is kept. The implementation plan of pricing policy can beadjusted accordingly.4.2 AnalysisIn this section, we still first look at the optimal assortment decision and then studythe pricing decision.4.2.1 Assortment DecisionTheorem 4.1. By selecting as many delivery options as possible in the service assortment, theseller can achieve the highest expected profit. That’s to say, S∗ = A.44Proof. We begin by assuming the seller can achieve some profit pi0 by finding an as-sortment S0 ⊂ A with prices p0i for i ∈ S0. Then we add another option k ∈ A \ S0to form a new assortment S = S0 ∪ {k}. We can prove that even if we keep the p0i fori ∈ S0 unchanged, we would be able to increase the expected profit.For the simplicity of notation, we first denote vi ≡ eα−βpi−γti and v0i ≡ eα−βp0i−γti .Then we can write the original profit aspi0 = ∑i∈S0(p0i − ci)λ0i = ∑i∈S0(p0i − ci)v0i1 +∑j∈S0 v0j. (4.4)After we bring in the option k and form a new assortment S, if we keep the pricesin previous assortment S0 unchanged, the new expected profit pi would bepi = ∑i∈S(pi − ci)λi= (pk − ck)λk + ∑i∈S0(p0i − ci)λi= (pk − ck) vk1 +∑j∈S vj + ∑i∈S0(p0i − ci)v0i1 +∑j∈S vj= (pk − ck) vk1 + vk +∑j∈S0 v0j+ ∑i∈S0(p0i − ci)v0i1 + vk +∑j∈S0 v0j= (pk − ck) vk1 + vk +∑j∈S0 v0j+ ∑i∈S0(p0i − ci)v0i1 +∑j∈S0 v0j1 +∑j∈S0 v0j1 + vk +∑j∈S0 v0j= (pk − ck) vk1 + vk +∑j∈S0 v0j+1 +∑j∈S0 v0j1 + vk +∑j∈S0 v0j ∑i∈S0(p0i − ci)v0i1 +∑j∈S0 v0j= (pk − ck) vk1 + vk +∑j∈S0 v0j+1 +∑j∈S0 v0j1 + vk +∑j∈S0 v0jpi0.(4.5)in which the last equality follows from (4.4).45According to (4.5), we havepi − pi0 = (pk − ck) vk1 + vk +∑j∈S0 v0j+1 +∑j∈S0 v0j1 + vk +∑j∈S0 v0jpi0 − pi0= (pk − ck) vk1 + vk +∑j∈S0 v0j+ −vk1 + vk +∑j∈S0 v0jpi0= vk1 + vk +∑j∈S0 v0j(pk − ck − pi0).As it’s clear that vk1+vk+∑j∈S0 v0j > 0, if we set the price of option k so that pk >ck + pi0, then we would have pi > pi0. That’s to say, by adding a new option inthe assortment and pricing this new option high enough, we can surely increase theexpected profit.Note this finding is same as in Aydin and Ryan (2000), Hopp and Xu (2005), andWang (2012), but we show that the conclusion can be drawn even before we obtainany structural property of optimal pricing decisions.4.2.2 Pricing DecisionAccording to Theorem 4.1, the seller’s problem now is to set the vector of bundleprices p = (p1, p2, · · · , pn) optimally for all the n items in A:maxpi pi(p) = ∑i∈A(pi − ci)λi.Concavity of profit functionPreviously, it has been found by Hanson and Martin (1996) for similar problemthat the profit function is not jointly concave in price vector p. However, it has beennoticed in some similar problems with standard MNL model and nested logit model(e.g., Dong et al. 2009, Song and Xue 2007, and Li and Huh 2011) that the profit func-tion is jointly concave in the vector of “market share”, which is λ = (λ1,λ2, · · · ,λn)in our case.46In the next part, we will show that after the bundle price transformation, the sell-er’s problem is a special case of the pricing problem studied in Li and Huh (2011) andthe profit function pi is jointly concave for vector λ.According to (3.2) and (4.3), we should have:∑i∈Aλi = ∑i∈A eα−βpi−γti1 +∑j∈A eα−βpj−γtj.Then we have1− ∑i∈Aλi =11 +∑j∈A eα−βpj−γtj.Combining the equation above with (3.2) and (4.3) once again, we should haveλi1−∑j∈A λj = eα−βpi−γti .Take the logarithm on both sides of the equation above, we havelogλi − log(1− ∑j∈Aλj) = α− βpi − γtiwhich leads topi = α− γtiβ −1β[logλi − log(1− ∑j∈Aλj)]. (4.6)Therefore, the seller’s profit function can be expressed as:pi(λ) = ∑i∈A(pi(λ)λi − ciλi)= ∑i∈Aα− βci − γtiβ λi −1β ∑i∈A λi[logλi − log(1− ∑j∈Aλj)].(4.7)Using Lemma 2 in Li and Huh (2011), we know that the second term is concavewhile it’s clear that the first term is linear in λi. Therefore, since the concavity is pre-served by linear combination, we may conclude that the profit function is concave inthe λ vector.In the following part, we show that the optimal price p∗i and optimal expectedprofit pi∗ can be found through the searching of an optimal proportion of buyers Λ∗(Λ ≡ ∑i∈S λi).47Optimization through the proportion of buyersUsing First-Order-Condition, from (4.7) we know that∂pi(λ)∂λi= α− βci − γtiβ −1β[logλi − log(1− ∑i∈Aλi) +11−∑i∈A λi]= 0which leads tologλi = α− βci − γti + log(1− ∑i∈Aλi)−11−∑i∈A λi .Take the exponential on both sides of the equation above and use Λ to denote∑i∈A λi, we getλi = eα−βci−γti− 11−Λ (1−Λ). (4.8)Take the summation of i ∈ S on both hands and rearrange, we can getΛ1−Λ = e− 11−Λ ∑i∈Aeα−βci−γti . (4.9)It’s not hard to see that the optimal Λ∗ can be found by one-dimensional search.Then according to (4.8), (4.6), and (4.4) we can represent the optimal customerproportion, optimal price, and optimal profit asλ∗i = (1−Λ∗)eα−11−Λ∗−βci−γtip∗i =1β(1−Λ∗) + cipi∗ = Λ∗β(1−Λ∗) .(4.10)We note that the “constant mark-up” property in the pricing equation is consistentwith findings in previous studies on similar problems.Next, we show mathematically that the optimal mark-up here would be the sameas the optimal price p∗ in previous problem. In other words, the customers wouldpay the same total price in both problems.48Rearranging (4.9), we can get( 11−Λ − 1)e( 11−Λ−1) = ∑i∈Aeα−1−βci−γti .Therefore we have11−Λ − 1 = W(∑i∈Aeα−1−βci−γti)which leads to11−Λ = 1 + W(∑i∈Aeα−1−βci−γti).Combining with (4.10), we have:p∗i =1β[1 + W(∑i∈Aeα−1−βci−γti)]+ cip∗i − ci =1β[1 + W(∑i∈Aeα−1−βci−γti)],whose right-hand-side equals to p∗ in Corollary 3.1.It’s not hard to see that the optimal expected profit pi∗ would be the same as inthe previous problem as well. Therefore, it confirms that the seller should include allthe available options in his assortment because W(·) is increasing in (0,∞). On theother hand, if the seller faces cardinality constraint for some reason, a simple sortingalgorithm would help to find the optimal assortment. We can sort all the items byki = −βci − γti and the first K options with highest kis should be selected in the sell-er’s assortment if he can only choose up to K options.Although the seller now has more flexibility to differentiate the listed surchargefor each delivery option, his highest expected profit cannot exceed the one in previousproblem in which his power in pricing is restricted to the product only. Because ofthe “constant mark-up” property in the optimal pricing policy, higher flexibility inpricing doesn’t actually bring higher profitability for the seller.49On the other hand, since the market share of each option doesn’t change in thesetwo problems (recall that the customers’ payment on each option is invariant in thesetwo problems), it’s clear that the revenue of each DSP won’t be affected by the changein the seller’s pricing flexibility. Although granting such pricing flexibility to the sellerwould seemingly hurt the DSPs, our analysis shows that the “constant mark-up”property guarantees that the DSPs should be indifferent between the two schemes.4.2.3 Numerical ExampleIn this section we present the following example to compare the different resultsin different assortments.Example 4.1. We set α = 5, β = 0.15, γ = 0.2, A = {1, 2, 3}, (t1 = 3, c1 = 5), (t2 = 2,c2 = 8), and (t3 = 6, c3 = 10).We can calculate the optimal mark-up mi, the market share of each delivery optionλi, the optimal proportion of buyers Λ = ∑i∈S λi, the total revenue for the DSPs(SerRev = ∑i∈S λici), customers’ average payment (AvgPay = 1/Λ∑i∈S λi(p + ci)),customers’ total payment (TotPay = ∑i∈S λi(p + ci)), customers’ average mean utilityu¯ = 1/Λ∑i∈S λiui, and the optimal profit pi. We also provide the results under otherassortments (when S 6= A) in the following table in descending order of profit.4Table 4.1: Optimal results in different assortments in Example 4.1S mi λ1 λ2 λ3 Λ SerRev AvgPay TotPay u¯ pi{1, 2, 3} 23.0753 0.3489 0.2717 0.0905 0.7111 4.8229 29.8577 21.2316 -0.0786 16.4087{1, 2} 22.4341 0.3951 0.3077 NA 0.7028 4.4373 28.7476 20.2047 0.1754 15.7674{1, 3} 20.8412 0.5401 NA 0.1400 0.6801 4.1007 26.8706 18.2752 0.2459 14.1745{2, 3} 19.9742 NA 0.4999 0.1664 0.6662 5.6627 28.4737 18.9702 0.1292 13.3075{1} 19.8087 0.6634 NA NA 0.6634 3.3172 24.8087 16.4593 0.6787 13.1420{2} 18.7191 NA 0.6439 NA 0.6439 5.1509 26.7191 17.2032 0.5921 12.0524{3} 14.3699 NA NA 0.5361 0.5361 5.3607 24.3699 13.0639 0.1445 7.70324All the results in the solutions are computed in R, which is developed by R Core Team (2015). Thecode is provided in Appendix.50As expected, pi would be the highest when the seller includes all available deliveryoptions. From this table we also can see that mi as well as Λ would be the highestwhen all the options are included in S, and that while pi increases across the assort-ments, both mi and Λ are increasing. This shows that while the trade-off betweenhigher unit profit and more buyers is necessary for the optimization problem withineach assortment, we can actually improve both of them simultaneously with betterassortment decision at the beginning.For the DSPs, each of them has the lowest market share when the seller obtainsthe highest profit, but the total service revenue flow to the DSPs under such circum-stance is not the lowest. It tells us that the delivery service industry as a whole wouldnot necessarily suffer the most when the seller selects his optimal assortment. Buton the other hand, if everyone has a slice of the cake (when all the delivery optionsare included in the assortment), this group as a whole may not be able to enjoy thehighest total service revenue due to competition.On the customers’ side, both the average payment and the total payment wouldbe the highest when the seller gets the highest profit. But in other assortments, ahigher profit is not necessarily associated with a higher payment. More importantly,although there is no perfect correlation between u¯ and pi, we can see that u¯ is thelowest when the seller obtains the highest profit. This reflects the seller’s monopolypower in the market. By providing all available options and charging a constant mark-up for all the options, the seller makes the customers less elastic about the price. Thenthis constant mark-up could be high enough to extract more values from the market.We shouldn’t be too surprised to observe a negative average mean utility in thisexample. First of all, we note the negative utility can also be observed in numericalexamples in related studies, e.g., Li and Huh (2011) and Gallego and Wang (2014),if we explicitly calculate it. Then recall that we normalize the no-purchase utility to51be zero. That to say, we may also choose other values for the purpose of normaliza-tion and the structure of solutions won’t be affected (if we want to keep the relativescale consistent, we need to change the model parameters accordingly, though). Thenwhat really matters here is to understand why the mean utility could be below theno-purchase utility. Equation (3.2) just provides the answer: no-purchase option isnothing but “another option”, thus it is still governed under the structure of the prob-abilistic choice model. That’s to say, this option cannot absorb all the customers withmean utility under the no-purchase utility and is still compared against all the otheroptions in a probabilistic manner. Then it’s easy to understand that even if the meanutility of one option is below the no-purchase utility, it still has positive probability tobe chosen. Hence, the average mean utility could be negative (below the no-purchaseutility).52Chapter 5C: Pricing Product with CustomerRating5.1 Model FormulationIn this problem, we go back to the set-up in the basic problem (Problem A) buttake into account the influence of customer rating. Customers would be able to pro-vide feedback about their experience of delivery service via rating on the website.At the beginning, the seller selects the assortment of delivery options. This isa long-term strategic decision that cannot be altered later. In each of the followingperiods, the seller decides the base price for the product, and then the customersobserve the rating generated in previous period in addition to what they can also seein Problem A and make the purchasing decision. Those who buy can rate the deliveryservice quality after receiving the product. The rating is aggregated at the end of eachperiod and the average value becomes available for prospective customers in the nextperiod to view. The time-line can be visualized in Figure 5.1.53Figure 5.1: Time-line of eventsAlthough some platforms such as Taobao.com provide customers with the chanceto rate on more than one dimension of their complete shopping experience, we willonly focus on the rating for delivery service quality, which is most relevant to our re-search. While the other factors may indeed influence customers’ purchase decisions,it’s sufficient for us to hold them constant across all the four problems in this researchsince we are most interested in the comparison among them. Particularly, we maythink the impact of such factors (other ratings or related information) is absorbed inparameter α and it doesn’t change across the four problems we study.We consider the situation in which the rating is associated with the seller, insteadof each delivery option. We rarely see ratings specific to one delivery option in reality,but the rating associated with a seller (e.g., Taobao.com) or a particular product (e.g.,Amazon.com) is often seen in practice.5 It is easier to implement and the numberswould be comparable among the counterparts. On the other hand, the option-specificrating case is not hard to investigate since we can show that with some easy trans-formation, it would be reduced to a problem very similar to Problem A, which we’vestudied in Chapter 3. Since the true reliability of each option is disguised by the ag-gregated rating, we would like to examine whether the seller can make a profit out of5In our setting, these two schemes would be equivalent.54the information distortion resulting from the rating aggregation.We assume the total market size is approximately the same over the periods.6Following previous two problems, it is normalized to one in each period. The problemfor the seller is to select the appropriate delivery options and then price the product,to maximize the expected profit in each period in a steady state. But apart fromthe quoted delivery time and surcharge information, the seller would also be able toknow some information regarding the service quality of each delivery option.5.1.1 Customers’ ChoiceThe customers make the purchasing decision according to MNL model and thetheory in standard MNL model suggests a customer chooses option i with the proba-bility following equation (3.2).Now the customers’ mean utility associated with each delivery option i wouldbecomeui = α− β(p + ci)− γti + δr(·), (5.1)in which δ is the customers’ sensitivity for rating and r(·) is a function representingthe aggregated rating. We will talk about its details in the next part. The other partsof (5.1) are exactly the same as in Problem A.Following Ho and Zheng (2004), a linear relationship between service quality (rat-ing) and quoted lead time is adopted here. While there are also other possibilities tomodel the interactions between them, we believe this is a reasonable choice becausethis linear formation is a very general form that can not only reflect the link betweenthem, but also capture the possible connection between price and service quality aswe discussed in the literature review.6This would not influence our analysis for this problem since we will show later that the “steadiness”of the rating would enable us to consider the optimization problem for each period separately.555.1.2 Customers’ RatingAmong all the rating systems, we consider the binary rating system. Despite itssimplicity for handling, we note some other studies adopted this system or similarsetting. Kuksov and Xie (2010) use the binary rating system to reflect the customer’soverall experience for online purchasing. Ho and Zheng (2004) use a similar binarysatisfaction setting to consider the delivery service performance, which is a represen-tation of the service quality. On the other hand, as we will show later, this binarysetting will enable us to incorporate the reliability level of each delivery service op-tion in our model directly.If we want to allow for a spectrum of ratings, we may need to assume the cus-tomers can rationally form a belief about the distribution of the actual delivery timeof each option so that they may assess the performance accordingly. However, cus-tomers in reality hardly have access to complete information, may not be sophisticatedenough to evaluate the rating so accurately, and maybe simply don’t bother to makethis simple task so complicated.The interpretation of the numeric value of the aggregated rating should not be aconcern compared with other rating systems, since we can convert the rating from a[0, 1] scale to other scales, such as [1, 5] in the five-star rating system, through simplelinear transformation.It’s noted that the rating is usually used to reflect the overall purchasing experi-ence, but in our setting we’re only considering the rating on delivery service explicitlyand treating the other ratings to be identical, and we also note that the price informa-tion is not lost in the model.In such a binary rating system, a customer indexed j choosing to buy the productwith delivery option i would rate rji ∈ {1, 0} to represent whether the delivery is on-56time after observing the actual delivery time Tji. When Tji 6 ti, the exogenous quotedlead time for option i, the customer is satisfied and a rating of 1 would be offered.Otherwise, a rating of 0 would be given. In other words,rji = 1(Tji 6 ti)in which 1(·) is the indicator function.As we would not consider behavioral components in customer rating, we assumeall the buyers rate truthfully, which means every single customer who bought theproduct provides a rating based on actual on-time delivery performance only. It help-s to get rid of the influence biased weight on certain customers7 and reinforce ourfocus on the delivery service quality.Denoting