Online Appendix Appendix A: Numerical Examples

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1 Aticle submitte to Management Science; manuscipt no. MS Online Appenix Appenix A: Numeical Examples We constuct two instances of the ecycling netwok (RN) base on the EPR implementation in Washington state in the US an the Japanese implementation of the Home Appliance Recycling Law(HARL). We analyze whethe an when these two instances satisfy the esign-einfocing conition. Example 2 (EPR implementation in the Washington State). To constuct an instance of the RN moel in this pape, we fist nee to benchmak pouces access to ecycling capacity if they wee to opeate inepenently. An example in pactice whee pouces have mae an effot to establish inepenent ecycling capacities is the following: Washington state s e-waste law manates all pouces selling in the state to eithe paticipate in a plan un by the Washington Mateials Management & Financing Authoity (WMMFA), o establish an opeate thei own inepenent plans (subject to egulatoy appoval) to fulfill thei collection an ecycling obligations. Two such inepenent plans wee submitte in the past: One by HP who opeates a lage scale in-house ecycling facility in nothen Califonia, the othe by a goup of TV manufactues (Panasonic, Shap an Toshiba, which we efe to as PST below), who woul wok with pocessos though contactual aangements. In this case, the question is whethe thee exists a cost allocation that incentivizes these pouces to take pat in a collective system that woks with all available pocessos while guaanteeing at least as high esign incentives as if they wee to opeate thei inepenent plans. To aess this question, we aopt the RN famewok an moel the two popose inepenent plans by two epesentative pouce noes A an B. In aition, note that if HP an PST opeate thei inepenent plans, the emaining pouces can still paticipate in the plan un by WMMFA. This motivates us to moel a thi pouce noe C in the RN instance that epesents all othe pouces whose poucts ae ecycle by the pocessos contacte by the WMMFA. Capacities: The pocessing capacity at HP s ecycling facility in Califonia is estimate to be 18 million lbs pe yea 5. Howeve, given that thee ae thee states on the west coast that have e-waste laws (CA, OR an WA), we o not have soli ata on how much of that capacity woul be use to ecycle the etun volume fom Washington state. Similaly, we o not have soli capacity ata fom the contactos that PST woks with. Hence, we shall enote these capacity paametes as k (A) an k (B) in the instance an late iscuss the ange of these paametes une which the esign-einfocing conition is met. Fo pouce C, it has access to the total capacity contacte by WMMFA in Washington, which we assume to be the same as the total volume pocesse in the plan un by WMMFA in 215 oune up to the neaest million (i.e., 43 million lbs). Retun volumes: We calculate the etun volume of the thee epesentative pouces base on the Washington opeating ata in 215, incluing the total volume pocesse (42,585,847 lbs 6 ) an each pouce s etun shae 7. We summaize the esults in Table (page 8)

2 2 Aticle submitte to Management Science; manuscipt no. MS Annual capacity in an Annual etun Aveage cost Pouce Retun shae iniviual system (million lbs) volume (million lbs) (cents/lb) A k (A) 5.52% B k (B) 21.1% C % Table 4 Summay of the RN instance base on the Washington state EPR implementation Pocessing technology efficiency: To obtain a easonable estimate of the oveall pocessing technology efficiency at the pocessos, we aopt the cost stuctue popose by Gui et al. (216) 8. Base on a minimum ecycling ate of 9% an a 7:3 volume atio between TVs/monitos an IT poucts 8, we can calculate the aveage unit ecycling cost incue at each pocesso noe (see Table 4). The RN instance illustate in Table 4 satisfies the esign-einfocing conition ientifie in Theoem 1 (une which the maginal contibution base cost allocation ensues goup incentive compatibility an a supeio esign outcome in the collective system) if k (A) +k (B) <31.25 (1) This conition tanslates to a total capacity fo pouces A an B that is less than 2.75 times of thei total volume, which is ealistic consieing pactice. Fo example, the conition is met if HP shaes its capacity at the Califonia ecycling facility equally among the thee states on the west coast that have e-waste legislation (i.e., k (A) =18/3=6), an the capacity level at the contactos that PST woks with equals twice PST s etun volume. Moeove, it can be futhe calculate that when k (A) <8.98 an k (A) +k (B) <31.25 both hol, the RN instance also satisfies the esign-einfocing conition ientifie in Poposition 1 une which the collective system leas to a supeio esign outcome given the ual-base cost allocation. This conition is also ealistic consieing that k (A) =6 when HP equally shaes the ecycling capacity among the states of CA, OR an WA, which by itself is a high estimate as CA typically has a lage e-waste volume than OR an WA. Recall that ou main analysis in the pape assumes that pouct ecyclability an pocess technology efficiency ae substitutes. Hence, the above iscussion inicates that it is plausible fo a collective implementation in the Washington state to enhance incentives fo esign-fo-ecyclability attibutes that substitute pocessing technology efficiency (e.g., using less scews, moulaity esign, using less toxic mateials). Example 3 (Japanese HARL). The Japanese HARL implementation is known fo incentivizing esignfo-ecyclability impovements that ae tailoe fo the ecycling technology use at the pocessos. Below we map HARL implementation ata to ou moel an show the esign-einfocing conition is satisfie when pouct ecyclability an pocessing technology efficiency ae complements. Capacities: A easonable way to moel pouces access to ecycling capacities when opeating inepenently is to assume that they will use thei in-house facilities if thee ae sufficient capacities to pocess thei own volumes. The Japanese HARL implementation povies soli infomation in this ega. In paticula, its Goup B collective system is mainly opeate base on ecycling plants built by joint ventues among 8 L. Gui, A. Atasu, Ö. Egun, L. B. Toktay (216) Efficient Implementation of Collective Extene Pouce Responsibility Legislation. Management Science 62(4): (Table 2)

3 Aticle submitte to Management Science; manuscipt no. MS pouces, of which the capacity an the shaehole infomation is publicly available. We summaize this infomation in Table 5, whee the majo shaehole is efine as one with a significantly highe shae than the est of the shaeholes 9. Obseving that fo some ecycling plants, thee ae two pouces with simila shaes that ae much highe than the othe shaeholes 1, we list both pouces as majo shaeholes in that case. Recycling plant Annual capacity (million units) Majo shaehole(s) Kansai Recycling Systems Co., Lt Shap an Mitsubishi 12 Hokkaio Eco Recycling Systems Co., Lt Hitachi an Mitsubishi 13 East Japan Recycling Systems Co., Lt Mitsubishi 14 Kantou Eco Recycle Co., Lt Hitachi 15 Copoation Hype Cycle Systems Co., Lt Mitsubishi 16 Tokyo Eco Recycle Co., Lt Hitachi 17 Fuji Eco Cycle Co., Lt Fujitsu 18 Geen Cycle Co., Lt Sony 19 Table 5 Pouce-built ecycling plants in the Goup B collective system une the Japanese HARL implementation We assume that the capacity at the ecycling plants in Table 5 will be accessible by its majo shaehole if it opeates inepenently. In cases whee two majo shaeholes exist, we assume that the capacity will be equally ivie if both shaeholes opeate inepenently. Accoingly, we can calculate the amount of ecycling capacity each of these majo pouces has access to in an iniviual system. We summaize the esulting capacity levels as well as the publicly available etun volume infomation of these majo pouces in Table 6. 9 A epesentative example is the East Japan Recycling Systems Co., Lt, whee Mitsubishi hols 82.18% an each of the othe manufactues hols 3.58% of the total shaes. 1 A epesentative example is the Kansai Recycling Systems Co., Lt., whee Shap an Mitsubishi each hol 43.3%, an each of the othe manufactues hols 3.3% of the total shaes. 11 Waste electical an electonic equipment (WEEE): innovating novel ecovey an ecycling technologies in Japan. DTI Global Watch Mission Repot (page 75)

