Channel structure and evolutionary stability analysis between traditional and green service supply chains

This paper aims to explore the optimal pricing and green service decisions and discuss the evolutionary stability strategy (ESS) of vertical channel structure strategic interaction between the traditional and green service supply chains (TSSC and GSSC). Considering these two supply chains could choose between the centralized (C) and decentralized (D) channel structures, current research establishes four channel models, namely, Models DD, DC, CD and CC, wherein Model DD(CC) means that both supply chains adopt channel D(C) and Model DC(CD) refers to GSSC adopting channel D(C) while TSSC using channel C(D). Furthermore, an evolutionary game is developed to explore the ESSs of the dynamic competitive system. The research results show that the stronger the integration between upstream and downstream firms of GSSC is, the higher green service the supply chain would provide when TSSC adopts channel D. Besides, when the market competition is sufficiently low, only point (0,0) is the ESS; when it is moderate, there exist two ESSs, i.e., ESS (0,0) and ESS (1,1); when it is extremely high, only point (1,1) is the ESS. The numerical examples show that the green service level increases in market competition while some retailing and wholesale prices under specific models would not be affected by it or show an inverted U shape, and the initial states of two supply chains’ channel strategies significantly impact the system’s ESSs.


Introduction
With the continuous deterioration of the global ecological environment, the highest temperature in many parts of the world exceeded the historical one in the summer of 2022, promoting people all over the world to pay high attention to environmental protection. Nowadays, most enterprises are challenged to develop and provide green services/products to meet the standard of sustainable development (Tian et al. 2014;Zhu and He 2017;Ma and He 2022), especially driven by the environmental legislations promulgated by countries around the world. For example, China has proposed the goal of ''Carbon Peak, Carbon Neutralization'' in May 2021, and Japan also has promised to achieve carbon neutrality by 2050. Many works show that firms engaging in green activities would generate a core competitiveness enhancement and capture more market shares (Xing et al. 2017;He et al. 2018). Nevertheless, some companies may be hesitant to participate in the green program due to the high cost of implementing green innovations. Consequently, competition between traditional and green service firms is not uncommon in the early stage of promoting sustainable development. With the popularity of global supply chain, the competition among enterprises has increasingly become a chain-to-chain competition (Li and Li 2016;Wang et al. 2020). Given this, this paper will explore the chain-to-chain competition problem between a traditional service supply chain (TSSC) and a green service supply chain (GSSC). Meanwhile, it is well known that channel structure could be used as a strategic tool to fight against competitors and seize larger market shares (Liu et al. 2021). For instance, catalog sales and some giant retailers (e.g., Amazon and JD.com) frequently sell their own store brands through centralized channel structures to stick up for sales force and maintain close contact with customers. By contrast, some pure manufacturers or transnational corporations often cooperate with independent distributors to sell products for capturing more market shares. Hence, it is of great significance to investigate the strategic interaction of channel structure choices between GSSC and TSSC. Concurrently, some scholars have studied the vertical channel structure selection under chain-to-chain competition from a static perspective (Xiao et al. 2014;Huang et al. 2018;Moradinasab et al. 2018). They have underlined that the competition between supply chains plays a vital role in participants' decision-makings of pricing and channel structure choices. In contrast, this paper focuses on exploring the evolutionary stability strategy (ESS) of a dynamic competitive system comprised of one TSSC and one GSSC. Specifically, this study differs from the previous research in two ways. On the one hand, we consider two asymmetric service supply chains where the GSSC provides not only basic products but also green services (e.g., carbon emission reduction and pollutant discharge pretreatment), which can meet the environmental needs of consumers. By comparison, the TSSC only sells basic products without any environmental protection service. On the other hand, we discuss different channel structure combinations of two supply chains by considering the effects of green service and time on demands. In this context, the following research questions arise.
(1) What are the optimal pricing and green service decisions for two supply chains? (2) When each supply chain should adopt a centralized or decentralized channel structure? (3) How does the market competition influence the pricing decisions, demands and profits of two supply chains, as well as the ESSs of the dynamic competitive system?
To shed light on these issues, we develop four game models and solve the optimization problems to compare the optimal green service and profits of two supply chains. We further employ evolutionary game theory and numerical simulation methods to analyze the dynamic competitive system's evolutionary trends, evolution speed and the last ESSs. The research findings indicate that for the GSSC, no matter which channel structure the TSSC selects, there always exists a key threshold of market competition that can distinguish her optimal channel structure choice. Specifically, when the market competition is relatively low, channel C is better for the GSSC, whereas channel D performs better under a stronger competitive situation. Different from traditional wisdom, it is worthy of noting that when the GSSC adopts channel C, the TSSC always prefers channel C. From a long-term perspective, the ESSs of the dynamic competitive system vary with the market competition. Particularly, the initial states of two supply chains' channel strategies would largely affect the evolution paths and ESSs when the market competition is moderate. In this setting, when both the GSSC and TSSC have lower initial probabilities to utilize channel D, this system would lastly evolve to ESS (0,0), and vice versa.
The rest of this paper is organized as follows. The next section discusses the literature background. Section 3 describes the research problem and demand functions. The model development, optimal solutions and comparative analysis under the one-shot game are conducted in Sect. 4. In Sect. 5, we further consider the repeated game case and identify the ESSs, followed by Sect. 6, which illustrates some main results and examines the effects of market competition. Section 7 concludes the limitations and future research avenues.