ri as the aggregated rating for delivery option i, we should be able tosee immediately thatri = Fi(ti)in which Fi(t) is the CDF of delivery time for option i.We assume that seller is not informed by the DSPs about the complete Fi(t) func-tion, but only the reliability level θi (0 < θi 6 1) of each option i under the quotedlead time term, i.e. θi = Fi(ti). This would be more realistic considering the DSPsmay not be able to provide complete CDF function or may not be willing to sharesuch detailed information, but they should have good understanding about the ser-vice reliability under the terms they quote and they have to prepare this informationfor their customer (the seller in our case) to make decision. Under such circumstance,announcing a lead time other than the one quoted by the DSP would not be an optionfor the seller if he intends to do so, since he doesn’t have enough information to de-termine the optimal plan. We also assume such reliability information is not available7Our model can accommodate, however, one kind of behavioral scenario in which a certain fractionof satisfied customers would not rate. Provided that such fraction is exogenous and constant, it wouldbe equivalent to decrease the value of δ accordingly.57to the customers. If yes, then it would be the same as the option-specific rating casethat we’ve talked about at the beginning of this chapter.It was proved by So and Song (1998) and later confirmed in many other subse-quent papers that the constraint on reliability level is binding in optimal solutionwhen the DSPs make their decisions, so we would only consider the situation wherethe quoted reliability level is exactly achieved by the DSPs.Therefore, the aggregated r, which also represents the proportion of buyers whoreceive on-time delivery, is the weighted average of ri across all i ∈ S with the weightbeing the proportion of buyers choosing option i:r = ∑i∈Sri λi∑i∈S λi =∑i∈S θiλi∑i∈S λi , (5.2)in which θi is reliability for each delivery option i.Since two consecutive periods are connected via the aggregated rating (rating inprevious period would influence λi via ui in the next period), we are interested in itsproperty over the periods.Lemma 5.1. The aggregated rating r in one period is not affected by the rating in previousperiod or the product price p in current period. Thus, it should be steady across periodsprovided that S doesn’t change.Proof. Let rk be the aggregated rating generated in period k and denote vi ≡ eui .According to the definition in (5.2), in period k we haverk = ∑i∈S θiλi∑i∈S λi =∑i∈S θi vi1+∑j∈S vj∑i∈S vi1+∑j∈S vj= ∑i∈S θivi∑i∈S vi= ∑i∈S θieα−βp−βci−γti+δrk−1∑i∈S eα−βp−βci−γti+δrk−1= ∑i∈S θie−βci−γti∑i∈S e−βci−γti.58As we can see, rk is not influenced by rk−1 or p. On the other hand, since p doesn’tinfluence rk, it cannot influence period k + 1 via rk. Therefore, holding the assortmentof delivery options stable, the aggregated rating r is steady across periods.According to Lemma 5.1, we haver(S) = ∑i∈S θie−βci−γti∑i∈S e−βci−γti. (5.3)It reveals that when the seller can only price the product, he cannot effectivelyalter the relative proportions of buyers who choose different delivery options, thus hecan only use different assortments of options to influence the aggregated rating. Onthe other hand, we should also notice that the product value would not change theaggregated rating, which reflects the idea in our assumption that the rating should bebased on the delivery service only and is irrelevant to the product itself.5.1.3 Seller’s ProblemAs we see in previous section, the rating would be stable over periods given theassortment is constant. Because we assume the assortment cannot be changed fromthe beginning, we can simply consider the maximization problem in one period insteady state.The objective of the seller is to maximize its expected profit from the customers inone period in steady state, and it can be written asmaxS,p ∑i∈S pλi = maxS,p ∑i∈S peα−β(p+ci)−γti+δr(S)1 +∑j∈S eα−β(p+ci)−γti+δr(S)in which p is the base price for the product.5.2 AnalysisAccording to Lemma 5.1 and equation (5.3), we know the aggregated rating rwould be a constant once the assortment is decided. We may look at the pricingproblem first and then study the assortment problem.595.2.1 Pricing DecisionFollowing Corollary 3.1, it is not hard to tell that the structure of the solutionwould be the same as in Problem A and the optimal price p∗ should bep∗ = 1β[1 + W(∑i∈Sexp(α− 1− βci − γti + δ∑i∈S θie−βci−γti∑i∈S e−βci−γti))]. (5.4)This shows that higher product value (higher α) and better delivery service (lowerci and ti, and higher rating r) would give the seller more power to charge a higherproduct price, which is consistent with our intuition and our finding in Problem Aand Problem B.5.2.2 Assortment DecisionIn Problem A, we proved that the seller should include all the available deliveryoptions in his assortment in order to get the highest expected profit. However, in thisproblem, “the more, the better” policy may not be optimal because of the impact ofcustomer rating. We firstly provide an example to illustrate this idea and remark onsome observations, then formally present the assortment problem, and look at someconventional algorithms at last.A numerical exampleExample 5.1. We set α = 5, β = 0.15, γ = 0.2, δ = 2.5, A = {1, 2, 3}, (t1 = 3, c1 = 5,θ1 = 0.8), (t2 = 2, c2 = 8, θ2 = 0.9), and (t3 = 6, c3 = 10, θ3 = 0.4).According to our solution, we can calculate the optimal price p, the market shareof each delivery option λi, the optimal proportion of buyers Λ = ∑i∈S λi, the to-tal revenue for DSPs (SerRev = ∑i∈S λici), customers’ average payment (AvgPay =1/Λ∑i∈S λi(p + ci)), customers’ total payment (TotPay = ∑i∈S λi(p + ci)), customer-s’ average mean utility u¯ = 1/Λ∑i∈S λiui, the aggregated rating r, and the optimalprofit pi under different assortment conditions. We summarize these related resultsin the following table ordered by profit.60Table 5.1: Optimal results in different assortments in Example 5.1S p λ1 λ2 λ3 Λ SerRev AvgPay TotPay u¯ r pi{1, 2} 33.0622 0.4488 0.3495 NA 0.7984 5.0404 39.3756 31.4359 0.6907 0.8438 26.3955{1, 2, 3} 33.0352 0.3916 0.3050 0.1015 0.7982 5.4137 39.8176 31.7822 0.3958 0.7873 26.3685{1, 3} 29.5957 0.6152 NA 0.1595 0.7747 4.6712 35.6251 27.6002 0.7269 0.7177 22.9290{1} 29.4685 0.7738 NA NA 0.7738 3.8688 34.4685 26.6707 1.2297 0.8000 22.8019{2} 29.4685 NA 0.7738 NA 0.7738 6.1902 37.4685 28.9920 1.2297 0.9000 22.8019{2, 3} 29.3405 NA 0.5798 0.1930 0.7728 6.5683 37.8400 29.2421 0.6620 0.7751 22.6738{3} 18.2926 NA NA 0.6356 0.6356 6.3555 28.2926 17.9814 0.5561 0.4000 11.6259As we can see, the biggest distinction from Example 4.1 is that when the sellerincludes all of the three delivery options, his profit is no longer the highest and hewould be better off if he eliminates Option 3 in his assortment. Since the seller canonly price the product, such restriction limits the seller’s power on dealing with thenegative impact of including Option 3, and the loss of including this “dominated”option outweighs the benefits of including it.From this example we can see that it would be dangerous for the DSPs to assumethat everyone would be included in the competition when they make strategic deci-sions. While some DSPs are expecting low attractiveness and low reliability wouldonly lead to low market share, this example warns those DPSs that they could be to-tally left out from the competition because of the intervention of the seller. Therefore,the setting in Problem C would dramatically reshape the competition game amongthe DSPs.61The combinatorial optimization problemFollowing Corollary 3.2 and Lemma 5.1, we should know that the structure of thesolution would be the same and the optimal profit pi∗ would bepi∗ = 1βW(∑i∈Sexp(α− 1− βci − γti + δ∑i∈S θie−βci−γti∑i∈S e−βci−γti))= 1βW(exp(α− 1 + δ∑i∈S θie−βci−γti∑i∈S e−βci−γti)∑i∈Sexp (−βci − γti))= 1βW(eα−1 exp(δ∑i∈S θie−βci−γti∑i∈S e−βci−γti)∑i∈Se−βci−γti).Since W(·) is increasing in (0,∞) and eα−1 is constant, the maximization of pi∗ isequivalent tomaxSyˆ(S) = exp(δ∑i∈S θiyi∑i∈S yi)∑i∈Syi, (5.5)in which yi = e−βci−γti , and we may call yi as “attractiveness” to represent a compre-hensive attribute and distinguish it with reliability, the other attribute in the equation.This would help us understand why including Option 3 is bad for the seller inExample 5.1: although Option 3 would help to increase ∑i∈S yi, its negative impacton exp(δ∑i∈S θiyi∑i∈S yi)surpasses that increase and leads to overall decrease in the value ofyˆ(S). Then we know that when δ increases, the seller should benefit more from theelimination of Option 3 and we may expect his optimal expected profit can increasemore after he excludes Option 3.Conventional sorting algorithmsIn this part, we show that an algorithm based on conventional sorting and stop-ping policy, similar to simplified revenue-ordered policy in the literatures, may lead tosuboptimal solutions. In each example, we firstly sort the options by some attributes,and add each option sequentially into the assortment S to check the value of yˆ(S), andstop adding if it decreases. The x-axis in result plot indicates the number of optionsin the assortment, and y-axis shows the value of yˆ for the current assortment.62Example 5.2. Suppose we have the following A sorted by θ and then by y in descend-ing order (δ = 2.5):Table 5.2: Options in A in Example 5.2i 1 2 3 4 5 6 7 8 9 10 11θi 1 1 1 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3yi 0.75 0.6 0.3 0.5 0.4 0.3 0.2 0.1 0.8 0.7 0.7We would have the following graph:0 2 4 6 8 10 12101520Number of options in Syˆ(S)Figure 5.2: Plot of yˆ in Example 5.2 when adding options sequentiallyConsidering θ has very important role in yˆ, we sort the options according to θfirst. But this algorithm would not work very well since if we follow it, we wouldstop adding new option after we include Option 3. However, it turns out that yˆ startsto increase if we add more items down the list and we could eventually have a larger yˆif we include all the options inA compared with only including the first three options.In the next example, we switch to order the options according to y first:63Example 5.3. Suppose we have the following A sorted by y and then by θ in descend-ing order (δ = 2.5):Table 5.3: Options in A in Example 5.3i 1 2 3 4 5 6 7 8 9 10 11yi 0.8 0.75 0.6 0.6 0.5 0.35 0.3 0.3 0.1 0.05 0.05θi 0.8 0.6 0.8 0.2 0.1 0.4 0.5 0.3 0.6 0.3 0.1We would have the following graph:0 2 4 6 8 10 126810121416Number of options in Syˆ(S)Figure 5.3: Plot of yˆ in Example 5.3 when adding options sequentiallyAs we can see, this algorithm would not work well either since it advises us tostop adding new option after we include Option 3. However, yˆ would become largerafter we include the 7th option and it would be even larger after we add more optionsdown the list.Then we would like to see whether sorting the options based on θiyi initiallywould be better. Let’s see the next example:64Example 5.4. Suppose we have the following A sorted by θ · y and then by θ indescending order (δ = 2.5):Table 5.4: Options in A in Example 5.4i 1 2 3 4 5 6 7 8 9 10 11θiyi 0.75 0.6 0.3 0.24 0.21 0.21 0.2 0.16 0.12 0.08 0.04θi 1 1 1 0.3 0.7 0.3 0.4 0.4 0.4 0.4 0.4yi 0.75 0.6 0.3 0.8 0.3 0.7 0.5 0.4 0.3 0.2 0.1We would have the following graph:0 2 4 6 8 10 12101520Number of options in Syˆ(S)Figure 5.4: Plot of yˆ in Example 5.4 when adding options sequentiallyThis graph shows a “local maximum” after the first peak: when adding Option 4,yˆ decreases, but it increases after adding Option 5, and decreases again when addingOption 6. It fails too.Lastly we would like to check whether switching to ordering by y subsequentlywould provide better performance. Let’s see the next example:65Example 5.5. Suppose we have the following A sorted by θ · y and then by y indescending order (δ = 2.5):Table 5.5: Options in A in Example 5.5i 1 2 3 4 5 6 7 8 9 10 11θiyi 0.64 0.48 0.45 0.14 0.12 0.12 0.12 0.06 0.05 0.02 0.005yi 0.8 0.6 0.75 0.7 0.6 0.4 0.2 0.1 0.5 0.1 0.05θi 0.8 0.8 0.6 0.2 0.2 0.3 0.6 0.6 0.1 0.2 0.1The result graph shows similar decrease-increase pattern in previous examples:0 2 4 6 8 10 126810121416Number of options in Syˆ(S)Figure 5.5: Plot of yˆ in Example 5.5 when adding options sequentiallyAll of these failing examples demonstrate that the conventional sorting policiesmay not help us find the optimal solution.5.2.3 Structure of an Optimal AssortmentBecause we cannot use the conventional sorting algorithm, we need to find otheruseful algorithm to help to solve the assortment problem efficiently. So in this partwe present a few observations on yˆ to stimulate some ideas.66Small δFirst of all, we can notice that if δ is very small, the seller should care more aboutthe second term ∑i∈S yi. Thus, he would tend to want to include as many options aspossible. In the special case of δ = 0, it would reduce to Problem A.This shows that if the customers are not very sensitive to the rating, there isn’tmuch for the seller to do to increase their mean utility.Identical θIt’s clear that if all the θis are identical, the seller should include as many optionsas possible, becauseyˆ(S) = exp(δθ∑i∈S yi∑i∈S yi)∑i∈Syi = eδθ ∑i∈Syi. (5.6)This may correspond to the situation where there is an industry-wide commonreliability level, which is the shared assumption in many literatures we reviewed inChapter 2.According to Lemma 5.1, the rating is fixed and there is no way for the seller toimprove it, thus he doesn’t really have the incentive to eliminate any unfavorable op-tion. In this way, it’s reasonable for the DSPs to assume that everyone is included inthe competition.This also suggests that if somehow a really low industry reliability is sustained,the seller would not eliminate any option because leaving out any option on one handwould not help to increase the rating and on the other hand would force him to giveup some niche market.67General caseWe observe that it’s impossible for an optimal assortment S to include a set ofoptions with identical reliabilities (θi = θ ∀i ∈ S) while leaving an option with ahigher reliability (θj > θ) out of the assortment.Proof. Suppose we selected a set S of options with identical θi = θ and we then add anew option j with θj > θ.We firstly haveyˆ(S) = exp (δθ)∑i∈Syi.After adding the new option, we haveyˆ(S′) = exp(δθjyj + θ∑i∈S yiyj +∑i∈S yi)(yj +∑i∈Syi).Sinceδθjyj + θ∑i∈S yiyj +∑i∈S yi > δθyj + θ∑i∈S yiyj +∑i∈S yi = δθandyj +∑i∈Syi > ∑i∈Syi,we should have yˆ(S′) > yˆ(S).Similarly, it’s easy to prove that we should also include other option m such thatθm = θj.Lastly, we can show that all the options with the highest θs should always beincluded in the assortment.Proof. Suppose we know θˆ = max{θi} for i ∈ A and then we selected an assortment Swhich doesn’t include all the options with θˆ. In other words, ∃i ∈ A such that θi = θˆand i 6∈ S. We then add one of such an option j with θj = θˆ into S to form a newassortment S′.68We firstly haveyˆ(S) = exp(δ∑i∈S θiyi∑i∈S yi)∑i∈Syi.After adding the new option, we haveyˆ(S′) = exp(δθjyj +∑i∈S θiyiyj +∑i∈S yi)(yj +∑i∈Syi).Since θj = θˆ, we knowθj ∑i∈Syi > ∑i∈Sθiyiin which the equality holds only when θi = θˆ ∀i ∈ S.Then we haveyj(θj ∑i∈S yi −∑i∈S θiyi)(yj +∑i∈S yi)∑i∈S yi> 0,since the denominator is surely greater than 0.Then we haveθjyj ∑i∈S yi − yj ∑i∈S θiyi(yj +∑i∈S yi)∑i∈S yi> 0θjyj ∑i∈S yi − yj ∑i∈S θiyi +∑i∈S θiyi ·∑i∈S yi −∑i∈S yi ·∑i∈S θiyi(yj +∑i∈S yi)∑i∈S yi> 0(θjyj +∑i∈S θiyi)∑i∈S yi −(yj +∑i∈S yi)∑i∈S θiyi(yj +∑i∈S yi)∑i∈S yi> 0θjyj +∑i∈S θiyiyj +∑i∈S yi −∑i∈S θiyi∑i∈S yi > 0.That’s to say, we haveδθjyj +∑i∈S θiyiyj +∑i∈S yi > δ∑i∈S θiyi∑i∈S yi .As it’s clear thatyj +∑i∈Syi > ∑i∈Syi,then we know yˆ(S′) > yˆ(S).Therefore, we should include option j in our assortment. Thus, all items with thehighest θs should be included in our assortment.695.2.4 Two Heuristic AlgorithmsAlgorithm 1Considering our observations above, we may start with the subset of options withthe highest θ and use a greedy policy to keep adding one option at a time into themto try to increase the value of yˆ. But we still need to consider the impact of the com-bination of more than one option, so the remaining options still need to go throughthe normal enumeration process. So we propose the following algorithm:Firstly divide all options into two groups so that the first group contains all theoptions with the highest reliability. Then find the most “beneficial” option in thesecond group that can generate the largest yˆ when adding into the first group, andactually include it in the first group if the new yˆ is higher than its previous value.Repeat the last step until there is no element in the second group or the new yˆ is nothigher than its previous value. If there is any option left in the second group afterthe previous process, the remained options should be added according to the normalenumeration algorithm until all the combinations are exhausted. This algorithm usesa greedy policy to include as many options as possible in part 1 and minimize thenumber of options that have to go through the enumeration process in part 2. It canbe written as:70Algorithm 1: Greed-EnumerationInput: δ, A, y, θOutput: Assortment S∗1 begin2 θˆ ←− Max(θ)3 S∗ ←− A[θ = θˆ] . Obtain index of θi such that θi = θˆ4 Ss ←− A \ S∗5 if Ss 6= ∅ then6 yˆ←− CalHatY(δ, S∗, y, θ) . Function CalHatY calculates yˆ usingequation (5.5)7 repeat8 a←− Ss[1] . Obtain the first item in Ss9 Ss ←− Ss \ {a}10 M←− ∑i∈S∗ yi11 N ←− ∑i∈S∗ yiθi12 for b ∈ Ss do13 l ←− δ( yaθa+Nya+M −ybθb+Nyb+M)14 r ←− log( yb+Mya+M)15 if l < r then a←− b16 S′ ←− S∗ ∪ {a}17 yˆ′ ←− CalHatY(δ, S′, y, θ)18 if yˆ′ > yˆ then19 yˆ←− yˆ′20 S∗ ←− S′21 Ss ←− A \ S∗22 until yˆ′ 6 yˆ or Ss = ∅23 Ss ←− A \ S∗24 if |Ss| > 1 then25 S′ ←− S∗26 foreach I ⊂ Ss and |I| > 1 do27 S′′ ←− S′ ∪ I28 yˆ′ ←− CalHatY(δ, S′′, y, θ)29 if yˆ′ > yˆ then30 yˆ←− yˆ′31 S∗ ←− S′′32 return S∗71Algorithm 2Recall in the four examples for conventional sorting algorithms, we notice that al-though we would fail to obtain the optimal assortment if we stop adding the