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Green strategies: producers' competition and cooperation in sustainability

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Green strategies: producers' competition and cooperation in sustainability

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Content GREENSTRATEGIES:PRODUCERS’COMPETITIONANDCOOPERATIONINSUSTAINABILITY by FangTian ADissertationPresentedtothe FACULTYOFTHEGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (BUSINESSADMINISTRATION) May2015 Copyright 2015 FangTian To my family: Thisthesisisdedicatedtomyfamily. Myparents,Tian,WeiandShi,Shuqin, gavemeastablefoundationofmylife—theytaughtmeinvaluablelessonsaboutwork, marriage,parenting,relationships,andlife. Myhusband,Ma,Jingran,ismybestfriendand partner—wekeepnosecretsfromeachotherandwehavethesameopiniononeverybig decision. Myparents-in-law,Ma,ZhiyuanandBai,Shuhua,providedmegreatsupportafter ourmarriage,especiallyduringmyPh.D.stage. Mytwodaughters,Ma,IrisandMa,Doris, bothbornduringmyPh.D.stage,aremymostprecioustreasure;theyarethebiggest motivationofmyhard-working. ii Acknowledgments I,FangTian,amgratefultotheGodforleadingthewayofcompletingmydegree. Foremost, I would like to express my sincere gratitude to the chair of my committee, Professor Soˇ si´ c, Greys. She is my Ph.D. advisor: through the six years of my study, Professor Soˇ si´ c trained me with not onlyprofessionalknowledgebutalsoskillstoconductgoodresearch. Meanwhile,sheismyco-authorwho provided me solid help whenever I encountered difficulties in research. Professor Soˇ si´ c has also been very supportiveinmyprofessionallife,encouragingmetocloselyconnectwithpeerscholars. Besides,Iwouldliketothankmyotherco-author,ProfessorDebo,Laurens. Throughthethreeprojects we co-worked, Professor Debo’s several sharp and valuable ideas taught me to identify interesting and insightfulresearchdirections. My thanks also go to the rest of my committee, Professor Zhu, Leon and Professor Wiltermuth, Scott. Withouttheirhelp,mythesiscannotbeapplicabletogeneralreaders. Lastbutnotleast,IwishtotakethisopportunitytoexpressmygratitudetotheOperationsManagement groupoftheDepartmentofDataScienceandOperationsatUSCMarshallSchoolofBusiness. Growingup inthisenvironment,Ibecomeanindependentscholar. iii Contents Acknowledgments iii Contents iv ListofTables vii ListofFigures viii Abstract ix 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 ResearchQuestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 ApproachtoResearchQuestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 BriefModelDescriptionandMainContent . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 ResultPreview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 LiteratureReview 11 3 RecyclingofSymmetricProductsunderEgalitarianCostAllocation 15 3.1 SymmetricProductModelSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 SocialProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 iv 3.3 ExternalityProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 ResponsibilityProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4.1 CoalitionalStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4.2 FarsightedStabilityinRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 GovernmentalDecision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5.1 TaxandSubsidy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5.2 AchievingSPOptimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5.3 ComparingSocialWelfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 RecyclingofAsymmetricProductsunderEgalitarianCostAllocation 46 4.1 Single-MarketManufacturing: [3;3]Model . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1.1 ExternalityProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.2 SocialProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.3 ResponsibilityProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Single-MarketandCross-MarketManufacturing: [2;3]Model . . . . . . . . . . . . . . . . 58 4.2.1 ResponsibilityProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Cross-MarketManufacturing: [2;4]Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5 RecyclingofAsymmetricProductsunderProportionalCostAllocation 70 5.1 Single-MarketandCross-MarketManufacturing: [2;3]Model . . . . . . . . . . . . . . . . 70 5.2 Cross-MarketManufacturing: [2;4]Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 v 6 Conclusion 83 6.1 ModelReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.2 ResultReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.3 Contributions,InsightsandShortcomings . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 References 88 A TechnicalAppendixtoChapter3 93 B TechnicalAppendixtoChapter4 134 C TechnicalAppendixtoChapter5 144 vi ListofTables 4.1 TotalRecyclingCostandSPSocialWelfareunderEachCoalitionStructure(ThreeProducts) 49 4.2 Firms’OptimalPayoffsunderEachCoalitionStructure([3;3]Model) . . . . . . . . . . . . 52 B.1 TotalRecyclingCostandSPSocialWelfareunderEachCoalitionStructure(FourProducts) 142 vii ListofFigures 1.1 AsymmetricProductsModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1 UnitCostConvergenceRate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 OptimalStructureofSocialProblem(12Firms) . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 OptimalStructureofExternalityProblem(12Firms) . . . . . . . . . . . . . . . . . . . . . 26 3.4 StableStructureofResponsibilityProblem(4Firms) . . . . . . . . . . . . . . . . . . . . . 34 3.5 OptimalSizeoftheMajorityCoalition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1 [3;3]ModelRPStableStructures(EgalitarianCostAllocation) . . . . . . . . . . . . . . . . 58 4.2 [2;3]ModelRPStableStructures(EgalitarianCostAllocation) . . . . . . . . . . . . . . . . 64 4.3 [2;4]ModelRPStableStructures(EgalitarianCostAllocation) . . . . . . . . . . . . . . . . 67 5.1 [2;3]ModelRPStableStructures(ProportionalCostAllocation) . . . . . . . . . . . . . . . 73 5.2 [2;4]ModelRPStableStructures(ProportionalCostAllocation) . . . . . . . . . . . . . . . 80 viii Abstract In the context of the Extended Producer Responsibility (EPR), governments and producers are making efforts to recycle consumer products at their end of life. In the electronics industry, several local govern- ments have introduced legislations such as the WEEE (Waste Electrical and Electronic Equipment) Direc- tive to impose the recycling responsibility to producers. Implementations of these EPR-type legislations includetwotypesofmechanisms: producersjointlyrecycleproductstogether,whichfollowstheCollective Producer Responsibility (CPR); producers individually recycle their own products, which follows the Indi- vidual Producer Responsibility (IPR). Discussions on the CPR and IPR mainly focus on the economies of scale and product heterogeneity. Following CPR, producers get the economies of scale, but pay a higher unit recycling-related cost, as the product heterogeneity increases the difficulty of collection, separation, disassembly,andmaterialprocess. FollowingIPR,producerscanavoidtheunitcostincreasecausedbythe productheterogeneity,buttheyalsolosetheeconomiesofscale. This paper addresses the following four questions: (1) What is the trade-off between recycling guided by governmental EPR-type legislations while organized by producers, and recycling directly organized by thegovernment? (2)IncompliancewithEPR-typelegislationssuchastheWEEE,shouldproducersfollow the CPR and jointly recycle their products, or should they follow the IPR and set up individual recycling programs? (3) When some producers make multiple products across different markets, how should they organizerecycling? (4)Whataretheimpactsofthecostsharingmechanism? To cover these questions, on one hand, this paper discusses three scenarios. (a) The government plans for production and recycling, in order to maximize the social welfare, which a benchmark model and is referred to as Social Problem (SP). (b) Producers compete in product quantities in order to maximize their revenues (excluding the recycling cost), while the government organizes recycling in order to minimize the ix total recycling cost, which is referred to as Externality Problem (EP). (c) Producers organize both produc- tionandrecycling,inordertomaximizetheirprofits(consideringtherecyclingcost),whichisreferredtoas ResponsibilityProblem(RP).Particularly,inRP,buildingupastructureofrecyclingcoalitionsisacooper- ativedecisionofallproducersbuteveryproducerhasitsownobjective. Hence,welookforstablestructures underwhichnoproducerhastheincentivetochangeitsdecision. Ontheotherhand,thispaperfocusesontwotypesofproductmodels. First,eachproducerproducesone product; products are symmetrically heterogeneous; all products belong to a general market; such products are considered as symmetric products. Second, some products are homogeneous and belong to the same market; some products are heterogeneous and belong to different markets; some producers make multiple products across multiple markets; such products are considered as asymmetric products. We also consider differentcostallocationmechanismsforthesetwomodels. Withsymmetricproducts,weconsidertheegal- itarian cost allocation, in which the fixed cost for all products that are recycled together are the same. With asymmetric products, we consider both the egalitarian cost allocation and the proportional cost allocation, inwhichtheeconomiesofscaledependthetotalquantityandaresharedaccordingtomarketshare. Our analysis shows that when products are symmetric, only forming coalitions with similar sizes is optimal in EP, but is not always optimal/stable in SP and RP. Based on these results, the stable structure of RP can generate higher social welfare than the optimal structure of EP; but if the government imposes taxes to producers, EP may achieve the SP optimal but RP cannot. When products are asymmetric, if producersareunequalexante(e.g.,aproducermakesproductsinonemarketandtheotherproducermakes products across different markets), to increase the competitive advantage, producers may adopt the firm- based recycling strategy. That is, to recycle their own products regardless of the product types, although it limits the economies of scale and increases the unit recycling cost (due to product heterogeneity). If producers are equal ex ante (e.g., both producers make products in both markets), the firm-based recycling strategy cannot be adopted by both producers. Moreover, under egalitarian cost allocation, the firm-based recycling strategy may be adopted by only one producer; under proportional cost allocation, the firm-based recyclingstrategyisneveranoption,consideringthestabilityofarecyclingstructure. Keywords: ExtendedProducerResponsibility(EPR),firm-basedrecycling,economiesofscale,product heterogeneity x Chapter1 Introduction 1.1 Background As, historically, governments bore the brunt of recycling-related costs, they have been active in proposing newsolutionsinordertoreducetheirfinancialburden. In1990,theExtendedProducerResponsibility(EPR; see, e.g., European Parliament 2012, Lifset et al. 2013) was introduced as a policy tool to reduce the waste streams generated by increasing volumes and varieties of consumer products. It financially encourages, motivates, or requires producers to take the environmental responsibility for the products they bring to the marketthroughouttheirproductlifecycles. Whileproducersstilldeterminethequantityoftheirproductsin themarket,EPRimplementationre-balancesthemarketcompetitionunderthenewcoststructure. Currently, EPR-type strategies are widely deployed in different parts of the world. As early as 2002, the European Union led the way in collecting, recycling and recovering electrical and electronic products through the WasteElectricalandElectronicEquipmentDirective2002/96/EC(WEEE),whichimposestheresponsibility of disposing electronic wastes on their producers. This directive has become European law and has been implemented in all EU member countries by now. In the US, legislations similar to the WEEE have not been approved by the federal government yet, but 25 states have passed legislations requiring the statewide recyclingofe-waste(see,e.g.,Souza2013). 1 Forinstance,Texaspassedacomputertake-backlawin2007, whichrequiresproducerssellingnewcomputerequipmentinTexastoofferconsumersafreeandconvenient recyclingprogram;in2011,asimilarlawwasalsoapprovedfortelevisions(seeChapa2012). One of the biggest obstacles to the recycling of consumer goods is the high cost of material separation. Therecoveredvaluefromrecyclingdecreasesasthedisassemblyofthediscardedproductismoredifficult. For example, recovery of high-value reusable components from a car, television set or cell phone may be toolaborintensive,makingrecyclingcompaniesforgothatoptionandsimply“grindtheproduct”torecover less valuable raw materials such as steel, precious metals or plastic instead of more valuable components. 1 SeeElectronicsTakebackCoalition: http://www.electronicstakeback.com/promote-good-laws 1 The heterogeneity of the material streams is a primary determinant of the unit (recycling) cost (including separation) (see, e.g., Dahmus and Gutowski 2007, Gutowski and Dahmus 2005). Indeed, upon examining theunitpriceschargedbyrecyclersthatfocusonlyoncertaintypesofproductsandcomparingthemtoprices charged for recycling of miscellaneous products, we find that heterogeneous waste streams tend to exhibit higherunitrecyclingcost. Forinstance,inEarthworksRecycling,Inc.,arecyclingcompanyinWashington State,weobserve: i)computermonitors,CPUs,televisions,laptopsande-readers/e-booksarerecycledfree ofcharge;ii)refrigerators,freezers,airconditionersandanyotherappliancesthatcontainFreonarerecycled at $0.10 per pound; and iii) miscellaneous electronics are recycled at $0.30 per pound. One approach to overcome this obstacle would be to recycle products at the level where they are more homogeneous, i.e., at the level of the individual producer (see, e.g., Dempsey et al. 2010). Such policy mechanism is called the Individual Producer Responsibility (IPR; see, e.g., Tojo 2003, Dempsey et al. 2010), which is an category of EPR. For example, Samsung has an independent recycling system designed to take back only Samsung productsinallstatesoftheUSthathaveEPR-typelegislations. Asecondmajorobstacletorecyclingistheeconomiesofscale. Processesoftakingbackproductsfrom consumers involves: i) setting up a mail back program; ii) organizing collection events; iii) establishing recyclingsites;iv)contractingwithlocalgovernments,retailers,andnon-profitorganizations;v)conducting community collection events; vi) educating consumers; vii) certifying participating recyclers’ processes; viii)administeringfinancialflowsbetweenconsumers,producersandrecyclers;etc. Theseactivitiesusually require high fixed setup costs and complicated procedures, and individual producers may not have enough volume to make recycling break even. In order to achieve high enough volumes to defray fixed costs, some producers share recycling resources and collectively recycle their products. Such policy mechanism is called the Collective Producer Responsibility (CPR; see, e.g., Atasu and Subramanian 2012), which is another EPR category. For instance, the MRM is an electronic producer recycling company sponsored by Mitsubishi, Panasonic, Sanyo, Sharp and Toshiba; it recycles televisions, monitors and laptops for all of its members. The implementation of the WEEE Directive follows both the IPR and CPR. In Europe, governments compel producers to set up Producer Compliance Schemes (PCS) (see European Commission 2009). For example, originally set up by Gillette, Braun, Electrolux, HP and Sony, the European Recycling Platform (ERP) is a pan-European scheme for electronic waste (see Stevens and Goosey 2008). In the Netherlands, there are three PCSs for electronic waste: NVMP for household electronic goods, ICT Milieu for IT, and 2 Stichting Lightrec for lighting equipment (see Savage et al. 2006). In the UK, the Environment Agency approved over 40 PCSs (Valpak, REPIC, etc.) to ensure proper disposal of electronic waste (see UK Envi- ronment Agency 2015). Those PCSs contain different numbers of producers, and producers are free to join anexistingPCSortoestablishanewPCS(see,e.g.,UKBISDepartment2014,Sachs2006). IntheUnited States, state governments require producers to either join a statewide program or operate independent pro- gramsontheirown. Forexample,producersinWashingtonStatehavetworecyclingplanoptions,Standard Plan and Independent Plan (see Washington Materials Management Financing Authority 2013); in New York State, producers have to participate in either an individual electronic waste-acceptance program or a collectiveelectronicwaste-acceptanceprogram(seeNewYorkState2010). 1.2 ResearchQuestion Discussions on the EPR mainly focus on the following problems, which are also research topics of this paper. 1. What is the trade-off between recycling guided by governmental EPR-type legislations while orga- nized by producers, and recycling directly organized by the government? As discussed above, in an industrywithcompleteEPR-typelegislations,suchaselectronicsindustrywiththeWEEEDirective, producers take the responsibility of recycling their own products. However, before introducing the EPR-type legislation to a certain industry, the government usually takes charge of full responsibility for recycling of all producers’ products, including encouraging consumers to bring their end-of-life productstocollectingcenters,buildingupreverselogisticschannels,contractingwithrecyclingfacil- ities, and certifying recycling processes. Following this way, the California government has been the leaderofrecyclingbeveragecontainersintheUnitedStates. ThroughCalifornia’sBeverageContainer Recycling Program, in 2013, the Department of Resources Recycling and Recovery, or CalRecycle, recycled 18.2 billion beverage containers from 21 billion sold in California. 2 Intuitively, because of financial burdens, governments may prefer to be responsible for products that are cheaper to recycle, while let producers be responsible for products that are more expensive to recycle. However, such financial burden may be relieved via taxes and subsidies. In this paper, we analyze factors that play importantrolesinchoosingbetweenthesetwo(recycling)responsibilities. 2 seeBeverageContainerRecycling: http://www.calrecycle.ca.gov/bevcontainer/ 3 2. In compliance with EPR-type legislations such as the WEEE, should producers follow the CPR and jointlyrecycletheirproducts,orshouldtheyfollowtheIPRandsetupindividualrecyclingprograms? Let us discuss the recycling of TV sets in the United States for example. 1) Some TV producers, Toshiba, Panasonic, Sanyo, Mitsubishi, and Sharp, cooperate and recycle their products together, through the MRM, a company of Manufacturer (producer) Recycling Management for televisions, desktops, laptops, and monitors (see Gui et al. 2013). 2) Sony and LG are members of the Waste Management, a company for recycling of general products, which also recycles TV sets of Sony and LG. 3 Finally,3)Samsungusesitsownprivaterecyclingsystem(seeSamsungElectronics2009);with this system, all Samsung brand products are recycled. From the observation on these coalitions, why producers form coalitions of different sizes is of our first interest. In addition, an important feature of producers following EPR is, they are free to switch between the IPR and CPR. For instance, Vizio used to be one of the participating producers of the MRM, but recently it has been removed from the MRM website. We wonder whether any other ”switching” may happen, and in what circumstance thereisnofurther”switching”. 3. When some producers make multiple products across different markets, how should they organize recycling? Inotherwords,shouldallproductsberecycledtogether(all-inclusive),orshouldproducts made by the same producer be recycled together, regardless of product types (firm-based), or should products from the same market be recycled together, regardless of product brands (market-based)? For example, Panasonic and Samsung are both producers of televisions, monitors, laptops, and toner cartridges. As one of the MRM founders, Panasonic recycles its televisions, monitors, and laptops through the MRM. However, it uses a separate take back program to recycle its toner cartridges; that is, Panasonic adopts market-based recycling strategy. On the other hand, Samsung adopts firm-based recyclingstrategy: itrecyclesallSamsungbrandproductsthroughitsowntakebackprogram. 4. What are the impacts of the cost sharing mechanism? One mechanism is the egalitarian cost allo- cation, under which producers pays the same amount of cost (see Gui 2012). Another typical cost allocation mechanism is return share or market share (which is considered as proportional), respec- tively, whereby each producer is allocated a portion of the cost that is equivalent to its share in the total waste volume returned or sold. For example, in states such as New Jersey and Wisconsin, pro- ducers with different market share pays the same amount of registration fee; while in states such as 3 SeeWasteManagement: http://www.wm.com/residential/takeback-program.jsp 4 Connecticut and Maine, producers’ registration fee depend on their market share. 4 We are interested inthedifferencebetweenthesetwomechanisms. 1.3 ApproachtoResearchQuestion Inthefollowingtext,wetakethegovernmentasasocialplanner,andtakeproducersasfirms. ToaddressResearchQuestion1,wefirstconsiderthesocialwelfaremaximizationproblemofthesocial planner. We refer to it as the Social Problem (SP), and use it as a benchmark model. In SP, for a given industry setting (number of firms, degree of heterogeneity, etc.) and cost structure (fixed and variable costs of recycling), the social planner determines the production volume for each firm and designs a recycling networkstructuretorecycleallproducts,inordertomaximizethetotalsocialwelfare(marketsurplusminus the total recycling costs). Then, we study two commonly adopted recycling approaches, the Externality Problem (EP) and the Responsibility Problem (RP). The EP mimics the following situation: before EPR- type legislation is introduced, recycling is the external business of firms. On one hand, firms compete with eachotherintheprimarymarketwithquantitiesignoringallcostsrelatedtorecycling,inordertomaximize their revenues. On the other hand, a social planner sets up a recycling network and properly disposes all products that firms brought to the market, in order to minimize total recycling costs. The RP mimics the following situation: after EPR-type legislation is introduced, recycling is the extended responsibility of firms. In RP, to maximize payoff, firms compete with quantities in the primary market; meanwhile they recycletheirproductsthroughdifferenttypesofjoint-recyclingcoalitions. Toaddress Research Question 2, it is important to understand whyfirms form recyclingcoalitions. In a simple economy with identical firms competing with differentiated products in the primary market (which are considered as Symmetric Products), absence of variable-cost considerations would naturally lead to an all-inclusive coalition to share fixed costs. An absence of fixed-cost considerations would naturally lead to no coalitions whatsoever, in order to minimize variable costs. In the presence of both fixed- and variable- cost considerations, endogenous formation of coalitions is less clear. As every firm is free to join or leave a coalition, or to form a new coalition with others at their will, we identify stable recycling structures in which no firm has an incentive to change its decision. The decisions are based on firms’ payoffs, and take into account that any decision by one firm may cause reactions from other firms. We capture this setting 4 seeStateElectronicsRecyclingPrograms: http://www.mrmrecycling.com/states.htm 5 by using dynamic/farsighted stability concepts. In this paper, we use the Largest Consistent Set (LCS, see Chwe 1994) and the Equilibrium Process of Coalition Formation (EPCF, see Konishi and Ray 2003). The LCS considers players’ payoffs in the final (stable) structure; while the EPCF considers players’ infinite horizondiscountedpayoffs. We address Research Question 3 by analyzing structures for recycling of products from multiple mar- kets, e.g., laptopsandrefrigerators. Productsfromdifferentmarketscanbecollectivelyrecycled,savingon the fixed cost, but at a higher unit recycling cost, due to the heterogeneity in material streams. We consider thecasewhereinsomeproductsbelongtothesamemarket,whilesomeproductsbelongtodifferentmarkets (which are considered as Asymmetric Products). As we are interested in joint recycling based on product typesversusjointrecyclingbasedonproductbrands,weconsiderfirmsmakingdifferenttypesofproducts. That is, products in the same market may be made by different firms, and products made by the same firm mayhaveacrossmultiplemarkets. We address Research Question 4 by considering two cost allocation mechanisms. In some models, we assume each recycling infrastructure pays a fixed setup cost, and it is equally shared among participating firms. We consider such mechanism Egalitarian Cost Allocation, in which the economies of scale have the sameimpactstofirmsinthesamecoalition. Inanothermodel,weignorethefixedsetupcostbutinstead,we assume a more general form of economies of scale. It depends on the total product quantity of a recycling infrastructure, and are shared according to participating firms’ market share. We consider such mechanism Proportional Cost Allocation, in which the economies of scale have different impacts to firms in the same coalition. 1.4 BriefModelDescriptionandMainContent In Chapter 2, we summarize related literature on features of closed-loop supply chain, implementations of IPR and CPR, impacts of product heterogeneity on the unit cost, and stability concepts within the field of gametheory. InChapter3,westudytherecyclingofsymmetricproductsunderegalitariancostallocation. Toassume symmetricproducts,wesupposethateachfirmmakesoneproduct;allproductsbelongtothesamemarket; there is a uniform degree of heterogeneity between any two products. To assume the egalitarian cost allo- cation, we suppose that when products are jointly recycled, their firms equally share the fixed setup cost. 6 ThischaptermainlyaddressesResearchQuestions1and2—itstudiesSP,EPandRP;especiallyforRP,it discussesthedynamic/farsightedstabilityoffirms’coalitions. In Section 3.1, we introduce one of the most important assumptions throughout of the paper: product heterogeneity increases the unit recycling-related cost. In Section 3.2, we study the benchmark model, Social Problem, in which the social planner determines production quantities in the primary market and recycling structures in the recycling market, to maximize the social welfare. In Section 3.3, we discuss the model without EPR-type legislations introduced, the Externality Problem, in which firms determine their quantities in the primary market to maximize their revenues, while the social planner chooses the recycling structure in the recycling market to minimize the total recycling cost. In Section 3.4, we study the model followingEPR-typelegislations,theResponsibilityProblem,inwhichfirmscompetitivelychoosetheirown product quantities and cooperatively determine a recycling structure. We use the Largest Consistent Set (LCS) to identify recycling structures that are stable from the dynamic/farsighted perspective. In Section 3.5, we discuss the governmental decision making, e.g., imposing taxes/subsidies to firms, and choosing fromEPandRPbycomparingthehighestsocialwelfaregeneratedfromthem. InChapter4,westudytherecyclingofasymmetricproductsunderegalitariancostallocation. Toassume asymmetric products, we suppose that firms may produce multiple products across different markets; dif- ferent firms may compete in the same market; products from different markets are with great heterogeneity whileproductsfromthesamemarketdoesnothaveheterogeneity;productscanbejointlyrecycled,but,not necessarily so. This chapter mainly addresses Research Question 3. It analyzes conditions under which (1) all products should be recycled together (we refer to it as all-inclusive recycling), (2) products should be jointly recycled based on their markets (we refer to it as market-based recycling), and (3) products should bejointlyrecycledbasedontheirfirms(werefertoitas firm-based recycling). In Section 4.1, we start with a simple case with three firms each making one product (in total three products), which we denote by [3;3] (first number denotes the number of firms, and second the number of products). For instance, Dell and Lenovo both produce laptops, and GE produces refrigerators. More generally,saythatfirmsA,B andC makethreeproducts, 1,2and3,respectively;products1and2belong to the same market and are quite similar (i.e., exhibit a low level of heterogeneity); product 3 belongs to an independent market and has higher heterogeneity level with respect to products 1 and 2; this is illustrated in Figure 1.1a. In Section 4.2, we reduce the number of firms (from three to two), while keeping the same products from the [3;3] model. For instance, Dell produces laptops, and Samsung produces laptops and 7 refrigerators. Thismodelcanbe viewedasfirmsB andC (ofthe [3;3] model)mergetogether andproduce bothproducts2and3. Inotherwords,onefirm(A)makesproducts(1)inonemarket,whileanother(larger) firm (B) makes products (2 and 3) across different markets. We denote this model by [2;3], which we illustrate in Figure 1.1b. In Section 4.3, we increase the number of products (from three to four), while keeping the same firms from the [2;3] model. For instance, Samsung and LG both produce laptops and refrigerators. This model can be viewed as firmA (of the [2;3] model) introduces a new product, say, 4, which is independent of products 1 and 2, but is substitutable with product 3. That is, firmsA andB are equalexante. Wedenotethismodelby[2;4],whichweillustrateinFigure1.1c. (a)[3;3]Model (b)[2;3]Model (c)[2;4]Model Figure1.1: AsymmetricProductsModels In Chapter 5, we study the recycling of asymmetric products under proportional cost allocation. To assume the proportional cost allocation, we suppose that (1) the economies of scale depend on the quantity of products that are recycled together; and (2) if different firms cooperate and jointly recycle their products together, they share the economies of scale according to their market share. This chapter mainly addresses Research Question 4. We are interested in under what conditions the all-inclusive recycling, market-based recycling and firm-based recycling can emerge as the stable recycling structure. As firm-based recycling is notanoptionfordiscussioninthe[3;3]model,weonlyfocusonthe[2;3]modeland[2;4]model,discussed inSections5.1and5.2,respectively. InChapter6,weconcludethefindings,contributionsandinsightsofthiswork. 8 1.5 ResultPreview Resultsofsymmetricproducts: Facingwithsymmetricallyheterogeneousproductsinageneralmarket, we attribute the competition in that market to the product heterogeneity. Then, we find that the fixed cost and the product heterogeneity are the two determinants of our analysis. When the fixed cost is high (low), the social planner and firms have more (less) incentive to jointly recycle products. When products are with great heterogeneity, the unit cost of joint recycling increase, but the market competition is less intense. In addition,duetotheproductheterogeneity,asmorefirmsjoinacoalition,theunitcostincreases. Therateof thisincreaseisanotherfactorthatcanleadtodifferentresultsinouranalysis. As a results, for the Social Problem, although products are symmetric and indifferent to the social planner,thesocialplannermightnotalwayschoosecoalitionsofsimilarsizes. Whentheunitcostincreases quickly in the coalition size, it is optimal for the social planner to create some small recycling coalitions and one large coalition. When the unit cost increases slowly, then only choosing coalitions of similar sizes maximizes the social welfare. For the Externality Problem, the optimal recycling structure contains only similarly-sizedcoalitions. ResultsfortheResponsibilityProblemistheoppositeoftheSPresults: whenthe unit cost increases quickly, the stable structure contains only similarly-sized coalitions; when the unit cost increases slowly, there exists an optimal coalition size such that firms form coalitions of this size as many aspossible,andleavingtheremainingfirmstoformonesmallcoalition. RP can generate a higher social welfare when the recycling task is “challenge” — (i) the fixed cost is high,and(ii)therearemanyproducers,and/or(iii)productsarenothomogeneous,and/or(iv)thebasicunit (recycling) cost is not low. With the consideration of imposing taxes and subsidies, EP can achieve the SP optimumifprovidedwithatax,whileRPcanachievetheSPoptimumifprovidedwithasubsidy. However, when there are many producers, this subsidy can be negligible. In general, RP is more beneficial for the government. Resultsofasymmetricproducts: Facing with asymmetric products across multiple markets (under both egalitarianandproportionalcostallocations),weseparatethetwoconceptsofmarketcompetitionandprod- uctheterogeneity. Forproductsbelongingtothesamemarket,thereexiststhemarketcompetition,butwith noproductheterogeneity;forproductsbelongingtodifferentmarkets,thereexiststheproductheterogeneity, 9 but with no market competition. Then, the economies of scale, product heterogeneity and market compe- tition are the three determinants of our analysis. When the economies of scale are high (low), firms have more(less)incentiveforjointrecycling. Whenproductsfromdifferentmarketsarewithgreatheterogeneity, firms have more incentive to jointly recycle products based on their markets. When products in the same market are with great competition intensity, firms have more incentive to jointly recycle products based on theirfirms. As a result, whenever the social planner is responsible for recycling (in the EP and SP), depending on the level of economies of scale, either all products should be recycled together regardless of the market (whentheeconomiesofscalearehigh),whichisreferredtoasthe all-inclusiverecycling,orproductsfrom the same market should be recycled together regardless of their firms (when the economies of scale are low), which is referred to as the market-based recycling. When firms are responsible for recycling (in the RP), if the market competition is intense, competing firms may choose not to cooperate in the all-inclusive recyclingormarket-basedrecycling. Instead,theymaychoosetorecycletheirownbrandproductstogether, regardlessoftheproducttype. Werefertoitasthefirm-basedrecycling,andthepurposeistoincreasefirms’ competitiveadvantage. Ouranalysisshowsthat: whenfirmsareunequalexante(onefirmmakesproductsin onemarketwhileanotherfirmmakesproductsacrosstwomarkets),inabroaderrangeofparametervalues, the firms-based recycling can emerge as stable; when firms are equal ex ante (both firms make products acrossthetwomarkets),thefirm-basedrecyclingcannotbeadoptedbybothfirms. Egalitarianandproportionalcostallocationshavesimilarconditionsunderwhichtheall-inclusiverecy- cling and market-based recycling can emerge as stable. However, compared with the egalitarian cost allo- cation, the proportional cost allocation increases the competition between firms. As a result, when the two firms are unequal ex ante, the firm-based recycling is more likely to be adopted under proportional cost allocationthanegalitariancostallocation. Whenthetwofirmsareequalexante,ifonefirmadoptsthefirm- basedrecyclingwhiletheotherfirmdoesnot,thetwofirmsareunequalexpost. Wefindthatsuchrecycling structure can be stable in a limited range of parameter values under egalitarian cost allocation, but is never stable under proportional cost allocation. It seems counter-intuitive but the reason is: under proportional costallocation,thepenaltyofbeing”unequalexpost”issevererthanunderegalitariancostallocation. 10 Chapter2 LiteratureReview Therearefourmajorliteraturestreamsrelatedtoissuesraisedinthisthesis. Thisthesisfitswellintheclosed-loopsupplychainliterature. Fleischmann(2003)andEsenduranetal. (2012) classify reverse logistics networks based on the form of reprocessing (remanufacturing versus recy- clingversusreuse),thedriverforproductrecovery(economicsversuslegislation)andtheownerofrecovery processes (producer versus third-party). A variety of papers consider different combinations of these three characteristics. Forinstance,Savaskanetal.(2004)discussaneconomics-drivenmodelinwhichaproducer either performs remanufacturing by itself or subcontracts remanufacturing to a third party. Toyasaki et al. (2011) introduce a model in which two producers outsource the WEEE-driven recycling to two recyclers; recyclers either directly contract with producers or negotiate with a non-profit organization. The main part of this thesis focuses on recycling driven by EPR-type legislations and organized by producers; the rest focusesonrecyclingorganizedbythegovernment,withoutconsideringdifferentlegislations. Thestreamofliteratureonlegislation-drivenproducerrecyclingfocusesonimplementationsoftheIPR and CPR. Atasu et al. (2009) study how to implement the IPR in a model with a single product made by multiple producers. The authors suggest that besides the product quantities, when implementing IPR, one should also consider the recycling treatment cost, the market competition intensity, the product environ- mental impact and customers’ willingness-to-pay for the decrease in the environmental impact. This thesis considersallfactorsexceptthelasttwo,astheyaremainfeatureoftheIPRmodel;weconsiderbothIPRand CPRmodels. Gui(2012)studieshowtoimplementtheCPRthroughacostallocationmechanisminalarge collection and recycling network, which consists of multiple producers with multiple products. The author arguesthat,whenimplementingtheCPR,theegalitariancostallocationmechanismismorelikelytoinduce cooperation of all producers within a single network than the proportional cost allocation mechanism. The author uses the cooperative game-theoretical methodology to analyze the stability of networks/coalitions. There are two common interests between Gui (2012) and this thesis. First, this thesis also compares the egalitarian cost allocation mechanism and the proportional cost allocation mechanism, but we focus on the 11 impact of different mechanisms on the recycling structure. Second, this thesis also uses the cooperative game-theoretical framework, but we consider both the case in which all producers join a single network andthecaseinwhichproducersformseveralnetworks. AtasuandSubramanian(2012)studyhowtoselect between the IPR and CPR in a recycling model with two products made by two producers. After compar- ing the case in which both producers follow the IPR and individually recycle with the case in which both followtheCPRandcollectivelyrecycle,theauthorsconcludethatthetrade-offisthereductionofrecycling costs through improved design in the IPR and the operational cost-efficiency under the CPR. This thesis considers both sides of the trade-off, but we also allow for simultaneous existence of IPR-based and CPR- based recycling. Esenduran and Kemahlıo˘ glu-Ziya (2014) discuss producers’ choice between the IPR and CPR based on cost structures. The authors identify the size of the CPR coalition, the coalition composition (large/small producers), and the financial benefit from environmentally-friendly product design as the key decision making factors. While they discuss the effect of a producer’s defection on the CPR coalition, they donotconsiderotherstabilityissues,suchasdefectionsofmultipleproducers. Thisthesisalsoconfirmsthe importance of the first two factors; moreover, we provide a more comprehensive discussion of the stability ofCPRcoalitions. A third category related to this thesis is literature on the product heterogeneity. A number of engineer- ing papers study the relationship between product heterogeneity and unit (recycling) costs. Dahmus and Gutowski (2007) and Gutowski and Dahmus (2005) propose to measure the unit cost by material dilution, implyingthatthecomplexproductdesignresultsinincreasedunitcosts. Oguchietal.