Alliance formation in assembly systems with quality-improvement incentives

Alliance formation in assembly systems with quality-improvement incentives

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Alliance formation in assembly systems with quality-improvement incentives Tingting Li, Junlin Chen PII: DOI: Reference:

S0377-2217(20)30179-X https://doi.org/10.1016/j.ejor.2020.02.041 EOR 16361

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European Journal of Operational Research

Received date: Accepted date:

1 February 2019 21 February 2020

Please cite this article as: Tingting Li, Junlin Chen, Alliance formation in assembly systems with quality-improvement incentives, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.02.041

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Highlights • We study externalities of suppliers’ quality improvements in assembly systems. • We study the farsighted stability of coalition formation among suppliers. • The grand coalition is stable under the equal and the proportional allocations. • The grand coalition is conditionally stable under the Shapley value allocations. • Coalitions with lower quality efficiencies could free ride on others.

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Alliance formation in assembly systems with quality-improvement incentives Tingting Lia , Junlin Chenb,∗ a

School of Management Science and Engineering, Dongbei University of Finance and Economics, Dalian 116025, China b Department of Management Science, Central University of Finance and Economics, Beijing, 100081, China

Abstract We study an assembly system where n upstream complementary suppliers produce components and sell them to a downstream manufacturer. The manufacturer assembles all the components into final products and sells them in the final market. The demand for final products is assumed to be deterministic and sensitive to both the selling price set by the manufacturer and the quality-improvement effort levels of all suppliers. The suppliers may form coalitions to better coordinate their wholesale pricing and qualityimprovement effort decisions. We analyze the stability of coalition structures by adopting farsighted stability concepts. To characterize supplier’s profit allocation in a coalition, we consider three allocation rules, including the equal allocations, the proportional allocations and the Shapley value allocations. The results show that the grand coalition is always stable under both the equal allocations and the proportional allocations. However, under the Shapley value allocations, the grand coalition is stable only when the suppliers’ quality efficiencies have relatively small differences. Conditions under which the suppliers will not act independently are presented as well. Because of the positive externalities of quality improvements, we demonstrate that coalitions of suppliers with lower quality efficiencies could benefit from free-riding on the investments of coalitions of suppliers with higher quality efficiencies in the system. ∗

Corresponding author. Tel.: +86 10 6228 8622; Fax: +86 10 6228 8622 Email address: [email protected] (Junlin Chen)

Preprint submitted to Elsevier

February 28, 2020

Keywords: game theory, alliance formation, decentralized assembly system, farsighted stability, externality 1. Introduction In this article, we consider a decentralized assembly system that comprises a downstream manufacturer (assembler) and n upstream suppliers providing complementary components. Examples for decentralized assembly systems span industries from automobiles and various electronic and high-tech sectors to retail and service settings (Nagarajan and Soˇsi´c, 2009). In a decentralized assembly system, the manufacturer buys components from the suppliers and assembles final products to meet market demand. As opposed to the traditional approach in which the assembler procures and assembles individual components, we consider the modular assembly approach, in which the assembler may obtain pre-assembled modules from a reduced base of supplier alliances. The modular assembly approach is usually welcomed by assemblers (Bernstein and DeCroix, 2004). For example, Mercedes/Swatch designed and built the Smart car with around 25 module suppliers while a typical vehicle manufacturer (assembler) is likely to deal with around 200-300 suppliers (Des, 2002). Nissan used a handful of outside suppliers who delivered assembled vehicle sections for the final assembly line of Mississippi-based truck factory (Chappell, 2001). Suppliers voluntarily form alliances and produce a bigger payoff pie than in non-cooperation, which increase their benefits from cooperation under an agreed allocation rules (Cabral and Pacheco-de-Almeida, 2019). Lavie (2007) investigated 367 public traded software firms, and reported that on average each focal firm joined a number of 56.7 alliances to enhance corporate performance. Alliances would take various forms, including complementary resources (Miotti and Sachwald, 2003; Arranz and de Arroyabe, 2008), collaborative R&D (Lavie, 2007), operational cost saving, risk pooling, and even ¨ market power improving (Chen and Chen, 2002; Ozen et al., 2012; Huang et al., 2016). In the decentralized assembly system we considered in this paper, complementary suppliers would engage in alliances to provide price coordination and joint investment on component quality improvement in the modular assembly system. Some studies have considered price coordination (Nagarajan and Soˇsi´c, 2009; Yin, 2010; He and Yin, 2015) and price/production coordination (Granot and Yin, 2008) when studying supplier coalitions in assembly systems. 3

(We use alliances and coalitions interchangeably.) These studies demonstrated that price coordination would eliminate the horizontal decentralization among component suppliers, thereby leads to a lower joint selling price and a higher order quantity from the downstream assembler. In reality, the market demand is affected by not only prices but also product quality levels. For products with the same price, the higher-quality ones usually have higher market demand. In assembly systems, the quality of final products is directly determined by the qualities of sourced components, as final products are assembled with components provided by complementary suppliers. When suppliers exert efforts to improve the qualities of components, the quality of final products will be improved correspondingly. Therefore, the market demand of final products, as well as the order quantity from the assembler, could be enhanced. The quality-improvement efforts of the suppliers contain product upgrading, applying new equipments with high reliability, organizational training, and other innovation operations (Gurnani and Erkoc, 2008). Suppliers could charge high wholesale prices for high-quality components, however, manufacturers still favor component suppliers who drive innovation to enhance component qualities (Wang and Shin, 2015). For example, Ningbo Rongbai, a famous anode material supplier in China, developed a high-quality high nickel ternary anode material in 2017. By using the new material, the downstream battery producer CATL company in China, approached battery products with much better performances on stability and capacity, which allowed CATL to obtain an additional €4 billion order from the automaker BMW on July 9th, 2018. Ningbo Rongbai in turn opened up several new production lines to meet CATL’s new order demand. In assembly systems, when one of the suppliers makes efforts to improve the component quality, all the other suppliers benefit from the enhanced quality of the final products. On the basis of this observation, we investigate how the positive externalities of suppliers’ quality improvement would affect alliance formation among suppliers and the equilibrium solutions of the assembly system. We establish a two-level game as follows. In the first level, suppliers form coalitions and determine the coalition structure. The second level is constructed as a two-stage game. In the first stage, the assembler moves first and determines the profit margin. In the second stage, suppliers in the same coalition offer a single joint wholesale price for the module of components and decide a vector of quality-improvement effort levels for components produced by the coalition. For the allocation of the coalitional 4

profit, we consider three allocation rules, including the equal allocations, the proportional allocations, and the Shapley value allocations. Due to the positive externalities of suppliers’ quality-improvement efforts, the profit generated by a coalition not only depends on the decisions of suppliers in this coalition, but also depends on the decisions of suppliers in other coalitions. Therefore, we adopt the approach based on a cooperative game in partition function form, which was first introduced by Thrall and Lucas (1963), to analyze the stability of coalition structures. Our analysis shows that, due to the positive externalities of quality improvement, coalitions with lower quality efficiencies could benefit from free-riding on the investments of coalitions with higher quality efficiencies who exert more efforts on quality improvement. To analyze the stability of coalition structures, we consider two solution concepts, including the core concept and the nucleolus concept. It is shown that, when the quality efficiency of the supplier system is high, the core is non-empty. When the core is non-empty, the nucleolus always falls into the core. We then investigate three profit allocation rules for suppliers in a coalition, including the equal allocations, the proportional allocations and the Shapley value allocations. We obtain the conditions under which the equal allocations and the proportional allocations are in the core. It is worth noting that, the Shapley value allocations are always in the core. Then, we study the farsighted stability of coalition formation among suppliers by adopting the notation of the largest consistent set (LCS). We demonstrate that, when the quality efficiencies of the suppliers are symmetric, the grand coalition is always stable under the three allocation rules. Furthermore, the independent structure is unstable when the quality efficiency of the supplier system is relatively high. When the quality efficiencies of the suppliers are asymmetric, the grand coalition is still stable under the equal allocations and the proportional allocations, however, under the Shapley value allocations, the grand coalition is stable only when the suppliers’ quality efficiencies have relatively small differences. The structure of this paper is as follows. In the next section, we provide a literature review. We describe our model in Section 3. Section 4 conducts equilibrium analysis under a given coalition structure and Section 5 conducts stability analysis of coalition structures. We conclude the paper and point out future research directions in Section 6.