optionsonce the value yˆ decreases, the sorting may help and we may still have a good chanceto get a high yˆ if we don’t stop but continue adding the options. The proposed algo-rithm is:We firstly sort all the options by reliability and then by attractiveness in the de-scending order (reliability is the primary attribute, and if there is a tie in reliability,check the attractiveness to try to break the tie). Starting from the group of optionswith the highest reliability, we keep adding in the options down the list accordingto the order and compare the yˆ value in all cases to choose the assortment with thelargest yˆ value. This is another analogue of the revenue-ordered policy widely men-tioned in the literatures and can be written as:Algorithm 2: Reliability-Attractiveness-OrderedInput: δ, A, y, θOutput: Assortment S∗1 begin2 n←− |A|3 L←− Sort(A,θ,y,DESCEND) . Sort by θ and y descendingly andthen pass the ordered index to L4 θˆ ←− Max(θ)5 S∗ ←− A[θ = θˆ] . Obtain index of θi such that θi = θˆ6 Ss ←− A \ S∗7 k←− |S∗|8 if k < n then9 yˆ←− CalHatY(δ, S∗, y, θ) . Function CalHatY calculates yˆ usingequation (5.5)10 for i = (k + 1) to n do11 S′ ←− L[1 : i] . Pass index of first i items in list L to S′12 yˆ′ ←− CalHatY(δ, S′, y, θ)13 if yˆ′ > yˆ then14 yˆ←− yˆ′15 S∗ ←− S′16 return S∗72Numerical experimentsIn this part, we run 18 numerical experiments to test the performance of proposedtwo algorithms as well as the greedy policy (from Line 1 to 22) within Algorithm 1.First of all, we form a set of all possible options P consisting of 343 options witht ∈ {2, 4, 6, · · · , 14}, c ∈ {5, 7.5, 10, · · · , 20}, and θ ∈ {0.4, 0.5, 0.6, · · · , 1}. The set ofavailable options A that consists of n (n = 10, 12, 15) items is sampled randomly fromP without replacement. In each experiment, 20000 cases of A are generated. Theparameters for the base case are: α = 5, β = γ = 0.2, and δ = 2.5, and we have sixtreatments in total. The treatments are as follows:Table 5.6: Treatment on numerical experiments on proposed algorithmsTreatment I II III IV V VIChange inparametersBase Case δ = 5 β = 2 γ = 2 β = γ = 1δ = 5β = γ = 0.02We may first check the results for the greedy policy within Algorithm 1 and lookat the following measurements:• the percentage of cases in which greedy policy obtains the optimal yˆ: yˆG = yˆO,which is defined as 100/20000 ∗∑20000j=1 1(yˆGj = yˆOj)• the average ratio of yˆ obtained by greedy policy over the optimal yˆ: Mean yˆG/yˆO,which is defined as 100/|S| ∗∑j∈S yˆGj/yˆOj and we would see the average for allcases (All: S = {1, 2, · · · , 20000}) and the average for non-optimal cases only(Non-opt: S = {j|yˆGj 6= yˆOj})• the number of options that need to be processed with enumeration procedure:N. We would check both mean (N¯) and median (median(N)) of N as well asN¯/n and median(N)/n.73Table 5.7: Summary of numerical experiments on greedy policyTreatment n yˆG = yˆO(%)Mean yˆG/yˆO(%) N¯ N¯/n(%) median(N)median(N)/n(%)All Non-optIBase Case10 98.335 99.9717 98.3023 0.3328 3.3275 0 0.000012 98.110 99.9676 98.2870 0.3474 2.8946 0 0.000015 97.645 99.9720 98.8109 0.4057 2.7043 0 0.0000IIδ = 510 93.400 99.5723 93.5193 3.6561 36.5605 4 40.000012 92.090 99.5540 94.3613 4.5325 37.7704 5 41.666715 90.830 99.5670 95.2780 5.8787 39.1910 6 40.0000IIIβ = 210 99.855 99.9971 97.9782 0.5934 5.9340 0 0.000012 99.820 99.9966 98.0942 0.6944 5.7863 0 0.000015 99.725 99.9965 98.7402 0.8720 5.8130 0 0.0000IVγ = 210 99.925 99.9984 97.8716 0.5682 5.6815 0 0.000012 99.850 99.9976 98.3832 0.6915 5.7621 0 0.000015 99.815 99.9966 98.1780 0.8467 5.6447 0 0.0000Vβ = γ = 1δ = 510 99.540 99.9667 92.7620 2.5396 25.3955 2 20.000012 99.515 99.9738 94.6057 3.1609 26.3404 3 25.000015 99.325 99.9569 93.6160 4.1341 27.5603 4 26.6667VIβ = γ = 0.0210 96.980 99.9319 97.7454 0.1677 1.6765 0 0.000012 96.570 99.9391 98.2230 0.1808 1.5063 0 0.000015 96.370 99.9460 98.5125 0.1898 1.2653 0 0.0000We can see from the first three columns that the performance of greedy policyalone is not bad as it can achieve more than 90% in all the treatments. Its perfor-mance decreases obviously when n or δ increases but the increase in β and γ wouldhelp to improve it. The last four columns show that when δ is high, the contributionof single option to yˆ may not be much and when δ is not high enough, the greedypolicy would be able to determine that almost all options should be included in theassortment.74Then let’s turn to the results for the two proposed algorithms. We would look atthe following measurements and all results are presented in percentages (%):• the percentage of cases in which Algorithm i (i = 1, 2) obtains the optimal yˆ:yˆAi = yˆO, which is defined as 100/20000 ∗∑20000j=1 1(yˆAij = yˆOj)• the average ratio of yˆ obtained by Algorithm i (i = 1, 2) over the optimal yˆ:Mean yˆAi/yˆO, which is defined as 100/|S| ∗∑j∈S yˆAij/yˆOj and we would see theaverage for all cases (All: S = {1, 2, · · · , 20000}) and the average for non-optimalcases only (Non-opt: S = {j|yˆAij 6= yˆOj})• the average ratio of program processing time of Algorithm i (i = 1, 2) overthat of the normal enumeration algorithm: Mean ptAi/ptO, which is defined as100/20000 ∗∑20000j=1 ptAij/ptOj.8The results are summarized on the next page (“NA” denotes there is no non-optimal cases).The first two columns show that both algorithms work pretty well consideringthey find the optimal yˆ in more than 99% of all the cases. The performance of Algo-rithm 1 is considerably steady across all the treatments. The performance of Algorith-m 2, on the other hand, drops a little when β and γ are large.The second set of two measurements reveal that although these algorithms some-times cannot obtain the optimal yˆ, they still achieve reasonably good yˆ on averagefor all cases. Even if we only look at the non-optimal cases, in the majority of thetreatments they may still get 100% of yˆO on average after rounding.8This one may not be accurate because the actual implementation in R (R Core Team, 2015) is a littlebit different than the exact Algorithm 1 and Algorithm 2 in order to keep the same value precision andrecord the measurements for intermediate steps. These would make it a bit longer than the actual timethe algorithms require. Also note that all the functions are compiled with compiler package, and it mayinfluence the actual processing time.75Table 5.8: Summary of numerical experiments on proposed algorithmsTreatment n yˆA1 = yˆO(%)yˆA2 = yˆO(%)Mean yˆA1/yˆO(%)Mean yˆA2/yˆO(%) MeanptA1/ptO(%)MeanptA2/ptO(%)All Non-opt All Non-optIBase Case10 100.000 100.000 100.0000 NA 100.0000 NA 6.7429 1.124112 100.000 100.000 100.0000 NA 100.0000 NA 2.4967 0.307715 100.000 100.000 100.0000 NA 100.0000 NA 0.4973 0.0470IIδ = 510 100.000 100.000 100.0000 NA 100.0000 NA 8.7085 1.065712 99.995 100.000 99.9994 88.1672 100.0000 NA 3.9267 0.309615 100.000 100.000 100.0000 NA 100.0000 NA 1.2272 0.0486IIIβ = 210 100.000 99.835 100.0000 NA 100.0000 100.0000 6.8648 1.118712 99.990 99.765 100.0000 100.0000 100.0000 100.0000 2.5105 0.321515 100.000 99.720 100.0000 NA 100.0000 100.0000 0.5056 0.0471IVγ = 210 100.000 99.980 100.0000 NA 100.0000 100.0000 6.9288 1.094512 100.000 99.995 100.0000 NA 100.0000 100.0000 2.5065 0.319415 100.000 99.985 100.0000 NA 100.0000 100.0000 0.5072 0.0468Vβ = γ = 1δ = 510 100.000 99.910 100.0000 NA 100.0000 100.0000 7.9293 1.114712 100.000 99.910 100.0000 NA 100.0000 100.0000 3.3561 0.309315 100.000 99.920 100.0000 NA 100.0000 100.0000 0.9548 0.0479VIβ = γ = 0.0210 100.000 100.000 100.0000 NA 100.0000 NA 6.7629 1.069812 100.000 100.000 100.0000 NA 100.0000 NA 2.4860 0.308715 100.000 100.000 100.0000 NA 100.0000 NA 0.4947 0.0486The last but one column presents that large δ would increase the processing timeof Algorithm 1. The reason behind it is that large δ would make the seller more ‘se-lective’ and more options need to go through the normal enumeration operation. Butlarge β and γ would make it better. On the other hand, the time it takes to executeAlgorithm 2 is always lower than Algorithm 1 and not influenced by the model pa-rameters, as we can expect from its structure.Therefore, if we aim at minimizing the processing time, we may choose Algorithm2 as it can obtain fairly good results and its processing time is lower between the two76and steady under all circumstances. But if we want to secure better performance, wemay choose Algorithm 1 as it has steady and good performance while its processingtime is higher and subject to changes in model parameters.If the seller faces cardinality constraint and can only offer up to K options, al-though it may still lead to suboptimal results because it ignores the influence of com-binations of options, we would expect a greedy policy can help to identify a set ofoptions that deserve our attention and help to obtain a relatively good performance,and we may build on it to modify Algorithm 1. On the other hand, the way to mod-ify Algorithm 2 is not very clear since it may lead to similar problems as we sawin Examples 5.2 to 5.5. Besides them, another idea is to combine our current find-ings with some elimination-based algorithms such as the one proposed by Li andRusmevichientong (2014) for NL model to come up with some new algorithms.5.3 Further Remarks on Example 5.1We may notice the impact of Option 3 on Option 1 and 2 are different if we consid-er the choice of adding Option 3 to each of them, and we would find it’s not alwaysoptimal to exclude the unreliable Option 3: adding Option 3 to 1 would increase theseller’s expected profit but adding Option 3 to 2 would decrease his expected prof-it. Note Option 1 is more attractive (−βc1 − γt1 > −βc2 − γt2) while Option 2 ismore reliable (θ2 > θ1). So adding this dominated option to an attractive option mayactually bring net positive effect but adding it to a reliable option might hurt the seller.The aggregated rating r is not the highest when the seller obtains the highest prof-it, which shows that the seller’s interest in maximizing the profit doesn’t guaranteethe customers enjoy the best service assortment. He is more willing to add someattractive but less reliable options to please a larger range of customers than to servea smaller group of customers with better service quality. Although the customers’interests in service quality are not quite consistent with the seller’s priority, they do77protect the customers from some of the worst delivery assortments. We may expectthat when δ is extremely high, only the most reliable option(s) would be included, be-cause the seller would want to ensure the VIP-level service when faced with a groupof demanding clients who have strict requirements in service quality. They care lessabout the money and the seller can take this chance to increase the price.For the customers, we notice that both the average payment and the total paymentwould not be the highest when the seller obtains the highest profit, which is differentfrom the case in Example 4.1. At the same time, we can see that u¯ is not the lowest anylonger under the seller’s optimal assortment decision. They show that the customerswould benefit in many aspects from the seller’s exclusion of Option 3.Note the DSPs providing the two non-dominated options would be better off whenOption 3 is eliminated because of reduced competition, but the increase in buyer pro-portion is less than the decrease of λ3. On the other hand, the total service revenuesfrom the market when including only Option 3 could be higher than that in the sit-uation where two or three options are included. If the DSPs aim to maximize theirrevenues only, these signs show the possibility for collusion among the DSPs. In thefollowing two subsections, we would temporarily stray from our assumption that alldelivery options are exogenously given and the DSPs have no direct interactions, andlook at what might happen if we allow those possibilities.5.3.1 Possible Collusion among the DSPsFor the purpose of description, we will refer to the DSPs offering Option 1, 2,and 3 as DSP1, DSP2, and DSP3, respectively. Throughout this part, we consider themarket condition is unchanged, which means the model parameters are the same asin Example 5.1.We mentioned earlier that there exists possibility for collusion among the DSPs.Specifically, DSP3 would have incentive to bribe DSP1 and DSP2: considering the fact78that his option would be eliminated from the seller’s assortment, DSP3 may offersome contract in advance to encourage DSP1 and DSP2 to offer worse service optionsto the seller in order to induce the seller to include all three options. Let’s look at thefollowing example in which the reliabilities of the first two options are decreased by0.1 compared with Example 5.1:9Example 5.6. Anticipating Option 3 would be eliminated from the seller’s assortment,before the DSPs provide the information of their options to the seller, DSP3 proposes acontract to the other two DSPs so that they would provide the following informationto the seller: A = {1, 2, 3}, (t1 = 3, c1 = 5, θ1 = 0.7), (t2 = 2, c2 = 8, θ2 = 0.8),and (t3 = 6, c3 = 10, θ3 = 0.4). DSP3 would compensate for the loss in revenues ofthe other two DSPs (if any) and provide some reward e that is arbitrarily small butpositive to each of DSP1 and DSP2.Then the seller’s optimization problem would have the following results:Table 5.9: Optimal results in different assortments in Example 5.6S p λ1 λ2 λ3 Λ SerRev AvgPay TotPay u¯ r pi{1, 2, 3} 31.8793 0.3881 0.3022 0.1006 0.7909 5.3641 38.6617 30.5767 0.3510 0.7001 25.2126{1, 2} 31.7385 0.4441 0.3459 NA 0.7900 4.9873 38.0520 30.0591 0.6392 0.7438 25.0718{1, 3} 28.5756 0.6089 NA 0.1578 0.7667 4.6227 34.6049 26.5316 0.6813 0.6382 21.9089{2, 3} 28.3789 NA 0.5740 0.1911 0.7651 6.5028 36.8784 28.2151 0.6187 0.7001 21.7123{1} 28.1874 0.7635 NA NA 0.7635 3.8174 33.1874 25.3381 1.1719 0.7000 21.5207{2} 28.1874 NA 0.7635 NA 0.7635 6.1079 36.1874 27.6286 1.1719 0.8000 21.5207{3} 18.2926 NA NA 0.6356 0.6356 6.3555 28.2926 17.9814 0.5561 0.4000 11.6259The results show that now the seller would want to include all the three options.Compared with Example 5.1, λ1 and λ2 decrease by 0.0608 and 0.0473, respectively,which means DSP1 and DSP2 would lose 0.0608 ∗ 5 + 0.0473 ∗ 8 = $0.6824 for theirtotal revenues. But DSP3 would obtain 0.1006 ∗ 10 = $1.0060 and can easily fullycompensate the loss of the other two DSPs, provide the rewards, and still gain net9We limit our discussions in this section to the case where the quoted surcharges cannot be deceitfullyreported, considering they may be observed from other information channels. This is to acknowledgethe fact that there should be some public information in the market and to make the comparison easier.79benefit about $0.3236.This would be a win-win contract for the DSPs because DSP3 now enjoys positiverevenue while it brings twofold benefits to the other two DSPs: on one hand theymay get more than what they can obtain in Example 5.1, and on the other hand, theycould divert some important resources to provide better service for their other clientsor save some costs from not having to maintain a higher reliability.Note that this contract is compensation-based and the loss of market share isassessed by comparing a hypothetical state in which the collusion doesn’t happen(should-be) and the state described in the contract (would-be). It’s possible that DSP3doesn’t have accurate information about the other two options that are going to beprovided to the seller but only anticipates that Option 3 would be eliminated becauseof its low quality. Under such circumstance, expecting this possible bribe of DSP3,the other two DSPs would have the incentive to “bluff” about their actual capabilitiesin the hypothetical state to claim a higher loss from the collusion with DSP3 in orderto lure him to offer higher compensation. Let’s look at the following example inwhich DSP1 and DSP2 falsely report their reliabilities to be 0.1 higher than theiractual reliabilities in Example 5.1:Example 5.7. Anticipating DSP3 is going to offer a bribe contract as in 5.6 and DSP3doesn’t know the exact service terms they are going to quote to the seller, the othertwo DSPs would claim that they are going to provide the following delivery optionsto the seller: (t1 = 3, c1 = 5, θ1 = 0.9), (t2 = 2, c2 = 8, θ2 = 1), and they know Option3 would still be (t3 = 6, c3 = 10, θ3 = 0.4).Then the seller’s optimization problem would have the following results in thishypothetical situation:80Table 5.10: Optimal results in different assortments in Example 5.7S p λ1 λ2 λ3 Λ SerRev AvgPay TotPay u¯ r pi{1, 2} 34.3994 0.4532 0.3530 NA 0.8062 5.0899 40.7128 32.8226 0.7401 0.9438 27.7327{1, 2, 3} 34.2013 0.3950 0.3076 0.1024 0.8051 5.4603 40.9837 32.9950 0.4391 0.8746 27.5346{2} 30.7662 NA 0.7833 NA 0.7833 6.2665 38.7662 30.3660 1.2851 1.0000 24.0995{1} 30.7662 0.7833 NA NA 0.7833 3.9166 35.7662 28.0161 1.2851 0.9000 24.0995{1, 3} 30.6262 0.6213 NA 0.1611 0.7823 4.7169 36.6555 28.6764 0.7708 0.7971 23.9595{2, 3} 30.3114 NA 0.5852 0.1948 0.7801 6.6301 38.8109 30.2748 0.7040 0.8502 23.6447{3} 18.2926 NA NA 0.6356 0.6356 6.3555 28.2926 17.9814 0.5561 0.4000 11.6259The results show that, as we expect, the seller would not be willing to includeOption 3, which gives DSP3 the reason to bribe the other two DSPs. Comparingthe results in the contract in Example 5.6 and in the hypothetical case Example 5.7,λ1 and λ2 would hypothetically decrease by 0.0652 and 0.0508, respectively, whichmeans DSP1 and DSP2 would hypothetically lose around $0.7319 in total.10 As DSP3would obtain $1.0060, he can still fully compensate the “loss” of the other two DSPs,provide the rewards, and gain net benefit about $0.2741.The task of bluffing for the first two DSPs is costless. They don’t need to actuallyprovide the delivery options as in hypothetical case Example 5.7 since they know theywould only need to execute contract in Example 5.6.If we consider the bluffing of DSP1 and DSP2, we would naturally ask whetherDSP3 would also want to “counter-bluff”, in order to pay less compensation to DSP1and DSP2. But it won’t be hard to see that