(2011)focusesonthe data of 14 electronics products providing information on the material dilution in CRT, PDP and LCD TVs, concludingthattheunitcostincreasesinthedegreeofheterogeneity. Basedonthisliterature,weassumein this thesis that the unit cost increases in the size of the coalition as separation becomes more expensive and incentivestohomogenizewastestreamsarediluted. Theeffectisstrongerinproductsthataremorehetero- geneous. Another group of papers (e.g., Sosale et al. 1997, Gu and Sosale 1999, Gershenson et al. 2003) studytherelationshipbetweentheproductmodularityandtheunit(recycling)cost,concludingthatmodular designhasapositiveeffectonproductlifecycle. Fromtheoperationsmanagementside,severalpapers(e.g., Desai et al. 2001, Fleischmann et al. 2000, Alumur et al. 2012) analyze the impact of component common- ality,andshowthatitreducestheunitmanufacturingandrecyclingcost. Accordingtothesepapers,therate at which the unit cost increases in the coalition size can be inferred by product design. However, none of 12 thepapersinthiscategorystudiestheconsequencesofcollaborativeproductrecycling. Thisthesisfillsthis gap. The last important category of literature studies different notions of stability within the field of game theory. The earliest stability concepts in multilateral games are the von Neumann-Morgenstern stable set (see Morgenstern and Von Neumann 1953) and the Nash equilibrium (see Nash 1950), which consider the instant payoff after an action as the incentive for that action. These stability concepts are myopic/static, as they only consider immediate consequences of players’ actions. In addition, they consider players’ com- petitive behavior, as each player makes independent decisions and receives corresponding payoffs. Unlike thesetwoconcepts,thecore(seeGillies1959)andthecoalition-proofNashequilibrium(seeBernheimetal. 1987)allowplayerstoformcoalitionsandcooperateintheirdecisions,butstillunderamyopic/staticsetting. More recently, researchers acknowledged that players consider how others react to their actions, and devel- oped stability concepts of a more dynamic/farsighted nature. The bargaining set (see Aumann 1961) only considerstwosteps,objectionandcounter-objection,andthusallowsforpartialfarsightedness. Morerecent work allows that players look further into future, and includes the canonical set (see Harsanyi 1974), the largest consistent set (LCS, see Chwe 1994), the optimistic or conservativestable standard of behavior(see Xue 1998), the equilibrium process of coalition formation (EPCF, see Konishi and Ray 2003), the largest cautiousconsistentset(seeMauleonandVannetelbosch2004),andthesubgameperfectconsistentstability (see Granot and Hanany 2013). Majority of dynamic/farsighted concepts assume that players only receive their payoffs in the final (stable) structure; in the EPCF, players receive their payoffs after every move and base their decisions on infinite horizon discounted payoffs. Several papers in the operations management area study coalition formation and stability in a non-myopic sense; that is, by assuming that agents, when makingtheirdecisions, takeintoaccounthowotheragentsmakereacttotheirmoves. Forexample,Granot and Soˇ si´ c (2005), Soˇ sic (2006), and Nagarajan and Soˇ sic (2007) study horizontal cooperation among sev- eral retailers; Granot and Yin (2008), Nagarajan and Bassok (2008), and Nagarajan and Soˇ sic (2009) study horizontalcooperationinassemblymodels;Kemahlioglu-ZiyaandBartholdiIII(2011)studiescooperation amongretailerswhoorderfromacommonsupplier;Soˇ si´ c(2010)studyverticalcooperationinathree-level supply chain. However, applications on producer take-back programs are quite few so far. Gui (2012) uses the core to analyze the stability of the CPR coalition consisting of all producers. This thesis use both the LCS to analyze the stability of recycling coalitions for producers of symmetric products, and use the EPCF to analyze the stability of recycling coalitions for producers of asymmetric products. To the best of our 13 knowledge, it is the first work to apply the dynamic/farsighted stability approach in the area of sustainable operations. 14 Chapter3 RecyclingofSymmetricProductsunder EgalitarianCostAllocation Inthischapter,westudytherecyclingofheterogeneousproductsfromageneralmarket. InSection3.1,we analyze the basic elements of such a general market and its recycling-related cost structure. In Section 3.2, we study a benchmark problem of the social planner, which optimizes production quantities and recycling structurestomaximizethesocialwelfare. InSection3.3,wediscussthecaseinwhichthereisnoEPR-type legislation introduced. Instead, the social planner organizes recycling. In Section 3.4, we study the case in whichtheEPR-typelegislationhasbeenintroducedandfirmsendogenouslyformcoalitionstorecycletheir products. In Section 3.5, we discuss governmentalsolutions to achievea higher social welfare: one issue is tointroducetheEPR-typelegislation,theotherissueistoimposetaxes/subsidies. 3.1 SymmetricProductModelSetup In this section, we explain the market surplus and the cost structure of a market in which products have a uniformlevelofheterogeneity. Market surplus: We consider an economy withN ∈Z + ex ante equal firms. LetN = {1,...,N} be the set of firms. Suppose that each firm produces a single type of product. We useq i > 0 to denote the output of firmi,i∈N, and use q = (q 1 ,...,q N ) to denote the vector of quantities produced by thoseN firms. Following Singh and Vives (1984), we assume that theseq products create a market surplusU(q) in theprimarymarket: U(q) = N ∑ i=1 q i − 1 2   β N ∑ i=1 q 2 i +γ N ∑ i=1 N ∑ l=1;l̸=i q i q l   , (3.1) whereβ > 0andγ∈ [0,β]. 15 To understand the effects ofβ andγ, consider the production quantitiesq i andq j of two firmsi andj (̸=i),respectively. Considernowthetwoextremevaluesofγ:γ = 0andγ =β. • Forγ = 0, since @U(q) @q i = 1−βq i and @ 2 U(q) @q i @q j = 0, the marginal contribution of firmi’s product to the market surplus does not depend on firmj’s production quantity. In other words, the products of firmsi andj are independent of each other. We interpretγ = 0 as the case in which theN firms are independentwithheterogeneousproducts. • For γ = β, when the market surplus is maintained at a certain level, we have dU(q) dq i = @U(q) @q i + @U(q) @q j dq j dq i = 0. That is, dq j dq i = − @U(q)=@q i @U(q)=@q j = − 1− ∑ N l=1 q l 1− ∑ N l=1 q l = −1. Hence, in order to have the same market surplus, if we increase firm i’s production quantity q i by one unit, we must decrease firmj’s production quantityq j by one unit. In other words, firmi’s and firmj’s products are perfect substitutes. We interpretγ = β as the case in which theN firms are substitutes with homogeneous products. We introduce x = 1− ∈ [0,1] as a measure of the degree of product heterogeneity: When x = 0, we consider products as perfectly homogeneous; when x = 1, we consider products as completely heterogeneous. Without loss of generality (WLOG), we assume that the unit manufacturing cost is zero. In this thesis, wefocusonlyonthecostsincurredafterconsumptionoftheqproducts. Weassumetheusedproductcannot be used anymore and must be discarded in an environmentally friendly way, as typically required by ERP legislation. Werefertothisprocessas“recycling,”andconsidertherelatedcosts. Structure outcomes: In our model, the recycling costs (fixed and variable) depend on how recycling is organized. To that end, we introduce in Definition 3.1.1 a recycling coalition, or in short, a coalition, A, which is a set of firms; a coalition structure,A, which is a partition of the set of all firms; and a structure outcome,n,whichisthesetofthesizesofallcoalitionsinacoalitionstructure. Definition3.1.1. We callA a coalition ifA⊆N ={1,...,N},N ∈Z + . We call|A| the size of coalition A. We callA = {A 1 ,...,A J } a coalition structure if it is a partition ofN: A j ⊆ N,A i ∩A j = ∅ for i,j = 1,...,J, 1 6 J 6 N and i ̸= j, and for J ∪ j=1 A j = N. We calln = {n 1 ,...,n J } a structure outcome, ifA ={A 1 ,...,A J } is a coalition structure andn j =|A j | forj = 1,...,J. Finally, we calln j the size of thej-th coalition. 16 1. WLOG,weassumen 1 6···6n J —coalitionsareorderedbyascendingsize. 2. 16 n j 6 N forj = 1,...,J, and ∑ J j=1 n j = N — no coalition contains more thanN firms, and theN firmsareefficientlydividedintotheJ coalitions. 3. |A| =|n| — a coalition structure and its corresponding structure outcome indicate the same number ofcoalitions. 4. WhenacoalitionA j hasmorethanonemember,n j > 1,wesaythatthecoalitionisnontrivial. WhenN = 10, for example, there are ten firms, 1,...,10;A ={4,6,8} is a coalition, which contains three firms, 4, 6 and 8; A = {{4,6,8},{1,2,3,5,7,9,10}} is a coalition structure, which includes two coalitions,{4,6,8} and{1,2,3,5,7,9,10}; and suchA corresponds to the structure outcomen ={3,7}, which indicates that there are two coalitions, the first one contains three firms and the second one contains sevenfirms. Next,wediscusshowthestructureoutcomedeterminesthefixedandunitrecyclingcosts. Fixedrecyclingcost: Productsaretakenbackthroughreverselogistics,anditisusuallycostlytoestablish arecyclinginfrastructure;thisprocessincludesbuildinguplocalrecyclingcenters,certifyingrecyclers,and arranging the transportation of the collected end-of-life products. We assume that every coalitionA j ∈ A incurs a fixed cost,K > 0, to set up a recycling infrastructure to be used by all members of the coalition, regardless of what products, or how many products, are being recycled. For a coalition structure, or a structureoutcome,includingJ coalitions,thetotalfixedrecyclingcostsareJK. The structure outcome determines the unit (recycling) cost in a more intricate way. The unit cost is relatedto(i)howmanyfirmsparticipateinthecoalitionand(ii)howheterogeneoustheseproductsare. We discussthesetwoimpactsbelow. Unit recycling cost, impact of the coalition size: As more firms join a coalition, recyclers have to deal with more product types, which increases the difficulty of collecting, separating, disassembling and recy- clingeachproductunit. Whenn(> 1)firmsproducingcompletelyheterogeneousproductsjoinacoalition, weassumetheunitcost, ¯ c(n),tobeafunctionofnwiththefollowingproperties. 1. ¯ c(1) = 0: WLOG,wenormalizetheunitcostforanindividualfirmtozero. 17 2. ∆¯ c(n) = ¯ c(n+1)−¯ c(n)> 0: Asmorefirmsjoinacoalition,thecostofseparatingoneproductunit fromthepoolof(diverse)productsincreases. 3. ∆¯ c(n+1)< ∆¯ c(n): Themarginalcostdecreaseswithincreasingcoalitionsize. 4. lim n→+∞ ¯ c(n) = c 0 : Because of the limitation of available resources, the unit cost is bounded, and convergestoabasicunitcost,c 0 . Tomakesurefirmsalwayshavenonnegativequantitiesandprofits, we assume 0 < c 0 6 ˜ c. Equation (A.9) in the proof of Lemma A.0.6 in the Appendix shows the expressionfor ˜ c. More specifically, we assume the following functional form for the unit (recycling) cost for completely heterogeneous products: ¯ c(n) = c 0 (1−θ 1−n ). The parameterθ ∈ (1,+∞) is a measure of how fast the unitcostincreasesorconvergesinthenumberoffirmsparticipatingintherecyclingcoalition. Asshownin Figure3.1,asmaller(larger)θ leadstoslower(faster)unitcost, ¯ c(n),convergencetothebasicunitcost,c 0 . Thus, values ofθ close to 1 cause the unit cost to increase slowly, and almost linearly as the coalition size grows. High values ofθ cause the unit cost to increase quickly to nearlyc 0 for small recycling coalitions andthenincreaseslowlyforlargerecyclingcoalitions. Hence,θ canbeconsideredaconvergencerate. Figure3.1: UnitCostConvergenceRate Unit recycling cost, impact of the product heterogeneity: Due to the similarity of materials, compo- nents, bills of material, etc. for products more homogeneous, the unit cost is lower than that for products thataremoreheterogeneous. WLOG,wenormalizetheunitcostofperfectlyhomogeneousproducts(x = 0) tozeroandsettheunitcostforcompletelyheterogeneousproducts(x = 1)to ¯ c(n). Forintermediatelevels ofheterogeneity,weassumethattheunitcostis ¯ c(n)x =c 0 x(1−θ 1−n ). 18 Unitrecyclingcost,interpretation: Theunitcost ¯ c(n)xcouldbeinterpretedviatheproductheterogene- ity. The engineering literature links the unit cost to the product differentiation via the concept of “entropy” (see,e.g.,DahmusandGutowski2007,GutowskiandDahmus2005): Asmore(diverse)productsarerecy- cled together, the unit cost increases. The operations management literature links unit cost to product com- monality: Product commonality reduces the product’s unit manufacturing and recycling cost (see, e.g., Desai et al. 2001, Fleischmann et al. 2000, Alumur et al. 2012). The convergence rate, θ, could be inter- pretedthroughproductmodularity: Ononehand,theproductmodularityreducesthedifficultyofrecycling complex products (see,, e.g., Sosale et al. 1997, Gu and Sosale 1999, Gershenson et al. 2003); on the other hand, according totheDesign forEnvironmentmethodology, productsdesigned withhighmodularity have more flexibility to be reused, remanufactured, and even recycled (see, e.g., Ishii et al. 1994). This leads to one possible interpretation of the convergence rate,θ: If products are designed with high modularity,θ is low — the unit cost increases slowly as the number of firms in a recycling coalition grows; if products are designed with low modularity,θ is high — the unit cost increases steeply to nearlyc 0 , the basic unit cost, forsmallcoalitions. In summary, when firms produce q products and organize recycling via a coalition structure A = {A 1 ,...,A J }whichcorrespondstoastructureoutcomen ={n 1 ,...,n J },thetotalrecyclingcostis C(q,n) = J ∑ j=1   ¯ c(n j )x ∑ i∈A j q i +K   . (3.2) With the above model setting, we capture the following: When multiple firms join a coalition, on one hand,theirshareofthefixedcostisreduced;ontheotherhand,theunit(recycling)costisincreased. There- fore, the social planner and firms are balancing the trade-off between the fixed-cost saving and increasing unitcost. Inotherwords,theyconsider(1)howmanycoalitionsshouldbeformedand(2)howtodistribute firmsacrossthecoalitions. Next, we discuss the answer based on different scenario settings: In Section 3.2, the social planner determinesboththeproductionquantitiesandrecyclingcoalitions’structureoutcome. InSection3.3,firms determine their production quantities ignoring recycling costs, while the social planner manages proper disposal of the products manufactured by all firms. In Section 3.4, firms take the full responsibility for recycling,determiningtheirownproductionquantitiesandendogenouslyformingrecyclingcoalitions. 19 3.2 SocialProblem In this section, we determine the optimal structure outcome when the social planner determines both the productionquantitiesandtherecyclingcoalition’sstructureoutcome. SPobjective: For a given structure outcomen ={n 1 ,...,n J }, the social welfare created byq products is obtained by subtracting the total recycling-relatedcost,C(q,n), from the market surplus generated from theseproducts,U(q): W(q,n) =U(q)−C(q,n). (3.3) Thesocialplanner’sobjectiveistomaximizesuchsocialwelfare,bydeterminingtheproductionquantitiesq andstructureoutcomen: max q;n W(q,n). Weconsiderthismaximizationproblemasthebenchmarkproblem, andwerefertothisproblemasthe Social Problem(SP). SP optimal quantities: To solve such an optimization problem with two sets of variables, q andn, we first fix the structure outcome to an arbitrary n, and solve the corresponding problem with respect to q: max q W(q,n). As W(q,n) is concave in q i , firm i’s optimal quantity is easily obtained from the first- order conditions (FOC) — the expression of the optimal quantity is given by equation (A.2) in the proof of Lemma A.0.1 in the Appendix. Letting q SP i (n) be firm i’s optimal quantity, and letting q SP (n) = (q SP 1 (n),...,q SP N (n))bethevectorofallfirms’optimaloutputs,foragivenstructureoutcomen,thesocial welfareiswrittenasw SP (n) =W ( q SP (n),n ) . Wethenconsidertheoptimalsocialwelfareamongdifferent structure outcomes: max n w SP (n). We usen SP to denote the optimal structure outcome, and use J SP = |n SP |todenotetheoptimalnumberofcoalitions. To illustrate the behavior ofn SP , we first look numerically atn SP with different combinations of the fixedcostK anddegreeofheterogeneityx,whichmotivateustodevelopanalyticalresultsforspecialcases. Figure3.2displaysthefollowingqualitativebehaviorforaneconomywithtwelvefirms(N = 12): 1. When the fixed cost is significant (K is high) and products more homogeneous (x is low), the all- inclusivecoalition,{12},isoptimal,sinceittakesadvantageofeconomiesofscaletoreducethefixed costwithoutincreasingtheunitcost. 20 Figure3.2: OptimalStructureofSocialProblem(12Firms) 2. When the fixed cost is insignificant (K is low) and products more heterogeneous (x is high), each firm acts independently ({1,...,1} is optimal), since the economies of scale are negligible, but the unitcostmayincreasesignificantlyifdifferentproductsarerecycledtogether. 3. As products become more heterogeneous (x increases) or the fixed cost becomes more insignificant (K decreases), more and more coalitions should be formed (J SP increases). For example, {12} containsonecoalition;{6,6},{5,7},{4,8}and{3,9}containtwocoalitions;{4,4,4},{3,3,6}and {2,2,8}containthreecoalitions,andsoon. 4. For the optimal structure outcomes in some regions, coalitions have similar sizes: in{12}, {6,6}, {4,4,4},{3,3,3,3},{2,...,2},and{1,...,1}. Otheroptimalstructureoutcomesfeatureonelarger coalition and several smaller coalitions:{5,7},{3,3,6}, and{2,2,2,6}. This is interesting because allproductsaresymmetric,buttheyhavetoberecycledviarecyclingcoalitionsofdifferentsizes. The above observations are obtained when the unit (recycling) cost increases at a moderate speed (θ is moderate). When the results when the unit cost increases slowly (θ is small) or rapidly (θ is large), the firstthreefindingscarryover. Weobservesomedifferenceswithrespecttoourlastfinding: Onlycoalitions withsimilarsizesareoptimalwhenθ issmall,andonlyonelargecoalitionandseveralindependentfirmsis optimal whenθ is large, e.g.,{1,1,1,9}. We next discuss the properties of the optimal structure outcomes 21 in these two extreme cases (small and largeθ) in Definitions 3.2.1 and 3.2.2. Then, we explain the results forthesetwoextremecasesinPropositions3.2.1and3.2.2. Similarly-sizedandright-polarizedproperties: Whencoalitionshavesimilarsizes,werefertoitasthe similarly-sizedproperty,e.g.,{5,5,5}and{2,2,3,3}. Definition3.2.1formallydefinesthesimilarly-sized structure. Definition 3.2.1. Ifn = {n 1 ,...,n J } satisfies max j=1;:::;J n j − min j=1;:::;J n j 6 1, we sayn is similarly-sized, denoted S(N,J) ={ ⌊ N J ⌋ ,..., ⌊ N J ⌋ | {z } J−mod(N;J) , ⌈ N J ⌉ ,..., ⌈ N J ⌉ | {z } mod(N;J) }, whereN ∈Z + is the number of firms andJ ∈{1,...,N} is the number of coalitions. WhenN is given, S(N,J) is shortened asS(J). The set of all similarly-sized structure outcomes is defined as the similarly-sized family: S ={S(N,J) :N ∈Z + andJ ∈{1,2,...,N}}. Forexample,foraneconomywithtenfirms(N = 10),therearetensimilarly-sizedstructureoutcomes: S(1) = {10}, S(2) = {5,5}, S(3) = {3,3,4}, S(4) = {2,2,3,3}, S(5) = {2,2,2,2,2}, S(6) = {1,1,2,2,2,2}, S(7) = {1,1,1,1,2,2,2}, S(8) = {1,...,1,2,2}, S(9) = {1,...1,2}, andS(10) = {1,...,1}. Inanotherspecialcasewehaveonelargecoalition,whichwecallexceptional,andtheremainingcoali- tions with similar (smaller) sizes, e.g., {3,3,3,11} and{4,5,11}. We say that this structure outcome is polarizedtotheright. Definition3.2.2formallydefinesrightpolarization. Definition3.2.2. Ifn ={n 1 ,...,n J }satisfiesn\ { max j=1;:::;J n j } ∈S,wesaynisright-polarized,denoted PR(N,J;n J ) =            {⌊ N−n J J−1 ⌋ ,..., ⌊ N−n J J−1 ⌋ | {z } J−1−mod(N−n J ;J−1) , ⌈ N−n J J−1 ⌉ ,..., ⌈ N−n J J−1 ⌉ | {z } mod(N−n J ;J−1) ,n J } , ifJ ̸= 1 {N}, ifJ = 1 , 22 whereN ∈Z + is the number of firms,J ∈{1,...,N} is the number of coalitions, andn J = max j=1;:::;J n j ∈ {⌈ N J ⌉ ,...,N−J +1 } is the size of the exceptional coalition. WhenN is given,PR(N,J;n J ) is short- ened asPR(J;n J ). The set of all right-polarized structure outcomes is defined as the right-polarized family PR = { PR(N,J;n J ) :N ∈Z + , J ∈{1,...,N} andn J ∈ {⌈ N J ⌉ ,...,N−J +1 }} . Now, we discuss some special right-polarized structure outcomes. For a given number of coalitions,J, inPR(J;N −J + 1) = {1,...,1 | {z } J−1 ,N −J + 1}, the exceptional coalition contains the most firms; and inPR ( J; ⌈ N J ⌉) = {⌊ N J ⌋ ,..., ⌊ N J ⌋ | {z } J−mod(N;J) , ⌈ N J ⌉ ,..., ⌈ N J ⌉ | {z } mod(N;J) } , the exceptional coalition contains the fewest firms. We notice thatPR ( J; ⌈ N J ⌉) = S(J). That is, the right-polarized structure outcome whose exceptional coalition has the smallest size becomes similarly-sized. We then conclude that all similarly-sized structure outcomesarealsoright-polarized,S ⊆PR. For example, when N = 10 and J = 4 (ten firms forming four coalitions), there are five right- polarized structure outcomes: PR(4;7) = {1,1,1,7},PR(4;6) = {1,1,2,6},PR(4;5) = {1,2,2,5}, PR(4;4) ={2,2,2,4},andPR(4;3) ={2,2,3,3} =S(4). ItisobviousthatinPR(4;7)theexceptional coalitioncontainsthemostfirms;andinPR(4;3)theexceptionalcoalitioncontainsthefewestfirms. SP optimal results: From our numerical analysis, we observe that a small θ leads to a special case of analysis — the exceptional coalition contains the most firms for a given number of coalitions. A large θ leads to another special case — the exceptional coalition contains the fewest firms for a given number of coalitions. Finally, a moderateθ leads to some intermediate results. We therefore consider the two extreme cases,θ→ 1 + andθ→ +∞,toobtaininsightsintheoptimalstructureoutcomen SP . Proposition 3.2.1. Whenθ → 1 + ,n SP =S ( J SP ) ∈S;J SP is non-increasing inK and non-decreasing inx. TheproofofProposition3.2.1,alongwithotherproofs,isintheAppendix. According to Proposition 3.2.1, when the unit (recycling) cost ¯ c(n)x increases slowly in the coalition size n (i.e., θ is close to 1), for similar coalitions, the unit cost is almost the same. Since products are 23 symmetric, the social planner would allocate similar numbers of firms to different coalitions. The question that remains is how many coalitions to form. On one hand, as the fixed cost increases, firms tend to form (fewer) larger coalitions to share the high fixed cost. On the other hand, as the product heterogeneity increases, between two coalitions with different sizes, we observe greater difference in their unit costs. Inthiscase,alargernumberofsmallercoalitionsispreferredtoavoidthehighunitcost. Proposition3.2.2. Whenθ→ +∞,n SP =PR ( J SP ;N−J SP +1 ) ∈PR;J SP is non-increasing inK and non-decreasing inx. Proposition 3.2.2 is interesting because it states that in the optimal structure, coalitions need not have similarsizes,eventhoughproductsaresymmetric. Whentheunit(recycling)cost,¯ c(n)x,convergesquickly (i.e.,θ is large; see, e.g., the right panel of Figure 3.1), we observe significant differences in the unit cost betweencoalitionsofsize1and2,butnodifferencebetweencoalitionsofsizenandn+1forsomen> 1 (recall ¯ c(1) = 0 and lim n→∞ ¯ c(n) = c 0 ). Therefore, if firms recycle individually, they save the unit cost. However, since every coalition incurs a fixed cost, there is a restriction on the total number of coalitions. Hence, after determining the optimal number of coalitions, the social planner would let several firms stay independent,andhaveallotherfirmsjoinasinglecoalition. We are also able to provide analytical results for some special cases of product heterogeneity, perfect homogeneityandperfectheterogeneity. Proposition3.2.3. Forx = 0,n SP ={N}; forx = 1,n SP = argmax n={1;:::;1};{N} w SP (n) whenθ→ +∞. For perfectly homogeneous products, there is no unit (recycling) cost. According to Proposition 3.2.3, all firms tend to form the all-inclusive coalition and share the fixed cost, as long as this cost is positive. Next,considertheproductsthatarecompletelyheterogeneous,whentheunitcostincreasesverysteeply. If a coalition has more than one member, the unit cost is close to the basic unit cost, c 0 . This is due to two factors. On one hand, there is no unit cost discount caused by the product homogeneity; on the other hand, the unit cost converges to its limit for all nontrivial coalitions. Hence, the social planner would either form a single coalition to share the fixed cost, or let firms stay independent, to leverage the potentially high unit cost. As noted above, Proposition 3.2.1 states that whenθ→ 1 + , the optimal structure outcome is similarly- sized,n SP ∈ S, and Proposition 3.2.2 states that whenθ → +∞, the optimal structure outcome is right- polarized,n SP ∈ PR. Since the similarly-sized structure outcome is a special case of the right-polarized 24 one (S ⊆ PR), we conclude that the optimal structure outcome is always right-polarized for both cases (when θ → 1 + and when θ → +∞). From our numerical analysis (see, e.g., Figure 3.2), the optimal structure outcome for intermediateθ is also right-polarized. The difference is, whenθ → +∞, there are as many firms as possible in the exceptional coalition,n SP J = N −J SP + 1; whenθ → 1 + , there are as fewfirmsaspossibleintheexceptionalcoalition,n SP J = ⌈ N J SP ⌉ ;andwhenθ isintermediate,thenumberof firmsintheexceptionalcoalitionisintermediate, ⌈ N J SP ⌉ 6n SP J 6N−J SP +1. Now we have discussed our question — Under what structure outcomes do the end-of-life products get recycled? — when we assume that the social planner allocates the societal resources to achieve the highest social welfare. However, SP is a benchmark problem since the production quantities are firms’ endogenous decisions. Next, we discuss two prevalent models under a more realistic scenario — when firms determine the production quantities by themselves. In Section 3.3, we assume that the social planner takes the responsibility for recycling — such a system could exist prior to EPR legislation. In Section 3.4, we assume that firms take the responsibility for recycling and endogenously form recycling coalitions — suchasystemcouldbemotivatedbyEPRlegislation. 3.3 ExternalityProblem Inthissection,wediscussthecasewhenfirmscompetewithoutconsideringrecyclingandthesocialplanner minimizesthetotalrecyclingcost. EP objective: When there is no EPR legislation in place, firms may focus on competing in the primary market, leaving the social planner with the task of proper disposal of all consumer products. We refer to thisproblemastheExternalityProblem(EP),inwhichfirmsconsiderrecyclingastheexternalbusinessand thereforeignorealldisposal-relatedcosts. We consider a two-stage game in which theN firms first compete in quantities in the primary market, and the social planner then sets up (optimal) recycling infrastructures to dispose all products of these N firms. That is, firms maximize individual profits: max q i P i (q)q i . As the profit is concave in q i , firm i’s equilibrium production quantity is easily obtained from the FOC. The expression of this quantity is given by equation (A.5) in the proof of Lemma A.0.4 in the Appendix, from which we know that all firms have the same equilibrium quantity, which we denote q EP . Letting q EP = (q EP ,...,q EP ) be the vector of 25 all firms’ equilibrium quantities, for a given structure outcome n, the total recycling cost is written as C EP (n) = C(q EP ,n). We then consider the optimal recycling cost among different structure outcomes: min n C EP (n). We usen EP to denote the optimal structure outcome and use J EP = |n EP | to denote the optimalnumberofcoalitions. EPoptimalresults: WhennumericallyanalyzingtheoptimalEPstructureoutcomes, weconsiderdiffer- ent combinations of the fixed costK and product heterogeneityx. For instance, whenN = 12, Figure 3.3 displaysthefollowingqualitativebehavior. Figure3.3: OptimalStructureofExternalityProblem(12Firms) 1. SimilarlytothediscussionsinSP,whenthefixedcostissignificant(K ishigh)andproductsaremore homogeneous(xislow),theall-inclusivecoalition,{12},isstable. Whenthefixedcostisinsignificant (K islow)andproductsaremoreheterogeneous(xishigh),eachfirmactsindependently({1,...,1} isstable). 2. Coalitionsintheoptimalstructureoutcomehavesimilarsizes. 3. As products become more heterogeneous (x increases) or the fixed cost becomes more insignificant (K decreases),moreandmorecoalitionsshouldbeformed(J SP increases). 26 Next, we confirm analytically these observations when the unit (recycling) cost increases slowly or steeply(θ6e 2 N orθ>e 2 ). Proposition3.3.1. • When θ 6 e 2 N , n EP = S(J EP ) ∈ S; J EP is non-increasing in K and non- decreasing inx. • When θ > e 2 , n EP = argmin n={1;:::;1};{N} C EP (n); as K decreases and x increases, n EP changes from {N} to{1,...,1}. Proposition 3.3.1 states that when the unit cost increases at a slow or steep rate (θ 6 e 2 N orθ > e 2 ), theoptimalstructureoutcomeofrecyclingcoalitionsissimilarly-sized(notethatasproductsaresymmetric, theyhavethesamequantity). Basedonthesimilarly-sizedresults,asthefixedcostincreases,(fewer)larger coalitions should be formed to share the high fixed cost. As the products become more heterogeneous, a greaternumberofsmallercoalitionsshouldbeformedtoavoidthehighvariablecost. In this section, we discuss a product takeback system in which firms ignore recycling while the social planner organizes disposing the end-of-life consumer products. Next, we discuss another existing takeback systeminwhichfirmstaketherecycling-relatedcostsintoconsiderationandorganizerecyclingviarecycling coalitions. 3.4 ResponsibilityProblem In this section, we discuss the case when firms are fully responsible for taking back the consumer products they bring to the market. As recycling is taken as the extended responsibility of firms, we refer to this problemasthe Responsibility Problem(RP). RP objective: In RP, a firm’s payoff is obtained by subtracting the recycling-related costs from its profit in the primary market. To calculate the recycling-related costs, we assume that the fixed cost of setting up a recycling infrastructure is equally split among all members of the firm’s coalition, which is referred to as the egalitarian cost allocation. Hence, under a given coalition structure, A = {A 1 ,...,A J }, and its corresponding structure outcome, n = {n 1 ,...,n J }, the recycling-related costs of firm i ∈ A j are ¯ c(n j )xq i + K n j . To calculate the profit in the primary market, we need to develop the product price first. 27 Following Singh and Vives(1984), by taking the derivativeof the market surplusU(q) withrespect to firm i’sproductionquantityq i ,weobtainitsprice P i (q) = ∂U(q) ∂q i = 1−βq i −γ N ∑ l=1;l̸=i q l . (3.4) In summary, firmi’s payoff is a function of the production quantities, q, and the size of the coalition that firmibelongsto,n j : Π i (q,n j ) =P i (q)q i −¯ c(n j )xq i − K n j , i∈A j . (3.5) Each firm’s objective is to maximize its payoff by determining its production quantity and also by endoge- nouslyformingrecyclingcoalitions. RP equilibrium quantities: To maximize the payoff, for any arbitrary structure outcome,n = {n 1 ,..., n J }, each firm first competes in a Cournot competition with the optimization problem max q i Π i (q,n j ). The resultingCournotequilibriumquantityisafunctionofallfirms’coalitionsizes,orthestructureoutcomen. Weletq RP i (n)befirmi’sequilibriumquantity—sinceΠ i (q,n j )isconcaveinq i ,q RP i (n)iseasilyobtained fromtheFOC,asshowninequation(A.7)intheproofofLemmaA.0.6intheAppendix. Since products are symmetric, and we assume the egalitarian cost allocation, firms that belong to the samecoalitionhavethesameequilibriumquantities. Letq RP (n) = (q RP 1 (n),...,q RP N (n))bethevectorof all firms’ equilibrium outputs for a givenn. Then, firmi’s payoff is a function of the structure outcomen anditscoalitionsizen j : π(n,n j ) = Π i ( q RP (n),n j ) , i∈A j . (3.6) Hence,thepayoffsoffirmsthatbelongtothesamecoalitionarethesame: Theydependonlyonthecoalition size, noton specificfirms. In other words,π(n,n j ) isthe payofffor anyfirmin coalitionA j . Forexample, when there are four firms, n = {1,3} is a structure outcome containing a three-firm coalition and an individualfirm. Then,π({1,3},1)isthepayoffofthesinglefirm,andπ({1,3},3)isthepayoffofanyfirm inthethree-firmcoalition. Based on the payoffs under each structure outcome, firms endogenously form recycling coalitions to make their payoffs as high as possible. Since the coalition formation requires cooperation among firms, we discussthe“equilibrium”structureoutcomesinthenextsubsection. 28 3.4.1 CoalitionalStability InSP,astheonly“agent,”thesocialplannersolvesanoptimizationprobleminwhichthestructureoutcome isoptimallydetermined. InRP,therearemultiple“agents”(firms)whosolvemultipleoptimizationproblems that depend on the structure outcome; yet, the structure outcome is the result of firms’ decisions to join or leaveacoalition. Sinceallfirmsarefreetomakedecisionssimultaneously,eachfirm’sstrategyisinfluenced by others’ decisions, and each firm’s decision influences others’ strategies. In the end, firms are expected to reach a stable state, from which no firms would want to move. This changes our main question: What structure outcomes are likely to emerge as stable? To answer this question, we first need to address the conceptof“stability.” StabilityExamplesandIssues If there is a structure outcome that Pareto dominates all others, the answer to the above question is obvious — the Pareto dominating structure is stable. For example, for an economy with four firms (N = 4), there areseveralpossiblestructureoutcomes: theall-inclusivecoalition,{4};threefirmsformacoalitionandone firm acts independently,{1,3}; no coalitions are formed,{1,1,1,1}; etc. Suppose the payoff preferences fordifferentfirmsindifferentstructureoutcomesaregivenby π({4},4)> max{π({1,3},3),π({2,2},2),π({1,1,2},2),π({1,3},1),π({1,1,2},1),π({1,1,1,1},1)}. Then, a single coalition containing all firms,{4}, maximizes each firm’s payoff, so it emerges as a stable structureoutcome. Inmanycases,theParetodominancedoesnotexist,soweneedtoconsideralternativestabilityconcepts. Wetacklethisissuebyapplyingthegame-theoreticalmethodology. Inabroadsense,astructureoutcomeis taken to be unstable if there exist some firm(s) who have the incentive to deviate from the status quo. If we againconsidertheN = 4example,butnowassumethatfirms’payoffpreferencesaregivenby π({1,3},3)>π({4},4)>π({1,3},1) > max{π({2,2},2),π({1,1,2},2),π({1,1,2},1),π({1,1,1,1},1)}, (3.7) 29 then firms in a three-firm coalition in the structure outcome{1,3} make the highest possible payoff. If, for instance, the current status quo is a coalition of all four firms,{4}, then any three firms would benefit by defecting from the status quo and leaving a single firm behind. Thus, in this instance it appears that{1,3} islikelytoemergeasstable. However, the analysis can become much more complex when the firms’ preferences for different struc- ture outcomes become more complicated. Once again, we consider N = 4 with the following payoff preferences: π({1,3},1)>π({4},4)>π({1,1,1,1},1)>π({1,3},3) > max{π({2,2},2),π({1,1,2},2),π({1,1,2},1)}. We can follow similar reasoning as before to conclude that {4} is not stable — since π({1,3},1) > π({4},4), a single firm prefers to leave the other three firms and stay alone. But we can also notice that π({1,1,1,1},1) > π({1,3},3), so the three firms who remain in a coalition would want to change the current status quo into a structure outcome in which there are no coalitions whatsoever. However, this new structure outcome,{1,1,1,1}, cannot be stable becauseπ({4},4) > π({1,1,1,1},1). That is, all firms would benefit if they join the all-inclusive coalition. Now recall that, as mentioned above,{4} itself is not stable — we have a situation in which we cannot identify any stable structure outcome by following the traditional(staticormyopic)reasoning. In order to deal with environments in which traditional stability fails to identify stable structure out- comes,weturntotheideaofdynamicorfarsightedstability. Inthissetting,firmsconsiderpossiblereactions ofotherfirmsorcoalitionsbeforecommittingtoacertaincourseofaction. Inourlastexample,wecancon- sider, forinstance, the sequence ofdeviationsgivenby{4}→{1,3}→{1,1,2}→{1,1,1,1}→{4}→ ···. If firms are farsighted and the current status quo is{4}, they know that if any one of them deviates by leaving the all-inclusive coalition, the possible sequence of defections of the remaining three firms would lead to{1,1,1,1}, in which the initial defector is worse off. Therefore, no firm has an incentive to deviate, and{4}canbeconsidereddynamic/farsightedstable. Furthermore,asπ({1,1,1,1},1)>π({1,3},3),we canseethat{1,3}cannotbestable,and{4}istheonlystablestructureoutcome. 30 Another interesting example occurs when we have multiple dynamic/farsighted stable structure out- comes. WestillconsiderN = 4,withthefollowingpayoffpreferences: π({1,3},1)>π({4},4)>π({1,3},3)>π({1,1,1,1},1) > max{π({2,2},2),π({1,1,2},2),π({1,1,2},1)}. (3.8) Now, if the current status quo is {4}, we can again look at a deviation sequence, {4} → {1,3} → {1,1,2} → {1,1,1,1} → {4} → ···, to show that{4} is stable. However, under this scenario{4} is not the only stable structure outcome, due to the fact thatπ({1,3},3)>π({1,1,1,1},1). In order to see this,supposethatthecurrentstatusquois{1,3};notethatthesinglefirmdoesnotwanttomove,asitspay- offcanonlydecrease. Theonlywayforthethreefirmsinthecoalitiontoincreasetheirpayoffisbymoving to{4}orbybecomingthesinglefirmin{1,3};bothofthesewouldrequireparticipationofthesinglefirm. Consequently, a defection is unlikely to occur, in which case both{4} and{1,3} are dynamic/farsighted stable. LargestConsistentSet(LCS) Theaboveanalysisprovidesexamplesoftheconceptofthedynamic/farsightedstability. Now,wedefinethe conceptmorerigorouslyusingthelargestconsistentset(LCS)introducedbyChwe(1994). Beforeformally introducingtheLCS,weneedtodefinethetermsofdeviation(seeDefinition3.4.1),preferencerelation(see Definition 3.4.2) and dominance relation (see Definition 3.4.3). To show how different structure outcomes areformedfromfirms’moves,wefirstintroducetheterm“deviation.” Definition3.4.1. [Deviation]Assumen ={n 1 ,...,n J }isthestructureoutcomecorrespondingtoacoali- tion structureA = {A 1 ,...,A J }, wheren j = |A j | forj = 1,...,J. LetB = J ∪ j=1 B sub j ⊆ N be a set of defectors:B sub j ⊆A j isthesetofdefectorsfromcoalitionA j ,where|B sub j | =b j ∈ [0,n j ]forj = 1,...,J. We sayB deviates fromn tom, denoted byn⇀ B m, ifm ={m 1 ,...,m J+1 } wherem k =n k −b k for k = 1,...,J andm J+1 = ∑ J j=1 b j . Note: WLOG, we refine suchm immediately after it is formed: We first eliminate all zero elements or emptycoalitions,andthenascendinglyreordertheremainingcoalitionssuchthatm 1 6···6m |m| . 