Literature review
This study is closely related to three streams of existing literature, i.e., chain-to-chain competition, green supply chain management and channel structure management.

Chain-to-chain competition
When it comes to chain-to-chain competition, a considerable number of papers examine the influences of price and/ or quality competition (Jafarian et al. 2019;Nematollahi et al. 2021). Li and Li (2016) investigate product sustainability between two symmetrical reverse supply chains. They uncover that vertical integration is not an equilibrium strategy unless two sustainable supply chains are completely independent. Wu and Chen (2016) focus on two supply chains' channel choices under uncertain demand. They demonstrate that the dual-decentralized structure is the equilibrium for substitutable products, whereas the dual-integrated structure is the equilibrium for complementary products. Goodarzi et al. (2017) explain why the cash flow bullwhip effect happens, and its influences on the integrated or decentralized supply chains' performance. Wang and Liu (2019) explore the vertical contract selections of competitive shipping supply chains between the wholesale price and revenue-sharing contracts and identify the condition for different contract equilibrium. Li et al. (2020a) examine the effects of partial centralization on the performance of competitive supply chains. They show that this channel structure would be the equilibrium unless the product substitutability is very high. Some other papers study chain-to-chain competition by considering more factors or strategies such as vertical and horizontal information sharing (Chen et al. 2019), product sustainability strategy (Deng et al. 2020), carbon emission , technology upgradation and financing risk (Wu and Kung 2020) and clean development mechanisms (Liu et al. 2021).

Green supply chain management
The research on green supply chain management is very extensive (Chan et al. 2016;Hong et al. 2018;Liu and Zhang 2022). Most previous works confirm that research and development of a green products could enhance firms' core competitiveness and expand their market shares. Chu and Chung (2016) develop an integrated balanced score card by combining the analytic network process model to establish the indicators for green tourism supply chain, which can help tour firms to find the key factors and balance the revenues and their environmental protection responsibility. Hafezalkotob (2015) and Sinayi and Rasti-Barzoki (2018) investigate the effects of government intervention on supply chain members' pricing, greening and social welfare policies. Song and Gao (2018) indicate that the revenue-sharing contract plays a significant role in the benefit distribution between chain members, and it indeed improves the whole green supply chain's performance. Hong and Guo (2019) compare three coordination contracts, namely, price only, green marketing cost-sharing, and two-part tariff contracts in a green supply chain. They find a counterintuitive result that it is always profitable for the manufacturer to share the retailer's greening cost but not for the retailer. Ma et al. (2020) take the uncertain information into account and propose an alternative decision rule based on the firms' confidence level to coordinate the green supply chain with a cost-sharing contract. Some scholars take the service into green supply chain management (Laari et al. 2018;He et al. 2019;Chen et al. 2021;Hong and Liu 2022;Nouri-Harzvili et al. 2022). Particularly, Tseng et al. (2018) employ the Fuzzy Delphi method and analytical network process to design a framework to examine the sustainable service supply chain's performance by considering uncertainty. Ma et al. (2021) and Ma and He (2022) study the green tourism service supply chain and explore the tourism firms' green technology investment, pricing and advertising decisions. They show that the consumers' green tourism preference and tourism experience significantly affect the tour firms' optimal decisions.