he doesn’t really have the chance to in-fluence the hypothetical state. There are only two possible outcomes resulting fromDSP3’s bluffing: (1) Option 3 is still not good enough for the seller and it won’tchange the outcome that only the first two options would be included in the assort-ment, which means the hypothetical state won’t be impacted and DSP3 cannot make10Due to rounding, the actual calculation should be 0.065168 ∗ 5 + 0.050753 ∗ 8 ≈ 0.7319, instead of0.0652 ∗ 5 + 0.0508 ∗ 8 = 0.7324. But it doesn’t influence the conclusion.81DSP1 and DSP2 “lose” less. (2) Option 3 looks very good, then the seller would in-clude it in the assortment, which eliminates the necessity to bribe DSP1 and DSP2 andmakes the proposal from DSP3 suspicious to the other two DSPs (if we consider thatonly the possible elimination of Option 3 would trigger DSP3 to propose a collusioncontract). This also suggests that whether DSP1 and DSP2 know the details of Option3 in advance may not be important for them.From the discussion above, we may expect the net gain of DSP3 would be mini-mized because of the bluffing of the other two DPSs, but we can show that if DSP1and DSP2 are not coordinated, the competition between them might make the life ofDSP3 easier. Let’s look at the following example in which DSP1 and DSP2 falselyreport their reliabilities to the maximum and the quote lead time to the minimum:Example 5.8. Anticipating DSP3 is going to offer a bribe contract as in 5.6 and DSP3doesn’t know the exact service terms they are going to quote to the seller, the othertwo DSPs would claim that they are going to provide the following delivery optionsto the seller: (t1 = 1, c1 = 5, θ1 = 1), (t2 = 1, c2 = 8, θ2 = 1), and they know Option 3would still be (t3 = 6, c3 = 10, θ3 = 0.4).Then the seller’s optimization problem would have the following results in thishypothetical situation:Table 5.11: Optimal results in different assortments in Example 5.8S p λ1 λ2 λ3 Λ SerRev AvgPay TotPay u¯ r pi{1, 2} 36.8805 0.5003 0.3190 NA 0.8192 5.0531 43.0486 35.2669 0.8427 1.0000 30.2138{1, 2, 3} 36.6455 0.4516 0.2880 0.0785 0.8181 5.3467 43.1812 35.3256 0.5830 0.9424 29.9789{1} 34.2093 0.8051 NA NA 0.8051 4.0256 39.2093 31.5682 1.4186 1.0000 27.5426{1, 3} 33.8777 0.6843 NA 0.1189 0.8032 4.6106 39.6179 31.8217 0.9872 0.9112 27.2110{2} 31.8154 NA 0.7905 NA 0.7905 6.3237 39.8154 31.4724 1.3277 1.0000 25.1488{2, 3} 31.3933 NA 0.6190 0.1687 0.7876 6.6385 39.8217 31.3652 0.7913 0.8715 24.7267{3} 18.2926 NA NA 0.6356 0.6356 6.3555 28.2926 17.9814 0.5561 0.4000 11.6259Comparing the results in the contract in Example 5.6 and in the hypothetical caseExample 5.8, λ1 and λ2 would hypothetically decrease by 0.1122 and 0.0168, respec-82tively, which means DSP1 and DSP2 would hypothetically lose around $0.6951 intotal.11 As DSP3 would obtain $1.0060, he can still fully compensate the “loss” ofthe other two DSPs, provide the rewards, and gain net benefit about $0.3109, whichwould be even higher than the bluffing case in Example 5.7, but still lower than theno-bluffing case in Example 5.1.This shows that a DSP with cost advantage would gain more from bluffing andthe DSP offering the worst option (and proposing the bribe contract) somehow losesless in this competitive bluffing game because he pays a lower unit price for the largermarket share loss. Note that if the constraint on the bluffing changes, we may evenobserve he might be able to get more in the case of bluffing than in the case of nobluffing.5.3.2 Possible Actions of the SellerIn previous discussions, we consider the situation in which the seller doesn’t knowanything about the underlying negotiations among the DSPs. Now let’s look at howthe seller’s involvement in the game might change the outcome.Recall in Example 5.6, ironically, in order to prevent the seller from finding outthe collusion via the rating number, DSP1 and DSP2 would have to actually decreasetheir service reliability instead of only quoting a lower reliability when making theoffer. Even if the seller knows the DSPs collude, he can do nothing after the servicereliabilities of the first two options actually drop, because his best response would beincluding all three options in such case. Therefore, the collusion may sustain and theseller incurs a loss in profit for sure.As we see in previous discussions, the seller would always suffer from the col-lusion of the DSPs, now let’s see what might happen if the seller knows some in-11Due to rounding, the actual calculation should be 0.112200 ∗ 5 + 0.016759 ∗ 8 ≈ 0.6951, instead of0.1122 ∗ 5 + 0.0168 ∗ 8 = 0.6954. But it doesn’t influence the conclusion.83formation in advance. Anticipating DSP3 would propose the contract as in Example5.6, before the proposal, the seller can negotiate with DSP3 to avoid the collusion bypaying DSP3 $0.3236, which equals the net benefit DSP3 can get from the collusionwithout bluffing. He would be willing to do so because his loss would be $1.1828if the collusion is successful, and DSP3 would be willing to accept that because thepossible risk for the other two DSPs’ bluffing is removed and his gain is maximizedgiven the terms in contract in Example 5.6.However, actually the seller has a better solution: simply including Option 3 in hisassortment in the first place in Example 5.1. This offer is better for both the seller andDSP3 because now the loss of the seller is only $0.0270, which is smaller than $0.3236,and on the other hand DSP3’s revenue can dramatically increase to $1.0153. Even ifwe consider the possible benefit from the uncoordinated bluffing of DSP1 and DSP2when the constraint on bluffing changes, this revenue can almost never be achievedfor DSP3 under a compensation-based contract as in Example 5.1.Therefore, the solution for the seller to deter such collusion and avoid greater lossis sacrificing some profit by including Option 3 in his assortment. These examplesdemonstrate that if the interactions among the DSPs and those between the DSPs andthe seller are considered, the optimal assortment might still be the one that includesall the available options.Note Example 5.6 may not be the best contract available for the DSP3 to proposeto the other two DSPs, while Example 5.7 and Example 5.8 only represent two pos-sible bluffing scenarios. We were not arguing that those examples reveal the exactequilibrium, but on the other hand demonstrating some states are unstable by show-ing profitable unilateral deviation from those states are possible, so we cannot simplyconclude from those examples that including all options is always optimal for theseller.84Since the focus of this research is not to make strategic decisions for the DSPs,we would like to re-emphasize our original assumption regarding exogeneity andwe will not formally explore further on how DSPs should respond to the seller’sintervention or the possible interactions among them. We will mention some possibleways to extend on this topic in the section of Future Research within Chapter 8. Forthe analysis and discussions below, our focal point is still on the situation in whichthe delivery options are exogenously given.85Chapter 6D: Pricing Product Bundle withCustomer RatingIn this problem, we consider a similar setting to Problem C but increase the sell-er’s pricing flexibility so that he can not only price the product but also list a differentsurcharge other than the one quoted by the DSP. We want to check how this changewould impact the seller’s assortment and pricing decision.Following the previous chapter, we assume the total market size is relatively con-stant over the periods. The problem for the seller is still to select the appropriatedelivery options and to price them as well as the product, to maximize the expectedprofit in each period in the steady state.The timeline is the same as in Figure Model Formulation6.1.1 Customers’ ChoiceThe customers make the purchasing decision according to MNL model and thetheory in standard MNL model suggests a customer chooses option i with the proba-86bility following equation (3.2).Now the customers’ mean utility associated with each delivery option i wouldbecomeui = α− β(p0 + p˜i)− γti + δr(·),in which p0 is the base price of the product, p˜i is the listed surcharge for deliveryoption i. The others are exactly the same as in Problem C except the form of r(·)would change, as we will talk about in the following part.Again, we may combine p0 and p˜i terms together as in Problem B. Then we mayrewrite ui asui = α− βpi − γti + δr(·), (6.1)in which pi is the price for the bundle of product plus delivery option i. We only needto implement the pricing decision in the way as we discussed in Chapter Customers’ RatingWe still consider a binary rating system in which all buyers truthfully rate aboutthe actual on-time delivery performance. We also assume the reliability informationof each option is provided to the seller but is not observable to the prospective cus-tomers.Therefore, the aggregated r, which also represents the proportion of buyers whoreceive on-time delivery, is the weighted average rating. Following (5.2), we can getthis lemma regarding its property over the periods.Lemma 6.1. The aggregated rating r in one period is not affected by the rating in previousperiod but is influenced by the bundle prices pis in current period.Proof. Let rk be the aggregated rating generated in period k and denote vi ≡ eui , then87in period k we haverk = ∑i∈S θiλi∑i∈S λi =∑i∈S θi vi1+∑j∈S vj∑i∈S vi1+∑j∈S vj=11+∑j∈S vj ∑i∈S θivi11+∑j∈S vj ∑i∈S vi= ∑i∈S θivi∑i∈S vi= ∑i∈S θieα−βpi−γti+δrk−1∑i∈S eα−βpi−γti+δrk−1= ∑i∈S θie−βpi−γti∑i∈S e−βpi−γti.As we can see, rk is not influenced by rk−1 but is influenced by pis. On the otherhand, since pis can influence rk, it will influence period k + 1 via rk.According to Lemma 6.1, we haver(S, p) = ∑i∈S θie−βpi−γti∑i∈S e−βpi−γti. (6.2)Comparing with Problem C, we know that the seller can enjoy more power oncontrolling the aggregated rating, which is a direct benefit of having pricing flexibility.6.1.3 Seller’s ProblemSince the total market size is relatively constant over the periods, in the steadystate, the price vector p = (p1, p2, · · · , pn) is the same across different periods.If we consider the market size changes and the seller is able to know in advance,then the seller would be motivated to optimize over periods. For example, anticipat-ing the market size would increase in the second period, the seller may be willing todecrease the price in the first period to boost the rating and charge higher prices inthe second period to take advantage of high rating. On the other hand, however, ifwe include strategic customers, some customers in the second period may be able toswitch to the first period to enjoy a lower price, which might negatively impact the88seller’s plan. But as we assume the total market size is relatively constant, we don’tneed to consider the intertemporal optimization problem.The objective of the seller is to maximize its expected profit from the customers inone period in the steady state, and it can be written asmaxS,p ∑i∈S(pi − ci)eui1 +∑j∈S euj = maxS,p ∑i∈S(pi − ci)eα−βpi−γti+δr(S,p)1 +∑j∈S eα−βpj−γtj+δr(S,p), (6.3)in which pi is the price for the bundle of product and delivery option i.6.2 AnalysisIn this section, we will first study the pricing decision and then look at the assort-ment decision.6.2.1 Pricing DecisionConstant reliability-adjusted mark-upIn this part, we first show that any two optimal mark-ups can be linked by theirreliabilities and we further quantify such relationship in subsequent steps.Theorem 6.1. At optimality, the mark-up of option j, mj, and mark-up of option k, mk, satisfymj −mk = δ(θk − θj)x,in which θi represents the reliability of option i and x is some constant that doesn’t depend onmi.Proof. Using First-Order-Condition, from (6.3) we know that∂pi(p)∂pi =−eα+δr(−βv˜i + δ ∂r∂pi ∑j∈S v˜j)∑j∈S v˜j(pj − cj)(1 + eα+δr ∑j∈S v˜j)2+(1 + eα+δr ∑j∈S v˜j) [v˜i − βv˜i(pi − ci) + δ ∂r∂pi ∑j∈S v˜j(pj − cj)](1 + eα+δr ∑j∈S v˜j)2= 089in which v˜i ≡ e−βpi−γti .Then we have:− eα+δr(−βv˜i + δ ∂r∂pi ∑j∈S v˜j)∑j∈Sv˜j(pj − cj)+(1 + eα+δr ∑j∈Sv˜j)[v˜i − βv˜i(pi − ci) + δ ∂r∂pi ∑j∈S v˜j(pj − cj)]= 0.We observe that there are many terms which don’t have the index i. To simplifythe notation, let’s denote A ≡ eα+δr, B ≡ ∑j∈S v˜j, and C ≡ ∑j∈S v˜j(pj − cj). Then wehave:−AC(−βv˜i + δB ∂r∂pi)+ (1 + AB)[v˜i − βv˜i(pi − ci) + δC ∂r∂pi]= 0.And it leads to:δC ∂r∂pi + v˜i [1 + AB + βAC− β(1 + AB)(pi − ci)] = 0. (6.4)According to (6.2), we have∂r∂pi = βv˜i∑j∈S v˜jθj − θi ∑j∈S v˜j(∑j∈S v˜j)2 .Let’s denote D ≡ ∑j∈S v˜jθj and substitute it back to (6.4), following the samenotation that B ≡ ∑j∈S v˜j and C ≡ ∑j∈S v˜j(pj − cj), then we haveβδv˜iC D− θiBB2 + v˜i [1 + AB + βAC− β(1 + AB)(pi − ci)] = 0.Rearranging the equation above, we can getv˜i[βδC D− θiBB2 + 1 + AB + βAC− β(1 + AB)(pi − ci)]= 0.So we either have v˜i = 0, which requires pi = ∞ or let the term in the squarebracket to be 0, which leads to1 + AB + βAC = β(1 + AB)(pi − ci)− βδC D− θiBB2 .90Note the equation above should be true for all i. Therefore, for option j and optionk, we should haveβ(1 + AB)(pj − cj)− βδCD− θjBB2 = β(1 + AB)(pk − ck)− βδCD− θkBB2 .Rearranging the equation above we haveβ[B(1 + AB)(pj − cj − pk + ck) + δC(θj − θk)]B = 0.Since β 6= 0, we have(pj − cj)− (pk − ck) = (θk − θj)δ CB(1 + AB) .Let mj = pj − cj, mk = pk − ck, and x = CB(1+AB) , we have:mj −mk = (θk − θj)δx.We notice that the LHS is just the difference between the mark-up for option jand k while the RHS is the (negative) difference between their reliabilities times δ andsome constant x that doesn’t depend on index j or k.Recall that the optimal mark-up is constant for all options in the assortment inprevious three problems. However, this theorem tells us this would only happen inrare cases for this problem. As we will show later that x 6= 0, there is only one chanceto allow we have mj = mk: θk = θj. This corresponds to the industry-level standardscenario in which every option has identical reliability. But we note this would not betrue for a more general market and we will show how a high reliability would bringcompetitive advantage for the DSP.Gallego and Wang (2014) introduce the idea of adjusted mark-up, which is definedas price minus cost minus the reciprocal of price sensitivity and prove this adjustedmark-up is constant for all items in a nest within a NL model. If we look from anotherperspective, we may present a comparable concept in the following corollary:91Corollary 6.1. The reliability-adjusted mark-up, which is defined asmi − (θ1 − θi)δx, or pi − ci − (θ1 − θi)δx,is constant at optimality for all options i ∈ S.Proof. According to Theorem 6.1, anchoring the mark-up for any option would allowus to represent the other mark-ups at optimality. Specifically, if we anchor on Option1,12 the mark-up of any item i ∈ S should satisfy:mi −m1 = (θ1 − θi)δx.Set m1 = m, then the mark-up of any option i ∈ S can be presented as:mi = (θ1 − θi)δx + m.In the special case of i = 1, it would be m1 = 0 · δx + m = m, which is constantwith our notation. Denoting θ˜i ≡ (θ1 − θi)δx, we have:mi − θ˜i = m. (6.5)Therefore, if we call mi − θ˜i reliability-adjusted mark-up, then at optimality, all theitems in the assortment should have the same reliability-adjusted mark-up.We will provide more remarks on the reliability-adjusted mark-up once we iden-tify the value of x, which is what we are going to do in the second half of this part.According to (6.5), we may write the price of each item as:pi = mi + ci = m + θ˜i + ci. (6.6)Then we may have the following lemma:Lemma 6.2. At optimality, the aggregated rating r∗ in the steady state is a constant thatdoesn’t depend on the constant reliability-adjusted mark-up m.12It’s easy to show that we can anchor on any mark-up and the result would be similar. For consistency,throughout this thesis we anchor on Option 1, and we will show later that we can obtain a generalrepresentation that doesn’t require any anchoring.92Proof. According to (6.2) and (6.6), we have:r∗ = ∑i∈S θie−βpi−γti∑i∈S e−βpi−γti= ∑i∈S θie−β(m+θ˜i+ci)−γti∑i∈S e−β(m+θ˜i+ci)−γti= e−βm ∑i∈S θie−β(θ˜i+ci)−γtie−βm ∑i∈S e−β(θ˜i+ci)−γti= ∑i∈S θie−β(θ˜i+ci)−γti∑i∈S e−β(θ˜i+ci)−γti.Before obtaining the value of x, we will need the following lemma:Lemma 6.3. The x in Theorem 6.1 equals seller’s profit multiplying the proportion of non-buyers divided by the proportion of buyers at optimality, i.e.,x = 1−Λ∗Λ∗ pi∗,in which Λ∗ is the optimal proportion of buyers and pi∗ is the optimal profit.Proof. Let’s go back to the term x = C/(B(1 + AB)). Substitute the original termsback, we haveCB(1 + AB) =∑j∈S v˜j(pj − cj)∑j∈S v˜j(1 + eα+δr ∑j∈S v˜j)= ∑j∈S e−βpj−γtj(pj − cj)∑j∈S e−βpj−γtj(1 + eα+δr ∑j∈S e−βpj−γtj)=eα+δr ∑j∈S e−βpj−γtj(pj − cj)eα+δr ∑j∈S e−βpj−γtj(1 +∑j∈S eα−βpj−γtj+δr)= 1∑j∈S eα−βpj−γtj+δr∑j∈S(pj − cj)eα−βpj−γtj+δr1 +∑j∈S eα−βpj−γtj+δr= 1∑j∈S eα−βpj−γtj+δrpi.in which the last equality follows (6.3) and (6.1).93On the other hand, according to (3.2) and (6.1), we should have:Λ ≡ ∑i∈Sλi = ∑i∈S eα−βpi−γti+δr1 +∑j∈S eα−βpj−γtj+δr.Then we have1−Λ = 11 +∑j∈S eα−βpj−γtj+δr.That’s to say1∑j∈S eα−βpj−γtj+δr= 1−ΛΛ . (6.7)Therefore, we know at optimality in Theorem 6.1:CB(1 + AB) =1−Λ∗Λ∗ pi∗.Then we have:x = 1−Λ∗Λ∗ pi∗.Now we show the value of x:Theorem 6.2. The x in Theorem 6.1 equals 1 over customers’ price sensitivity, i.e., x = 1/β.Proof. Using (6.6), we may re-write the seller’s profit function as:pi(m) = ∑i∈S(m + θ˜i) eα−β(m+θ˜i+ci)−γti+δr1 +∑j∈S eα−β(m+θ˜j+cj)−γtj+δr= ∑i∈S(m + θ˜i) eα−βm+δre−β(θ˜i+ci)−γti1 + eα−βm+δr ∑j∈S e−β(θ˜j+cj)−γtj= eα−βm+δrm∑i∈S e−β(θ˜i+ci)−γti + eα−βm+δr ∑i∈S θ˜ie−β(θ˜i+ci)−γti1 + eα−βm+δr ∑j∈S e−β(θ˜j+cj)−γtj.Noticing some terms don’t depend on m, we can simplify the notation by denotingV ≡ ∑i∈S e−β(θ˜i+ci)−γti and Z ≡ ∑i∈S θ˜ie−β(θ˜i+ci)−γti , and then we may have:pi(m) = eα−βm+δrmV + eα−βm+δrZ1 + eα−βm+δrV .94By First-Order-Condition, with the help of Lemma 6.2, we have:∂pi(m)∂m =Veα−βm+δr [Veα−βm+δr − βZ/V − (βm− 1)](1 + Veα−βm+δr)2= 0.That’s to say, we haveVeα−βm+δr − βZ/V − (βm− 