31 For example, for an economy with eight firms (N = 8), assumingA ={{1},{2,3,4},{5,6,7,8}} is the initial coalition structure,n ={1,3,4} is the corresponding structure outcome. If firm 1 from the first coalition,firm4fromthesecondcoalition,andfirms5,6and7fromthethirdcoalitionwouldliketodeviate (B sub 1 ={1},B sub 2 ={4} andB sub 3 ={5,6,7}), thenB ={1,4,5,6,7} is the defector set, andb 1 = 1, b 2 = 1andb 3 = 3. Fromthedeviationn⇀ B m,theresultingstructureoutcomeis{0,2,1,5},whichafter therefinementbecomes{1,2,5}. Next,toshowplayers’incentivesfordeviations,weintroducetheterm“preferencerelation.” Definition 3.4.2. [Preference Relation] LetB ⊆ N be a defector set, andn = {n 1 ,...,n J } andm = {m 1 ,...,m J ′} be the structure outcomes corresponding toA ={A 1 ,...,A J } andA ′ ={A ′ 1 ,...,A ′ J ′ }, respectively. Denoted by ≺ i , firm i’s strong preference relation is described as follows. n ≺ i m ⇔ π(n,n j )<π(m,m j ′) ifi∈A j ,i∈A ′ j ′ ,j = 1,...,J andj ′ = 1,...,J ′ . Denoted by≺ B , coalitionB’s strong preference relation is described as follows:n≺ B m⇔n≺ i m for alli∈B. For example, ifA = {{1},{2,3,4}} andA ′ = {{2,3},{1,4}},n = {1,3} andm = {2,2}. Firm 1 belongs to the first coalition in A and the second coalition in A ′ : 1 ∈ A 1 = {1} and n 1 = 1; 1 ∈ A ′ 2 = {1,4} and m 2 = 2. Firm 4 belongs to the second coalition inA and the second coalition inA ′ : 4∈A 2 ={2,3,4} andn 2 = 3; 4∈A ′ 2 ={1,4} andm 2 = 2. If firm 1 receives higher payoff inA ′ than inA,π(n,1) < π(m,2), then firm 1 prefersm ton,n≺ 1 m. If firm 4 receives higher payoff inA ′ than inA,π(n,3)<π(m,2),thenfirm4prefersmton,n≺ 4 m;ifbothstatementsholdtrue,thenthesecond coalitioninA ′ prefersmton,n≺ {1;4} m. Finally, we show how preference relations motivate firms to deviate by introducing “dominance rela- tion.” Definition 3.4.3. [Dominance Relation] We say thatn is directly dominated bym, denoted byn <m, if thereexistsadefectorsetB,suchthatn⇀ B mandn≺ B m. Wesaythatnisindirectlydominatedbym, denoted byn≪m, if there exist structure outcomesn 1 ,...,n k−1 and defector setsB 1 ,...,B k , such that n =n 0 ⇀ B 1 n 1 ⇀ B 2 ···⇀ B l n l =m andn k−1 ≺ B k m for allk = 1,...,l. Note that direct dominance is a special case of the indirect dominance. More specifically, direct domi- nance considers whether an immediate defection makes a firm better off, while indirect dominance is con- cerned with the final structure outcome — it does not worry about immediate payoffs during intermediate 32 steps. In other words, members defecting from a coalition have to prefer the final structure outcome to the current status quo — we requiren k−1 ≺ B k m, notn k−1 ≺ B k n k . We now define the dynamic/farsighted stabilityconceptthatweuseinthisthesis—theLCS. Definition3.4.4. [Largest Consistent Set] A setY is called consistent if and only ifn∈Y and, for allm andB,n⇀ B m, there is an ′ ∈Y, wherem =n ′ orm≪n ′ such thatn⊀ B n ′ . The LCS is the largest consistent set. In other words, a defection from a structure outcome is deterred if it could trigger a sequence of further defections that eventually leads to a structure outcome in which the original defecting parties are not better offthanintheinitialstructureoutcome. Asaresult,astructureoutcomewhichinitiallyseemedunstablemay actually prove to be stable. The opposite can also happen — an initial defection that leads to an immediate lower payoff may eventually result in a structure outcome in which the parties that triggered the sequence of moves are better off. Under this scenario, a structure outcome that seems stable from a myopic or static viewpointbecomesunstablewhenthefirmsarefarsighted. Since every coalition considers the possibility that, once it acts, another coalition may react, and then yet another, and so on, the LCS incorporates dynamic or farsighted coalitional stability. Besides, it has been shown that for finite number of firms, the LCS is nonempty (see Chwe 1994). Hence, in the next subsection, we use the LCS to define the dynamic/farsighted stability in RP. That is, when we say that a structureoutcomeis(dynamic/farsighted)stable,weimplythatitisintheLCS. 3.4.2 FarsightedStabilityinRP SimilarlytothediscussioninSP,wefirstlooknumericallyatstablestructureoutcomesthatareintheLCS, with different combinations of the fixed cost, K, and degree of heterogeneity, x. Figure 3.4 displays the followingqualitativebehaviorforaneconomywithfourfirms(N = 4). 1. When the fixed cost is significant (K is high) and products more homogeneous (x is low), the all- inclusive coalition, {4}, is stable; when the fixed cost is insignificant (K is low) and products are moreheterogeneous(xishigh),eachfirmactsindependently:{1,1,1,1}isstable. 2. When products are completely heterogeneous (x = 1), the LCS contains a unique element. For instance,asdiscussedabove,whenthepreferenceisgivenbyequation(3.7),{1,3}isuniquelystable. 33 3. For general cases, the LCS may contain multiple elements. For example, {4} and {1,3} are both stablewhenthepreferenceisgivenbyequation(3.8),asdiscussedabove. Figure3.4: StableStructureofResponsibilityProblem(4Firms) Withasmallnumberoffirms,suchasN = 4,weareabletoanalyticallyidentifyallstablestructureout- comes. However,whenthenumberoffirmsbecomesarbitrary,identifyingallelementsintheLCSbecomes NP-hard(seeNagarajanandSoˇ sic2011)becauseweneedtoconsiderallpossibledeviationsequencesfrom all structure outcomes. Nevertheless, we are able to identify element(s) in the LCS, and we use n RP to denote them. The easiest case occurs when there is a single structure outcome that dominates all others, as describedinthefollowingLemma. Lemma 3.4.1. If a structure outcome n ∗ satisfies: for any other structure outcome n̸=n ∗ , we have n≪n ∗ , thenn ∗ is in the LCS. Lemma 3.4.1 states that if we can find a structure outcome that indirectly dominates all other structure outcomes, then this element is in the LCS. Following this approach, we are able to identify stable structure outcomes when “θ is small” (θ6e 2 N ) and “θ is large” (θ>e 2 ). We first discuss the property of the stable structureoutcomesinthecasewhen“θ issmall”inDefinition3.4.5. Then,weexplainourfindingsforthese twocasesinPropositions3.4.1and3.4.2. 34 Left-polarization property: We consider a structure outcome in which several coalitions have the same size, which we call the majority coalition, eventually leaving at most one smaller coalition, such as in {4,4,4,4} and {3,5,5,5}. We say that this structure outcome is polarized to the left. Definition 3.4.5 formallydefinesleftpolarization. Definition3.4.5. Ifn ={n 1 ,...,n J }satisfiesn 1 6n 2 =··· =n J ,wesayn isleft-polarized,denotedby PL(N;n J ) = { N−n J (⌈ N n J ⌉ −1 ) ,n J ,...,n J | {z } ⌈ N n J ⌉ −1 } , whereN ∈Z + isthenumberoffirmsandn J ∈{1,...,N}isthesizeofthemajoritycoalition. WhenN is given,PL(N;n J ) is shortened asPL(n J ). The set of all left-polarized structure outcomes is defined as the left-polarized family: PL ={PL(N;n J ) :N ∈Z + andn J ∈{1,...,N}}. Note that our definition of left polarization is different from that of right polarization. First, in right polarization, there is just one larger coalition, but there are several smaller coalitions; in contrast, with left polarization, there is just one smaller coalition and there are several larger coalitions. Second, right polar- ization is “looser” — it allows all coalitions but the exceptional one to have similar sizes. In comparison, our definition of left polarization is more restrictive — all majority coalitions must have exactly the same size. Third,rightpolarizationisdeterminedbyboththenumberofcoalitionsandthesizeoftheexceptional coalition,whileleftpolarizationdependsonlyonthesizeofthemajoritycoalition. Forexample,whenN = 10(tenfirms),therearetenleft-polarizedstructureoutcomes:PL(10) ={10}, PL(9) ={1,9},PL(8) ={2,8},PL(7) ={3,7},PL(6) ={4,6},PL(5) ={5,5},PL(4) ={2,4,4}, PL(3) ={1,3,3,3},PL(2) ={2,2,2,2,2}, andPL(1) ={1,...,1}. Compared with the ten similarly- sized structure outcomes for the same N listed after Definition 3.2.1, it is easy to show that neither the similarly-sizedfamilynorleft-polarizedfamilyisasubsetoftheother:S*PLandPL*S. RPstableresults: Next,wediscussthestablestructureoutcomes. Wefirstconsiderthecaseinwhichthe unit(recycling)costincreasesslowly:θ6e 2 N . 35 Proposition3.4.1. Whenθ6e 2 N ,thereexists ˜ ˜ csuchthatn RP =PL(n RP J )∈PL,wheren RP J = argmax n J =1;:::;N π(PL(n J ),n J ), if • mod(N,n RP J ) = 0; or • mod(N,n RP J ) =n RP 1 > 0 andπ(PL(n RP J ),n RP 1 )>π(PL(n RP J ),n RP J ); or • mod(N,n RP J ) =n RP 1 > 0,π(PL(n RP J ),n RP 1 )<π(PL(n RP J ),n RP J ) andc 0 6 ˜ ˜ c. Proposition 3.4.1 states that when the unit cost increases slowly (θ 6 e 2 N ), there is a left-polarized structure outcome which is stable. In addition, this stable structure is determined by firms in majority coalitions: wecomparethepayoffsofthosefirmsunderallleft-polarizedstructureoutcomes. Forexample, forfourfirms,wechoosefromPL(1) ={1,1,1,1},PL(2) ={2,2},PL(3) ={1,3}andPL(4) ={4}. Thatis,wecompareπ({1,1,1,1},1),π({2,2},2),π({1,3},3)andπ({4},4): Figure3.5showstheoptimal size for different combinations of the fixed cost,K, and the product heterogeneity,x. Whenx = 0.95 and K = 1800 ,n RP J = 3,andFigure3.4showsthat{1,3}isuniquelystable—itisthecaseinwhichthepayoff preferences are given by equation (3.7). Whenx = 0.4 andK = 300 ,n RP J = 4. Correspondingly, Figure 3.4showsthat{4}isstable—itisthecaseinwhichthepayoffpreferencesaregivenbyequation(3.8). Figure3.5: OptimalSizeoftheMajorityCoalition Once we determine the optimal size of the majority coalition, we explain our findings in Proposition 3.4.1as follows. Considering the factthat withsymmetric products, when allfirms can be split into several n RP J -sized coalitions, the similarly-sized structure outcome with coalitions of sizen RP J emerges as stable. When the number of firms is not divisible byn RP J , we need to consider what happens with the “leftover” firmswhoarenotinthen RP J -sizedcoalitions. Sincetheunitcostincreasesslowly(i.e.,θ6e 2 N ), theeffect 36 of sharing the fixed cost becomes important, and as the size gets closer ton RP J , the individual payoff also getsclosertothemaximumlevel. Therefore,iftheremainingfirmsformonecoalition,theindividualpayoff is higher than that if they form several smaller coalition. As a result, the left-polarized structure outcome emergesasstable. Next,weanalyzethecaseinwhichtheunitcostincreasessteeply:θ>e 2 . Proposition 3.4.2. Whenθ > e 2 ,n RP = argmax n={1;:::;1};{N} π ( n, N |n| ) ; asK decreases andx increases,n RP changes from{N} to{1,...,1}. Forahighconvergencerate,θ,oftheunit(recycling)cost,thebigincreaseintheunitcostoccurswhen firms move from acting alone to joining another firm in a coalition, while the difference between unit costs incoalitionswithtwo,three,ormoremembersbecomesnegligible. Asaresult,Proposition3.4.2statesthat wearelikelytoseecaseswithnocoalitionswhatsoever,ifsetupcostsarelowandthebenefitsfromsharing itarenotsignificant,andcasesinwhichallfirmsjoininordertoreceivelowshareofthesetupcost. FromPropositions3.4.1and3.4.2,weknowthatthestablestructureinRPisleft-polarized(n RP ∈PL) when θ 6 e 2 N , and is similarly-sized (n RP ∈ S) when θ > e 2 . In addition, we obtain the following analyticalresultswhenproductsarecompletelyhomogeneous(x = 0)orcompletelyheterogeneous(x = 1). Proposition3.4.3. • Whenx = 0,n RP ={N} is uniquely in the LCS. • Whenx = 1, iff(n) = (1− c(n)) 2 4 − K n has a unique maximum point,n RP = n rp (N) is uniquely in the LCS, where n rp (M) =              { n rp J (M),...,n rp J (M) | {z } ⌊ M n rp J (M) ⌋ } ∪n rp (mod(M,n rp J (M))), ifM > 0 ϕ, ifM = 0 andn rp J (M) = argmax n=1;:::;M { (1− c(n)) 2 4 − K n } . Whentheproductsareperfectlyhomogeneous(i.e.,xisclosetozero),theunit(recycling)costdoesnot increase in the coalition size. Hence, the all-inclusive coalition is socially optimal as it “spreads” the fixed costofformingarecyclingcoalitionoverthemaximumnumberoffirms. Whentheproductsexhibitahigh level of heterogeneity (i.e.,x is close to one andγ is close to zero), the firms’ payoffs depend only on their own coalitional behavior, or, more specifically, the size of their own coalition. That is, firms only need to 37 consider moves and reactions from their coalitional partners, rather than from all firms. Since products are symmetric and firms are equal ex ante, once an rp J (M)-sized coalition is formed, no firm in this coalition hasanincentivetodeviate. Asaresult,therewouldbeasmanyn rp J (M)-sizedcoalitionformedaspossible, leavingtheremainingfirms,ifany,tolookforthenewconditionallyoptimalsize. NotethatPropositions3.4.1and3.4.2applywhentheunitcostconvergesslowlyorquickly,θ6e 2 N or θ> e 2 , but for general product heterogeneity,x. The LCS may contain multiple elements and we identify oneofthem. Proposition3.4.3applieswhenproductsarecompletelyhomogeneousorheterogeneous,x = 0 or x = 1, but for general unit-cost convergence, θ. The LCS has a unique element and we are able to identify it. We combine all the above findings in Proposition 3.4.4: The stable structure outcome identified inPropositions3.4.1and3.4.2areuniqueintheLCS. Proposition3.4.4. Whenx = 1, iff(n) = (1− c(n)) 2 4 − K n has a unique maximum, • forθ>e 2 ,n RP = argmax n={1;:::;1};{N} π ( n, N |n| ) is uniquely in the LCS; • forθ6e 2 N ,n RP =PL(n RP J ) is uniquely in the LCS, wheren RP J = argmax n=1;:::;N f(n) is non-decreasing inK. Proposition3.4.4statesthatwhentheunit(recycling)costincreasesslowly,θ6e 2 N ,evenforcompletely heterogeneous products, the effect of the unit cost is insignificant, compared with that of the fixed cost. Hence,asthefixedcostincreases,largersizeisoptimalforthemajoritycoalitiontoleveragethehighfixed cost. So far, we have focused on the optimal and stable structure outcomes based on three scenario settings. In Section 3.2, we assumed the social planner determines both the production quantities and the structure outcome. In Section 3.3, we assumed that firms determine their production quantities while the social planner determines the structure outcome. In this section, we assume that firms determine their production quantities and endogenously form recycling coalitions. Next, we compare the three models in terms of the socialwelfare,inordertoobtainsomeinsightsthatmaybehelpfultothesocialplanner. 38 3.5 GovernmentalDecision-Making As mentioned earlier, the social planner’s objective is to maximize the social welfare. In this section, we compare the social welfare generated in the EP model under the optimal structure outcome with the social welfare generated in the RP model under the stable structure outcome. Since SP is a benchmark model thatgeneratesthehighestsocialwelfare,wefirstconsidertheconditionsunderwhichtheSPoptimalsocial welfare is achieved. Then, if such conditions are difficult (or impossible) to implement, we directly com- pare the social welfare in EP and RP, to see which model generates greater social welfare. To simplify expressions, we define the social welfare generated from the equilibrium quantities and optimal and stable structure outcome in EP and RP asw EP (n EP ) = W(q EP ,n EP ) andw RP (n RP ) = W(q RP (n RP ),n RP ), respectively. Recallfromequation(3.3),thesocialwelfaredependsontheproductionquantities,q,andthestructure outcome, n. Hence, in EP or RP, we can achieve the SP optimal social welfare if firms produce the SP optimal quantities, q SP (n SP ), and recycling infrastructures are organized as the SP optimal structure,n SP . Accordingly,weconsiderinducingfirmsviataxesorsubsidiestoproduceq SP (n SP )products,andconsider thepossibilitiesoforganizingrecyclinginfrastructuresasn SP . 3.5.1 TaxandSubsidy Sincefirmsareequalexanteandproductsaresymmetric,assumethatthesocialplannerimposesanidenti- cal tax/subsidy for each unit of product, regardless of the specific firm. We uset to denote this tax/subsidy: Positive t represents a tax, zero t represents no tax/subsidy, and negative t represents a subsidy. Taking such tax/subsidy into consideration, the expressions of some variables introduced in previous sections are changed. For distinction, we add hats on the variable notations in the above sections to represent the corre- spondingvariablesunderthetax/subsidysetting. Next,wediscusstheeffectsofthetax/subsidyasfollows. • The market surplus with taxes/subsidies, ˆ U(q), is reduced by the total tax or increased by the total subsidy from that without taxes/subsidies, U(q): ˆ U(q) = U(q)−t ∑ N i=1 q i . Correspondingly, the productprice(receivedbythefirm)withtaxes/subsidies, ˆ P i (q),isreducedbythetaxorincreasedby the subsidy from that without taxes/subsidies,P i (q): ˆ P i (q) =P i (q)−t. Comparing the two prices, 39 we find that without taxes/subsidies, the maximum price is 1; with a tax/subsidy oft, the maximum pricebecomes1−t. • Thetotalrecyclingcostandsocialwelfareremainthesameafterimposingtaxes/subsidies: ˆ C(q,n) = C(q,n)and ˆ W(q,n) = ˆ U(q)− ˆ C(q,n)+t ∑ N i=1 q i =U(q)−C(q,n) =W(q,n). • In EP, each firm’s objective is to maximize the profit under the new price, ˆ P i (q): max q i ˆ P i (q)q i , with theequilibriumquantities^ q EP = (ˆ q EP ,...,ˆ q EP ), ˆ q EP = (1−t)q EP (seeequation(A.10)intheproof of Proposition 3.5.1 in the Appendix). To determine the optimal structure,^ n EP , the social planner’s objective is to minimize the total cost for recycling those ^ q EP products: min n ˆ C EP (n) = C(^ q EP ,n). From the proofs in Section 3.3, the value of ˆ q EP does not change the results presented in Proposition 3.3.1, but only affects the exact ^ n EP for a specific combination of the fixed cost, K, and product heterogeneityx. For instance, when the unit (recycling) cost increases slowly,θ 6 e 2 N , the optimal structure is similarly-sized: n EP ,^ n EP ∈ S. Besides, as the fixed cost becomes insignificant, K decreases,andproductsbecomemoreheterogeneous,xincreases,moreandmore(smaller)coalitions becomeoptimal. Itispossiblethattwosimilarly-sizedcoalitionsareoptimalifthereisnotax/subsidy, whiletheall-inclusivecoalitionbecomesoptimalafterthetax/subsidyisintroduced:n EP =S(2)and ^ n EP =S(1). Thesocialwelfaregeneratedfrom ^ q EP and^ n EP is ˆ w EP (^ n EP ) =W(^ q EP ,^ n EP ). • In RP, each firm’s objective is to maximize the payoff under the new price, ˆ P i (q): max q i ˆ Π i (q,n j ) = ˆ P i (q)q i − ¯ c(n j )xq i − K n j , i ∈ A j , leading to the equilibrium quantities ^ q RP (n) = (ˆ q RP 1 (n),...,ˆ q RP N (n)). Comparing the expressions forq RP i (n) and ˆ q RP i (n) (see equation (A.11) in the proof of Proposition 3.5.1 in the Appendix), ˆ q RP i (n) is obtained by replacing the maximum price of1inq RP i (n)with1−t. Thetax/subsidyisusedasaninstrumentto“push”thestablestructureofthe endogenouscoalitionformationclosertothesociallyoptimalstructure. Todeterminethestablestruc- ture,^ n RP ,firmsconsiderthepayoff ˆ π(n,n j ) = ˆ Π i (^ q RP (n),n j ),i∈A j . FromtheproofsinSection 3.4, the maximum price does not change the results presented in Propositions 3.4.1, 3.4.2, 3.4.3 and 3.4.4; it only affects the exact^ n RP for a specific combination of the fixed cost,K, and product het- erogeneityx. For instance, when the unit cost increases steeply,θ>e 2 , the stable structure contains either an all-inclusive coalition or no coalitions at all. As the fixed cost becomes insignificant, K decreases, and products become more heterogeneous,x increases, the stable structure switches from the all-inclusive coalition, {N}, to the structure with no coalitions, {1,...,1}. It is possible that the latter is stable if there is no tax/subsidy, while the all-inclusive coalition becomes stable after the 40 tax/subsidy is introduced: n RP = {1,...,1} and^ n RP = {N}. The social welfare generated from ˆ q RP i (n)and^ n RP is ˆ w RP (^ n RP ) =W(^ q RP (^ n RP ),^ n RP ). 3.5.2 AchievingSPOptimum WefirstusetheresultsinSections3.3and3.4toidentifytheconditionsunderwhichtheSPoptimalstructure is formed: ^ n EP =n SP and^ n RP =n SP . Then, if these conditions are satisfied, we discuss the tax/subsidy thatinducesfirmstoproducetheSP-optimalquantities: ^ q EP = q SP (n SP )and ^ q RP (^ n RP ) = q SP (n SP ). Themostobviouscasesoccurwhentheoptimalandstablestructurecontainstheall-inclusivecoalition, {N}, or when it contains no coalitions,{1,...,1}. From Propositions 3.2.1, 3.2.2, 3.3.1, 3.4.2 and 3.4.4, whenthefixedcostissignificant,largeK,orproductsarehomogeneous,smallx,theall-inclusivecoalition isoptimalandstableinallthreeproblems: ^ n EP =^ n RP =n SP ={N}; whenK issmallandxislarge,no coalition is optimal and stable in all three problems: ^ n EP = ^ n RP =n SP = {1,...,1}. In both cases, we canconsidertheeffectofataxorsubsidyontheproductionquantities. Proposition3.5.1. • When^ n EP =^ n RP =n SP ={N}, – ˆ w EP (^ n EP ) =w SP (n SP ) ift = ¯ c(N)x− 1− c(N)x N−(N−1)x ; asN −→ +∞,t =c 0 x. – ˆ w RP (^ n RP ) =w SP (n SP ) ift =− 1− c(N)x N−(N−1)x ; asN −→ +∞,t = 0. • When^ n EP =^ n RP =n SP ={1,...,1}, ˆ w EP (^ n EP ) = ˆ w RP (^ n RP ) =w SP (n SP ) ift =− 1 N−(N−1)x ; asN −→ +∞,t = 0. Proposition3.5.1statesthatwhentheall-inclusivecoalitionorthestructurewithnocoalitionsisoptimal and stable, RP always requires a subsidy to achieve the SP optimum. However, as the number of firms increases, the subsidy shrinks to zero. Intuitively, since firms pay for recycling, their cost increases, which discourages production. To encourage production, the social planner has to provide subsidies. With a large number of firms, each firm’s market share in both RP and SP is insignificant. Therefore, the difference between the production quantities in RP and SP becomes negligible, so the social planner does not need to providesubsidiestofirms. Proposition3.5.1alsostatesthatwhennocoalitionisformedinanoptimalandstablestructure,EPand RP require the same amount of subsidy to achieve the SP optimum. Intuitively, if there are no recycling 41 coalitions and the end-of-life products are recycled independently, the unit (recycling) cost is zero. Hence, it does not matter whether the social planner or the firms pay for recycling: to achieve the SP optimum, EP has the same requirement as RP, a subsidy. In addition, when the all-inclusive coalition is optimal, EP may require either a tax or a subsidy: it is a tax if the unit cost is significant, ¯ c(N)x is large. Especially, with a large number of firms, EP requires a tax which compensates for the actual unit cost,c 0 x. Intuitively, as the social planner pays for recycling in SP, he prefers that firms do not compete with too many products; while inEP,firmscompetewiththefullproductioncapacitysincetheydonotpayforrecycling. Hence,iftheunit (recycling)costishigh,thesocialplannerhastoimposeataxonfirmstoreducetheproductquantities. The above result provides insights to the social planner on how to achieve the SP optimum for special combinations of fixed cost,K, and product heterogeneity,x. Next, we discuss the issues for general com- binations ofK andx, when the unit cost increases slowly,θ → 1 + , and when it increases steeply,θ>e 2 , respectively. WefirstchecktheconditionsforachievingtheSPoptimumwhentheunitcostincreasesslowly,θ→ 1 + . Recall from Propositions 3.2.1, 3.3.1 and 3.4.1 that whenθ → 1 + , the optimal structure outcomes in SP and EP are both similarly-sized,n SP ,^ n EP ∈S; while the stable structure outcome in RP is left-polarized, ^ n RP ∈ PL. Since the relationship between the similarly-sized family and the left-polarized family is indeterminate,S*PLandPL*S,theRPstablestructureoutcomeisgenerallynotoptimalforthesocial planner:^ n RP ̸=n SP . Hence,itisgenerallydifficulttoachievetheSPoptimalsocialwelfareinRP.Wethen compare the optimal structure in EP with that in SP,^ n EP withn SP , and the corresponding social welfare, ˆ w EP (^ n EP )withw SP (n SP ). Proposition3.5.2. Whenθ→ 1 + , asN −→ +∞,^ n EP =n SP ; ˆ w EP (^ n EP ) =w SP (n SP ) ift = ¯ c(n SP j )x. Proposition 3.5.2 states that with a large number of firms, when the unit (recycling) cost increases at a slow rate,θ → 1 + , the optimal structures in EP and SP are the same. In addition, the SP optimum can be achieved if the social planner charges a tax which compensates the unit cost. Intuitively, if firms incur the samecostinEP(informofatax)astheydoinSP(informoftheunitcost),theybringthesamequantityto themarket. Next,wechecktheconditionsforachievingtheSPoptimumwhentheunitcostincreasessteeply,θ>e 2 . Recall from Propositions 3.2.2, 3.3.1 and 3.4.2 that when θ > e 2 , the optimal and stable structure in EP and RP contains either an all-inclusive coalition or no coalitions at all, ^ n EP ,^ n RP ∈ {{1,...,1},{N}}, 42 while the optimal structure outcome in SP is right-polarized, n SP ∈ PR. Since the right-polarized family contains other structures besides the all-inclusive coalition and the structure with no coalitions, PR\{{1,...,1},{N}} ̸= ϕ, in most cases, neither ^ n EP nor ^ n RP is optimal for the social planner: ^ n EP ̸= n SP and ^ n RP ̸= n SP . Hence, generally, it is difficult to achieve the SP optimal social welfare underbothEPandRP. 3.5.3 ComparingSocialWelfare Wenowcomparethesocialwelfarewithouttaxesorsubsidies,w EP (n EP )andw RP (n RP ). Proposition3.5.3. Whenθ>e 2 ,lettingΩ = [N +2−(N−1)x]c 0 ( 1−θ 1−N ) x,thereexist ¯ K and ¯ K RP with ¯ K RP < ¯ K, such that • w EP (n EP ) =w RP (n RP ) ifK6 ¯ K RP ; • w EP (n EP )6w RP (n RP ) ifK> ¯ K andΩ> 2; • w EP (n EP )>w RP (n RP ) otherwise. From Proposition 3.5.3, when the unit (recycling) cost increases at a steep rate, θ > e 2 , if the fixed cost is low, then EP and RP generate the same social welfare; if the fixed cost is high and Ω > 2, then RP generates a higher social welfare than EP; otherwise, the EP social welfare is higher. Note that when Ω> 2, (1) there are many firms (N is large), (2) products are not homogeneous (x is not small) or (3) the basic unit cost,c 0 , is not low. In other words, Ω> 2 implies that recycling is challenging. To conclude the discussionofProposition3.5.3,ifthefixedcost,K,islow,ouranalysisinSections3.3and3.4impliesthat the optimal and stable structure in either EP or RP does not have any nontrivial coalition, in which case, the unit recycling cost is zero. Hence, EP and RP perform the same. If recycling is costly (K > ¯ K) and challenging(Ω> 2),thesocialplannerhasmoreincentivetoshifttherecyclingburdentofirms. Inthissection,wetakeSPasabenchmarktodiscusstheperformanceofEPandRP.Fromouranalysis, if the optimal and stable structures are known to be the same for the three problems (i.e., the all-inclusive coalitionortheno-coalitionstructure)RPalwaysrequiresasubsidy,whichbecomesnegligiblewhenalarge number of firms join the market. If the optimal and stable structures differ across the three problems, when the unit cost increases slowly and we have a large number of firms, EP can achieve the SP optimum via a 43 tax, while RP cannot; when the unit cost increases steeply, neither EP nor RP is guaranteed to achieve the SPoptimum,butRPgenerateshighersocialwelfarethanEPwhenrecyclingismorechallenging. 3.6 Summary In this chapter, we study the recycling of symmetric products under egalitarian cost allocation. In the primarymarket,weassumethatfirmsareequalexanteandproductsaresymmetric: (1)eachfirmproduces one product; (2) all firms compete in the same market; and (3) there is a uniform degree of heterogeneity between any two products. In the recycling market, we assume that when firms form a coalition, their products are recycled together. On one hand, due to the economies of scale, forming a coalition reduce the fixed setup cost — firms equally share the fixed cost, which is the egalitarian cost allocation. On the otherhand, duetothe productheterogeneity, forminga coalitionincreasethe unitcost. That is, theproduct heterogeneityandthefixedcostaretwodeterminantsthatinfluenceourresults. Facingthesetwoimportant factors, decision makers (the social planer and firms) are interested in the coalitional structure. In other words,howmanycoalitionsshouldbeformed,andhowmanyfirmsshouldformacoalition? We study three scenarios. The first scenario is a benchmark model. The social planner plans for pro- duction and recycling, with the objective of maximizing the social welfare. We refer to this problem as the Social Problem (SP). Although products are symmetric, the social planner might not always choose coali- tionsof similar sizes — it depends on howfastthe unit cost increases withthe coalition size. If it increases very slow, all coalitions have similar sizes; when it increases very fast, it is optimal to set up several small coalitions, leaving the remaining firms forming a large coalition. The second scenario is the case if there is noEPR-typelegislationintroduced. Firmsorganizeproductiontomaximizetheirrevenues,whilethesocial plannerorganizesrecyclingtominimizethetotalrecyclingcost. WerefertothisproblemastheExternality Problem (EP). The optimal coalition structure in EP only contains coalitions of similar sizes. The third scenario is the case if the EPR-type legislation is introduced. Firms organize production and endogenously formrecyclingcoalitions,tomaximizetheirprofits. WerefertothisproblemastheResponsibilityProblem (RP). Firms in RP finally achieve a stable coalition structure in which no firm has the incentive or power to change its decision. Such stable coalition structure may not contain coalitions of similar sizes only. If the unit cost increases very slow, the fixed cost and product heterogeneity determine an optimal coalition 44 size, firms form as many as possible coalitions of such size, leaving the remaining firms forming a small coalition. Providedwiththeaboveresults,itisthesocialplanner’sdecisiontochoosefromEPandSP,inorderto maximize the social welfare. From our analysis, for challenging recycling tasks (i.e., those with high fixed cost and heterogeneous products), RP brings about a higher social welfare than EP. In addition, the social planner can also impose tax/subsidy to increase the social welfare. When the social planner recycles, firms should always be taxed; when firms recycle, whether subsidies are necessary depends on the total number of firms in the market — a small number of firms necessitates subsidies, but a large number of firms does not. In this chapter, although products are symmetric, they are not always recycled via coalitions of similar sizes. Inthenextchapter,wewilldiscussthecasewhenproductsarenotsymmetric: productsarefromtwo independentmarkets;withineachmarket,productsarehomogeneous. 45 Chapter4 RecyclingofAsymmetricProductsunder EgalitarianCostAllocation In the previous chapter, we study the recycling of heterogeneous products from a general market, in which products are symmetric. In this chapter, we study the recycling of both heterogeneous products and homo- geneousproducts. Westartwiththreefirmsproducingthreeproducts,twoofwhichcompeteinonemarket, and the other one is in an independent market (see Section 4.1 and Figure 1.1a). Then, we let one firm produce two products in two independent markets respectively; another firm produce one product, which alsobelongstooneofthetwomarkets(seeSection4.2andFigure1.1b). InSection4.3,welettwofirmsbe equalexante: eachmaketwoproductsintwoindependentmarketsrespectively. 4.1 Single-MarketManufacturing: [3;3]Model We start our analysis with a model in which each firm participates in only one market, with a total of three products made by three firms. Two products are competing in a market, and the other product is in another independentmarket. Forexample,DelllaptopsandLenovolaptopsaresubstitutes,andGErefrigeratorsare independentofthem. Market surplus: We assume that products 1, 2 and 3 are made by firmsA, B andC, respectively. All three firms have the same demand-price effect,β > 0. Products 1 and 2 are related (substitutable) with a crossdemand-priceeffect,γ∈ (0,β),andproduct3isindependent. ThemodelisillustratedinFigure1.1a. We useq i > 0 to denote the output of producti,i = 1,2,3. Following Singh and Vives (1984), we assume thatthemarketsurplusbroughtbythethreeproductsis U =q 1 +q 2 +q 3 − 1 2 [ β(q 2 1 +q 2 2 +q 2 3 )+2γq 1 q 2 ] . (4.1) 46 By taking the partial derivatives of the market surplus with respect to the product outputs, we obtain prod- ucts’prices(inversedemands): P i = ∂U ∂q i = 1−βq i −γq j , i,j = 1,2, i̸=j; andP 3 = ∂U ∂q 3 = 1−βq 3 . (4.2) Recycling structures: A product can either be recycled individually, or be recycled collectively (with other products). Individual recycling assumes that each product is recycled independently, in which case it occupiesanexclusiverecyclingchannel,includingcollectionsites,reverselogisticsandprocessingfacilities. Collective recycling assumes that each product is recycled jointly with other products, in which case all products share a recycling channel. We use {·} to denote products that share a recycling channel. For instance, if product 1 is recycled independently, we denote it by {1}; if products 2 and 3 are recycled together, we denote it by {23}. We use the recycling structure to describe how these three products get recycled. In total, there are fivepossible recyclingstructures for three products:{123},{1}{23},{13}{2}, {12}{3}and{1}{2}{3}. Wereferto: • {123}asthe all-inclusiverecycling,becauseallproductsarerecycledtogether; • {12}{3}asthemarket-basedrecycling,becauseproductsfromthesamemarketarerecycledtogether; • {1}{23}and{13}{2}asthecross-marketrecycling,becauseproducti,i∈{1,2}isrecycledwithits independentproduct,3,whileitsrelatedproduct,3−i,isrecycledseparately; • {1}{2}{3}asthe product-based recycling,becauseeachproductsisrecycledseparately. Cross-market recycling is the most counter-intuitive one (e.g., Dell laptops and GE refrigerators are recycled together, while Lenovo laptops are recycled independently). We denote the set of all recycling structuresbyX. Cost structure: As our main concern is the impact of recycling costs, without loss of generality, we normalize manufacturing costs to zero. For every recycling channel, there is a fixed cost,K > 0, incurred for establishing and maintaining a recycling channel. We assume that the cost for individually recycling a unit of products 1 and 2 isc 1 ∈ (0,1), and for product 3 it isc 3 ∈ (0,1). When products 1 and 2 (with a positive substitutability level) are recycled together (that is, in market-based recycling), we assume that 47 theirunitrecyclingcostsremainthesame. Whenproduct3(whichisindependentof1and2)iscollectively processedandrecycledwithproducts1and/or2(i.e.,inthecaseofall-inclusiveandcross-marketrecycling), their unit costs are increased toλc 3 andλc 1 (λ> 1), respectively. λ is the increase rate of the unit cost; to makesurethatfirmsmakeapositivepayoff,weassumethatc 1 ,c 3 < 1 +1 . Wenoteherethatourassumptions aboutunitcostchangesaremorelikelytoholdinsettingswithmorecompleterecyclinglegislation,wherein implementation is more regulated; in environments in which compliance is more difficult to achieve, any increase in the number of firms that jointly recycle, regardless of product substitutability, is likely to lead a unitcostincrease. Before discussing our results, we introduce some notation that will be used throughout the thesis. To simplifyourexpressions,wedefineconstants µ 1 = (λ−1)c 1 2β +γ , ν 1 = 2−(λ+1)c 1 2β +γ andµ 3 = (λ−1)c 3 2β . Notethat0<c 1 < 1and1<λ< 1 c 1 implyµ 1 ,ν 1 ,µ 3 > 0.Wefurtherdefine ρ = c 3 [2−(λ+1)c 3 ] c 1 [2−(λ+1)c 1 ] , σ = 2−2c 1 2−(λ+1)c 1 andx = γ β . ρmeasuresthedifferencebetweenunitrecyclingcostsinthetwomarkets: sincec 1 ,c 3 < 1 +1 ,ρisincreas- ing inc 3 and decreasing inc 1 . Hence, high (resp., low)ρ indicates that independent recycling of product 3 is more (resp., less) expensive than independent recycling of products 1 and 2;ρ close to one indicates that unit recycling costs of all products are similar. Note thatρ also depends on the rate of unit cost increase,λ: asλincreases,avalueofρ> 1decreases(resp.,ρ< 1increases).σ measurestheincreaseinunitrecycling costofproducts1and2duetocross-marketrecycling—σ isincreasinginc 1 andλ. Inotherwords,largeσ indicatesthateitherproduct-basedrecyclingofproducts1and2iscostly(c 1 ishigh),orcross-marketrecy- cling (including all-inclusive recycling) significantly increases unit costs (λ is large). Since 0<c 1 ,c 3 < 1 and 1 < λ < 1 c 1 , 1 c 3 , we haveρ > 0 and 1 < σ < 2. x measures how substitutable products 1 and 2 are: higherx implies that products 1 and 2 are more substitutable. Since 0 < γ < β, we have 0 < x < 1; to makesurethatallquantitiesarenon-negative,werequire2β(1−λc 1 )−γ(1−c 1 )> 0,whichisequivalent tox< 4 −2. Hence,0<x< min { 1, 4 −2 } . 48 Westillconsiderthreescenarios: (1)thesocialplannerrecyclesproductsthatareexogenouslymadeby different firms (SP); (2) the social planner organizes production and recycling (EP); and (3) firms endoge- nouslymanageproductionandrecycling(RP). 4.1.1 ExternalityProblem We first consider the scenario without EPR-type legislations, under which a social planner takes charge of the disposal of all three products. In Section 3.3, we refer to this problem as the Externality Problem (EP). Under this scenario, firms select product outputs that maximize their revenues: max q i P i q i , i = 1,2,3. The equilibrium quantities areq 1 = q 2 = 1 2+ andq 3 = 1 2 (q 3 > q 1 = q 2 ), regardless of the recycling structures for product take-back. However, the total recycling cost varies. We useC X to denote the total recycling cost in the system under the recycling structureX ∈X. For example, in{1}{23}, the total cost isC {1}{23} =c 1 q 1 +λc 1 q 2 +λc 3 q 3 +2K. Thecompletelistofthetotalrecyclingcostunderallrecycling structuresisshowninTable4.1. Table4.1: TotalRecyclingCostandSPSocialWelfareunderEachCoalitionStructure(ThreeProducts) RecyclingStructure TotalRecyclingCost SPSocialWelfare all-inclusive{123} λc 1 q 1 +λc 1 q 2 +λc 3 q 3 +K (1−c 1 ) 2 + + (1−c 3 ) 2 2 −K cross-market{i}{3−i,3} c 1 q i +λc 1 q 3−i +λc 3 q 3 +2K (1−c 1 ) 2 +(1−c 1 ) 2 −2 (1−c 1 )(1−c 1 ) 2( 2 − 2 ) + (1−c 3 ) 2 2 −2K market-based{12}{3} c 1 q 1 +c 1 q 2 +c 3 q 3 +2K (1−c 1 ) 2 + + (1−c 3 ) 2 2 −2K product-based{1}{2}{3} c 1 q 1 +c 1 q 2 +c 3 q 3 +3K (1−c 1 ) 2 + + (1−c 3 ) 2 2 −3K After observing product quantities, the social planner determines the optimal recycling structure, by minimizing the total recycling cost: min X∈X C X (the social planner’s objective is maximizing the social wel- fare, which is obtained the difference between the market surplusU and the total recycling cost from; as production quantities are fixed,U is also fixed and social welfare maximization equals to minimization of thetotalrecyclingcost). Comparingofthetotalrecyclingcostsindifferentstructuresleadstothefollowing result. Proposition4.1.1. Consider the EP in [3;3] model, and letK EP = 2µ 1 +µ 3 ; then, all-inclusive recycling is optimal forK >K EP and market-based recycling is optimal forK <K EP . 49 All the proofs are given in the technical Appendix. Proposition 4.1.1 suggests that the social planner shouldeitheradoptall-inclusiverecycling,{123},whenthefixedcostishigh(K >K EP )ormarket-based recycling,{12}{3},whenthefixedcostislow(K <K EP );itshouldneitherusetheproduct-basedrecycling ({1}{2}{3})noradoptcross-marketrecycling({1}{23}or{13}{2}). Inotherwords,thetwosubstitutable products,1and2,shouldalwaysberecycledtogether. Thereasoningbehindthisisstraightforward: • The positive fixed cost is the main driver behind joint recycling, so it is always beneficial to recycle substitutableproductstogether. • All-inclusive recycling increases unit costs, and therefore it may not always be optimal; sometimes, market-basedrecyclingmayworkbetter. Themagnitudeofthefixedcostdetermineswhethertoadopt all-inclusiveormarket-basedrecycling. • If two products are recycled together, it is always cheaper to chose substitutable products; indeed, cross-marketrecyclingiscounter-intuitiveanditdoesnotemergeintheEP[3;3]model. Next,weconsiderthecaseinwhichthesocialplannerorganizesbothproductionandrecycling. 