4 4 Aticle submitte to Management Science; manuscipt no. MS Pouce Annual capacity in an iniviual system Annual etun volume (million units) (million units) Mitsubishi Hitachi Sony Fujitsu Shap Table 6 Summay of the ecycling capacity an the etun volume of the majo pouces in the Goup B collective system une the Japanese HARL implementation. Retun volumes: Note that in Table 6, the etun volumes of Hitachi an Fujitsu ae calculate base on the epote figues of.6 an.147 million tons, espectively (fom the efeences cite in the coesponing footnotes). The convesion ate is assume to be 1 ton fo units, calculate base on the total ecycling volume ata in Japan: A total of million units an.468 million tons of appliances wee ecycle in 212 accoing to the AEHA (Association fo Electic Home Appliances in Japan) annual epot 25. Moeove, to ensue ata accuacy, we also estimate the pouces etun volumes base on (i) the annual ecycling volume of each type of appliance covee une HARL (i.e., ai conitiones, CRT TVs, LCD TVs, efigeatos, an washing machines) epote by AEHA 25, (ii) the maket shae of the pouces in an The esulting etun volume numbes ae close to those epote in Table 6. It can be obseve fom Table 6 that except Shap, all othe pouces ae capable of opeating inepenently using the capacity at the ecycling plants whee they ae the majo shaehole. Moeove, it can be obseve that if Shap wee to opeate inepenently, it nees to wok with Mitsubishi since no othe pouce o pouce goup has sufficient capacity to pocess its own volume plus Shap s volume. To captue these two obsevations, we constuct an RN instance consisting of fou pouce noes that epesent the Mitsubishi-Shap subgoup, an Hitachi, Sony an Fujitsu by themselves, espectively. Pocessing technology efficiency: In tems of pocessing technology efficiency levels between the pouces capacity, note that the Mitsubishi goup has a membe company (Mitsubishi Mateials) that has the expetise in electonic mateials/components ecycling an metal pocessing. Hence, it is easonable to assume that the capacity associate with the Mitsubishi-Shap subgoup is the most efficient. In aition, thee exists evience that Hitachi an Sony have ha initiatives to impove the ecycling technology at thei facilities 28. Since Hitachi has moe volume an theefoe potentially benefits moe fom scale economies, we assume Hitachi s capacity is the secon most efficient, an Sony s the thi most efficient. A summay of the RN instance: Motivate by all the above, we constuct an RN instance consisting of fou epesentative pouces {MS,H,S,F} with etun volume equal to π(ms) = =2.45, π(h) =1.454, π(s) =.474 an π(f) =.341 (all numbes in this example ae in million units). We assume Tomoaki Shimaa an Luk N. Van Wassenhove, Close- Loop Supply Chain Activities in Japanese Home Appliance/Pesonal Compute Manufactues: A Case Stuy, Inten. Jounal of Pouction Economics, http: //x.oi.og/1.116/j.ijpe Waste electical an electonic equipment (WEEE): innovating novel ecovey an ecycling technologies in Japan. DTI Global Watch Mission Repot. 25

5 Aticle submitte to Management Science; manuscipt no. MS these pouces have access to sufficient capacity fo thei own volumes at fou iffeent pocessos, espectively, an thei capacity levels equal k (MS) = =3.135, k (H) =1.595, k (S) =.85 an k (F) =.56. The efficiency anking of these fou pocessos is τ (MS) <τ (H) <τ (S) <τ (F). It can be calculate that this RN satisfies the esign einfocing conition ientifie in Poposition 5 of the pape when pouct ecycability an pocessing technology efficiency ae complementay, i.e., π(h) + π(s) + π(f) <k (MS) ; π(s) + π(f) <k (H) ; π(f) <k (S) (2) Appenix B: Two Aitional Extensions B.1. Extension to Pouct-iffeentiate Pocesso Efficiency In pactice, the ecycling efficiency at a given pocesso may be iffeentiate by pouct. This can happen when the pocessing technology equie fo iffeent poucts vaies, an pocessos specialize in pocessing cetain pouct types. Fo example, Apple s Liam obot cuently specializes in ecycling iphone 6 evices (Reutes 216) 29, an pocessos equippe with toxic mateial sepaation an ecycling capabilities specialize in pocessing electonics that contain these mateials. To captue this heteogeneity, we consie a moe geneal vesion of the multiplicative moel of the unit ecycling cost consiee in the main analysis. That is, in this subsection, we allow the efficiency measue of a pocesso to epen on the pouct π an enote it as τ π. Thus unit ecycling cost c π equals c π =λ π τ π + c π π Π R. (3) Poposition 7. Given any RN instance whee the unit ecycling cost is moele as in (3), if thee exists a linea esign-base allocation x l that is goup incentive compatible an ensues a supeio equilibium esign pofile compae to an iniviual system, i.e., Λ ne l Λ in, then the RN instance satisfies π(i) k (j) i fo which {j :τ π(i) (j) <τπ(i) } 2. (i) j:τ π(i) (j) <τπ(i) (i) Poof of Poposition 7. Please efe to Appenix C.4.1 fo the poof. Note that the set {j :τ π(i) (j) <τπ(i) (i) }, which is pouct epenent, euces to {j :τ (j)<τ (i) } when the cost efficiency of a given pocesso is not iffeentiate by pouct, as in the main analysis. Hence, the conition ientifie above genealizes the low-synegy conition in Theoem 1 to the case with pouct-epenent heteogeneity in pocessing efficiency. We conclue that the esign-einfocing chaacteization eive in the main analysis can be extene to this moe geneal setting. B.2. Extension to Retun Volume Uncetainty In this extension, we analyze the impact of etun volume uncetainty. To o so, we moel each volume paamete π(i) as an inepenent anom vaiable whose pobability istibution is known to the pouces. We exten the efinition of an RN instance to inclue these pobabilistic paametes, an call it an RN instance une etun volume uncetainty. Accoingly, we genealize the efinition of esign outcomes une EPR as follows: The optimal esign pofile inuce by an iniviual system is the esign pofile that minimizes pouces expecte ecycling costs plus thei esign investments. The equilibium esign pofile 29 Reutes Apple s obot ips apat iphones fo ecycling. us-apple-poucts-ecycling-iuskcnwn1y.