1) = 0.Then we have:Veα+δr−βm = βm + β ZV − 1. (6.8)On the other hand, applying the same notation to (6.7), we may have:1−ΛΛ =1∑j∈S eα−βpj−γtj+δr= 1eα−βm+δr ∑j∈S e−β(θ˜i+ci)−γtj= 1eα−βm+δrV . (6.9)Combining with the equation of pi(m), we know:1−ΛΛ pi =1eα−βm+δrVeα−βm+δrmV + eα−βm+δrZ1 + eα−βm+δrV= 1VmV + Z1 + eα−βm+δrV= m +ZV1 + eα−βm+δrV .Combining with (6.8), we know that at optimality:1−Λ∗Λ∗ pi∗ = m +ZV1 + βm− 1 + βZV= m +ZVβm + βZV= 1β .We then know x = 1/β by Lemma 6.3.According to Theorem 6.2 and (6.5), we have:θ˜i =δβ (θ1 − θi),mi = δβ (θ1 − θi) + m,(6.10)95and by Lemma 6.2, we have:r∗ = ∑i∈S θie−β(θ˜i+ci)−γti∑i∈S e−β(θ˜i+ci)−γti= ∑i∈S θie−βci−γti+δθi∑i∈S e−βci−γti+δθi. (6.11)Note that in Gallego and Wang (2014), the “adjustment” is based on the price sen-sitivity only, but (6.10) tells us the “adjustment” in our case is based on two otherfactors apart from the price sensitivity: rating (or service) sensitivity and the differ-ence between the delivery service reliabilities of any option and the anchoring option(Option 1 in our case). Specifically, higher service sensitivity and larger differencebetween the two reliabilities would mean higher magnitude of the adjustment whiletheir comparative relationship decides the direction of the adjustment — higher (com-pared to Option 1) reliability would mean lower (compared to Option 1) non-adjustedmark-up. The impact of price sensitivity is similar in the two cases. On the otherhand, (6.10) also shows the value of the product, the time sensitivity, and the originalquoted surcharge would not influence the adjustment.This suggests that the seller should differentiate the bundle price of each option sothat the options with higher reliabilities would eventually have lower mark-ups, andthe more the customers care about the service quality and the less the customers careabout the money they spend, the more divergent the mark-ups should be. In addition,a pair of two options with larger gap in reliabilities should have larger difference intheir mark-ups compared with another pair of two options with smaller gap. Thisclearly shows the seller’s interests in exploiting the available resources for mark-updifferentiation. However, the quoted surcharges should not influence the differencein mark-ups.Unimodal profit function and optimal mark-upsNow the pricing problem is greatly reduced to a one-dimension search for theoptimal m∗ in pi(m). In this part, we will prove the unimodal property of the profitfunction and derive the closed-form solution for the optimal mark-up. Firstly, wepresent the following theorem:96Theorem 6.3. Letting m1 = m and setting the item with lowest reliability to be Option 1,i.e., θ1 = min{θi} ∀i ∈ S, then the profit function with respect to m is strictly unimodal onits domain [0,∞).Proof. Recall that∂pi(m)∂m =Veα−βm+δr [Veα−βm+δr − βZ/V − (βm− 1)](1 + Veα−βm+δr)2.It’s clear that Veα−βm+δr > 0, then the sign of ∂pi(m)∂m only depends on the term:Veα−βm+δr − β ZV − (βm− 1).Let m˜ ≡ βm, then we may transform the mentioned term into:h(m˜) = Veα−m˜+δr − β ZV + 1− m˜.With the help of (6.10), we may write it as:h(m˜) = Veα−m˜+δr − β ZV + 1− m˜= eα−m˜+δr ∑i∈Se−βci−γti+δθi−δθ1 − βδ/β∑i∈S(θ1 − θi)e−βci−γti+δθi−δθ1∑i∈S e−βci−γti+δθi−δθ1+ 1− m˜= e−m˜eα+δr−δθ1 ∑i∈Se−βci−γti+δθi − δ∑i∈S(θ1 − θi)e−βci−γti+δθi∑i∈S e−βci−γti+δθi+ 1− m˜.Denoting ki ≡ e−βci−γti+δθi (it’s clear that ki > 0), we may simplify the notationand rewrite-it as:h(m˜) = e−m˜eα+δr−δθ1 ∑i∈Ski − δ∑i∈S(θ1 − θi)ki∑i∈S ki + 1− m˜= e−m˜eα+δr−δθ1 ∑i∈Ski + δ∑i∈S(θi − θ1)ki∑i∈S ki − m˜ + 1.Recall that although we let m1 = m, we didn’t impose any restriction on Op-tion 1, so any option i ∈ S could be treated as Option 1. Without loss of generality,now we set the option with the lowest reliability to be Option 1. That’s to say nowθ1 = min{θi} ∀i ∈ S.97To study the property of h(m˜), we would like to find out suitable domain ofm˜ first. According to (6.5), we know that m should be the highest among mi, i.e.,m = max{mi} ∀i ∈ S, because now θ˜i 6 0. Then we must have m > 0. Otherwise, wewould have mi < 0 and the seller would have negative profit, but we know under nocircumstance the seller should set mi < 0 ∀i ∈ S, considering he can always set mi = 0∀i ∈ S and obtains zero profit, which is higher than the negative profit. Hence, forthe domain of pi(m˜), we only need to focus on m˜ > 0 considering m˜ = βm and β > 0.We notice:∂h(m˜)∂m˜ = −e−m˜eα+δr−δθ1 ∑i∈Ski − 1 < 0.That’s to say, h(m˜) is strictly decreasing. On the other hand, it’s easy to see thatlimm˜→∞ h(m˜) = −∞ and we also know thath(0) = eα+δr−δθ1 ∑i∈Ski + δ∑i∈S(θi − θ1)ki∑i∈S ki + 1 > 0,because θi − θ1 > 0 leads to ∑i∈S(θi−θ1)ki∑i∈S ki > 0 with the equality holding only whenθi = θ1 ∀i ∈ S and the other terms are positive.Therefore, we can conclude that h(m˜) has positive value when m˜ is small and asm˜ increases, h(m˜) decreases to negative values. There is only one root m˜∗ for theequation h(m˜) = 0.Hence, we showed that when m < m˜∗/β, pi(m) is increasing and when m > m˜∗/β,pi(m) is decreasing. That’s to say, function pi(m) is unimodal in m.This lemma suggests that First-Order-Condition guarantees global optimality. Thenwe may present the following theorem.Theorem 6.4. The optimal reliability-adjusted mark-up m anchoring at Option 1 (i.e., m1 =m) should be:m = 1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]− δβ (θ1 − r∗) ,98in which W(·) is the Lambert W function and r∗ is determined as in (6.11).Proof. According to (6.8), we should have:Veα+δr+βZ/V−1 = eβm−1+βZ/V(βm− 1 + βZV).By using Lambert W function again, we have:βm− 1 + βZV = W(Veα+δr+βZ/V−1).Therefore, we havem = 1β[W(Veα+δr+βZ/V−1)+ 1]− ZV .Note the value of r, V, and Z are all known and we can further simplify thenotation. Firstly, let’s focus on the term δr + βZ/V. Using the form in Lemma 6.2, wehave:r = ∑i∈S θie−β(θ˜i+ci)−γtiV .Then we have:δr + βZ/V = δ∑i∈S θie−β(θ˜i+ci)−γti + β∑i∈S θ˜ie−β(θ˜i+ci)−γtiV= ∑i∈S(δθi + βθ˜i)e−β(θ˜i+ci)−γtiV=∑i∈S[δθi + β δβ (θ1 − θi)]e−β(θ˜i+ci)−γtiV= ∑i∈S [δθi + δ(θ1 − θi)] e−β(θ˜i+ci)−γtiV= δθ1 ∑i∈S e−β(θ˜i+ci)−γtiV= δθ1VV= δθ1.That’s to say, we will also have:ZV =δβ (θ1 − r) .99Therefore, we knowm = 1β[W(Veα+δr+βZ/V−1)+ 1]− ZV= 1β[W(Veα+δθ1−1)+ 1]− δβ (θ1 − r)= 1β[W(eα+δθ1−1 ∑i∈Se−β(θ˜i+ci)−γti)+ 1]− δβ (θ1 − r)= 1β[W(∑i∈Seα+δθ1−1−βθ˜i−βci−γti)+ 1]− δβ (θ1 − r)= 1β[W(∑i∈Seα+δθ1−1−β δβ (θ1−θi)−βci−γti)+ 1]− δβ (θ1 − r)= 1β[W(∑i∈Seα+δθ1−1−δθ1+δθi−βci−γti)+ 1]− δβ (θ1 − r)= 1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]− δβ (θ1 − r) .The next corollary specifies the optimal mark-up and the optimal price for everyitem i ∈ S:Corollary 6.2. The optimal mark-up mi for option i ∈ S is:mi = 1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]+ δβ (r∗ − θi) ,and the optimal price for option i ∈ S is:pi = 1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]+ δβ (r∗ − θi) + ci,in which W(·) is the Lambert W function and r∗ is determined as in (6.11).100Proof. According to Theorem 6.4 and (6.10), we have:mi = δβ (θ1 − θi) + m= δβ (θ1 − θi) +1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]− δβ (θ1 − r∗)= 1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]− δβ (θ1 − r∗ − θ1 + θi)= 1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]+ δβ (r∗ − θi) .Then according to (6.6), we have:pi = 1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]+ δβ (r∗ − θi) + ci.Recall in our previous three problems, the optimal bundle mark-up or base prod-uct price should always be positive but Corollary 6.2 shows that not all the optimalmark-ups should necessarily be positive in this problem. Let’s look at the followingexample:Example 6.1. We set α = 5, β = 0.3, γ = 1.8, δ = 5, S = {1, 2, 3}, (t1 = 5, c1 = 5,θ1 = 0.4), (t2 = 6, c2 = 7, θ2 = 0.9), and (t3 = 4, c3 = 5, θ3 = 0.6).The optimal results in different assortments are:Table 6.1: Optimal results in different assortments in Example 6.1S m1 m2 m3 λ1 λ2 λ3 Λ SevRev u¯ r pi{1, 2, 3} 7.9458 -1.2208 4.2792 0.0089 0.0126 0.1612 0.1826 0.9384 -1.9400 0.6110 0.7449{2, 3} NA -1.0532 4.4468 NA 0.0128 0.1638 0.1766 0.9087 -1.7992 0.6217 0.7151{1, 3} 7.5112 NA 3.8445 0.0091 NA 0.1650 0.1741 0.8703 -1.7620 0.5896 0.7024{3} NA NA 4.0053 NA NA 0.1678 0.1678 0.8388 -1.6016 0.6000 0.6719{1, 2} 8.8166 -0.3500 NA 0.0127 0.0181 NA 0.0308 0.1902 -4.1265 0.6933 0.1060{2} NA 3.3963 NA NA 0.0185 NA 0.0185 0.1298 -3.9689 0.9000 0.0630{1} 3.3780 NA NA 0.0132 NA NA 0.0132 0.0661 -4.3134 0.4000 0.0446101The results show that when the seller includes all three options in his assortmen-t, Option 2 should have negative mark-up while the other two should have positivemark-ups. On the other hand, if the seller eliminates Option 2, his optimal expectedprofit in the assortment S = {1, 3} would decrease about 5.70% compared to includ-ing all three options.Note that although Option 2 has the lowest mark-up, it doesn’t have the highestmarket share, because the other attributes are also very important in determining theoverall attractiveness.Gallego and Wang (2014) show that in a NL model, only when the nest coefficientis higher than 1, the optimal adjusted mark-up can be negative while when the nestcoefficient is no more than 1, the optimal adjusted mark-up must be strictly positive.Our standard MNL model is equivalent to a NL model with only one nest that has thecoefficient of 1. But our reliability-adjusted mark-up in this problem is not directlycomparable to their adjusted mark-up here because it would change in accordancewith the anchoring option according to our definition. In Example 6.1, if we anchorat Option 3, our reliability-adjusted mark-up would be positive, but if we anchor atOption 2, it would be negative. However, we note that if we anchor at the option withthe lowest reliability (e.g., Option 1 in Example 6.1), it should always be non-negative(positive if we consider the seller always obtains positive profit), as we discussed inthe proof of Theorem 6.3.More importantly, the results of Gallego and Wang (2014) also indicate that whenthe nest coefficient is no more than 1, all the margins (non-adjusted mark-ups) shouldalways be strictly positive as well. Our results show that the mark-up could be neg-ative in this problem when the nest coefficient equals 1. The different outcomes mayresult from the different roles that the “loss-leaders” play in two different problems.They comment that the “loss-leader” phenomenon arises because the products102with negative adjusted mark-ups or even negative margins help to attract more at-tention to the nest. In our problem, the “loss-leaders” on one hand help to attractcustomers and decrease no-purchase probability, and on the other hand encouragemore customers to choose options with higher reliability, which would increase theweight of those options in the aggregated rating and thus improve the rating val-ue. This also can help to explain why Option 2 in Example 6.1 should have negativemark-up.When all the options are not very good, the no-purchase probability could be high.When the nest coefficient is no more than 1, in the problem where service quality andrating is not considered, turning some options into “loss-leaders” would actually de-crease the profit, although it increases the proportion of buyers, because it decreasesthe market share of those options with positive mark-ups and it contributes negativeprofit directly. However, the rating in our problem matters, so turning some optionswith high reliability option into “loss-leaders” may help even when the nest coeffi-cient is no more than 1. On one hand, the market share of those options with positivemark-ups may not decrease or even increase because the improved rating increasesthe attractiveness of all options. On the other hand, it would help to weaken the po-tential negative impact on the aggregated rating that results from increasing numberof customers who choose low-reliability options.Note that when the reliabilities of all options are relatively close, these two benefitswould not be very prominent and the usage of “loss-leaders” may not help much.Corollary 6.2 shows that when the reliabilities of all options are relatively close, thechance to obtain a negative mark-up would be low (since r∗ and θi would be close),which supports our explanation.6.2.2 Assortment DecisionIn order to find out the optimal assortment, it’s helpful to find out the representa-tion for the seller’s profit. We present this result in the following corollary:103Corollary 6.3. The seller’s optimal profit can be represented as:pi = 1βW(∑i∈Seα−1−βci−γti+δθi),in which W(·) is the Lambert W function.Proof. By Lemma 6.3 and Theorem 6.2, we have:1−Λ∗Λ∗ pi∗ = 1β .Then we know at optimality:pi = 1βΛ∗1−Λ∗ .According to (6.9), we have:Λ∗1−Λ∗ = eα−βm+δrV.So we know:pi = 1β eα−βm+δrV.Using (6.8) again, we havepi = 1β eα−βm+δrV = 1β(βm− 1 + βZV)= m− 1β +ZV .Using Theorem 6.4, we havepi = 1β[W(∑i∈Seα−1−βci−γti+δθi)+ 1]− ZV −1β +ZV= 1βW(∑i∈Seα−1−βci−γti+δθi)+ 1β −1β= 1βW(∑i∈Seα−1−βci−γti+δθi).According to this corollary, since W(·) is increasing in (0,∞), it’s clear that theseller should include as many options as possible. That’s to say, in all optimal results104we obtained in this problem, we should set S = A. Therefore, “the more, the better”policy is still optimal in this problem.Note that if the seller faces cardinality constraint for some reason, a simple sortingalgorithm would suffice to find the optimal assortment. We can sort all the items byki = −βci − γti + δθi and the first K options with highest kis should be selected in theseller’s assortment if he can only choose up to K options.Although the optimal profits in Problem C and Problem D are not directly com-parable, we know that with the pricing flexibility, the seller can effectively mimic thepricing strategy in the setting of Problem C, where he has only the product price todecide, if he finds it’s optimal to set a constant mark-up for all the bundles. Sincethe optimal solution shows he should not do so, we can infer that the seller’s optimalprofit in Problem D cannot be lower than that in Problem C.6.2.3 Numerical ExampleLet’s look at the following example.Example 6.2. We set α = 5, β = 0.15, γ = 0.2, δ = 2.5, A = {1, 2, 3}, (t1 = 3, c1 = 5,θ1 = 0.8), (t2 = 2, c2 = 8, θ2 = 0.9), and (t3 = 6, c3 = 10, θ3 = 0.4).We can find the optimal mark-ups mi, the market share of each delivery optionλi, the total revenue for the DSPs (SerRev = ∑i∈S λici), customers’ average payment(AvgPay = 1/Λ∑i∈S λi pi), customers’ total payment (TotPay = ∑i∈S λi pi), customers’average mean utility u¯ = 1/Λ∑i∈S λiui, the aggregated rating r, and the seller’s profitpi under different assortment conditions. We summarize these related results in thefollowing table ordered by profit for the purpose of comparison.105Table 6.2: Optimal results in different assortments in Example 6.2S m1 m2 m3 λ1 λ2 λ3 SevRev TotPay u¯ r pi{1, 2, 3} 33.8436 32.1770 40.5103 0.3818 0.3818 0.0364 5.3282 32.0132 0.5403 0.8295 26.6850{1, 2} 33.9368 32.2701 NA 0.3993 0.3993 NA 5.1910 31.6277 0.6845 0.8500 26.4368{1, 3} 29.3591 NA 36.0257 0.7097 NA 0.0677 4.2250 27.4979 0.9545 0.7652 23.2728{2, 3} NA 29.2140 37.5473 NA 0.7097 0.0677 6.3540 29.6268 0.9545 0.8565 23.2728{1} 29.4685 NA NA 0.7738 NA NA 3.8688 26.6707 1.2297 0.8000 22.8019{2} NA 29.4685 NA NA 0.7738 NA 6.1902 28.9920 1.2297 0.9000 22.8019{3} NA NA 18.2926 NA NA 0.6356 6.3555 17.9814 0.5561 0.4000 11.6259As we can see, the biggest distinction compared with Problem C is that when theseller includes all the available delivery options, his profit is the highest and now it’salso higher than the optimal profit in Example 5.1. Comparing with the S = {1, 2} as-sortment scenario, we notice that the mark-ups of Option 1 and 2 actually drop whenthe seller includes Option 3, which is different from the case in Problem C where theprice increases when the seller’s profit increases.In this example, the seller would be better off by including the 3rd delivery op-tion in his assortment despite the fact that it’s not a preferable option. The pricingflexibility offers the seller more power on controlling the aggregated rating, whichis dramatically different from the case in Problem C where the seller can deal witha limited number of ratings because of limited number of combinations of options.Now the seller can charge high enough on Option 3 to prevent a large number ofcustomers choosing Option 3 while take advantage of the increased availability ofchoices for customers. The simultaneous decreasing in mark-ups for Option 1 andOption 2 coordinates with this strategy.As a side note, we would like to remark that there is still possibility for collusionin this example. Although all three options would be included anyway, the first twoDSPs can still deliberately decrease the quality of the first two options in order to letDSP3, who has higher unit surcharge, acquire more market share and their loss could106be compensated by the gain of DSP3 while three parties would still obtain positivebenefit from the collusion. For example, a possible contract could be: (t1 = 8, c1 = 5,θ1 = 0.4), (t2 = 7, c2 = 8, θ2 = 0.5), and (t3 = 6, c3 = 10, θ3 = 0.4)6.3 Summary of Optimal Assortment and Pricing DecisionsIn this section, we summarize optimal assortment and pricing decisions in all fourproblems. Let’s look at the summary for optimal assortment decisions first.Table 6.3: Comparison of optimal assortment decisionsPricing flexibility Rating availabilityNo rating Aggregated ratingRestricted A arg maxS⊂Ayˆ(S)Flexible A AAs we can see, it is generally optimal for the seller to include all available optionsin his assortment, except when aggregated customer rating is available and he canonly price the product. However, we need to note that the possible collusion amongthe DSPs may force the seller to include all the options if we consider the interactionsamong them, as we discussed in Chapter 5.Then let’s see the summary for optimal pricing decisions.Table 6.4: Comparison of optimal pricing decisionsPricing flexibility Rating availabilityNo rating Aggregated ratingRestricted (p) 1β[1 + W (∑i∈A eα−1−βci−γti)] 1β[1 + W (∑i∈S