4.1.2 SocialProblem IntheEP,productionandrecyclingareseparatelymanagedtoachievetwodifferentobjectives: maximizing individual firms’ revenues and minimizing the total recycling cost, respectively. In this subsection, we consider the case in which both production and recycling are organized with the goal of maximizing the social welfare. In Section 3.2, we refer to this problem as the Social Problem (SP). For a given recycling structureX ∈X,wedenotethesocialwelfaregeneratedfromthethreeproductsbyW X = max q 1 ;q 2 ;q 3 U−C X . Ourobjectiveistoselectarecyclingstructurethatmaximizesthesocialwelfare, max X∈X W X . Forinstance,in cross-marketrecycling{1}{23},quantitiesq 1 ,q 2 ,q 3 generate U−C {1}{23} =q 1 +q 2 +q 3 − 1 2 [ β(q 2 1 +q 2 2 +q 2 3 )+2γq 1 q 2 ] −[c 1 q 1 +λc 1 q 2 +λc 3 q 3 +2K]. From the first order conditions, we have q 1 = (1−c 1 )− (1−c 1 ) 2 − 2 , q 2 = (1−c 1 )− (1−c 1 ) 2 − 2 and q 3 = 1−c 3 . NotethatunliketheEPcase,thesequantitiesareendogenousandvarywithrecyclingstructures. Thesocial welfare in this case isW {1}{23} = (1−c 3 ) 2 2 − 2K + 1 2 (1−c 1 ) 2 +(1−c 1 ) 2 −2 (1−c 1 )(1−c 1 ) 2 − 2 . The complete 50 listofthesocialwelfareexpressionsunderallrecyclingstructuresisshowninTable4.1. Comparisonofthe socialwelfareexpressionsunderdifferentrecyclingstructuresleadstothefollowingresult. Proposition 4.1.2. Consider the SP in [3;3] model, and let K SP = (2 +x) 2 ( 1 1+x + 2 ) βµ 1 ν 1 ; then, all-inclusive recycling is optimal forK >K SP and market-based recycling is optimal forK <K SP . Proposition4.1.2isalsointuitive. Ononehand,asthepositivefixedcostinducesjointrecycling,substi- tutableproducts(products1and2)shouldalwaysberecycledtogetherandthefixedcostdetermineswhether toadoptall-inclusiveormarket-basedrecycling. Ontheotherhand,itisalwayscheaperandeasiertojointly recycle substitutable products (products 1 with 2) than independent products (product 3 with product 1 or 2);thus,theformerbringsahighersocialwelfare. Propositions4.1.1and4.1.2showsomecommonfeaturesoftheEPandSP,thatwesummarizebelow. Corollary 4.1.1. In [3;3] model, when the social planner organizes recycling, products from the same market(products1and2)shouldalwaysberecycledtogether;itisneveroptimaltorecyclethemseparately (i.e., cross-market recycling). Furthermore, whether or not to include the independent product (product 3) depends on the level of fixed costs. We now compare the fixed cost thresholds between market-based and all-inclusive recycling in the EP andSPmodels. Corollary 4.1.2. In [3;3] model, there existsλ 0 = 2c 1( 2 2+x −c 1)+(1+x)c 3 (1−c 3 ) 2c 1 +(1+x)c 3 such thatK EP ≷ K SP for λ≷λ 0 . Corollary 4.1.2 indicates that if cross-market recycling significantly increases the unit recycling cost (λ > λ 0 ), it is more likely to organize market-based recycling,{12}{3}, in the EP than in the SP; if rate of the unit cost increase is small (λ < λ 0 ), it is more likely to organize all-inclusive recycling,{123}, in the EP than in the SP. Intuitively, as product quantities are fixed in the EP, when cross-market recycling is expensive, reducing the share of the fixed cost might not offset the increase in unit cost stemming from all- inclusiverecycling. However,asquantitiesintheSParedeterminedwiththeconsiderationofunitrecycling costs,all-inclusiverecyclingmaystillbeattractive. Next,westudyrecyclingstructuresthatemergewhenfirmsmakerecyclingdecisionsontheirown. 51 4.1.3 ResponsibilityProblem We now discuss the scenario under which EPR-type legislations are introduced and firms are required to take the responsibility for recycling of their own products. In section 3.4, we refer to this problem as the ResponsibilityProblem(RP).Weassumetheegalitariancostallocation. Thatis,K isequallysharedamong products. To benefit from a lower share of the fixed setup cost, firms may recycle products together. We useπ X i to denote firmi’s payoff under the recycling structureX ∈ X,i = A,B,C. Firms first calculate the optimal payoffs they generate under each recycling structure X: max q i π X i ; then, firms cooperatively determine and establish recycling structures that optimize their payoffs. For instance, if firm A and C recycletheirproductstogether,whilefirmB individuallyrecyclesitsproducts,theirpayoffsare π {13}{2} A =P 1 q 1 −λc 1 q 1 − K 2 , π {13}{2} B =P 2 q 2 −c 1 q 2 −K andπ {13}{2} C =P 3 q 3 −λc 3 q 3 − K 2 . From the first order conditions, we haveq 1 = 2(1−c 1 )− (1−c 1 ) 4 2 − 2 ,q 2 = 2(1−c 1 )− (1−c 1 ) 4 2 − 2 andq 3 = 1−c 3 2 . Thus,optimalpayoffsaregivenby π {13}{2} A =β [ 2β(1−λc 1 )−γ(1−c 1 ) 4β 2 −γ 2 ] 2 − K 2 , π {13}{2} B =β [ 2β(1−c 1 )−γ(1−λc 1 ) 4β 2 −γ 2 ] 2 −K andπ {13}{2} C =β ( 1−λc 3 2β ) 2 − K 2 . The list of optimal payoffs under all recycling structures is shown in Table 4.2. Note that firmsA andB receive the same payoffs in all recycling structures except for cross-market recycling: payoff generated by firmAin{1}{23}correspondstoB’spayoffin{13}{2},andviceversa. Table4.2: Firms’OptimalPayoffsunderEachCoalitionStructure([3;3]Model) RecyclingStructure FirmA’sOptimalPayoff FirmC’sOptimalPayoff all-inclusive{123} β ( 1−c 1 2+ ) 2 − K 3 β ( 1−c 3 2 ) 2 − K 3 cross-market{1}{23} β [ 2(1−c 1 )− (1−c 1 ) 4 2 − 2 ] 2 −K β ( 1−c 3 2 ) 2 − K 2 cross-market{13}{2} β [ 2(1−c 1 )− (1−c 1 ) 4 2 − 2 ] 2 − K 2 β ( 1−c 3 2 ) 2 − K 2 market-based{12}{3} β ( 1−c 1 2+ ) 2 − K 2 β ( 1−c 3 2 ) 2 −K product-based{1}{2}{3} β ( 1−c 1 2+ ) 2 −K β ( 1−c 3 2 ) 2 −K 52 Unlike the EP and SP, the RP is an endogenous coalition formation problem. Depending on payoffs under different recycling structures, every firm has its most preferred structure. However, in many cases, there is an inconsistency among structures that are most preferred by different firms. Thus, some firm(s) maynotbeabletoformcoalitionsattheirwill,becauseother(s)mightnotwanttojointhemintheirdesired structures,inwhichcasevarioussequencesofdefectionsmightoccur. Asaresult,thefinalrecyclingstruc- ture has to show a level of farsightedness: every firm considers possible reactions (by others) to its actions. Weadoptadynamic/farsightedstabilityconceptcalledtheequilibriumprocessofcoalitionformation,intro- ducedinmoredetailinthenextsubsection. EquilibriumProcessofCoalitionFormation(EPCF) Konishi and Ray (2003) propose a dynamic/farsighted approach to stability in multilateral coalition forma- tion games, which they call the equilibrium process of coalition formation (EPCF). In this subsection, we introducetheEPCFinoursetting. LetN denotethesetofallfirmsandXdenotethesetofallpartitionsofN,alsoreferredtoasstructures. For every firm i ∈ N, let π X i denote i’s payoff under structure X ∈ X, and let δ i ∈ [0,1] denote the discount factor fori’s future payoffs. Then,i’s payoff from a sequence of structures{X t } can be written as ∑ ∞ t=0 δ t i π Xt i . Whenδ i = 0, we consider only immediate payoffs, which corresponds to myopic/static stabilityconceptssuchasthecore. Sinceweareinterestedinfarsightedresults,weconsiderδ i closertoone. A process of coalition formation (PCF) is a transition probability ψ : X×X → [0,1] such that ∑ Y∈X ψ(X,Y) = 1 for ∀X ∈ X. A PCF ψ induces a value function Π i for every firm i, which rep- resentsi’s infinite horizon payoff starting from the structureX underψ and is the unique solution to the equation Π i (X,ψ) =π X i +δ i ∑ Y∈X ψ(X,Y)Π i (Y,ψ). Denote the defection made by a set of firms S ⊆ N from a structure X to another structure Y by X ⇀ S Y. We say that a defectionX ⇀ S Y is profitable underψ if Π i (Y,ψ)≥ Π i (X,ψ) for∀i∈S; we further say that the defection is strictly profitable if the above inequality is strict. We say that the defection is efficient if there is no other move X ⇀ S Z such that Π i (Z,ψ) > Π i (Y,ψ) for ∀i ∈ S. Then, the equilibriumPCF(EPCF)isdefinedasfollows. APCFψ isanequilibriumPCFifthefollowingholds: 53 • wheneverψ(X,Y) > 0 for someY ̸= X, there existsS ⊆ N such thatX ⇀ S Y is profitable and efficient; • ifthereisastrictlyprofitabledefectionfromX,thenψ(X,X) = 0andthereexistsastrictlyprofitable andefficientdefectionX ⇀ S Y suchthatψ(X,Y)> 0. Thus, a defection from one structure to another occurs only if all members of the deviating set agree to move and they cannot find a strictly better alternative structure. In addition, a defection from a structure must occur if there is a strictly profitable move. Note that this definition does not require that every strictly profitablemovehasapositiveprobability,andallowsdefectionsinwhichtheinitialdeviatingsetisindiffer- ent to the change. Konishi and Ray (2003) show that an equilibrium process of coalition formation always existsforcountableX. We say that a PCFψ is deterministic ifψ(X,Y)∈{0,1} for∀X,Y ∈X. A structureX is absorbing ifψ(X,X) = 1, while a PCFψ is absorbing if, for every structureY, there is some absorbing structureX such thatψ (k) (X,Y) > 0, whereψ (k) denotes thek-step transition probability. Konishi and Ray (2003) show that the set of all absorbing states, under all deterministic absorbing EPCFs, is a subset of the largest consistent set (LCS) 1 introduced by Chwe (1994). Thus, the absorbing states of the EPCF may provide a refinementoftheLCS.Inaddition,astheEPCFandLCSareconceptuallyratherdifferent—playersinthe LCSonlyreceivetheirpayoffsinthefinal(stable)structure,whileplayersintheEPCFreceivetheirpayoffs aftereverydefectionandbasetheirdecisionsontheirinfinitehorizonpayoffs—byadoptingtheEPCF,our resultsshowalevelofrobustness. The relationship between the EPCF and LCS is useful in determining the stability of a particular struc- ture,asitmaybeeasiertocheckifastructureisanabsorbingstatethanifitbelongstotheLCS.Inthenext subsection, we use this fact to analyze stability of recycling structures: we first identify conditions under whichastructureisanabsorbingstateoftheEPCF,andthenconcludethatunderthesameconditionsitalso belongstotheLCS. 1 The LCS assumes that a defection from a structure is deterred if it might eventually lead to a stable structure (a member of the LCS), under which some initially defecting firms are not better off. Since every firm considers the possibility that, once it takes an action, some other firm(s) may react, and then yet some other(s), and so on, the LCS is a dynamic solution concept that incorporates farsighted stability. The LCS “rules out with confidence”: if a structureX is not contained in the LCS,X cannot be stable. However, if we start from an arbitrary structure, the LCS does not predict which of the stable structures is most likely to occur. Instead,theLCSgivesasetofstructuresfromwhichdefectionsarenotlikelytohappen. 54 FarsightedStabilityinRP Weusetheabovedefinitionstoidentifyallstablerecyclingstructuresinthe[3;3]model. Proposition 4.1.3. Consider the RP in [3;3] model, and let K RP = max { 6βµ 1 ν 1 , 3 8 (2+x) 2 ρβµ 1 ν 1 } . Then: 1. all-inclusive recycling is stable forK >K RP and market-based recycling is stable forK <K RP ; 2. in addition, if 5 4 <σ < √ 2,x> 2σ−2 √ σ 2 −1 andρ< 4 (2−x)x (4−x 2 ) 2 , then cross-market recycling is also stable formax { 4 2−x (2−x) 2 βµ 1 ν 1 , 1 2 (2+x) 2 ρβµ 1 ν 1 } <K < 2 (2−x)x (2−x) 2 βµ 1 ν 1 . Proposition4.1.3.1indicatesthatthecaseinwhichfirmsendogenouslyformcoalitionstomaximizetheir individual payoffs is consistent with the case in which the social planner organizes recycling to maximize the social welfare: when the fixed cost is high (K >K RP ), firms engage in all-inclusive recycling,{123}, andwhenthefixedcostislow(K <K RP ),firmschosemarket-basedrecycling,{12}{3}. However,Proposition4.1.3.2showsthatthecounter-intuitivechoiceofcross-marketrecycling,{1}{23} and{13}{2},canalsoemergeasstable. Forinstance,considerthefollowingorderingoffirms’preferences (basedontheirpayoffrealizations)underdifferentrecyclingstructures: π {1}{23} A >π {12}{3} A >π {123} A >π {13}{2} A >π {1}{2}{3} A , π {13}{2} B >π {12}{3} B >π {123} B >π {1}{23} B >π {1}{2}{3} B , (4.3) π {123} C >π {1}{23} C =π {13}{2} C >π {12}{3} C =π {1}{2}{3} C . WecanusethedeterministicPCFψ definedbythefollowingprobabilitytransitionmatrix: X\Y {123} {1}{23} {13}{2} {12}{3} {1}{2}{3} {123} 0 1 0 0 0 {1}{23} 0 1 0 0 0 {13}{2} 0 0 0 0 1 {12}{3} 0 0 0 0 1 {1}{2}{3} 0 1 0 0 0 55 Then, the following is an EPCF with {1}{23} as the absorbing state: {123} ⇀ {A} {1}{23}; {1}{23} ⇀ {B;C} {1}{23};{13}{2} ⇀ {A} {1}{2}{3};{12}{3} ⇀ {A} {1}{2}{3};{1}{2}{3} ⇀ {B;C} {1}{23}. Intuitively, firm A (resp., B) maximizes its payoff if it recycles individually. It is then better for firmsB (resp.,A) andC to collaborate, which leads to{1}{23} (resp.,{13}{2}), than to act individu- ally, which leads to{1}{2}{3}, becauseπ {1}{23} B >π {1}{2}{3} B ,π {1}{23} C >π {1}{2}{3} C . Thus, cross-market recyclingmayemergeasstable. Moregenerally,cross-marketrecyclingisstablewhenequation4.3holds: • productsinthesamemarketarehighlysubstitutable(xishigh); • it is more expensive to recycle substitutable products than the independent product (c 1 is high,c 3 is low)—thatis,ρislow; • the unit cost increases significantly for cross-market recycling (λ is large, which decreases ρ < 4 (2−x)x (4−x 2 ) 2 < 1); • thefixedcost(K)isinthemoderatelyhighrange. Whenproductsaremoresubstitutable(xishigher),themarketcompetitionismoreintense. Especially, if recycling the two substitutes is costly (c 1 is high), the competition intensity increases even further. Thus, firmshavemoreincentivestoincreasetheirownoutputandrestraintheircompetitor’soutput. Toachievethis target, a firm may chose to recycle individually, in which case its competitor may have to engage in cross- marketrecycling(in{13}or{23});ifthissignificantlyincreasesthecompetitor’sunitcost(λislarge),then the incentive is even more obvious. However, if recycling of the independent product is also expensive (c 3 ishigh), theindependentfirmmaynotwanttoparticipateincross-marketrecycling(neither{13}nor{23} is formed). In addition, the fixed cost can neither be too high (otherwise, all-inclusive recycling is stable) nortoolow(otherwise,market-basedrecyclingisstable). 56 Because 2 (2−x)x (2−x) 2 βµ 1 ν 1 < 6βµ 1 ν 1 6 K RP , the range of parameter values for which cross-market recyclingisstableiscontainedwithintherangeforwhichmarket-basedrecycling,{12}{3},isstable. Con- sider again preferences described by equation (4.3), but now consider the deterministic PCFψ defined by thefollowingprobabilitytransitionmatrix: X\Y {123} {1}{23} {13}{2} {12}{3} {1}{2}{3} {123} 0 0 0 1 0 {1}{23} 0 0 0 0 1 {13}{2} 0 0 0 0 1 {12}{3} 0 0 0 1 0 {1}{2}{3} 0 0 0 1 0 The following EPCF has {12}{3} as its absorbing state: {123} ⇀ {A;B} {12}{3}; {1}{23} ⇀ {B} {1}{2}{3};{13}{2} ⇀ {A} {1}{2}{3};{12}{3} ⇀ {A;B} {12}{3}{1}{2}{3} ⇀ {A;B} {12}{3}. Intu- itively, because firm B would rather cooperate with firm A than with firm C, π {12}{3} B > π {1}{23} B , if B declinestojoinC,firmAmayreconsiderjoiningB informing{12}{3},asπ {12}{3} A >π {1}{2}{3} A . Thatis, {12}{3}alsoemergesasstable(jointrecyclingbyAandB doesnotrequirecooperationoffirmC). Thus, itispossibletohavebothcross-marketandmarket-basedrecyclingaspotentiallystableoutcomesforsome parametervalues. WenowshowsomenumericalresultsinFigure4.1. InFigure4.1b,astheincreaseinunitcostofproduct 1 or 2 in cross-market recycling is significant (λ is large andc 1 is high), firmA (orB) has the incentive to recycle individually and force its competitor,B (orA), to jointly recycle with firmC. By doing so, firmA (orB) obtains a higher market share as its competitor’s cost is significantly increased. At the same time, because recycling of product 3 is inexpensive, firmC wants to cooperate with other firms, as it can share thefixedcostwithoutasignificantincreaseinitsunitcost. Therefore,whenthefixedcostisinthemoderate range, cross-market recycling may be stable. In Figure 4.1a, cross-market recycling only slightly increases the unit costs (λ is small), and firmsA andB do not have enough incentives to recycle individually (recall thatafirmthatrecyclesindividuallypaystheentirefixedcost)andpushtheircompetitorstocooperatewith firmC. Asaresult,cross-marketrecyclingisnotstable. InFigure4.1c,cross-marketrecyclingleadstohigh unit cost of product 3 (λ is large andc 3 is high), and firmC does not want to cooperate with firmA orB. Hence,cross-marketrecyclingisagainnotstable. 57 (a)c 1 = 0.3,c 3 = 0,λ = 1.25 (b)c 1 = 0.3,c 3 = 0,λ = 2.25 (c)c 1 = 0.3,c 3 = 0.3,λ = 2.25 Figure4.1: [3;3]ModelRPStableStructures(EgalitarianCostAllocation) In the above discussion, each firm only makes a product in one market, and the decision maker (social planer or firms) choose among all-inclusive recycling, market-based recycling, cross-market recycling, and product-basedrecycling. Whenthesocialplannerorganizesrecycling(intheEPandSP),productsfromthe samemarketshouldalwaysberecycledtogether;inaddition,thefixedcost,K,determineswhethertoadopt all-inclusive (K is high) or market-based recycling (K is low). When firms are responsible for recycling andendogenouslyformrecyclingcoalitions(intheRP),resultsforall-inclusiverecyclingandmarket-based recyclingaresimilar. Moreover,forarangeofparametervalues(whenthecompetitionbetweenfirmsAand B isintense,recyclingofproducts1and2iscostlywhilethatofproduct3itislessexpensive,theunitcost increase for cross-market recycling is large, and the fixed cost is in the moderate range), a counter-intuitive outcomeofcross-marketrecyclingmayalsobestable. In the next section, we study the case in which one firm makes a product in one market, while the other firm makes products across multiple markets. This require introduction of another recycling model: firm-basedrecycling. 4.2 Single-MarketandCross-MarketManufacturing: [2;3]Model In the preceding section, we discuss three firms that make three products ([3;3] model): two products are substitutable in one market while the third product is independent in another market. In this section, we let two firms from the [3;3] model (one from each market) merge into a single cross-market firm, yielding the 58 [2;3] model. In other words, two firms are duopolists in one market, and one of them is also a monopolist inthe other market. Forexample, Dell and Samsung compete with their laptops; meanwhile, Samsung also producesrefrigerators. Bulowetal.(1985)assumesuchmodelsettinganddiscusshowdoesthemonopolist firm’s action in the monopoly market change its competitor’s strategy in the duopoly market by affecting the monopolist firm’s marginal cost in the duopoly market. We assume the same model setting and discuss bothfirms’strategieswhenonefirmcanmakedecisionsinbothmarkets. AssumethatfirmB mergeswithfirmC;thatis,firmAstillmakesproduct1,andfirmB makesproducts 2and3. ThismodelisillustratedinFigure1.1b. Observethatunderthisassumptionstructures{1}{23}and {13}{2},previouslyreferredtoascross-marketrecycling(inthe[3;3]model),ceasetobeindistinguishable — in {1}{23}, products made by the same firm are recycled together, and we refer to it as firm-based recycling, while{13}{2}, in which products from different markets made by different firms are recycled together,isreferredtoascross-market/firmrecycling. Thatis,inthe[2;3]model,weconsiderthefollowing fiverecyclingstructures: all-inclusive:{123} market-based:{12}{3} firm-based:{1}{23} product-based:{1}{2}{3} cross-market/firm:{13}{2} Recall that cross-market recycling in the [3;3] model (e.g., recycling together a Dell laptop and a GE refrigerator,whileLenovolaptopisrecycledseparately)israthercounter-intuitive. Nowinthe[2;3]model, when one firm makes product across markets, cross-market/firm recycling (e.g., recycling together a Dell laptop and a Samsung refrigerator, while Samsung laptop is recycled separately) remains counter-intuitive, but firm-based recycling (e.g, recycling together a Samsung laptop and a Samsung refrigerator, while Dell laptopisrecycledseparately)isadoptedforawiderrangeofparametervalues. We carry over all other assumptions from the [3;3] model, including (1) all products have the same price-demand effectβ; (2) products 1 and 2 are related with the cross price-demand effectγ, and product 3 is independent; (3) in the product-based and in market-based recycling, the unit recycling cost for products 1 and 2 isc 1 , and for product 3 it isc 3 ; (4) in cross-market and in all-inclusive recycling, the increased unit recyclingcostforproducts1and/or2isλc 1 ,andforproduct3itisλc 3 ;and(5)foreveryrecyclingchannel, thefixedcostK isequallysharedamongproductsunderegalitariancostallocation. We observe that in the EP and SP, the production quantities do not depend on the firms of products 2 and 3 (firmB only vs. firmsB andC). Therefore, we omit the discussions of the EP and SP, because they 59 do not change from the [3;3] model. However, in the RP, when firms both manufacture and recycle their products, we observe different results in several cases, because the firm that manufactures products across markets(firmB)mayoffsetlowerprofitononeproductbyahigherprofitontheotherproduct. Weanalyze theRPinthenextsubsection. 4.2.1 ResponsibilityProblem Similar to our approach in the [3;3] model, we first calculate firms’ optimal payoffs under each recycling structure. Because we still consider the same products as the [3;3] model, firmA’s optimal payoffs carry over, while firmB’s optimal payoffs can be obtained by adding payoffs of firmsB andC from the [3;3] model. WeagainusetheEPCFtoidentifystablerecyclingstructures. Proposition4.2.1. Inthe[2;3]model,thereexistK RP ,K RP , ˇ K RP and ˆ K RP (functionsofµ 1 ,ν 1 ,ρ,σ and x) satisfyingK RP <K RP and ˇ K RP < ˆ K RP , such that the following holds for the RP: 1. all-inclusive recycling is stable forK >K RP , and market-based recycling is stable forK <K RP ; 2. ifρ< max { 2 (7−6)x−2 (4−x 2 ) 2 ,8 −x 2 +3x−2 (4−x 2 ) 2 } , then firm-based recycling is stable for ˇ K RP <K < ˆ K RP ; 3. inaddition,if 5 4 <σ< 3 2 ,x> 2 4−3 andρ< 4 (2−x)x (4−x 2 ) 2 ,thencross-market/firmrecyclingisstablefor 4 2−x (2−x) 2 βµ 1 ν 1 <K < 12 5 2−(2−)x (2−x) 2 βµ 1 ν 1 . The exact expressions for K RP , K RP , ˇ K RP and ˆ K RP are shown in equations (B.1) and (B.2) in the Appendix. Proposition4.2.1.1and4.2.1.3showsimilarfeaturestothe[3;3]model(Proposition4.1.3): • whenthefixedcostishigh(K >K RP ),firmsadoptall-inclusiverecycling,{123}; • whenthefixedcostislow(K <K RP ),firmsadoptmarket-basedrecycling,{12}{3}; • when products 1 and 2 are highly substitutable (x is high), it is more expensive to recycle products 1 and2thanproduct3(c 1 ishighandc 3 islow,andconsequentlyρislow),theincreaserateofunitcost (λ) is high, and the fixed costK is in the moderate range, firms adopt cross-market/firm recycling, {13}{2}. 60 Proposition 4.2.1.2 introduces a structure that does not exist in the [3;3] model: firm-based recy- cling. When it is more expensive to recycle products 1 and 2 than product 3, firm-based recycling is used when fixed cost, K, is moderately high and products 1 and 2 are highly substitutable (note that as both 8 −x 2 +3x−2 (4−x 2 ) 2 and 2 (7−6)x−2 (4−x 2 ) 2 increase in x for 0 < x < min { 1, 4 −2 } , it is easy to prove that ρ< max { 2 (7−6)x−2 (4−x 2 ) 2 ,8 −x 2 +3x−2 (4−x 2 ) 2 } is equivalent to imposing thatx has to be above a certain threshold). ThishappenseitherduetofirmB’spreferenceforfirm-basedrecycling,orbecausefirmAchoosestorecy- cle individually, which makes it more profitable for firmB to recycle its products together than separately. Forinstance,if π {12}{3} A >π {1}{23} A >π {123} A >π {13}{2} A >π {1}{2}{3} A , π {123} B >π {13}{2} B >π {1}{23} B >π {12}{3} B >π {1}{2}{3} B , market-based recycling ({12}{3}) is firm A’s most preferred outcome, but firm B favors firm-based recycling, π {1}{23} B > π {12}{3} B . Therefore, anticipating that market-based recycling is unlikely to hap- pen, firm A decides to recycle individually, and firm B then recycles its products together, because π {1}{23} B > π {1}{2}{3} B . Consider the deterministic PCF ψ defined by the following probability transition matrix: X\Y {123} {1}{23} {13}{2} {12}{3} {1}{2}{3} {123} 0 1 0 0 0 {1}{23} 0 1 0 0 0 {13}{2} 0 0 0 0 1 {12}{3} 0 0 0 0 1 {1}{2}{3} 0 1 0 0 0 Then,thefollowingEPCFhas{1}{23}asitsabsorbingstate:{123}⇀ A {1}{23};{1}{23}⇀ B {1}{23}; {13}{2} ⇀ A {1}{2}{3}; {12}{3} ⇀ B {1}{2}{3}; {1}{2}{3} ⇀ B {1}{23}. Observe that firm B chooses firm-based recycling only when the unit recycling cost of product 3 is low, as otherwise it cannot afford the high increase in unit costs of its products (2 and 3) stemming from cross-market recycling. In addition, firmB benefits from firm-based recycling when the market competition is intense and the fixed costisnotlowbymakingitscompetitor,A,paytheentirefixedcost;whenthefixedcostislow,thisstrategy maynotbenefitfirmB. 61 Recall that in the [3;3] model both{1}{23} and{13}{2} represented cross-market recycling; in the [2;3] model,{13}{2} is cross-market/firm, while{1}{23} denotes firm-based recycling. We now compare conditions under which firm-based and cross-market/firm recycling are stable — it is easy to verify that firm-based recycling emerges as stable over a wider range of parameter values. This confirms our intuition andfitswithreal-lifeobservation—cross-market/firmrecyclingislessintuitive,whilefirm-basedrecycling canmoreoftenbeseeninpractice. Next,wecompareresultsinthe[3;3]model(Proposition4.1.3)and[2;3]model(Proposition4.2.1)and observethefollowingdifferencesbetweenthetwomodels. TRANSITION BETWEEN ALL-INCLUSIVE AND MARKET-BASED RECYCLING: Whileinthe[3;3]model there is a fixed boundary, K RP , between all-inclusive recycling and market-based recycling, in the [2;3] model there is a range of parameter values (i.e., moderately high fixed cost,K RP <K <K RP ) for which both all-inclusive and market-based recycling are stable. In this range, one firm prefers one structure while theotherfirmpreferstheremainingstructure;forinstance,if π {12}{3} A >π {123} A >π {1}{23} A >π {1}{2}{3} A >π {13}{2} A , π {123} B >π {13}{2} B >π {12}{3} B >π {1}{23} B >π {1}{2}{3} B , market-basedrecycling,{12}{3},isfirmA’smostpreferredoutcome,whilefirmBhasthehighestpayoffin all-inclusiverecycling,{123}. FirmAmayopttorecycleonitsown,whichwouldleadtoeitherfirm-based recycling ({1}{23}) or product-based recycling ({1}{2}{3}), the least preferred outcomes forB. Conse- quently,firmB maycompromiseandagreetoadopt{12}{3}(becauseπ {12}{3} B >π {1}{23} B >π {1}{2}{3} B ), which would make market-based recycling stable. On the other hand, becauseπ {123} A > π {1}{23} A , firmA may want to join firm B in all-inclusive recycling in order to prevent the emergence of firm-based recy- cling. The EPCF with{12}{3} as the absorbing state is given by{123} ⇀ {A} {1}{23};{1}{23} ⇀ {B} {1}{2}{3};{13}{2} ⇀ {A} {1}{2}{3};{12}{3} ⇀ {A;B} {12}{3};{1}{2}{3} ⇀ {A;B} {12}{3}, while theEPCFwith{123}astheabsorbingstateisgivenby{123}⇀ {A;B} {123};{1}{23}⇀ {B} {1}{2}{3}; {13}{2}⇀ {A} {1}{2}{3};{12}{3}⇀ {B} {1}{2}{3};{1}{2}{3}⇀ {A;B} {123}. RANGE OF STABILITY FOR ALL-INCLUSIVE/MARKET-BASED RECYCLING: It is easy to verify that K RP >K RP >K RP ;thatis,market-basedrecyclingismorelikelytoemergeasstableinthe[3;3]model, while all-inclusive recycling is more likely to emerge as stable in the [2;3] model. We can conclude that 62 it is easier to implement all-inclusive recycling when fewer firms are involved, and market-based recycling whenthenumberoffirmsincreases. RANGE OF STABILITY FOR CROSS-MARKET/FIRM RECYCLING: It is easy to verify that the range of parameter valuesfor which cross-market/firmrecycling ({13}{2}) emerges as stable is broader in the [2;3] model than in the [3;3] model. In the [3;3] model, the firm manufacturing the independent product (firm C) only makes one product (product 3), while in the [2;3] model, that firm (firmB) makes two products (products2and3). Asaresult,therearemorecombinationsofparametervaluesunderwhichfirmB cansee anincreaseinitspayoffasaresultofcross-market/firmrecycling(adecreaseinprofitabilityofoneproduct maybeoffsetbyanincreaseinprofitabilityoftheotherproductmadebyB). SOURCES OF STABILITY FOR CROSS-MARKET/FIRM-BASED RECYCLING: It is easy to verify that the range of parameter values for which{1}{23} emerges as stable is broader in the [2;3] model than [3;3] model. In the [3;3] model, cross-market recycling ({1}{23}) is stable because it is the most preferred outcomeforfirmA(itisneverfavoredbyfirmsB andC becauseitincreasestheirunitcost,andatthesame time induces a higher share of the fixed cost than all-inclusive recycling). In the [2;3] model, stability of thefirm-basedrecycling({1}{23})mayalsostemfromfirmB’spreferences,asitspayoffcomesfromboth products2and3. Therefore,itispossiblethatfirm-basedrecyclingisfirmB’smostpreferredoutcome. We show some numerical results in Figure 4.2. Following an analysis similar to the one depicted in the [3;3] model (Figure 4.1), we observe that cross-market/firm recycling,{13}{2}, is stable over a small rangeofparametervaluesinFigure4.2b,andisneverstableinFigures4.2aand4.2c. Incontrast,withhigh substitutability between products 1 and 2 (largex), for fixed cost (K) in a moderate range, we observe that firm-basedrecycling,{1}{23},isstableinFigures4.2aand4.2b;especially,inFigure4.2bbothfirm-based andcross-market/firmrecyclingcanbestable,andtherangeoverwhichtheformerisstablestrictlycontains therangeoverwhichthelatterisstable. WenowcompareFigure4.1([3;3]model)andFigure4.2([2;3]model)toillustrateourresultsabove: • InFigure4.1([3;3]model),all-inclusiverecycling,{123},andmarket-basedrecycling,{12}{3},can never be stable in the same range. On the other hand, in every graph of Figure 4.2 ([2;3] model), we canobservearangeinwhichbothstructuresarestablewhenthefixedcostismoderatelyhigh. 63 (a)c 1 = 0.3,c 3 = 0,λ = 1.25 (b)c 1 = 0.3,c 3 = 0,λ = 2.25 (c)c 1 = 0.3,c 3 = 0.3,λ = 2.25 Figure4.2: [2;3]ModelRPStableStructures(EgalitarianCostAllocation) • All-inclusive recycling is stable over a wider range of parameters in Figure 4.2 ([2;3] model) than in Figure4.1([3;3]model),whiletheoppositeistrueformarket-basedrecycling. • Cross-market/firmrecycling,{13}{2},isstableoverawiderrangeofparametersinFigure4.2b([2;3] model)thaninFigure4.1b([3;3]model). • The recycling structure{1}{23} is stable over a wider range of parameters in Figure 4.2 (firm-based recyclinginthe[2;3]model)thaninFigure4.1(cross-marketrecyclinginthe[3;3]model). In the above discussion, we consider the scenario in which one firm makes products across multiple markets, while the other firms makes a single product in one of those markets. Many of our findings are consistentwiththeresultsinthe[3;3]model(conditionsunderwhichall-inclusive,market-basedandcross- market/firm, {13}{2}, recycling are stable). More importantly, in [2;3], a new result is identified: when the competition between substitutable products (products 1 and 2) is intense (largex) and when the fixed cost(K)ismoderatelyhigh,firm-basedrecycling,{1}{23},mayemergeasastableoutcome;notethatthis outcomeisnevertheoptimalchoiceforthesocialplanner. In [3;3] model, one firm is a monopolist in a market and the other two firms are duopolists in another market. In[2;3]model,onefirmmakesproductsinasinglemarketandtheotherfirmmakesproductsacross twomarkets. Inthesetwomodels,wesaythatfirmsarenotequalexante. Inthenextsection,westillfocus 64 on asymmetric products, but we let firms be equal ex ante. We are interested in whether the two firms are stillequalexpost,withstablerecyclingstrategiesadopted. 4.3 Cross-MarketManufacturing: [2;4]Model InSection4.2,wediscusstwofirmsthatmakethreeproducts([2;3]model): onefirmmakesproductsacross two different markets, while the other firm makes a product in one of those markets. In this section, we assumethatthelatterfirmfromthe[2;3]modelintroduceanewproductandthuscompeteswiththeformer firm in both markets, yielding the [2;4] model. That is, two firms make a total of four products in two independent markets: every firm makes a product in each of the markets, and every product made by one firm is substitutable with a product made by the other firm. For example, Samsung and LG compete with bothlaptopsandrefrigerators. Notethatinthissetting,thetwofirmsareequalexante. Weassumethatthenewproduct,say4,isindependentfromproducts1and/or2,andsubstitutablewith product3. FirmAmakesproducts1and4andfirmB makesproducts2and3,asillustratedinFigure1.1c. We assume the same cross demand-price effect,γ, to hold between products 1 and 2 and between products 3and4. FollowingSinghandVives(1984),themarketsurplusthenbecomes U =q 1 +q 2 +q 3 +q 4 − 1 2 [ β(q 2 1 +q 2 2 +q 2 3 +q 2 4 )+2γq 1 q 2 +2γq 3 q 4 ] , (4.4) andproductpricesaregivenby P i = ∂U ∂q i = 1−βq i −γq j , i,j = 1,2, i̸=j, ori,j = 3,4, i̸=j. (4.5) The[2;4]modelhasfifteenrecyclingstructures: all-inclusive:{1234} fullmarket-based:{12}{34} fullfirm-based:{14}{23} fullcross-market/firm:{13}{24} fullproduct-based:{1}{2}{3}{4} half-inclusive-1:{1}{234},{134}{2} half-inclusive-3:{124}{3},{123}{4} halfmarket-based-1:{1}{2}{34} halfmarket-based-3:{12}{3}{4} halffirm-based:{1}{23}{4},{14}{2}{3} halfcross-market/firm:{1}{24}{3},{13}{2}{4} 65 ItiseasytoobtainoptimalrecyclingstructuresfortheEPandSPmodels. Proposition 4.3.1. In [2;4] model, letK ′EP = 2µ 1 + 4 2+x µ 3 andK ′SP = (2+x) 2 1+x (1+ρ)βµ 1 ν 1 ; then, for the EP (resp., SP), all-inclusive recycling is optimal for K > K ′EP (resp., K ′SP ) and full market-based recycling is optimal forK <K ′EP (resp.,K ′SP ). Proposition 4.3.1 shows that our earlier results from the [3;3] and [2;3] models carry over when the social planner organizes recycling: it should either adopt all-inclusive recycling, {1234}, when the fixed cost is high (K > K ′EP in the EP and K > K ′SP in the SP model), or full market-based recycling, {12}{34}, when the fixed cost is low (K < K ′EP in the EP andK < K ′SP in the SP model). When we compare boundaries between market-based recycling and all-inclusive recycling in the three-product case ([3;3] and [2;3] models) and the four-productcase ([2;4] model), it is easy to verifythatK ′EP >K EP and K ′SP >K SP . In other words, with more products in the market, it is less likely that all-inclusive recycling will take place. The intuition behind this is straightforward: with more products in the market, the positive impactonall-inclusiverecyclingoftheeconomiesofscalediminishesthenegativeimpactofincreasedunit recyclingcosts. Thequestionis: fortheRP,doourearlierresultsfromthe[3;3]and[2;3]modelscarryoveraswell? For instance,whenthefixedcostishigh(resp.,low),isall-inclusiverecycling,{1234}(resp.,fullmarket-based recycling,{12}{34}), stable? As with four products we have to consider fifteen different recycling struc- tures, identifying all stable outcomes under all parameter values becomes intractable. We first analytically eliminate recycling structures that cannot be stable, and then obtain insights for other recycling structures fromnumericalexamples. Proposition 4.3.2. In the [2;4] model under egalitarian cost allocation, for the RP, full product-based recycling,{1}{2}{3}{4}, full cross-market/firm recycling,{13}{24}, full firm-based recycling,{14}{23}, and half market-based recycling,{1}{2}{34} and{12}{3}{4}, are never stable. ItiseasytoshowthatrecyclingstructuresmentionedinProposition4.3.2arealwaysdominatedforboth firmsbyeitherall-inclusiveorfullmarket-basedrecycling,{1234}or{12}{34}. Forinstance,observethat the increase in the unit recycling cost is the same in full firm-based and all-inclusive recycling, but firms leverage the economies of scale with smaller product volume. Hence, full firm-based recycling will not be adopted. 66 In what follows, we will not discuss some counter-intuitive recycling structures (half cross-market/firm recycling, {1}{24}{3} and{13}{2}{4}), as they are less likely to be adopted in practice (see numerical examplesbelow). (a)c 1 = 0.3,c 3 = 0,λ = 1.25 (b)c 1 = 0.3,c 3 = 0,λ = 2.25 (c)c 1 = 0.3,c 3 = 0.3,λ = 2.25 Figure4.3: [2;4]ModelRPStableStructures(EgalitarianCostAllocation) ThroughnumericalexamplesgiveninFigure4.3,weillustratethefollowingresults: • Stability areas for all-inclusive recycling,{1234}, and full market-based recycling,{12}{34}, carry overfromthe[3;3]and[2;3]models: whenthefixedcostishigh(resp.,low),all-inclusive(resp.,full market-based)recyclingisstable. • If the unit cost in one market is very low compared to the other market, e.g., we assume thatc 1 > 0 and c 3 = 0 (see Figures 4.3a and 4.3b). When the market competition is intense and fixed cost is in the moderate range, a firm, say, A, may prefer to individually recycle its product from the high- unit-cost market, i.e., product 1. The other firm, B, may be willing to jointly recycle one of its competitor’s products from the low-unit-cost market, i.e., product 4. As a result, the half-inclusive-1 recycling,{1}{234},isstable. Bydoingthis,firmAavoidstheincreaseintheunitcostofproduct1; meanwhile,AgetssomeeconomiesofscalefromcooperatingwithfirmB torecycleproduct4. Note thatAisnotconcernedwiththeincreaseintheunitcostofproduct4,whichisnegligibleasc 3 isvery low. On the other hand, the incentive for firmB to cooperate withA is the economies of scale. Note 67 that jointly recycling product 4 does not increase firmB’s unit costs: as long asB recycles products 2and3together,ithastopayhighunitcosts,nomatterwhetherproduct4isalsotogether. • If the unit costs in both markets are similar, e.g., assume thatc 1 = c 3 (see Figure 4.3c). When the market competition is intense and fixed cost is in the moderate range, a firm, say, A, may prefer to individually recycle its two products, 1 and 4. The other firm,B, may prefer to adopt the firm-based recycling strategy. As a result, the half firm-based recycling,{1}{23}{4}, is stable. By doing this, firmA avoids the increase in its unit costs; firmB gets the economies of scale. However, as firmA concerns with losing the economies of scale and firmB is worried about the unit cost increase, such rangeofparametervaluesisverylimited. In the above discussion, we analyze the model with two ex ante equal firms that make products across multiplemarkets,andcompeteineachofthemarkets. Whileourearlierresultsforall-inclusiveandmarket- based recycling carry over, the results for firm-based recycling cease to hold in most cases. Although with limited range of parameter values firms may adopt some recycling strategies that make them not equal ex post (e.g., half-inclusive recycling or half firm-based recycling), there is always a structure under which firmscankeepequalexpost(e.g.,all-inclusiveormarket-based). 4.4 Summary In this chapter, we study the recycling of asymmetric products under egalitarian cost allocation. In the primary market, some products belong to the same market, while others are from an independent market. Besides,onefirmmakesmultipleproducts,whiletheotherfirmmaymakeasingleproduct. Intherecycling market,similarlytoChapter3,whenmultipleproductsarerecycledtogether,duetotheeconomiesofscale, their producers equally share the fixed cost, which is considered as the egalitarian cost allocation. On the other hand, because of the asymmetry of products, when products from the same market are recycled together,theirunitcostsremainthesame;whenproductsfromdifferentmarketsarerecycledtogether,their unitcostsincrease. Westillconsidertheproductheterogeneityandeconomiesofscaleastwodeterminants, andlookforanswerto: howshouldtheproductsberecycledtogether? Westudythreescenarios. Inthefirstscenario,threefirmsmakethreeproducts,oneeach(werefertothis problemasthe[3;3]model),twoofwhichcompeteinthesamemarket,andtheotheroneisindependentin 68 another market. In the second scenario, two firms produce three products (we refer to this problem as the [2;3] model): one firm makes two products in two markets; the other firm makes the third product in one of the two markets. In the third scenario, two firms produce four products (we refer to this problem as the [3;3] model): each firm makes two products in two markets; each market contains two products made by twofirms. We still consider Social Problem (SP), Externality Problem (EP) and Responsibility Problem (RP). For allthreescenarios,theoptimalrecyclingstructuresforSPandEParesimilar: eitherallproductsarerecycled together, which is referred to as all-inclusive recycling, or products from the same market are recycled togetherregardlessoftheirproducers,whichisreferredtoasmarket-basedrecycling. ForRP,besidessimilar results of all-inclusive and market-based, when the market competition is intense, some other recycling structures are also possible to be stable. The most common structure is products made by the same firm are recycled together, regardless of their markets, which is referred to as firm-based recycling. When firms arenotequalexante,e.g.,inthe[2;3] model,firmschoosethefirm-basedrecyclingtotakethecompetitive advantage. Whenfirmsareequalexante,e.g.,inthe[2;4]model,fullfirm-basedrecyclingcannotbestable, sincethecompetitiveadvantagedisappears. In the previous two chapters, due to the economies of scale, for products that are recycled together, we assumethefixedsetupcostisequallysplitamongthoseproducts,whichisconsideredastheegalitariancost allocation. In the next chapter, we will consider a more general form of the economies of scale, which is sharedbyfirms’marketshare(productionquantities). 