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2. Literature Review There are three streams of literature relevant to this paper. The first stream is the literature using noncooperative game theory to study operations problems in decentralized assembly systems. The second stream is the literature adopting cooperative game theory to analyze the coalition formation among complementary or substitutable suppliers. The third stream is research on price and quality dependent demand. For papers using noncooperative game theory to study decentralized assembly systems, most discuss equilibrium decisions with exogenous demand, including pricing and production (Wang, 2006; Kalkancı and Erhun, 2012), inventory (Bernstein and Decroix, 2006; Gurnania, 2007; DeCroix, 2013), lead time (Bollapragada et al., 2004; Fang et al., 2008), capacity (Wang and Gerchak, 2003; Bernstein and DeCroix, 2004; Bernstein et al., 2007; Roels and Tang, 2017), and financing schemes (Deng et al., 2018). Some papers extend the above research to incorporate price-sensitive demand. For instance, Jiang and Wang (2010) discuss direct competition among suppliers for producing the same component in a decentralized assembly supply chain with price-sensitive and uncertain demand. Leng and Parlar (2010) analyze a decentralized assembly system with a random price-dependent demand. The authors investigate both simultaneous-move and leader-follower games to determine respectively the Nash and Stackelberg equilibria of complementary suppliers’ production quantities and assembler’s retail price. This stream of literature neglects alliance formation among suppliers, and our work is partially related to them in the way of modeling operational interactions between upstream and downstream firms in assembly supply chains. The second stream of relevant literature uses cooperative game theory to examine alliance formation in decentralized assembly systems. One branch of the related literature uses the Nash stability concept to study the stability of alliances. In a Nash stable coalition structure, a player could not obtain a higher profit if she makes a unilateral defection from her current coalition. Granot and Yin (2008) use the Nash stability concept to discuss alliance formation among complementary suppliers in pull and push assembly systems, respectively, where the exogenous demand has a power distribution. Yin (2010) adopts the Nash stability concept to study how market demand conditions drive coalition formation among complementary suppliers. The author characterizes the stable coalitions considering both price-sensitive deterministic demand and multiplicative stochastic demand, and shows that 6

the introduction of a multiplicative stochastic element in demand has an insignificant impact on stable coalitions. He and Yin (2015) study both supplier competition and retailer competition in an assembly system. In their model, there are two complementary components a and b, where component a has a monopolist supplier (supplier a) and component b has multiple partial substitutable suppliers, which causes the supply-level competition. Suppliers of component b could form alliances with supplier a. The retail competition is caused by decentralization among retailers. Huang et al. (2016) consider alliance formation strategy among suppliers facing an exogenous random shock that may result in an order default, where the suppliers can either be complementary or substitutable. Each supplier has access to a recourse fund that can mitigate this risk. The suppliers can share the fund resources within an alliance, but they need to allocate the profits of the alliance equitably among the partners. The other branch adopts farsighted stability concepts to characterize stable coalition structures that discourage defections (Chwe, 1994). Under such concepts, players are able to anticipate future deviations by other players brought about by their own deviations (i.e., the players are farsighted). They seem to better fit real-life situations than the myopic static stability concepts, as they take into account possible dynamics in coalition formation. There have been a few papers that apply farsighted stability concepts in supply chain setting. Nagarajan and Bassok (2008) analyze alliance formation among farsighted suppliers and mainly focus on bargaining process and neglect specific operational decisions such as pricing or ordering quantities. The paper most relevant to ours is Nagarajan and Soˇsi´c (2009), which studies a decentralized assembly supply chain with n suppliers and an assembler, where each supplier sells a complementary component to the downstream assembler facing market demand characterized by a linear function of retail price. The authors discuss three modes of competition where the suppliers have dominant power and move first, the assembler has dominant power and moves first, and both parties move simultaneously. They characterize coalition structures that are farsighted stable, and use Shapley value as the payoff allocation rule among suppliers in an alliance. Different from Nagarajan and Soˇsi´c (2009), our contribution in this study is to address the externalities of complementary suppliers’ quality-improvement efforts in assembly systems. It comes from the fact that if one supplier increases the effort level on quality improvement, other suppliers would benefit from the increased demand as well, because the assembler would order more from all 7

complementary suppliers. We try to shed light on how the price and quality coordination among suppliers can be reached to reduce or eliminate the inefficiency due to decentralization of complementary suppliers. In addition, Soˇsi´c (2011) studies the impact of demand uncertainty on stability results obtained in Nagarajan and Soˇsi´c (2009). Nagarajan et al. (2019) extend the results in Nagarajan and Soˇsi´c (2009) by assuming that k components have multiple competing suppliers and that the remaining n − k suppliers are monopolists. In the literature of cooperative game theory, externality is an important feature of many situations that are of interest to researchers. The situation that the value of a player in an alliance is allowed to depend on the alliance structure outside, is referred as the situation with externalities (AlvarezMozos and Ehlers, 2017). For instance, Belleflamme and Bloch (2004) discuss externalities in oligopolistic markets and procurement auctions, and analyze market sharing agreements. They characterize stable complete alliances when firms and markets are symmetric. Navarro (2007) proposes a component efficient and fair allocation rule when the value of the network is allowed to exhibit any type of externalities across its components. This rule is found closely related to an extension of the Shapley value. Esteban and Dinar (2013) discuss two types of externalities in the cooperation on groundwater resources among n users, which implies that water extractions in one sub-aquifer impact water levels in neighboring sub-aquifers, and decisions in each sub-aquifer affect the health of the ecosystem. The authors empirically and theoretically demonstrate that both extraction and environmental externalities interact in affecting the likelihood of cooperation among the users. Fang and Cho (2020) consider a supply chain where a supplier delivers parts to a set of competing manufacturers, who may form alliances in two forms, namely joint auditing and information sharing. The authors highlight the crucial role of externalities in manufacturer’s incentives to cooperate. In assembly systems, the introduction of externalities will lead to different predictions about stable coalition structures. To the best of our knowledge, the research on coalition structures in assembly systems with externalities is missing in the literature. The current paper attempts to fill in this gap by examining coalition formations with externalities of quality-improvement efforts among suppliers under different allocation rules. In the third stream of literature, researchers have taken the product quality as an important influential factor of demand. Many papers propose longterm models for price- and quality-dependent demand. Teng and Thompson 8