6 6 Aticle submitte to Management Science; manuscipt no. MS in a collective implementation is a esign pofile such that no pouce can euce the sum of its expecte ecycling cost an its esign investment by unilateal eviation. We show that given an RN instance une etun volume uncetainty, whethe a goup incentive compatible allocation leas to supeio esign incentives fo a pouce citically hinges on the pobability of the RN ealization being esign-einfocing. To see this in the context of the two-pouce setting, ecall that supeio esign incentives (une no etun volume uncetainty) equie π(2) k (1). This conition can be elaxe an still ensue supeio esign incentives fo pouce 1 une etun volume uncetainty. That is, π(2) can take a value smalle than k (1) as long as the associate pobability is sufficiently low. To see this, note that in the eteministic case, the ate of change of pouce 1 s ual-base cost allocation can be stictly highe than the ate of change of its stan-alone cost une a esign-einfocing RN (e.g., in the fist RN instance consiee in Example 1; see Table 2). Hence, une etun volume uncetainty, whethe pouce 1 s expecte cost allocation changes at a faste o slowe ate in esign epens on the elative pobability of a esign-einfocing ealization of the RN vesus a non-esign-einfocing ealization. In fact, fo a two-pouce RN instance, we can show that a sufficient conition to ensue supeio esign incentives fo pouce 1 une the ual-base cost allocation is P(π(2) <k (1) ) P( π(2) k (1) ) E(π(1) ) k (1), whee E( π(1) ) is the expectation of π(1) s etun volume (see Appenix C.4.2 fo the eivation). The above iscussion inicates that uncetainty in RN paametes may incease the likelihoo of some pouces choosing a supeio esign in a collective implementation compae to une an iniviual system. Nevetheless, in geneal, if supeio esign incentives ae equie fo all pouces in conjunction with goup incentive compatibility, then we nee the conition that fo all i such that {j :τ (j) <τ (i) } 2, the sample space of the anom vaiable π(i) is containe in the inteval [ j:τ (j) <τ (i) k (j), ) (see Poposition 1 in Appenix C.4.2). This genealizes Theoem 1, implying that the funamental insights egaing the esign-stability taeoff continue to apply une RN uncetainty. Appenix C: Poofs an Technical Analysis Fo convenience, in this appenix (except fo in the poof of Poposition 7 whee the pocesso efficiency is pouct-iffeentiate), we assume that the inices of the pouces ae aange accoing to the efficiency levels of the capacities that they have access to in an iniviual system. Specifically, i < j implies that τ (i) <τ (j), i.e., (i) is moe efficient. C.1. Technical Details in 4 In this section, we pesent the equilibium analysis of the esign outcome une the ual-base cost allocation. We fist stuy RNs with two pouces, an pesent Poposition 8 that explains the equilibium analysis in Example 1. We then exten the analysis to RNs with n pouces an povie the etaile poof of Poposition 1. C.1.1. RNs with Two Pouces Poposition 8. Consie an RN instance with two pouces. Une the ual-base cost allocation mechanism x, 1. when π(2) k (1), i.e., the capacity at (1) is no lage than pouce 2 s etun volume, thee exists a unique equilibium esign pofile Λ ne that is supeio to the esign pofile inuce by an iniviual system, i.e., Λ ne Λ in.

7 Aticle submitte to Management Science; manuscipt no. MS when π(2) <k (1), if an equilibium esign pofile exists, it is infeio to the esign pofile inuce by an iniviual system, i.e., Λ ne Λ in ; othewise, thee exists a mixe-stategy equilibium une which any esign pofile that can occu with a positive pobability is infeio to Λ in. Poof of Poposition 8. The poof of Poposition 8 is base on the following thee-step poceue. Step 1 (compute the best esponse functions): Fo each pouce i = 1, 2, we calculate the ecyclability level of π(i) that minimizes the sum of i s ual-base cost allocation an its esign investment, i.e., x i +Q i. We calculate this esign choice fo i as a function of the othe pouce s esign stategy. This function, enote as λ π(i), is calle pouce i s best esponse function to j s esign choice. Step 2 (fin the equilibium Λ ne ): We calculate the equilibium esign pofile by solving the two best esponse functions simultaneously. This can be one by fining the intesection point of the two functions. Step 3 (compae Λ ne inuce by an iniviual system (i.e., Λ in). with Λ in): We compae the equilibium esign pofile with the esign pofile Below we explain the calculation etails in each step une the two RN cases escibe in Poposition 8. Case 1: When the RN satisfies π(2) k (1) Step 1: To compute the best esponse functions, we fist analyze the ual-base cost allocation to each pouce. This cost allocation is calculate base on the optimal ual solution of the centalize allotment poblem (C), an thus epens on the socially optimal allotment f. It can be obseve that when π(2) k (1), f vaies epening on the elative ecyclability of the poucts (as epicte in Figue 4(b) an (c)). Accoingly, we can calculate pouce 1 s ual-base cost allocation x 1 as follows. { π(1) c π(1) x 1 (2) = k (1) (c π(2) (2) cπ(2) ) if (1) λπ(1) < π(1) π(1) [c(1) +(cπ(2) (2) cπ(2) )] k (1) (1) (c π(2) (2) cπ(2) ) if (4) (1) λπ(1) Recall that in ou main moel, c π =λ π τ + c π fo each pai of pouct π an pocesso. Hence, it can be obseve that given pouce 2 s pouct esign, fomula (4) is a piecewise linea function of pouce 1 s own esign vaiable, an the beak point occus when =. Since the investment function Q 1 is convex with espect to, we can show that pouce 1 s total cost x 1 +Q 1 is a piecewise convex function of, whose shape epens on the given an is as epicte in Figue 6. It can be obseve base on the figue that the minimize of x 1 +Q 1 (which is by efinition pouce 1 s best esponse λ π(1) ) can occu at two iffeent values efine as below. l 1. =(Q 1 ) 1( π(1) τ (2) ) ; l2. =(Q 1 ) 1( π(1) τ (1) ) It can be obseve that l 1 < l 2 since τ (1) < τ (2) (i.e., (1) is the moe cost efficient pocesso) an the investment function Q 1 is convex eceasing. Futhe calculation inicates that pouce 1 s best esponse λ π(1) is a two-piece step function base on l 1 an l 2. { λ π(1) l 2 = l 1 if if > whee is a constant efine by the fomula Q1 (l 2 )+ π(1) τ(2) l 2 Q 1 (l 1 ) π(1) τ(1) l 1. It can be veifie that is a constant between l 1 an l 2. π(1) (τ (2) τ (1) ) (5) (6)