69 Chapter5 RecyclingofAsymmetricProductsunder ProportionalCostAllocation In Chapter 3, we consider the recycling of symmetric products in a general market. In Chapter 4, we consider the recycling of asymmetric products across different markets. However, in those two chapters, to get the economies of scale, we assume that the fixed setup cost is equally shared, and we consider it as theegalitariancostallocation. Inthischapter,westillconsidertherecyclingofasymmetricproductsacross differentmarkets. However,weuseamoregeneralformoftheeconomiesofscale,accordingtoproduction quantities, and we call it the proportional cost allocation. As we focus on the trade-off between the all- inclusiverecycling, market-basedrecyclingand firm-based recycling, we skip the [3;3] model, in which no firmhasmultipleproductstoallowthefirm-basedrecycling. InSection5.1,westudythescenarioinwhich two firms make three products, or the [2;3] model. In Section 5.2, we study the scenario in which two ex anteequalfirmsmakefourproducts,orthe[2;4]model. 5.1 Single-MarketandCross-MarketManufacturing: [2;3]Model In the [2;3] model, two firms, A andB, make three products, 1, 2 and 3. FirmA makes product 1; firm B makes products 2 and 3. Products 1 and 2 belong to the same market, and product 3 belongs to an independent market. β > 0 is the own price-demand effect of all three products, and γ ∈ [0,β] is the crossprice-demandeffectbetweenproducts1and2. Thequantitiesofthethreeproductsareq 1 ,q 2 ,andq 3 , respectively. Following Singh and Vives (1984), the market surplus brought by the three products is given byequation(4.1). A product can either be recycled independently, or be recycled together with other product(s). {·} denotes a product that is recycled independently or some products that are recycled together. For instance, if product 1 is recycled independently, it is denoted by{1}; if products 2 and 3 are recycled together, it is 70 denotedby{23}. Arecyclingstructuredescribeshowthesethreeproductsgetrecycled. Possiblestructures include: all-inclusive{123}, market-based{12}{3}, firm-based{1}{23}, cross-market/firm{13}{2}, and product-based{1}{2}{3}. If product 1 or 2 is recycled independently, its unit cost for collection, separation, disassembly, and recycling is c 1 > 0; if product 3 is recycled independently, the unit recycling-related cost is c 3 > 0; if products 1 and 2 are recycled together, their unit recycling costs are stillc 1 ; if products 1 and/or 2 are recycledtogetherwithproduct3,theirunitcostsareλc 1 andλc 3 ,respectively,where16λ6 2,λc 1 6 0.5, andλc 3 6 0.5. Proportionalcostallocation: Due to the economies of scale, firms benefit from recycling large numbers of products. In previous two chapters, we assume that different products (made by different firms or from different markets) that are recycled together equally share the fixed setup cost of establishing/maintaining the recycling channel. In other words, firms share for each product the same amount of fixed cost when the productsarerecycledtogether;werefertothismechanismasthe egalitarian cost allocation. However, as products are asymmetric — some belong to the same market and some products belong to different markets — products may have different quantities when they are recycled together. To take this into account, firms may apportion the economies of scale by their market share. In this chapter, we study such a model, and refer to this mechanism as the proportional cost allocation. Following Amir (2003), we assume that the recycling related costs are reduced quadratically with the quantity of products, and we use κ ∈ [0,κ] to denote the factor of the quadratic form (κ is the upper bound ofκ, and it depends onc 1 and c 3 ). We also assume that when products are recycled together, the economies of scale are allocated among productsproportionallyaccordingtotheirproductquantities. Tosummarizetheexpressionoftherecycling cost, we let c 2 = c 1 , and let Z i be the set of jointly-recycled products that includes product i. In other words,Z i is a set of products that are jointly recycled, andi∈Z i . Then, the cost for recycling producti is λc i q i − q i ∑ j∈Z i q j κ ( ∑ j∈Z i q j ) 2 ,whereλ =      1,ifZ i ={1},{2},{3},or{12} λ,otherwise . Next, we first consider the benchmark model, Social Problem (SP), in which the social planner plans for both production and recycling. On one hand, the social planer determines production quantities for all firms; on the other hand, the social planner chooses a structure to recycle all products. The objective is to 71 maximize the social welfare. After comparing the social welfare generated under each recycling structure, wehavethefollowingresults. Proposition5.1.1. In[2;3], for the SP, there existsκ SP ∈ (0,κ) such that all-inclusive recycling is optimal forκ>κ SP and market-based recycling is optimal forκ<κ SP . Results of Proposition 5.1.1 show consistency with results for the SP under egalitarian cost allocation (Proposition 4.1.2). When the economies of scale are large, all three products should be recycled together. When the economies of scale are small, products from the same market should be recycled together. That is,differentcostallocationmechanismsdonotinfluencethebenchmarkmodel. Next,westudythemodelinwhichEPR-typelegislationshavebeenintroduced,ResponsibilityProblem. In RP, the social planner requires firms to take the responsibility of recycling all products they bring to the market. On one hand, firms compete in the primary market; on the other hand, firms may cooperate in the recycling market, to get the economies of scale. By taking the partial derivatives of the market surplus with respect to the product quantities, products’ prices are given by equation (4.2). Note that to focus on recycling, the production costs and fixed costs are normalized to zero. Hence, for producti∈Z i , it brings thefollowingpayofftoitsproducer: π Z i i =P i q i −   λc i − q i ∑ j∈Z i q j κ   ∑ j∈Z i q j   2   , (5.1) whereλ =      1,ifZ i ={1},{2},{3},or{12} λ,otherwise . Notethatc 1 =c 2 . Then,firmA’spayoffisπ Z 1 1 andfirm B’spayoffisπ Z 2 2 +π Z 3 3 . Firmsfirstcooperativelydeterminetherecyclingstructure,andthencompetitivelychoosetheproduction quantities, in order to receive higher payoffs. The quantities are chosen to achieve the equilibrium; the recycling structure is determined to achieve a stable status in which no firms have the incentive or power to change. Similar to Chapters 3 and 4, it is an endogenous coalition formation problem, and we study the farsightedstability. In the previous two chapters, we use the Largest Consistent Set (LCS, see Chwe 1994) and the Equi- librium Process of Coalition Formation (EPCF, see Konishi and Ray 2003) to study the farsighted stability, 72 respectively. The LCS assumes that a defection from a structure is deterred if it might eventually lead to a stable structure (a member of the LCS), under which some initially defecting firms are not better off. According to such definition, players only consider their final (stable) payoffs. Hence, the LCS can give a setofstructuresfromwhichdefectionsarenotlikelytohappen,butifwestartfromanarbitrarystructure,it cannotpredictwhichofthestablestructuresismostlikelytooccur. The EPCF provides a refinement of the LCS. According to the definition of the EPCF given in Section 4.1.3, players receive their payoffs after every defection and base their decisions on their infinite horizon discountedpayoffs. Inaddition,thesetofallabsorbingstates,underalldeterministicabsorbingEPCFs,isa subsetoftheLCS.Basedontheaboveconsiderations,inthischapter,weusetheEPCFtoidentifyallstable structures,sinceitiseasiertocheckifastructureisanabsorbingstatethanifitbelongstotheLCS. Next,wefirststudythestablerecyclingstructurethroughsomenumericalexamples. Then,weshowthe stability results when the two parameters,κ (economies of scale factor) andγ (cross price-demand effect), reachtheirboundaryvalues(notethatκ∈ [0,κ]andγ∈ [0,β]). Numericalstudyonthestablerecyclingstructures: Numericalsimulationsindicatethatthestablerecy- cling structure depends on the relationship ofc 1 andc 3 , e.g.,c 1 < (>)c 3 . We discuss three typical cases, c 1 <c 3 ,c 1 =c 3 ,andc 1 >c 3 depictedinFigure5.1. (a)c 1 = 0,c 3 = 0.15,λ = 1.5 (b)c 1 = 0.15,c 3 = 0.15,λ = 1.5 (c)c 1 = 0.15,c 3 = 0,λ = 1.5 Figure5.1: [2;3]ModelRPStableStructures(ProportionalCostAllocation) 73 By comparing Figures 4.2 and 5.1, we observe some similarities between the recycling of asymmetric productsunderegalitariancostallocationandunderproportionalcostallocation. • ALL-INCLUSIVE RECYCLING: When the economies of scale (represented by K in Chapter 4 and by κ is this chapter) are large, the all-inclusive recycling, {123}, is uniquely stable. In this case, large economies of scale are the incentive of joint recycling for both firms. That is, the all-inclusive recyclingistheParetodominantstructure. • MARKET-BASED RECYCLING: When the economies of scale are small,{12}{3}, the market-based recycling, is uniquely stable. Although both firms still prefer the joint recycling, considering the increasedunitcostwhenproductsfromdifferentmarketsarerecycledtogether,firmswouldnotrecy- cle all three products together. Instead, they prefer to cooperate within the same market without increasingtheunitcost. Asaresult,themarket-basedrecyclingisstable(preferredbybothfirms). • TRANSITION BETWEEN ALL-INCLUSIVE RECYCLING AND MARKET-BASED RECYCLING: When the economies of scale are moderate, both the all-inclusive recycling and the market-based recycling canemergeasstable. Inthiscase,firmAprefersthemarket-basedrecycling,whilefirmB prefersthe all-inclusiverecycling. Thereasonsarelistedasfollows. – InChapter4,underthemarket-basedrecycling({12}{3}),A’sfixedcostis 1 2 K,whileB’sfixed costis 3 2 K. Undertheall-inclusiverecycling({123}),A’sfixedcostis 1 3 K whileB’sfixedcost is 2 3 K. Thatis,frommarket-basedrecyclingtoall-inclusiverecycling,A’sfixedcostisreduced by 1 6 K whileB’sfixedcostisreducedby 5 6 K. Hence,B hasmoreincentivefortheall-inclusive recycling. – In this chapter, under the market-based recycling, firmsA andB have the same market share. Under the all-inclusive recycling, firmA’s market share decreases and firmB’s increases. As theeconomiesofscalearesharedaccordingtofirms’marketshare,firmAhastheincentivefor themarket-basedrecyclingwhilefirmB hastheincentivefortheall-inclusiverecycling. As a result, if firmA compromises with firmB, the all-inclusive recycling is stable; if firmB com- promiseswithfirmA,themarket-basedrecyclingisstable. • CROSS-MARKET/FIRM RECYCLING: When the economies of scale are moderate, and the market competition is intense ( is large), forc 3 = 0, the cross-market/firm recycling,{13}{2}, can emerge 74 as stable, but it cannot be uniquely stable. In this case, firm A prefers the market-based recycling, whilefirmBpreferstheall-inclusiverecycling. Inaddition,firmAprefers{13}{2}to{123},because of a larger market share. Meanwhile, due to the intense market competition, firmB prefers{13}{2} to {12}{3}: note that firm A’s unit cost in {13}{2} is increased to λc 1 while firm B’s unit costs arec 1 and 0, respectively; in other words, firmB gets the competitive advantage. Hence, if neither firmA nor firmB would like to make a compromise,{13}{2} may emerge as stable, although it is a counter-intuitive structure. However, it is also possible that firms compromise, in which case other structure(s)mayemergeasstable. • FIRM-BASED RECYCLING: When the economies of scale are moderate, and the market competition is intense, for c 3 = 0, the firm-based recycling, {1}{23}, can emerge as stable. In this case, firm A prefers the market-based recycling, while firmB prefers the all-inclusive recycling. In addition, firmAprefers{1}{23}to{123}: theunitcostincreasedominatestheeconomiesofscaleforfirmA; in other words,A gets the competitive advantage. Meanwhile, firmB prefers{1}{23} to{12}{3}: the economies of scale dominates the unit cost for firmB (note that{1}{23} only increases the unit cost of product 2, not product 3). As a result, if neither firm A nor firm B would like to make a compromise,{1}{23}mayemergeasstable. Meanwhile,wealsoobservesomedifferencesbetweenFigures5.1and4.2. • Fromγ = 0toγ =β,theslopesofthetransitionbetweentheall-inclusiverecyclingandmarket-based recyclingaredifferent. – DOWNSLOPE: In Figure 4.2, the transition is downslope. In other words, higher fixed cost with less intense market competition is equivalent to lower fixed cost with more intense market competition. When the fixed cost is high, less intense market competition can increase the revenueandoffsetthecost,andviceversa. – UPSLOPE: InFigure5.1,thetransitionisupslope. Inotherwords,lowereconomiesofscalewith less intense market competition are equivalent to higher economies of scale with more intense marketcompetition. Whentheeconomiesofscalearelow,firms’revenueisreduced,whichcan besavedbyamarketwithlessintensecompetition,andviceversa. 75 • Whentheeconomiesofscalearelargeandthemarketcompetitionisintense,{1}{23},thefirm-based recycling, is uniquely stable in Figure 5.1; while there is no such stability under similar conditions in Figure 4.2. In Chapter 4, the fixed cost is equally shared among the three products, regardless of theirquantities. Inthischapter,theeconomiesofscalearesharedbyfirms’marketshare,whichgives firms more incentive to increase their competitive advantage. As a result, the firm-based recycling, {1}{23},becomesaParetodominantstrategyoffirmB. However,asthe{1}{23}increasesfirmB’s unitcost,whenc 1 =c 3 ,theincentiveofincreasingthecompetitiveadvantageislimited,and{1}{23} isstableinforasmallerrangeofparametervalues. Incontrast,whenc 1 = 0orc 3 = 0,suchincentive islesslimited,and{1}{23}isstableforalargerrangeofparametervalues. • Whentheeconomiesofscalearemoderate,andthemarketcompetitionisintense,forc 3 = 0,thefirm- basedrecycling,{1}{23},isuniquelystableinFigure5.1;whileitisnotuniquelystableundersimilar conditions in Figure 4.2. As analyzed above, firms have more incentive to increase their competitive advantage. Compared to the firm-based recycling,{1}{23}, the product-based,{1}{2}{3}, may be more preferred by firmA, as firmB loses the economies of scale in{1}{2}{3}. As a result, as firm B prefers{1}{23}to{12}{3},{12}{3}cannotbestable. AsfirmAprefers{1}{2}{3}to{1}{23}, andprefers{1}{23}to{123},{123}cannotbestable. Therefore,{1}{23}isuniquelystable. Analyticresults: Next,weshowthestabilityresultsontheboundariesinabovefigures(notethatthecross price-demand effectγ ∈ [0,β] and the economies of scale factorκ ∈ [0,κ]), to verify our findings in the numericalanalysis. Proposition5.1.2. Whenκ = 0,{12}{3} is stable for06γ6β. The result of Proposition 5.1.2 applies in general and does not depend on the relationship betweenc 1 andc 3 . For the remaining boundaries, analyzing the stability for general values ofc 1 andc 3 is intractable. Hence, we still study the three typical cases: c 1 <c 3 ,c 1 =c 3 , andc 1 >c 3 . First, let us determineκ in the threecasesaccordingly. Lemma 5.1.1. Whenc 1 = c 3 > 0, orc 1 = 0 andc 3 > 0,κ = 1 4 β ensures that the equilibrium quantities are non-negative forx∈ [0,1]; whenc 1 > 0 andc 3 = 0,κ = 1 5 β ensures the non-negative quantities. Proposition 5.1.3. Forκ = κ, there existsγ 0 < β such that{1}{23} is uniquely stable forγ > γ 0 . For γ <γ 0 , 76 • whenc 1 =c 3 orc 3 = 0,κ = 1 4 β. {123} and{12}{3} are possible to emerge as stable; they may be stable together. • whenc 1 = 0,κ = 1 5 β.{123} is uniquely stable. Proposition5.1.4. Whenc 1 =c 3 , orc 1 = 0, orc 3 = 0, forγ = 0, there existκ 1 <κ 2 such that{12}{3} is stable forκ6κ 2 and{123} is stable forκ>κ 1 . Note that it is possibleκ 1 ,κ 2 > κ. Ifκ 2 > κ,{12}{3} is stable for 06 κ < κ 1 ; and both{123} and {12}{3}arestableforκ 1 6κ6κ. Ifκ 1 >κ,{12}{3}isstablefor06κ6κ. Proposition 5.1.5. Forγ = β, there existsκ 0 < κ such that{1}{23} is uniquely stable forκ> κ 0 . For κ<κ 0 , • whenc 1 = c 3 orc 1 = 0,{123} and{12}{3} are possible to emerge as stable; they may be stable together. • whenc 3 = 0,{123},{12}{3},{1}{23} and{13}{2} are possible to emerge as stable;{1}{23} is always uniquely stable;{13}{2} cannot be uniquely stable. Note that for κ > κ 0 , the firm-based recycling, {1}{23}, is uniquely stable because it is the Pareto dominant structure. Forκ < κ 0 ,{1}{23} is uniquely stable because firmB prefers{1}{23} to{12}{3}, while firm A prefers {1}{2}{3} and {1}{23} to {123} and {2}{13}. As a result, {1}{23} emerges as uniquelystable. From Propositions 5.1.2, 5.1.3, 5.1.4, and 5.1.5, we know that in all three cases, the market-based recycling,{12}{3},isuniquelystableforsmalleconomiesofscale;andthefirm-basedrecycling,{1}{23}, isuniquelystableforlargeeconomiesofscaleandintensemarketcompetition( islarge). Besides, • when c 1 = c 3 , the all-inclusive recycling, {123}, is uniquely stable for large economies of scale (if the upper boundary of the market-based recycling is belowκ); for moderate economies of scale, both the all-inclusive recycling and market-based recycling are stable (if the upper boundary of the market-basedrecyclingisbelowκ). • whenc 1 = 0,theall-inclusiverecyclingisstableforlargeeconomiesofscale(certainly);formoderate economiesofscale,boththeall-inclusiverecyclingandmarket-basedrecyclingarestable(certainly). 77 • whenc 3 = 0, the all-inclusive recycling,{123}, is stable for large economies of scale (if the upper boundary of the market-based recycling is belowκ); for moderate economies of scale, both the all- inclusive recycling and market-based recycling are stable (if the upper boundary of the market-based recycling is belowκ). Besides, when the market competition is intense and the economies of scale are moderate, the firm-based recycling is uniquely stable, and cross-market/firm recycling,{13}{2}, maybestabletogetherwithotherstructures(e.g.{123}). Through showing the stability for the boundary values of parameters, our analytic results verify the numerical examples. Both the analytic and numerical analysis indicates that: in the [2;3] model, when asymmetric products are recycled under proportional cost allocation, the firm-based recycling is stable in a broader range of parameter values than under egalitarian cost allocation. Even more, once it is stable, it is uniquely stable (be adopted with the probability of 100%). Recall that under egalitarian cost allocation, the firm-based recycling is never stable uniquely. We conclude that under proportional cost allocation, the firm-basedrecyclingismuchmorelikelytobeadopted. Next,weconsiderthe[2;4]modeltocheckwhetherwecanmakeasimilarconclusion. 5.2 Cross-MarketManufacturing: [2;4]Model In this section, we study the [2;4] model, in which two identical firms, say, A andB, each produces two heterogeneous products. Assume that firmA produces products 1 and 4, firmB produces products 2 and 3, products 1 and 2 belong to one market, and products 3 and 4 belong to another (independent) market. All four products useβ as the own price-demand effect, and both markets useγ as the cross price-demand effect. When products from different markets are recycled together, their unit recycling costs are increased bythefactorofλ. Inthe[2;4]model,weconsiderthefollowingstructures: all-inclusive:{1234} fullmarket-based:{12}{34} fullfirm-based:{14}{23} fullcross-market/firm:{13}{24} fullproduct-based:{1}{2}{3}{4} half-inclusive-1:{1}{234},{134}{2} half-inclusive-3:{124}{3},{123}{4} halfmarket-based-1:{1}{2}{34} halfmarket-based-3:{12}{3}{4} halffirm-based:{1}{23}{4},{14}{2}{3} halfcross-market/firm:{1}{24}{3},{13}{2}{4} 78 According to Section 4.3, following Singh and Vives (1984), the market surplus brought by the four products is given by equation (4.4). We still assume the cost allocation mechanism is proportional. Letting c 2 = c 1 and c 4 = c 3 , the cost for recycling product i ∈ Z i is λc i q i − q i ∑ j∈Z i q j κ ( ∑ j∈Z i q j ) 2 , where λ =      1,ifZ i ={1},{2},{3},{4},{12},or{34} λ,otherwise . Based on such market surplus and recycling cost, it iseasytoobtainoptimalrecyclingstructuresforSPmodel. Proposition 5.2.1. In [2;4] model, letκ ′SP = √ P 2 Q 2 +(M 2 +N 2 )[(M 2 +N 2 )−(P 2 +Q 2 )]−PQ 4(M 2 +N 2 )+4 √ P 2 Q 2 +(M 2 +N 2 )[(M 2 +N 2 )−(P 2 +Q 2 )]−4PQ (β + γ); then, for the SP, all-inclusive recycling is optimal for κ > κ ′SP and full market-based recycling is optimal forκ<κ ′SP . Proposition 5.2.1 shows that our earlier results under egalitarian cost allocation carry over when the social planner organizes recycling: it should either adopt all-inclusive recycling, {1234}, when the economies of scale are high (κ>κ ′SP ), or full market-based recycling,{12}{34}, when the economies of scalearelow(κ<κ ′SP ). Next, we consider the results for RP. For producti ∈ Z i , considering its price (P i ) given by equation (4.5) andλ =      1,ifZ i ={1},{2},{3},{4},{12},or{34} λ,otherwise , the expression of its payoff (π Z i i ) follows equation (5.1). Note thatc 1 = c 2 andc 3 = c 4 . Then, firmA’s payoff isπ Z 1 1 +π Z 4 4 and firmB’s payoff is π Z 2 2 +π Z 3 3 . Similarly to the methodology in Section 4.3, based on the above payoffs, we first analytically elimi- nate recycling structures that cannot be stable, and then obtain insights for other recycling structures from numericalexamples. Proposition 5.2.2. In the [2;4] model under proportional cost allocation, full product-based recycling, {1}{2}{3}{4}, full cross-market/firm recycling,{13}{24}, full firm-based recycling,{14}{23}, and half market-based recycling,{1}{2}{34} and{12}{3}{4}, are never stable. It is easy to show that full firm-based recycling is dominated by all-inclusive recycling, {1234}; full product-based recycling, full cross-market/firm recycling, and half market-based recycling are dominated byfullmarket-basedrecycling,{12}{34}. OurnumericalstudyisbasedonFigure5.2. 79 (a)c 1 = 0.15,c 3 = 0.15,λ = 1.5 (b)c 1 = 0.15,c 3 = 0,λ = 1.5 Figure5.2: [2;4]ModelRPStableStructures(ProportionalCostAllocation) WealsocompareFigures4.3and5.2toanalyzethestabilityresults. • In both models, when the economies of scale (fixed costs) are high, all-inclusive recycling,{1234}, is stable; when the economies of scale are low, full market-based recycling, {12}{34}, is stable. However,thereisnotransitionregioninwhichbothstructuresemergeasstable. Itisbecauseinboth models, the two firms are equal ex post with these two recycling structures adopted. In other words, π {1234} A =π {1234} B andπ {12}{34} A =π {12}{34} B in both models. Therefore, one structure always Pareto dominatestheotherstructure. • Next, suppose that the unit cost in one market is very low compared to the other market, e.g., we assume thatc 1 > 0 andc 3 = 0 (see Figures 4.3a, 4.3b and 5.2b). When the market competition is intense and fixed cost is in the moderate range, a firm, say,A, may prefer to individually recycle its product from the high-unit-cost market, i.e., product 1. The other firm,B, may be willing to jointly recycle one of its competitor’s products from the low-unit-cost market, i.e., product 4. As a result, the half-inclusive-1 recycling,{1}{234}, is stable. By doing this, firmA avoids the increase in the unit cost of product 1; meanwhile,A gets some economies of scale from cooperating with firmB to recycleproduct 4. Note thatA is not concerned with the increase in the unit cost of product 4, which is negligible as c 3 is very low. On the other hand, the incentive for firm B to cooperate with A is the economies of scale. Note that jointly recycling product 4 does not increase firmB’s unit costs: 80 as long asB recycles products 2 and 3 together, it has to pay high unit costs, regardless of product 4 beingrecycledjointlyornot. • Now, suppose that the unit costs in both markets are similar, e.g., assume thatc 1 = c 3 . Under egal- itarian cost allocation, the half firm-based recycling, {1}{23}{4} and{14}{2}{3}, are possible to emergeasstablewhenthemarketcompetitionisintense(seeFigure4.3c). Incontrary,underpropor- tionalcostallocation,thefirm-basedrecyclingwillnotbeadoptedbyanyfirm. Moreover,bothfirms alwaysadoptthesamestrategy(eithertheall-inclusiveorthemarket-based)(seeFigure5.2a). These results seem counter-intuitive: under egalitarian cost allocation, firms may adopt different strategy whileunderproportionalcostallocation,firmsadoptthesamestrategy. However,theyresultfromthe following reasons. When the cost allocation is egalitarian (Chapter 4), if products that are recycled individually may not harm other products too much, as they just equally share the fixed cost. Hence, the half firm-based structure is possible to be stable. When the cost allocation is proportional (this chapter),productsthatarerecycledindividuallyaremoreharmfultootherproducts,astheeconomies of scale depend on the total quantities. For this reason, the balance between the two ex ante equal firms is more difficult to be broken. Otherwise, once the two ex ante equal firms become not equal ex post, a structure that one firm prefers the most may be hated the most by the other firm. In other words,thetwofirmsprefertobeequalallthetimeandchoosethesamestrategy. 5.3 Summary In this chapter, we study the recycling of asymmetric products under proportional cost allocation. In the primary market, firms compete with their products across multiple markets: some products are in the same market,whileothersareinindependentmarkets. Inaddition,somefirmsmakemultipleproductsandothers make only one product. Hence, products in this chapter are asymmetric. In the recycling market, we keep the assumption that product heterogeneity increases the unit recycling cost. In Section 4.3, the fixed setup cost is shared equally, so all products that are jointly recycled pay the same cost, which we refer to as the egalitarian cost allocation. In this chapter, the economies of scale are apportioned based on the market share, so products that are jointly recycled may pay different cost; we refer to this as the proportional cost allocation. Our question is: what is the difference between the two cost allocation mechanisms, egalitarian andproportional,regardingtotheirimpactsonthestablestructures? 81 As we focus on the all-inclusive recycling, market-based recycling and firm-based recycling, we only considerthe[2;3]modeland[2;4]model. Notethatinthe[3;3]model,nofirmproducesmultipleproducts that allows the firm-based recycling strategy. In the [2;3] model, most conditions under which the all- inclusive recycling structure, {123}, and the market-based recycling structure, {12}{3}, are stable carry over. That is, when the economies of scale (κ) are large, the all-inclusive recycling emerges as stable; when the economies of scale are small, the market-based recycling emerges as stable; there is a transition rangeinbetween,whereboththeall-inclusiveandmarket-basedstructurescanemergeasstable. Moreover, whenc 3 = 0,thefirm-basedrecycling,{1}{23},emergesastheuniquelystablestructureinintensemarket competition, for both very large economies of scale and moderate economies of scale. Recall that under egalitarian cost allocation, the firm-based recycling emerges as stable only when the market competition is intenseandtheeconomiesofscalearemoderate,anditisneveruniquelystable. Thatis,underproportional cost allocation, the firm-based recycling structure is stable in a broader range of parameter values. It is because under the proportional cost allocation, the competition between firms is enhanced. As a result, to increasetheircompetitiveadvantage,firmsaremorelikelytoadoptthefirm-basedrecyclingstrategy. In the [2;4] model, two firms are equal ex ante. Conditions for the all-inclusive recycling, {1234}, and full market-based recycling,{12}{34}, to be stable still carry over. However, the firm-based recycling strategyisneveradoptedbyanyfirm. Recallthatwhenallproductshavethesameunitcost,itseemscounter- intuitive: firmsthatequalanteadoptdifferentstrategies(e.g.,halffirm-basedrecycling,{1}{23}{4})under egalitarian cost allocation but adopt the same strategy (i.e., all-inclusive and market-based recycling) under proportional cost allocation. The reason is: under proportional cost allocation the penalty of changing ex anteequalfirmstoexpostnotequalfirmsaremoreseverethanunderegalitariancostallocation. 82 Chapter6 Conclusion 6.1 ModelReview This thesis focuses on problems related to governments’ and producers’ responsibilities and implementa- tions in recycling of consumer products at their end of life. Instead of organizing recycling themselves, followingtheExtendedProducerResponsibility(EPR),governmentshavebeenmakinglegislationssuchas theWEEEDirective,requiringproducerstoberesponsibleforallproductstheybringtothemarket. Produc- erscaneithercooperateandjointlyrecycletheirproducts,followingtheCollectiveProducerResponsibility (CPR), or individually recycle its own products, following the Individual Producer Responsibility (IPR). Interesting questions involved in this process and motivating our study include: (1) What is the trade-off betweenrecyclingguidedbygovernmentalEPR-typelegislationsbutorganizedbyproducers,andrecycling directly organized by the government? (2) In compliance with EPR-type legislations such as the WEEE, shouldproducersfollowtheCPRandjointlyrecycletheirproducts,orshouldtheyfollowtheIPRandsetup individualrecyclingprograms? (3)Whensomeproducersmakemultipleproductsacrossdifferentmarkets, howshouldtheyorganizerecycling? And(4)whataretheimpactsofthecostsharingmechanism? The first research question investigates the basic setting of our study: what is the impact of introducing EPR-type legislations to producers. To address it, we compare three scenarios. (a) Local governments plan for production and set up a recycling network that recycles all consumer products, in order to maximize the social welfare. This scenario is a benchmark model, and is referred to as Social Problem (SP). (b) Pro- ducers compete with quantities in the primary market while local governments set up a recycling network that recycles consumer products that are made by all producers. This scenario mimics the situation if no EPR-type legislations are introduced, and is referred to as Externality Problem (EP). And (c) producers not only compete with quantities in the primary market, but also cooperatively set up a recycling network that 83 contains several producer-recycling coalitions. This scenario mimics the situation with EPR-type legisla- tionsintroduced,andisreferredtoasResponsibilityProblem(RP).Thesethreescenarios,especiallySPand RP,arestudiesthroughoutthisthesis. To address the remaining questions, we focus on two types of product models. First of all, we consider symmetricproducts—producersmakeexanteequalproductsinageneralmarket(oneeach),andthereisa uniformlevelofheterogeneitybetweenanytwoproducts. Forsymmetricproducts,weassumeaegalitarian costallocationmechanism—becauseoftheeconomiesofscale,productsthatarerecycledtogetherequally share the fixed setup cost. Then, we consider asymmetric products — products belong to two independent markets,atleastoneofwhichcontainsmultipleproductsandhasmarketcompetition;productsinthesame market are homogeneous but products from different markets are heterogeneous. For asymmetric products, we study both the egalitarian cost allocation mechanism, and a proportional cost allocation mechanism — the economies of scale depend on the total production quantity and are apportioned among participating producersaccordingtotheirmarketshare. 6.2 ResultReview Throughourstudy,theeconomiesofscale,productheterogeneityandmarketcompetitionareconsideredas important factors that influence governments’ or producers’ choices on different recycling structures. The economies of scale are the incentive for joint recycling; the product heterogeneity is the incentive for joint recycling of products belonging to the same market; and the market competition is the incentive for joint recyclingofproductsmadebythesameproducer. When products symmetrically belong to a general market, we attribute the their substitutability to the productheterogeneity. Inotherwords,themarketcompetitionandproductheterogeneityarecorrelated. As a result, in EP, it is optimal to only set up recycling coalitions of similar sizes; while in SP and RP, it may notbeoptimal(stable)todoso. Inparticular,inSP,whentheunitcostincreasesinthecoalitionsizequickly (duetotheproductheterogeneity),itisoptimalforthegovernmenttocreatesomesmallrecyclingcoalitions and one large coalition. In RP, when the unit cost increases in the coalition size slowly (due to the product heterogeneity), the economiesofscale andproduct heterogeneitydetermine anoptimalcoalition size; most producers endogenously form coalitions of this optimal size, leaving the remaining producers (if any) to formasmallercoalition. 84 We use these results to provide decision support to the government. After comparing the optimal result in EP and stable result in RP, we conclude that RP can generate a higher social welfare when the recycling taskis“challenging”—(i)thefixedcostishigh,and(ii)therearemanyproducers,and/or(iii)productsare not homogeneous, and/or (iv) the basic unit (recycling) cost is not low. With the consideration of imposing taxes and subsidies, EP can achieve the SP optimum if taxes are imposed, while RP can achieve the SP optimumifsubsidiesareawarded. However,whentherearemanyproducers,thissubsidycanbenegligible. Ingeneral,RPismorebeneficialforthegovernment. When products asymmetrically cross two markets, we separate the market competition and product heterogeneity. We assume that on one hand, there is no market competition between products from dif- ferent markets, but these products are highly heterogeneous, which increases products’ unit recycling cost. On the other hand, products that belong to the same market have low product heterogeneity that has neg- ligible impact on products’ unit recycling cost, but there exists market competition. Out result indicate that in both SP and EP, when the fixed costs under egalitarian cost allocation or economies of scale under proportional cost allocation are high, products should be recycled all together, which is referred to as the all-inclusive recycling structure. When the fixed costs or economies of scale are low, products should be recycled according to the product type (market) regardless of their producers, which is referred to as the market-based recycling structure. In RP, results of the all-inclusive recycling and market-based recycling carry over. In addition, if producers are not equal ex ante, e.g., one producer makes products in a single market while the other producer makes products across multiple markets, producers may choose to jointly recycletheirownproductsregardlessoftheproducttypeinacompetitionintensemarket,inordertogetthe competitiveadvantage;thisrecyclingstructureisreferredtoasfirm-based. However,ifproducersareequal ex ante, e.g., both producers make products in both markets, the firm-based recycling cannot be adopted in mostcases. Itisbecausewhenproducersareequalexante,ifbothofthemchoosethefirm-basedrecycling, theycannotgetthecompetitiveadvantage. The main difference between the egalitarian cost allocation and proportional cost allocation focuses on the firm-based recycling. If producers are not equal ex ante, producers under proportional cost allocation are more likely to adopt the firm-based recycling, as proportional cost allocation enhances the competition between different producers. If producers are equal ex ante, in very limited range of parameter values, one producer under egalitarian cost allocation may adopt the firm-based recycling; however, no producer under proportionalcostallocationwouldadoptthefirm-basedrecycling. Itisbecauseifonlyoneproduceradopts 85 the firm-based recycling, producers become not equal ex post. In the proportional cost allocation, such situationisevenmoreunbalanced,asoneproducerbenefitsthemostbuttheotherproducerlosesthemost. 6.3 Contributions,InsightsandShortcomings This thesis has five contributions to the literature. First, this thesis focuses on two existing recycling sys- tems, one before EPR legislation (EP) and the other under EPR legislation (RP). Through the analysis of recycling structures in each system, we compare them with a benchmark system (SP), and provide sugges- tions for their implementations. To the best of our knowledge, this thesis is the first work addressing these two systems from the perspective of recycling structures. Second, we use game-theoretical methodology to study the implementation (CPR and IPR) of EPR-type legislations. In RP, producers choose between recycling individually or collectively, which involves endogenous formation of coalitions. We study the dynamic/farsighted stability of recycling structures in order to better capture possible actions and reactions of multiple producers involved in joint recycling. To the best of our knowledge, this thesis is the first work thatintroducethedynamic/farsightedstabilitytotheareaofsustainableoperations. Third,weanalyzefirm- based and market-based recycling. In our model, we consider different producers competing in the same market,andproducersmanufacturingacrossdifferentmarkets. Thisisthesimplestmodelthatenablesusto studyimpactsthatcompetitionbetweenmultipleproducersandmanufacturingacrossmultiplemarketshave onjointrecycling,andevensuchasimplemodelisnotentirelytractable. Tothebestofourknowledge,this thesisisthefirstworkthatanalyzesthistypeofeffects. Lastbutnotleast,weconsiderimpactsofeconomies ofscale,productheterogeneity,andmarketcompetitionontherecyclingstructures. Theeconomiesofscale areconsideredastheincentiveofjointrecycling;theproductheterogeneityisconsideredastheincentiveof market-based recycling; and the market competition is considered as the incentive of firm-based recycling. This thesis fills out the gap that no paper captures consequences (coalition formation) of these impacts in thesustainableoperationsfield. This thesis provides insights to both governments and producers. For the government, we help identify appropriate environmental policies. Our suggestion for the selection between EP and RP is based on the characteristics of the economy, such as the fixed cost, unit (recycling) cost, and product heterogeneity. If EPRlegislationhasnotbeenintroduced,wesuggestthegovernmentconstructrecyclinginfrastructures,and charge a tax from producers; otherwise, we suggest providing a subsidy to producers to help achieve the 86 optimal social welfare. For producers, we identify stable recycling structures in both a general setting of symmetric products, and two detailed settings of asymmetric products. Our analysis of factors (economies of scale, market competition and product heterogeneity) that lead to different recycling structures can help producerschoosethestablebeneficialstructure. Our work still has some shortcomings, which we plan to address in our future work. 