(1996) establish long-term price-quality decision models in which dynamic demand is a function of price level, quality level, and cumulative sales. Voros (2006) considers price- and quality- dependent demand and sets up a dynamic model to characterize the optimal price, quality, and investment decisions. Chenavaz (2012) investigates dynamic pricing, product and process investment policies under price- and quality-dependent demand under an additive separable demand function and a multiplicative separable demand function, respectively. Liu et al. (2015) formulate an inventory model for perishable foods in which the demand depends on both the price and the quality. They determine a joint dynamic pricing and preservation technology investment strategy. Some other papers discuss models in one period setting. Smith (1986) and Riordan (1986) are early works in supply chains that assume quality sensitive market demand. Xu (2009) assumes that customers become more willing to purchase the product and less sensitive to the product price as the product quality is improved and focuses on equilibrium analysis in direct-sell and indirect-sell channels, respectively. Zhu et al. (2007) study a supply chain where both supplier and retailer have an incentive to invest in quality-improvement effort. Glock et al. (2012) also consider a priceand quality-dependent demand, and they treat sustainability as a quality attribute that is measured in terms of the levels of scrap and emissions generated in the supply chain. Different from the above literature examining pricing and quality issues simultaneously, this paper uses cooperative game theory to study complementary suppliers’ alliance formation and their willingness to exert effort to improve quality under the positive externality of demand in assembly systems. 3. The Model We consider a decentralized assembly system which consists of a single assembler and n suppliers. Denote N as the set of suppliers. Each supplier sells a complementary component to the downstream assembler, who assembles all the components into a final product and sells it in the final market. Suppose that a final product requires a single unit of each component. Denote the unit production cost of supplier i as ci (i = 1, · · · , n) and the unit assembly cost of the assembler as c0 . Assume that the market demand is deterministic and sensitive to both the retail price and the quality-improvement effort (“quality effort” for short) levels of all suppliers. Referring to Gurnani and Erkoc (2008) and Ma et al. 9

(2013), we assume that the demand information is common knowledge and the demand function of the final product is as follows: D =a−p+

n X

δi e i ,

(1)

i=1

P where a (≥ 0) is the base market size, p (≥ c0 + ni=1 ci ) is the retail price, ei (≥ 0) is supplier i’s quality-improvement effort level, and δi (≥ 0) measures the impact of supplier i’s quality-improvement effort on demand. We refer to δi as the quality effect factor. Suppose supplier i’s cost at qualityimprovement effort level ei is ki e2i /2 (“quality cost” for short). Refer to the coefficient in the quality cost ki (≥ 0) as the quality cost factor. The quadratic form of the quality cost suggests an increasing marginal cost for effort investment and, as a result, diminishing returns on unit effort investment to improve the quality. Note that, when one of the suppliers makes an effort to improve the quality of her component, all the other suppliers benefit from the enhanced quality of the final products, as the assembler orders more products from all the suppliers. This is the positive externality of quality improvement. The higher δi is, the higher the degree of positive externality is. Before interacting with the assembler, the suppliers could form alliances freely among themselves. Thereby, we use a two-level model to investigate the game between the assembler and the suppliers. Level I models the coalition formation among the suppliers. That is, the suppliers form a coalition structure which is a partition on N . Level II is concerned with the interaction between the assembler and the coalitions by using Stackelberg game model. In Level I, we identify coalition structures (partition on N ) that deter suppliers from deviation in equilibrium, which are defined as stable coalition structures. In Level II, we formulate a two-stage game. In the first stage, the assembler determines the profit margin (Nagarajan and Soˇsi´c, 2009). In the second stage, suppliers in the same coalition offer a single joint wholesale price (coalitional wholesale price) and a vector of quality-improvement effort levels for the bundle of the components produced by this coalition. In practice and literature, both assembler and suppliers could act as the market leader (Nagarajan and Soˇsi´c, 2009; Nagarajan et al., 2019). In this study, we discuss the final product’ demand is sensitive to both quality and price. It is more reasonable for the assembler to act as the first mover, e.g., automakers (Des, 2002), as the assembler is close to the market and 10

has the incentive to choose the suppliers who invest in component quality improvement. This kind of system where a big assembler has dominant power and benefits from the first-mover advantage in the price setting is also referred as pull contract systems in literature (Pan and So , 2016). For convenience, some of the notation used in this paper is summarized as follows: N : Set of suppliers, N = {1, · · · , n} wi : Wholesale price of supplier i, i ∈ N ei : Quality-improvement effort level of supplier i m: Profit margin of the assembler p: Retail price of the final product ci : Unit production cost of supplier i c0 : Unit assembly cost of the assembler S B: The set of coalition structures, B = {B|B = {B1 , · · · , Bl }, where li=1 Bi = N, Bh ∩ Bk = ∅, for h 6= k} l: Number of coalitions in a coalition structure B B ∗ : Grand coalition structure, i.e., B ∗ = {N } B: Independent coalition structure, i.e., B = {{1}, {2}, · · · , {n}} P WBj : Coalitional wholesale price of coalition Bj ∈ B, i.e., WBj = (wi : i ∈ Bj ) P W : Total wholesale price of all suppliers or coalitions, i.e., W = ni=1 wi = Pl j=1 WBj P CBj : Coalitional production cost of coalition Bj ∈ B, i.e., CBj = (ci : i ∈ Bj ) P C: Total production cost of all suppliers or coalitions, i.e., C = ni=1 ci = Pl j=1 CBj Q: Order quantity of the assembler Πsj : Profit of supplier j ΠBj : Profit of coalition Bj ΠT C : Total profit of all coalitions (suppliers) ΠA : Profit of the assembler ΠC : Total profit of the suppliers and the assembler 4. Equilibrium Analysis Under a Given Coalition Structure The stability analysis of coalition structures in Level I necessitates an equilibrium analysis in Level II under a given coalition structure. Thus, we use backward induction to solve these two levels of problems. 11

Considering a coalition structure B = {B1 , · · · , Bj , · · · , Bl }, denote |Bj | as the cardinality of coalition Bj and suppose the cardinality of the coalitions are n1 , n2 , · · · , nl , respectively, where n1 + n2 + · · · + nl = n. We use double-subscript ij to represent the ith supplier in coalition Bj . In the grand coalition structure and the independent coalition structure, we use single-subscript i to represent the ith supplier. For a given coalition structure B, the assembler first determines the profit margin m (defined as p − W − c0 ), and then each coalition Bj offers a coalitional wholesale price WBj and a vector of quality-improvement effort levels eBj = (e1j , e2j , · · · , enj ,j ). Denote eB = (eB1 , · · · , eBl ). Coalition Bj ’s profit function is nj 1X kij e2ij . (2) ΠBj = (WBj − CBj )D − 2 i=1 P Pnj δij eij . where D = a − p + lj=1 i=1 For the sake of convenience, denote tij =

δij2 /2kij , TBj

=

nj X i=1

tij , T =

l X

TBj .

j=1

We could interpret tij from the following two aspects. First, it is obvious that higher values of tij indicate higher values of quality effect factor δij or lower values of quality cost factor kij , which implies higher efficiency of quality improvement (“quality efficiency” for short). Thereby, we refer to tij as the quality efficiency factor of the ith supplier in coalition Bj . The further interpretation of tij in the analysis of Corollary 1 shows that higher values of tij not only indicate higher efficiency of quality improvement but also indicate higher cost incurred by quality improvement. Second, tij captures the ratio between the degree of positive externality generated by the quality improvement of supplier i in coalition j and the cost of providing such quality improvement. Therefore, the higher tij is, the more beneficial quality improvement is for the assembler. The interpretation of TBj and T are similar to that of tij . Similarly, we refer to TBj as the quality efficiency factor of the jth coalition, and T as the quality efficiency factor of the supplier system (i.e., composed of all suppliers), respectively. Furthermore, TBj captures the ratio between the degree of positive externalities generated by the quality improvements of suppliers in coalition j and the cost of providing such quality improvements. 12