8 8 Aticle submitte to Management Science; manuscipt no. MS Figue 6 This figue shows pouce 1 s total cost function x 1 +Q 1 when π(2) k (1). In the figue, l 1 an l 2 ae as efine in (5), an is a constant in (l 1,l 2), calculate base on the given RN instance. x 1 +Q1 x 1 +Q1 x 1 +Q1 x 1 +Q1 l 1 l 2 l 1 l 2 l 1 l 2 l 1 l 2 (a) When <l 2 (b) Whenl 1 (c) When < l 2 () When >l 2 Figue 7 This figue shows the equilibium esign pofile Λ ne une the ual-base allocation when π(2) k (1). In this figue, l 1, l 2 an q 1 ae as efine in (5) an (8), an is the same constant as shown in Figue 6. The vetical an hoizontal lines epesent the best esponse functions of pouce 1 an 2 espectively. q 1 Λ ne q 1 Λ ne l 1 l 2 l 1 l 2 (a) When q 1 (b) When q 1 > We continue to compute the best esponse function fo pouce 2, which is moe staightfowa. This is because we can calculate that pouce 2 s ual-base cost allocation equals x 2 π(2) = π(2) c(2) (7) une both pattens of the socially optimal allotment f. Hence, we conclue that when the RN satisfies π(2) k (1), x 2 is inepenent of pouce 1 s esign choice. Accoingly, it can be calculate that pouce 2 s best esponse equals a constant q 1 efine as q 1. =(Q 2 ) 1( π(2) τ (2) ). (8) Step 2: We locate the equilibium esign pofile Λ ne at the intesection the two best esponse functions calculate above (Figue 7). Base on this figue, we conclue that when the RN satisfies π(2) k (1), thee exists a unique equilibium esign pofile une the ual-base cost allocation, an it equals eithe {l 1,q 1 } o {l 2,q 1 } epening on the specification of the RN consiee. Step 3: We have shown in 3.2 that fo each pouce i, the pouct esign inuce by an iniviual system is λ π(i) in = (Q i ) 1( π(i) τ (i) ). Compaing this fomula with (5) an (8), we immeiately obtain that l 2 = in an q 1 = in. Moeove, as we have shown l 1 <l 2, we know that l 1 < in, inicating the possibility of a stictly supeio esign of the pouct π(1) une the ual-base cost allocation compae to that in an iniviual system. This completes the poof of the fist esult in Poposition 8.

9 Aticle submitte to Management Science; manuscipt no. MS Figue 8 This figue shows pouce 1 s total cost function x 1 +Q 1 when π(2) <k (1) < π(1) + π(2). In the figue, l 2 an l 3 ae as efine in (1). x 1 +Q1 x 1 +Q1 x 1 +Q1 l 2 l 3 l 2 l 3 l 2 λ l π(1) 3 (a) When l 2 (b) When l 2 < <l 3 (c) When l 3 Case 2: When the RN satisfies π(2) <k (1) We continue to pove the secon esult in Poposition 8, i.e., the ual-base cost allocation leas to infeio pouct esigns than those inuce in an iniviual system when π(2) <k (1). We analyze two sub-cases as follows. Case 2(a): We consie the sub-case whee the capacity at (1) is sufficient to pocess the etun volume of π(2) but not all poucts etune, i.e., π(2) <k (1) < π(1) + π(2). We follow the thee-step poceue intouce at the beginning of the poof to analyze this sub-case. Step 1: We fist obseve that when π(2) <k (1) < π(1) + π(2), simila to Case 1, thee ae two possible pattens of the socially optimal allotment epening on the elative ecyclability of the poucts (Figue 4() an (e)). Accoingly, we can calculate pouce 1 s ual-base cost allocation x 1 as follows. { π(1) c π(1) x 1 (2) = k (1) (c π(1) (2) cπ(1) ) if (1) λπ(1) < π(1) π(1) [c ] k (1) (1) (c π(2) (2) cπ(2) ) if (9) (1) λπ(1) (1) +cπ(2) (2) cπ(2) It is easy to see that x 1 is a piecewise linea function of pouce 1 s esign vaiable given any value of, just as in Case 1. Howeve, the iffeence is that x 1 efine in (9) is convex in as oppose to being concave in Case 1. This esults in a iffeent functional stuctue of pouce 1 s total cost x 1 +Q 1 (Figue 8), whose minimum may be attaine at the following two values of. l 2. =(Q 1 ) 1( π(1) τ (1) ) ; l3. =(Q 1 ) 1( π(1) τ (2) +k (1) (τ (2) τ (1) ) ). (1) Since we can calculate that π(1) τ (2) k (1) (τ (2) τ (1) ) π(1) τ (1), we conclue that l 3 >l 2 as Q 1 is convex eceasing. Hence, it can be shown that the best esponse of pouce 1, λ π(1), is no longe a step function as in Case 1. Instea, it equals l 2 if l 2 λ π(1) = λ π(2) if (l 2,l 3 ) (11) l 3 if l 3 We continue to compute the best esponse of pouce 2 in this sub-case. We can calculate that in this case, pouce 2 s ual-base cost allocation equals { π(2) [c π(2) x 2 (1) = +cπ(1) (2) cπ(1) ] if (1) λπ(1) < π(2) (12) π(2) c(2) if Following the same agument that we use to eive pouce 1 s best esponse in Case 1, we can show that λ π(2) is the following two-piece step function: { λ π(2) q 2 = q 1 if if > (13)

10 1 Aticle submitte to Management Science; manuscipt no. MS Figue 9 This figue shows the equilibium esign pofile Λ ne une the ual-base allocation when π(2) <k (1) < π(1) + π(2). In this figue, l 2, l 3, q 1 an q 2 ae as efine in (1) an (14) espectively, an is the constant use in fomula (13). The best esponse function of pouce 1 is epesente by the hoizontal lines, an that of pouce 2 s is epesente by the vetical an the iagonal lines. Note that in panel (b), no q 2 = pue-stategy equilibium esign pofile exists. q 2 = q 2 Λ ne = q 1 Λ ne q 1 q 1 λ π(1) l 2 l 3 l 2 π(1) λ l 3 l 2 l 3 λ π(1) (a) When l 2 (b) When l 2 < <l 3 (c) When l 3 whee q 1 an q 2 equal q 1. =(Q 2 ) 1( π(2) τ (2) ) ; q2. =(Q 2 ) 1( π(2) τ (1) ). (14) an is a constant efine by the fomula Q2 (q 1 )+ π(2) τ(2) q 1 Q 2 (q 2 ) π(2) τ(1) q 2. It can be veifie that π(2) (τ (2) τ (1) ) (q 1,q 2 ) (it is easy to show that q 1 <q 2 as τ (2) >τ (1) an Q 2 is convex eceasing). Step 2: We fin the equilibium esign pofile at the intesection of the two best esponse functions calculate above (Figue 9). Note that in this sub-case, a pue-stategy equilibium oes not exist if the RN instance leas to the situation shown in Figue 9(b). In that situation, we show that a mixe-stategy equilibium always exists (see Lemma 5 pesente at the en of Step 3). Step 3: Base on the calculations aleay pesente in this poof, we have l 3 >l 2 = in an q 2 >q 1 = in. Hence, we conclue that when π(2) < k (1) < π(1) + π(2), all possible values of the best esponse function of each pouce i shown in fomula (11) an (13) ae no bette than i s esign choice inuce by an iniviual system. Hence, x leas to an infeio equilibium esign pofile if such an equilibium exists (Figue 9(a) an (c)). We futhe show that this esult can be extene to the mixe-stategy equilibium in the situation shown in Figue 9(b). Definition 4 (Mixe-stategy Equilibium). Consie an RN instance an a cost allocation mechanism x. A mix-stategy eisgn pofile {f i (λ π(i) ), i N} is a mixe-stategy equilibium if no pouce can euce the sum of its expecte esign investment an ecycling cost allocation by unilateal eviation, i.e., fo each pouce i,... [xi ({,...,λ π(n) }) + Q i (λ π(i) )] n j=1 fj (λ π(j) )λ π(j)... [xi ({,...,λ π(n) })+Q i (λ π(i) )] n j i fj (λ π(j) )λ π(j) f i (λ π(i) )λ π(i) hols fo any pobability istibution f i of the λ π(i) vaiable. Base on Definition 4, we pove the following esult. Lemma 5. Given an RN instance such that π(2) <k (1) < π(1) + π(2) an l 1 < <l 3, thee always exists a mixe-stategy equilibium esign pofile {f 1 ( ),f 2 ( )}. Moeove, une this equilibium, i=1,2 an λ π(i) such that f i (λ π(i) )>, the inequality λ π(i) λ π(i) in is satisfie.