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Forexample,ifn ={1,3,5,7,9},thens(n) ={2,3,5,7,8}. Lemma A.0.1. When θ → 1 + , for any arbitrary structure outcome n = {n 1 ,...,n J } satisfying N = ∑ J j=1 n j > 4,26J6N−2,n 1 6···6n J andn J −n 1 > 2,w SP (s(n))>w SP (n). Proof. The proof contains two steps. First, we give the expression of w SP (n). Then, we show that for x = 0,w SP (s(n)) =w SP (n);forx> 0, lim →1 + { w SP (s(n))−w SP (n) } > 0. Step1: Fromequations(3.1),(3.2)and(3.3), W(q,n) = N ∑ i=1 q i − 1 2   β N ∑ i=1 q 2 i +γ N ∑ i=1 N ∑ l=1;l̸=i q i q l   − |n| ∑ j=1   ¯ c(n j )x ∑ i∈A j q i +K   = |n| ∑ j=1 (1−¯ c(n j )x) ∑ i∈A j q i | {z } TermA − 1 2 [ (β−γ) N ∑ i=1 q 2 i |{z} TermB +γ ( N ∑ i=1 q i ) 2 | {z } TermC ] −|n|K. (A.1) SincesuchW(q,n)isconcaveinq i ,theoptimalquantityisobtainedfromtheFOC: q SP i (n) =      [N−(N−1)x](1− c(n j )x)−(1−x) ∑ |n| k=1 n k (1− c(n k )x) x[N−(N−1)x] , ifx> 0 1+ ∑ |n| k=1 n k c(n k )−N c(n j ) N , ifx = 0 , i∈A j . (A.2) 93 • When x = 0 and β = γ, equation (A.1) simplifies to w SP (n) = ∑ |n| j=1 n j q SP i (n) i∈A j − 2 ( ∑ N i=1 q SP i (n) ) 2 − |n|K. From equation (A.2), n j q SP i (n) = 1 N ( n j +n j ∑ |n| k=1 n k ¯ c(n k )−Nn j ¯ c(n j ) ) ,i∈A j ,so |n| ∑ j=1 n j q SP i (n) i∈A j = ∑ |n| j=1 n j + ( ∑ |n| j=1 n j ) ∑ |n| k=1 n k ¯ c(n k )−N ∑ |n| j=1 n j ¯ c(n j ) Nβ = N +N ∑ |n| k=1 n k ¯ c(n k )−N ∑ |n| k=1 n k ¯ c(n k ) Nβ = 1 β , and ( ∑ N i=1 q SP i (n) ) 2 = ( ∑ |n| j=1 n j q SP i (n) i∈A j ) 2 = 1 2 . Therefore, w SP (n) = 1 β − β 2 1 β 2 −|n|K = 1 2β −|n|K. (A.3) • Whenx> 0,fromequation(A.2),wecalculatethethreetermsinequation(A.1)asfollows. TermA: ∑ |n| j=1 (1−¯ c(n j )x) ∑ i∈A j q SP i (n) = ∑ |n| j=1 n j (1−¯ c(n j )x)q SP i (n) i∈A j . Since n j (1−¯ c(n j )x)q SP i (n) = [N−(N−1)x]n j (1−¯ c(n j )x) 2 −(1−x)n j (1−¯ c(n j )x) ∑ |n| k=1 n k (1−¯ c(n k )x) βx[N−(N−1)x] , i∈A j ,weobtain |n| ∑ j=1 n j (−¯ c(n j )x)q SP i (n) i∈A j = [N−(N−1)x] ∑ |n| j=1 n j (1−¯ c(n j )x) 2 −(1−x) [ ∑ |n| j=1 n j (1−¯ c(n j )x) ] ∑ |n| k=1 n k (1−¯ c(n k )x) βx[N−(N−1)x] = [N−(N−1)x] ∑ |n| k=1 n k (1−¯ c(n k )x) 2 −(1−x) [ ∑ |n| k=1 n k (1−¯ c(n k )x) ] 2 βx[N−(N−1)x] . TermB: ∑ N i=1 q SP i (n) 2 = ∑ |n| j=1 n j q SP i (n) 2 i∈A j . From n j q SP i (n) 2 = [N−(N−1)x] 2 n j (1−¯ c(n j )x) 2 β 2 x 2 [N−(N−1)x] 2 + (1−x) 2 n j [ ∑ |n| k=1 n k (1−¯ c(n k )x) ] 2 β 2 x 2 [N−(N−1)x] 2 − 2[N−(N−1)x](1−x)n j (1−¯ c(n j )x) ∑ |n| k=1 n k (1−¯ c(n k )x) β 2 x 2 [N−(N−1)x] 2 , i∈A j , 94 weobtain |n| ∑ j=1 n j q SP i (n) 2 i∈A j =− 2[N−(N−1)x](1−x) [ ∑ |n| j=1 n j (1−¯ c(n j )x) ] ∑ |n| k=1 n k (1−¯ c(n k )x) β 2 x 2 [N−(N−1)x] 2 + [N−(N−1)x] 2 ∑ |n| j=1 n j (1−¯ c(n j )x) 2 β 2 x 2 [N−(N−1)x] 2 + (1−x) 2 ( ∑ |n| j=1 n j )[ ∑ |n| k=1 n k (1−¯ c(n k )x) ] 2 β 2 x 2 [N−(N−1)x] 2 =− 2[N−(N−1)x](1−x) [ ∑ |n| k=1 n k (1−¯ c(n k )x) ] 2 β 2 x 2 [N−(N−1)x] 2 + [N−(N−1)x] 2 ∑ |n| k=1 n k (1−¯ c(n k )x) 2 β 2 x 2 [N−(N−1)x] 2 + N(1−x) 2 [ ∑ |n| k=1 n k (1−¯ c(n k )x) ] 2 β 2 x 2 [N−(N−1)x] 2 = [N−(N−1)x] 2 ∑ |n| k=1 n k (1−¯ c(n k )x) 2 −[N−(N−2)x](1−x) [ ∑ |n| k=1 n k (1−¯ c(n k )x) ] 2 β 2 x 2 [N−(N−1)x] 2 . TermC:( ∑ N i=1 q SP i (n)) 2 = ( ∑ |n| j=1 n j q SP i (n) i∈A j ) 2 . From n j q SP i (n) = [N−(N−1)x]n j (1−¯ c(n j )x)−(1−x)n j ∑ |n| k=1 n k (1−¯ c(n k )x) βx[N−(N−1)x] , i∈A j , weobtain |n| ∑ j=1 n j q SP i (n) i∈A j = [N−(N−1)x] ∑ |n| j=1 n j (1−¯ c(n j )x)−(1−x)( ∑ |n| j=1 n j ) ∑ |n| k=1 n k (1−¯ c(n k )x) βx[N−(N−1)x] = [N−(N−1)x] ∑ |n| k=1 n k (1−¯ c(n k )x)−N(1−x) ∑ |n| k=1 n k (1−¯ c(n k )x) βx[N−(N−1)x] = x ∑ |n| k=1 n k (1−¯ c(n k )x) βx[N−(N−1)x] , and ( ∑ |n| j=1 n j q SP i (n) i∈A j ) 2 = x 2 [ ∑ |n| k=1 n k (1− c(n k )x) ] 2 2 x 2 [N−(N−1)x] 2 . 95 CombiningTermsA,BandC,wecalculatew SP (n)asfollows: w SP (n) = [N−(N−1)x] ∑ jnj k=1 n k (1−¯ c(n k )x) 2 −(1−x) [ ∑ jnj k=1 n k (1−¯ c(n k )x) ] 2 βx[N−(N−1)x] −γ x 2 [ ∑ jnj k=1 n k (1−¯ c(n k )x) ] 2 2β 2 x 2 [N−(N−1)x] 2 −(β−γ) [N−(N−1)x] 2 ∑ jnj k=1 n k (1−¯ c(n k )x) 2 −[N−(N−2)x](1−x) [ ∑ jnj k=1 n k (1−¯ c(n k )x) ] 2 2β 2 x 2 [N−(N−1)x] 2 −|n|K = [N−(N−1)x] ∑ jnj k=1 n k (1−¯ c(n k )x) 2 −(1−x) [ ∑ jnj k=1 n k (1−¯ c(n k )x) ] 2 βx[N−(N−1)x] − (1−x)x [ ∑ jnj k=1 n k (1−¯ c(n k )x) ] 2 2βx[N−(N−1)x] 2 − [N−(N−1)x] ∑ jnj k=1 n k (1−¯ c(n k )x) 2 2βx[N−(N−1)x] + [N−(N−2)x](1−x) [ ∑ jnj k=1 n k (1−¯ c(n k )x) ] 2 2βx[N−(N−1)x] 2 −|n|K = [N−(N−1)x] ∑ jnj k=1 n k (1−¯ c(n k )x) 2 −(1−x) [ ∑ jnj k=1 n k (1−¯ c(n k )x) ] 2 2βx[N−(N−1)x] −|n|K = [N−(N−1)x] ∑ jnj k=1 n k (1−cx+cxθ 1n k ) 2 −(1−x) [ ∑ jnj k=1 n k (1−cx+cxθ 1n k ) ] 2 2βx[N−(N−1)x] −|n|K = c 2 0 x(1−x) 2β[N−(N−1)x]    ( N + x 1−x ) jnj ∑ k=1 n k θ 22n k −   jnj ∑ k=1 n k θ 1n k   2 + 2(1−c 0 x) c 0 (1−x) jnj ∑ k=1 n k θ 1n k    + N(1−c 0 x) 2 2β[N−(N−1)x] −|n|K. To simplify suchw SP (n), we introduce functionsg I (n) =nθ 1−n andg II (n) =nθ 2−2n , and param- etersλ = c 2 0 x(1−x) 2[N−(N−1)x] > 0andΛ = N(1−c 0 x) 2 2[N−(N−1)x] . Then, w SP (n) =λ   ( N + x 1−x ) |n| ∑ k=1 g II (n k )−   |n| ∑ k=1 g I (n k )   2 + 2(1−c 0 x) c 0 (1−x) |n| ∑ k=1 g I (n k )   −|n|K+Λ. (A.4) Step2: Wethencomparew SP (s(n))andw SP (n). Whenx = 0 andβ = γ, since|s(n)| = |n|, from equation (A.3), w SP (s(n)) = w SP (n). As this equalityistrueforanyθ,itistrueforθ→ 1 + . 96 Whenx > 0, we considerw SP (s(n))−w SP (n) asθ → 1 + . Recall thatn = {n 1 ,...,n J } where n J −n 1 > 2;s(n) ={s 1 ,...,s J }wheres 1 =n 1 +1,s J =n J −1,ands k =n k fork = 2,...,J−1. Defining∆g I and∆g II asfollows: ∆g I = |s(n)| ∑ k=1 g I (s k )− |n| ∑ k=1 g I (n k ) = ( g I (n 1 +1)+g I (n J −1) ) − ( g I (n 1 )+g I (n J ) ) = (n 1 +1)θ −n 1 +(n J −1)θ 2−n J −n 1 θ 1−n 1 −n J θ 1−n J = (1−θ −1 )θ −n 1 (n J θ n 1 +2−n J −n 1 θ+1+θ −1 +···+θ n 1 +3−n J | {z } n J −n 1 −2 ), and ∆g II = |s(n)| ∑ k=1 g II (s k )− |n| ∑ k=1 g I (n k ) = ( g II (n 1 +1)+g II (n J −1) ) − ( g II (n 1 )+g II (n J ) ) = (n 1 +1)θ −2n 1 +(n J −1)θ 4−2n J −n 1 θ 2−2n 1 −n J θ 2−2n J = (1−θ −1 )(1+θ −1 )θ −2n 1 (n J θ 2n 1 +4−2n J −n 1 θ 2 +1+θ −2 +···+θ 2n 1 +6−2n J | {z } n J −n 1 −2 ), fromequation(A.4),wecanwrite w SP (s(n))−w SP (n) =λ    ( N + x 1−x ) ∆g II −∆g I   |s(n)| ∑ k=1 g I (s k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      . We now show that lim →1 + { w SP (s(n))−w SP (n) } > 0. From the definitions of ∆g I and ∆g II , we have lim →1 + ∆g I 1−θ −1 =n J −n 1 +1+···+1 | {z } n J −n 1 −2 = (2n J −2n 1 −2), and lim →1 + ∆g II 1−θ −1 = (1+1)[n J −n 1 +1+···+1 | {z } n J −n 1 −2 ] = 2(2n J −2n 1 −2). Since|s(n)| =|n|<N = ∑ J j=1 n j = ∑ J j=1 s j ,neithernnors(n)is{1,...,1}. Therefore, |n| ∑ k=1 g I (n k ) = |n| ∑ k=1 n k θ 1−n k = N ∑ l=1 θ 1−n k < N ∑ l=1 θ 1−1 =N 97 and |s(n)| ∑ k=1 g I (s k ) = |s(n)| ∑ k=1 s k θ 1−s k = N ∑ l=1 θ 1−s k < N ∑ l=1 θ 1−1 =N. Hence, lim →1 + w SP (s(n))−w SP (n) 1−θ −1 =λ lim →1 +    ( N + x 1−x ) ∆g II 1−θ −1 − ∆g I 1−θ −1   |s(n)| ∑ k=1 g I (s k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      =λ(2n J −2n 1 −2)    2 ( N + x 1−x ) −   |s(n)| ∑ k=1 g I (s k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      >λ(2n J −2n 1 −2) { 2 ( N + x 1−x ) − [ N +N− 2(1−c 0 x) c 0 (1−x) ]} (n J −n 1 > 2andλ> 0) = c 0 x(2n J −2n 1 −2) β[N−(N−1)x] > 0 (n J −n 1 > 2, 0<x6 1andc 0 > 0). Sinceθ isalwaysgreaterthan1,1−θ −1 > 0. Therefore, lim →1 + { w SP (s(n))−w SP (n) } > 0. Proof of Proposition 3.2.1. The proof contains two parts. First, we show that when θ → 1 + , n SP = S(J SP )∈S. Then,weshowthatJ SP isnon-increasinginK andnon-decreasinginx. PartI:Assumingn SP ={n SP 1 ,...,n SP J },weconsiderthefollowingcases. • IfN = 1,thenn SP ={1} =S(1)∈S,whereJ SP = 1. • IfN = 2, thenn SP = {2} = S(1) ∈ S, whereJ SP = 1, orn SP = {1,1} = S(2) ∈ S, where J SP = 2. • IfN = 3, thenn SP = {3} = S(1) ∈ S, whereJ SP = 1, orn SP = {1,2} = S(2) ∈ S, where J SP = 2,orn SP ={1,1,1} =S(3)∈S,whereJ SP = 3. • IfJ SP = 1,thenn SP ={N} =S(1)∈S. • IfJ SP =N−1,thenn SP ={1,...,1,2} =S(N−1)∈S. • IfJ SP =N,thenn SP ={1,...,1} =S(N)∈S. 98 • Ifn SP J −n SP 1 = 1,thenn SP =S(J SP )∈S. • Otherwise,fromDefinitionA.0.1andLemmaA.0.1,thecorrespondings(n SP )exists,suchthatwhen θ→ 1 + ,w SP (s(n SP ))>w SP (n SP ). Therefore,n SP isnotoptimal—wehaveacontradiction. Inconclusion,whenθ→ 1 + ,n SP =S(J SP )∈S. PartII: We now show that whenθ → 1 + ,J SP is non-increasing inK and non-decreasing inx. Since n SP = S(J SP ) maximizesw SP (n), we haveJ SP = argmax J=1;:::;N w SP (S(J)). To developw SP (S(J)), consid- ering the symmetric property ofS(J), we first approximately write ∑ |S(J)| k=1 g II (n k ) = ∑ J k=1 g II (n k ) ≈ Nθ 2−2 N J and ∑ |S(J)| k=1 g I (n k ) = ∑ J k=1 g I (n k )≈ Nθ 1− N J . Then, for equation (A.4), we do a Taylor series approximationaroundθ = 1. Thus, w SP (S(J)) =λ   ( N + x 1−x )|S(J)| ∑ k=1 g II (n k )−   |S(J)| ∑ k=1 g I (n k )   2 + 2(1−c 0 x) c 0 (1−x) |S(J)| ∑ k=1 g I (n k )   −|S(J)|K +Λ ≈λ [( N + x 1−x ) Nθ 2−2 N J − ( Nθ 1− N J ) 2 + 2(1−c 0 x) c 0 (1−x) Nθ 1− N J ] −JK +Λ(symmetricapproximation) =Nλ [ x 1−x θ 2−2 N J + 2(1−c 0 x) c 0 (1−x) θ 1− N J ] −JK +Λ ≈Nλ [ x 1−x + 2(1−c 0 x) c 0 (1−x) ] −JK +Λ +Nλ [ x 1−x ( 2−2 N J ) + 2(1−c 0 x) c 0 (1−x) ( 1− N J )] (θ−1)(Taylorapproximation) =− [ JK + 2N 2 λ(θ−1) c 0 (1−x)J ] +Nλ 2θ−c 0 x c 0 (1−x) +Λ. LettingJ ∗ = √ 2N 2 (−1) c 0 (1−x)K =N √ c 0 (−1) [ N x −(N−1)]K ,then J SP =              1, ifJ ∗ 6 1 N, ifJ ∗ >N J ∗ , otherwise . It is obvious that J ∗ decreases in K because @J ∗ @K = − N 2 √ c 0 (−1) [ N x −(N−1)]K 3 < 0, and J ∗ increases in x because @J ∗ @K = N 2 2x 2 √ c 0 (−1) K[ N x −(N−1)] 3 > 0. SinceJ SP is non-decreasing inJ ∗ , we have thatJ SP is non- increasinginK andnon-decreasinginx. 99 DefinitionA.0.2. For any structure outcomen ={n 1 ,...,n J } that satisfiesN = ∑ J j=1 n j > 4, 26J 6 N−2,n 1 6···6n J andn J−1 > 1, we can construct more polarized structure outcomes: • If n 1 > 2, we let p a (n) = {p 1 ,...,p J } where p 1 = n 1 − 1, p J = n J + 1, and p k = n k for k = 2,...,J−1. • Otherwise, assumingn 1 =··· =n i = 1<n i+1 6···6n J ,16i6J−2, we – let p b (n) = {p 1 ,...,p J } where p i+1 = 1, p J = n J +n i+1 − 1, and p k = n k for k = 1,...,i,min{i+2,J−1},...,J−1 and – letp c (n) ={p 1 ,...,p i+1 }, wherep i+1 = ∑ J k=i+1 n k , andp k =n k fork = 1,...,i. Note: WLOG, we assume suchp a (n),p b (n) andp c (n) are automatically reordered immediately after beingformed,suchthatp 1 6···6p i+1 (6p i+2 6···6p J ). For example, ifn = {3,4,5,6,7}, thenp a (n) = {2,4,5,6,8}. Ifn = {1,1,4,5,6}, thenp b (n) = {1,1,1,5,9}andp c (n) ={1,1,15}. Lemma A.0.2. Whenθ → +∞, for any arbitrary structure outcomen = {n 1 ,...,n J } satisfyingN = ∑ J j=1 n j > 4, 26J 6N −2,n 1 6···6n J andn J−1 > 1, at least one of the following holds true: a) w SP (p a (n))>w SP (n); b)w SP (p b (n))>w SP (n); or c)w SP (p c (n))>w SP (n). Proof of Lemma A.0.2. When x = 0, we use equation (A.3) to show that w SP (p c (n)) > w SP (p b (n)) = w SP (p a (n)) = w SP (n). When x > 0, we use equation (A.4) to show that lim →+∞ { w SP (p a (n))−w SP (n) } > 0 if n 1 > 2 and that lim →+∞ { w SP (p b (n))−w SP (n) } > 0 or lim →+∞ { w SP (p c (n))−w SP (n) } > 0ifn 1 = 1. Case I: Whenx = 0, since|p c (n)| < |p b (n)| = |p a (n)| = |n|, from equation (A.3),w SP (p c (n)) > w SP (p b (n)) =w SP (p a (n)) =w SP (n). Asthisinequalityistrueforanyθ,itistrueforθ→ +∞. Case II: When x > 0, if n 1 > 2, we consider w SP (p a (n))−w SP (n) as θ → +∞; if n 1 = 1, we considerw SP (p b (n))−w SP (n)andw SP (p c (n))−w SP (n)asθ→ +∞. 100 • Ifn 1 > 2, recall thatn ={n 1 ,...,n J };p a (n) ={p 1 ,...,p J }, wherep 1 = n 1 −1,p J = n J +1 andp k =n k fork = 2,...,J−1. Defining∆g aI and∆g aII as ∆g aI = |p a (n)| ∑ k=1 g I (p k )− |n| ∑ k=1 g I (n k ) = ( g I (n 1 −1)+g I (n J +1) ) − ( g I (n 1 )+g I (n J ) ) = (n 1 −1)θ 2−n 1 +(n J +1)θ −n J −n 1 θ 1−n 1 −n J θ 1−n J =θ 2−n 1 [ (n 1 −1)+(n J +1)θ n 1 −2−n J −n 1 θ −1 −n J θ n 1 −1−n J ] and ∆g aII = |p a (n)| ∑ k=1 g II (p k )− |n| ∑ k=1 g II (n k ) = ( g II (n 1 −1)+g II (n J +1) ) − ( g II (n 1 )+g II (n J ) ) = (n 1 −1)θ 4−2n 1 +(n J +1)θ −2n J −n 1 θ 2−2n 1 −n J θ 2−2n J =θ 4−2n 1 [ (n 1 −1)+(n J +1)θ 2n 1 −4−2n J −n 1 θ −2 −n J θ 2n 1 −2−2n J ] , fromequation(A.4),wecanwrite w SP (p a (n))−w SP (n) =λ    ( N + x 1−x ) ∆g aII −∆g aI   |p a (n)| ∑ k=1 g I (p k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      . We now show that lim →+∞ { w SP (p a (n))−w SP (n) } > 0. From the definitions of ∆g aI and ∆g aII , sincen J +2>n J +1>n J >n 1 ,wehave lim →+∞ ∆g aI θ 2−n 1 = lim →+∞ { θ 2−n 1 θ 2−n 1 [ (n 1 −1)+(n J +1)θ n 1 −2−n J −n 1 θ −1 −n J θ n 1 −1−n J ] } =n 1 −1 and lim →+∞ ∆g aII θ 2−n 1 = lim →+∞ { θ 4−2n 1 θ 2−n 1 [(n 1 −1)+(n J +1)θ 2n 1 −4−2n J −n 1 θ −2 −n J θ 2n 1 −2−2n J ] } = (n 1 −1)θ 2−n 1 . Sincep 1 =n 1 −1> 1,andp J =n J +1>n J >p J−1 =n J−1 >···>p 2 =n 2 >n 1 > 2, |n| ∑ k=1 g I (n k ) = |n| ∑ k=1 n k θ 1−n k = N ∑ l=1 θ 1−n k 6 N ∑ l=1 θ 1−2 =Nθ −1 101 and |p a (n)| ∑ k=1 g I (p k ) = |p a (n)| ∑ k=1 p k θ 1−p k = N ∑ l=1 θ 1−p k <θ 1−1 + N ∑ l=2 θ 1−2 = 1+(N−1)θ −1 . Hence, lim →+∞ w SP (p a (n))−w SP (n) θ 4−2n 1 =λ lim →+∞    ( N + x 1−x ) 1 θ 2−n 1 ∆g aII θ 2−n 1 − 1 θ 2−n 1 ∆g aI θ 2−n 1   |p a (n)| ∑ k=1 g I (p k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      =λ(n 1 −1) lim →+∞    N + x 1−x − 1 θ 2−n 1   |p a (n)| ∑ k=1 g I (p k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      >λ(n 1 −1) lim →+∞ { N + x 1−x − 1 θ 2−n 1 [ 1+Nθ −1 +(N−1)θ −1 − 2(1−c 0 x) c 0 (1−x) ]} (n 1 > 2andλ> 0) =λ(n 1 −1) { N + x 1−x + 1 θ 2−n 1 [ 1− 2(1−c 0 x) c 0 (1−x) ]} = (n 1 −1)c 0 x 2β[N−(N−1)x] [ Nc 0 (1−x)+c 0 x+ 2−c 0 (1+x) θ 2−n 1 ] > 0 (n 1 > 2, 0<x6 1and0<c 0 6 1). Since θ is always greater than 1, θ 4−2n 1 > 0. Therefore, if n 1 > 2, lim →+∞ { w SP (p a (n))−w SP (n) } > 0. • If n 1 = 1, we consider the lower bounds of lim →+∞ { w SP (p b (n))−w SP (n) } and lim →+∞ { w SP (p b (n))−w SP (n) } ,andshowthatunderanycircumstance,atleastoneofthetwolower boundsisnonnegative. – Forp b (n)), recall thatn = {n 1 ,...,n J } where n 1 = ··· = n i = 1 < n i+1 6 ··· 6 n J , 16i6J−2,p b (n) ={p 1 ,...,p J } wherep i+1 = 1,p J =n J +n i+1 −1, andp k =n k for k = 1,...,i,min{i+2,J−1},...,J−1. Defining ∆g bI = |p b (n)| ∑ k=1 g I (p k )− |n| ∑ k=1 g I (n k ) = ( g I (1)+g I (n J +n i+1 −1) ) − ( g I (n i+1 )+g I (n J ) ) = 1+(n J +n i+1 −1)θ 2−n J −n i+1 −n i+1 θ 1−n i+1 −n J θ 1−n J 102 and ∆g bII = |p b (n)| ∑ k=1 g II (p k )− |n| ∑ k=1 g II (n k ) = ( g II (1)+g II (n J +n i+1 −1) ) − ( g II (n i+1 )+g II (n J ) ) = 1+(n J +n i+1 −1)θ 4−2n J −2n i+1 −n i+1 θ 2−2n i+1 −n J θ 2−2n J , fromequation(A.4),wecanwrite w SP (p b (n))−w SP (n) =λ    ( N + x 1−x ) ∆g bII −∆g bI   |p b (n)| ∑ k=1 g I (p k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      . We now get the lower bound of lim →+∞ { w SP (p b (n))−w SP (n) } . From the definitions of ∆g bI and∆g bII ,sincen J >n i+1 > 1,wehave lim →+∞ ∆g bI = lim →+∞ ∆g bII = 1. Sincep J = n J +n i+1 −1 > n J > n J−1 = p J−1 >···> n i+1 > 2 andp i+1 =n i =··· = n 1 =p 1 = 1, |n| ∑ k=1 g I (n k ) = |n| ∑ k=1 n k θ 1−n k = N ∑ l=1 θ 1−n k 6 i ∑ l=1 θ 1−1 + N ∑ l=i+1 θ 1−2 =i+(N−i)θ −1 and |p b (n)| ∑ k=1 g I (p k ) = |p b (n)| ∑ k=1 p k θ 1−p k = N ∑ l=1 θ 1−p k < i+1 ∑ l=1 θ 1−1 + N ∑ l=i+2 θ 1−2 =i+1+(N−i−1)θ −1 . 103 Hence, lim →+∞ { w SP (p b (n))−w SP (n) } =λ lim →+∞    ( N + x 1−x ) ∆g bII −∆g bI   |p b (n)| ∑ k=1 g I (p k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      =λ lim →+∞    N + x 1−x −   |p b (n)| ∑ k=1 g I (p k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      >λ lim →+∞ { N + x 1−x − [ i+1+(N−i−1)θ −1 +i+(N−i)θ −1 − 2(1−c 0 x) c 0 (1−x) ]} (λ> 0) =λ [ N + x 1−x −2i−1+ 2(1−c 0 x) c 0 (1−x) ] . – For p c (n), assume η ∈ [1,J −i] is the number of coalitions with the size of n i+1 in n = {n 1 ,...,n J }:n 1 =··· =n i = 1<n i+1 =··· =n i+ <n i++1 6···6n J ,16i6J−2. Recall thatp c (n) ={p 1 ,...,p i+1 }, wherep i+1 = ∑ J k=i+1 n k andp k = n k fork = 1,...,i. Defining ∆g cI = |p c (n)| ∑ k=1 g I (p k )− |n| ∑ k=1 g I (n k ) =g I ( J ∑ k=i+1 n k ) − J ∑ k=i+1 g I (n k ) = ( J ∑ k=i+1 n k ) θ 1− ∑ J k=i+1 n k − J ∑ k=i+1 n k θ 1−n k =θ 1−n i+1   ( J ∑ k=i+1 n k ) θ − ∑ J k=i+2 n k − J ∑ k=i++1 n k θ n i+1 −n k −ηn i+1   and ∆g cII = |p c (n)| ∑ k=1 g II (p k )− |n| ∑ k=1 g II (n k ) =g II ( J ∑ k=i+1 n k ) − J ∑ k=i+1 g II (n k ) = ( J ∑ k=i+1 n k ) θ 2−2 ∑ J k=i+1 n k − J ∑ k=i+1 n k θ 2−2n k =θ 2−2n i+1   ( J ∑ k=i+1 n k ) θ −2 ∑ J k=i+2 n k − J ∑ k=i++1 n k θ 2n i+1 −2n k −ηn i+1   , 104 fromequation(A.4),wecanwrite w SP (p c (n))−w SP (n) =λ    ( N + x 1−x ) ∆g cII −∆g cI   |p c (n)| ∑ k=1 g I (p k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)      +(J−i−1)K. We now get the lower bound of lim →+∞ w SP (p c (n))−w SP (n) 1−1n i+1 . From the definitions of ∆g cI and ∆g cII ,sincen J >···>n i++1 >n i+1 > 0, lim →+∞ ∆g cI θ 1−n i+1 = lim →+∞ { θ 1−n i+1 θ 1−n i+1 [( J ∑ k=i+1 n k ) θ − ∑ J k=i+2 n k − J ∑ k=i++1 n k θ n i+1 −n k −ηn i+1      =−ηn i+1 and lim →+∞ ∆g cII θ 1−n i+1 = lim →+∞ { θ 2−2n i+1 θ 1−n i+1 [( J ∑ k=i+1 n k ) θ −2 ∑ J k=i+2 n k − J ∑ k=i++1 n k θ 2n i+1 −2n k −ηn i+1      = 0. Sincen J >···>n i+1 andp i =n i =··· =p 1 =n 1 = 1, |n| ∑ k=1 g I (n k ) = |n| ∑ k=1 n k θ 1−n k = N ∑ l=1 θ 1−n k > i ∑ l=1 θ 1−1 + N ∑ l=i+1 θ 1−n J =i+(N−i)θ 1−n J and |p c (n)| ∑ k=1 g I (p k ) = |p c (n)| ∑ k=1 p k θ 1−p k = N ∑ l=1 θ 1−p k = i ∑ l=1 θ 1−1 + N ∑ l=i+1 θ 1−p i+1 =i+(N−i)θ 1−p i+1 . 105 Hence, lim →+∞ w SP (p c (n))−w SP (n) θ 1−1n i+1 = lim →+∞    λ ( N + x 1−x ) ∆g cII θ 1−n i+1 −λ ∆g cI θ 1−n i+1   |p c (n)| ∑ k=1 g I (p k ) + |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)   + (J−i−1)K θ 1−n i+1    = lim →+∞    ληn i+1   |p c (n)| ∑ k=1 g I (s k )+ |n| ∑ k=1 g I (n k )− 2(1−c 0 x) c 0 (1−x)   + (J−i−1)K θ 1−n i+1    > lim →+∞ { ληn i+1 [ i+(N−i)θ 1−p i+1 +i+(N−i)θ 1−n J − 2(1−c 0 x) c 0 (1−x) ] + (J−i−1)K θ 1−n i+1 } (λ> 0) =ληn i+1 [ 2i− 2(1−c 0 x) c 0 (1−x) ] + (J−i−1)K θ 1−n i+1 ( p i+1 = J ∑ k=i+1 n k >n J >n i+1 > 1 ) >ληn i+1 [ 2i− 2(1−c 0 x) c 0 (1−x) ] (i<J−1andK > 0) By now we have obtained the lower bounds of lim →+∞ {w SP (p b (n)) − w SP (n)} and lim →+∞ w SP (p c (n))−w SP (n) 1−1n i+1 : lim →+∞ { w SP (p b (n))−w SP (n) } >λ [ N + x 1−x −2i−1+ 2(1−c 0 x) c 0 (1−x) ] and lim →+∞ w SP (p c (n))−w SP (n) θ 1−1n i+1 >ληn i+1 [ 2i− 2(1−c 0 x) c 0 (1−x) ] . Then, our proposed statement, lim →+∞ { w SP (p b (n))−w SP (n) } > 0 or lim →+∞ { w SP (p c (n))−w SP (n) } > 0,holdstrueinthefollowingtwocases. ♢ For0<x< 1,λ> 0. If2i6 2(1−c 0 x) c 0 (1−x) +N−1+ x 1−x ,then lim →+∞ { w SP (p b (n))−w SP (n) } > 0; and if 2i > 2(1−c 0 x) c 0 (1−x) , then lim →+∞ w SP (p c (n))−w SP (n) 1−1n i+1 > 0. Sinceθ is always greater than 1, θ 1−1n i+1 > 0. Therefore, if 2i > 2(1−c 0 x) c 0 (1−x) , then lim →+∞ { w SP (p c (n))−w SP (n) } > 0. We knowthatatleastoneofthetwoconditions,2i6 2(1−c 0 x) c 0 (1−x) +N−1+ x 1−x and2i> 2(1−c 0 x) c 0 (1−x) ,is alwayssatisfied. Therefore,atleastoneofthetworesults, lim →+∞ { w SP (p b (n))−w SP (n) } > 0 and lim →+∞ { w SP (p c (n))−w SP (n) } > 0,holdstrue. 106 ♢ Forx = 1,since0<c 0 6 1, lim →+∞ { w SP (p b (n))−w SP (n) } >λ [ N + x 1−x −2i−1+ 2(1−c 0 x) c 0 (1−x) ] = c 0 x(2−c 0 x) 2β[N−(N−1)x] > 0. Proof of Proposition 3.2.2. The proof contains two parts. First, we show that when θ → +∞, n SP = PR(J SP ;N−J SP +1)∈PR. Then,weshowthatJ SP isnon-increasinginK andnon-decreasinginx. PartI:Assumingtheoptimaln SP ={n SP 1 ,...,n SP J },weconsiderthefollowingcases. • IfN = 1,thenn SP ={1} =PR(1;1)∈PR,whereJ SP = 1. • IfN = 2,thenn SP ={2} =PR(1;2)∈PR,whereJ SP = 1,orn SP ={1,1} =PR(2;1)∈PR, whereJ SP = 2. • IfN = 3,thenn SP ={3} =PR(1;3)∈PR,whereJ SP = 1,orn SP ={1,2} =PR(2;2)∈PR, whereJ SP = 1,orn SP ={1,1,1} =PR(3;1)∈PR,whereJ SP = 3. • IfJ SP = 1,thenn SP ={N} =PR(1;N)∈PR. • IfJ SP =N−1,thenn SP ={1,...,1,2} =PR(N−1;2)∈PR. • IfJ SP =N,thenn SP ={1,...,1} =PR(N;1)∈PR. • Ifn SP J−1 = 1,thenn SP ={1,...,N−J SP +1} =PR(J SP ;N−J SP +1)∈PR. • Otherwise, if n SP 1 > 2, then from Definition A.0.2 and Lemma A.0.2, the correspondingp a (n SP ) exists, such that whenθ → +∞,w SP (p a (n SP ))>w SP (n SP ). Therefore,n SP is not optimal — we haveacontradiction. • Otherwise, we assume n SP 1 = ··· = n SP i = 1 < n SP i+1 6 ··· 6 n SP J , 1 6 i 6 J SP − 2. From DefinitionA.0.2,thecorrespondingp b (n SP )andp c (n SP )exist. FromLemmaA.0.2,whenθ→ +∞, n SP isnotoptimal,because – if2i6 2(1−c 0 x) c 0 (1−x) +N−1+ x 1−x ,thenw SP (p b (n))>w SP (n)and – if2i> 2(1−c 0 x) c 0 (1−x) ,thenw SP (p c (n))>w SP (n). 107 Inconclusion,whenθ→ +∞,n SP =PR(J SP ;N−J SP +1)∈PR. Part II: We now show that when θ → +∞, J SP is non-increasing in K and non- decreasing in x. Since n SP = PR(J SP ;N − J SP + 1) maximizes w SP (n), we have J SP = argmax J=1;:::;N { lim →+∞ w SP (PR(J;N−J +1)) } . Fromequation(A.4), w SP (PR(J;N−J+1)) =λ {( N + x 1−x ) [ J−1+(N−J +1)θ 2J−2N ] − [ J−1+(N−J +1)θ J−N ] 2 + 2(1−c 0 x) c 0 (1−x) [ J−1+(N−J +1)θ J−N ] } −JK +Λ. Therefore,forJ <N, lim →+∞ w SP (PR(J;N−J +1)) =λ [( N + x 1−x ) (J−1)−(J−1) 2 + 2(1−c 0 x) c 0 (1−x) (J−1) ] −JK +Λ =−λ(J−1) 2 + [ λN +λ 2−c 0 x c 0 (1−x) −K ] (J−1)−K +Λ. LettingJ ∗ = 1 2 [ N + 2−c 0 x c 0 (1−x) − K ] +1 = 1 2 [ N + 2c 0 −c 2 0 x−2K c 2 0 (1−x) − 2NK c 2 0 x ] +1,wehave J SP =              1, ifJ ∗ 6 1 N, ifJ ∗ >N J ∗ , otherwise . SinceJ SP is non-decreasing inJ ∗ , alternatively, we only need to show thatJ ∗ is non-increasing inK and non-decreasing in x. It is obvious that J ∗ decreases in K because @J ∗ @K = − c 2 0 (1−x) − N c 2 0 x < 0. Since @J ∗ @x = 1 2c 2 0 [ c 0 (2−c 0 )−2K (1−x) 2 + 2NK x 2 ] ,toanalyzeJ ∗ andx,wehavethefollowingresults. Whenc 0 (2−c 0 )> 2βK,J ∗ increasesinxsince @J ∗ @x > 0. When2βK >c 0 (2−c 0 ), J ∗ = 1 2 { N + c 0 x(2−c 0 x)−2βK[x+N(1−x)] c 2 0 x(1−x) } +1 < 1 2 { N + c 0 x(2−c 0 x)−c 0 (2−c 0 )[x+N(1−x)] c 2 0 x(1−x) } +1 = 3 2 − N(2−c 0 −c 0 x) 2c 0 x 6 3 2 . 108 As a result, J SP = argmax J=1;2 { lim →+∞ w SP (PR(J;N−J +1)) } . By comparing lim →+∞ w SP (PR(J;N−J +1))whenJ = 1andJ = 2: lim →+∞ w SP (PR(J;N−J +1)) J=2 =λ Nc 0 (1−x)+2−c 0 c 0 (1−x) −2K +Λ = c 0 x[Nc 0 (1−x)+2−c 0 ] [N(1−x)+x] 1 2β −2K +Λ< c 0 x[Nc 0 (1−x)+2−c 0 ] [N(1−x)+x] x 2 −N (1−x) 2 c 0 (2−c 0 )x 2 K−2K +Λ = N(1−x) c 0 x 2−c 0 +x N(1−x)+x x 2 −N (1−x) 2 x 2 K−2K +Λ6−K +Λ = lim →+∞ w SP (PR(J;N−J +1)) J=1 . WehaveJ SP = 1,whichisnon-decreasinginx. Proof of Proposition 3.2.3. CaseI:Whenx = 0,fromequation(A.3),w SP (n) = 1 2 −|n|K. Tomaximize suchw SP (n),weminimize|n|. Therefore,n SP ={N}. Case II: Whenx = 1 andθ → +∞, from Proposition 3.2.2,n SP = PR ( J SP ;N−J SP +1 ) where J SP = argmax J=1;:::;N w SP (PR(J;N−J +1)). Then,equation(A.4)reducesto w SP (PR(J;N−J+1)) = c 2 0 2β [J−1+(N−J+1)θ 2J−2N ]+ c 0 (1−c 0 ) β [J−1+(N−J+1)θ J−N ]−JK+ N(1−c 0 ) 2 2β . Bytakingthesecondderivative ∂ 2 w SP (PR(J;N−J +1)) ∂J 2 = c 0 θ J−N lnθ β [2c 0 θ J−N +2c 0 (N−J+1)θ J−N lnθ+2(1−c 0 )+(1−c 0 )(N−J+1)lnθ], weknowthat @ 2 w SP (PR(J;N−J+1)) @J 2 > 0andw SP (PR(J;N−J+1))isconvexinJ. Therefore,tomaximize suchw SP (PR(J;N−J +1)),J SP = 1orN. ToprovethemainresultsofSection3.3,weneedsomelemmasandadefinition. Lemma A.0.3. Letg(n) = nθ 1−n . Whenθ6 e 2 N ,g(n) is concave onn ∈ [1,N]; whenθ> e 2 ,g(n) is convex onn∈ [1,N]. Proof of Lemma A.0.3. Thisproofisstraightforwardafterobservingthat @ 2 g(n) @n 2 = (nlnθ−2)θ 1−n lnθ. Lemma A.0.4. When θ 6 e 2 N , for any arbitrary structure outcome n = {n 1 ,...,n J } satisfying N = ∑ J j=1 n j > 4,26J6N−2,n 1 6···6n J andn J −n 1 > 2,C EP (s(n))6C EP (n). 109 Notethats(n)isdefinedinDefinitionA.0.1. Proof of Lemma A.0.4. Sincefirmscompetetomaximizetheindividualprofit,P i (q)q i ,whichisconcavein q i ,theequilibriumquantityisobtainedfromtheFOC: q EP = 1 β[N +1−(N−1)x] . (A.5) Fromequation(3.2), C EP (n) =Nc 0 xq EP −c 0 xq EP J ∑ j=1 g(n j )+JK, (A.6) whereg(n) =nθ 1−n . FromLemmaA.0.3,whenθ6e 2 N ,g(n)isconcaveon[1,N]. Hence, C EP (s(n))−C EP (n) =c 0 xq EP (g(n 1 )+g(n J )−g(n 1 +1)−g(n J −1))6 0. DefinitionA.0.3. For any structure outcomen ={n 1 ,...,n J } that satisfiesN = ∑ J j=1 n j > 3, 26J 6 N−1 andn 1 6···6n J , we construct some transformative structure outcomes • ifn J−1 > 1, we lett(n) = {t 1 ,...,t J } wheret J−1 = 1,t J = n J +n J−1 − 1, andt k = n k for k = 1,...,J−2; • otherwise, we lett + (n) ={1,...,1,N−J} andt − (n) ={1.,,,.1,N−J +2}. Note: WLOG, we assume sucht(n),t + (n) andt − (n) are automatically reordered immediately after beingformed,suchthatt 1 6···6t J−1 (6t J 6t J+1 ). For example, if n = {3,4,5,6,7}, then t(n) = {1,3,4,5,12}. If n = {1,1,15}, then t + (n) = {1,1,1,14}andt − (n) ={1,16}. Lemma A.0.5. When θ > e 2 , for any arbitrary structure outcome n = {n 1 ,...,n J } satisfying N = ∑ J j=1 n j > 3,26J6N−1andn 1 6···6n J ,atleastoneofthefollowingholdstrue: a)C EP (t(n))6 C EP (n); b)C EP (t + (n))6C EP (n); and c)C EP (t − (n))6C EP (n). 110 Proof of Lemma A.0.5. FromLemmaA.0.3,g(n)inequation(A.6)isconvexon[1,N]. • Ifn J−1 > 1, C EP (t(n))−C EP (n) =c 0 xq EP (g(n J−1 )+g(n J )−g(1)−g(n J +n J−1 −1))6 0. • Otherwise,n ={1,...,1,N−J+1},t + (n) ={1,...,1,N−J}andt − (n) ={1.,,,.1,N−J+2}. Wethenhave C EP (t + (n))−C EP (n) =c 0 xq EP (g(N−J +1)−g(1)−g(N−J))+K and C EP (t − (n))−C EP (n) =c 0 xq EP (g(N−J +1)+g(1)−g(N−J +2))−K. Since (C EP (t + (n))−C EP (n)) + (C EP (t − (n))−C EP (n)) = c 0 xq EP (2g(N −J + 1)−g(N − J)−g(N−J +2))6 0,weknowthatmin{C EP (t + (n)),C EP (t − (n))}6C EP (n). Proof of Proposition 3.3.1. The proof contains four parts. First, we show that when θ 6 e 2 N , n EP = S(J EP ) ∈ S. Second, we show thatJ EP is non-increasing inK and non-decreasing inx whenθ 6 e 2 N . Third, we show that whenθ> e 2 ,n EP ∈{{1,...,1},{N}}. Fourth, we show that asK decreases andx increases,n EP changesfrom{N}to{1,...,1}forθ>e 2 . PartI:Weneedtoshowthatwhenθ6e 2 N ,n EP =S(J EP )∈S. FollowingLemmaA.0.4,theproofis similartothatofProposition3.2.1,and,therefore,omitted. Part II: We show that when θ 6 e 2 N , J EP is non-increasing in K and non-decreasing in x. Since n EP = S(J EP ) minimizesC EP (n), we haveJ EP = argmin J=1;:::;N C EP (S(J)). Due to the symmetric property ofS(J),C EP (S(J))≈Nc 0 xq EP ( 1−θ 1− N J ) +JK. BytakingthepartialderivativesofC EP (S(J))with respecttoJ, ∂C EP (S(J)) ∂J =K− N 2 J 2 c 0 xq EP θ 1− N J lnθ 111 and ∂ 2 C EP (S(J)) ∂J 2 = ( 2− N J lnθ ) N 2 J 3 c 0 xq EP θ 1− N J lnθ, we find that forJ > 1, @ 2 C EP (S(J)) @J 2 > 0, sinceθ6e 2 N and 2− N J lnθ> 2− 2 J > 0. That is, @C EP (S(J)) @J increasesinJ,orC EP (S(J))isconvexon[1,+∞). Considerthat lim J−→+∞ @C EP (S(J)) @J =K > 0: • If K > N 2 c 0 x 1−N ln [N+1−(N−1)x] or x < (N+1)K (N−1)K+N 2 c 0 1−N ln , then @C EP (S(J)) @J J=1 > 0. Hence, @C EP (S(J)) @J > 0 andC EP (S(J)) increases inJ on [1,+∞). In this case, we haveJ EP = 1, which is non-increasinginK andnon-decreasinginx. • If K 6 N 2 c 0 x 1−N ln [N+1−(N−1)x] or x > (N+1)K (N−1)K+N 2 c 0 1−N ln , then @C EP (S(J)) @J J=1 6 0. Hence, @C EP (S(J)) @J = 0 has a unique solution J ∗ ∈ [1,+∞) which satisfies K c 0 ln [ N+1 x − (N − 1)] = N 2 J ∗2 θ 1− N J ∗ . We notice that the right-hand side is non-increasing in J ∗ since @ @J ∗ N 2 J ∗2 θ 1− N J ∗ = ( N J lnθ−2 ) N 2 J 3 θ 1− N J lnθ6 0 when J > 1 and θ 6 e 2 N . As the left-hand side increases in K anddecreasesinx,J ∗ isnon-increasinginK andnon-decreasinginx. WithsuchJ ∗ , J EP =              1, ifJ ∗ 6 1 N, ifJ ∗ >N J ∗ , otherwise , whichisalsonon-increasinginK andnon-decreasinginx. Part III: We show that whenθ> e 2 ,n EP ∈ {{1,...,1},{N}}. Assumingn EP = {n EP 1 ,...,n EP J }, weconsiderthefollowingcases. • IfN = 1,thenn EP ={1}. • IfN = 2,thenn EP ={2}orn EP ={1,1}. • IfJ EP = 1,thenn EP ={N}. • IfJ EP =N,thenn EP ={1,...,1}. • Otherwise, from Definition A.0.3 and Lemma A.0.5, ifn J−1 > 1, the correspondingt(n EP ) exists, such thatC EP (t(n EP )) 6 C EP (n EP ). Otherwise, the correspondingt + (n EP ) andt − (n EP ) exist, 112 suchthatmin{C EP (t + (n EP )),C EP (t − (n EP ))}6C EP (n EP ). Therefore,n EP isnotoptimal—we haveacontradiction. Inconclusion,whenθ>e 2 ,n EP ∈{{1,...,1},{N}}. Part IV: We show that as K decreases and x increases, C EP ({1,...,1}) > C EP ({N}) at first, and thenC EP ({1,...,1})<C EP ({N}). Fromequation(A.6), C EP ({1,...,1})−C EP ({N}) = (N−1)K−Nc 0 xq EP ( 1−θ 1−N ) . Itisobviousthat, whenK > (<) N N−1 c 0 xq EP (1−θ 1−N ),C EP ({1,...,1})−C EP ({N})> (<) 0, so {N}({1,...,1})isoptimal. Besides, by taking the partial derivative with respect tox, we find thatC EP ({1,...,1})−C EP ({N}) isdecreasinginx: ∂{C EP ({1,...,1})−C EP ({N})} ∂x =− Nc 0 x(1−θ 1−N )(N−1) β[N +1−(N−1)x] 2 − Nc 0 (1−θ 1−N ) β[N +1−(N−1)x] < 0. Proof of Lemma 3.4.1. We consider an arbitrary deviation fromn ∗ :n ∗ ⇀ B n. Sincen≪n ∗ , andn ∗ ⊀ B n ∗ ,fromDefinition3.4.4,n ∗ isinallconsistentsets. Therefore,n ∗ isintheLCS. ToprovethemainresultsofSection3.4,weintroducesomelemmasfirst. LemmaA.0.6. Whenn j is fixed,π(n,n j ) is non-increasing with ∑ |n| k=1 n k θ 1−n k . Proof of Lemma A.0.6. Since Π i (q,n) is concave in q i , firm i’s equilibrium quantity q RP i (n) is obtained fromtheFOCofequation(3.5): q RP i (n) = [N +1−(N−1)x](1−¯ c(n j )x)−(1−x) ∑ |n| k=1 n k (1−¯ c(n k )x) β(1+x)[N +1−(N−1)x] =D+E·θ 1−n j −F |n| ∑ k=1 n k θ 1−n k , i∈A j , (A.7) 113 whereD = (1−c 0 x) [N+1−(N−1)x] ,E = c 0 x (1+x) andF = (1−x)c 0 x (1+x)[N+1−(N−1)x] > 0. Therefore,q RP i (n) is non- increasingwith ∑ |n| k=1 n k θ 1−n k . Fromequations(3.4),(3.5)and(3.6),wehave π(n,n j ) =   1−βq RP i (n)−γ N ∑ l=1;l̸=i q RP l (n)   q RP i (n)−¯ c(n j )xq RP i (n)− K n j = [ 1−¯ c(n j )x−(β−γ)q RP i (n)−γ N ∑ l=1 q RP l (n) ] q RP i (n)− K n j . FromtheFOCofequation(3.5),q RP i (n)satisfies1−¯ c(n j )x−(β−γ)q RP i (n)−γ ∑ N l=1 q RP l (n) =βq RP i (n). Therefore, π(n,n j ) =βq RP i (n) 2 − K n j , i∈A j , (A.8) whichimpliesthatπ(n,n j )isincreasingwithq RP i (n)ifq RP i (n)> 0. Now we show that q RP i (n) > 0. In Section 3.2, we assume c 0 6 ˜ c to make sure firms’ quan- tities and profits are nonnegative. That is, for any given structure outcome n, q RP i (n) > 0; and for any given quantity q > 0, P i (q) > 0. We observe that q RP i (n) > 0 is equivalent to c 0 6 1+x x{1+x−[N+1−(N−1)x] 1−n j +(1−x) ∑ |n| k=1 n k 1−n k} , and P i (q) > 0 is equivalent to c 0 6 1. Therefore, we make ˜ c = min    1, min n;j=1;:::;|n| 1+x x { 1+x−[N +1−(N−1)x]θ 1−n j +(1−x) ∑ |n| k=1 n k θ 1−n k }    . (A.9) Inconclusion,fromourassumptionthatc 0 6 ˜ c,π(n,n j )isincreasingwithq RP i (n)andnon-increasing with ∑ |n| k=1 n k θ 1−n k . Lemma A.0.7. Assumef(x) is convex andg(x) is concave on [a,b), and one of them is strictly convex or concave on(a,b). Then,f(x) andg(x) have at most two intersecting points on[a,b). Proof of Lemma A.0.7. Assume there exist at least three intersecting points forf(x) andg(x): x 1 ,x 2 and x 3 ,wherea6x 1 <x 2 <x 3 <b,x 2 =αx 1 +(1−α)x 3 andα∈ (0,1). Iff(x)isstrictlyconvexon(a,b), thenf(x 2 )<αf(x 1 )+(1−α)f(x 3 ). Fromtheconcavityofg(x),wehaveg(x 2 )>αg(x 1 )+(1−α)g(x 3 ). 114 Sincef(x i ) =g(x i ) fori = 1,2,3, we have a contradiction. A similar contradiction can be shown ifg(x) isstrictlyconcavewhilef(x)isconvex. Notethatinthefollowingtext,weuse“valley”todenotethepointatwhichafunctionattainsaninterior localminimum,and“peak”todenotethepointatwhichthefunctionattainsaninteriorlocalmaximum. Lemma A.0.8. If 1 < θ 6 e 2 N ,AB > 0 andK > 0, thenf(n) = (A +Bθ 1−n ) 2 − K n has at most two local extrema, one valley pointn v and one peak pointn p on [1,N]. Besides, ifn v exists, thenn v > N 2 ; if bothn p andn v exist, thenn p <n v . Proof of Lemma A.0.8. Theproofiscomposedofthreeparts. Wefirstshowthatf(n)hasatmostonevalley pointandonepeakpoint,andtwolocalextremaon[1,N]. Thenweshowthatthepeakpointissmallerthan thevalleypointifbothexist. Finally,weshowthatthereisnovalleypointon [ 1, N 2 ] . PartI:Bylettingg(n) = (A+Bθ 1−n ) 2 andh(n) = K n ,wehavef(n) =g(n)−h(n). Wethenshow thatf ′ (n) = 0org ′ (n) =h ′ (n)hasatmosttwosolutionson[1,N]. Aftertakingthederivativeofg(n)and h(n),wehave • g ′ (n) =−2(A+Bθ 1−n )Bθ 1−n lnθ =−(aX 2 +bX),wherea = 2B 2 θ 2 lnθ> 0,b = 2ABθlnθ> 0,andX =θ −n ∈ [ θ −N , 1 ] ; • h ′ (n) =− K n 2 =−KY 2 ,whereY = 1 n ∈ [ 1 N ,1 ] . Alternatively,weconsiderthenumberofsolutionstosimultaneousequations    −(aX 2 +bX) =g ′ (n) =h ′ (n) =−KY 2 ln 1 X ln =n = 1 Y or          Y 1 =Y 2 Y 1 = √ aX 2 +bX K Y 2 =− ln lnX . BytakingthesecondderivativeofY 1 withrespecttoX, Y ′′ 1 =− (2aX +b) 2 4 ( aX 2 +bX K )3 2 K 2 + a ( aX 2 +bX K )1 2 K =− b 2 4K 2 ( aX 2 +bX K ) − 3 2 6 0, 115 weknowthatY 1 isconcavewithX> 0. Meanwhile,thesecondderivativeofY 2 withrespecttoX is Y ′′ 2 =− 2lnθ (lnX) 3 X 2 − lnθ (lnX) 2 X 2 = lnθ (lnX) 2 X 2 ( 2 ln 1 X −1 ) . Sinceθ6e 2 N , ln 1 X =nlnθ6 2n N . That is,Y ′′ 2 = ln (lnX) 2 X 2 ( 2 ln 1 X −1 ) > ln (lnX) 2 X 2 ( N n −1 ) . Hence,Y 2 is convex withθ −N 6X < 1, and strictly convex withθ −N <X < 1. From Lemma A.0.7, we know that Y 1 andY 2 haveatmosttwointersectingpointsonX ∈ [θ −N ,1). Hence,f ′ (n) = 0 org ′ (n) = h ′ (n) has at most two solutions onn ∈ [1,N]. Therefore,f(n) has at mostonepeakpointn p andonevalleypointn v onn∈ [1,N]. Iff(n)hasatleastthreelocalmaxima,then it has at least two valley points, which is a contradiction. Therefore,f(n) has at most two local extrema on n∈ [1,N]. PartII:Ifbothn p andn v exist,letX p =θ −np andX v =θ −nv . Sincef ′ (n p ) =f ′ (n v ) = 0,X p andX v are intersecting points ofY 1 andY 2 . SinceY 1 is concave andY 2 is convex inX, whenX < min{X p ,X v } andX > max{X p ,X v },Y 1 <Y 2 ;whenmin{X p ,X v }<X < max{X p ,X v },Y 1 >Y 2 . WhenY 1 = √ aX 2 +bX K > (<)Y 2 =− ln lnX = 1 n =Y,aX 2 +bX > (<)KY 2 . Equivalently,g ′ (n)< (>)h ′ (n),orf ′ (n)<(>)0,sof(n)decreases(increases)inn. Tosummarizetheabove, whenn > max{n p ,n v } (orX < min{X p ,X v }), andn < min{n p ,n v } (orX > max{X p ,X v }), f(n)increaseswithn; when min{n p ,n v }<n< max{n p ,n v } (or min{X p ,X v }<X < max{X p ,X v }),f(n) decreases withn. Therefore,n p <n v . PartIII: ConsideringY 1 andY 2 atX =θ − N 2 , we show thatf(n) has no valley point onn∈ [ 1, N 2 ] in thefollowingtwocases. 116 ⋆ IfY 1 6Y 2 atX =θ − N 2 ,wefirstconsiderY ′ 1 andY ′ 2 . Sinceb> 0andK > 0, Y ′ 1 X= − N 2 = √ aθ −N +bθ − N 2 K ·θ N 2 − b 2K √ a −N +b − N 2 K 6 √ aθ −N +bθ − N 2 K ·θ N 2 . From Y 1 | X= − N 2 = √ a −N +b − N 2 K and Y 2 | X= − N 2 = 2 N , we have 2 N > √ a −N +b − N 2 K . Since θ6e 2 N , 1 ln > N 2 . Asaresult, Y ′ 2 X= − N 2 = 2 N · 2 N · 1 lnθ ·θ N 2 > √ aθ −N +bθ − N 2 K · 2 N · N 2 ·θ N 2 = √ aθ −N +bθ − N 2 K ·θ N 2 . Hence,Y ′ 1 6Y ′ 2 atX =θ − N 2 . Then, we use Y ′ 1 | X= − N 2 6 Y ′ 2 | X= − N 2 to show our statement. From the concavity of Y 1 , Y ′ 1 decreases in X ∈ [ θ − N 2 , 1 ] , and from the strict convexity of Y 2 , Y ′ 2 strictly increases in X ∈ [ θ − N 2 , 1 ] . Since Y ′ 1 6 Y ′ 2 at X = θ − N 2 , Y ′ 1 < Y ′ 2 on X ∈ ( θ − N 2 , 1 ] . That is, as X increases on ( θ − N 2 , 1 ] ,Y 1 increases strictly slower thanY 2 . As a result, ifY 1 6Y 2 atX =θ − N 2 ,Y 1 <Y 2 on X ∈ ( θ − N 2 , 1 ] . From Part II we know thatf(n) increases inn∈ [ 1, N 2 ) — there is no valley point on [ 1, N 2 ] . ⋆ IfY 1 >Y 2 atX =θ − N 2 ,since lim X→1 − Y 1 = √ a+b K < lim X→1 − Y 2 = +∞,Y 1 andY 2 haveanoddnumber of intersecting points on X ∈ [ θ − N 2 ,1 ) . Besides, since Y 1 and Y 2 have at most two intersecting points,Y 1 andY 2 haveonlyoneintersectingpointX ∈ [ θ − N 2 ,1 ) . † IfX ∈ [ θ − N 2 , 1 ] : forX < X 6 1 or 16 n6− lnX ln ,Y 1 < Y 2 , andf(n) increases inn; for θ − N 2 6X <X or− lnX ln <n6 N 2 ,Y 1 >Y 2 ,andf(n)decreasesinn. Hence,X ∈ [ θ − N 2 , 1 ] correspondstoapeakpoint—thereisnovalleypointn∈ [ 1, N 2 ] . † IfX ∈ ( 1 ,1),thenY 1 >Y 2 onX ∈ [ θ − N 2 , 1 ] . Correspondingly,f(n)decreasesinn∈ [ 1, N 2 ] —thereisnovalleypointinthisregion. Proof of Proposition 3.4.1. From Lemma 3.4.1, for any arbitrary structure outcome n = {n 1 ,...,n |n| } (assumingn 1 6···6n |n| ),ifwecanshowthatn≪PL(n RP J ),thenthepropositionstatementholdstrue. 117 Todothis,wefirstconstructadeviationsequencefromntoPL(n RP J ). Then,weshowthatcomparedwith thecorrespondingpayoffsinPL(n RP J ),everydeviationalongthissequenceisprofitableforthedefector(s). Thesequencecanbeconstructedbythefollowingsteps. Step1: Iftherearenontrivialcoalitionsinn,amemberofalargestcoalition,A |n| ,defects: ifn |n| > 1,then n⇀ i∈A |n| n ′ ={1,n 1 ,...,n |n| −1},andgotoStep2;else,gotoStep3. Step2: Letn =n ′ ,reordernsuchthatn 1 6···6n |n| ,andgotoStep1. Step3: Allfirmsformtheall-inclusivecoalition:n⇀ N n ′ ={N},andgotoStep4. Step4: Letn =n ′ ,andgotoStep5. Step5: If there is a coalition with more thann RP J members, somen RP J firms defect: ifn |n| > n RP J , then n⇀ B⊂A |n| ;|B|=n RP J n ′ =n\{n |n| }∪{n |n| −n RP J ,n RP J },andgotoStep6;else,gotoStep7. Step6: Letn =n ′ ,reordernsuchthatn 1 6···6n |n| ,andthengotoStep5. Step7: Letn RP =n,thenwehavealeft-polarizedstructureoutcomen RP =PL(n RP J ). FromStep1toStep7,wehaveconstructedadeviationsequencefromntoPL(n RP J ). Intherestofthe proof,weshowthatalldeviationsalongthissequenceareprofitableforthedefectors. Wefirstshowthatthis argument holds true for deviations in Step 1. Then, we use the result of Step 1 to show that the argument alsoholdstruefordeviationsinSteps3and5. Proof for Step 1: In each deviation of Step 1, the defector is a firm in the largest coalition, and her payoff isπ(n,n |n| ) before the deviation. Based on whether firms would obtain undifferentiated payoffs in the final structure outcome,PL(n RP J ), our discussion includes two cases: I) mod ( N,n RP J ) = 0, in which all firms receive the payoffπ ( PL(n RP J ),n RP J ) , and II) mod ( N,n RP J ) = n RP 1 > 0, in which some firms receivethepayoffπ ( PL(n RP J ),n RP 1 ) whileothersreceiveπ ( PL(n RP J ),n RP J ) . CASE I: When mod ( N,n RP J ) = 0, all firms would end up in coalitions with the sizen RP J , so every firm receives the same payoff π ( PL(n RP J ),n RP J ) in PL(n RP J ). Therefore, we only need to show that π(n,n |n| ) 6 π ( PL(n RP J ),n RP J ) . To do this, we first construct a new sequence from n to PL(n |n| ), and show that for members in the largest coalition, their payoffs keep increasing along the sequence: π(n,n |n| ) 6 π(PL(n |n| ),n |n| ). Notice that along the sequence, we keep the size of the largest coali- tion the same. Next, we getπ(PL(n |n| ),n |n| )6 π ( PL(n RP J ),n RP J ) from the definition ofn RP J : n RP J = 118 argmax n J =1;:::;N π(PL(n J ),n J ). Therefore, we haveπ(n,n |n| )6π(PL(n |n| ),n |n| )6π ( PL(n RP J ),n RP J ) . The stepsofconstructingthissequenceandshowingtheincreasingpayoffareasfollows. S1. We “polarize” two coalitions with smaller size difference to two with greater size difference: If there exist n i ,n j ∈ n such that n i ,n j < n |n| and n i +n j > n |n| , let n ′ = n\{n i ,n j }∪{n i +n j − n |n| ,n |n| }, and go to step S2; else, go to step S3. From Lemma A.0.3, g(n) = nθ 1−n is concave on n ∈ [1,N] whenθ 6 e 2 N , son i θ 1−n i +n j θ 1−n j > n |n| θ 1−n |n| + (n i +n j −n |n| )θ 1−(n i +n j −n |n| ) . That is, ∑ |n| k=1 n k θ 1−n k n > ∑ |n ′ | k=1 n k θ 1−n k n ′ . From Lemma A.0.6, π(n,n |n| ) is non-increasing in ∑ |n| k=1 n k θ 1−n k . Therefore,π(n,n |n| ) 6 π(n ′ ,n |n ′ | ) — in this move, the size of the largest coalition remainsthesame,n |n| =n |n ′ | ,andthepayoffofafirminthelargestcoalitionincreasesinn ′ . S2. Letn =n ′ ,andgotostepS1. S3. Wecombinetwosmallercoalitionsintoonelargercoalition: ifthereexistn i ,n j ∈nsuchthatn i ,n j < n |n| and n i +n j 6 n |n| , let n ′ = n\{n i ,n j }∪{n i +n j }, and go to step S2; else, go to step S4. Sincen i θ 1−n i +n j θ 1−n j >n i θ 1−(n i +n j ) +n j θ 1−(n i +n j ) , ∑ |n| k=1 n k θ 1−n k n > ∑ |n ′ | k=1 n k θ 1−n k n ′ . In addition, it follows from Lemma A.0.6 thatπ(n,n |n| ) 6 π(n ′ ,n |n ′ | ) — in this move, the size of the largestcoalitionremainsthesame,n |n| =n |n ′ | ,andthepayoffofafirminthelargestcoalitionincreases inn ′ . S4. Wehavealeft-polarizedstructureoutcome,n =PL(n |n| ). We have shown above that when firms are restructuring (as described in S1) and merging (as described in S3) while keeping the size of the largest coalition unchanged, the payoff of the firms in the largest coalitionincreases. Asaresult,thatpayoffismaximizedintheleft-polarizedstructureoutcome. Therefore, π(n,n |n| )6π(PL(n |n| ),n |n| )6π ( PL(n RP J ),n RP J ) . CASE II: When mod ( N,n RP J ) = n RP 1 > 0, some n RP 1 firms would end up in a coalition of size n RP 1 <n RP J , with the payoffπ ( PL(n RP J ),n RP 1 ) ; and the otherN −n RP 1 firms would end up in coalitions of sizen RP J , with the payoffπ ( PL(n RP J ),n RP J ) . Hence, we need to determine what firms would end up in what coalitions when we construct the sequence from Step 1 to Step 7. Assuming the initial structure outcomen includesn 0 independent firms andN −n 0 nonindependent firms,n 0 = 0,...,N − 2,N, we proposethefollowingrules. 