Channel structure management
As for channel structure management, many researchers focus on investigating the merits and demerits of decentralized or centralized channel structure in different types of supply chains (Schmitt et al. 2015;Li et al. 2020b;Bendadou et al. 2021). Most prior studies indicate that vertical cooperation can reduce the double marginalization effect, while some argue that decentralization may be a better strategy when market competition is relatively high (Peng et al. 2018;Heydari et al. 2021;Yang et al. 2022). Particularly, Ghosh and Shah (2012) discuss the problem of how channel structures impact the greening levels, prices and profits and examine the roles of greening costs and consumer sensitivity toward green apparel. Xiao et al. (2014) establish a Retailer Stackelberg pricing model to explore the manufacturer's product variety and channel strategies. They show that higher marginal cost of variety, marginal selling cost of the retailer and cost of customer fit would encourage the manufacturer to choose dual channels. Zhu and He (2017) investigate the problems of how supply chain structures, green product types and competition types impact supply chains' decisions on product greenness. They indicate that price competition can increase products' green level, while the greenness competition would reduce it. Huang et al. (2018) extend the chain-to-chain competition to the channel structure and quality selections. They find that the follower can strategically decentralize its channel structure to change the leader's quality choice. Yang and Yu (2019) analyze how integrated logistics and procurement services provided by a third-party logistics company affect a supply chain's operational performance. Further, He et al. (2022) propose a partial logistics integrated strategy for a platform supply chain, which would perform better than the completely integrated or decentralized channel structure from the whole supply chain aspect.
Different from the previous studies, which generally show that the completive supply chains should adopt an appropriate channel structure, selling mode or cooperation contract, and make optimal marketing decisions according to different market situations (Li and Li 2016;Wang and Liu 2019;He et al. 2022;Wang et al. 2022), this paper first extends the symmetric chain-to-chain competition issue to the asymmetric situation by considering the GSSC's green service decisions, and highlights that considering asymmetric green service would enrich the traditional wisdom as the TSSC would just prefer channel C if the GSSC utilizes channel C. Additionally, the prior literature on green supply chain management confirms that government intervention plays an important role in encouraging green transformation of enterprises (Hafezalkotob 2015;Sinayi and Rasti-Barzoki 2018;. This study takes the factor of time and learning ability into consideration, and aims to explore the ESSs of strategic channel structure interaction between the TSSC and GSSC from a repeated game perspective. The research result suggests that encouraging green firms to use channel C would benefit the environment if there are relatively many traditional enterprises using channel D. Last but not least, we examine how supply chain competition and the initial states of different types of supply chains' channel structure strategies affect the evolution path, evolution speed and the last ESSs of the dynamic competitive service system.

Model setting
This paper considers two competitive TSSC and GSSC, both of which could utilize a decentralized or centralized vertical channel structure (channel D or C). The GSSC would provide environmentally friendly (green) service products. By contrast, the TSSC just provides non-green service products. We focus on discussing four possible channel structure models, namely, Model DD in which both supply chains adopt channel D; Model DC(CD) where the GSSC adopts channel D(C) while the TSSC uses channel C(D); Model CC where both the GSSC and TSSC adopt channel C. Under different channel combinations, manufacturers or retailers make pricing and/or service decisions to maximize their individual profits. It is supposed that when both supply chains adopt channel D, the manufacturer acts as the leader and the retailer acts as the follower, engaging in a Stackelberg game within the same supply chain, while the two supply chains play a Nash game. Additionally, we assume that information is apparent for all members who are risk neutral He et al. 2022).
Following the majority of works (Li et al. 2019b;Zhang and Li 2020;Fan et al. 2022), we model the demands as linear functions of the retail price p i ði ¼ 1; 2Þ and service level s i . We regard the service of supply chain 2 as the green service benchmark (i.e., s 2 ¼ 0). Accordingly, the demand functions for two supply chains can be given as follows: where in a and b represent the basic market share and the market competition between two supply chains, respectively. As for green service cost, we employ a quadratic function ks 2 1 =2 to reflect the feature of marginal cost efficiency diminishing, wherein k denotes the service cost sensitivity coefficient (Dan et al. 2012). Without loss of generality, we follow the prior studies to normalize k to 1 Li et al. 2019a). We employ notation P ti ; t ¼ r; m; sc and i ¼ 1; 2 to denote the profits of retailers, manufacturers and the whole supply chains, and w i to denote the wholesale prices. The superscript ''*'' and l ¼ DD; DC; CD; CC refer to the optimal solutions under Model l.

One-shot game
We next develop and solve four channel models under different channel structure combinations based on the oneshot game.

Model DD
We start by modeling channel DD where both the GSSC and TSSC utilize channel D. In each supply chain, the retailer sets its retail price p i under the manufacturer's wholesale price w i and the green service level s 1 . According to the demand functions, the optimization problems of two retailers under Model DD can be expressed as follows: It can be derived that the profitability function P ri is concave in p i due to o 2 P r1 =op 2 1 ¼ o 2 P r2 =op 2 2 ¼ À2\0. So, solving the first-order conditions with respect to p i ; p j ; i ¼ 1; 2 and j ¼ 3 À i can yield the following optimal response functions.
Afterward, the manufacturers set the corresponding wholesale prices, and green service level simultaneously to maximize their own profits. Hence, the optimization problems of two manufacturers are given by: By plugging Eq. 4 into Eq. 5 and Eq. 6, we can obtain o 2 P m2 =ow 2 2 ¼ À2ð2 À b 2 Þ=ð4 À b 2 Þ\0 and the Hessian matrix of P m1 as shown in Eq. 7 by taking the secondorder derivatives of P mi with respect to w 1 , s 1 and w 2 .
Hence, we can get the optimal wholesale prices and service level based on the first-order conditions. Further, we can take them into the retailers' response functions to obtain the optimal retail prices, demands and profits. Since the solving processes of other models are similar to Model DD, we omit concrete proofs to save space and just summarize the optimal outcomes in Table 1.

Model DC
In this situation, the GSSC adopts channel D while the TSSC uses channel C. The timeline of this game can be described as follows. The green manufacturer first determines the wholesale price and green service level. Afterward, the green retailer and the TSSC decide the retail prices simultaneously. We can obtain their optimization problems as below:

Model CD
Compared to Model DC, the GSSC uses channel C while TSSC adopts channel D. In this setting, the traditional manufacturer first determines his wholesale price. Then, the GSSC and the traditional retailer set retail prices and green service level simultaneously. The optimization problems under Model CD can be given by:

Model CC
Under this case, both supply chains use channel C. The timing of this game is: The GSSC determines her retail price and green service level, and the TSSC decides his retail price simultaneously. We can obtain the optimization problems as follows:

Equilibrium analysis
Based on the aforementioned optimal solutions, we next conduct equilibrium analyses. Firstly, we compare the optimal green service levels under four models. 1 . Proposition 1 shows that channel structures of two supply chains play significant roles in green service decisions. Particularly, the GSSC would set the highest green service level when she adopts the centralized channel structure, and the TSSC uses the decentralized one, followed by, channel CC and then by channel DD. When the GSSC chooses channel D, she would set the lowest green service level as long as the TSSC uses channel C. Put it differently, the lower the integration between upstream and downstream enterprises in the TSSC, the higher the green service level the GSSC with channel C needs to provide so that she can maintain her competitive advantage and obtain more benefits. By contrast, when the GSSC adopts channel D, she would like to provide a higher green service level when the rival also uses channel D. Consequently, it suggests that local governments first encourage the highly integrated green firms to implement green service innovations in the face of decentralized traditional competitors can better promote the implementation of green development policies.
Proposition 2 For the GSSC, there exist thresholds b 1 and b 2 : (1) When the TSSC adopts channel D, if 0\b b 1 , the GSSC prefers channel C, otherwise channel D is preferred if b 1 \b\1; (2) When the TSSC adopts channel C, if 0\b b 2 , the GSSC prefers channel C, otherwise channel D is preferred if b 2 \b\1.
Proposition 2 indicates that no matter which channel structure the TSSC adopts, there always exists a key threshold for the GSSC to make better channel choices. Namely, the GSSC can gain more from a specific channel structure under different market environments. Specifically, when the market competition between two supply chains is relatively low, the GSSC would like to adopt channel C. On the contrary, channel D performs better when the market competition is relatively high. This implies that when two types of supply chains compete fiercely, channel D is beneficial to the GSSC, which is similar to some traditional research results. Moreover, it is worth noting that the first threshold is smaller than the second (i.e., b 1 \b 2 ), which means that the rival's adoption of channel C would decrease the GSSC's incentive to adopt channel D. This suggests that the GSSC members should cooperate with each other to implement green service innovation when the market competition is not fierce. In what follows, we will analyze the channel preference of the TSSC by comparing his profits under given the GSSC's channel structure as shown in Proposition 3, wherein b 3 can be solved by f 3 b ð Þ ¼ 6 À 14b 2 þ 8b 4 À b 6 uniquely.
Proposition 3 For the TSSC, there exists a key threshold b 3 : (1) When the GSSC adopts channel D, if 0\b b 3 , the TSSC prefers channel C, otherwise channel D is preferred if b 3 \b\1; (2) When the GSSC adopts channel C, the TSSC always prefers channel C.
Proposition 3 shows that the market competition also significantly affects the TSSC's channel preferences. Similarly, it can be found from Proposition 3(1) that when the market competition is relatively low, the TSSC prefers the centralized channel structure more than the decentralized one, and vice versa. In contrast to the GSSC's channel preference, when the GSSC uses channel C, there is no motivation for the TSSC to adopt channel D. This implies that when a supply chain faces competition from a GSSC, the advantage of channel D disappears completely. This is because the TSSC is relatively weak without providing green service, so that he has to use channel C to enhance market competitiveness and strive for more consumers when facing a stronger GSSC. It suggests that TSSCs should introduce green services to enhance the chance of keeping channel D. Furthermore, we can obtain that b 1 \b 3 \b 2 , which means the GSSC can widen the advantage region of channel D compared to the TSSC. Note that although the GSSC can keep the advantage of channel D under Model DC, it would not be too high. Namely, this advantage cannot be over more than the TSSC's decentralized channel merit under Model DD.

Evolutionary stability strategy analysis
In reality, enterprise managers often make decision-makings not once but constantly adjust their strategies by keeping learning and imitating their competitors' tactics over time through repeated game. Hence, we employ the evolutionary game to explore the long-term stable strategy of this system. Based on the above two supply chains' optimal profits, we can obtain the following payoff matrix.
It is assumed that the proportion of GSSC adopting channel D is x, then that of GSSC utilizing channel C is ð1 À xÞ. Similarly, we employ y and ð1 À yÞ to represent the proportion of TSSC adopting channel D and C, respectively. According to Yi and Yang (2017) and He et al. (2019), the increasing rate ðdx=dtÞ=x of GSSC adopting channel D equals to the difference between earnings a Á A 1 Á y; 1 À y ð Þ T and the average revenue of two supply chains x; 1 À x ð ÞÁA 1 Á y; 1 À y ð Þ T , wherein a ¼ ð1; 0Þ indicates that the GSSC adopts channel D with a probability of 1, and A 1 denotes the profit matrix ! of the GSSC. Therefore, the GSSC's replicator dynamic equation can be given by: To facilitate the following statements, we hereafter define Dm P DDÃ SC1 þ P CCÃ SC1 À P DCÃ SC1 À P CDÃ SC1 and Dn P DCÃ SC1 À P CCÃ SC1 , so that DL Dm þ Dn. Similarly, and DL Dm þ Dn. Furthermore, we can obtain the replicator dynamic equations of the GSSC and TSSC based on Tables 1 and 2 as below: From Eq. 15, we can derive the competitive system's Jacobi matrix J of replicator dynamic equations, as well as the TrJ and DetJ as below: TrJ From Eq. 17, we can deduce the system's ESSs under different conditions in terms of b, which are presented in Proposition 4. To save space, we omit some proofs in the main text and just provide the representative conditions of Proposition 4(3) in Table 3.
Proposition 4 demonstrates that the ESSs of the competitive service supply chain system. In particular, when the competition between two types of supply chains is sufficiently low (i.e., b b 3 ), only point (0,0) is the ESS; when it is moderate, both points (0,0) and (1,1) may be the ESS (i.e., b 3 \b b 2 ); when it is extremely high, only point (1,1) is the ESS (i.e., b 2 \b\1). In other words, both the GSSC and TSSC would adopt the centralized channel structure when the market competition is extremely weak. On the contrary, when two supply chains compete heavily, channel DD is the stable equilibrium channel structure over time. The reason behind this is that supply chain internal integration often increases their competitiveness. Accordingly, when the market competition is fierce, both supply chains can reduce their chain-to-chain competition by decreasing the degree of supply chain integration. It is noteworthy that when the market competition is moderate, the ESS of this system may be ESS (0,0) or ESS (1.1), which is largely affected by the initial states of two supply chains adopting channel C. Particularly, when two supply chains have lower initial willingness to adopt the decentralized channel structure, the dynamic competitive system will lastly evolve to ESS (0,0), otherwise to ESS (1,1). Namely, it is more likely for service firms in the same supply chain to achieve stable cooperation in highly centralized industries.  Channel structure and evolutionary stability analysis between traditional and green service supply chains 2471 6 Numerical example

Sensitivity analysis
In this subsection, the effects of market competition on individual or supply chains' optimal prices, green service levels, demands and profits under different situations are examined numerically. We take the tourism service market as an example and set the basic market share a ¼ 100 and let b change over from 0 to 1. Then, the effects of b can be illustrated by Figs. 1, 2, 3 and 4. Figure 1 shows that the green service level under each model is always increasing with the increase in market competition. Moreover, it is verified that the ranking relationship of green service levels is in line with the prior finding (i.e., s CDÃ . By contrast, the effects of b on wholesale prices are not all monotonously increasing (see Fig. 1b). Particularly, it can be found that the two supply chains' wholesale prices under Model DD increase in market competition, while the wholesale price of GSSC under Model DC first increases and then decreases as b increases, showing an inverted U-shaped structure. Note that the market competition would not affect the wholesale price of TSSC under Model CD.
As shown in Fig. 2, almost all retail prices of two supply chains are increasing as the market competition increases, with the exception of p CDÃ 2 and p CCÃ 2 , which is unaffected by it. More interestingly, there is a general trend that two supply chains' retail prices under Model DD are the highest but the lowest under Model CC. The reason behind this may be that the double marginalization effect under Model DD is the fiercest, while it is the weakest under Model CC.
Two supply chains with channel C would set lower retail prices due to no price increment by upstream enterprises. It is noteworthy that the retail prices of the two supply chains under Model CD are higher than that under Model DC when the market competition is relatively low, and vice versa.
From Fig. 3, we can find that most demands of the two supply chains increase in b except for d CDÃ 2 and d CCÃ 2 , which cannot be influenced by it. Furthermore, it can be found that the demands of GSSC under four models always satisfy the relationship of d CDÃ when market competition is relatively high. This implies that the supply chain with channel C can acquire more market shares than that with channel D, especially when its rival has a decentralized channel.
From Fig. 4, it shows that two supply chains' profits are increasing along with the market competition except for P CDÃ SC2 and P CCÃ SC2 , which cannot be influenced by b. Besides, these illustrations verify the previous propositions that there exist key thresholds for two supply chains to make better channel decisions. Furthermore, when market competition is relatively high, channel DD is more beneficial for two supply chains. This implies that supply chains reduce their integration degree to increase their retail prices so that they can obtain more consumer surplus values, even if this behavior would compress their total market shares. Fig. 1 The effects of b on green service levels and wholesale prices

Evolutionary stability analysis
This subsection employs numerical simulation to observe the evolution trend of the competitive system. To illustrate the evolution path of two supply chains' channel strategies more intuitively, we normalize the basic market a ¼ 1 and let b change in internal [0, 1] under different initial states of the dynamic competitive system, resulting in Fig. 5, wherein the blue, red and green lines represent the three cases of 0\b\b 3 , b 3 \b\b 2 and b 2 \b\1, respectively.
It can be seen from the above figure that both the market competition and initial states of two supply chains' channel strategies play significant roles in the evolution trends and the ESSs of the dynamic competitive system. Particularly, when the market competition is relatively low, the competitive system lastly evolves to point (0, 0) regardless of the initial states (see the blue lines). Namely, both supply chains will eventually adopt channel C. However, it would evolve to point (1, 1) when the market competition is sufficiently high regardless of the initial states (see the green lines). More importantly, the initial states will make significant sense when the market competition is moderate. In this setting, the dynamic competitive system would lastly evolve to point (0, 0) when the initial states are very low, and vice versa (see red lines in Fig. 5(a)). It is also (a) (b) Fig. 2 The effects of b on two supply chains' retail prices (a) (b) Fig. 3 The effects of b on two supply chains' demands Channel structure and evolutionary stability analysis between traditional and green service supply chains 2473 generally concluded that the system evolves to ESS (0, 0) faster when the initial states of two supply chains using channel D are lower, while it evolves to ESS (1, 1) faster with higher initial states.

Conclusions and future directions
In this paper, a joint decision problem of channel structure, price and green service for TSSC and GSSC is studied. Considering the possible channel structure combinations, we develop four mathematical models to investigate the channel choice, pricing and green service decisions of participants under the one-shot game. Further, a repeated game is considered to identify the ESSs of the dynamic competitive system. The main research results are summarized as follows.
Under the one-shot game, when given the TSSC's channel structure, the GSSC prefers channel C under a lower market competition, while channel D is preferred under a higher one. By contrast, it shows that the TSSC always prefers channel C when the GSSC adopts channel C. Additionally, the GSSC would like to provide the highest green service level when she adopts channel C while the TSSC uses channel D. Moreover, the wholesale price of the GSSC under Model DC first increases and then decreases as the market competition rises. Finally, we find that the ESSs of the dynamic competitive system are dependent of the competition degree between two supply chains and the initial states of their channel strategies. Therefore, it can be suggested that managers should adopt different channel structures according to the market competition intensity of their industry and select the appropriate time node to implement large-scale green service investments so that they can obtain stronger market competitiveness.
This research could be extended from the following aspects. From the theoretical aspect, we follow the hypothesis theory of the economist to assume that all players aim to maximize their own profits. Hence, taking the behavioral preference theory (e.g., corporate social responsibility, fairness preference, etc.) into the current research framework would be an interesting research direction. Moreover, considering information asymmetry may be another interesting future research direction as some firms would have private costs and/or demand information. Lastly, this paper focuses on the decentralized and centralized channel structure choices of two different service supply chains. Similarly, it would be interesting to study the online and offline channel choices of different types of supply chains from a long-term perspective.

Appendix
Proof for Proposition 1 Comparing the optimal green service levels under different models yields the following results.
Proof for Proposition 4 The conditions for different ESSs are presented in Tables 4, 5 and 6.