11 Aticle submitte to Management Science; manuscipt no. MS Poof of Lemma 5. We follow a two-step poceue to pove Lemma 5. STEP 1: We popose the following mix-stategy esign pofile {f 1 ( ),f 2 ( )}. The pobability istibution f 1 ( ) is efine such that f 1 ( )=1 whee is the beak point efine in fomula (13). This is equivalent to pouce 1 choosing a fixe ecyclability level of (λ π(1) The pobability istibution f 2 ( ) is efine such that f 2 (q 1 ) =. Q1 ) a 1 a 2 a 1 an f 2 (q 2 ) =1. f 2. (q 1 ), whee q 1 =(Q 2 ) 1( π(2) τ ). (2) an q2 =(Q 2 ) 1( π(2) τ ) (1) ae as efine in fomula (13), an a 1 an a 2 ae two constants such that a 1. = π(1) τ (2) k (1) (τ (2) τ (1) ) an a 2. = π(1) τ (1). That is, pouce 2 anomizes between two ecyclability levels q 1 an q 2. We nee to pove that in the RN instance whee pue-stategy equilibium oes not exist, i.e., in the RN instance epicte in Figue 9(b), f 2 (q 1 ) (,1) so that f 2 is inee a pobability istibution an {f 1,f 2 } is inee a mix-stategy esign pofile. To see this, ecall that the RN instance in Figue 9(b) satisfies l 2 < <l 3. Since the esign investment function Q 1 is convex eceasing, we know that Q 1 (l 2 )<Q 1 ( )< Q 1 (l 3 ). Recall that l 2. =(Q 1 ) 1( π(1) τ (1) ) an l3. =(Q 1 ) 1( π(1) τ (2) +k (1) (τ (2) τ (1) ) ), it can be calculate that Q 1 (l 2 ) = π(1) τ (1) = a 2 an Q 1 (l 3 ) = π(1) τ (2) +k (1) (τ (2) τ (1) ) = a 1. Hence, Q 1 (l 2 )<Q 1 ( )<Q 1 (l 3 ) is equivalent to. a 2 <Q 1 ( )< a 1 (15) whichimpliesthat< Q 1 ( ) a 1 <a 2 a 1.Theefoe,weconcluethatf 2 (q 1 ) =. Q1 π(1) (λ Hence, f 2 is a pobability istibution, an {f 1,f 2 } is a mix-stategy esign pofile. ) a 1 a 2 a 1 (,1). STEP 2: Next, we show that (f 1,f 2 ) is an equilibium esign pofile. To o this, we evaluate each pouce s expecte total cost (i.e., esign investment plus its ual-base cost allocation) une this esign pofile. Given that pouce 1 chooses its own esign accoing to f 1 (i.e., selects the ecyclability level ), then accoing to Figue 9(b), pouce 2 s total cost x 2 +Q 2 is minimize at eithe =q 1 o =q 2. Hence, any mixe stategy that anomizes between q 1 an q 2 also minimizes pouce 2 s total cost. Since f 2 is an example of such mixe stategies, pouce 2 s total cost is also minimize une f 2, that is, f 2 is a best esponse fo pouce 2 when pouce 1 follows f 1. Giventhatpouce2choosesitsownesign accoingtof 2,wecalculatepouce1 sexpecte ual-base cost allocation (enote as E f2 [x 1 ]) as follows. Fo notational bevity, in this fomula, we enote the unit ecycling cost of π(2) at any pocesso when its ecyclability level equals =q as c π(2) (q). = q τ + c π(2) π(1) π(1) c k (2) (1) (c π(1) (2) cπ(1) ) if (1) λπ(1) q [ 1 ] f 2 π(1) (q 1 ) π(1) [c(1) +cπ(2) (q (2) 1) c π(2) (q (1) 1)] k (1) [c π(2) (q (2) 1) c π(2) (q (1) 1)] [ ] E f2 [x 1 ]= +f 2 π(1) (q 2 ) π(1) c k (2) (1) (c π(1) (2) cπ(1) ) (1) if (q 1,q 2 ) (16) [ ] π(1) π(1) c(1) +f2 (q 1 ) ( π(1) k (1) ) [c π(2) (q (2) 1) c π(2) (q (1) 1)] [ ] +f 2 (q 2 ) ( π(1) k (1) ) [c π(2) (q (2) 2) c π(2) (q (1) 2)] if q 2 Replacing c π with λ π τ + c π in the above fomula, we can calculate that E f2 [x 1 ] equals the following piecewise linea function of pouce 1 s own esign vaiable. a 1 +b 1 if q 1 E f2 [x 1 ]= (f 2 (q 1 ) a 2 +f 2 (q 2 ) a 1 ) +(f 2 (q 1 ) b 1 2+f 2 (q 2 ) b 1 ) if (q 1,q 2 ) (17) a 2 +(f 2 (q 1 ) b 1 2+f 2 (q 2 ) b 2 2) if q 2

12 12 Aticle submitte to Management Science; manuscipt no. MS whee a 1 an a 2 ae the two constants intouce in the efinition of the pobability istibution f 2, an the othe thee constants equal b 1 = π(1) c π(1), b 1 2 = π(1) c π(1) +( π(1) k (1) ) q 1 (τ (2) τ (1) )], an b 2 2= π(1) c π(1) +( π(1) k (1) ) q 2 (τ (2) τ (1) )]. Accoingly, we can calculate the eivative of E f2 [x 1 ]+Q 1 with espect to as follows. [ E f2 (x 1 )( )+Q 1 ( ) ] a 1 +Q 1 ( ) if q 1 = (f 2 (q 1 ) a 2 +f 2 (q 2 ) a 1 )+Q 1 ( ) if (q 1,q 2 ) (18) a 2 +Q 1 ( ) if q 2 Accoing to fomula (15), a 1 +Q 1 ( )< hols. This inicates that when q 1, the eivative of E f2 [x 1 ] + Q 1 with espect to is negative. Hence, pouce 1 s total expecte cost eceases in when (,q 1 ], an is minimize at q 1 in this inteval. Similaly, fomula (15) also inicates that Q 1 ( ) + a 2 >. Hence, in the inteval whee [q 2, ), pouce 1 s expecte total cost E f2 [x 1 ]+Q 1 inceases in an thus is minimize at q 2. Theefoe, the global minimize of the cost function E f2 [x 1 ]+Q 1 must esie in the inteval [q 1,q 2 ], which can be obtaine by solving the equation [E f2 (x 1 )(λπ(1) )+Q 1 ( )] =(f 2 (q 1 ) a 2 +f 2 (q 2 ) a 1 )+Q 1 ( )=. The solution to this equation is = (Q 1 ) 1 ( f 2 (q 1 ) a 2 f 2 (q 2 ) a 1 ), which equals ( ) (Q 1 ) 1( ) f 2 (q 1 ) a 2 (1 f 2 (q 1 )) a 1 =(Q 1 ) 1 Q 1 ( )+a 1 (a 2 a 1 ) a 1 a 2 a 1 =(Q 1 ) 1 (Q 1 ( ))= (19) Hence, we conclue that if pouce 2 aopts the mixe stategy f 2, = minimizes pouce 1 s expectetotalcostanisthebestesponsefopouce1.wethenconcluethat(f 1,f 2 )isamixe-stategy equilibium esign pofile base on Definition 4. This completes the poof of Lemma 5. The above lemma inicates that une the mixe-stategy equilibium {f 1,f 2 }, any pouct esign outcome that can occu with positive pobability is infeio to the one inuce by an iniviual system. Case 2(b): We finally consie the othe sub-case whee (1) has enough capacity to pocess the entie volume, i.e., k (1) π(1) + π(2). In this case, the socially optimal allotment f sens all poucts to the cheape pocesso (1) (Figue 4(f)) egaless of the esign pofile. Because of this, the optimal esign choices of the two pouces ae not coelate in the collective system, an thus the equilibium analysis becomes much simple. Specifically, we can calculate that the ual-base cost allocation of each pouce i equals x i π(i) = π(i) c. (2) (1) Accoingly,theesignchoicefoithatminimizesitstotalcostx i +Q i equalsλ π(i) =(Q i ) 1( π(i) τ (1) ), which is no smalle than λ π(i) in. Moeove, fo pouce 2, we can calculate that λ π(2) > in since τ (1) <τ (2) an the investment function Q 2 is convex eceasing. That is, paticipating in a collective system inuces pouce 2 to aopt a stictly wose esign in this case of the RN. Combining the above esults, we conclue that Poposition 8 hols.

13 Aticle submitte to Management Science; manuscipt no. MS C.1.2. RNs with n Pouces Poof of Poposition 1 In this online appenix, we povie the etaile poof of Obsevation 1, which leas to Poposition 1 accoing to the sketch of the poof pesente in the appenix of the pape. Recall that Obsevation 1 states the following. Obsevation 1 Une the low-synegy conition π(i) k j<i (j) i N, each pouce i s ual-base cost allocation in the oiginal RN instance G is equivalent to that in a sub-netwok G i that consists of pouce i an all othe pouces whose capacity is less efficient than (i). Poof of Obsevation 1 WeshowthatObsevation1holsintwosteps.InStep1,wepovieanalgoithm to compute the socially optimal allotment in any RN with a given esign pofile of the poucts; this enables us to come up with close-fom expessions of the optimal ual solution [β π,α] as functions of the esign pofile. In Step 2, we show that une the low-synegy conition, the optimal ual solution that coespons to i s pouct an capacity is inepenent of any othe pouce with moe efficient capacity (i.e., any pouce j <i). STEP 1: The algoithm we popose to compute the socially optimal allotment is a geey one (efee to as the geey allotment algoithm), escibe in Algoithm 2 shown on the next page. Algoithm 3: A geey pouct allotment algoithm in an n-pouce RN Input: An RN instance G Output: A pouct allotment solution f 1. Rank the poucts in eceasing oe of its ecyclability level. Let σ(π) enote the inex of pouct π une the esulting anking. Let Π = Π an R = R equal the set of poucts an pocessos in G, espectively. Let f [π,] = an π. Let t =. while Π t o 1. Let π. = agmax π Πt λ π be the pouct with the wost esign in Π t. Let. = agmin Rt τ be the most efficient pocesso in R t. 2. Let f [ π, ] = min{ π,k }. Let π = π f [ π, ] an k = k f [ π, ]. 3. Let Π t+1. = Πt \{π Π t : π = }, an R t+1. = Rt \{ R t : k = }. Let t = t+1. en Output the pouct allotment f. The optimality of the f solution geneate by the geey allotment algoithm (Algoithm 3) can be poven base on uality theoy. Claim 1. The outing f compute by the geey allotment algoithm (Algoithm 3) is optimal to the centalize tanspotation poblem (C). Poof of Claim 1 Fo convenience, we assume in this poof that f is nonegeneate; the poof also applies to the egeneate case if we focus on one of the multiple potential ual solutions. We pove the optimality of f base on uality theoy, i.e., we show the existence of a feasible ual solution of(c)thatsatisfiesthecomplementay slacknessconstaintswithespectto f. To thisen,wefistfomulate the ual poblem of (C) in the n-pouce case. (D) max π Π π β π + k α s.t. α +β π c π π Π, R; α R (21) R

14 14 Aticle submitte to Management Science; manuscipt no. MS Below, we show that the ual solution efine by fomula (A.1)-(A.2) in the appenix of the pape is feasible to (D) an satisfies complementay slackness with espect to the allotment solution f compute by Algoithm 3. Fo iscussion convenience, we uplicate the fomulas below: Let R π. ={ : f[π,] >} be the set of pocessos whee pouct π is pocesse une f. We also enote the least an the most cost efficient pocesso in R π as π. =max{j:j R π} an π. =min{j:j R π} espectively. Fo each pouce i N, we fist efine β π(i) as β π(i) =c π(i) π(i) (c π(j) π(j) c π(j) π(j) ). (22) j:λ π(j) <λ π(i) Then consie the pocesso (i). Assume thee exists t N such that (i) R π(t) \ π(t), i.e., (i) pocesses pouct π(t) une f yet (i) is not the least cost efficient pocesso to o so. In othe wos, π(t) is the most ecyclable pouct pocesse at une f (which we enote as π in the appenix of the pape). We efine α (i) as follows. α (i) =(c π(t) (i) cπ(t) π(t) )+ (c π(j) π(j) c π(j) π(j) ) (23) j:λ π(j) <λ π(t) Notethatα (i) iswell-efine,sincetheecannotexistt 1 t 2 suchthat(i)belongstobothsetsr π(t1 )\ π(t1 ) an R π(t1 )\ π(t2 ) une the geey outing algoithm. In case that (i) / n i=1 R π(i)\ π(i), we efine α =. We show that [α,β π ] is a feasible solution to (D) that satisfies complementay slackness with espect to f. Dual feasibility: Fo each i N, by efinition, π(i) is the least cost efficient pocesso in R π(i). Thus c π(i) c π(i) π(i) hols fo all R π. This inicates that α R. Consie any pouct π(i) an any pocesso. Assume that R π(t) \ π(t) fo some t N. We can calculate that α +β π(i) c π(i) = (λ j:λ π(j) λ π(j ) ) (τ π(t) <λ π(j) λ π(i) π(j) τ ) if λ π(t) <λ π(i) if t=i (24) (λ π(j) λ π(j ) ) (τ j:λ π(i) <λ π(j) λ π(t) τ π(j) ) if λ π(t) >λ π(i) In the above fomula, π(j ) enotes the pouct that is the least ecyclable among those with a bette esigncompaetoπ(j).hence,weknowthat λ π(j) >λ π(j ).Futhemoe,sincetissuchthat R π(t) \ π(t), we know that π(t) is pocesse at. Since une the geey outing algoithm, less ecyclable poucts ae pocesse using moe efficient capacity, we conclue that (i) when λ π(t) <λ π(j), τ π(j) τ hols, an (ii) when λ π(t) >λ π(j), τ π(j) τ hols. Accoing to fomula (24), this inicates that the ual constaint α +β π(i) c π(i) is satisfie fo evey pai of pouct π(i) an pocesso such that R π(t) \ π(t) fo some t. We then show that α +β π(i) c π(i) emains vali in the case whee / n R i=1 π(i)\ π(i). In the non-egeneate case, we know that if / n i=1 R π(i)\ π(i), then must be only patially use o not use at all une f. Let be the pocesso that is patially use (thee is only one such pocesso une the geey outing algoithm). Note that if / n i=1 R π(i)\ π(i), then must be no moe cost efficient than, i.e., c π(i) c π(i). Hence, we can calculate that α +β π(i) c π(i) =β π(i) c π(i) c π(i) π(i) c π(i) = (c π(j) π(j) c π(j) π(j) ) j:λ π(j) <λ π(i) j:λ π(j) <λ π(i) (λ π(i) λ π(j) ) (τ π(j) τ π(j) ) (25)

15 Aticle submitte to Management Science; manuscipt no. MS (D). Combining the above obsevations, we conclue that [α,β π ] is a feasible solution to the ual poblem Complementay slackness: Any pocesso that is not fully use une f is not in the set n i=1 R π(i) \ π(i), an theefoe is assigne a ual value α =. Consie any pouct π(i) an any pocesso. Assume that f [π(i),] >. Then by efinition R π(i). If R π(i) \ π(i), then by fomula (24) we know that α +β π(i) c π(i) =. If is the pocesso π(i), we consie two situations. In the fist situation, we assume that π(i) is the most ecyclable pouct in Π. Then we know that = π(i) is actually the patially use pocesso. Hence, by fomula (25), α +β π(i) c π(i) =c π(i) cπ(i) =. In the secon situation, π(i) is not the most ecyclable pouct. Consie the set S of poucts pocesse at = π(i). Accoing to the geey outing algoithm, we know that π(i) is the least ecyclable pouct in S. In aition, if we let π(t) enote the most ecyclable pouct in S, then all pouct π(j) such that λ π(i) <λ π(j) <λ π(t) ae in S an ae entiely pocesse at, inicating that π(j) = π(i), which is exactly. Hence, we know that τ π(j) =τ j :λ π(i) <λ π(j) <λ π(t). Thus by fomula (24), α +β π(i) c π(i) = as well. Combining the above esults, we pove that [α,β π ] is a feasible ual solution to (C) that satisfies complementay slackness with espect to f. By uality theoy, this inicates that f is a socially optimal outing in the RN. This completes the poof of Claim 1. STEP 2: In this step, we pove that, une the low-synegy conition π(i) j<i k (j) i N, fo each pouce i, the optimal ual solution [β π(i),α π(i) ] is inepenent of any pouce j <i. We fist show this fo β π(i). Fo convenience, we ewite fomula (A.1) that calculates β π(i) as follows. β π(i) =c π(i) π(i) + }{{} 1 π(j):λ π(j) <λ π(i) an j>i (c π(j) π(j) c π(j) π(j) ) } {{ } 2 + π(j):λ π(j) <λ π(i) an j<i (c π(j) π(j) c π(j) π(j) ) } {{ } 3 In the following points (i) (iii), we show that each tem in the above fomula is inepenent of both the pouct an the capacity of any pouce j > i, espectively. (i) Pat 1 in (26) is inepenent of pouces j <i: This is ue to the fact that fo any i, π(i) coespons to a pocesso (l) whee l i. To illustate this, consie an example RN instance with fou pouces that satisfies π(i) > j<i k (j) 3. Assuming the esign pofile is such that λ π(3) > > >λ π(4), the socially optimal allotment is as epicte in Figue 1. We can make the following obsevations. In geneal RNs, we can show that fo any pouct π(i), eithe its last unit is allotte to (i) (as in the case of π(3) an π(4) in this example), o it is entiely allotte to some (j) whee j >i (as in the case of π(1) an π(2) in this example). Hence, π(i) always coespons to a pocesso (l) whee l i. (ii) Pat 2 in (26) is inepenent of pouces j <i: This is the case as we can show that fo any π(j) such that λ π(j) < λ π(i), both π(j) an π(j) coespon to (l) s whee l i. In othe wos, π(j) is ecycle at pocessos less efficient than (i) une the socially optimal allotment. To see this, note (26) 3 Fo convenience, we consie a non-egeneate vesion of the low-synegy conition. The agument can be genealize to egeneate cases as well.

16 16 Aticle submitte to Management Science; manuscipt no. MS π(3) π(1) π(2) π(4) Figue 1 (1) (2) (3) (4) An RN example Fo pouct π(3), ue to the low-synegy conition π(3) >k (1) +k (2), its last unit will be allotte to (3). This is also the case fo pouct π(4): Since π(4) >k (1) +k (2) +k (3), the last unit of π(4) will be allotte to (4). Fo pouct π(1) an π(2), we can see that (1) an (2) ae aleay satuate by π(3) when these two poucts ae allotte in the geey algoithm. Moeove, ue to the moeling assumption, we know that (1), (2), (3) combine have sufficient capacity to pocess π(1), π(2), π(3). Hence, π(1) an π(2) ae both allote to (3) only. that such a pouct π(j) will be allote afte π(i) in the geey algoithm an thus will only be allote to pocessos that ae less efficient than π(i), which we know fom point (i) is less o equally efficient compae to (i). (iii) Pat 3 in (26) equals zeo: This is because fo any π(j) such that λ π(j) <λ π(i) an whose inex j <i, it must be ecycle at only one pocesso une the socially optimal allotment, i.e., π = π. We have shown this in point (i), illustate by the allotment of pouct π(1) an π(2) in the RN in Figue 1. Combining the above obsevations, we conclue that β π(i) is inepenent of any pouce j <i. We continue to show that α(i) is also inepenent of any pouce j <i. Simila to the above analysis to β π(i), we also ewite fomula (A.2) that calculates α(i) as the summation of thee pats. α (i)=(c π (i) c π (i) (i) ) + (c π(j) π(i) π(j) c π(j) π(j) ) + }{{} π(j):λ π(j) <λ π (i) an j>i π(j):λ π(j) <λ π (i) an j<i 1 } {{ } 2 (c π(j) π(j) c π(j) π(j) ) } {{ } 3. (27) We also show that each pat in this fomula is inepenent of eithe the pouct o the capacity of any pouce j < i. The poof is also base on the agument in (i)-(iii) above, plus some aitional obsevations. (iv) Pat 1 in (27) is inepenent of pouces j <i: This is because fo any i, π (i) coespons to a pouct π(l) whee l i. To see this, note that accoing to point (i) above, if thee exists a pouct π(j) whee j >i an that is less ecyclable than (i) (i.e., allotte befoe π(i) in the geey algoithm), then ue to the low-synegy conition, (i) will be satuate by π(j), i.e., π (i) =π(j); this is the case fo i=1,2 in the example shown in Figue 1. Othewise, we show that the last unit of π(i) is allotte to (i), an the emaining capacity at (i) is sufficient to pocess the poucts that follow (i) in the geey algoithm until a pouct with a lage inex appeas; this is the case fo i=3 in the example, an thus π (3) =4. (v) Pat 2 in (27) is inepenent of pouces j <i an pat 3 thee equals zeo. This can be shown by combining points (iv) with the aguments in points (ii) an (iii), espectively. Combining the above obsevations, we conclue that α (i) is inepenent of any pouce j < i. This completes the poof of Obsevation 1. Hence Poposition 1 follows accoing to the poof pesente in the appenix of the pape. C.2. Technical Details in 5 In this section, we povie the technical etails in the analysis of the maginal contibution-base cost allocation x m.

17 Aticle submitte to Management Science; manuscipt no. MS Poof of Poposition 2 Recall in the poof pesente in the appenix of the pape, we have shown that within an inteval of λ π(i) whee f {j:j i} is unchange, we can calculate that the local minimum of the function x i m+q i must satisfy the equation below: (x i m+q i ) λ π(i) = n f[π(i),(t)] {j:j i} cπ(i) (t) t=i λ π(i) +Qi = n t=i f {j:j i} [π(i),(t)] τ (t)+q i = (28) an thus equals (Q i ) 1 ( n t=i f {j:j i} [π(i),(t)] τ (t)). A key technical step that follows is that the global minimize of the function x i m+q i is achieve at such a local minimum une some f {j:j i} solution. We povie the etails of this step in this online appenix. Assume this is not the case, an the global minimize of x i m +Q i equals some othe value λ π(i). Then we know that thee exist intevals [c 1,λ π(i) ] an [λ π(i),c 2 ], whee the function x i m +Q i stictly eceases an inceases, espectively. Assume c 1 an c 2 ae sufficiently close to λ π(i) such that when pouce i s esign choice vaies within each of the two intevals, the optimal pouct allotment in the coalition {j : j i} is not change. Denote the coesponing optimal pouct allotment solutions as f {j:j i} 1 an f {j:j i} 2. Since x i m+q i stictly eceases on [c 1,λ π(i) ], we know that λ π(i) <(Q i ) 1 ( n t=i (f {j:j i} 1 ) [π(i),(t)] τ (t) ). Similaly, we can conclue that λ π(i) >(Q i ) 1 ( n t=i (f {j:j i} 2 ) [π(i),(t)] τ (t) ). This leas to the inequality (Q i ) 1 ( n t=i (f {j:j i} 1 ) [π(i),(t)] τ (t) )>(Q i ) 1 ( n t=i (f {j:j i} 2 ) [π(i),(t)] τ (t) ). Howeve, this cannot be the case: The optimal allotment changes fom f {j:j i} 2 to f {j:j i} 1 afte pouce i impoves its own esign (i.e., the value of λ π(i) is smalle in the inteval [λ π(i),c 2 ] compae to in the inteval [c 1,λ π(i) ]). Accoing to the geey allotment algoithm, pouct π(i) will be allotte to less efficient pocessos une f {j:j i} 1 than f {j:j i} 2. This means that n t=i f {j:j i} 1 ) [π(i),(t)] τ (t) n t=i (f {j:j i} 2 ) [π(i),(t)] τ (t), an thus (Q i ) 1 ( n t=i (f {j:j i} 1 ) [π(i),(t)] τ (t) ) shoul be no lage than (Q i ) 1 ( n t=i (f {j:j i} 2 ) [π(i),(t)] τ (t) ). This leas to a contaiction. Thus we conclue that the global minimum of x i m + Q i equals (Q i ) 1 ( n t=i f {j:j i} [π(i),(t)] τ (t)) une some f {j:j i} solution. Poof of Poposition 3 In oe to show that x m is iniviually ational (i.e., x i m=v({j :j i}) v({j : j >i}) v(i) hols fo each pouce i), we show that the PA game is sub-aitive. Recall that we use f S to enote the socially optimal allotment within the sub-coalition S. Let S 1 an S 2 enote any two mutually exclusive sub-coalitions in N. Consie thei union S 1 S 2 an the following pouct allotment within this combine coalition: Fo all poucts manufactue by pouces in S 1 an S 2, it is allotte accoing to f S 1 an f S 2, espectively. This is obviously a feasible allotment solution an the total cost incue equals v(s 1 )+v(s 2 ). Since v(s 1 S 2 ) is the minimum total cost that can be achieve in S 1 S 2, we know that v(s 1 S 2 ) v(s 1 )+v(s 2 ). Since this inequality hols fo any mutually exclusive sub-coalitions S 1 an S 2, by efinition, the PA game is sub-aitive. This means that fo any pouce i, v(i {j :τ (j) >τ (i) }) v({j : τ (j) >τ (i) })+v(i). Since x i m is calculate as the iffeence, v(i {j :τ (j) >τ (i) }) v({j :τ (j) >τ (i) }), we conclue that x i m v(i) hols fo each pouce i, i.e., x m is by efinition iniviually ational. Poof of Poposition 4. We povie the complete poof of Poposition 4 in this online appenix. Consie an RN instance that satisfies the conition π(i) j<i k (j) i>2. In oe to show weak convexity of the PA game, we pove that given any esign pofile Λ, fo each pouce i, v(t {i}) v(t) v(s {i}) v(s) hols S T {i+1,...,n}. We o so by stuying two cases of the sub-coalition T. In each case, we fist