119 Rulei): Ifn 0 >N−n RP 1 , we select arbitraryN−n RP 1 independent firms that would form then RP J -sized coalitionandreceivethepayoffπ ( PL(n RP J ),n RP J ) . Meanwhile,welettheothern 0 − ( N−n RP 1 ) independentfirmsjointheN−n 0 nonindependentfirmstoformthen RP 1 -sizedcoalitionandreceive the payoffπ ( PL(n RP J ),n RP 1 ) . As a result, we haveN −n RP 1 firms ending up in then RP J -sized coalitionandn 0 − ( N−n RP 1 ) +(N−n 0 ) =n RP 1 firmsendingupinthen RP 1 -sizedcoalition. Ruleii): Ifn 0 < N −n RP 1 , afterN −n RP 1 −n 0 iterations of Step 1 and 2, we haveN −n RP 1 −n 0 new independentfirms. Letthesen 0 + ( N−n RP 1 −n 0 ) =N−n RP 1 independentfirmsformthen RP J - sizedcoalition,andlettheremainingN−n 0 − ( N−n RP 1 −n 0 ) =n RP 1 firmsformthen RP 1 -sized coalition. Followingtheaboverules,wediscusswhetherthedeviationinStep1isprofitableifa)thedefectorends upinthen RP J -sizedcoalitionandb)thedefectorendsupinthen RP 1 -sizedcoalition. • CASE II-a: If the defector ends up in the n RP J -sized coalition and receives the payoff π(PL(n RP J ),n RP J ), we need to show π(n,n |n| ) 6 π(PL(n RP J ),n RP J ). The proof is the same as thatinCASEI:π(n,n |n| )6π(PL(n |n| ),n |n| )6π ( PL(n RP J ),n RP J ) . • Ifthedefectorendsupinthen RP 1 -sizedcoalition,wewanttoshowπ(n,n |n| )6π ( PL(n RP J ),n RP 1 ) . – (CASE II-b-1:) If π ( PL(n RP J ),n RP 1 ) > π ( PL(n RP J ),n RP J ) , from the proof in CASE I, we know that π(n,n |n| ) 6 π(PL(n |n| ),n |n| ) 6 π ( PL(n RP J ),n RP J ) . Hence, π(n,n |n| ) 6 π(PL(n RP J ),n RP 1 ). – If π ( PL(n RP J ),n RP 1 ) < π ( PL(n RP J ),n RP J ) , we first claim that n |n| 6 n RP 1 . Otherwise, if n |n| >n RP 1 , assume the currentn includesn ′ 0 > n 0 independent firms. Sincen ′ 0 +n |n| 6N, n 0 6n ′ 0 6N−n |n| <N−n RP 1 —werefertoRuleii). Obviously,thedefectorwouldbecome the (n ′ 0 +1) -th independent firm. Sincen ′ 0 +16 N −n RP 1 , the defector would end up in the n RP J -sizedcoalition—wehaveacontradiction. Next, we discuss the relationship betweenπ(n,n |n| ) andπ ( PL(n RP J ),n RP 1 ) . From equations (A.7) and (A.8), π(n,n |n| ) = β(D + Eθ 1−n |n| − F ∑ |n| k=1 n k θ 1−n k ) 2 − K n |n| , where D = 120 (1−c 0 x) [N+1−(N−1)x] ,E = c 0 x (1+x) > 0 andF = (1−x)c 0 x (1+x)[N+1−(N−1)x] > 0. Sincen 1 6···6n |n| 6 n RP 1 <n RP J ,θ 1−n 1 >···>θ 1−n |n| >θ 1−n RP 1 >θ 1−n RP J . Hence, π(n,n |n| ) =β [ D+Eθ 1−n |n| −F(θ 1−n 1 +···+θ 1−n 1 | {z } n 1 +···+θ 1−n |n| +···+θ 1−n |n| | {z } ) n |n| ] 2 − K n |n| <β [ D+Eθ 1−n |n| −F(θ 1−n RP 1 +···+θ 1−n RP 1 | {z } n RP 1 +θ 1−n RP J +···+θ 1−n RP J | {z } N−n RP 1 ) ] 2 − K n |n| =β { D+Eθ 1−n |n| −F [ n RP 1 θ 1−n RP 1 + ( N−n RP 1 ) θ 1−n RP J ]} 2 − K n |n| . Wealsoknowthat π ( PL ( n RP J ) ,n RP 1 ) =β [ D+Eθ 1−n RP 1 −F ( n RP 1 θ 1−n RP 1 + ( N−n RP 1 ) θ 1−n RP J )] 2 − K n RP 1 and π ( PL ( n RP J ) ,n RP J ) =β [ D+Eθ 1−n RP J −F ( n RP 1 θ 1−n RP 1 + ( N−n RP 1 ) θ 1−n RP J )] 2 − K n RP J . Referring to Lemma A.0.8, let A = √ β { D−F [ n RP 1 θ 1−n RP 1 +(N−n RP 1 )θ 1−n RP J ]} and B = √ βE > 0. We then have π(n,n |n| ) < f(n |n| ), π ( PL(n RP J ),n RP 1 ) = f ( n RP 1 ) and π ( PL(n RP J ),n RP J ) = f ( n RP J ) . Hence,π ( PL(n RP J ),n RP 1 ) < π ( PL(n RP J ),n RP J ) is equiva- lenttof ( n RP 1 ) <f ( n RP J ) . Moreimportantly,ifwecanshowthatf(n |n| )6f ( n RP 1 ) ,wehave π(n,n |n| )<π ( PL(n RP J ),n RP 1 ) . To show thatf(n |n| ) 6 f(n RP 1 ), we need to apply Lemma A.0.8, which requiresA > 0. By letting ˜ ˜ c = min { ˜ c, 1+x x{1+x+(1−x)[n RP 1 1−n RP 1 +(N−n RP 1 ) 1−n RP J ]} } , we have that A > 0 when c 0 6 ˜ ˜ c. We then apply the results of Lemma A.0.8 here: f(n) has at most one valley point n v and one peak pointn p on [1,N]; ifn v exists, thenn v > N 2 ; if bothn p andn v exist, then n p <n v . ∗ Whenf(n)ismonotonicon[1,N],wehavethefollowingcases. (CASEII-b-2:) Iff(n) is increasing on [1,N], sincen |n| 6n RP 1 ,π(n,n |n| )<f(n |n| )6 f ( n RP 1 ) =π ( PL(n RP J ),n RP 1 ) —thedeviationinStep1isprofitableforthedefector. (CASE II-b-3:) Iff(n) is decreasing on [1,N], sincen RP 1 < n RP J , we havef ( n RP 1 ) > f ( n RP J ) ,whichcontradictsourassumptionthatf ( n RP 1 ) <f ( n RP J ) . 121 ∗ When f(n) is has a peak point n p but no valley point on [1,N], we have the following cases. (CASE II-b-4:) Ifn p ∈ [ n RP 1 ,N ] , thenn p 6 n RP 1 < n RP J . Sincef(n) is decreasing on [n p ,N],f ( n RP 1 ) >f ( n RP J ) ,whichcontradictsourassumptionthatf ( n RP 1 ) <f ( n RP J ) . (CASE II-b-5:) If n p ∈ [ 1,n RP 1 ) , then n |n| 6 n RP 1 < n p . Since f(n) is increasing on [1,n p ],f(n |n| )6f ( n RP 1 ) —thedeviationinStep1isprofitableforthedefector. ∗ When f(n) has a valley point n v ∈ ( N 2 ,N] but no peak point on [1,N], we have the followingcases. (CASE II-b-6:) Ifn RP J ∈ [ 1, N 2 ) , thenn RP 1 < n RP J < N 2 < n v . Sincef(n) is decreas- ing on [1,n v ], f ( n RP 1 ) > f ( n RP J ) , which contradicts our assumption that f ( n RP 1 ) < f ( n RP J ) . (CASEII-b-7:) Ifn RP J ∈ [ N 2 ,N ] , then sincef(n) has a valley pointn v but no peak point on( N 2 ,N],n RP J = N 2 orN. Wenoticethatinbothcases,n RP 1 = 0. Thatis,wearebackto CASEI. ∗ Whenf(n) has a peak pointn p ∈ [ 1, N 2 ) , and a valley pointn v ∈ ( N 2 ,N], we have the followingcases. (CASE II-b-8:) Ifn RP J ∈ [ 1, N 2 ) , sincef(n) has a peak pointn p but no valley point on [1, N 2 ), the proof is similar to those in CASE II-b-4 and II-b-5: n p ∈ [ n RP 1 , N 2 ) results in a contradiction,andn p ∈ [ 1,n RP 1 ) impliesthatf(n |n| )6f ( n RP 1 ) . (CASEII-b-9:) Ifn RP J ∈ [ N 2 ,N ] ,theproofisthesameasthatinCASEII-b-7:n RP J = N 2 orN;andwearebacktoCASEIinbothcases. ∗ Whenf(n) has a peak pointn p ∈ [ N 2 ,N ] and a valley pointn v ∈ ( N 2 ,N], we have the followingcases. (CASEII-b-10:) Ifn RP J ∈ [ 1, N 2 ) , sincef(n) is increasing on [1, N 2 )⊂ [1,n p ], the proof issimilartothatinCASEII-b-2:f(n |n| )6f ( n RP 1 ) . (CASE II-b-11:) Ifn RP J ∈ ( N 2 ,N ) , n |n| 6 n RP 1 = mod ( N,n RP J ) = N −n RP J < N 2 . Sincef(n)isincreasingon [ 1, N 2 ] ⊆ [1,n p ],f(n |n| )6f ( n RP 1 ) . (CASE II-b-12:) ifn RP J = N 2 orN, the proof is the same as that in CASE II-b-7: we are backtoCASEIinbothcases. 122 In summary, under the proposition conditions, every deviation in Step 1 is profitable for the defector because π ( n,n |n| ) 6π ( PL(n RP J ),n RP J ) holdstrueforallcases; if mod ( N,n RP J ) = n RP 1 > 0, π(n,n |n| ) 6 π ( PL(n RP J ),n RP 1 ) holds true when π(PL(n RP J ),n RP 1 )>π ( PL(n RP J ),n RP J ) ;orπ ( PL(n RP J ),n RP 1 ) <π ( PL(n RP J ),n RP J ) andc 0 6 ˜ ˜ c. Proof for Step 3: In Step 3,n |n| = 1 andn ={1,...,1}. Therefore, all defectors are in the coalition of sizen |n| and face the same payoffπ(n,n |n| ). From the proof for Step 1, for the firms who would end up in a coalition of sizen RP J , we know thatπ(n,n |n| )6π ( PL(n RP J ),n RP J ) ; for the firms who would end up in a coalition of sizen RP 1 if mod ( N,n RP J ) =n RP 1 > 0, we know thatπ(n,n |n| )6π(PL(n RP J ),n RP 1 ) whenπ ( PL(n RP J ),n RP 1 ) > π ( PL(n RP J ),n RP J ) , or whenπ ( PL(n RP J ),n RP 1 ) < π ( PL(n RP J ),n RP J ) and c 0 6 ˜ ˜ c. Hence,underthepropositionconditions,alldeviationsareprofitableforthedefectors. ProofforStep5: In Step 5, the defectors are from the largest coalition with the payoffπ(n,n |n| ), and would formn RP J -sized coalitions with the payoffπ(PL(n RP J ),n RP J ). From the proof for Step 1, we know thatπ(n,n |n| )6π ( PL(n RP J ),n RP J ) . Hence,alldeviationsareprofitableforthedefectors. Proof of Proposition 3.4.2. The proof contains two parts. First, we show that when θ > e 2 , n RP ∈ {{1,...,1},{N}}. Then, we show that as K decreases and x increases, n RP changes from {N} to {1,...,1}. Part I: We show thatn RP = arg n={1;:::;1};{N} π ( n, N |n| ) . Similar to the proof of Proposition 3.4.1, from Lemma 3.4.1, for any arbitrary structure outcomen = {n 1 ,...,n |n| } (assumingn 1 6 ···6 n |n| ), if we can show thatn≪ arg n={1;:::;1};{N} π ( n, N |n| ) , then the proposition statement holds true. To do this, we first construct a deviation sequence fromn to arg n={1;:::;1};{N} π ( n, N |n| ) . Then, we show that compared with the corresponding payoffs in arg n={1;:::;1};{N} π ( n, N |n| ) , every deviation along this sequence is profitable for the defector(s). Thesequencecanbeconstructedbythefollowingsteps. Step1: Iftherearenontrivialcoalitionsinn,amemberofalargestcoalition,A |n| ,defects: ifn |n| > 1,then n⇀ i∈A |n| n ′ ={1,n 1 ,...,n |n| −1},andgotoStep2;else,gotoStep3. Step2: Letn =n ′ ,reordernsuchthatn 1 6···6n |n| ,andgotoStep1. 123 Step3: Firms choose between being independent and all cooperating together: if π({1,...,1},1) > π({N},N),thenletn RP ={1,...,1};else,n⇀ N n ′ ={N}andletn RP ={N}. We need to show that every deviation in Step 1 is profitable for the defector, and ifπ({1,...,1},1) < π({N},N), the deviation in Step 3 is profitable for all firms. For Step 3, the proof is straightforward. For Step 1, in each deviation, the defector is a firm in the largest coalition, and her payoff is π(n,n |n| ) before the deviation. Therefore, we need to show thatπ(n,n |n| ) 6 max n={1;:::;1};{N} π ( n, N |n| ) . To do this, we first construct a new sequence fromn toPR (⌈ N n |n| ⌉ ;n |n| ) , and show that for members in the largest coalition, their payoffs keep increasing along the sequence π(n,n |n| ) 6 π ( PR (⌈ N n |n| ⌉ ;n |n| ) ,n |n| ) . Notice that along the sequence, we keep the size of the largest coalition the same. Then, we show that π ( PR (⌈ N n |n| ⌉ ;n |n| ) ,n |n| ) 6 max n={1;:::;1};{N} π ( n, N |n| ) . The steps of constructing this sequence and showingtheincreasingpayofffollow. S1. We “balance” two coalitions with a larger size difference into two with a smaller size difference: If there exist n i ,n j ∈ n such that n i + 1 < n j < n |n| , letn ′ = n\{n i ,n j }∪{n i + 1,n j − 1}, and go to step S2; else, go to step S3. From Lemma A.0.3,g(n) = nθ 1−n is convex onn ∈ [1,N] when θ > e 2 ,n i θ 1−n i +n j θ 1−n j > (n i +1)θ 1−(n i +1) +(n j − 1)θ 1−(n j −1) . That is, ∑ |n| k=1 n k θ 1−n k n > ∑ |n ′ | k=1 n k θ 1−n k n ′ . From Lemma A.0.6,π(n,n |n| ) is non-increasing with ∑ |n| k=1 n k θ 1−n k . Therefore, π(n,n |n| )6π(n ′ ,n |n ′ | )—inthismove,thesizeofthelargestcoalitionremainsthesame,n |n| =n |n ′ | , andthepayoffofafirminthelargestcoalitionincreasesinn ′ . S2. Letn =n ′ ,andgotostepS1. S3. Wehavearight-polarizedstructureoutcome,n =PR (⌈ N n |n| ⌉ ;n |n| ) . We have shown above when firms are restructuring (as described in S1) while keeping the size of the largestcoalitionunchanged,thepayoffofthefirmsinthelargestcoalitionincreases. Asaresult,thatpayoff ismaximizedintheright-polarizedstructureoutcome. Thatis,π(n,n |n| )6π ( PR (⌈ N n |n| ⌉ ;n |n| ) ,n |n| ) . Next, we show that π ( PR (⌈ N n |n| ⌉ ;n |n| ) ,n |n| ) 6 max n={1;:::;1};{N} π ( n, N |n| ) . From equations (A.7) and (A.8), π(n,n |n| ) = β ( D+Eθ 1−n |n| −F ∑ |n| k=1 n k θ 1−n k ) 2 − K n |n| , where D = (1−c 0 x) [N+1−(N−1)x] , F = (1−x)c 0 x (1+x)[N+1−(N−1)x] > 0 andE = c 0 x (1+x) > NF. Sincen 1 6 ···6 n |n| ,θ 1−n 1 > ···> θ 1−n |n| . Hence, π ( PR (⌈ N n |n| ⌉ ;n |n| ) ,n |n| ) 6β[D+(E−NF)θ 1−n |n| ] 2 − K n |n| . 124 Wealsoknowthat π({1,...,1},1) =β(D+E−NF) 2 −K and π({N},N) =β[D+(E−NF)θ 1−N ] 2 − K N , Let ˜ f(n) = β[D + (E − NF)θ 1−n ] 2 − K n . Then, π ( PR (⌈ N n |n| ⌉ ;n |n| ) ,n |n| ) 6 ˜ f(n |n| ), π({1,...,1},1) = ˜ f(1),andπ({N},N) = ˜ f(N). From ˜ f(n |n| )− ˜ f(1) =β[(E−NF)(θ 1−n |n| +1)+2D](E−NF)(θ 1−n |n| −1)+ n |n| −1 n |n| K and ˜ f(n |n| )− ˜ f(N) =β[(E−NF)(θ 1−n |n| +θ 1−N )+2D](E−NF) ( θ 1−n |n| −θ 1−N ) − N−n |n| n |n| N K, bylettingg(n) =nθ 1−n ,wehave (N−n |n| )( ˜ f(n |n| )− ˜ f(1))+N(n |n| −1)( ˜ f(n |n| )− ˜ f(N)) = (N−n |n| )β[(E−NF)(θ 1−n |n| +1)+2D](E−NF)(θ 1−n |n| −1) +N(n |n| −1)β[(E−NF)(θ 1−n |n| +θ 1−N )+2D](E−NF) ( θ 1−n |n| −θ 1−N ) 6 (N−n |n| )β[(E−NF)(θ 1−n |n| +1)+2D](E−NF)(θ 1−n |n| −1)+N(n |n| −1)β[(E−NF)(θ 1−n |n| +1) +2D](E−NF) ( θ 1−n |n| −θ 1−N ) (n |n| > 1andE−NF > 0) =β[(E−NF)(θ 1−n |n| +1)+2D](E−NF) [ (N−n |n| )(θ 1−n |n| −1)+N(n |n| −1) ( θ 1−n |n| −θ 1−N )] =β[(E−NF)(θ 1−n |n| +1)+2D](E−NF)(N−1) [ g(n |n| )− ( N−n |n| N−1 g(1)+ n |n| −1 N−1 g(N) )] 6 0 (D> 0,andg(n)isconvexonn∈ [1,N]whenθ>e 2 ). Thatis,atleastoneofthefollowingtwostatementsholdstrue: 1) ˜ f(n |n| )− ˜ f(1)6 0,and2) ˜ f(n |n| )− ˜ f(N)6 0. Therefore,thedeviationinStep1isprofitableforthedefector: π(n,n |n| )6π ( PR (⌈ N n |n| ⌉ ;n |n| ) ,n |n| ) 6 ˜ f(n |n| )6 max{ ˜ f(1), ˜ f(N)} = max n={1;:::;1};{N} π ( n, N |n| ) . 125 Part II: We show that asK decreases andx increases, ˜ f(1) < ˜ f(N) at first, and then ˜ f(1) > ˜ f(N). FromPartI, ˜ f(1)− ˜ f(N) =β(1−θ 1−N )(E−NF) [ 2D+ ( 1+θ 1−N ) (E−NF) ] − N−1 N K. ItisobviousthatwhenK >(<) N N−1 β(1−θ 1−N )(E−NF)[2D+(1+θ 1−N )(E−NF)], ˜ f(1)− ˜ f(N)< (>)0and{1,...,1}({N})isstable. Besides,bytakingthepartialderivativewithrespecttox,weknowthat ˜ f(1)− ˜ f(N)increasesinx: ∂{ ˜ f(1)− ˜ f(N)} ∂x = 2c 0 (1−θ 1−N )(1−c 0 x+c 0 xθ 1−N ) β[N +1−(N−1)x] 2 + 2(N−1)c 0 x ( 1−θ 1−N )( 2−c 0 x+c 0 xθ 1−N ) β[N +1−(N−1)x] 3 > 0. Proof of Proposition 3.4.3. Case I: When x = 0, from equations (A.7) and (A.8), for any i ∈ A j , π(n,n j ) = 1 (N+1) 2 − K n j <π({N},N) = 1 (N+1) 2 − K N . Therefore,wehavethefollowingresults. • {N}isintheLCS:fromanystructureoutcomen̸={N},alldefectorswouldbenefitfromthemove n⇀ N {N}. Thatis,n<{N}. FromLemma3.4.1,{N}isintheLCS. • There is no other structure outcome indirectly dominating {N}: Assume there exists a structure outcomen ′ such that{N}≪n ′ , and assume in the corresponding deviation sequence from{N} to n ′ ,theinitialdefectorsareB. Then,everyfirminB shouldreceiveahigherpayoffinn ′ thanin{N}, butweknowthatallfirmsreceivethehighestpayoffin{N}—wehaveacontradiction. • {N} is uniquely in the LCS: Assuming there exists another structure outcome m ̸= {N} in the LCS, we consider the deviation m⇀ N {N}. Since there is no other structure outcome indirectly dominating{N}, and{N} is in the LCS, we just need to compare every defector’s payoff inm and in{N}. Since all defectors receive higher payoffs from this deviation, this move cannot be deterred. Therefore,misnotintheLCS. CaseII:Whenx = 1,fromequations(A.7)and(A.8),foranyi∈A j ,π(n,n j ) = (1−c 0 +c 0 1−n j ) 2 4 − K n j , whichdependsonlyonn j . Next,wecheckwhetherwehavesimilarresultsforx = 1tothoseforx = 0. 126 • Isn rp (N) in the LCS? From any structure outcomen̸=n rp (N), we construct a deviation sequence fromnton rp (N),andshowthatcomparedwiththecorrespondingpayoffsinn rp (N),everydeviation is profitable for the defectors. The steps of constructing this sequence and showing the increasing payoffareasfollows. Step1: LetM =N,andgotoStep2. Step2: Firms form ann rp J (M)-sized coalition if possible: if there existn j 1 ,...,n j l ∈ n, such that n j k ̸=n rp J (M),k = 1,...,landn rp J (M)6 ∑ l k=1 n j k 6M,thenn⇀ B⊆∪ l k=1 A j k ;|B|=n rp J (M) n ′ and go to Step 3; else, go to Step 4. Since n rp J (M) = argmax b=1;:::;M { (1− c(n)) 2 4 − K n } = argmax n;16n j k 6M π(n,n j k ), for the defector i ∈ A j k ∪ B, π(n,n j k ) < π(n ′ ,n rp J (M)) = π(n rp (N),n rp J (M)). That is, compared with the corresponding payoff inn rp (N), this move isprofitableforthedefectors. Step3: Letn =n ′ ,reordernsuchthatn 1 6···6n |n| ,andgotoStep2. Step4: Ifmod(M,n rp J (M))> 0,thenletM = mod(M,n rp J (M)),andgotoStep2;else,gotoStep 5. Step5: Wehaven =n rp (N). Therefore, for an arbitrary structure outcomen ̸= n rp (N), we can construct a sequence fromn to n rp (N), with every step profitable for the defectors. That is, n≪n rp (N). From Lemma 3.4.1, n rp (N)isintheLCS. • Isn rp (N) uniquely in the LCS? Assume there exists another structure outcomem̸=n rp (N) in the LCS. – Ifthereexistn j 1 ,...,n j l ∈m,suchthatn j k ̸=n rp J (N),k = 1,...,landn rp J (N)6 ∑ l k=1 n j k , weconsiderthedeviationm⇀ B⊆∪ l k=1 A j k ;|B|=n rp J (N) m ′ . Foranystructureoutcomem ′′ which indirectly dominatesm ′ , the firms inB would still stay in a coalition inm ′′ , since 1) the firms inB would not receive better payoffs from further deviations (n rp J (N) = argmax n;16n j k 6N π(n,n j k )); and 2) no deviations of firms that are not inB would affect the payoff of firms that are inB. Therefore, for the defectori∈A j k ∪B,π(m,n j k )<π(m ′ ,n rp J (N)) =π(m ′′ ,n rp J (N)). That is,thismovecannotbedeterred—misnotintheLCSandwehaveacontradiction. 127 – Otherwise, if mod(N,n rp J (N)) = 0, thenm = n rp (N) and we have a contradiction; else, let M 0 =N andM = mod(N,n rp J (N)),andconsiderthefollowingcases. (a) If there exist n j 1 ,...,n j l ∈ m, such that n j k ̸= n rp J (M), k = 1,...,l and n rp J (M) 6 ∑ l k=1 n j k 6M, we consider the deviationm⇀ B⊆∪ l k=1 A j k ;|B|=n rp J (M) m ′ . For any struc- tureoutcomem ′′ whichindirectly dominatesm ′ , thefirmsinB wouldstillstayin acoali- tion in m ′′ , since 1) the M 0 −M firms who are in the n rp J (M 0 )-sized coalitions would not move; 2) given the fact claimed in 1), the firms inB would not receive better payoffs from further deviations (n rp J (M) = argmax n;16n j k 6M π(n,n j k )); and 3) no deviations of firms that are not inB would affect the payoff of firms that are inB. Therefore, for the defector i∈ A j k ∪B,π(m,n j k ) < π(m ′ ,n rp J (M)) = π(m ′′ ,n rp J (M)). That is, this move cannot bedeterred—misnotintheLCSandwehaveacontradiction. (b) Otherwise,ifmod(M,n rp J (M)) = 0,thenm =n rp (M)andwehaveacontradiction;else, letM 0 =M andM = mod(M,n rp J (M)),anditerativelyreconsiderthecases(a)and(b). After several iterations, we show that there always exists a contradiction: eitherm is not in the LCS orm =n rp (N). Hence,n rp (N)isuniquelyintheLCS. Proof of Proposition 3.4.4. Theproofcontainstwoparts. First,weshowthattheLCShasauniqueelement. Then,weshowthatn RP J isnon-decreasinginK. Part I: We first show that the LCS contains a unique element which is proposed in the proposition statement. From Proposition 3.4.3, we know that when x = 1, if f(n) = (1− c(n)) 2 4 − K n has a unique maximumpoint,theLCShasasingleelement. FromPropositions3.4.1and3.4.2,PL(n RP J ),wheren RP J = argmax n=1;:::;N f(n), is in the LCS forθ6e 2 N , and argmax n={1;:::;1};{N} π ( n, N |n| ) is in the LCS forθ>e 2 . As a result, theseidentifiedstablestructureoutcomesareuniquelyintheLCS. PartII: We then show that whenθ6e 2 N ,n RP J is non-decreasing inK. Referring to Lemma A.0.8, by lettingA = 1−c 0 2 √ > 0 andB = c 0 2 √ > 0, we apply the Lemma proof and result here. In the Lemma proof, we know thatY 1 = √ aX 2 +bX K is concave forX> 0,Y 2 =− ln lnX is convex onX ∈ [ θ −N ,1 ) , andY 1 and Y 2 haveatmosttwointersectingpointsonX ∈ [ θ −N ,1 ) . Since lim X→1 − Y 1 = √ a+b K < lim X→1 − Y 2 = +∞,as 128 we increaseK,Y 1 andY 2 have one (whenK < K), two (whenK 6 K < K), one (whenK = K) and zero(whenK >K)intersectingpoint(s). • When K < K, we use X ∈ [ θ −N ,1 ) to denote the unique intersecting point, corresponding to n∈ (0,N]. Then,Y 2 <Y 1 whenX <X orn>n, andY 2 >Y 1 whenX >X orn<n (note that from the Lemma proof, whenY 1 > (<)Y 2 ,f(n) decreases (increases) inn). Therefore,f(n) only hasapeakpointnandn RP J = max{1,n}. We now show thatn increases inK. To do this, we considerK u andK v which satisfyK u <K v < K, and the corresponding intersecting points under them, X u andX v : Y 1 (X u )| Ku = Y 2 (X u ) and Y 1 (X v )| Kv = Y 2 (X v ). Applying the fact Y 2 > Y 1 when X > X, on K u and X u , we know that Y 2 (X)>Y 1 (X)| Ku whenX >X u . SinceY 1 decreasesinK,Y 2 (X v ) =Y 1 (X v )| Kv <Y 1 (X v )| Ku . Therefore,X v >X u . Thatis,X decreasesinK,ornincreasesinK. Asn RP J isnon-decreasinginn,itisnon-decreasinginK. • WhenK 6K <K, we useX andX to denote the two intersecting points, corresponding ton and n,respectively. ThenY 2 <Y 1 whenX <X <X (n<n<n),andY 2 >Y 1 whenX >X (n<n) andX < X (n > n). Therefore, n is a peak point off(n) andn is a valley point. We then have n RP J = argmax n J =max{1;n};N f(n J ). We first need to show that max{1,n} is non-decreasing inK. The proof is similar to the case when K <K,and,therefore,omitted. Next,weshowthatn RP J increasesinK. Todothis,weconsiderK u andK v whichsatisfyK6K u < K v <K, and the corresponding peak points,n u andn v (note thatn u <n v ). Considerf(n)| Ku and f(n)| Kv : sinceK u < K v ,f(n)| Ku > f(n)| Kv forn∈ (0,N]. Asn u is the peak point off(n)| Ku , we have f(n u )| Ku > f(n v )| Ku > f(n v )| Kv . Besides, since f(n v )| Ku −f(n v )| Kv = Kv−Ku nv > Kv−Ku N =f(N)| Ku −f(N)| Kv , f(n u )| Ku −f(n v )| Kv = (f(n u )| Ku −f(n v )| Ku )+(f(n v )| Ku −f(n v )| Kv )>f(N)| Ku −f(N)| Kv . Hence, if f(n u )| Ku < f(N)| Ku , then f(n v )| Kv < f(N)| Kv . That is, n RP J is non-decreasing in max{1,n}. Therefore,n RP J isalsonon-decreasinginK. 129 • When K = K, we use X 0 ∈ [ θ −N ,1 ) to denote the unique intersecting point, corresponding to n 0 ∈ (0,N]. Then,Y 2 > Y 1 forX ̸= X 0 (n̸= n 0 ) andY 2 = Y 1 atX = X 0 (n = n 0 ). Therefore, f(n)isincreasinginn,andn RP J =N whichisnon-decreasinginK. • WhenK >K, since there is no intersecting point,Y 2 >Y 1 . Therefore,f(n) is increasing inn, and n RP J =N whichisnon-decreasinginK. Proof of Proposition 3.5.1. Similartoequations(A.5)and(A.7),theequilibriumquantitiesafterintroducing taxesorsubsidiestoEPandRPareobtainedfromtheFOC: ˆ q EP = 1−t β[N +1−(N−1)x] (A.10) and ˆ q RP i (n) = [N +1−(N−1)x](1−t−¯ c(n j )x)−(1−x) ∑ |n| k=1 n k (1−t−¯ c(n k )x) β(1+x)[N +1−(N−1)x] , i∈A j . (A.11) • When ^ n RP = n SP = {N}, from equations (A.2) and (A.11), q SP i (n SP ) = 1− c(N)x [N−(N−1)x] and ˆ q RP i (^ n RP ) = 1−t− c(N)x [N+1−(N−1)x] . Since^ n EP = n SP , to make ˆ w EP ( ^ n EP ) = W ( ^ q EP ,^ n EP ) = w SP ( n SP ) = W ( q SP ( n SP ) ,n SP ) , we only need to make ˆ q EP = q SP i ( n SP ) , which requires 1−t [N+1−(N−1)x] = 1− c(N)x [N−(N−1)x] or t = ¯ c(N)x− 1− c(N)x N−(N−1)x . Itisobviousthat lim N−→+∞ t =c 0 x. Since ^ n RP = n SP , to make ˆ w RP ( ^ n RP ) = W ( ^ q RP ( ^ n RP ) ,^ n RP ) = w SP ( n SP ) = W ( q SP ( n SP ) ,n SP ) , we only need to make ˆ q RP i ( ^ n RP ) = q SP i ( n SP ) , which requires 1−t− c(N)x [N+1−(N−1)x] = 1− c(N)x [N−(N−1)x] ort =− 1− c(N)x N−(N−1)x . Itisobviousthat lim N−→+∞ t = 0. • When ^ n RP = n SP = {1,...,1}, from equations (A.2) and (A.11), q SP i (n SP ) = 1 [N−(N−1)x] and ˆ q RP i (^ n RP ) = 1−t [N+1−(N−1)x] = ˆ q EP . Since^ n EP = ^ n RP = n SP , to make ˆ w EP (^ n EP ) = ˆ w RP (^ n RP ) = w SP (n SP ), we only need to make ˆ q EP = ˆ q RP i (^ n RP ) =q SP i (n SP ),whichrequirest =− 1 N−(N−1)x . Itisobviousthat lim N−→+∞ t = 0. 130 Proof of Proposition 3.5.2. FromtheproofofProposition3.2.1,weknowthatwhenθ→ 1 + , J SP =              1, ifJ SP∗ 6 1 N, ifJ SP∗ >N J SP∗ , otherwise , whereJ SP∗ =N √ c 0 x(−1) [N−(N−1)x]K . From Proposition 3.3.1, we know that when θ 6 e 2 N , ^ n EP = S( ˆ J EP ) ∈ S. Hence, ˆ J EP = argmin J=1;:::;N ˆ C EP (S(J)) = Nc 0 xˆ q EP ( 1−θ 1− N J ) +JK. When θ → 1 + , we do the Taylor series approxi- mationfor ˆ C EP (S(J)): ˆ C EP (S(J))≈JK + N 2 c 0 xˆ q EP (θ−1) J −Nc 0 xˆ q EP (θ−1). Letting ˆ J EP∗ = √ N 2 c 0 x^ q EP (−1) K =N √ c 0 x(−1)(1−t) [N+1−(N−1)x]K ,then ˆ J EP =              1,if ˆ J EP∗ 6 1 N,if ˆ J EP∗ >N ˆ J EP∗ ,otherwise . SinceJ SP = ˆ J EP orn SP =^ n EP requirest =− 1 N−(N−1)x , and, ingeneral,q SP i ( n SP ) = ˆ q EP requires t = N+1−(N−1)x N−(N−1)x ¯ c ( n SP j ) x− 1 N−(N−1)x >− 1 N−(N−1)x , it is difficult to makew SP ( n SP ) = ˆ w EP ( ^ n EP ) . However, asN → +∞,J SP = ˆ J EP orn SP = ^ n EP also holds for finitet; andq SP i ( n SP ) = ˆ q EP requires t = ¯ c ( n SP j ) x. Therefore, as N → +∞, n SP = ^ n EP ; especially, when t = ¯ c ( n SP j ) x, w SP ( n SP ) = ˆ w EP ( ^ n EP ) . Proof of Proposition 3.5.3. From the proof of Proposition 3.3.1, we knowthat{N} ({1,...,1})is optimal inEPif K > (<) N N−1 c 0 x(1−θ 1−N ) β[N +1−(N−1)x] := ¯ K EP . FromtheproofofProposition3.4.2,weknowthat{N}({1,...,1})isstableinRPif K > (<) N N−1 c 0 x(1−θ 1−N ) β[N +1−(N−1)x] 2−c 0 x(1−θ 1−N ) 2+(N−1)(1−x) := ¯ K RP < ¯ K EP . 131 Recalling thatq EP = q RP i ({1,...,1}) = 1 [N+1−(N−1)x] , andq RP i ({N}) = 1− c(N)x [N+1−(N−1)x] , we con- siderthefollowingthreecases: 1)n EP =n RP ={1,...,1};2)n EP =n RP ={N};3)n EP ={1,...,1} andn RP ={N}. 1. WhenK6 ¯ K RP ,n EP =n RP ={1,...,1}. FromequationA.1,wehave w EP ( n EP ) =Nq EP − β 2 N[N−(N−1)x]q EP2 −NK and w RP ( n RP ) =Nq RP i ({1,...,1})− β 2 N[N−(N−1)x]q RP i ({1,...,1}) 2 −NK. Sinceq EP =q RP i ({1,...,1}),w EP ( n EP ) =w RP ( n RP ) . 2. WhenK > ¯ K EP ,n EP =n RP ={N}. Wehave w EP ( n EP ) =N(1−¯ c(N)x)q EP − β 2 N[N−(N−1)x]q EP2 −K and w RP ( n RP ) =N(1−¯ c(N)x)q RP i ({N})− β 2 N[N−(N−1)x]q RP i ({N}) 2 −K. If we let h(q) = N(1− ¯ c(N)x)q − 2 N[N − (N − 1)x]q 2 − K, then w EP ( n EP ) = h ( q EP ) and w RP ( n RP ) = h(q RP i ({N})). Assuming the maximum point of h(q) is q ∗ , we have q ∗ = 1− c(N)x [N−(N−1)x] . Since q RP i ({N}) < q ∗ and q RP i ({N}) < q EP , h ( q EP ) 6 h(q RP i ({N})) if and only if q ∗ − q RP i ({N}) 6 q EP − q ∗ . That is, w EP ( n EP ) 6 w RP ( n RP ) if and only if 26 [N +2−(N−1)x]c 0 (1−θ 1−N )x = Ω. 3. When ¯ K RP <K6 ¯ K EP ,n EP ={1,...,1}andn RP ={N}. Wehave w EP ( n EP ) =Nq EP − β 2 N[N−(N−1)x]q EP2 −NK and w RP ( n RP ) =N(1−¯ c(N)x)q RP i ({N})− β 2 N[N−(N−1)x]q RP i ({N}) 2 −K. 132 Itisobviousthatw EP ( n EP ) 6w RP ( n RP ) ifandonlyif K> N N−1 [N +2−(N−1)x][2−c 0 x(1−θ 1−N )]c 0 x(1−θ 1−N ) 2β[N +1−(N−1)x] 2 := ¯ K > ¯ K RP . Wefurthercompare ¯ K and ¯ K EP : ¯ K EP > ¯ K ifandonlyifΩ> 2. Therefore, - If Ω > 2, we have ¯ K RP < ¯ K 6 ¯ K EP : when ¯ K RP < K < ¯ K, w EP ( n EP ) > w RP ( n RP ) ; when ¯ K6K6 ¯ K EP ,w EP ( n EP ) 6w RP ( n RP ) . - If Ω< 2, we have ¯ K RP < ¯ K EP < ¯ K: w EP ( n EP ) >w RP ( n RP ) since ¯ K RP <K6 ¯ K EP < ¯ K. Inconclusion, • WhenK6 ¯ K RP ,w EP ( n EP ) =w RP ( n RP ) . • WhenΩ> 2andK > ¯ K,w EP ( n EP ) 6w RP ( n RP ) . • WhenΩ< 2andK > ¯ K RP ,orΩ> 2and ¯ K RP <K < ¯ K,w EP ( n EP ) >w RP ( n RP ) . 133 AppendixB TechnicalAppendixtoChapter4 ThroughoutallproofsofChapter4,wewillusethefollowingnotations. K 1 = 6βµ 1 ν 1 , K 2 = 3 2−(2−σ)x (2−x) 2 βµ 1 ν 1 , K 3 = 2 (2σ−x)x (2−x) 2 βµ 1 ν 1 , K 4 = 2 2+x 2−x βµ 1 ν 1 , K 5 = 4 2−σx (2−x) 2 βµ 1 ν 1 , K 6 = 3 2 βµ 1 ν 1 , K 7 = 1 2 (2+x) 2 ρβµ 1 ν 1 , K 8 = 3 8 (2+x) 2 ρβµ 1 ν 1 , K 9 = 3 10 [ 4+(2+x) 2 ρ ] βµ 1 ν 1 , K 10 = 12 5 2−(2−σ)x (2−x) 2 βµ 1 ν 1 , K 11 = 3 16 [ 4+(2+x) 2 ρ ] βµ 1 ν 1 , K 12 = 1 2 [ 8 2−σx (2−x) 2 +(2+x) 2 ρ ] βµ 1 ν 1 , K 13 = 1 4 [ 8 2−σx (2−x) 2 +(2+x) 2 ρ ] βµ 1 ν 1 ,and K 14 = 1 2 [ (2+x) 2 ρ−4 (2σ−x)x (2−x) 2 ] βµ 1 ν 1 . ItiseasytoshowthatK 1 >K 4 >K 2 >{K 3 ,K 6 },K 12 >K 9 >K 11 andK 12 >K 13 . Proof of Proposition 4.1.1. We consider Table 4.1. Since K > 0, C {12}{3} < C {1}{2}{3} ; since λ > 1, C {12}{3} < C {1}{23} = C {13}{2} . Hence, {1}{2}{3}, {1}{23} and {13}{2} are not optimal. By comparingC {123} andC {12}{3} ,wefindK EP = 2µ 1 +µ 3 . Proof of Proposition 4.1.2. We consider Table 4.1. Because K > 0, W {12}{3} > W {1}{2}{3} . We now show that W {12}{3} > W {1}{23} (= W {13}{2} ). Let f(x) = x + 2x−(+1) 2(1−x) x 2 ; it is easy to verify that W {12}{3} >W {1}{23} ⇔f(c 1 )+ 1 2 (1+x)c 3 [2−(λ+1)c 3 ]> 0. Because 0<c 3 < 1 , 1 2 (1+x)c 3 [2− (λ+1)c 3 ]> 0;because0<c 1 < 1 ,wejustneedtoshowthatf(x)> 0for0<x< 1 . • Ifx> +1 2 ,wehave 2x−(+1) 2(1−x) > 0. Asf(0) = 0,wehavef(x)> 0forx> 0. • Ifx< +1 2 ,wehave 2x−(+1) 2(1−x) < 0. Asf(0) = 0 andf ( 1 ) = −1 2(1−x) 2 > 0,wehavef(x)> 0for 0<x< 1 . 134 Finally,bycomparingW {123} andW {12}{3} ,wefindK SP = (2+x) 2 ( 1 1+x + 2 ) βµ 1 ν 1 . Proof of Proposition 4.1.3. The proof includes five steps: first, we compare firms’ payoff preferences; sec- ond, we check the conditions under which {123} is stable; third, we check the conditions under which {12}{3} is stable; fourth, we check the conditions under which {1}{23} or {13}{2} is stable; fifth, we checktheconditionsunderwhich{1}{2}{3}isstable. Step1. AccordingtoTable4.2,wehave • π {123} A >π {13}{2} A ,π {12}{3} A >π {13}{2} A ,π {12}{3} A >π {1}{2}{3} A ,andπ {1}{23} A >π {1}{2}{3} A ; • π {123} B >π {1}{23} B ,π {12}{3} B >π {1}{23} B ,π {12}{3} B >π {1}{2}{3} B ,andπ {13}{2} B >π {1}{2}{3} B ; • π {123} C >π {1}{23} C =π {13}{2} C andπ {12}{3} C =π {1}{2}{3} C ;and • π {123} A > π {12}{3} A ⇔ π {123} B > π {12}{3} B ⇔ K > K 1 ,π {123} A > π {1}{23} A ⇔ π {123} B > π {13}{2} B ⇔ K > K 2 , π {12}{3} A > π {1}{23} A ⇔ π {12}{3} B > π {13}{2} B ⇔ K > K 3 , π {13}{2} A > π {1}{23} A ⇔ π {1}{23} B > π {13}{2} B ⇔ K > K 4 , π {13}{2} A > π {1}{2}{3} A ⇔ π {1}{23} B > π {1}{2}{3} B ⇔ K > K 5 , π {123} A > π {1}{2}{3} A ⇔ π {123} B > π {1}{2}{3} B ⇔ K > K 6 , π {1}{23} C = π {13}{2} C > π {12}{3} C = π {1}{2}{3} C ⇔K >K 7 ,andπ {123} C >π {12}{3} C =π {1}{2}{3} C ⇔K >K 8 . Step2. Wechecktheconditionsunderwhich{123}isstable. • When K < K 1 (i.e., π {12}{3} A > π {123} A and π {12}{3} B > π {123} B ), {123} cannot be stable as {123}⇀ {A;B} {12}{3}cannotbedeterred. • When K < K 8 (i.e., π {12}{3} C > π {123} C ), {123} cannot be stable because {123} ⇀ {C} {12}{3} cannotbedeterred. • When K > max{K 1 ,K 8 }, we have π {123} A > π {12}{3} A > { π {13}{2} A ,π {1}{2}{3} A ,π {1}{23} A } , π {123} B > π {12}{3} B > { π {1}{23} B ,π {1}{2}{3} B ,π {13}{2} B } , π {123} C > π {12}{3} C = π {1}{2}{3} C , and π {123} C >π {1}{23} C =π {13}{2} C . Thatis,{123}isParetodominantandthusstable. Inaword,{123}isstableforK > max{K 1 ,K 8 }. Step3. Wechecktheconditionsunderwhich{12}{3}isstable. 135 • When K > K 1 (i.e., π {123} A > π {12}{3} A and π {123} B > π {12}{3} B ) and K > K 8 (i.e., π {123} C > π {12}{3} C ), {12}{3} cannot be stable since all three firms would rather form {123} than stay in {12}{3}. • WhenK <K 8 , we haveπ {12}{3} C =π {1}{2}{3} C >π {123} C >π {1}{23} C =π {13}{2} C . {12}{3} is stable because it is firmC’s most preferred recycling structures, and firmsA andB prefer it to{1}{2}{3} (π {12}{3} A >π {1}{2}{3} A andπ {12}{3} B >π {1}{2}{3} B ). • WhenK <K 1 ,weconsidertwocases: – When K 3 < K < K 1 , we have π {12}{3} A > π {123} A > π {13}{2} A , π {12}{3} A > π {1}{23} A > π {1}{2}{3} A ,π {12}{3} B > π {123} B > π {1}{23} B , andπ {12}{3} B > π {13}{2} B > π {1}{2}{3} B . {12}{3} is stablebecauseitisthemostpreferredrecyclingstructureforfirmsAandB. – WhenK <K 3 , asK 3 <K 2 <K 4 <K 1 , we haveπ {1}{23} A >π {12}{3} A >π {123} A >π {13}{2} A , π {1}{23} A > π {12}{3} A > π {1}{2}{3} A ,π {13}{2} B > π {12}{3} B > π {123} B > π {1}{23} B , andπ {13}{2} B > π {12}{3} B >π {1}{2}{3} B . ThefollowingEPCFhas{12}{3}asitsabsorbingstate:{123}⇀ {A;B} {12}{3};{1}{23}⇀ {B} {1}{2}{3};{13}{2}⇀ {A} {1}{2}{3};{12}{3}⇀ {A;B} {12}{3}; {1}{2}{3}⇀ {A;B} {12}{3}. Thus,{12}{3}isstableforK < max{K 1 ,K 8 }. Recallingthat{123}isstableforK > max{K 1 ,K 8 }, weconcludethatK RP = max{K 1 ,K 8 }. Step4. Wechecktheconditionsunderwhich{1}{23}or{13}{2}isstable. Wefirstdiscussthestability of{1}{23};theanalysisof{13}{2}issymmetric. • WhenK > K 3 (i.e.,π {12}{3} A > π {1}{23} A ), sinceπ {12}{3} B > π {1}{23} B ,{1}{23} cannot be stable as {1}{23}⇀ {A;B} {12}{3}cannotbedeterred. • When K < K 3 , as K 3 < K 2 < K 4 < K 1 , we have π {1}{23} A > π {12}{3} A > π {123} A > π {13}{2} A , π {1}{23} A > π {12}{3} A > π {1}{2}{3} A , π {13}{2} B > π {12}{3} B > π {123} B > π {1}{23} B , and π {13}{2} B > π {12}{3} B >π {1}{2}{3} B . Further,weconsidertwocases: – WhenK <K 5 (i.e.,π {1}{2}{3} B >π {1}{23} B ) orK <K 7 (i.e.,π {1}{2}{3} C >π {1}{23} C ),{1}{23} cannotbestableas{1}{23}⇀ {B} {1}{2}{3}⇀ {A;B} {12}{3}cannotbedeterred. 136 – WhenK >K 5 (i.e.,π {1}{23} B >π {1}{2}{3} B ) andK >K 7 (i.e.,π {1}{23} C >π {1}{2}{3} C ), the fol- lowing EPCF has {1}{23} as its absorbing state: {123} ⇀ {A} {1}{23}; {1}{23} ⇀ {B;C} {1}{23}; {13}{2} ⇀ {A} {1}{2}{3}; {12}{3} ⇀ {A} {1}{2}{3}; {1}{2}{3} ⇀ {B;C} {1}{23}. NowwediscusstheconditionsthatK 3 >K 7 andK 3 >K 5 . WeknowthatK 3 >K 7 ⇔ρ< 4 (2−x)x (4−x 2 ) 2 andK 3 > K 5 ⇔ 2σ− 2 √ σ 2 −1 < x < 2σ + 2 √ σ 2 −1. Recalling that 0 < x < min { 1, 4 −2 } , we need 2σ−2 √ σ 2 −1< min { 1, 4 −2 } < 2σ +2 √ σ 2 −1 to have effective range ofx forK 3 >K 5 . As 2σ +2 √ σ 2 −1 > 1, 2σ−2 √ σ 2 −1 < 1⇔ σ > 5 4 , and 2σ−2 √ σ 2 −1 < 4 −2⇔ σ < √ 2, we just need 5 4 <σ < √ 2. To summarize, ifρ< 4 (2−x)x (4−x 2 ) 2 , 5 4 <σ < √ 2 andx> 2σ−2 √ σ 2 −1, then{1}{23} or{13}{2}isstableformax{K 5 ,K 7 }<K <K 3 . Step 5. We check the conditions under which {1}{2}{3} is stable. As π {12}{3} A > π {1}{2}{3} A and π {12}{3} B >π {1}{2}{3} B ,{1}{2}{3}cannotbestableas{1}{2}{3}⇀ {A;B} {12}{3}cannotbedeterred. Proof of Proposition 4.2.1. The proof includes six steps: first, we compare firms’ payoff preferences; sec- ond, we check the conditions under which {123} is stable; third, we check the conditions under which {12}{3} is stable and show thatK RP < K RP ; fourth, we check the conditions under which{1}{23} is stable and showthat ˇ K RP < ˆ K RP ; fifth, we check the conditions under which{13}{2} is stable; sixth, we checktheconditionsunderwhich{1}{2}{3}isstable. Step1. AccordingtoTable4.2,wehavethefollowingresults: • For firmA, it is obvious thatπ {12}{3} A > π {13}{2} A , π {12}{3} A > π {1}{2}{3} A , π {123} A > π {13}{2} A , and π {1}{23} A > π {1}{2}{3} A . Besides, it is easy to derive thatπ {123} A > π {12}{3} A ⇔ K > K 1 ,π {123} A > π {1}{23} A ⇔ K > K 2 , π {12}{3} A > π {1}{23} A ⇔ K > K 3 , π {13}{2} A > π {1}{23} A ⇔ K > K 4 , π {13}{2} A >π {1}{2}{3} A ⇔K >K 5 ,andπ {123} A >π {1}{2}{3} A ⇔K >K 6 . • ForfirmB,itisobviousthatπ {12}{3} B >π {1}{2}{3} B andπ {123} B >π {1}{23} B . Besides,itiseasytoderive thatπ {13}{2} B >π {12}{3} B ⇔ρ< 4 (2−x)x (4−x 2 ) 2 ,π {1}{23} B >π {13}{2} B ⇔K >K 4 ,π {123} B >π {12}{3} B ⇔ K > K 9 , π {123} B > π {13}{2} B ⇔ K > K 10 , π {123} B > π {1}{2}{3} B ⇔ K > K 11 , π {1}{23} B > π {12}{3} B ⇔K >K 12 ,π {1}{23} B >π {1}{2}{3} B ⇔K >K 13 ,andπ {13}{2} B >π {1}{2}{3} B ⇔K >K 14 . Step2. Wechecktheconditionsunderwhich{123}isstable. 137 • WhenK >K 1 (i.e.,π {123} A >π {12}{3} A ),then{123}isthemostpreferredrecyclingstructureforfirm A. Note thatK > K 1 > K 2 > K 6 indicates thatπ {123} A > π {1}{23} A andπ {123} A > π {1}{2}{3} A , and recallthatπ {123} B >π {1}{23} B . Wethendiscusstwocases. – When K > K 11 (i.e., π {123} B > π {1}{2}{3} B ), the following EPCF has {123} as its absorb- ing state: {123} ⇀ {A;B} {123}; {1}{23} ⇀ {A;B} {123}; {13}{2} ⇀ {A} {1}{2}{3}; {12}{3}⇀ {A} {1}{2}{3};{1}{2}{3}⇀ {A;B} {123}. – When K < K 11 (i.e., π {1}{2}{3} B > π {123} B ), {123} cannot be stable since {123} ⇀ {A;B} {1}{2}{3}cannotbedeterred. • WhenK 6 <K <K 1 ,thenπ {12}{3} A >π {123} A > { π {1}{2}{3} A ,π {13}{2} A } . Wediscusstwocases: – When K > K 9 (i.e., π {123} B > π {12}{3} B ), we have π {123} B > π {1}{2}{3} B (as K 9 > K 11 ) and π {123} B > π {1}{23} B . The following EPCF has {123} as its absorbing state: {123} ⇀ {A;B} {123};{1}{23} ⇀ {B} {1}{2}{3};{13}{2} ⇀ {A} {1}{2}{3};{12}{3} ⇀ {B} {1}{2}{3}; {1}{2}{3}⇀ {A;B} {123}. – WhenK <K 9 (i.e.,π {12}{3} B >π {123} B ),{123}cannotbestableasbothfirmswouldratherstay in{12}{3}thanin{123}. • WhenK <K 6 (i.e.,π {1}{2}{3} A >π {123} A ), sinceπ {1}{23} A >π {123} A (asK 6 <K 2 ),{123} cannot be stable: thedefection{123}⇀ {A} {1}{23}cannotbedeterred. Insummary,{123}isstableforK > max{K 1 ,K 11 }andformax{K 6 ,K 9 }<K <K 1 . Step3. Wechecktheconditionsunderwhich{12}{3}isstable. Ifρ> 4 (2−x)x (4−x 2 ) 2 ,wehaveπ {12}{3} B >π {13}{2} B . • When K > K 12 (i.e., π {1}{23} B > π {12}{3} B ), then {12}{3} cannot be stable since the defection {12}{3}⇀ {B} {1}{23}cannotbedeterred. • WhenK 9 < K < K 12 , thenπ {123} B > π {12}{3} B > { π {1}{23} B ,π {1}{2}{3} B ,π {13}{2} B } . We consider twocases: – WhenK >K 1 (i.e.,π {123} A >π {12}{3} A ),then{12}{3}cannotbestablesincebothfirmswould ratherstayin{123}thanin{12}{3}. 138 – When K < K 1 (i.e., π {12}{3} A > π {123} A ), we have π {12}{3} A > π {13}{2} A and π {12}{3} A > π {1}{2}{3} A . The following EPCF has{12}{3} as its absorbing state: {123} ⇀ {A} {1}{23}; {1}{23} ⇀ {B} {1}{2}{3}; {13}{2} ⇀ {A} {1}{2}{3}; {12}{3} ⇀ {A;B} {12}{3}; {1}{2}{3}⇀ {A;B} {12}{3}. • When K < K 9 (i.e., π {12}{3} B > π {123} B ), as π {12}{3} B > π {1}{2}{3} B , π {12}{3} B > π {13}{2} B (as ρ > 4 (2−x)x (4−x 2 ) 2 ), and π {12}{3} B > π {1}{23} B (as K 9 < K 12 ), {12}{3} is the most preferred recy- cling structure for firm B. The following EPCF has{12}{3} as its absorbing state: {123} ⇀ {B} {1}{23}; {1}{23} ⇀ {B} {1}{2}{3}; {13}{2} ⇀ {B} {1}{2}{3}; {12}{3} ⇀ {A;B} {12}{3} {1}{2}{3}⇀ {A;B} {12}{3}. Ifρ< 4 (2−x)x (4−x 2 ) 2 ,wehaveπ {13}{2} B >π {12}{3} B . • When K > K 12 (i.e., π {1}{23} B > π {12}{3} B ), {12}{3} cannot be stable since the defection {12}{3}⇀ B {1}{23}cannotbedeterred. • When K < K 12 , then π {13}{2} B > π {12}{3} B > { π {1}{23} B ,π {1}{2}{3} B } . Moreover, we have π {12}{3} A > π {13}{2} A , π {12}{3} A > π {1}{2}{3} A , and π {12}{3} A > π {123} A (as ρ < 4 (2−x)x (4−x 2 ) 2 < 4 3x 2 −(12−2)x+8 (4−x 2 ) 2 ,K 12 <K 1 ). ThefollowingEPCFhas{12}{3}asitsabsorbingstate:{123}⇀ {A} {1}{23}; {1}{23} ⇀ {B} {1}{2}{3}; {13}{2} ⇀ {A} {1}{2}{3}; {12}{3} ⇀ {A;B} {12}{3}; {1}{2}{3}⇀ {A;B} {12}{3}. In summary, ifρ > 4 (2−x)x (4−x 2 ) 2 ,{12}{3} is stable forK 9 < K < min{K 1 ,K 12 } and forK < K 9 ; if ρ< 4 (2−x)x (4−x 2 ) 2 ,{12}{3}isstableforK <K 12 . Recallingthat{123}isstableforK > max{K 1 ,K 11 }and for max{K 6 ,K 9 }<K <K 1 , we concludeK RP andK RP as follows, from which it is easy to verify that K RP <K RP :                              K RP =K 6 andK RP =K 12 , ifρ< 1 (2+x) 2 K RP =K 9 andK RP =K 12 , if 1 (2+x) 2 <ρ< 4 3x 2 −(12−2)x+8 (4−x 2 ) 2 K RP =K 9 andK RP =K 1 , if4 3x 2 −(12−2)x+8 (4−x 2 ) 2 <ρ< 16 (2+x) 2 K RP =K 1 andK RP =K 9 , if 16 (2+x) 2 <ρ< 28 (2+x) 2 K RP =K 11 andK RP =K 9 , ifρ> 28 (2+x) 2 . (B.1) 139 Step4. Wechecktheconditionsunderwhich{1}{23}isstable. • WhenK > K 2 (i.e.,π {123} A > π {1}{23} A ), sinceπ {123} B > π {1}{23} B ,{1}{23} cannot be stable: both firmswouldratherstayin{123}thanin{1}{23}. • WhenK 3 <K <K 2 ,wehaveπ {12}{3} A >π {1}{23} A >π {123} A >π {13}{2} A andπ {12}{3} A >π {1}{23} A > π {1}{2}{3} A . Weconsidertwocases. – When K > K 12 , as K 2 < K 4 , K 12 > K 13 and K 12 > K 9 > K 11 , we have { π {13}{2} B ,π {123} B } > π {1}{23} B > { π {1}{2}{3} B ,π {12}{3} B } . The following EPCF has{1}{23} as its absorbing state: {123} ⇀ {A} {1}{23}; {1}{23} ⇀ {B} {1}{23}; {13}{2} ⇀ {A} {1}{2}{3};{12}{3}⇀ {B} {1}{2}{3};{1}{2}{3}⇀ {B} {1}{23}. – WhenK < K 12 (i.e.,π {12}{3} B > π {1}{23} B ),{1}{23} cannot be stable since both firms would ratherstayin{12}{3}thanin{1}{23}. • WhenK < K 3 , asK 3 < K 2 < K 4 < K 1 , we haveπ {1}{23} A > π {12}{3} A > π {123} A > π {13}{2} A and π {1}{23} A > π {12}{3} A > π {1}{2}{3} A —{1}{23} is the most preferred recycling structure for firmA. Wethenconsidertwocases. – WhenK > K 13 (i.e.,π {1}{23} B > π {1}{2}{3} B ), the following EPCF has{1}{23} as its absorb- ing state: {123} ⇀ {A} {1}{23}; {1}{23} ⇀ {B} {1}{23}; {13}{2} ⇀ {A} {1}{2}{3}; {12}{3}⇀ {A} {1}{2}{3};{1}{2}{3}⇀ {B} {1}{23}. – When K < K 13 (i.e., π {1}{2}{3} B > π {1}{23} B ), {1}{23} cannot be stable as the defection of {1}{23}⇀ {B} {1}{2}{3}cannotbedeterred. In summary,{1}{23} is stable for max{K 3 ,K 12 }<K <K 2 and forK 13 <K <K 3 . AsK 2 >K 3 , K 2 >K 12 ⇔ρ< 2 (7−6)x−2 (4−x 2 ) 2 andK 3 >K 13 ⇔ρ< 8 −x 2 +3x−2 (4−x 2 ) 2 , to have effective range for{1}{23} to 140 be stable, we needρ< max { 2 (7−6)x−2 (4−x 2 ) 2 ,8 −x 2 +3x−2 (4−x 2 ) 2 } . Considering thatK 12 >K 13 andK 3 >K 12 ⇔ ρ< 4 −x 2 +4x−4 (4−x 2 ) 2 ,weconclude ˇ K RP and ˆ K RP asfollows,fromwhichitiseasytoverifythat ˇ K RP < ˆ K RP :                              ˇ K RP =K 13 and ˆ K RP =K 2 , ifρ< 4 −x 2 +4x−4 (4−x 2 ) 2 ˇ K RP =K 12 and ˆ K RP =K 2 , or ˇ K RP =K 13 and ˆ K RP =K 3 , if4 −x 2 +4x−4 (4−x 2 ) 2 <ρ< min { 2 (7−6)x−2 (4−x 2 ) 2 ,8 −x 2 +3x−2 (4−x 2 ) 2 } ˇ K RP =K 12 and ˆ K RP =K 2 , ifρ> min { 2 (7−6)x−2 (4−x 2 ) 2 ,8 −x 2 +3x−2 (4−x 2 ) 2 } andx< 5+6− √ 25 2 +60−60 8 ˇ K RP =K 13 and ˆ K RP =K 3 , ifρ> min { 2 (7−6)x−2 (4−x 2 ) 2 ,8 −x 2 +3x−2 (4−x 2 ) 2 } andx> 5+6− √ 25 2 +60−60 8 . (B.2) Step5. Wechecktheconditionsunderwhich{13}{2}isstable. • Ifρ > 4 (2−x)x (4−x 2 ) 2 (i.e.,π {12}{3} B > π {13}{2} B ), then asπ {12}{3} A > π {13}{2} A ,{13}{2} cannot be stable: bothfirmswouldratherstayin{12}{3}thanin{13}{2}. • IfK > K 10 (i.e., π {123} B > π {13}{2} B ), then asπ {123} A > π {13}{2} A ,{13}{2} cannot be stable: both firmswouldratherstayin{123}thanin{13}{2}. • Ifρ < 4 (2−x)x (4−x 2 ) 2 andK < K 10 , then{13}{2} is the most preferred recycling structure for firmB: π {13}{2} B >π {12}{3} B >π {1}{2}{3} B andπ {13}{2} B >π {123} B >π {1}{23} B . Weconsidertwocases: – WhenK < K 5 , we haveπ {1}{23} A > π {1}{2}{3} A > π {13}{2} A . {13}{2} cannot be stable as the defection{13}{2}⇀ {A} {1}{2}{3}cannotbedeterred. – WhenK > K 5 (i.e., π {13}{2} A > π {1}{2}{3} A ), the following EPCF has{13}{2} as its absorb- ing state: {123} ⇀ {B} {1}{23}; {1}{23} ⇀ {B} {1}{2}{3}; {13}{2} ⇀ {A;B} {13}{2}; {12}{3}⇀ {B} {1}{2}{3};{1}{2}{3}⇀ {A;B} {13}{2}. In summary,{13}{2} is stable whenρ< 4 (2−x)x (4−x 2 ) 2 andK 5 <K <K 10 . We now check the condition ofK 5 <K 10 ⇔x> 2 4−3 . Recallingthat 0<x< min{1, 4 −2},weneed 2 4−3 < 4 −2⇔σ< 3 2 and 2 4−3 < 1⇔ σ > 5 4 to have effective range ofx forK 5 < K 10 . Hence, ifρ< 4 (2−x)x (4−x 2 ) 2 , 5 4 <σ < 3 2 and x> 2 4−3 ,{13}{2}isstableforK 5 <K <K 10 . 