eα−1−βci−γti+δr)]Flexible (mi) 1β[1 + W (∑i∈A eα−1−βci−γti)] 1β[1 + W (∑i∈A eα−1−βci−γti+δθi)]+δβ (r∗ − θi)107When there is no aggregated rating, the seller’s pricing flexibility won’t changehis pricing decision: even if he can re-price the quoted surcharge, he would stillset a constant mark-up for all the options. But when there is aggregated customerrating, the seller should not charge a constant mark-up for all the options when heenjoys the pricing flexibility, and the differentiation is based on the reliabilities of eachoption. We also should note that the seller’s optimal pricing decision depends onthe assortment decision when he can only price the product and there is aggregatedcustomer rating.6.4 Value of Pricing FlexibilityIn order to understand the value of pricing flexibility, we need to investigate theoptimal profit in each problem. Corollary 6.3 shows a very important property: thestructure of the optimal profit equation is very similar to the one in Problem B (andProblem A, since the optimal profits are the same in those two problems), and theonly difference is that one more element eδθi is multiplied to each term summed to-gether within the Lambert W function.Then it’s useful to introduce the disaggregated rating case as an intermediate stepthat can help us compare and draw the conclusion. If we add the θi attribute as deliv-ery reliability of each option i and allow the customers to observe them directly in thesetting of Problem A and B in which the aggregated rating element is not available,we can expect to get exactly the same optimal profit here in Problem D, consideringthe structure of the solution. In fact, this would essentially be the option-specific (dis-aggregated) rating case in which the customers observe the actual reliability of eachoption via ratings given all the buyers truthfully rate on the actual on-time deliveryperformance.We may compare the seller’s optimal profit in the six scenarios:108Table 6.5: Comparison of optimal profits in six scenariosRating availability Pricing flexibilityRestricted FlexibleNo rating 1βW(∑i∈A eα−1−βci−γti) 1βW(∑i∈A eα−1−βci−γti)Aggregated rating 1βW(∑i∈S eα−1−βci−γti+δr) 1βW(∑i∈A eα−1−βci−γti+δθi)Disaggregated rating 1βW(∑i∈A eα−1−βci−γti+δθi) 1βW(∑i∈A eα−1−βci−γti+δθi)The comparison among those six problems suggests that:1. Rating aggregation cannot help the seller to obtain a higher profit. If the sell-er has a choice between displaying option-specific rating and aggregated rating, hewould not be better off by showing the aggregated rating. His highest expected profitwould be the same under these two different rating systems.2. When rating is aggregated, the lack of pricing flexibility would hurt the seller’sprofit but smart assortment strategy can help to weaken its negative impact. As wesee in Problem C where the seller can only price the product, the aggregated ratinghelps to enlarge the market share of those options with lower reliabilities but shrinkthe market share of those options with higher reliabilities. This would in turn hurt theoverall rating since the weights for low reliabilities increase while the weights for highreliabilities decrease. Then all the parties are hurt because of the reduced aggregatedrating. By selecting appropriate options in his assortment, the seller has a chance tominimize the negative impact.3. The market is optimal when all the customers know true attributes of eachoption. Trying to distort the information by hiding the option-specific attribute wouldnot benefit, if not hurt, the seller. The optimal pricing strategy for the seller in ProblemD is simply charging a “service quality premium” from those options benefit froman aggregated rating that is higher than their actual reliabilities and subsidizing a109“service quality bonus” to those options hurt because their reliabilities are actuallyhigher than the rating so that eventually the customers choose according to and payfor the actual attributes of those options. Intuitively speaking, the seller would wantto attract more customers to the options with high reliabilities and discourage toomany customers from choosing options with low reliabilities and the optimal balancehappens to be the state where the listed surcharge of each option fully reflects theircharacteristics and not a single option is benefiting from or hurt by the aggregatedrating. To understand this, we can write the customers’ mean utility for each option iunder the seller’s optimal pricing strategy:ui = α− β[1β +1βW(∑i∈Aeα−1−βci−γti+δθi)+ δβ (r∗ − θi) + ci]− γti + δr∗= α− β[1β +1βW(∑i∈Aeα−1−βci−γti+δθi)+ ci]− γti + δθi,and we can check it would be exactly the same as in the disaggregated rating casesin which the customers know the information for all attributes before making choices.4. The power of pricing flexibility lies in its ability to effectively disaggregate theaggregated information and remove information distortion, so if there is no distor-tion, such flexibility would not help and the seller should be indifferent between thetwo pricing schemes, just as in Problem A and B. In other words, pricing flexibilityonly helps to improve from sub-optimal state to optimal state, but it cannot help toreach a state that is better than the status where the information for options is dis-aggregated and accurate. On the other hand, the seller’s assortment adjustment inProblem C would relieve the negative influence of rating aggregation because it helpsto control the scope of information distortion.5. This conclusion can be extrapolated to other similar scenarios in which someinformation about option attributes is aggregated and customers generate ratings to“communicate”. We show that as an information exchange channel, aggregated cus-tomer rating is still very efficient so that the seller has to reduce the information110distortion through assortment adjustment when he doesn’t have pricing flexibility, orto reflect true attribute information through differentiated prices when he has pricingflexibility.111Chapter 7Numerical StudiesIn our previous numerical examples within each chapter, we compared optimaland non-optimal solutions within each problem, but we’re also interested in how thedifferent combinations of the model parameters would influence the optimal results.In this chapter, we will conduct numerical studies to investigate how the seller, theDSPs, and the customers are influenced by different model parameter combinations,and present the results with graphs as they can let us have a better understanding ofthe relationships.Since the focus of this thesis is on Problem C and D, we ignore Problem A andB in following studies. Recall that we demonstrate in Chapter 6 that if the reliabilityattribute is added into Problem A and B or we consider the case in which the rating isoption-specific, they would have exactly the same results as in Problem D, so there isno need to include that situation. Nonetheless, we add one hypothetical benchmarksituation denoted by “H” that represents a market in which there is no involvementof the seller and the product is sold at zero price to investigate the seller’s influence,including double marginalization effect on DSPs and the customers.1127.1 Set upWe consider α ∈ {0, 1, 2, · · · , 25}, β ∈ {0.2, 0.5, 1, 2}, γ ∈ {0.2, 0.5, 1, 2}, andδ ∈ {0.5, 1, 2.5, 5, 10}. Therefore, we will include 26× 4× 4× 5 = 2080 cases.For the delivery options, we consider A = {1, 2, 3}, (t1 = 3, c1 = 5, θ1 = 0.8),(t2 = 2, c2 = 8, θ2 = 0.9), and (t3 = 6, c3 = 10, θ3 = 0.4).Since the variation of γ doesn’t influence qualitative results much, we will onlypresent the results for γ = 0.5 situation if we don’t specifically emphasize otherwise.7.2 Impact on SellerFor this part, we will check the following three measurements: profit, rating, andproportion of buyers. The hypothetical scenario is not part of the first two compar-isons since there is no seller’s profit or rating in that scenario.First of all, let’s look at the total profit. Since the difference between C and D isvery small, we will check their difference directly in Figure 7.1.As we can see, piD is higher than piC in all cases, which confirms our earlier in-ference. In addition, the difference between the two cases is high when δ is high andβ is low, which shows that the more customers care about service quality and theless they care about the money they pay, the higher profit the seller could gain withpricing flexibility.Then let’s look at the comparison of rating and we will show the case for γ = 0.2in Figure 7.2.113Figure 7.1: Graphs for piD − piC when γ = 0.5114Figure 7.2: Graphs for r when γ = 0.2115We can see that the exclusion of Option 3 when δ increases to 2.5 leads to a jumpin the rating under the setting of Problem C. But since the rating in Problem C isdetermined by the assortment of the delivery options, after Option 3 is eliminated,the change in δ would not influence its value in all the cases we studied. That’s to say,even if customers are becoming more concerned about service quality, the seller can-not adjust to it accordingly, which creates loss of business opportunity. On the otherhand, since it is not very flexible, we can see that the rating in Problem C can actuallybe higher than that in Problem D (for example, see the β = 0.2, δ = 2.5 panel). Thisshows that there would be some “waste” since this resource is not fully exploited bythe seller.For Problem D, however, the rating is adjusted along with the changes in market-ing conditions so that the preference of the customers can be satisfied. This illustratesthat when rating is aggregated, the seller’s flexibility in pricing can help to accommo-date customers’ changing desires.Lastly, let’s look at the proportion of buyers. Since the difference between the Cand D scenarios is very small, we will again check the difference directly in Figure 7.3.We can see clearly that the seller can attract more buyers with more pricing flexi-bility in Problem D, but there is a peak in the difference. It peaks when α is low andincrease of β would move the peak point to the right hand side while the increase ofδ would push it to the left side and increase the altitude of the peak.7.3 Impact on DSPsIn this part, we will look at two measurements: market share and total servicerevenue. The first measurement will help us understand the impact on individualDSP while the later one helps us know the overall situation for the delivery industry(if we only consider the DSPs included here).116Figure 7.3: Graphs for ΛD −ΛC when γ = 0.5117First of all, let’s compare the market shares of three options and they are presentedin Figure 7.4, Figure 7.5, and Figure 7.6, respectively.Figure 7.4: Graphs for λ1 when γ = 0.5118Figure 7.5: Graphs for λ2 when γ = 0.5119Figure 7.6: Graphs for λ3 when γ = 0.5120It’s clear that the increase of β would contribute to higher λ1, and lower λ2 andλ3, because lower quoted surcharge can give Option 1 more competitive advantagewhen customers care more about the money they pay.A very interesting finding is that for Option 1 and Option 3, their market sharescan be even higher than the benchmark case in the setting of Problem C where therating is aggregated and the seller can only price the product. That’s to say, the inter-vention of the seller doesn’t necessarily only bring the DSPs negative impacts that areexemplified by the double marginalization effect. But for Option 2, his market shareis always lower in the setting of Problem C than in Problem D, and never exceeds thatin the benchmark case.The reason behind them is intuitive: as an option with the highest reliability, Op-tion 2 suffers from the aggregated rating but benefits other options in the assortment.Option 3 can gain from this when δ is low but it would be kicked out of the assortmentwhen δ becomes large. And at the same time, Option 1 even benefits more because itsreliability is high enough to be kept in the assortment while a competitor is excludedfrom the market. Note that Option 2 still suffers even when Option 3 is eliminated,as we may expect.These findings show how distorted information may bring unfair competitive ad-vantages to certain players in this market and also highlight the importance of deter-mining a suitable reliability level for DSPs in such settings: one that is too low maylead to elimination from the competition while one that is too high would make thefirm suffer.On the other hand, we should notice that if Option 1 and Option 3 want to obtaina market share higher than that in the benchmark case, α should be high enough.This provides new finding from another perspective: the product quality would also121play an important role in deciding the DSPs’ market share, although the DSPs mayoverlook such connection. We also note that the threshold α it requires to allow themarket share of Option 1 (or Option 3) to be higher than that in the benchmark caseis getting closer to zero when β decrease and δ increases.Now let’s turn to the total service revenues.As we may see in Figure 7.7, due to the double marginalization effect, the totalservice revenues cannot exceed the benchmark case with the intervention of the seller.But if the information is not distorted (under setting of Problem D), the industry as awhole would be better off (than under the setting of Problem C).7.4 Impact on CustomerFor this part, we would look at two measurements: one is average payment (totalpayment is similar to average payment and we will omit it) and the other is averagemean utility.First let’s look at the average payment. As the difference between the two is verysmall, we will look at the difference between the average payment in Problem D andC directly in Figure 7.8.As we can see, customers always pay more in the setting of Problem D than in C,and such gap is increasing when δ increases and β decreases.Then let’s turn to the average mean utility in Figure 7.9 .We can see the lines for Problem C and Problem D are very close and always low-er than the benchmark line. The overall pattern is that u¯ increases while α increases,and it’s not hard to see when they are less than zero, both the u¯s in the setting of122Figure 7.7: Graphs for total service revenues when γ = 0.5123Figure 7.8: Graphs for AvgPayD −AvgPayC when γ = 0.5124Figure 7.9: Graphs for u¯ when γ = 0.5125Problem C and Problem D closely follow the increase of the benchmark line but oncethey reach zero, the lines become flat. This shows that the seller’s involvement in themarket always hurts the customer welfare, and the seller is more disciplined whenthe potential customers have low purchasing intentions but would exploit resourcesto extract values from the market so that even if the value of the product to the cus-tomers increases, the customer cannot gain all the benefits.Since the plot for Problem C and Problem D are very close, we can focus on thecomparison between them. Let’s look at the graphs for u¯D − u¯C in Figure 7.10.As we can see, growing δ would increase |u¯D − u¯C|, no matter β is small or large.But u¯D tends to be higher than u¯C when β is small, while u¯C is higher than u¯D when βis large. This shows that although customers can have better service in the setting ofProblem D (compared with Problem C), they may not actually enjoy it because theywould need to pay more, as well see in Figure 7.8. We notice that when the customersdon’t care much about money (β is small), the benefit of increased service qualitydominates the loss in money and the customers would have a higher average meanutility with better service quality (in Problem D). But when the customers care a lotabout money (β is large), the loss in money dominates the benefit of increased servicequality so that the customers would actually have a higher average mean utility withlower service quality (in Problem C). This presents how customers’ price sensitivitywould influence their final welfare, although all customers would prefer higher ser-vice quality when price is the same.On the other hand, we notice that the exclusion of Option 3 greatly increases u¯C,and this confirms our intuition that the customers would be better off if the inferioroption is eliminated from the market with information asymmetry.126Figure 7.10: Graphs for u¯D − u¯C when γ = 0.5127Chapter 8Conclusion and Future Work8.1 Discussion and ConclusionExcept for Problem C in which the seller has to solve a combinatorial optimizationproblem with non-linear objective function, the optimal assortment strategy is simplyto include all available delivery options. Even if the seller faces cardinality constraint,all the options can be evaluated by a single aggregated attribute and ordered by it,and the first K options should be selected for the assortment when only K optionsare allowed: the optimal policy is still “the more, the better”. For Problem C, weshow that although conventional sorting method cannot guarantee optimal solution,algorithms based on this kind of philosophy can still provide good results, but theperformance would not be that good under certain circumstances. As a side note, thefirst algorithm may have good performance under cardinality constraint, too.We show that constant mark-up is not always an optimal choice for the seller.When delivery service quality rating is aggregated, he should leverage his pricingflexibility to differentiate the mark-ups of all options in the assortment according totheir reliabilities.We demonstrate that the seller’s direct benefit of obtaining pricing flexibility is to128have more power on controlling the aggregated rating, but we also provide anotherway to explain the power of pricing flexibility from the perspective of informationcommunication. If some information regarding the options’ attribute is aggregated,the seller’s pricing flexibility would enable him to get rid of the information distortionand achieve a result in which as if all the customers actually know the true attributesof each option and make the choices accordingly. But if there is no such flexibility,like in Problem C, the impact would be on all the options and careful selection of as-sortment is necessary for the seller to control the scope of the information distortionand soften the negative influence. For Problem A and B, since there is no aggregatedinformation and customers know the actual attributes of each option, the flexibility inpricing doesn’t actually bring higher profitability to the seller.Displaying the customer rating would help to increase the seller’s profit becausecustomers have more information about service quality and the perceived shoppingexperience would be better. In addition, we should note that disaggregated ratingwould be better than the aggregated rating since rating aggregation is more like aconstraint for the seller. Then it’s clear that if he is restrained in one of these twodimensions (rating and pricing), the seller needs to pursue the flexibility in the oth-er dimension to ensure he still can obtain the highest expected profit. When he isconstrained in both dimensions, he has to consider dropping off some options tominimize his loss in expected profit.The settings in Problem C highlight DSPs’ possible mistake of making decisionsbased on the assumption that everyone is included in the competition. Problem Cshows that it’s possible that some options would be eliminated and thus some DSPsare not quite part of the game.This study also sheds light upon a potential area for future research and providessome insights for the DSPs. If we consider the interactions between the DSPs, theremight be collusions among them and sometimes the seller has to include all the op-129tions in order to let everyone have a piece of cake and deter the collusion, in orderto make little sacrifice to safeguard major interests. However, it is not guaranteedthat the DSPs won’t come up with other contracts to coordinate. On the other hand,if there is an industry-wide reliability standard, then “the more, the better” policywould be optimal in Problem C. But the DSPs may want to deviate from this kindof standard when making decisions because their competitors might be kicked out ofthe game and their market shares would increase. Such kind of deviation is costly forthem because increasing reliability would mean increasing cost while the benefit isnot guaranteed or paralleled with the effort because their competitors may still stayin the competition and make them hurt more due to rating aggregation. Then thoseforces may help to sustain an industry-standard that is high enough to eliminate allDSPs with low capacity on one hand, and to encourage DSPs with similar character-istics to coordinate on a standard on the other hand. But in the setting of Problem D,the coordination may not be a concern for the DSPs.The impact on the customers depends on their characteristics. If we use averagemean utility to assess how “happy” the customers are, we cannot conclude whetherthey are happier in Problem C or Problem D. Although usually the average servicelevel is higher in Problem D, the seller charges more accordingly. So it’s possiblethat the customers’ concern over money outweighs the pleasure from higher servicelevel and they actually cannot enjoy better service. But if they are not sensitive aboutmoney, they should be able to appreciate it and are happier on average.8.2 Future ResearchIn this section, we list a few ideas on possible approaches to continue and extendthe research in this thesis.1308.2.1 DSPs’ Strategic Decisions and InteractionsIn this study, we qualitatively characterize some potential reactions of the DSPswhen facing with the strategies adopted by the seller in the online retailing market.It would be interesting to quantitatively check how those DSPs would make differentsurcharge, lead time, and reliability decisions after knowing how the seller wouldbehave.The other area that can be explored is the game among the DSPs and the seller.Recall we restrict that the surcharge cannot be altered or falsely reported in the exam-ples we mentioned in Section 5.3, but if we relax this constraint, we may enlarge thestrategy space of the DSPs. For example, when the first two DSPs evaluate the choiceof bluffing, they may even claim a lower surcharge to further increase the marketshare in the hypothetical state but the reduced surcharge may negatively impact theoverall revenue in that state, so the trade-off still needs careful consideration. On theother hand, DSP3 may use a decreased surcharge in the contract (as we discussed hecannot influence the hypothetical state) to reduce the impact of the other two DSPs’bluffing, but when and how to use this tool needs to be carefully evaluated becausethe competitive bluffing between the other two DSPs is not always bad for him andsacrificing on the unit surcharge may hurt his overall revenue.We only talked about the compensation-based contract in previous discussions.However, it’s possible to introduce other contract types, such as revenue sharing, andwe may compare the decisions of different parties under different contract structures.For example, we may examine how DSP3 can maximize his advantage from the com-peting bluffing of the other two DSPs. These would provide some insights from theperspective of mechanism design.Another interesting topic to look at is the impact of ownership on the coordinationand competition. In the examples we mentioned, each option is owned by a differ-131ent DSP. If we change this setting and allow a DSP to offer multiple options, we canstudy how this would influence the coordination and competition among the DSPs.For example, we can study when Option 1 and Option 2 are offered by a single DSP,how this DSP should bluff in order to induce more benefits and how the DSP offeringOption 3 should figure out his own moves.Lastly, the structure of the problem also offers a chance to investigate the role andvalue of information in this industry. As we discussed, different information set anddifferent sequence of events may lead to different results in the game. If we considerinformation asymmetry, it’s possible that the seller and the DSPs would get differentinformation because of their different roles in the industry. It would be interestingto look at how the information quantity and quality would influence the interactionsbetween different parties. For example, we may analyze whether including all optionsis always optimal for the seller and whether he can take advantage of his special rolein the industry to participate in such a bribe game by providing some informationverification to certain DSPs or collaborating with some DSPs in order to maximize hisprofit or minimize his loss. The last few topics may provide important implicationsin the supply chain field.8.2.2 Multiple ProductsIn this research, we consider the seller only sells one product to the customers.It’s possible that we are interested in the impact of offering more than one productwith different characteristics, such as value, on the aggregated rating and expectedprofit. One important characteristic is that the seller may have to offer exactly thesame assortment of delivery service for all the products. It would be interesting tolook at how the features of different products influence each other and how theserelationships interact with the assortment and pricing decisions under such setting.We may consider independent, substitutive, and complementary products. For in-dependent products, we may allow them to have different market sizes and optimize132on each market separately, but the relationship among the market sizes needs to bediscussed. For substitutive products, a nested logit model might be needed to allowfor competition between all the products offered by the seller. For complementaryproducts, we may consider modifying the mean utility function to incorporate suchfeatures.Note that one possible obstacle is that even if there are only two products, theaggregated rating’s properties that we derived in this thesis may not hold any more.8.2.3 Competing SellersIn a multiple-seller B2C or C2C platform, the competition between similar sellersis inevitable. We note that if a competitive market is indeed considered, the tradi-tional nested logit model may not be a good choice. The fundamental problem isthat under such setting, the customers would observe the base price of the productand select one seller according to the price as well as the rating of the seller (whenavailable). The choice for delivery option would be a second layer decision problemand the “attractiveness” of those options usually cannot influence the choice on thefirst layer — the selection on the seller. That’s to say, the customers’ utility in the twostages would have different structures. Thus we may need to modify the nested logitmodel to reflect such specifications. The work by Bitran et al. (2006) might providesome inspirations.It would be interesting to examine the influence of different available sets to thecompeting sellers, just as in Besbes and Saure´ (2015), to see whether the equilibriumwould be different with and without overlapping delivery options available to thesellers.If we consider that the competing sellers can provide more than one product atthe same time, a three-layer modified nested logit model might be necessary. Li et al.(2015) would be an important reference to look at then.1338.2.4 Flexible Lead TimeWe are allowing the flexibility of deciding the surcharge for the delivery servicein Problem B and Problem D. Similarly, we may also consider the scenarios in whichthe seller can alter the quoted lead times and examine whether the seller should de-liberately announce a longer lead time than the quoted one from the DSP under somespecific conditions. However, we note that the simple reliability approximation forthe actual on-time delivery rate would not be sufficient in such setting and the sellerwould have to get distribution information from the Fi(t) functions. This may causeproblems since in practice the DSPs may not be able to or not be willing to providesuch information.On the other hand, when lead time is considered, it’s often viewed only as athreshold line. However, the lead time may be provided as a range in reality. Thissetting may capture more behavioral aspects of customer decision and facilitate ratingsystems that are more complex.8.2.5 Other Possibilities in Customer Rating FrameworkCustomer rating provides a very open and free framework to incorporate variousbehavioral aspects of customer decision.In our research, we assume all the customers would truthfully report the actualexperience about the delivery service. In reality, it’s possible that more behavioralaspects and dynamics in rating would also play important roles in the process of dis-seminating the quality information. It would be interesting to uncover the impact ofalternative rating mechanisms on the involved parties.We may look at what the rating represents, how the rating is formed, how therating evolves, and how the rating information is perceived by the customers whenlearning or making decisions. Each step provides many chances and the combina-134tions of them would offer more opportunities.In addition, a game scheme can also be considered to look at how different partiesrespond to each other and whether an equilibrium can be formed.135BibliographyAkc¸ay, Yalc¸in, Harihara Prasad Natarajan, Susan H Xu. 2010. Joint dynamic pricingof multiple perishable products under consumer choice. Management Science 56(8)1345–1361.Anderson, Simon P, Andre De Palma, Jacques Franc¸ois Thisse. 1992. 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International Journal of Production Economics 164 14–23.142AppendixAppendix A: R Code for All Functions1 ##Note that some intermediate variables might be replaced by data with different data typesin later steps within the same function23 A.solution <- function(par , option , total=nrow(option), lambda=FALSE){4 require(LambertW)56 alpha <- par[1]7 beta <- par[2]8 gamma <- par[3]910 n <- total1112 u <- alpha - 1 - beta * option$c - gamma * option$t13 k <- sum(sapply(u, FUN=exp))14 price <- 1/beta * (1 + W(k))1516 opt_u <- alpha - beta * (price + option$c) - gamma * option$t17 option$lambda <- exp(opt_u)/(1+sum(sapply(opt_u, FUN=exp)))1819 buyer <- sum(option$lambda)20 ser_rev <- sum(option$c * option$lambda)2122 avg_pay <- sum((price + option$c) * option$lambda)/buyer23 tot_pay <- sum((price + option$c) * option$lambda)24 avg_uti <- sum(opt_u * option$lambda)/buyer2526 profit <- price * buyer2728 if (lambda == TRUE) {29 lambda_frame <- as.data.frame(t(rep(NA, total)))30 colnames(lambda_frame) <- paste("lambda", c(1: total), sep="")3132 if(is.null(option$num)){33 lambda_frame[1, c(1: total)] <- option$lambda34 }else{35 lambda_frame[1, option$num] <- option$lambda36 }3714338 return(data.frame(cbind(price , lambda_frame , buyer , ser_rev , avg_pay , tot_pay , avg_uti ,profit)))3940 }else{41 return(data.frame(price , buyer , ser_rev , avg_pay , tot_pay , avg_uti , profit))42 }43 }444546 B.solution <- function(par , option , total=nrow(option), lambda=FALSE){47 require(LambertW)4849 alpha <- par[1]50 beta <- par[2]51 gamma <- par[3]5253 u <- alpha - 1 - beta * option$c - gamma * option$t54 k <- sum(sapply(u, FUN=exp))5556 buyer <- W(k)/(1 + W(k))5758 option$price <- 1/beta * (1 + W(k)) + option$c59 opt_u <- alpha - beta * option$price - gamma * option$t60 option$lambda <- exp(opt_u)/(1+sum(sapply(opt_u, FUN=exp)))61 ser_rev <- sum(option$c * option$lambda)6263 avg_pay <- sum(option$price * option$lambda)/buyer64 tot_pay <- sum(option$price * option$lambda)65 avg_uti <- sum(opt_u * option$lambda)/buyer6667 profit <- sum( (option$price - option$c) * option$lambda)6869 price_frame <- as.data.frame(t(rep(NA, total)))70 colnames(price_frame) <- paste("p", c(1: total), sep="")7172 if(is.null(option$num)){73 price_frame[1, c(1: total)] <- option$price74 }else{75 price_frame[1, option$num] <- option$price76 }7778 if (lambda == TRUE) {79 lambda_frame <- as.data.frame(t(rep(NA, total)))80 colnames(lambda_frame) <- paste("lambda", c(1: total), sep="")8182 if(is.null(option$num)){83 lambda_frame[1, c(1: total)] <- option$lambda84 }else{85 lambda_frame[1, option$num] <- option$lambda86 }8788 return(data.frame(cbind(price_frame , lambda_frame , buyer , ser_rev , avg_pay , tot_pay , avg_uti , profit)))8990 }else{91 return(data.frame(cbind(price_frame , buyer , ser_rev , avg_pay , tot_pay , avg_uti , profit)))92 }93 }144949596 C.solution <- function(par , option , total=nrow(option), lambda=FALSE , opt.search=TRUE){97 require(LambertW)9899 alpha <- par[1]100 beta <- par[2]101 gamma <- par[3]102 delta <- par[4]103104 n <- nrow(option)105 total <- total106107 hat_y <- 0108109 if(opt.search == TRUE){110 option$y <- exp(-beta * option$c - gamma * option$t)111112 cal.hat_y <- function(index , options , delta.par){113 assortment <- options[index ,]114 M <- sum(assortment$y)115 N <- sum(assortment$y * assortment$theta)116 hat_y <- exp(delta.par * N/M) * M117 }118119 for (i in 1:n){120 combination <- combn(seq(1,n), i)121 cNum <- ncol(combination)122 for (j in 1:cNum){123 select <- combination[,j]124 hat_y_new <- cal.hat_y(index=select , options=option , delta.par=delta)125126 if (hat_y_new > hat_y){127 hat_y <- hat_y_new128 select_star <- select129 }130 }131 }132 ori_option <- option133 select <- select_star134 option <- option[select ,]135 }else{136 select <- c(1: total)137 }138139 r_u <- - beta * option$c - gamma * option$t140 r <- sum(option$theta * sapply(r_u, FUN=exp))/sum(sapply(r_u, FUN=exp))141142 u <- alpha - 1 - beta * option$c - gamma * option$t + delta * r143 cap_k <- sum(sapply(u, FUN=exp))144145 price <- 1/beta * (1 + W(cap_k))146147 opt_u <- alpha - beta * (price + option$c) - gamma * option$t + delta * r148149 option$lambda <- exp(opt_u)/(1+sum(sapply(opt_u, FUN=exp)))150 buyer <- sum(option$lambda)151 ser_rev <- sum(option$c * option$lambda)145152153 avg_pay <- sum((price + option$c) * option$lambda)/buyer154 tot_pay <- sum((price + option$c) * option$lambda)155 avg_uti <- sum(opt_u * option$lambda)/buyer156157 rating <- r158 profit <- price * buyer159160 if (lambda == TRUE) {161 lambda_frame <- as.data.frame(t(rep(NA, total)))162 colnames(lambda_frame) <- paste("lambda", c(1: total), sep="")163164 if(is.null(option$num)){165 lambda_frame[1, select] <- option$lambda166 }else{167 lambda_frame[1, option$num] <- option$lambda168 }169170 outcome <- data.frame(cbind(price , lambda_frame , buyer , ser_rev , avg_pay , tot_pay , avg_uti , rating , profit))171172 }else{173 outcome <- data.frame(cbind(price , buyer , ser_rev , avg_pay , tot_pay , avg_uti , rating ,profit))174 }175176177 if(opt.search == TRUE){178 assortment <- paste(as.character(select_star), collapse = ’, ’)179 if(is.null(option$sn) == FALSE){180 assortment_sn <- paste(as.character(option$sn), collapse = ’, ’)181 return(data.frame(cbind(assortment , assortment_sn, outcome)))182 }else{183 return(data.frame(cbind(assortment , outcome)))184 }185 }else{186 return(outcome)187 }188 }189190191 D.solution <- function(par , option , total=nrow(option), lambda=FALSE){192 require(LambertW)193194 alpha <- par[1]195 beta <- par[2]196 gamma <- par[3]197 delta <- par[4]198199 n <- nrow(option)200201 r_u <- - beta * option$c - gamma * option$t + delta * option$theta202 r <- sum(option$theta * sapply(r_u, FUN=exp))/sum(sapply(r_u, FUN=exp))203204 u_m <- alpha - 1 - beta * option$c - gamma * option$t + delta * option$theta205 k <- sum(sapply(u_m, FUN=exp))206 m_core <- W(k)207146208 option$price <- 1/beta*(m_core +1) + delta/beta*(r - option$theta) + option$c209 u <- alpha - beta * option$price - gamma * option$t + delta * r210 option$lambda <- exp(u)/(1+ sum(sapply(u, FUN=exp)))211 buyer <- sum(option$lambda)212 ser_rev <- sum(option$c * option$lambda)213214 avg_pay <- sum(option$price * option$lambda)/buyer215 tot_pay <- sum(option$price * option$lambda)216 avg_uti <- sum(u * option$lambda)/buyer217218 rating <- r219 profit <- sum( (option$price - option$c) * option$lambda)220221 price_frame <- as.data.frame(t(rep(NA, total)))222 colnames(price_frame) <- paste("p", c(1: total), sep="")223224 if(is.null(option$num)){225 price_frame[1, c(1: total)] <- option$price226 }else{227 price_frame[1, option$num] <- option$price228 }229230 if (lambda == TRUE) {231 lambda_frame <- as.data.frame(t(rep(NA, total)))232 colnames(lambda_frame) <- paste("lambda", c(1: total), sep="")233234 if(is.null(option$num)){235 lambda_frame[1, c(1: total)] <- option$lambda236 }else{237 lambda_frame[1, option$num] <- option$lambda238 }239240 return(data.frame(cbind(price_frame , lambda_frame , buyer , ser_rev , avg_pay , tot_pay , avg_uti , rating , profit)))241242 }else{243 return(data.frame(cbind(price_frame , buyer , ser_rev , avg_pay , tot_pay , avg_uti , rating ,profit)))244 }245 }246247248 vary <- function(pro , par , option , alpha=par[1], beta=par[2], gamma=par[3], delta=par [4]){249 funs <- list(250 A = A.solution ,251 B = B.solution ,252 D = D.solution253 )254255 no <- nrow(option)256 outcome <- c()257258 if (pro == "A" || pro == "B"){259 par_list <- list(alpha , beta , gamma)260 }else{261 par_list <- list(alpha , beta , gamma , delta)262 }263 par_grid <- expand.grid(par_list)147264 n <- nrow(par_grid)265 outcome <- c()266267 for(i in 1:n){268 parameter <- unname(unlist(par_grid[i,]))269270 if (pro != "C"){271 opt <- funs[[pro]]( par=parameter , option=option , total=no, lambda=TRUE)272 }else {273 opt <- C.solution(par=parameter , option=option , total=no, lambda=TRUE , opt.search=TRUE)274 }275276 if (pro == "A" || pro == "B"){277 para_matrix <- as.data.frame(matrix(rep(parameter ,each =1),ncol =3))278 }else{279 para_matrix <- as.data.frame(matrix(rep(parameter ,each =1),ncol =4))280 }281 opt <- cbind(para_matrix , opt)282 outcome <- rbind(outcome , opt)283 }284 colnames(outcome)[1:4] <- c("alpha","beta","gamma","delta")285 return(outcome)286 }287288289 combinatory <- function(pro , par , option , order=TRUE){290 funs <- list(291 A = A.solution ,292 B = B.solution ,293 C = C.solution ,294 D = D.solution295 )296297 n <- nrow(option)298 outcome <- c()299300 for (i in n:1){301 combination <- combn(seq(1,n), i)302 cNum <- ncol(combination)303 for (j in 1:cNum){304 select <- combination[,j]305306 assortment <- option[select ,]307 assortment$num <- select308309 if (pro == "C"){310 opt <- funs[[pro]]( par=par , option=assortment , total=n, lambda=TRUE , opt.search=FALSE)311 }else{312 opt <- funs[[pro]]( par=par , option=assortment , total=n, lambda=TRUE)313 }314315 opt$assortment <- paste(as.character(select), collapse = ’, ’)316317 if(is.null(option$sn) == FALSE){318 opt$assortment_sn <- paste(as.character(assortment$sn), collapse = ’, ’)319 }320148321 outcome <- rbind(outcome , opt)322 }323 }324325 outcome <- outcome[c(ncol(outcome), 1:( ncol(outcome) -1))]326 if(is.null(option$sn) == FALSE){327 outcome <- outcome[c(ncol(outcome), 1:( ncol(outcome) -1))]328 }329 if (order == TRUE){330 outcome <- outcome[order(outcome$profit , decreasing = TRUE), ]331 }332 return(outcome)333 }334335336 compare <- function(par , option){337 require(plyr)338339 if (is.null(option$sn)){340 s <- 2341 }else{342 s <- 3343 }344345 C_outcome <- C.solution(par=par , option=option , lambda=TRUE , opt.search=TRUE)346 C_outcome <- C_outcome[, s:ncol(C_outcome)]347 D_outcome <- D.solution(par=par , option=option , lambda=TRUE)348349350 noproduct <- function(para=par , options=option){351 alpha <- para [1]352 beta <- para [2]353 gamma <- para [3]354 delta <- para [4]355 n <- nrow(options)356357 u <- alpha - beta * options$c - gamma * options$t + delta * options$theta358359 options$lambda <- exp(u)/(1+ sum(sapply(u, FUN=exp)))360 buyer <- sum(options$lambda)361 ser_rev <- sum(options$c * options$lambda)362363 avg_pay <- sum(options$c * options$lambda)/buyer364 tot_pay <- sum(options$c * options$lambda)365 avg_uti <- sum(u * options$lambda)/buyer366367 price_frame <- as.data.frame(t(rep(NA, n)))368 colnames(price_frame) <- paste("p", c(1:n), sep="")369 price_frame[1, ] <- options$c370 lambda_frame <- as.data.frame(t(rep(NA, n)))371 colnames(lambda_frame) <- paste("lambda", c(1:n), sep="")372 lambda_frame[1, ] <- options$lambda373374 return(data.frame(cbind(price_frame , lambda_frame , buyer , ser_rev , avg_pay , tot_pay , avg_uti)))375 }376 H_outcome <- noproduct ()377149378 outcome <- rbind.fill(H_outcome , C_outcome , D_outcome)379 outcome <- cbind(pro=c("H","C","D"), outcome)380 return(outcome)381 }382383384 compare.vary <- function(par , option , alpha=par[1], beta=par[2], gamma=par[3], delta=par [4]){385386 par_list <- list(alpha , beta , gamma , delta)387 par_grid <- expand.grid(par_list)388 n <- nrow(par_grid)389 outcome <- c()390391 for(i in 1:n){392 parameter <- unname(unlist(par_grid[i,]))393 step_outcome <- compare(par=parameter , option=option)394 para_matrix <- as.data.frame(matrix(rep(parameter ,each =3),ncol =4))395 step_outcome <- cbind(para_matrix , step_outcome)396 outcome <- rbind(outcome , step_outcome)397 }398399 colnames(outcome)[1:4] <- c("alpha","beta","gamma","delta")400 return(outcome)401 }402403 #####################################################404 ## Numerical experiments for algorithm performance405 #####################################################406 require(compiler)407 enableJIT (3)408409 cal.hat_y <- function(index , options , delta.par){410 assortment <- options[index ,]411 M <- sum(assortment$y)412 N <- sum(assortment$y * assortment$theta)413 hat_y <- exp(delta.par * N/M) * M414 }415416417 C.assortment <- function(par , option , sort){418 beta <- par[2]419 gamma <- par[3]420 delta <- par[4]421422 outcome_list <- c()423424 n <- nrow(option)425 all <- c(1:n)426 option$y <- exp(-beta * option$c - gamma * option$t)427428 if(sort==TRUE){429 option <- option[with(option , order(theta ,y,decreasing = TRUE)), ]430 }431432 max_theta <- option$theta[which.max(option$theta)]433 initial_index <- which(option$theta == max_theta)434 hat_y <- cal.hat_y(index=initial_index , options=option , delta.par=delta)435150436 search_index <- all [! all %in% initial_index]437 assortment_index <- initial_index438439 if(length(search_index) != 0){440 repeat{441442 a <- search_index [1]443 search_index <- search_index [-1]444445 for (b in search_index){446 hat_y_a <- cal.hat_y(index=c(assortment_index , a), options=option , delta.par=delta) #the exact implementation should not use cal.hat_y function here447 hat_y_b <- cal.hat_y(index=c(assortment_index , b), options=option , delta.par=delta)448 if (hat_y_a < hat_y_b){449 a <- b450 }451 }452 new_index <- c(assortment_index , a)453 new_hat_y <- cal.hat_y(index=new_index , options=option , delta.par=delta)454455 if(new_hat_y > hat_y){456 hat_y <- new_hat_y457 assortment_index <- new_index458 search_index <- all [! all %in% assortment_index]459 if(length(search_index) == 0){break}460 }else{461 break462 }463 }464465 part1_hat_y <- hat_y466 step1_index <- assortment_index [! assortment_index %in% initial_index]467 search_index <- all [! all %in% assortment_index]468 n2 <- length(search_index)469 if (n2!=0 && n2!=1){470 assortment_star <- assortment_index471 for (i in 1:n2){ # the exact implementation should be: for (i in 2:n2)472 combination <- combn(search_index , i)473 cNum <- ncol(combination)474 for (j in 1:cNum){475 select <- combination[,j]476 new_index <- c(assortment_star , select)477 hat_y_new <- cal.hat_y(index=new_index , options=option , delta.par=delta)478 if (hat_y_new > hat_y){479 hat_y <- hat_y_new480 assortment_index <- new_index481 }482 }483 }484 }485 }486 step2_index <- assortment_index [! assortment_index %in% c(initial_index , step1_index)]487 outcome_list [1] <- list(paste(as.character(c(initial_index , 0, step1_index , 0, step2_index)), collapse = ’, ’))488 outcome_list [2] <- list(hat_y)489 outcome_list [3] <- list(part1_hat_y)490 return(outcome_list)491 }151492493494 C.assortment2 <- function(par , option , sort=TRUE){495 beta <- par[2]496 gamma <- par[3]497 delta <- par[4]498499 outcome_list <- c()500501 n <- nrow(option)502 all <- c(1:n)503 option$y <- exp(-beta * option$c - gamma * option$t)504505 option <- option[with(option , order(theta ,y,decreasing = TRUE)), ]506 max_theta <- option$theta[which.max(option$theta)]507 initial_index <- which(option$theta == max_theta)508 start <- max(initial_index)509510 hat_y <- cal.hat_y(index=initial_index , options=option , delta.par=delta)511 select_star <- initial_index512 if(start < n){513 for(i in c((start +1):n)){514 select <- c(1:i)515 hat_y_new <- cal.hat_y(index=select , options=option , delta.par=delta)516 if (hat_y_new > hat_y){517 hat_y <- hat_y_new518 select_star <- select519 }520 }521 }522523 outcome_list [1] <- list(paste(as.character(select_star), collapse = ’, ’))524 outcome_list [2] <- list(hat_y)525 return(outcome_list)526 }527528529 C.origin <- function(par , option , sort){530 beta <- par[2]531 gamma <- par[3]532 delta <- par[4]533534 outcome_list <- c()535536 n <- nrow(option)537 hat_y <- 0538 option$y <- exp(-beta * option$c - gamma * option$t)539540 if(sort==TRUE){541 option <- option[with(option , order(theta ,y,decreasing = TRUE)), ]542 }543544 for (i in 1:n){545 combination <- combn(seq(1,n), i)546 cNum <- ncol(combination)547 for (j in 1:cNum){548 select <- combination[,j]549 hat_y_new <- cal.hat_y(index=select , options=option , delta.par=delta)152550551 if (hat_y_new > hat_y){552 hat_y <- hat_y_new553 select_star <- select554 }555 }556 }557558 outcome_list [1] <- list(paste(as.character(select_star), collapse = ’, ’))559 outcome_list [2] <- list(hat_y)560 return(outcome_list)561 }562563564 compare.algo <- function(par , n, size , sort=FALSE , check =0){565566 all_option <- expand.grid(t=seq(2,14,by=2), c=seq(5,20,by=2.5) , theta=seq(0.4,1,by=0.1))567 no <- nrow(all_option)568 all_option$sn <- c(1:no)569570 if (!(0 %in% check)){571 beta <- par[2]572 gamma <- par[3]573 outcome <- list()574 j <- 1575 for(i in 1:n){576 set <- all_option[sample(no, size , replace=FALSE), ]577578 if (i %in% check){579 if (sort==TRUE){580 set$y <- exp(-beta * set$c - gamma * set$t)581 outcome[j] <- list(set[with(set , order(theta ,y,decreasing = TRUE)), ])582 }else{583 outcome[j] <- list(set)584 }585 j <- j + 1586 }587 }588 }else{589590 algo_order <- c()591 algo2 <- c()592 best <- c()593594 algo_time <- c()595 algo2_time <- c()596 best_time <- c()597598 part1_hat_y <- c()599 algo_hat_y <- c()600 algo2_hat_y <- c()601 best_hat_y <- c()602603 for(i in 1:n){604 set <- all_option[sample(no, size , replace=FALSE), ]605606 ptm <- proc.time()607 C.algo <- C.assortment(par=par , option=set , sort=sort)153608 time <- proc.time()-ptm609 algo_time[i] <- unname(time [3])610611 ptm <- proc.time()612 C.algo2 <- C.assortment2(par=par , option=set)613 time <- proc.time()-ptm614 algo2_time[i] <- unname(time [3])615616 ptm <- proc.time()617 C.best <- C.origin(par=par , option=set , sort=sort)618 time <- proc.time()-ptm619 best_time[i] <- unname(time [3])620621 algo_order[i] <- C.algo [[1]]622 algo2[i] <- C.algo2 [[1]]623 best[i] <- C.best [[1]]624625 part1_hat_y[i] <- C.algo [[3]]626 algo_hat_y[i] <- C.algo [[2]]627 algo2_hat_y[i] <- C.algo2 [[2]]628 best_hat_y[i] <- C.best [[2]]629 }630631 outcome <- data.frame(algo_order , algo2 , best , algo_time , algo2_time , best_time , part1_hat_y, algo_hat_y, algo2_hat_y, best_hat_y, stringsAsFactors = FALSE)632 }633 return(outcome)634 }635636637 process.result <- function(data , size){638 assortment_list <- strsplit(data$algo_order , ", ")639640 number <- function(x){return(size - (tail(which(x==0) ,1) -2))}641 data$number <- unlist(lapply(assortment_list , number))642643 part1_assort.order <- function(x){644 x <- as.numeric(x[1: max(which(x==0))]); x <- sort(x);645 y <- x [! x %in% c(0)]; return(paste(y, collapse = ’, ’))}646 data$part1_eq <- lapply(assortment_list , part1_assort.order) == data$best647648 assort.order <- function(x){649 x <- sort(as.numeric(x)); y <- x [! x %in% c(0)]; return(paste(y, collapse = ’, ’))}650 data$algo_eq <- lapply(assortment_list , assort.order) == data$best651 data$algo2_eq <- data$algo2 == data$best652653 select <- which(data$part1_eq==TRUE)654 data$part1_hat_y[select] <- data$best_hat_y[select]655 select <- which(data$algo_eq==TRUE)656 data$algo_hat_y[select] <- data$best_hat_y[select]657 select <- which(data$algo2_eq==TRUE)658 data$algo2_hat_y[select] <- data$best_hat_y[select]659 data$part1_hat_y_eq <- data$part1_hat_y == data$best_hat_y660 data$algo_hat_y_eq <- data$algo_hat_y == data$best_hat_y661 data$algo2_hat_y_eq <- data$algo2_hat_y == data$best_hat_y662 return(data)663 }154Appendix B: R Code for Numerical Examples and Experiments1 # Examples and experiments are listed in the order of occurrence in the thesis234 #Example 4.1####5 par.eg4.1 <- c(5, 0.15, 0.2)6 option.eg4.1 <- as.data.frame(cbind(t=c(3,2,6),c=c(5,8,10)))7 eg4.1 <- combinatory(pro="B", par=par.eg4.1, option=option.eg4.1)89 #Example 5.1####10 par.eg5.1 <- c(5, 0.15, 0.2, 2.5)11 option.eg5.1 <- as.data.frame(cbind(t=c(3,2,6),c=c(5,8,10),theta=c(0.8, 0.9, 0.4)))12 eg5.1 <- combinatory(pro="C", par=par.eg5.1, option=option.eg5.1)1314 #Example 5.5####15 par.eg5.5 <- c(5, 0.15, 0.2, 2.5)16 option.eg5.5 <- as.data.frame(cbind(t=c(3,2,6),c=c(5,8,10),theta=c(0.7, 0.8, 0.4)))17 eg5.5 <- combinatory(pro="C", par=par.eg5.5, option=option.eg5.5)1819 #Example 5.6####20 par.eg5.6 <- c(5, 0.15, 0.2, 2.5)21 option.eg5.6 <- as.data.frame(cbind(t=c(3,2,6),c=c(5,8,10),theta=c(0.9, 1, 0.4)))22 eg5.6 <- combinatory(pro="C", par=par.eg5.6, option=option.eg5.6)2324 #Example 5.7####25 par.eg5.7 <- c(5, 0.15, 0.2, 2.5)26 option.eg5.7 <- as.data.frame(cbind(t=c(1,1,6),c=c(5,8,10),theta=c(1, 1, 0.4)))27 eg5.7 <- combinatory(pro="C", par=par.eg5.7, option=option.eg5.7)2829 ##Algorithm performance experiments for Problem C####30 #Part I31 algo_test_par <- c(5, 0.2, 0.2, 2.5)32 set.seed (9876) #Seeds are specified to make the results replicable33 algo_test_partI_1 <- compare.algo(par=algo_test_par , n=20000 , size=10, sort=TRUE)34 set.seed (1234)35 algo_test_partI_2 <- compare.algo(par=algo_test_par , n=20000 , size=12, sort=TRUE)36 set.seed (2722)37 algo_test_partI_3 <- compare.algo(par=algo_test_par , n=20000 , size=15, sort=TRUE)38 algo_test_partI_1 <- process.result(algo_test_partI_1, size =10)39 algo_test_partI_2 <- process.result(algo_test_partI_2, size =12)40 algo_test_partI_3 <- process.result(algo_test_partI_3, size =15)4142 #Part II43 algo_test_par2 <- c(5, 0.2, 0.2, 5)44 set.seed (4785)45 algo_test_partII_1 <- compare.algo(par=algo_test_par2 , n=20000 , size=10, sort=TRUE)46 set.seed (97278)47 algo_test_partII_2 <- compare.algo(par=algo_test_par2 , n=20000 , size=12, sort=TRUE)48 set.seed (1848)49 algo_test_partII_3 <- compare.algo(par=algo_test_par2 , n=20000 , size=15, sort=TRUE)50 algo_test_partII_1 <- process.result(algo_test_partII_1, size =10)51 algo_test_partII_2 <- process.result(algo_test_partII_2, size =12)52 algo_test_partII_3 <- process.result(algo_test_partII_3, size =15)5354 #Part III55 algo_test_par3 <- c(5, 2, 0.2, 2.5)15556 set.seed (8025)57 algo_test_partIII_1 <- compare.algo(par=algo_test_par3 , n=20000 , size=10, sort=TRUE)58 set.seed (62938)59 algo_test_partIII_2 <- compare.algo(par=algo_test_par3 , n=20000 , size=12, sort=TRUE)60 set.seed (67033)61 algo_test_partIII_3 <- compare.algo(par=algo_test_par3 , n=20000 , size=15, sort=TRUE)62 algo_test_partIII_1 <- process.result(algo_test_partIII_1, size =10)63 algo_test_partIII_2 <- process.result(algo_test_partIII_2, size =12)64 algo_test_partIII_3 <- process.result(algo_test_partIII_3, size =15)6566 #Part IV67 algo_test_par4 <- c(5, 0.2, 2, 2.5)68 set.seed (32946)69 algo_test_partIV_1 <- compare.algo(par=algo_test_par4 , n=20000 , size=10, sort=TRUE)70 set.seed (89788)71 algo_test_partIV_2 <- compare.algo(par=algo_test_par4 , n=20000 , size=12, sort=TRUE)72 set.seed (1848)73 algo_test_partIV_3 <- compare.algo(par=algo_test_par4 , n=20000 , size=15, sort=TRUE)74 algo_test_partIV_1 <- process.result(algo_test_partIV_1, size =10)75 algo_test_partIV_2 <- process.result(algo_test_partIV_2, size =12)76 algo_test_partIV_3 <- process.result(algo_test_partIV_3, size =15)7778 #Part V79 algo_test_par5 <- c(5, 1, 1, 5)80 algo_test_partV_1 <- compare.algo(par=algo_test_par5 , n=20000 , size=10, sort=TRUE)81 algo_test_partV_2 <- compare.algo(par=algo_test_par5 , n=20000 , size=12, sort=TRUE)82 algo_test_partV_3 <- compare.algo(par=algo_test_par5 , n=20000 , size=15, sort=TRUE)83 algo_test_partV_1 <- process.result(algo_test_partV_1, size =10)84 algo_test_partV_2 <- process.result(algo_test_partV_2, size =12)85 algo_test_partV_3 <- process.result(algo_test_partV_3, size =15)8687 #Part VI88 algo_test_par6 <- c(5, 0.02, 0.02, 2.5)89 set.seed (12406)90 algo_test_partVI_1 <- compare.algo(par=algo_test_par6 , n=20000 , size=10, sort=TRUE)91 set.seed (61788)92 algo_test_partVI_2 <- compare.algo(par=algo_test_par6 , n=20000 , size=12, sort=TRUE)93 set.seed (66962)94 algo_test_partVI_3 <- compare.algo(par=algo_test_par6 , n=20000 , size=15, sort=TRUE)95 algo_test_partVI_1 <- process.result(algo_test_partVI_1, size =10)96 algo_test_partVI_2 <- process.result(algo_test_partVI_2, size =12)97 algo_test_partVI_3 <- process.result(algo_test_partVI_3, size =15)9899100 #Example 6.1####101 par.eg6.1 <- c(5, 0.3, 1.8, 5.5)102 option.eg6.1 <- as.data.frame(cbind(t=c(5,6,4),c=c(5,7,5),theta=c(0.4, 0.9, 0.6)))103 eg6.1 <- combinatory(pro="D", par=par.eg6.1, option=option.eg6.1)104105 #Example 6.2####106 par.eg6.2 <- c(5, 0.15, 0.2, 2.5)107 option.eg6.2 <- as.data.frame(cbind(t=c(3,2,6),c=c(5,8,10),theta=c(0.8, 0.9, 0.4)))108 eg6.2 <- combinatory(pro="D", par=par.eg6.2, option=option.eg6.2)109110 #Example 6.2 collusion (results not listed in thesis)####111 par.eg6.2 <- c(5, 0.15, 0.2, 2.5)112 option.eg6.2C <- as.data.frame(cbind(t=c(8,7,6),c=c(5,8,10),theta=c(0.4, 0.5, 0.4)))113 eg6.2C <- combinatory(pro="D", par=par.eg6.2, option=option.eg6.2C)156114115116117118 ## Numerical Studies ####119 par1 <- c(5, 0.15, 0.2, 2.5)120 option1 <- as.data.frame(cbind(t=c(3,2,6),c=c(5,8,10),theta=c(0.8 ,0.9 ,0.4)))121 num_study1 <- compare.vary(par=par1 , option=option1 , alpha=seq(0, 25, by=1), beta=c(0.2 ,0.5 ,1 ,2), gamma=c(0.2 ,0.5 ,1 ,2), delta=c(0.5, 1, 2.5, 5, 10))157


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