T captures the ratio between the degree of positive externalities generated by the quality improvements of all suppliers and the cost of providing such quality improvements. It is obvious that, the higher TBj and T are, the more beneficial quality improvement is for the assembler. It can be shown that when TBj < 1, ΠBj is a joint concave function of WBj and eBj . Therefore, TBj < 1 assures that the solution satisfying the first order conditions of ΠBj is optimal. In fact, the assumption TBj < 1 can be regarded as a restriction on quality efficiency. Under the condition TBj < 1, coalition Bj will not excessively invest in quality improvement. Otherwise, the problem will become a trivial case with infinite demand (Tsay et al., 2000; Xiao and Yang, 2008). Therefore, we require that T < 1 throughout the paper to guarantee that all possible coalitions do not excessively invest in quality improvement. Taking into account coalition Bj ’s reactions WBj and eBj , the assembler chooses the optimal profit margin m to maximize his profit, which is given by m(a − c0 − C − m) . ΠA = mD = l + 1 − 2T Proposition 1 characterizes the equilibrium decisions and profits of the suppliers and the assembler under a given coalition structure. All proofs are given in appendix. Proposition 1. Under the assumption of T < 1, for any given coalition structure B = {B1 , · · · , Bl }, the equilibrium decisions and profits of the suppliers and the assembler are as follows: δ (a−c0 −C) a−c0 −C 0 )+(l+2−4T )C , e∗ij = 2kijij (l+1−2T , m∗ = a−c20 −C , W ∗ = l(a−c2(l+1−2T , (1) WB∗ j −CBj = 2(l+1−2T ) ) ) (2l+1−2T )a+(1−2T )(c0 +C) , where 2(l+1−2T ) 2 (1−TBj )(a−c0 −C) Π∗Bj = , Π∗T C 4(l+1−2T )2

p∗ =

j = 1, · · · , l and i = 1, · · · , nj ;

(2) j = 1, · · · , l.

=

(l−T )(a−c0 −C)2 , 4(l+1−2T )2

Π∗A =

(a−c0 −C)2 , 4(l+1−2T )

where

As stated by Part (1) of Proposition 1, WB∗ j − CBj (j = 1, · · · , l) equal to each other for all coalitions, irrespective of their production costs and quality costs. This is due to the fact that all components are perfectly complementary. Thus, they are equally important in the final products. This result is consistent with those in (Nagarajan and Soˇsi´c, 2009). We also have an intuitive result that the suppliers’ effort levels e∗ij increase in the value of quality effect factor δij and decrease in the value of quality cost factor kij . Furthermore, the assembler’s profit margin m∗ is fixed, independent of the number 13

of coalitions l and suppliers’ quality effect factor δij and quality cost factor kij . The reason is that the assembler has a first-mover advantage as being the leader of the two-stage game. Therefore, he can always charge a fixed profit margin no matter what the coalition structure is and how the values of quality effect and quality cost factors change. The sensitivity analysis of the equilibrium decisions and profits with respect to the two factors l and T will be presented in Corollaries 2 and 4, respectively. Part (2) of Proposition 1 presents the equilibrium profits of the coalitions and the assembler. Note that, a special case of our model is that, the quality efficiency of all suppliers equal to zero (i.e., T = 0). That is, the suppliers have no qualityimprovement effort decisions. It can be shown that the results of Proposition 1 for this special case are in line with those in Nagarajan and Soˇsi´c (2009). Corollary 1 is directly from Part (2) of Proposition 1. Corollary 1. Under the assumption of T < 1, (1) for any given coalition structure B = {B1 , · · · , Bl }, if TBi ≤ TBj , then we have ΠBi ≥ ΠBj , where i, j = 1, · · · , l and i 6= j; (2) in the independent coalition structure B, if ti ≤ tj , then we have ΠB si ≥ B Πsj , where i, j = 1, · · · , l and i 6= j. Part (1) of Corollary 1 shows that, in any coalition structure, a coalition Bj with higher value of TBj will obtain lower profit. Part (2) of Corollary 1 examines a special case of Part (1), i.e., the independent coalition structure, and indicates that suppliers with higher values of tj obtain lower profits. In the following, we will illustrate the result in Part (2) in detail. The interpretation of Part (1) is similar. One may intuitively expect that suppliers with higher quality efficiency obtain higher profits. However, the result shows the opposite. This is mainly because suppliers with higher quality efficiency incur higher quality cost, noting that supplier j’s quality cost is tj (a − c0 − C)2 /4(l + 1 − 2T )2 . Specifically, referring to the analysis of Part (1) in Proposition 1, the effort level of supplier j increases in δj and decrease in kj . Thus, suppliers with higher values of δj or lower values of kj exert higher effort levels, which incurs higher quality costs and lower profits. The underlying reason for the nonintuitive result in Corollary 1 is due to the positive externality of suppliers’ qualityimprovement effort. On account of the demand function shown in Equation (1), if one supplier increases her effort level on quality improvement, other suppliers also benefit from the increased demand, because the assembler will 14

order more from all complementary suppliers. Therefore, suppliers with lower quality efficiency could benefit from free-riding on the investment of suppliers with higher quality efficiency who exert more effort on quality improvement. Given a coalition structure, we are interested in how the number of coalitions would affect the performances of the whole system and its members. Corollary 2. For any given coalition structure B = {B1 , · · · , Bl }, (1)

∂e∗ij ∂l ∂Π∗B

≤ 0,

∗ ∂WB

j

∂l ∂Π∗A ∂l

≤ 0;

∂Π∗

∂Π∗

(2) ∂l j ≤ 0, ≤ 0, ∂lT C ≤ 0, ∂lC ≤ 0; ∗ ∂W ∗ (3) ∂l ≥ 0 and ∂p ≥ 0 if 0 ≤ T ≤ 1/2, ∂l ∗ 1/2 ≤ T < 1; ∂Q ≤ 0, where Q∗ = D(p∗ , e∗B ). ∂l

∂W ∗ ∂l

≤ 0 and

∂p∗ ∂l

≤ 0 if

Part (1) of Corollary 2 shows that, the effort levels and the coalitional wholesale prices in equilibrium decrease in the number of coalitions. The result stems from the horizontal decentralization of complementary suppliers. As the number of coalitions increases, the degree of decentralization increases, thus a higher proportion of the quality-improvement benefit is enjoyed by suppliers in other coalitions. Thus, the suppliers will decrease their effort levels. As a result, the suppliers will charge lower wholesale prices. Part (2) demonstrates that the profit of the assembler, the profit of each coalition, and the total profit of all coalitions decrease in the number of coalitions, which also stems from the horizontal decentralization. Moreover, the result in Part (2) implies that all these profits are maximized when the suppliers form a grand coalition. Part (3) indicates that the effect of the number of coalitions on the total wholesale price depends on the quality efficiency of the supplier system T . The reason is as follows. The derivative of the coalitional wholesale price WB∗ j with respect to the number of coalitions l is ∂WB∗ j /∂l = −(a − c0 − C)/2(l + 1 − 2T ). That is, when the number of coalitions increases by one, the coalitional wholesale price decreases by (a − c0 − C)/2(l + 1 − 2T ), which increases in T . Therefore, when the quality efficiency T is low (resp. high), the decrement of the coalitional wholesale price is small (resp. large). Thus, as the number of coalitions increases, the total wholesale price increases (resp. decreases). As the retail price p∗ = W ∗ + m + c0 , the impact of the number of coalitions on the retail price is similar to that of the total wholesale price. Lastly, we analyze the impact of the number of coalitions on the assembler’s order quantity, which is equal to the market demand. 15

Note that the derivative of the effort level e∗ij with respect to the number of coalitions l is ∂e∗ij /∂l = −δij (a − c0 − C)/2kij (l + 1 − 2T ). That is, when the number of coalitions increases by one, the effort level of the ith supplier in coalition Bj decreases by δij (a − c0 − C)/2kij (l + 1 − 2T ), which increases in T . For low quality efficiency T , as the number of coalitions increases, the retail price increases and the effort levels decrease, thus the market demand decreases. For high quality efficiency T , as the number of coalitions increases, even though the retail price decreases, the decrement of the effort levels is large, which significantly reduces the market demand. The following corollary is straightforward from Corollary 2 and states that the grand coalition B ∗ maximizes the total profit of all suppliers. For the special case with T = 0, Corollary is consistent with that in Nagarajan and Soˇsi´c (2009). Corollary 3. Let B = {B1 , · · · , Bl } be a coalition structure with at least B∗ two nonempty coalitions, then ΠB T C ≤ ΠT C . Next, we investigate how the quality efficiency factors affect the performances of the whole system and its members. Corollary 4. For any given coalition structure B = {B1 , · · · , Bl }, in equilibrium, (1)

∗ ∂WB

j

∂T ∂e∗ij ∂δij

≥ 0;

∂e∗

∂e∗

ij ≥ 0 and ∂kijij ≤ 0; ∂tkr ≥ 0 for 1 ≤ r, j ≤ l, 1 ≤ i ≤ nj , 1 ≤ k ≤ nr , (2) and r 6= j or k 6= i; ∗ ∗ ∗ (3) ∂W ≤ 0, ∂p ≤ 0, ∂Q ≥ 0, where Q∗ = D(p∗ , e∗B ); ∂T∗ ∂T∗ ∂T ∗ ∂ΠA ∂ΠT C ∂ΠC (4) ∂T ≥ 0, ∂T ≥ 0, ∂T ≥ 0;

(5)

∂Π∗B

j

∂TBk

if fBj ≤ 0, for 1 ≤ j ≤ l, where fBj

∂Π∗B

j

∂Π∗B

≥ 0 if fBj ≥ 0 and ∂TB j ≤ 0 ∂TBj j P = 3 − l + 2( lk=1,k6=j TBk − TBj ).

≥ 0 for 1 ≤ k, j ≤ l and k 6= j;

Part (1) of Corollary 4 shows that, as the quality efficiency of the supplier system T increases, the coalitional wholesale prices increase in equilibrium. Part (2) indicates that each supplier’s effort level increases in her own quality effect value and decreases in her own quality cost value. Therefore, each supplier’s effort level increases in her own quality efficiency. Furthermore, each supplier’s effort level increases in other suppliers’ quality efficiency. Parts (3) and (4) demonstrate that, as the quality efficiency of the supplier system T 16

increases, the total wholesale price and the retail price decrease, while the profit and order quantity of the assembler, the total profit of all suppliers, and the profit of the whole system increase. Part (5) shows that, if suppliers in coalition Bj increase their quality efficiency TBj , all other coalitions benefit, which is due to the positive externality of quality improvement. However, the impact of TBj on the profit of her own coalition Bj depends on the sign of fBj . Note that fBj > 0 (resp. < 0) indicates a small (resp. large) number of coalitions l and a relatively low (resp. high) quality efficiency TBj . Specifically, if TBj increases, the profit of coalition Bj increases only if the number of coalitions is small and the quality efficiency of coalition Bj is relatively low. The reason is that, if the number of coalitions is large or the quality efficiency of coalition Bj is high, as a supplier in coalition Bj increases her quality efficiency, a higher proportion of the quality-improvement benefit is enjoyed by suppliers in other coalitions, which results in a lower profit for coalition Bj . 5. Stability Analysis of Coalition Structures In this section, we analyze the stability of coalition structures. In Subsection 5.1, we consider two solution concepts used for analyzing stability of coalition structures, including the core concept and the nucleolus concept. In Subsection 5.2, we use the notation of the largest consistent set (LCS), introduced by Chwe (1994), to study farsighted stability of alliances. 5.1. Solution Concepts In this subsection, we consider two solution concepts–the core and the nucleolus. To introduce these two solution concepts for our cooperative game, we first characterize the structure of our game. According to the analysis of Section 4, due to the positive externalities of suppliers’ quality improvements, the profit generated by a coalition not only depends on the decisions of suppliers in that coalition, but also depends on the decisions of suppliers in other coalitions. Therefore, we adopt the approach based on a cooperative game in partition function form, which was first introduced by Thrall and Lucas (1963), to analyze the stability of coalition structures. Our game can be characterized by a triple (N, B, {vB }B∈B ), where B is the set of all coalition structures, B = (B1 , B2 , · · · , Bl ) is a coalition structure in B, and vB is a partition function that assigns to each coalition Bk (k = 1, · · · , l) to the profit it generates, vB (Bk ). A mapping Φ which associates every 17

cooperative game (N, B, vB∈B ) with a subset Φ(vB ) ∈ Rn is called a solution concept. Given a coalition structure B = (B1 , B2 , · · · , Bl ), an allocation is a payoff vector ϕ = {ϕ1 , ϕ2 , · · · , ϕn } which specifies how the profit generated by each coalition is distributed among the members P in that coalition. Under a coalition structure B, if an allocation satisfies i∈B Pk ϕi ≤ vB (Bk ) for all k, we say that it is feasible. An allocation is efficient if i∈N ϕi = v(N S). Denote B φ as the set of all feasible allocations under B, and define Ψ ≡ B∈B φB . 5.1.1. Core In this subsection, we adopt the definition of the core of games in partition function form, introduced in Fang and Cho (2020). In order to define the core, we first introduce a domination relation for two allocations. 0

Definition 1. (Fang and Cho , 2020) Suppose ϕ and ϕ are two allocations in Ψ and S is a coalition in N , 0 0 (1) P ϕ dominates ϕ via S, which is denoted as ϕ domS ϕ , if (i) i∈S ϕi ≤ vB (S) for all B where S ∈ B, 0 (ii) ϕi > ϕi for all i ∈ S; 0 0 (2) ϕ dominates ϕ , which is denoted as ϕ dom ϕ , if there exists S ⊆ N such 0 that ϕ domS ϕ . The definition of the core of games in partition function form in Fang and Cho (2020) is as follows. Definition 2. (Fang and Cho , 2020) The core is a set of feasible allocations that are not dominated by any other allocations, i.e., Co(N, B, {vB }B∈B ) = 0 0 {ϕ ∈ Ψ | @ ϕ ∈ Ψ , s.t. ϕ dom ϕ}. According to Corollary 3, the total profit of the suppliers is the highest when P they form the grand coalition. Thus, any allocation ϕ that satisfies i∈N ϕi > vB ∗ (N ) is not feasible and any allocation ϕ that satisfies P some feasible allocations. Therei∈N ϕi < vB ∗ (N ) is dominated via N byP fore, allocations in the core must satisfy i∈N ϕi = vB ∗ (N ). According to the definition of the core, allocations in the core are not dominated by any other allocations, so each member of the grand coalition has no incentives to deviate from the grand coalition under a core allocation. Proposition 2 presents the non-empty condition of the core in our game. When the quality efficiency of the supplier system is high (i.e., T ≥ (4n − 9)/(4n − 8)), the core is non-empty, which implies that, the grand coalition 18

is stable when the quality efficiency of the supplier system is high. Note that, (4n − 9)/(4n − 8) increases in n. Therefore, the stability of the grand coalition is less and less likely as the number of suppliers grow. Proposition 2. When n = 2, the core is always non-empty. When n ≥ 3, the core is non-empty if and only if T ≥ (4n − 9)/(4n − 8). 5.1.2. Nucleolus The nucleolus is a well-known point solution that is contained in the core when the core is non-empty (Schmeidler, 1969). Alvarez-Mozos and Ehlers (2017) extend the definition of nucleolus to games in partition function form. Before introducing the definition of nucleolus, we restate the structure of our game. Denote B(N ) as the set of partitions of N . A pair (S, B) where S ∈ N and B ∈ B(N \S) is called an embedded coalition of N . The set of all embedded coalitions of N is denoted as E C N . Our game can also be characterized by a triple (N, B, φ) where φ is a partition function from E C N to R, satisfying φ(∅, B) = 0, for every B ∈ B(N ). Denote X(N, B, φ) as the set ofPall efficient allocations of (N, B, φ), i.e., X(N, B, φ) = {x ∈ Rn : x(N ) = i∈N xi = φ(N, ∅)}. Given an embedded coalition (S, B) ∈ E C N , and an efficient allocation x ∈ X(N, B, φ), the excess of (S, B) at x is defined by e(S, B, x, φ) = φ(S, B) − x(S). (3) Denote c(n) = |E C N |, where n = |N |. Define e(x, φ) ∈ Rc(n) as the vector of all the excesses at x in φ, i.e., e(x, φ) = (e(S, B, x, φ))(S,B)∈E C N . For a given x ∈ X(N, B, φ), construct a vector θ(x, φ) that contains all the excesses, and the components of θ(x, φ) are arranged in non-increasing order as follows: c(n) c(n)

θ(x, φ) ∈ R

, where

[

i=1

c(n)

{θi (x, φ)} =

[

i=1

{ei (x, φ)},

θ1 (x, φ) ≥ θ2 (x, φ) ≥ · · · ≥ θc(n) (x, φ). Denote Rr≥ = {x ∈ Rr : x1 ≥ x2 ≥ · · · ≥ xr }. Denote - as the lexicographical ordering on Rr≥ : for any two vectors x, y ∈ Rr≥ , x - y if and only if either x = y or there is 1 ≤ k ≤ r, such that xi = yi for every 1 < i < k and xk < yk .

19

Definition 3. (Alvarez-Mozos and Ehlers, 2017) The nucleolus of a game in partition function is the set of efficient allocations which lexicographically minimize the ordered vector of excesses: η(N, φ) = {x ∈ X(N, φ) : θ(x, φ) - θ(y, φ) for all y ∈ X(N, φ)}. According to Definition 3, the nucleolus of our game is given in Proposition 3. Proposition 3. In the decentralized assembly system, the nucleolus is    1 − ti 1 1 n−T (a − c0 − C)2 + − , i ∈ N. (4) xi = 4 (3 − 2T )2 n 4(1 − T ) (3 − 2T )2 It can be observed from Equation (4) that, the allocated profit of each supplier under the nucleolus is composed of two parts: the first part is the supplier’s profit if she defects from the grand coalition, and the second part is the supplier’s gain (or loss) from participating in the grand coalition. Note that, if supplier i defects from the grand coalition to be independent, her 0 profit is Πi = (1 − ti )(a − c0 − C)/4(3 − 2T )2 . Thus, the total profit of each defecting P 0supplier if only one of them defects2 from the grand coalition 0 is ΠT C = Πi = (n − T )(a − c0 − C)/4(3 − 2T ) . It can be easily shown that, when T > (4n − 9)/(4n − 8), the profit of the grand coalition is higher 0 than ΠT C . Therefore, when T > (4n − 9)/(4n − 8), the second part of xi is supplier i’s gain from participating in the grand coalition; while when T ≤ (4n − 9)/(4n − 8), the second part of xi is supplier i’s loss from joining in the grand coalition. Corollary 5 shows that the nucleolus allocation is always in the core when the core is non-empty. Corollary 5. The nucleolus is in the core when the core is non-empty. We know that the core and the nucleolus allocations are applicable to the grand coalition. As we are interested in the stability of all coalition structures, we then examine three allocation rules and conduct stability analysis of coalition structures in Subsection 5.2.

20

5.2. Stable Coalition Structures In this subsection, we analyze the stable coalition structures. Before we introduce the stability concept we use, we will briefly describe the motivation of our framework. Most solution concepts used for analyzing stability of coalition structures (e.g., core, [Gillies 1959]; coalition structure core, [Aumann 1959]; strong Nash equilibrium, [Aumann and Dreze 1974]) are static. That is, they do not take into account the possibility that the deviations of some players will cause the subsequent deviations of other players. For instance, considering the stability of the grand coalition in our game, it can be easily verified that, each supplier has an incentive to defect from the grand coalition when T < (4n − 9)/(4n − 8). Under the static solution concepts, the grand coalition would be unstable. However, the defection of a supplier may cause the defection of another coalition, and further the defection of a third coalition, and so on. If the final result is not beneficial to the original supplier, she will not choose to defect in the first place. A solution concept that considers possible sequences of defections is the largest consistent set (LCS). In the following subsection, we briefly introduce LCS. 5.2.1. Largest Consistent Set Denote by ≺i the players’ strong preference relations, which is defined as 1 2 follows: for two coalition structures L1 and L2 , L1 ≺i L2 ⇔ ΠL < ΠL i i , L where Πi is player i’s profit in coalition structure L . If for all i ∈ S, L1 ≺i L2 holds, we write L1 ≺S L2 . Denote by *S the following relation: L1 *S L2 if coalition structure L2 is obtained when coalition S deviates from coalition structure L1 . We say that L1 is directly dominated by L2 , denoted by L1 < L2 , if there exists a coalition S, such that L1 *S L2 and L1 ≺S L2 . We say that L1 is indirectly dominated by Lm , denoted by L1  Lm , if there exist coalition structures L1 , L2 ,· · · , Lm , and coalitions S1 , S2 , · · · , Sm−1 , such that Li *Si Li+1 and Li *Si Lm for i = 1, 2, · · · , m − 1. A set Y is called consistent if L ∈ Y if and only if for all coalitions S and all coalition structures V such that L1 *S V , there is a B ∈ Y , where V = B or V  B, such that L ⊀S B. Roughly speaking, the LCS contains all coalition structures such that for each one of them all possible defections by all possible coalitions are deterred, as they may eventually lead to another coalition structure in the 21

LCS in which the players made the initial defection might be worse off (Soˇsi´c, 2011). 5.2.2. Allocation Rules for Coalitional Profit In order to analyze the coalition formation formation problem in Level I, we consider three rules for the allocation of coalitional profit. This includes, equal allocations, proportional allocations and Shapley Value allocations. Equal Allocations The simplest way to allocate the coalitional profit would be to give an equal portion to each supplier. The profit allocated to each supplier in coalition Bj is 1 (5) ΠB , ϕE j = |Bj | j

where ΠBj is the coalitional profit of coalition Bj and |Bj | is the number of suppliers in coalition Bj . Corollary 6 shows that, when the differences among the quality efficiency of the suppliers are relatively low, the equal allocations are in the core, which implies that the grand coalition is stable.

Corollary 6. The equal allocations are in the core when ti ≥ [1 − (3 − 2T )2 ]/4n(1 − T ), where i = 1, 2, · · · , n. Proportional Allocations Another simple and natural way to allocate the coalitional profit would be to distribute them proportionally to the qualityimprovement efficiency of the suppliers in the same coalition. The profit allocated to the ith supplier in coalition Bj is ϕPij =

tij ΠB , TBj j

(6)

where ΠBj is the coalitional profit of coalition Bj . Note that each player’s quality-improvement cost is proportional to her quality-improvement efficiency. Therefore, suppliers with higher quality cost are allocated higher profits, which is a relatively fair allocation rule. (“fair” means that suppliers with higher quality costs are allocated higher profits). Corollary 7 demonstrates that, when the differences among the quality efficiency of the suppliers are relatively low, the proportional allocations are in the core, which implies that the grand coalition is stable. Corollary 7. The proportional allocations are in the core when ti ≥ 4T (1 − T )/(9 − 8T ) for i = 1, 2, · · · , n. 22

Shapley Value Allocations Shapley value (Shapley, 1953) is a well-known allocation rule to distribute collective profits in cooperative games. It satisfies the properties of symmetry, null player, efficiency, and additivity. Due to these desirable properties, Shapley value is often regarded as one of the fairest ways to collective profits among players (Yin, 2010). In a given coalition structure B = {B1 , · · · , Bl }, define a characteristic function, vBj (Z), where Z ⊆ Bj , to represent the profit that the subset Z can obtain on its own as a group while all other players maintain their positions in B. Proposition 4 characterizes how the suppliers of the same coalition allocate the coalitional profit under Shapley value allocations. Proposition 4. In a given coalition structure B = {B1 , · · · , Bl }, the Shapley value that assigned to the ith supplier in coalition Bj is   TBj − |Bj |tij 1 − TBj (a − c0 − C)2 + , (7) φi (vBj ) = 4|Bj | (l + 1 − 2T )2 (l + 2 − 2T )2 where j = 1, · · · , l and i = 1, · · · , nj . From Equation (7), we observe that the allocated profit of a supplier in coalition Bj is composed of two parts: the first part is the average of the coalitional profit and the second part is determined by the difference between the supplier’s quality efficiency and the average of all suppliers’ quality efficiencies in coalition Bj . We also notice that the suppliers with lower quality efficiencies are allocated higher profits. The interpretation of this result is similar to that of Corollary 1. The reason is that a supplier’s investment on her own quality improvement has positive externality on other suppliers’ profits. Therefore, the suppliers with lower quality efficiencies could free-ride on the investments of those with higher quality efficiencies who invest more on quality improvement. Corollary 8 shows that the Shapely value allocations always fall into the core when the core is non-empty. Corollary 8. The Shapley value allocations are in the core when the core is non-empty. 5.2.3. Stable Outcomes In this subsection, we consider the stability of coalition structures in two cases: in the first case, the suppliers have equal quality efficiencies, which is 23

referred to as the symmetric efficiency case; in the second case, the suppliers have different quality efficiencies, which is referred to as the asymmetric efficiency case. In the symmetric efficiency case, it is obvious that the equal allocations, the proportional allocations and the Shapley value allocations are equivalent. Proposition 5 characterizes the stable coalition structures for a small number of suppliers. Proposition 5. In the symmetric efficiency case, under the three allocations, LCS = {B ∗ } when n < 5. With a small number of suppliers, the only stable coalition structure is the grand coalition. Note that, when T ≥ (4n − 9)/(4n − 8), a supplier is better off by defecting from the grand coalition. Thus, the grand coalition is myopically unstable when T ≥ (4n − 9)/(4n − 8). However, the grand coalition is farsighted stable. As the number of suppliers increases, we could not completely characterize the stable outcomes. But we could generate partial results for an arbitrary number of suppliers. Theorem 1 summarizes the results. Theorem 1. In the symmetric efficiency case, under the three allocations, the following statements hold: (1) the grand coalition B ∗ is always in the LCS; (2) when n ≥ 5, the independent coalition structure B is never in the LCS if T ≥ (n − 4)/(n − 3). Part (1) of Theorem 1 states the stability of the grand coalition. Although a supplier is always better off by defecting from the grand coalition when T ≥ (4n − 9)/(4n − 8), the fact that a supplier contemplating a move takes into account the following sequence of actions by other players guarantees the stability of the grand coalition. Part (2) presents the condition under which the independent coalition structure is unstable. It is demonstrated that when the quality efficiency of the supplier system is high, the independent coalition structure is unstable. In fact, we could show that the difference of the suppliers’ profits between the grand coalition and the independent coalition structure increases in the quality efficiency of the supplier system. Therefore, when the quality efficiency of the supplier system is high, the suppliers will not act independently. Note that, for the special case with 24

T = 0, Nagarajan and Soˇsi´c (2009) show that the grand coalition is always stable and the independent coalition structure is always unstable. Therefore, we conjecture that the independent coalition structure is always unstable for all value of T in our model. In the following, we analyze the asymmetric efficiency case, where the suppliers have different quality efficiencies. Without loss of generality, we assume that t1 ≤ t2 ≤ · · · ≤ tn . Theorem 2 presents our findings for the asymmetric case. Please refer to the proof of Theorem 2 in appendix for the formulation of F . Theorem 2. In the asymmetric efficiency case, (1) under both the equal allocations and the proportional allocations, the grand coalition B ∗ is always in the LCS; (2) under the Shapley value allocations, the grand coalition B ∗ is in the LCS if tn ≤ F . Theorem 2 shows that, when the suppliers are asymmetric, the grand coalition is stable under both the equal allocations and the proportional allocations. While under the Shapely value allocations, the grand coalition is stable if the differences among the suppliers’ quality efficiencies are relatively low. The interpretation of the result is as follows. The equal allocations and the proportional allocations are allocations that are more fair than the Shapley value allocations. Therefore, the suppliers are willing to form the grand coalition under both the equal allocations and the proportional allocations. While under the Shapely value allocations, due to positive externalities of quality-improvement effort, suppliers with low quality efficiencies are able to benefit from free-riding on the efforts of those with high quality efficiencies and to achieve even higher profits. Therefore, the suppliers with higher quality efficiencies only join in the grand coalition when the differences among their quality efficiencies are relatively low. Through the analysis of the three allocation rules, we show that, the grand coalition is stable under both the equal allocations and the proportional allocations, however, under the Shapley value allocations, the grand coalition is stable only when the suppliers’ quality efficiencies have relatively small differences. Thus, compared to the Shapley value allocations, the equal allocations and the proportional allocations facilitate better cooperation among the complementary suppliers. Furthermore, the equal allocations and the proportional allocations are allocations that are more fair than the Shapley 25

value allocations. Therefore, when there are positive externalities of suppliers’ quality-improvement efforts, it would be more suitable for the suppliers to use the equal allocations and the proportional allocations when allocating coalitional profit among them. Lastly, to examine the stability of the the coalition structures under the three allocations, we conduct three sets of numerical examples. The first set is for the symmetric efficiency case. The second and the third ones are for the asymmetric efficiency case, and they have moderate and substantial differences of suppliers’ quality efficiencies, respectively. Consider the case with five suppliers. The values of the parameters are as follows: a=50, c0 =2, C=10. For the symmetric case, the values of ti (i = 1, · · · , 5) are all equal to 0.3/2. We first calculate the profits of the suppliers in the independent coalition structure πiB . Then, we calculate the allocated profits of the suppliers ∗ in the grand coalition under the equal allocations πiE , the proportional al∗ ∗ locations πiP , and the Shapley value allocations πiS , respectively. Table 1 summarizes the results. The allocated profits of the suppliers under the three allocation rules are equal. Furthermore, the allocated profits of the suppliers in the grand coalition under the three allocations are higher than those in the independent coalition structure. According to Theorem 1, the grand coalition is always stable under the three allocations. Furthermore, it can be verified that the condition T ≥ (n − 4)/(n − 3) in Theorem 1 holds. Therefore, the independent coalition structure is unstable. That is, the suppliers will not act independently. Suppliers ti πiB ∗ πiE ∗ πiP ∗ πiS

1 0.3/2 15.15 72.20 72.20 72.20

2 3 4 5 0.3/2 0.3/2 0.3/2 0.3/2 15.15 15.15 15.15 15.15 72.20 72.20 72.20 72.20 72.20 72.20 72.20 72.20 72.20 72.20 72.20 72.20

Table 1: Results for Symmetric Efficiency Case

For the first setting of the asymmetric case, the values of ti (i = 1, · · · , 5) are t1 = 0.1/2, t2 = 0.2/2, t3 = 0.3/2, t4 = 0.4/2 and t5 = 0.5/2, respectively. Table 2 summarizes the results, where the notation has the same meaning as 26

that in Table 1. We observe that the allocated profits of the suppliers in the grand coalition under the three allocations are all higher than those in the independent coalition structure. According to Theorem 2, the grand coalition is stable under both the equal allocations and the proportional allocations. Furthermore, it can be verified that the stability condition tn ≤ F in Theorem 2 holds. Therefore, the grand coalition is also stable under the Shapley value allocations. Suppliers ti πiB ∗ πiE ∗ πiP ∗ πiS

1 0.1/2 16.94 72.20 24.07 88.24

2 3 4 5 0.2/2 0.3/2 0.4/2 0.5/2 16.04 15.15 14.26 13.37 72.20 72.20 72.20 72.20 48.13 72.20 96.27 120.33 80.22 72.20 64.18 56.16

Table 2: Results for the First Setting in Asymmetric Efficiency Case

For the second setting of the asymmetric case, the values of ti (i = 1, · · · , 5) are as follows: t1 = t2 = t3 = t4 = 0.04/2 and t5 = 1/2. Table 3 summarizes the results, where the notation has the same meaning as that in Table 1. It can be observed from Table 3 that, the allocated profits of the suppliers in the grand coalition under the equal allocations and the proportional allocations are both higher than those in the independent coalition structure. According to Theorem 2, the grand coalition is stable under both the equal allocations and the proportional allocations. We also notice that, the allocated profit of the fifth supplier under the Shapley value allocations is lower than that in the independent coalition. Furthermore, it can be verified that the stability condition tn ≤ F in Theorem 2 does not hold, which implies that the grand coalition may be unstable under the Shapley value allocations. In fact, the grand coalition is indeed unstable as the defection of the fifth supplier from the grand coalition to be independent cannot be deterred. Therefore, when the differences among the suppliers’ quality efficiencies are high, the grand coalition is unstable under the Shapley value allocations.

27

Suppliers ti πiB ∗ πiE ∗ πiP ∗ πiS

1 0.04/2 15.10 42.98 7.41 53.21

2 0.04/2 15.10 42.98 7.41 53.21

3 0.04/2 15.10 42.98 7.41 53.21

4 0.04/2 15.10 42.98 7.41 53.21

5 1/2 7.71 42.98 185.24 2.03

Table 3: Results for the Second Setting in Asymmetric Efficiency Case

6. Conclusion In this paper, we examine an assembly system with n complementary suppliers, where the suppliers could enhance demand by exerting qualityimprovement efforts. The component suppliers may form coalitions to better coordinate their wholesale pricing and quality-improvement effort decisions. Our analysis shows that, due to the positive externalities of suppliers’ quality improvement, coalitions with lower quality efficiencies could benefit from a free-ride on the quality-improvement efforts of coalitions with higher quality efficiencies who exert higher quality-improvement efforts. To analyze the stability of coalition structures, we first consider two solution concepts–the core and the nucleolus. Then, we use the notation of the largest consistent set, which is a farsighted stability concept, to study dynamic alliance formation among suppliers. For the allocation of coalitional profit, three allocation rules, i.e., the equal allocations, the proportional allocations, and the Shapley value allocations, are discussed. We provide conditions under which the grand coalition emerges as a tenable outcome and conditions under which the independent coalition structure is unstable. We find that, compared to the Shapley value allocations, the equal allocations and the proportional allocations are more fair to the suppliers and facilitate better cooperation among the suppliers. Therefore, when there are positive externalities of suppliers’ quality-improvement efforts, it may be more suitable for the suppliers to use the equal allocations and the proportional allocations when allocating coalitional profit among them. There are some limitations of our research. First, we do not consider the bargaining process between the suppliers and the assembler. Considering the bargaining process, it is obvious that the formation of coalitions among suppliers will impact their bargaining power in the interaction with the assem28

bler. Therefore, it would be interesting to take into account the bargaining power between the coalitions and the assembler. Second, the demand is deterministic with a linear demand function in our model. Stochastic demand or other forms of demand function deserve further investigation. Third, we have assumed that there is only one supply source for each component. In practice, there may be competing supply sources of the same component. Thus, it would be worthwhile to consider suppliers’ competition. Acknowledgements The work of Tingting Li was supported by the National Natural Science Foundation of China (NSFC), under grants 71971047, 71501030 and 91747105, and by the Postdoctoral Science Foundation project of China, under grant 2015M581341. The work of Junlin Chen was supported by the National Natural Science Foundation of China (NSFC), under grants 71761137004, 71401195 and 71871008, and by the Social Science Foundation of Beijing, China, under grant 17GLB036. References Alvarez-Mozos, M., Ehlers, L. (2017). Externalities and the nucleolus. Centre interuniversitaire de recherche en ´economie quantitative, CIREQ. Arranz, N., de Arroyabe, J.C.F. (2008). The choice of partners in R&D cooperation: An empirical analysis of Spanish firms. Technovation 28 (12), 88–100. Aumann, R.J. (1959). Acceptable points in general cooperative n-person games. Tucker, A.W., Luce, R.D., eds. Contributions to the Theory of Games IV. Princeton University Press, Princeton, NJ, 287-324. Aumann, R.J, Dreze,J.H. (1974). Cooperative games with coalition structures. International Journal of Game Theory 3 (4), 217–237. Basak, K., Feryal, E. (2012). Pricing games and impact of private demand information in decentralized assembly systems. Operations Research 60 (5), 1142–1156. Bernstein, F., DeCroix, G.A. (2004). Decentralized pricing and capacity decisions in a multitier system with modular assembly. Management Science 50 (9), 1293–1308. 29

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