141 Step 6. We check the conditions under which {1}{2}{3} is stable. As π {12}{3} B > π {1}{2}{3} B and π {12}{3} B > π {1}{2}{3} B ,{1}{2}{3} cannot be stable since both firms would rather stay in{12}{3} than in {1}{2}{3}. Proof of Proposition 4.3.1. ThetotalrecyclingcostintheEPandsocialwelfareintheSPareshowinTable B.1below. TableB.1: TotalRecyclingCostandSPSocialWelfareunderEachCoalitionStructure(FourProducts) CoalitionStructure TotalRecyclingCost SPSocialWelfare {1234} λc 1 q 1 +λc 1 q 2 +λc 3 q 3 +λc 3 q 4 +K (1−c 1 ) 2 +(1−c 3 ) 2 + −K {12}{34} c 1 q 1 +c 1 q 2 +c 3 q 3 +c 3 q 4 +2K (1−c 1 ) 2 +(1−c 3 ) 2 + −2K {13}{24} λc 1 q 1 +λc 1 q 2 +λc 3 q 3 +λc 3 q 4 +2K (1−c 1 ) 2 +(1−c 3 ) 2 + −2K {14}{23} λc 1 q 1 +λc 1 q 2 +λc 3 q 3 +λc 3 q 4 +2K (1−c 1 ) 2 +(1−c 3 ) 2 + −2K {1}{234} c 1 q 1 +λc 1 q 2 +λc 3 q 3 +λc 3 q 4 +2K (1−c 1 ) 2 +(1−c 1 ) 2 −2 (1−c 1 )(1−c 1 ) 2( 2 − 2 ) + (1−c 3 ) 2 + −2K {134}{2} λc 1 q 1 +c 1 q 2 +λc 3 q 3 +λc 3 q 4 +2K (1−c 1 ) 2 +(1−c 1 ) 2 −2 (1−c 1 )(1−c 1 ) 2( 2 − 2 ) + (1−c 3 ) 2 + −2K {124}{3} λc 1 q 1 +λc 1 q 2 +c 3 q 3 +λc 3 q 4 +2K (1−c 3 ) 2 +(1−c 3 ) 2 −2 (1−c 3 )(1−c 3 ) 2( 2 − 2 ) + (1−c 1 ) 2 + −2K {123}{4} λc 1 q 1 +λc 1 q 2 +λc 3 q 3 +c 3 q 4 +2K (1−c 3 ) 2 +(1−c 3 ) 2 −2 (1−c 3 )(1−c 3 ) 2( 2 − 2 ) + (1−c 1 ) 2 + −2K {1}{2}{34} c 1 q 1 +c 1 q 2 +c 3 q 3 +c 3 q 4 +3K (1−c 1 ) 2 +(1−c 3 ) 2 + −3K {1}{24}{3} c 1 q 1 +λc 1 q 2 +c 3 q 3 +λc 3 q 4 +3K (1−c 1 ) 2 +(1−c 1 ) 2 −2 (1−c 1 )(1−c 1 ) 2( 2 − 2 ) + (1−c 3 ) 2 +(1−c 3 ) 2 −2 (1−c 3 )(1−c 3 ) 2( 2 − 2 ) −3K {1}{23}{4} c 1 q 1 +λc 1 q 2 +λc 3 q 3 +c 3 q 4 +3K (1−c 1 ) 2 +(1−c 1 ) 2 −2 (1−c 1 )(1−c 1 ) 2( 2 − 2 ) + (1−c 3 ) 2 +(1−c 3 ) 2 −2 (1−c 3 )(1−c 3 ) 2( 2 − 2 ) −3K {14}{2}{3} λc 1 q 1 +c 1 q 2 +c 3 q 3 +λc 3 q 4 +3K (1−c 1 ) 2 +(1−c 1 ) 2 −2 (1−c 1 )(1−c 1 ) 2( 2 − 2 ) + (1−c 3 ) 2 +(1−c 3 ) 2 −2 (1−c 3 )(1−c 3 ) 2( 2 − 2 ) −3K {13}{2}{4} λc 1 q 1 +c 1 q 2 +λc 3 q 3 +c 3 q 4 +3K (1−c 1 ) 2 +(1−c 1 ) 2 −2 (1−c 1 )(1−c 1 ) 2( 2 − 2 ) + (1−c 3 ) 2 +(1−c 3 ) 2 −2 (1−c 3 )(1−c 3 ) 2( 2 − 2 ) −3K {12}{3}{4} c 1 q 1 +c 1 q 2 +c 3 q 3 +c 3 q 4 +3K (1−c 1 ) 2 +(1−c 3 ) 2 + −3K {1}{2}{3}{4} c 1 q 1 +c 1 q 2 +c 3 q 3 +c 3 q 4 +4K (1−c 1 ) 2 +(1−c 3 ) 2 + −4K Withoutlossofgenerality,weassumethatc 1 >c 3 . Theproofforthecasewithc 3 >c 1 issymmetric. 142 We first show the EP part. Since λ > 1, C {12}{34} < C {124}{3} = C {4}{124} 6 C {1}{234} = C {134}{2} < C {13}{24} = C {14}{23} and C {1}{2}{34} = C {12}{3}{4} < C {1}{24}{3} = C {1}{23}{4} = C {14}{2}{3} = C {13}{2}{4} . SinceK > 0,C {12}{34} < C {1}{2}{34} = C {12}{3}{4} < C {1}{2}{3}{4} . By comparingC {1234} andC {12}{34} ,wefindK ′EP = 2µ 1 + 4 2+x µ 3 . We then show the SP part. Since λ > 1, W {13}{24} = W {14}{23} < W {12}{34} . Since K > 0, W {12}{34} > W {1}{2}{34} = W {12}{3}{4} > W {1}{2}{3}{4} . AsW {1}{234} = W {134}{2} , W {124}{3} = W {123}{4} , and W {1}{24}{3} = W {1}{23}{4} = W {14}{2}{3} = W {13}{2}{4} , we only need to show that {124}{3},{1}{234} and{1}{24}{3} are not optimal. Letf(x) = x + 2x−(+1) 2(1−x) x 2 ; it is easy to verify that f(c 1 ) +c 3 [2− (λ + 1)c 3 ] > 0 ⇔ W {12}{34} > W {1}{234} , f(c 3 ) +c 1 [2− (λ + 1)c 1 ] > 0 ⇔ W {12}{34} > W {124}{3} , and f(c 1 ) +f(c 3 ) > 0 ⇔ W {1}{2}{34} > W {1}{24}{3} . As c 1 ,c 3 ∈ ( 0, 1 ) , Therefore,c 1 [2−(λ+1)c 1 ]> 0 andc 3 [2−(λ+1)c 3 ]> 0. To show our statement, we just need to show thatf(x)> 0for0<x< 1 ,whichisshownintheproofofProposition4.1.3. Finally,bycomparingW {1234} andW {12}{34} ,wefindK ′SP = (2+x) 2 1+x (1+ρ)βµ 1 ν 1 . 143 AppendixC TechnicalAppendixtoChapter5 Let 1−c 1 = M, 1−c 3 = N, 1−λc 1 = P, 1−λc 3 = Q, = y, = x, and M P = ω. Throughout all proofsofChapter5,wewillusethesenotations. Proof of Proposition 5.1.1. We compare the following social welfare generated under each recycling struc- ture. W {123} = 2(β−2κ)P 2 +8κPQ+(β +γ−4κ)Q 2 2[(β−2κ)(β +γ−4κ)−8κ 2 ] ; W {12}{3} = 2(β−2κ)M 2 +(β +γ−4κ)N 2 2(β−2κ)(β +γ−4κ) ; W {1}{23} = [(β−2κ) 2 −4κ 2 ]M 2 +(β−2κ) 2 P 2 +[(β−2κ) 2 −γ 2 ]Q 2 2(β−2κ)[(β−2κ) 2 −γ 2 −4κ 2 ] 4κ(β−2κ)PQ−2(β−2κ)γMP −4κγMQ 2(β−2κ)[(β−2κ) 2 −γ 2 −4κ 2 ] ; W {1}{2}{3} = 2(β−2κ)M 2 +(β +γ−2κ)N 2 2(β−2κ)(β +γ−2κ) . Letf(γ) = 2(β− 2κ) 2 (β +γ− 4κ)(M 2 −P 2 ) + (β− 2κ)(β +γ− 4κ) 2 (N 2 −Q 2 )− 16κ 2 (β− 2κ)M 2 −8κ(β−2κ)(β+γ−4κ)PQ−8κ 2 (β+γ−4κ)N 2 . Iff(κ)< 0,thenW {123} >W {12}{3} . Since f(κ)decreasesinκon[0,κ],f(0)> 0,andf(κ)< 0,thereexistsκ SP ∈ (0,κ)suchthatf(κ SP ) = 0. Proof of Proposition 5.1.2. Whenκ = 0,wehavethefollowingpayoffexpressions: βπ {123} A = P 2 (2+x) 2 andβπ {123} B = P 2 (2+x) 2 + Q 2 4 ; βπ {12}{3} A = M 2 (2+x) 2 andβπ {12}{3} B = M 2 (2+x) 2 + N 2 4 ; βπ {1}{23} A = (2M−xP) 2 (4−x 2 ) 2 andβπ {1}{23} B = (2P −xM) 2 (4−x 2 ) 2 + Q 2 4 ; 144 βπ {13}{2} A = (2P −xM) 2 (4−x 2 ) 2 andβπ {13}{2} B = (2M−xP) 2 (4−x 2 ) 2 + Q 2 4 ; βπ {1}{2}{3} A = M 2 (2+x) 2 > 0andβπ {1}{2}{3} B = M 2 (2+x) 2 + N 2 4 . Itiseasytocheckthatπ {13}{2} A <π {123} A <π {1}{2}{3} A =π {12}{3} A <π {1}{23} A andπ {1}{23} B <π {123} B < π {1}{2}{3} B =π {12}{3} B . Thatis,{12}{3}isstable. Proof of Lemma 5.1.1. From the first order conditions, firms’ equilibrium quantities under different struc- turesare q {123} 1 = [2−3y−(1−y)x]P +y(1−x)Q β[4−12y +6y 2 +2yx−(1−y)x 2 ] , q {123} 2 = [4−6y +3y 2 −2(1−y)x]P +y(4−3y−x)Q 2β[4−12y +6y 2 +2yx−(1−y)x 2 ] , q {123} 3 = 3y(2−y−x)P +(4−8y +3y 2 +2yx−x 2 )Q 2β[4−12y +6y 2 +2yx−(1−y)x 2 ] ; q {12}{3} 1 =q {12}{3} 2 = M β(2−3y +x) andq {12}{3} 3 = N 2β(1−y) ; q {1}{23} 1 = 2(1−2y)M−(1−y)xP −yxQ β(1−y)(4−8y−x 2 ) , q {1}{23} 2 = 2(1−y)P +2yQ−xM β(4−8y−x 2 ) , q {1}{23} 3 = 4y(1−y)P +[4(1−y) 2 −x 2 ]Q−2yxM 2β(1−y)(4−8y−x 2 ) ; q {13}{2} 1 = 2(1−y)P +yQ−xM β(4−8y +3y 2 −x 2 ) , q {13}{2} 2 = (4−8y +3y 2 )M−2(1−y)xP −yxQ 2β(1−y)(4−8y +3y 2 −x 2 ) , q {13}{2} 3 = 2y(1−y)P +[4(1−y) 2 −x 2 ]Q−yxM 2β(1−y)(4−8y +3y 2 −x 2 ) ; q {1}{2}{3} 1 =q {1}{2}{3} 2 = M β(2−2y +x) andq {1}{2}{3} 3 = N 2β(1−y) . 145 Correspondingpayoffsare: π X A = (β−κ)(q X 1 ) 2 andπ X B = (β−κ)(q X 2 ) 2 +(β−κ)(q X 3 ) 2 −2κq X 2 q X 3 , whenX ={123}and{1}{23}; π X A = (β−κ)(q X 1 ) 2 andπ X B = (β−κ)(q X 2 ) 2 +(β−κ)(q X 3 ) 2 , whenX ={12}{3},{13}{2}and{1}{2}{3}. Note that 0.5 6 P 6 M 6 1, 0.5 6 Q 6 N 6 1, 0 6 x 6 1, and 1 6 ω 6 1.5. It is easy to check that π X A ,π X B > 0 for all X = {123}, {12}{3}, {1}{23}, {13}{2}, and {1}{2}{3}. We then want to make sure that all equilibrium quantities are non-negative either, which requires 0 6 y < −x 2 −2x+12− √ x 4 +4x 3 +4x 2 −48x+48 12 and06y6 2M−xP 4M+x(Q−P) . Let us consider the lower bounds of −x 2 −2x+12− √ x 4 +4x 3 +4x 2 −48x+48 12 and 2M−xP 4M+x(Q−P) . First, it is easy to check that −x 2 −2x+12− √ x 4 +4x 3 +4x 2 −48x+48 12 increases inx on [0,1], so its lower bound is 3− √ 3 3 (when x = 0). As 2M−xP 4M+x(Q−P) = 1 2+ P+Q 2 M x −P decreasesinx, 2M−xP 4M+x(Q−P) > 2M−P 4M+Q−P (whenx = 1). • Whenc 1 =c 3 ,P =Q. 2M−P 4M+Q−P = 1 2 − 1 4! > 1 4 . • Whenc 1 = 0,M =P = 1. 2M−P 4M+Q−P = 1 3+Q > 1 4 . • Whenc 3 = 0,Q = 1. 2M−P 4M+Q−P = 1−(2−)c 1 4−(4−)c 1 = 1 4 + 3 4 −1 4 c 1 −(4−) > 1 5 (whenλ = 1andc 1 = 1 2 ,shown asbelow). – Ifλ> 4 3 , 1 4 + 3 4 −1 4 c 1 −(4−) increasesinc 1 . Hence, 1−(2−)c 1 4−(4−)c 1 > 1 4 (whenc 1 = 0). – If λ < 4 3 , 1 4 + 3 4 −1 4 c 1 −(4−) decreases in c 1 . Hence, 1−(2−)c 1 4−(4−)c 1 > 3−2 9−4 (when c 1 = 1 2 ). Since 3−2 9−4 = 1 3 − 2 27−12 increasesinλ,Hence, 3−2 9−4 > 1 5 (whenλ = 1). That is, forc 1 = c 3 orc 1 = 0, the lower bound of 2M−xP 4M+x(Q−P) is 1 4 ; forc 3 = 0, the lower bound is 1 5 . As 1 5 < 1 4 < 3− √ 3 3 , to make sure that all equilibrium quantities are non-negative,y 6 1 5 (orκ6 1 5 β) for c 3 = 0,andy6 1 4 (orκ6 1 4 β)forc 1 =c 3 orc 1 = 0. Proof of Proposition 5.1.3. • Ifc 1 = c 3 andκ = 1 4 β, leta = 2(98−56x−11x 2 −4x 3 +6x 4 ) (11+4x−6x 2 ) 2 ,b = 12 (5+4x) 2 + 1 3 , c = 12−4x 2 +x 4 3(2−x 2 ) 2 , d = 8x 3(2−x 2 ) 2 , e = 2x 2 3(2−x 2 ) 2 , f = 784x 2 +4(21−8x 2 ) 2 3(35−16x 2 ) 2 , g = 1960x+16x(21−8x 2 ) 3(35−16x 2 ) 2 , h = 1225+16x 2 3(35−16x 2 ) 2 , i = 12(3−2x) 2 (11+4x−6x 2 ) 2 , j = 12 (5+4x) 2 , k = 4x 2 3(2−x 2 ) 2 , l = 8x 3(2−x 2 ) 2 , m = 3 (3+2x) 2 , 146 n = 3 (3+2x) 2 + 1 3 ,o = 4 3(2−x 2 ) 2 ,p = 192x 2 (35−16x 2 ) 2 ,r = 588 (35−16x 2 ) 2 , ands = 672x (35−16x 2 ) 2 . We have the followingpayoffexpressions: β P 2 π {123} A =iand β P 2 π {123} B =a β P 2 π {12}{3} A =jω 2 and β P 2 π {12}{3} B =bω 2 β P 2 π {1}{23} A =k−lω +oω 2 and β P 2 π {1}{23} B =c−dω +eω 2 β P 2 π {13}{2} A =r−sω +pω 2 and β P 2 π {13}{2} B =f−gω +hω 2 β P 2 π {1}{2}{3} A =mω 2 and β P 2 π {1}{2}{3} B =nω 2 147 Itiseasytocheckthatπ {12}{3} A >{π {1}{23} A ,π {1}{2}{3} A }andπ {12}{3} B >π {1}{2}{3} B . Additionally, π {123} B >π {1}{23} B ⇔eω 2 −dω +(c−a)< 0⇔ω>W 1 (x) = d− √ d 2 −4e(c−a) 2e , π {12}{3} B >π {1}{23} B ⇔ (e−b)ω 2 −dω +c< 0⇔ω>W 2 (x) = d− √ d 2 −4(e−b)c 2(e−b) , π {13}{2} B >π {1}{23} B ⇔ (e−h)ω 2 −(d−g)ω +(c−f)< 0 ⇔ω>W 3 (x) = d−g− √ (d−g) 2 −4(e−h)(c−f) 2(e−h) , π {123} B >π {12}{3} B ⇔a>bω 2 ⇔ω<W 4 (x) = √ a b , π {123} A >π {12}{3} A ⇔i>jω 2 ⇔ω<W 5 (x) = √ i j , π {123} B >π {13}{2} B ⇔hω 2 −gω +(f−a)< 0⇔ω<W 6 (x) = g + √ g 2 −4h(f−a) 2h , π {123} A >π {13}{2} A ⇔pω 2 −sω +(r−i)< 0⇔ω>W 7 (x) = s− √ s 2 −4p(r−i) 2p , π {123} A >π {1}{23} A ⇔oω 2 −lω +(k−i)< 0⇔ω<W 8 (x) = l+ √ l 2 −4o(k−i) 2o , π {12}{3} B >π {13}{2} B ⇔ (h−b)ω 2 −gω +f < 0⇔ω>W 9 (x) = g− √ g 2 −4(h−b)f 2(h−b) , π {13}{2} B >π {1}{2}{3} B ⇔ (h−n)ω 2 −gω +f > 0⇔ω<W 10 (x) = g− √ g 2 −4(h−n)f 2(h−n) , π {123} B >π {1}{2}{3} B ⇔a>nω 2 ⇔ω<W 11 (x) = √ a n , π {123} A >π {1}{2}{3} A ⇔i>mω 2 ⇔ω<W 12 (x) = √ i m , π {12}{3} A >π {13}{2} A ⇔ (p−j)ω 2 −sω +r< 0⇔ω>W 13 (x) = s− √ s 2 −4(p−j)r 2(p−j) . – ω<W 1 :π {1}{23} B >π {123} B >{π {13}{2} B ,π {12}{3} B >π {1}{2}{3} B }.{1}{23}isstable. – W 1 < ω < W 2 : π {123} B > π {1}{23} B > {π {13}{2} B ,π {12}{3} B > π {1}{2}{3} B } and π {123} A > π {1}{23} A .{123}isstable. – W 2 <ω <W 3 : π {123} B >π {12}{3} B >{π {1}{2}{3} B ,π {1}{23} B >π {13}{2} B },{π {123} A ,π {12}{3} A }> {π {1}{23} A ,π {13}{2} A },andπ {12}{3} A >π {1}{2}{3} A . ∗ ω < W 5 : π {123} B > π {12}{3} B > {π {1}{2}{3} B ,π {1}{23} B > π {13}{2} B } and π {123} A > π {12}{3} A >{π {1}{23} A ,π {13}{2} A ,π {1}{2}{3} A }.{123}isstable. 148 ∗ ω > W 5 : π {123} B > π {12}{3} B > {π {1}{2}{3} B ,π {1}{23} B > π {13}{2} B },π {12}{3} A > π {123} A > {π {1}{23} A ,π {13}{2} A },andπ {12}{3} A >π {1}{2}{3} A . Both{123}and{12}{3}arestable. – W 3 <ω <W 4 : π {123} B >π {12}{3} B >π {1}{2}{3} B >π {13}{2} B >π {1}{23} B ,{π {123} A ,π {12}{3} A }> {π {1}{23} A ,π {13}{2} A },andπ {12}{3} A >π {1}{2}{3} A . ∗ ω<W 5 :π {123} B >π {12}{3} B >π {1}{2}{3} B >π {13}{2} B >π {1}{23} B andπ {123} A >π {12}{3} A > {π {1}{23} A ,π {13}{2} A ,π {1}{2}{3} A }.{123}isstable. ∗ ω > W 5 : π {123} B > π {12}{3} B > π {1}{2}{3} B > π {13}{2} B > π {1}{23} B ,π {12}{3} A > π {123} A > {π {1}{23} A ,π {13}{2} A },andπ {12}{3} A >π {1}{2}{3} A . Both{123}and{12}{3}arestable. – ω > W 4 : π {12}{3} B > {π {123} B ,π {1}{2}{3} B > π {13}{2} B } > π {1}{23} B andπ {12}{3} A > {π {123} A > π {13}{2} A ,π {1}{23} A ,π {1}{2}{3} A }.{12}{3}isstable. Inconclusion: – If 16 ω < W 1 (1), there exist 0 < x 1 6 1 such thatW 1 (x 1 ) = ω. Then,{123} is stable for x<x 1 ;{1}{23}isstableforx>x 1 . – IfW 1 (1)<ω<W 2 (x ∗ 2;5 ) =W 5 (x ∗ 2;5 ),{123}isstablefor06x6 1. – IfW 2 (x ∗ 2;5 ) = W 5 (x ∗ 2;5 ) < ω < W 2 (1), there exist 0 < x 5 < x 2 6 1 such thatW 2 (x 2 ) = W 5 (x 5 ) =ω. Then,{123}isstableforx<x 5 ;{123}and{12}{3}arestableforx 5 <x<x 2 ; {123}isstableforx>x 2 . – IfW 2 (1) < ω < W 4 (x ∗ 4 ) wherex ∗ 4 is the minimum point ofW 4 (x), there existsx 5 ∈ (0,1) such thatW 5 (x 5 ) = ω. Then,{123} is stable forx < x 5 ; {123} and{12}{3} are stable for x>x 5 . – IfW 4 (x ∗ 4 ) < ω < W 4 (1), there exist 0 < x 5 < x ′ 4 < x ′′ 4 6 1 such thatW 5 (x 5 ) = W 4 (x ′ 4 ) = W 4 (x ′′ 4 ) =ω. Then,{123}isstableforx<x 5 ;{123}and{12}{3}arestableforx 5 <x<x ′ 4 ; {12}{3}isstableforx ′ 4 <x<x ′′ 4 ;{123}and{12}{3}arestableforx>x ′′ 4 . – IfW 4 (1)<ω<W 5 (0), thereexist 06x 5 <x 4 < 1suchthatW 5 (x 5 ) =W 4 (x 4 ) =ω. Then, {123} is stable forx<x 5 ;{123} and{12}{3} are stable forx 5 <x<x 4 ;{12}{3} is stable forx>x 4 . – If W 5 (0) < ω < W 4 (0), there exists x 4 ∈ [0,1) such that W 4 (x 4 ) = ω. Then, {123} and {12}{3}arestableforx<x 4 ;{12}{3}isstableforx>x 4 . 149 – IfW 4 (0)<ω6 1.5,{12}{3}isstablefor06x6 1. • If c 1 = 0 and κ = 1 4 β, let a = 3(5−3x) 2 (11+4x−6x 2 ) 2 , b = 6(5−3x)(1−x) (11+4x−6x 2 ) 2 , c = 3(1−x) 2 (11+4x−6x 2 ) 2 , d = 3(43−24x) 2 64(11+4x−6x 2 ) 2 + 27(7−4x) 2 64(11+4x−6x 2 ) 2 − 3(43−24x)(7−4x) 32(11+4x−6x 2 ) 2 , e = 3(43−24x)(13−4x)+9(7−4x)(35+8x−16x 2 )−(43−24x)(35+8x−16x 2 )−3(7−4x)(13−4x) 32(11+4x−6x 2 ) 2 , f = 3(13−4x) 2 +3(35+8x−16x 2 ) 2 −2(13−4x)(35+8x−16x 2 ) 64(11+4x−6x 2 ) 2 , g = 12 (5+4x) 2 , h = (4−3x) 2 12(2−x 2 ) 2 , i = x(4−3x) 6(2−x 2 ) 2 , j = x 2 12(2−x 2 ) 2 , k = (3−2x) 2 6(2−x 2 ) 2 , l = 8(3−2x) 24(2−x 2 ) 2 , m = 3 (3+2x) 2 , n = 16[x 2 +(9−4x 2 ) 2 ] 3(35−16x 2 ) 2 , o = (9−4x 2 ) 2 −2(9−4x 2 )+9 48(2−x 2 ) 2 , p = 8[2(3−2x) 2 (3+2x)−x(35−24x)] 3(35−16x 2 ) 2 , r = 48(3−2x) 2 (35−16x 2 ) 2 , s = 48(3−2x) (35−16x 2 ) 2 , t = 12 (35−16x 2 ) 2 ,andu = (35−24x) 2 +4(3−2x) 2 3(35−16x 2 ) 2 . Wehavethefollowingpayoffexpressions: π {123} A =a+bQ+cQ 2 andπ {123} B =d+eQ+fQ 2 ; π {12}{3} A =g andπ {12}{3} B =g + 1 3β N 2 ; π {1}{23} A =h−iQ+jQ 2 andπ {1}{23} B =k+lQ+oQ 2 ; π {13}{2} A =r+sQ+tQ 2 andπ {13}{2} B =u+pQ+nQ 2 ; π {1}{2}{3} A =mandπ {1}{2}{3} B =m+ 1 3β N 2 . It is easy to check thatπ {123} A > π {12}{3} A > π {1}{2}{3} A , π {123} A > π {1}{23} A , π {123} B > π {12}{3} B > π {1}{2}{3} B ,π {123} B > π {13}{2} B , andπ {1}{23} B > π {13}{2} B . Hence,{12}{3},{1}{2}{3} and{13}{2} arenotstable. Inaddition, π {123} B >π {1}{23} B ⇔Q> l−e+ √ (e−l) 2 −4(f−o)(d−k) 2(f−o) =Q 0 . – Iff >o,wehaveQ>Q 0 ⇔π {123} B >π {1}{23} B .{123}isstable. – Iff <oandQ>Q 0 ,then{123}isstable. – Iff <oandQ<Q 0 ,then{1}{23}isstable. • If c 3 = 0 and κ = 1 5 β, let a = 5(7−4x) 2 (23+5x−10x 2 ) 2 , b = 10(1−x)(7−4x) (23+5x−10x 2 ) 2 , c = 5(1−x) 2 (23+5x−10x 2 ) 2 , d = 2(73−40x) 2 +18(9−5x) 2 −3(9−5x)(73−40x) 40(23+5x−10x 2 ) 2 , e = (73−40x)(17−5x) 10(23+5x−10x 2 ) 2 + 3(9−5x)(63+10x−25x 2 ) 10(23+5x−10x 2 ) 2 − (73−40x)(63+10x−25x 2 ) 40(23+5x−10x 2 ) 2 − 3(9−5x)(17−5x) 40(23+5x−10x 2 ) 2 , f = 2(17−5x) 2 +2(63+10x−25x 2 ) 2 −(17−5x)(63+10x−25x 2 ) 40(23+5x−10x 2 ) 2 , g = 20 (7+5x) 2 , h = 3 (12−5x 2 ) 2 , i = 15x (12−5x 2 ) 2 , j = 5x 2 4(12−5x 2 ) 2 , k = 3840−3000x 2 +625x 4 80(12−5x 2 ) 2 , 150 l = 20 (63−25x 2 ) 2 , m = 20 (8+5x) 2 , n = 25x(127−25x 2 ) 8(63−25x 2 ) 2 , o = 500x 2 (63−25x 2 ) 2 , p = 5[(64−25x 2 ) 2 +25x 2 ] 16(63−25x 2 ) 2 , r = 200x (63−25x 2 ) 2 ,s = 5(63 2 +25x 2 ) 16(63−25x 2 ) 2 ,andt = 20(1+25x 2 ) (63−25x 2 ) 2 . Wehavethefollowingpayoffexpressions: π {123} A =aP 2 +bP +candπ {123} B =dP 2 +eP +f; π {12}{3} A =gM 2 andπ {12}{3} B =gM 2 + 5 16β ; π {1}{23} A = 15hM 2 −iM−4iPM+16jP 2 +8jP+j andπ {1}{23} B = 15jM 2 −iM−4iPM+16hP 2 +8hP+k; π {13}{2} A =oM 2 −rM−8rPM+64lP 2 +16lP+l andπ {13}{2} B =sM 2 −nM−8rPM+tP 2 +16lP+p; π {1}{2}{3} A =mM 2 andπ {1}{2}{3} B =mM 2 + 5 16β . 151 It is easy to check thatg > m. Hence,π {12}{3} A > π {1}{2}{3} A ,π {123} B > π {12}{3} B > π {1}{2}{3} B , and π {123} B >π {13}{2} B . π {123} B >π {12}{3} B ⇔gM 2 −dP 2 −eP +( 5 16β −f) =F 1 < 0, π {123} A >π {13}{2} A ⇔oM 2 −rM−8rPM +(64l−a)P 2 +(16l−b)P +(l−c) =F 2 < 0, π {123} B >π {13}{2} B ⇔sM 2 −nM−8rPM +(t−d)P 2 +(16l−e)P +(p−f) =F 3 < 0, π {123} B >π {1}{23} B ⇔ 15jM 2 −iM−4iPM +(16h−d)P 2 +(8h−e)P +(k−f) =F 4 < 0, π {123} B >π {1}{2}{3} B ⇔mM 2 −dP 2 −eP +( 5 16β −f) =F 5 < 0, π {123} A >π {1}{23} A ⇔ 15hM 2 −iM−4iPM +(16j−a)P 2 +(8j−b)P +(j−c) =F 6 < 0, π {123} A >π {1}{2}{3} A ⇔mM 2 −aP 2 −bP −c =F 7 < 0, π {12}{3} A >π {13}{2} A ⇔ (o−g)M 2 −rM−8rPM +64lP 2 +16lP +l =F 8 < 0, π {12}{3} B >π {13}{2} B ⇔ (s−g)M 2 −nM−8rPM +tP 2 +16lP +(p− 5 16β ) =F 9 < 0, π {12}{3} B >π {1}{23} B ⇔ (15j−g)M 2 −iM−4iPM +16hP 2 +8hP +(k− 5 16β ) =F 10 < 0, π {1}{23} B >π {13}{2} B ⇔ (s−15j)M 2 −(n−i)M−(8r−4i)PM +(t−16h)P 2 +(16l−8h)P +(p−k) =F 11 < 0, π {1}{2}{3} B >π {13}{2} B ⇔ (s−m)M 2 −nM−8rPM +tP 2 +16lP +(p− 5 16β ) =F 12 < 0, π {13}{2} A >π {1}{23} A ⇔ (15h−o)M 2 −(i−r)M−(4i−8r)PM +(16j−64l)P 2 +(8j−16l)P +(j−l) =F 13 < 0, π {1}{2}{3} A >π {13}{2} A ⇔ (o−m)M 2 −rM−8rPM +64lP 2 +16lP +l =F 14 < 0, π {12}{3} A >π {1}{23} A ⇔ (15h−g)M 2 −iM−4iPM +16jP 2 +8jP +j =F 15 < 0, π {1}{2}{3} B >π {1}{23} B ⇔ (15j−m)M 2 −iM−4iPM +16hP 2 +8hP +(k− 5 16β ) =F 16 < 0, π {123} A >π {12}{3} A ⇔gM 2 −aP 2 −bP −c =F 17 < 0. – If F 4 > 0 or min{F 6 ,F 7 } > 0, then {123} is not stable. In other words, when max{F 4 ,min{F 6 ,F 7 }]< 0,{123}isstable. – If min{F 8 ,F 9 } > 0, or F 10 > 0, or−F 17 > 0, then{12}{3} is not stable. In other words, whenmax{min{F 8 ,F 9 },F 10 ,−F 17 }< 0,{12}{3}isstable. 152 – If max{F 4 ,F 6 } < 0, or max{F 10 ,F 15 } < 0, or F 16 < 0, or max{−F 11 ,F 13 } < 0, then {1}{23} is not stable. In other words, when min{max{F 4 ,F 6 },max{F 10 ,F 15 },max{−F 11 ,F 13 },F 16 }> 0,{1}{23}isstable. Proof of Proposition 5.1.4. Whenγ = 0,equilibriumquantitiesbecome q {123} 1 = (2−3y)P +yQ 2β(2−6y +3y 2 ) , q {123} 2 = (4−6y +3y 2 )P +y(4−3y)Q 4β(2−6y +3y 2 ) andq {123} 3 = (2−y)[3yP +(2−3y)Q] 4β(2−6y +3y 2 ) ; q {12}{3} 1 =q {12}{3} 2 = M β(2−3y) andq {12}{3} 3 = N 2β(1−y) ; q {1}{23} 1 = M 2β(1−y) , q {1}{23} 2 = (1−y)P +yQ 2β(1−2y) andq {1}{23} 3 = yP +(1−y)Q 2β(1−2y) ; q {13}{2} 1 = 2(1−y)P +yQ β(2−y)(2−3y) , q {13}{2} 2 = M 2β(1−y) andq {13}{2} 3 = yP +2(1−y)Q β(2−y)(2−3y) ; q {1}{2}{3} 1 =q {1}{2}{3} 2 = M 2β(1−y) andq {1}{2}{3} 3 = N 2β(1−y) . • Ifc 1 =c 3 ,wehavethefollowingpayoffexpressions: β P 2 π {123} A = (1−y) 3 (2−6y +3y 2 ) 2 and β P 2 π {123} B = (1−2y)(2−y) 2 2(2−6y +3y 2 ) 2 ; β P 2 π {12}{3} A = (1−y)ω 2 (2−3y) 2 and β P 2 π {12}{3} B = (1−y)ω 2 (2−3y) 2 + ω 2 4(1−y) ; β P 2 π {1}{23} A = ω 2 4(1−y) and β P 2 π {1}{23} B = 1 2(1−2y) ; β P 2 π {13}{2} A = 1−y (2−3y) 2 and β P 2 π {13}{2} B = ω 2 4(1−y) + 1−y (2−3y) 2 ; β P 2 π {1}{2}{3} A = ω 2 4(1−y) and β P 2 π {1}{2}{3} B = ω 2 2(1−y) . It is easy to check thatπ {12}{3} A > π {1}{23} A = π {1}{2}{3} A ,π {12}{3} A > π {13}{2} A ,π {123} A > π {13}{2} A , π {12}{3} B >π {1}{2}{3} B ,andπ {123} B >π {1}{23} B . Additionally,wehave π {1}{23} B >π {12}{3} B ⇔ 2(1−y)(2−3y) 2 (1−2y)[4(1−y) 2 +(2−3y) 2 ] =W 1 (y)>ω 2 , 153 π {13}{2} A >π {1}{23} A ⇔ 4(1−y) 2 (2−3y) 2 =W 2 (y)>ω 2 , π {123} A >π {12}{3} A ⇔ (1−y) 2 (2−3y) 2 (2−6y +3y 2 ) 2 =W 3 (y)>ω 2 , π {123} B >π {12}{3} B ⇔ 2(1−y)(1−2y)(2−y) 2 (2−3y) 2 (2−6y +3y 2 ) 2 [4(1−y) 2 +(2−3y) 2 ] =W 4 (y)>ω 2 , π {123} A >π {1}{23} A ⇔ 4(1−y) 4 (2−6y +3y 2 ) 2 =W 5 (y)>ω 2 , π {123} B >π {13}{2} B ⇔ 2(1−y)[(1−2y)(2−y) 2 (2−3y) 2 −2(1−y)(2−6y +3y 2 ) 2 ] (2−6y +3y 2 ) 2 (2−3y) 2 =W 6 (y)>ω 2 . Notethatfor06y6 1 4 ,16W 1 (y)<W 2 (y)<W 3 (y)<W 4 (y)<W 5 (y)<W 6 (y). – When ω 2 > W 4 (y), π {12}{3} A > { π {123} A ,π {1}{23} A =π {1}{2}{3} A } > π {13}{2} A , π {12}{3} B > π {1}{2}{3} B ,andπ {12}{3} B >π {123} B >π {1}{23} B .{12}{3}isstable. – When W 3 (y) < ω 2 < W 4 (y), π {12}{3} A > π {123} A > π {1}{23} A = π {1}{2}{3} A > π {13}{2} A , π {123} B >π {12}{3} B > { π {1}{2}{3} B ,π {1}{23} B } , andπ {123} B >π {13}{2} B . Both{123}and{12}{3} arestable. – When1<ω 2 <W 3 (y),π {123} A >π {12}{3} A > { π {13}{2} A ,π {1}{23} A =π {1}{2}{3} A } andπ {123} B > { π {1}{23} B ,π {13}{2} B ,π {12}{3} B >π {1}{2}{3} B } .{123}isstable. Note that bothW 3 (y) andW 4 (y) increase iny on [0, 1 4 ],W 3 (0) =W 4 (0) = 1,W 3 ( 1 4 )<W 4 ( 1 4 )< 1.5 2 . – If 1 < ω 2 6 W 3 ( 1 4 ), there existy 1 andy 2 satisfying 0 < y 2 < y 1 6 1 4 , such thatW 3 (y 1 ) = W 4 (y 2 ) = ω 2 . Then{123} is stable fory 1 6 y 6 1 4 ; both{123} and{12}{3} are stable for y 2 6y<y 1 ;{12}{3}isstablefor06y<y 2 . – IfW 3 ( 1 4 )<ω 2 6W 4 ( 1 4 ), there existy 2 satisfying 0<y 2 6 1 4 , such thatW 4 (y 2 ) =ω 2 . Then both{123}and{12}{3}arestablefory 2 6y6 1 4 ;{12}{3}isstablefor06y<y 2 . – Ifω 2 >W 4 ( 1 4 ),{12}{3}isstablefor06y6 1 4 . Let a = (1−y)(2−3y) 2 4(2−6y+3y 2 ) 2 , b = y(1−y)(2−3y) 2(2−6y+3y 2 ) 2 , c = y 2 (1−y) 4(2−6y+3y 2 ) 2 , d = (4−10y+3y 2 )(4−6y+3y 2 )+3y 2 (2−y)(2−3y) 16(2−6y+3y 2 ) 2 , e = y 2 (4−6y+3y 2 )+3y(2−y)(4−12y+7y 2 ) 8(2−6y+3y 2 ) 2 , f = 154 y 3 (4−3y)+(2−y)(2−3y)(4−12y+7y 2 ) 16(2−6y+3y 2 ) 2 , g = (1−y) (2−3y) 2 , h = 1 4(1−y) , i = 1−y 4(1−2y) , j = y 2(1−2y) , k = 4(1−y) 3 (2−y) 2 (2−3y) 2 ,l = 4y(1−y) 2 (2−y) 2 (2−3y) 2 , andm = y 2 (1−y) (2−y) 2 (2−3y) 2 . Note thate > j > b > l > f −h > i−h>c>mandd>a>g>f >i>k>h+m>h>j >b>l>f−h>i−h>c>m. • Ifc 1 = 0,wehavethefollowingpayoffexpressions: π {123} A =a+bQ+cQ 2 andπ {123} B =d+eQ+fQ 2 ; π {12}{3} A =g andπ {12}{3} B =g +hN 2 ; π {1}{23} A =handπ {1}{23} B =i+jQ+iQ 2 ; π {13}{2} A =k +lQ+mQ 2 andπ {13}{2} B =h+m+lQ+kQ 2 ; π {1}{2}{3} A =handπ {1}{2}{3} B =h+hN 2 . Itiseasytocheckthatπ {123} A >{π {12}{3} A ,π {13}{2} A }>π {1}{23} A =π {1}{2}{3} A ,π {123} B >π {1}{23} B > π {13}{2} B ,andπ {12}{3} B >π {1}{2}{3} B . Additionally, π {123} B >π {12}{3} B ⇔ (d−g)+eQ+fQ 2 >hN 2 , π {123} B >π {1}{2}{3} B ⇔ (d−h)+eQ+fQ 2 >hN 2 . – IfN < √ (d−g)+eQ+fQ 2 h ,π {123} A >{π {12}{3} A ,π {13}{2} A }>π {1}{23} A =π {1}{2}{3} A andπ {123} B > {π {1}{23} B >π {13}{2} B ,π {12}{3} B >π {1}{2}{3} B }.{123}isstable. – If √ (d−g)+eQ+fQ 2 h < N < √ (d−h)+eQ+fQ 2 h , π {123} A > {π {12}{3} A ,π {13}{2} A } > π {1}{23} A = π {1}{2}{3} A and π {12}{3} B > π {123} B > {π {1}{2}{3} B ,π {1}{23} B > π {13}{2} B }. Both {123} and {12}{3}arestable. – If N > √ (d−h)+eQ+fQ 2 h , π {123} A > {π {12}{3} A ,π {13}{2} A } > π {1}{23} A = π {1}{2}{3} A and π {12}{3} B >π {1}{2}{3} B >π {123} B >π {1}{23} B >π {13}{2} B .{12}{3}isstable. • Ifc 3 = 0,wehavethefollowingpayoffexpressions: π {123} A =aP 2 +bP +candπ {123} B =dP 2 +eP +f; π {12}{3} A =gM 2 andπ {12}{3} B =gM 2 +h; 155 π {1}{23} A =hM 2 andπ {1}{23} B =iP 2 +jP +i; π {13}{2} A =kP 2 +lP +mandπ {13}{2} B =hM 2 +mP 2 +lP +k; π {1}{2}{3} A =hM 2 andπ {1}{2}{3} B =hM 2 +h. It is easy to check that π {12}{3} A > π {1}{23} A = π {1}{2}{3} A , π {123} A > π {13}{2} A , π {123} B > π {1}{23} B , π {12}{3} B >π {1}{2}{3} B ,andπ {13}{2} B >π {1}{2}{3} B . Additionally, π {123} B >π {12}{3} B ⇔ dP 2 +eP +(f−h) g >M 2 , π {123} A >π {12}{3} A ⇔ aP 2 +bP +c g >M 2 , π {123} B >π {1}{2}{3} B ⇔ dP 2 +eP +(f−h) h >M 2 , π {123} A >π {1}{23} A ⇔ aP 2 +bP +c h >M 2 , π {12}{3} B >π {1}{23} B ⇔M 2 > iP 2 +jP +(i−h) g , π {123} B >π {13}{2} B ⇔ (d−m)P 2 +(e−l)P +(f−k) h >M 2 . – IfM 2 < aP 2 +bP+c g ,π {123} A >π {12}{3} A >π {1}{23} A =π {1}{2}{3} A ,π {123} A >π {13}{2} A ,π {123} B > π {1}{23} B ,andπ {123} B >{π {13}{2} B ,π {12}{3} B }>π {1}{2}{3} B .{123}isstable. – If aP 2 +bP+c g < M 2 < aP 2 +bP+c h , π {12}{3} A > π {123} A > {π {1}{23} A = π {1}{2}{3} A ,π {13}{2} A }, π {123} B >π {1}{23} B ,andπ {123} B >{π {13}{2} B ,π {12}{3} B }>π {1}{2}{3} B . ∗ IfM 2 < iP 2 +jP+(i−h) g ,π {12}{3} A >π {123} A >{π {1}{23} A =π {1}{2}{3} A ,π {13}{2} A },π {123} B > π {1}{23} B >π {12}{3} B >π {1}{2}{3} B ,andπ {123} B >π {13}{2} B >π {1}{2}{3} B .{123}isstable. ∗ If M 2 > iP 2 +jP+(i−h) g , π {12}{3} A > π {123} A > {π {1}{23} A = π {1}{2}{3} A ,π {13}{2} A }, π {123} B > π {13}{2} B > π {1}{2}{3} B , π {123} B > π {12}{3} B > π {1}{2}{3} B , and π {123} B > π {12}{3} B >π {1}{23} B . Both{123}and{12}{3}arestable. – If { aP 2 +bP+c g , iP 2 +jP+(i−h) g } < aP 2 +bP+c h < M 2 < { dP 2 +eP+(f−h) g , (d−m)P 2 +(e−l)P+(f−k) h } < dP 2 +eP+(f−h) h , π {12}{3} A > π {1}{23} A = π {1}{2}{3} A > π {123} A > π {13}{2} A , π {123} B > π {12}{3} B > π {1}{2}{3} B , π {123} B > π {12}{3} B > π {1}{23} B ,andπ {123} B >π {13}{2} B >π {1}{2}{3} B .{12}{3}isstable. 156 – If { aP 2 +bP+c g , iP 2 +jP+(i−h) g } < aP 2 +bP+c h < dP 2 +eP+(f−h) g < M 2 < dP 2 +eP+(f−h) h , π {12}{3} A > π {1}{23} A = π {1}{2}{3} A > π {123} A > π {13}{2} A , π {12}{3} B > π {123} B > π {1}{23} B , π {13}{2} B >π {1}{2}{3} B ,andπ {12}{3} B >π {123} B >π {1}{2}{3} B .{12}{3}isstable. – IfM 2 > dP 2 +eP+(f−h) h ,π {12}{3} A >π {1}{23} A =π {1}{2}{3} A >π {123} A >π {13}{2} A and{π {13}{2} B , π {12}{3} B }>π {1}{2}{3} B >π {123} B >π {1}{23} B .{12}{3}isstable. Proof of Proposition 5.1.5. Whenγ =β,equilibriumquantitiesbecome q {123} 1 = P 3β(1−y) , q {123} 2 = (2−4y +3y 2 )P +3y(1−y)Q 6β(1−y)(1−2y) andq {123} 3 = yP +(1−y)Q 2β(1−2y) ; q {12}{3} 1 =q {12}{3} 2 = M 3β(1−y) andq {12}{3} 3 = N 2β(1−y) ; q {1}{23} 1 = 2(1−2y)M−(1−y)P −yQ β(1−y)(3−8y) , q {1}{23} 2 = 2(1−y)P +2yQ−M β(3−8y) , q {1}{23} 3 = 4y(1−y)P +(1−2y)(3−2y)Q−2yM 2β(1−y)(3−8y) ; q {13}{2} 1 = 2(1−y)P +yQ−M β(3−8y +3y 2 ) , q {13}{2} 2 = (2−y)(2−3y)M−2(1−y)P −yQ 2β(1−y)(3−8y +3y 2 ) , q {13}{2} 3 = 2y(1−y)P +(1−2y)(3−2y)Q−yM 2β(1−y)(3−8y +3y 2 ) ; q {1}{2}{3} 1 =q {1}{2}{3} 2 = M β(3−2y) andq {1}{2}{3} 3 = N 2β(1−y) . • If c 1 = c 3 , let a = 13−8y 36(1−y)(1−2y) , b = 13 36(1−y) , c = 25−80y+64y 4(1−y)(3−8y) 2 , d = 4(1−2y) (1−y)(3−8y) 2 , e = 1−2y (1−y)(3−8y) 2 , f = (2−y) 2 +(3−6y+2y 2 ) 2 4(1−y)(3−8y+3y 2 ) 2 , g = (2−y) 2 (2−3y)+y(3−6y+2y 2 ) 2(1−y)(3−8y+3y 2 ) 2 , h = (2−y) 2 (2−3y) 2 +y 2 4(1−y)(3−8y+3y 2 ) 2 , i = 157 1 9(1−y) ,j = 1−y (3−2y) 2 ,k = 1 (1−y)(3−8y) 2 ,l = 4(1−2y) (1−y)(3−8y) 2 ,m = 1−y (3−2y) 2 + 1 4(1−y) ,n = 1−y (3−8y+3y 2 ) 2 , o = 4(1−2y) 2 (1−y)(3−8y) 2 ,p = 2(1−y)(2−y) (3−8y+3y 2 ) 2 ,andr = (1−y)(2−y) 2 (3−8y+3y 2 ) 2 . Wehavethefollowingpayoffexpressions: β P 2 π {123} A =iand β P 2 π {123} B =a; β P 2 π {12}{3} A =iω 2 and β P 2 π {12}{3} B =bω 2 ; β P 2 π {1}{23} A =k−lω +oω 2 and β P 2 π {1}{23} B =c−dω +eω 2 ; β P 2 π {13}{2} A =r−pω +nω 2 and β P 2 π {13}{2} B =f−gω +hω 2 ; β P 2 π {1}{2}{3} A =jω 2 and β P 2 π {1}{2}{3} B =mω 2 . 158 Itiseasytocheckthatπ {123} A >π {12}{3} A ,π {12}{3} A >{π {123} A ,π {1}{2}{3} B }andπ {12}{3} B >π {1}{2}{3} B . Additionally, π {123} B >π {1}{23} B ⇔eω 2 −dω +(c−a)< 0⇔ω> d− √ d 2 −4e(c−a) 2e =W 1 (y), π {12}{3} B >π {1}{23} B ⇔ (e−b)ω 2 −dω +c< 0⇔ω> d− √ d 2 −4(e−b)c 2(e−b) =W 2 (y), π {13}{2} B >π {1}{23} B ⇔ (e−h)ω 2 −(d−g)ω +(c−f)< 0 ⇔ω> d−g− √ (d−g) 2 −4(e−h)(c−f) 2(e−h) =W 3 (y), π {123} B >π {12}{3} B ⇔a>bω 2 ⇔ω< √ a b =W 4 (y), π {123} B >π {13}{2} B ⇔hω 2 −gω +(f−a)< 0⇔ω< g + √ g 2 −4h(f−a) 2h =W 6 (y), π {123} A >π {13}{2} A ⇔nω 2 −pω +(r−i)< 0⇔ω> p− √ p 2 −4n(r−i) 2n =W 7 (y), π {123} A >π {1}{23} A ⇔oω 2 −lω +(k−i)< 0⇔ω< l+ √ l 2 −4o(k−i) 2o =W 8 (y), π {12}{3} B >π {13}{2} B ⇔ (h−b)ω 2 −gω +f < 0⇔ω> g− √ g 2 −4(h−b)f 2(h−b) =W 9 (y), π {13}{2} B >π {1}{2}{3} B ⇔ (h−m)ω 2 −gω +f > 0⇔ω< g− √ g 2 −4(h−m)f 2(h−m) =W 10 (y), π {123} B >π {1}{2}{3} B ⇔a>mω 2 ⇔ω< √ a m =W 11 (y), π {123} A >π {1}{2}{3} A ⇔i>jω 2 ⇔ω< √ i j =W 12 (y), π {12}{3} A >π {13}{2} A ⇔ (n−i)ω 2 −pω +r< 0⇔ω> p− √ p 2 −4(n−i)r 2(n−i) =W 13 (y), π {12}{3} A >π {1}{23} A ⇔ (o−i)ω 2 −lω +k< 0⇔ω< l+ √ l 2 −4(o−i)k 2(o−i) =W 14 (y), π {1}{23} B >π {1}{2}{3} B ⇔ (e−m)ω 2 −dω +c> 0⇔ω< d− √ d 2 −4(e−m)c 2(e−m) =W 15 (y). – Ifω<W 1 ,π {1}{23} B >{π {13}{2} B ,π {123} B >π {12}{3} B >π {1}{2}{3} B }.{1}{23}isstable. – If W 1 < ω < W 2 , π {123} B > π {1}{23} B > {π {13}{2} B ,π {12}{3} B > π {1}{2}{3} B } and π {123} A > π {1}{23} A .{123}isstable. – If W 2 < ω < W 3 , π {123} B > π {12}{3} B > {π {1}{23} B > π {13}{2} B ,π {1}{2}{3} B } and π {12}{3} A > π {1}{2}{3} B >π {123} A >π {1}{23} A . Both{123}and{12}{3}arestable. 159 – If W 3 < ω < W 4 , π {123} B > π {12}{3} B > π {1}{2}{3} B > π {13}{2} B > π {1}{23} B and π {12}{3} A > π {1}{2}{3} B >π {123} A >π {1}{23} A . Both{123}and{12}{3}arestable. – If W 4 < ω < W 14 , π {12}{3} B > {π {1}{2}{3} B ,π {13}{2} B ,π {123} B > π {1}{23} B } and π {12}{3} A > {π {123} A ,π {1}{2}{3} B ,π {13}{2} A ,π {1}{23} A }.{12}{3}isstable. – ω >W 14 ,π {12}{3} B >π {1}{2}{3} B >π {13}{2} B >π {123} B >π {1}{23} B andπ {1}{23} A >π {12}{3} A > π {1}{2}{3} A >π {123} A >π {13}{2} A .{12}{3}isstable. Inconclusion, – IfW 4 (1)<ω6 1.5,{12}{3}isstablefor06y6 1 4 . – If W 2 (1) < ω < W 4 (1), there exists y 4 ∈ (0, 1 4 ] such that W 4 (y 4 ) = ω. Then, {12}{3} is stablefory<y 4 ;{123}and{12}{3}arestablefory>y 4 . – IfW 1 (1)<ω<W 2 (1),thereexists0<y 4 <y 2 6 1 4 suchthatW 2 (y 2 ) =W 4 (y 4 ) =ω. Then, {12}{3} is stable fory < y 4 ;{123} and{12}{3} are stable fory 4 < y < y 2 ;{123} is stable fory>y 2 . – If 1 6 ω < W 1 (1), there exists 0 6 y 4 < y 2 < y 1 6 1 4 such that W 1 (y 1 ) = W 2 (y 2 ) = W 4 (y 4 ) =ω. Then,{12}{3} is stable fory <y 4 ;{123} and{12}{3} are stable fory 4 <y < y 2 ;{123}isstablefory 2 <y<y 1 ;{1}{23}isstablefory>y 1 . • If c 1 = 0, let a = 1 9(1−y) , b = 4−8y+9y 2 36(1−y)(1−2y) , c = y 2(1−2y) , d = 1−y 4(1−2y) , e = 1 4(1−y) , f = (1−3y) 2 (1−y)(3−8y) 2 , g = 2y(1−3y) (1−y)(3−8y) 2 , h = y 2 (1−y)(3−8y) 2 , i = (1−2y) 3 (1−y)(3−8y) 2 , j = 4y(1−2y) 2 (1−y)(3−8y) 2 , k = 4y 2 +(3−8y)(1−2y)(3−2y) 4(1−y)(3−8y) 2 , l = (1−y)(1−2y) 2 (3−8y+3y 2 ) 2 , m = 1−y (3−2y) 2 , n = y 2 +(1−2y) 2 (3−2y) 2 4(1−y)(3−8y+3y 2 ) 2 , o = 2y(1−y)(1−2y) (3−8y+3y 2 ) 2 , p = y(1−8y+17y 2 −8y 3 ) 2(1−y)(3−8y+3y 2 ) 2 , r = y 2 (1−y) (3−8y+3y 2 ) 2 , and s = (2−6y+3y 2 ) 2 +y 2 (1−2y) 2 4(1−y)(3−8y+3y 2 ) 2 . Note thatb>{a,i}>l >m>s>f,{j,c}>g >o>{p,h>r}, andk >{n,d}>e. We have the followingpayoffexpressions: π {123} A =aandπ {123} B =b+cQ+dQ 2 ; π {12}{3} A =aandπ {12}{3} B =a+eN 2 ; π {1}{23} A =f−gQ+hQ 2 andπ {1}{23} B =i+jQ+kQ 2 ; π {13}{2} A =l+oQ+rQ 2 andπ {13}{2} B =s+pQ+nQ 2 ; 160 π {1}{2}{3} A =mandπ {1}{2}{3} B =m+eN 2 . It is easy to check that {π {123} A = π {12}{3} A ,π {13}{2} A } > {π {1}{2}{3} A ,π {1}{23} A }, π {12}{3} B > π {1}{2}{3} B ,and{π {123} B ,π {1}{23} B }>π {13}{2} B . Hence,{13}{2}and{1}{2}{3}arenotstable. Addi- tionally, π {123} B >π {1}{23} B ⇔ (k−d)Q 2 −(c−j)Q−(b−i)< 0⇔Q< c−j + √ (c−j) 2 +4(k−d)(b−i) 2(k−d) =Q 0 , π {123} B >π {12}{3} B ⇔ (b−a)+cQ+dQ 2 >eN 2 , π {12}{3} B >π {1}{23} B ⇔eN 2 > (i−a)+jQ+kQ 2 . – If Q < Q 0 , {π {123} A = π {12}{3} A ,π {13}{2} A } > {π {1}{2}{3} A ,π {1}{23} A }, π {12}{3} B > π {1}{2}{3} B , andπ {123} B >π {1}{23} B >π {13}{2} B . ∗ IfN 2 < (i−a)+jQ+kQ 2 e ,{π {123} A =π {12}{3} A ,π {13}{2} A }>{π {1}{2}{3} A ,π {1}{23} A },π {123} B > π {1}{23} B >π {13}{2} B ,andπ {123} B >π {1}{23} B >π {12}{3} B >π {1}{2}{3} B .{123}isstable. ∗ If (i−a)+jQ+kQ 2 e < N 2 < (b−a)+cQ+dQ 2 e , {π {123} A = π {12}{3} A ,π {13}{2} A } > {π {1}{2}{3} A ,π {1}{23} A }, π {123} B > π {12}{3} B > π {1}{2}{3} B , and π {123} B > π {12}{3} B > π {1}{23} B >π {13}{2} B .{123}isstable. ∗ If N 2 > (b−a)+cQ+dQ 2 e , {π {123} A = π {12}{3} A ,π {13}{2} A } > {π {1}{2}{3} A ,π {1}{23} A }, π {12}{3} B > π {1}{2}{3} B , and π {12}{3} B > π {123} B > π {1}{23} B > π {13}{2} B . {12}{3} is sta- ble. – If Q > Q 0 , {π {123} A = π {12}{3} A ,π {13}{2} A } > {π {1}{2}{3} A ,π {1}{23} A }, π {12}{3} B > π {1}{2}{3} B , andπ {1}{23} B >π {123} B >π {13}{2} B . ∗ If N 2 < (b−a)+cQ+dQ 2 e , {π {123} A = π {12}{3} A ,π {13}{2} A } > {π {1}{2}{3} A ,π {1}{23} A }, π {1}{23} B >π {123} B >π {12}{3} B >π {1}{2}{3} B , andπ {1}{23} B >π {123} B >π {13}{2} B . {1}{23} isstable. ∗ If (b−a)+cQ+dQ 2 e < N 2 < (i−a)+jQ+kQ 2 e , {π {123} A = π {12}{3} A ,π {13}{2} A } > {π {1}{2}{3} A ,π {1}{23} A }, π {1}{23} B > π {12}{3} B > π {1}{2}{3} B , and π {1}{23} B > π {12}{3} B > π {123} B >π {13}{2} B .{1}{23}isstable. 161 ∗ If N 2 > (i−a)+jQ+kQ 2 e , {π {123} A = π {12}{3} A ,π {13}{2} A } > {π {1}{2}{3} A ,π {1}{23} A }, π {12}{3} B > π {1}{2}{3} B , and π {12}{3} B > π {1}{23} B > π {123} B > π {13}{2} B . {12}{3} is sta- ble. • Ifc 3 = 0,leta = 1 36(1−y) ,b = 4−8y+9y 2 36(1−y)(1−2y) ,c = y 2(1−2y) ,d = 1−y 4(1−2y) ,e = 4(1−2y) 2 (1−y)(3−8y) 2 ,f = 1−y (3−8y) 2 ,g = 4(1−2y) (3−8y) 2 ,h = 4y(1−2y) (1−y)(3−8y) 2 ,i = 2y (3−8y) 2 ,j = y 2 (1−y)(3−8y) 2 ,k = 1−2y (1−y)(3−8y) 2 ,l = 4(1−y)(1−2y) (3−8y) 2 ,m = 1−y (3−2y) 2 ,n = y 2 +(1−2y) 2 (3−2y) 2 4(1−y)(3−8y+3y 2 ) 2 ,o = 8y(1−2y) (3−8y) 2 ,p = y(7−16y+7y 2 ) 2(1−y)(3−8y+3y 2 ) 2 ,r = 4y 2 +(1−2y)(3−2y)(3−8y) 4(1−y)(3−8y) 2 ,s = 1−y (3−8y+3y 2 ) 2 ,t = 4(1−y) 3 (3−8y+3y 2 ) 2 ,u = 4(1−y) 2 (3−8y+3y 2 ) 2 ,v = 2y(1−y) (3−8y+3y 2 ) 2 , w = 4y(1−y) 2 (3−8y+3y 2 ) 2 ,z = (1−y)(1+y 2 ) (3−8y+3y 2 ) 2 ,ϕ = y 2 (1−y) (3−8y+3y 2 ) 2 ,andζ = y 2 +(2−y) 2 (2−3y) 2 4(1−y)(3−8y+3y 2 ) 2 . Notethatg> u>l>e>t>ζ >r >{n,d}> 9a,g >u>l>e>t>ζ >f >k >z >s>b> 4a>m, andg >u>l >e>t>ζ >o>h>{w >v,c,i>v}>j >ϕ. We have the following payoff expressions: π {123} A = 4aP 2 andπ {123} B =bP 2 +cP +d; π {12}{3} A = 4aM 2 andπ {12}{3} B = 4aM 2 +9a; π {1}{23} A =eM 2 −hM−gPM+fP 2 +iP+j andπ {1}{23} B =kM 2 −hM−gPM+lP 2 +oP+r; π {13}{2} A =sM 2 −vM−uPM+tP 2 +wP+ϕandπ {13}{2} B =ζM 2 −pM−uPM+zP 2 +wP+n; π {1}{2}{3} A =mM 2 andπ {1}{2}{3} B =mM 2 +9a. 162 It is easy to check thatπ {12}{3} A >π {123} A ,π {12}{3} A >π {1}{2}{3} A , andπ {12}{3} B >π {1}{2}{3} B . Hence, {1}{2}{3}isnotstable. Inaddition, π {123} B >π {12}{3} B ⇔ 4aM 2 −bP 2 −cP +(9a−d) =F 1 < 0, π {123} A >π {13}{2} A ⇔sM 2 −vM−uPM +(t−4a)P 2 +wP +ϕ =F 2 < 0, π {123} B >π {13}{2} B ⇔ζM 2 −pM−uPM +(z−b)P 2 +(w−c)P +(n−d) =F 3 < 0, π {123} B >π {1}{23} B ⇔kM 2 −hM−gPM +(l−b)P 2 +(o−c)P +(r−d) =F 4 < 0, π {123} B >π {1}{2}{3} B ⇔mM 2 −bP 2 −cP +(9a−d) =F 5 < 0, π {123} A >π {1}{23} A ⇔eM 2 −hM−gPM +(f−4a)P 2 +iP +j =F 6 < 0, π {123} A >π {1}{2}{3} A ⇔mM 2 −4aP 2 =F 7 < 0, π {12}{3} A >π {13}{2} A ⇔ (s−4a)M 2 −vM−uPM +tP 2 +wP +ϕ =F 8 < 0, π {12}{3} B >π {13}{2} B ⇔ (ζ−4a)M 2 −pM−uPM +zP 2 +wP +(n−9a) =F 9 < 0, π {12}{3} B >π {1}{23} B ⇔ (k−4a)M 2 −hM−gPM +lP 2 +oP +(r−9a) =F 10 < 0, π {1}{23} B >π {13}{2} B ⇔ (ζ−k)M 2 −(p−h)M−(u−g)PM +(z−l)P 2 +(w−o)P +(n−r) =F 11 < 0, π {1}{2}{3} B >π {13}{2} B ⇔ (ζ−m)M 2 −pM−uPM +zP 2 +wP +(n−9a) =F 12 < 0, π {13}{2} A >π {1}{23} A ⇔ (e−s)M 2 −(h−v)M−(g−u)PM +(f−t)P 2 +(i−w)P +(j−ϕ) =F 13 < 0, π {1}{2}{3} A >π {13}{2} A ⇔ (s−m)M 2 −vM−uPM +tP 2 +wP +ϕ =F 14 < 0, π {12}{3} A >π {1}{23} A ⇔ (e−4a)M 2 −hM−gPM +fP 2 +iP +j =F 15 < 0, π {1}{2}{3} B >π {1}{23} B ⇔ (k−m)M 2 −hM−gPM +lP 2 +oP +(r−9a) =F 16 < 0. – IfF 1 > 0, or min{F 2 ,F 3 }> 0, orF 4 > 0, orF 5 > 0, or min{F 6 ,F 7 }> 0, then{123} is not stable. Thatis,whenmax{F 1 ,min{F 2 ,F 3 },F 4 ,F 5 ,min{F 6 ,F 7 }}< 0,{123}isstable. – If min{F 8 ,F 9 } > 0 or F 10 > 0, then {12}{3} is not stable. That is, when max{min{F 8 ,F 9 },F 10 }< 0,{12}{3}isstable. 163 – If max{F 4 ,F 6 } < 0, or max{F 10 ,F 15 } < 0, or max{−F 11 ,F 13 } < 0, or F 16 < 0, then {1}{23}isstable. Thatis,whenmin{max{F 4 ,F 6 },max{F 10 ,F 15 },max{−F 11 ,F 13 },F 16 }> 0,{1}{23}isstable. – If max{F 2 ,F 3 } < 0, or max{F 8 ,F 9 } < 0, or F 11 < 0, or F 12 < 0, or max{−F 13 ,F 14 } < 0, then {13}{2} is stable. That is, when min{max{F 2 ,F 3 },max{F 8 ,F 9 },F 11 ,F 12 ,max{−F 13 ,F 14 }}> 0,{13}{2}isstable. Proof of Proposition 5.2.2. Product quantities and firms’ payoffs under different recycling structures are listed below, from which it is easy to conclude that π {1234} i > π {14}{23} i and π {12}{34} i > {π {1}{2}{3}{4} i ,π {1}{2}{34} i ,π {12}{3}{4} i ,π {13}{24} i }fori =A,B. q {1234} 1 =q {1234} 2 = (2β +γ−3d)P +3dQ (2β +γ)(2β +γ−6d) andq {1234} 3 =q {1234} 4 = (2β +γ−3d)Q+3dP (2β +γ)(2β +γ−6d) , π {1234} A = (β−κ)(q {1234} 1 ) 2 +(β−κ)(q {1234} 4 ) 2 −2dq {1234} 1 q {1234} 4 , π {1234} B = (β−κ)(q {1234} 2 ) 2 +(β−κ)(q {1234} 3 ) 2 −2dq {1234} 2 q {1234} 3 ; q {12}{34} 1 =q {12}{34} 2 = M 2β +γ−3d andq {12}{34} 3 =q {12}{34} 4 = N 2β +γ−3d , π {12}{34} A = (β−κ)(q {12}{34} 1 ) 2 +(β−κ)(q {12}{34} 4 ) 2 andπ {12}{34} B = (β−κ)(q {12}{34} 2 ) 2 +(β−κ)(q {12}{34} 3 ) 2 ; q {1}{2}{34} 1 =q {1}{2}{34} 2 = M 2β +γ−2d andq {1}{2}{34} 3 =q {1}{2}{34} 4 = N 2β +γ−3d , π {1}{2}{34} A = (β−κ)(q {1}{2}{34} 1 ) 2 +(β−κ)(q {1}{2}{34} 4 ) 2 andπ {1}{2}{34} B = (β−κ)(q {1}{2}{34} 2 ) 2 +(β−κ)(q {1}{2}{34} 3 ) 2 ; q {14}{23} 1 =q {14}{23} 2 = (2β +γ−2d)P +2dQ (2β +γ)(2β +γ−4d) andq {14}{23} 3 =q {14}{23} 4 = (2β +γ−2d)Q+2dP (2β +γ)(2β +γ−4d) , 164 π {14}{23} A = (β−κ)(q {14}{23} 1 ) 2 +(β−κ)(q {14}{23} 4 ) 2 −2dq {14}{23} 1 q {14}{23} 4 , π {14}{23} B = (β−κ)(q {14}{23} 2 ) 2 +(β−κ)(q {14}{23} 3 ) 2 −2dq {14}{23} 2 q {14}{23} 3 ; q {13}{24} 1 =q {13}{24} 2 = (2β +γ−2d)P +dQ (2β +γ−κ)(2β +γ−3d) andq {13}{24} 3 =q {13}{24} 4 = (2β +γ−2d)Q+dP (2β +γ−κ)(2β +γ−3d) , π {13}{24} A = (β−κ)(q {13}{24} 1 ) 2 +(β−κ)(q {13}{24} 4 ) 2 andπ {13}{24} B = (β−κ)(q {13}{24} 2 ) 2 +(β−κ)(q {13}{24} 3 ) 2 ; q {1}{2}{3}{4} 1 =q {1}{2}{3}{4} 2 = M 2β +γ−2d andq {1}{2}{3}{4} 3 =q {1}{2}{3}{4} 4 = N 2β +γ−2d , π {1}{2}{3}{4} A = (β−κ)(q {1}{2}{3}{4} 1 ) 2 +(β−κ)(q {1}{2}{3}{4} 4 ) 2 , π {1}{2}{3}{4} B = (β−κ)(q {1}{2}{3}{4} 2 ) 2 +(β−κ)(q {1}{2}{3}{4} 3 ) 2 . Proof of Proposition 5.2.1. Wecomparethefollowingsocialwelfare: W {1234} = 1 2 Pq 1 + 1 2 Pq 2 + 1 2 Qq 3 + 1 2 Qq 4 whereq 1 =q 2 = (β +γ−4κ)P +4κQ (β +γ−4κ) 2 −16κ 2 andq 3 =q 4 = (β +γ−4κ)Q+4κP (β +γ−4κ) 2 −16κ 2 ; W {1}{234} = 1 2 Mq 1 + 1 2 Pq 2 + 1 2 Qq 3 + 1 2 Qq 4 whereq 2 = (β−2κ)(β +γ−4κ)P −(β +γ−4κ)γM +4κ(β−2κ)Q (β−2κ) 2 (β +γ−4κ)−(β +γ−4κ)γ 2 −8κ 2 (β−2κ) , q 1 = M−γq 2 β−2κ andq 3 = Q+2κq 2 β +γ−4κ ; W {12}{34} = 1 2 Mq 1 + 1 2 Mq 2 + 1 2 Nq 3 + 1 2 Nq 4 whereq 1 =q 2 = M β +γ−4κ andq 3 =q 4 = N β +γ−4κ ; W {13}{24} = 1 2 Pq 1 + 1 2 Pq 2 + 1 2 Qq 3 + 1 2 Qq 4 whereq 1 =q 2 = (β +γ−2κ)P +2κQ (β +γ−2κ) 2 −4κ 2 andq 3 =q 4 = (β +γ−2κ)Q+2κP (β +γ−2κ) 2 −4κ 2 ; 165 W {1}{2}{34} = 1 2 Mq 1 + 1 2 Mq 2 + 1 2 Nq 3 + 1 2 Nq 4 whereq 1 =q 2 = M β +γ−2κ andq 3 =q 4 = N β +γ−4κ ; W {1}{24}{3} = 1 2 Mq 1 + 1 2 Pq 2 + 1 2 Nq 3 + 1 2 Qq 4 whereq 2 = [(β−2κ) 2 −γ 2 ][(β−2κ)P −γM]+2κ(β−2κ)[(β−2κ)Q−γN] [(β−2κ) 2 −γ 2 ] 2 −4κ 2 (β−2κ) 2 , q 1 = M−γq 2 β−2κ , q 4 = [(β−2κ) 2 −γ 2 ][(β−2κ)Q−γN]+2κ(β−2κ)[(β−2κ)P −γM] [(β−2κ) 2 −γ 2 ] 2 −4κ 2 (β−2κ) 2 , andq 3 = N−γq 4 β−2κ ; W {1}{2}{3}{4} = 1 2 Mq 1 + 1 2 Mq 2 + 1 2 Nq 3 + 1 2 Nq 4 whereq 1 =q 2 = M β +γ−2κ andq 3 =q 4 = N β +γ−2κ . Itiseasytocheckthatmax{W {1234} ,W {12}{34} }> max{W {1}{234} ,W {13}{24} ,W {1}{2}{34} ,W {1}{24}{3} , W {1}{2}{3}{4} }. WethencompareW {1234} andW {12}{34} : W {1234} >W {12}{34} ⇔κ> √ P 2 Q 2 +(M 2 +N 2 )[(M 2 +N 2 )−(P 2 +Q 2 )]−PQ 4(M 2 +N 2 )+4 √ P 2 Q 2 +(M 2 +N 2 )[(M 2 +N 2 )−(P 2 +Q 2 )]−4PQ (β+γ). 166

Abstract (if available)

Abstract In the context of the Extended Producer Responsibility (EPR), governments and producers are making efforts to recycle consumer products at their end of life. In the electronics industry, several local governments have introduced legislations such as the WEEE (Waste Electrical and Electronic Equipment) Directive to impose the recycling responsibility to producers. Implementations of these EPR-type legislations include two types of mechanisms: producers jointly recycle products together, which follows the Collective Producer Responsibility (CPR)

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Asset Metadata

Creator Tian, Fang (author)

Core Title Green strategies: producers' competition and cooperation in sustainability

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 04/28/2015

Defense Date 03/23/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag economies of scale,EPR,extended producer responsibility,firm-based recycling,OAI-PMH Harvest,product heterogeneity

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sosic, Greys (committee chair), Debo, Laurens (committee member), Wiltermuth, Scott (committee member), Zhu, Leon (committee member)

Creator Email selenetian@gmail.com,tianfang@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-562575

Unique identifier UC11299000

Identifier etd-TianFang-3410.pdf (filename),usctheses-c3-562575 (legacy record id)

Legacy Identifier etd-TianFang-3410.pdf

Dmrecord 562575

Document Type Dissertation

Format application/pdf (imt)

Rights Tian, Fang

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA