Green Product Innovation Coordination in Aluminum Building Material Supply Chains with Innovation Capability Heterogeneity: A Biform Game-Theoretic Approach

Abstract

Green product innovation in aluminum building material supply chains is critical for sustainability, particularly amid growing economic and environmental pressures. However, effective coordination is challenged by the presence of multiple agents with divergent interests and heterogeneous innovation capacities. This study proposes coordination mechanisms based on a biform game that integrates both non-cooperative and cooperative elements. Key findings include the following: (1) Greater innovation capability heterogeneity promotes green innovation investment by the stronger manufacturer and enhances overall welfare, but reduce the supplier’s profit. (2) Biform game-based decision making supports the triple bottom line more effectively than decentralized models and offers greater flexibility than centralized ones. (3) A multi-perspective compensation contract, incorporating three decision-making modes, is developed within the biform game. Exogenous decision making helps resolve the endogenous game dilemma, improving coordination outcomes. (4) The coordination framework allows firms to dynamically adjust compensation parameters in response to environmental changes, thereby enhancing supply chain resilience. Our main contribution lies in applying a novel biform game approach to address coordination challenges in green product innovation under innovation capability heterogeneity. In addition, a multi-perspective contract coordination paradigm is proposed to support triple bottom line sustainability.

1. Introduction

China’s “dual-carbon” goals, which aim to peak carbon emissions by 2030 and achieve carbon neutrality by 2060, has placed pressure on the construction industry to reduce its carbon footprint. Aluminum building materials (e.g., alloy windows, curtain walls) are paradoxically both promoted as green solutions for their recyclability (>95% recovery rate) [1] and criticized for their carbon-intensive lifecycle (11.2t/t CO2e) [2]. This tension creates a critical decision-making challenge for innovators: how to leverage aluminum’s inherent sustainable advantages (lightweight, durability) while mitigating the environmental pollution resulting from production emissions. This challenge embodies the fundamental conflict between SDG 9 (Industry, Innovation and Infrastructure) and SDG 13 (Climate Action). Market realities exacerbate this dilemma—green building certifications (LEED v4.1) now require environmental product declarations [3], compelling manufacturers to urgently rethink product design and supply chain strategies to align with SDG 12 (Responsible Consumption and Production).

While large manufacturers have actively pursued green product innovation (GPI), such as incorporating recycled aluminum and developing coating-free aluminum profiles, to enhance environmental performance [4,5], challenges remain in promoting GPI. Compared to green process innovation, GPI is more technically feasible, as has been confirmed theoretically [6]. However, small- and medium-sized enterprises (SMEs) face systemic barriers, including limited financial resources, technological constraints, and uncertain returns on green investments [7,8]. These challenges discourage SMEs from proactively adopting GPI, resulting in a dual market failure: restricted upstream diffusion of green technologies and hampered downstream promotion of green buildings. This ultimately threatens coordinated decarbonization across the aluminum building materials supply chain (AlBSC).

Collaboration is becoming increasingly vital for promoting GPI [9]. In supply chain management (SCM), a key strategy for the focal firm lies in designing appropriate incentive mechanisms for its partners [10]. In the AlBSC, when a farsighted supplier recognizes that its cost-sharing strategy with manufacturers is a key driver for promoting GPI, it must determine the appropriate strategy to motivate manufacturers to innovate while minimizing harm to its own interests [11]. For example, the investments made by Yunnan Aluminum Co., Ltd. (Kunming, China) in clean energy and green industrial parks facilitate the reduction in green innovation costs for downstream manufacturers [12]. Manufacturers focus on pricing strategies to maximize profits amid competitors. Due to the interdependence between cooperation and competition, manufacturers cannot finalize their pricing (competition) strategies without knowing the supplier’s cost-sharing (cooperation) strategy, while suppliers must assess manufacturers’ GPI investment and bargaining power before determining their own. Hence, a novel decision-making mechanism is essential to capture the interdependent nature of GPI strategies in the AlBSC.

Biform games are particularly well suited to address this challenge as they can simultaneously optimize competition and cooperation strategies by integrating the noncooperative and cooperative game approaches into a unified framework. The non-cooperative component sets the strategic context, while the cooperative component reconstructs the profit function to derive equilibrium strategies [11]. It allows all agents within the supply chain to concurrently incorporate considerations of both competition and cooperation into their strategic decision making. The extant literature compares decentralized and centralized decision making in terms of efficiency, stability, and coordination, typically employing cooperative or non-cooperative game frameworks [13]. Some scholars have also verified the effectiveness of the biform game in the coordination of green supply chains [11,14]. In business practice, manufacturers exhibit substantial innovation capability heterogeneity (ICH), leading to significant disparities in the level of green GPI [15]. This heterogeneity is widely acknowledged in the market and is generally stable over time. According to the resource-based view (RBV), it reflects fundamental differences in manufacturers’ inherent innovation capabilities, shaped through the long-term accumulation of firm-specific resources [15,16]. Manufacturers’ innovation decisions are shaped by the ICH and competitive dynamics among peer manufacturers, thereby increasing the complexity of decision making [17]. Although ICH has been examined in decentralized decision contexts, existing biform game models generally assume homogeneous manufacturers. This simplification neglects ICH, limiting the models’ ability to capture its impact on strategic coordination and decision making. Consequently, the role of ICH in biform game settings remains underexplored, indicating a critical gap in the literature. To address these challenges, this study seeks to answer the following research questions:

RQ1 How does ICH influence GPI investment and AlBSC coordination?

RQ2 What are the advantages of biform game-based decision making, and how can it facilitate coordination in the AlBSC?

To answer these questions, this study develops a biform game-based decision-making model that integrates manufacturers’ ICH within a Hotelling competition framework. The Hotelling model describes how firms incorporate heterogeneity into their pricing strategies to compete for market share. We analyze how ICH affects strategic decision making and supply chain coordination under three governance structures: decentralized, centralized, and biform game-based modes. By comparing the outcomes across these modes, the study evaluates the effectiveness of the biform game in balancing competition and cooperation. Given that the supplier voluntarily bears the cost of GPI and its associated profit loss [11], we further propose a multi-perspective compensation contract within the biform game to facilitate coordination. This research aims to support suppliers and manufacturers in optimizing pricing and GPI strategies for more sustainable and collaborative development within the AlBSC.

The marginal contributions of this paper are as follows: (1) This study integrates ICH into a biform game framework to develop a coordinated decision-making model that simultaneously addresses both cooperative and competitive strategies, relaxing the no-externality assumption commonly made in cooperative games. (2) A novel multi-perspective promotion compensation contract is introduced within the biform game framework, providing a coordination mechanism for resilient AlBSC collaborations and offering a new paradigm for supply chain coordination. (3) Three key findings extend the existing literature. First, biform game-based decision making enables suppliers to incentivize downstream manufacturers to engage in GPI through cost-sharing contracts. Second, high ICH increases coordination complexity and constrains upstream supplier gains, providing a theoretical justification for multi-perspective contract mechanisms. Third, the contract with exogenously determined parameters leads to more effective coordination outcomes compared to decisions made endogenously.

The remainder of the paper is structured as follows: Section 2 reviews the relevant literature; Section 3 formulates and solves three decision-making models; Section 4 analyzes the equilibria; Section 5 extends a compensation contract to coordinate the AlBSC; Section 6 presents the numerical results; and Section 7 concludes with managerial implications and future research directions.

2. Literature Review

To clarify our contribution relative to existing research, we review the literature in three areas: decision making and coordination of supply chain on green innovation, the application of biform game in SCM, and the operational research on firm heterogeneity.

2.1. Decision Making and Coordination of Supply Chain on Green Innovation

Green innovation, which develops eco-friendly technologies and practices to allow organizations to meet market and regulatory demands [18], has been extensively studied across apparel [13], automotive [19], and food supply chains [20]. However, the unique characteristics of the AlBSC have received insufficient attention, despite its dual characteristics of being both resource-intensive and environmentally sensitive. Aluminum and its processed products have been extensively integrated into policy initiatives aimed at advancing green manufacturing and sustainable construction practices [21,22]. In recent years, leading firms such as Xingfa Aluminum and Zhongya Aluminum have invested in low-carbon aluminum smelting and innovation in architectural aluminum profiles, thereby contributing to the green transformation of the aluminum industry [23,24].

Existing research typically categorizes green technological innovation into GPI and process innovation [25]. In AlBSC, GPI primarily manifests in end-product innovations such as recyclable aluminum alloys and coating-free profiles, while process innovation focuses on production optimizations like low-carbon electrolysis and scrap aluminum refining. Notably, GPI is generally more readily implementable than process innovation in this sector, since product-level modifications (e.g., adjustments to alloy composition) can be adopted without costly overhauls of capital-intensive smelting infrastructure [26]. While the GPI pathway avoids the heavy capital investments required for process upgrades, it inevitably restructures supply chain cost mechanisms, thereby generating coordination demands.

To address supply chain coordination challenges arising from green innovation, scholars have predominantly employed game-theoretic approaches to develop incentive mechanisms. Non-cooperative game frameworks dominate this research stream, including contract designs such as promotion, wholesales price discount, cost sharing and revenue sharing [27,28]. However, their individual profit-maximization orientation often fails to achieve system-wide optimization. Recent studies have extended cooperative game-based coordination mechanisms in green supply chains by incorporating factors such as closed-loop structures [13], disappointment-aversion preferences [29], and fairness concerns [30], demonstrating their varying effectiveness relative to non-cooperative contracts. Although these studies provide methodological references for AlBSC coordination, they exhibit two unresolved limitations: (1) Static parameterization: contract terms (e.g., cost-sharing ratios) are either fixed exogenously or negotiated endogenously without external constraints, unable to adapt to market and policy dynamics. (2) Dichotomous game frameworks: existing models fails to capture the coexistence of competition and cooperation.

To address these limitations, this study adopts a biform game framework to analyze GPI coordination in AlBSC and design a multi-perspective contract based on it. This hybrid game emphasizes individual interests while transcending the assumption of perfect collaboration in pure cooperative games, thus providing a more realistic theoretical framework for the sustainability of AlBSC. The following section systematically reviews the application progress of biform games in supply chain research to establish the theoretical foundation for our modeling approach.

2.2. The Application of Biform Game in SCM

In real-world supply chains, competition and cooperation often coexist, and the AlBSC is no exception. Upstream suppliers often serve multiple downstream manufacturers, who both compete for market share and collaborate with upstream partners in areas such as GPI and product standardization [12,31]. Depending on whether a cooperative alliance is established, inter-agent relationships can be categorized as co-opetition based on either non-cooperative games or biform games. In the former, each agent makes decisions independently. Cooperation is typically limited to short-term issues formalized through pre-contracts, with profit-sharing determined through negotiation [13] or prevailing industry norms [17]. However, this model treats cooperation and competition as distinct processes, thus failing to capture their dynamic interplay in real-world supply chains. In contrast, the biform game approach better reflects their coexistence in practice. It enables agents to retain autonomy in non-cooperative areas while coordinating cooperative activities through centralized decision making, thereby accounting for the intertwined nature of these relationships [11,14]. In the context of GPI within the AlBSC, where investments are substantial, focal firms tend to establish long-term and stable cooperative partnerships while maintaining competitive relationships. This dual structure of simultaneous cooperation and competition is well captured by the theoretical framework of the biform game [31].

The biform game, introduced by Brandenburger and Stuart [32], provides a framework for companies to strategically shape competitive environments through two-stage decision making that integrates non-cooperative and cooperative game elements [33]. Its applications in SCM evolve along three research streams: (1) Competitive Operations: Stuart [34] modeled newsboy problems with random demand, using core-based benefit functions for price competition; (2) Vertical Coordination: Feess and Thun [35] analyzed investment incentives in supply chain alliances; (3) Sustainable SCM: Recent extensions study green R&D cost-sharing [36] and low-carbon investment considering power structure [37], demonstrating its relevance to environmentally driven decisions.

Prior studies exhibit divergence across three key methodological dimensions: (1) identifying the characteristic function of cooperative games, (2) determining optimal strategies in non-cooperative games, and (3) the coupling mechanisms between non-cooperative and cooperative games. For cooperative games, the characteristic function is commonly derived using two distinct approaches: one based on the core and confidence indices, which requires the introduction of exogenous variables [32]; and the other employing the Shapley value, which provides a more equitable allocation of payoffs among players [35]. In addition to satisfying the fairness axioms, the Shapley value also ensures monotonicity, which implies that a participant’s payoff decreases with a decline in marginal contribution. Compared to alternative allocation rules, it offers greater theoretical consistency and interpretability. For the non-cooperative game part, the Cournot model [34], the Nash game [36], and the Stackelberg game [37] are used to derive the equilibrium strategy. As for the coupling mechanism, it can be broadly classified into two types: discrete mechanisms involving finite pure strategy enumeration [32,34,35] and continuous mechanisms based on infinite strategy spaces [36,37].

Along with the extant literature, this study employed the Shapley value in the cooperative game part and a Nash equilibrium approach is adopted to model the strategic interactions among firms. Given that the products are homogeneous, this setting reflects the competitive nature of decision-making in such markets [11]. Since the strategies investigated in this study are continuous, the coupling mechanism for continuous strategies is adopted. However, a critical limitation of prior work is its reliance on the assumption of homogeneous players [11,14], which overlooks the role of firm heterogeneity in shaping equilibrium strategies. This gap motivates our exploration of firm heterogeneity in the next section.

2.3. The Operational Research on Firm Heterogeneity

The study of firm heterogeneity in supply chains has progressed along two distinct yet interrelated trajectories since Porter’s seminal work on competitive advantage. The first stream conceptualizes heterogeneity as a set of discrete strategies, emphasizing binary decisions regarding whether firms pursue differentiation strategies, such as obtaining environmental certification [38], selecting exclusive distribution channels [39], or adopting brand differentiation [40]. However, such an approach limits the ability to quantify how varying degrees of differentiation affect operational performance. More fundamentally, it implicitly assumes that firms can freely choose their market positions, thereby overlooking the prior capability constraints that may restrict strategic feasibility.

The second stream treats heterogeneity as an exogenous continuous variable and focuses on the impact of output heterogeneity (e.g., product quality [41] and brand differences [42]) on supply chain decision making and coordination. Although such studies confirm the market effects of apparent heterogeneity, they fall short of uncovering its capability-based foundations. According to the RBV, sustainable competitive advantage originates from firm-specific capabilities characterized by VRIN attributes [43]. In the context of green innovation, this advantage is reflected in persistent ICH. In the aluminum alloy building materials sector, significant ICH exists among manufacturers, as leading firms typically possess strong R&D capacity and green technology reserves that enable them to drive industry innovation, while SMEs often rely on industry standards and partnerships due to limited innovation resources [22,44]. Such ICH constrains the overall advancement of green transformation.

Although existing studies have begun to focus on the impact of innovation capabilities on supply chain decisions, such as partner selection [45] and green technology investment [46], there are still obvious limitations. (1) Most studies are restricted to simplified dyadic supply chain structures, focusing only on upstream and downstream interactions, thereby overlooking the influence of ICH where multiple manufacturers coexist. (2) Innovation capabilities are frequently conflated with other operational capabilities, such as production and marketing, particularly in studies on cooperative networks [47], strategy selection [48], and the design of pricing and incentive contracts [49]. While such approaches highlight the general impact of capability heterogeneity, they obscure the distinct mechanisms by which ICH influences supply chain outcomes.

This theoretical fragmentation is particularly prominent in GPI. On the one hand, the discrete strategies literature fails to explain why there are differences in firm performance under the same strategy; on the other hand, output difference studies have difficulty explaining the root causes of capabilities. This study bridges the two by introducing ICH. Methodologically, ICH is treated as an exogenous continuous parameter that simultaneously endogenizes firms’ GPI strategic choices. Theoretically, it echoes the core proposition of the RBV that capability differences determine strategic boundaries [50], thereby providing a deeper explanation for GPI in AlBSC decision making and coordination.

To systematically integrate existing literature and identify theoretical gaps,

Table 1. Summary of reviewed literature.

Ref.	Research Object	Supply Chain Structure	Coordinate Strategy	Methods	Heterogeneity Setting
[21]	Power battery recycling supply chain	1 manufacturer, 1 retailer	Promotion/wholesale price contracts	Non-cooperative game	No
[22]	General supply chain	1 manufacturer, 1 retailer	Revenue-sharing/cost-sharing contracts	Non-cooperative game	No
[13]	Green closed-loop supply chain	1 manufacturer, 1 recycler, 1 retailer	Cooperative alliance	Cooperative game	No
[29]	General supply chain	1 supplier, 2 manufacturers	Contracts and cooperative games	Non-cooperative and cooperative games	No
[11]	Eco-innovation supply chain	1 supplier, 2 manufacturers	Cooperative alliance	Biform game	No
[32]	General supply chain	1 supplier, 2 manufacturers	-	Biform game	No
[37]	Low-carbon supply chain	1 manufacturer, 1 retailer	Cooperative alliance	Biform game	No
[40]	Retail supply chain	Dual channels: 1 supplier, 1 retailer	-	Non-cooperative game	Brand
[41]	Retail supply chain	2 suppliers, 1 retailer	-	Non-cooperative game	Product quality
[42]	E-commerce logistics supply chain	1 supplier, 2 logistics service providers	-	Non-cooperative game	Brand

presents a categorized comparison of representative studies across key dimensions related to supply chain coordination and heterogeneity.

2.4. Statement of Our Contributions

Building on the research gaps highlighted in the reviewed literature (Table 1),

Table 2. Key features of this paper.

Item	Subitem	Extant Literature	This Paper
2.1 Decision making and coordination of supply chain on green innovation	Number of agents	2 or 3	3
	Methods	Non-cooperative game and/or cooperative game	Non-cooperative game and biform game
	Number of decision-making methods for contract parameters	1	3 (Section 5)
2.2 The application of biform game in SCM	Non-cooperative game	Profit maximization strategies	Profit maximization strategies
	Cooperative game	The core or Shapley value	Shapley value
2.3 The operational research on firm heterogeneity	Heterogeneous agents	Suppliers	Manufacturers
	Dimensions of heterogeneity	Product, quality or firm ability	Innovation capability

summarizes the key differences between this study and existing literature, along with the three main contributions of this study. (1) By incorporating contracts into the biform game framework, this study achieves a multi-perspective Pareto improvement. (2) A novel payment function is derived by applying the Shapley value in the cooperative game part to restructure the profit function of the three players in the non-cooperative game part. (3) To the best of our knowledge, this study is the first to integrate ICH into the biform game framework. Specifically, we employ extensive algebraic analysis to examine the impact of ICH on decision making and coordination.

3. Models

3.1. Assumptions and Notions

This paper develops an AlBSC comprising a shared supplier and two competing manufacturers, as shown in Figure 1a. This supply chain setup is commonly used in the literature [48,51]. The supplier provides standardized aluminum ingots to two manufacturers with ICH, who process them into building materials for the downstream construction and real estate markets. Yunnan Aluminum Co., Ltd. exemplifies this arrangement by supplying uniform ingots to firms manufacturing and distributing architectural aluminum products. In practice, numerous downstream manufacturers exhibit varying levels of ICH. For example, Xingfa Aluminum, with years of accumulated technological expertise and involvement in national standard-setting, demonstrates strong innovation capabilities [23], whereas Zhongya Aluminum, despite possessing moderate innovation capacity, has yet to establish a comprehensive innovation ecosystem [24]. Accordingly, this study categorizes manufacturers into two representative groups to capture their ICH and competitive relationships. To comply with government green development regulations and meet market green demand, the manufacturers are required to engage in GPI.

The main notations are presented in

Table 3. Notations of parameters, subscript, superscripts and variables.

Category	Notions	Descriptions
Parameters	$a$	The size of the market for the weak manufacturer, and $0<a<1$
	$\delta$	The demand-boosting effect of GPI
	$\sigma$	ICH
	$\mu$	The cost per unit of GPI for the strong manufacturer
	$w$	Wholesale price
Subscript	$i$	$i\in \left\{\mathrm{1,2}\right\}$, subscripts 1 and 2 distinguish the two manufactures.
Superscripts	$DF$	Decentralized decision making
	$CF$	Centralized decision making
	$BF$	Biform game-based decision making
	$PBF$	Biform game-based decision making under exogenous promotion compensation contract
	$CPBF$	Biform game-based decision making under cooperative endogenous promotion compensation contract
	$DPBF$	Biform game-based decision making under competitive endogenous promotion compensation contract
Decision variables	$t_i$	Manufacturer’s GPI efforts
	$p_i$	Manufacturer’s selling price
	${\lambda }_i$	Cost-sharing ratio for GPI investment
Variables	$q_i$	The manufacturer’s market demand
	$Q$	Total market demand
	${\pi }_S$	Supply chain’s profits
	${\pi }_{Sp}$	Supplier’s profits
	${\pi }_{{Mf}_i}$	Manufacturer’s profits
	$CS$	Consumer surplus

.

Assumption 1. 

The market demand is influenced by both manufacturers’ selling prices, ICH, and GPI efforts [11,36,42].

Assumption 2. 

To focus on the GPI incentives and to streamline the analysis, this study assumes that the market environment remains relatively stable in the short term, with no major policy or market shocks. This assumption is consistent with the implicit settings of [11,29]. The ‘strong manufacturer’ possesses relatively greater innovation capability, whereas the ‘weak manufacturer’ has relatively less. The strong manufacturer’s potential demand is normalized to 1, reflecting its greater innovation capability that results in more reliable product quality and leads consumers to prioritize its products, whereas the weak manufacturer’s potential demand is set to $a<1$ [41]. Based on this, the demand functions for the two manufacturers are given by Equations (1) and (2).

$$\begin{array}{c}{q}_1={\displaystyle \frac{1}{1+a}}\left(1-{\displaystyle \frac{p_1-p_2}{\sigma }}\right)+\delta t_1\end{array}$$

(1)

$$\begin{array}{c}{q}_2={\displaystyle \frac{1}{1+a}}\left(a-p_2+{\displaystyle \frac{p_1-p_2}{\sigma }}\right)+\delta t_2\end{array}$$

(2)

Assumption 3. 

The cost of GPI for the strong manufacturer is ${\displaystyle \frac{\mu }{2}}{t}_1^2$, a convex function of $t_1$ [11,36]. For the weak manufacturer, the cost of GPI is ${\displaystyle \frac{1}{2}}\mu (1+{\displaystyle \frac{\sigma }{1+\sigma }})t_2^2$, which is obtained from deriving base on ref. [17]. This cost function indicates the following: (1) An increase in ICH leads to a higher GPI cost for the weak manufacturer relative to that of the strong manufacturer. (2) As ICH increases, GPI cost rises at a diminishing rate, indicating slowing marginal cost growth [52]. This pattern reflects the nonlinear nature of innovation costs during the catch-up process: when ICH is high, improvements incur relatively lower incremental costs; however, as ICH narrows, enhancements require progressively greater investments. (3) The unit cost of GPI does not increase without bound as $\sigma \to +\infty$. Assume $\mu$ is sufficiently large so that the GPI cost motivates the supplier to share the burden with manufacturers [11].

Assumption 4. 

The supplier recognizes that enhancing the greenness of aluminum building materials can both facilitate the coordinated development of the economy and the environment while generating long-term economic benefits for itself. Consequently, it is willing to establish long-term and stable cooperative relationships with manufacturers. For instance, Yunnan Aluminum Co., Ltd. has concluded long-term strategic cooperation agreements with downstream partners [53]. The supplier indirectly shares the manufacturers’ GPI costs by providing green raw materials, technical support, and environmental certifications, and its indirect support is assumed to directly offset a proportion ${\lambda }_i$ of each manufacturer’s GPI cost.

Assumption 5. 

Both manufacturers purchase aluminum ingots from the supplier at a wholesale price of $w$ and subsequently sell the manufactured aluminum building materials to the downstream market at a price of $p_i$. To ensure feasible market demand and positive profits for both the supplier and manufacturers under standardized demand assumptions, it is assumed that $0<w<a$ and $p_i>w$ [11,41]. The unit production costs of the supplier and the manufacturers are assumed to be zero for analytical tractability, which does not alter the main analytical conclusions [11,27].

3.2. Decentralized AlBSC (Model DF)

To examine the effects of the cooperation mechanism under the biform game framework, Model DF is constructed as shown in Figure 1a. Manufacturers independently bear the costs of GPI and compete on price. Under this setting, the profit functions of each agent are given by Equations (3)–(5).

$$\begin{array}{c}{\pi }_{\mathrm{Sp}}^{DF}=w\left(q_1^{DF}+q_2^{DF}\right)\end{array}$$

(3)

$$\begin{array}{c}{\pi }_{M{f}_1}^{DF}=\left(p_1^{DF}-w\right)q_1^{DF}-{\displaystyle \frac{\mu }{2}}{t}_1^{DF2}\end{array}$$

(4)

$$\begin{array}{c}{\pi }_{M{f}_2}^{DF}=\left(p_2^{DF}-w\right)q_2^{DF}-{\displaystyle \frac{\mu }{2}}\left(1+{\displaystyle \frac{\sigma }{1+\sigma }}\right)t_1^{DF2}\end{array}$$

(5)

Given $w$, the Nash equilibrium between the two manufacturers can be derived by solving the profit maximization problem formulated in Equation (6).

$$\left\{\begin{array}{c}{\pi }_{M{f}_1}^{DF}\left(p_1^{*DF},t_1^{*DF},p_2^{*DF},t_2^{*DF}\right)\ge {\pi }_{M{f}_1}^{DF}\left(p_1^{DF},t_1^{DF},p_2^{*DF},t_2^{*DF}\right)\\ {\pi }_{M{f}_2}^{DF}\left(p_1^{*DF},t_1^{*DF},p_2^{*DF},t_2^{*DF}\right)\ge {\pi }_{M{f}_2}^{DF}\left(p_1^{*DF},t_1^{*DF},p_2^{DF},t_2^{DF}\right)\end{array}\right.$$

(6)

The manufacturers’ decisions and profits are derived by solving Equation (6). Due to the length of the expressions, the detailed derivation and results are presented in Appendix A (Appendix A and other subsequent appendices are included in the main manuscript).

3.3. Centralized AlBSC (Model CF)

To compare the impact of price competition under the biform game framework, this study constructs Model CF as shown in Figure 1b. In this scenario, the AlBSC is centrally managed, with decisions made to maximize collective profit. The profit function of the supply chain is given by Equation (7).

$${\pi }_S^{CF}=p_1^{CF}{q}_1^{CF}-{\displaystyle \frac{\mu }{2}}{t}_1^{CF2}+p_2^{CF}{q}_2^{CF}-{\displaystyle \frac{\mu }{2}}\left(1+{\displaystyle \frac{\sigma }{1+\sigma }}\right)t_2^{CF2}$$

(7)

Equation (7) characterizes the profit function with respect to the decision variables $p_1$, $p_2$, $t_1$ and $t_2$. The optimal solution is obtained by solving the first-order conditions of Equation (7). The full derivation steps and closed-form expressions are provided in Appendix B.

3.4. Biform Game-Based AlBSC (Model BF)

Due to the significant costs of GPI, the visionary supplier shares these costs with the manufacturers, as shown in Figure 1c. The structure and process of the biform game-based decision-making framework proposed in this study are illustrated in Figure 2, which demonstrates the integration of both decentralized and centralized decision-making features.

The profit functions for the three agents are formulated as Equations (8)–(10).

$$\begin{array}{c}{\pi }_{Sp}^{BF}=w(q_1^{BF}+q_2^{BF})-{\displaystyle \frac{\mu }{2}}{t}_1^{BF2}{\lambda }_1^{BF}-{\displaystyle \frac{\mu }{2}}\left(1+{\displaystyle \frac{\sigma }{1+\sigma }}\right)t_2^{BF2}{\lambda }_2^{BF}\end{array}$$

(8)

$$\begin{array}{c}{\pi }_{M{f}_1}^{BF}=\left(p_1^{BF}-w\right)q_1^{BF}-{\displaystyle \frac{\mu }{2}}{t}_1^{BF2}\left(1-{\lambda }_1^{BF}\right)\end{array}$$

(9)

$$\begin{array}{c}{\pi }_{M{f}_2}^{BF}=\left(p_2^{BF}-w\right)q_2^{BF}-{\displaystyle \frac{\mu }{2}}\left(1+{\displaystyle \frac{\sigma }{1+\sigma }}\right)t_2^{BF2}\left(1-{\lambda }_2^{BF}\right)\end{array}$$

(10)

The non-cooperative part: Two manufacturers determine their pricing strategies through the Nash game. Unlike in Model DF, the payoff function of the non-cooperative part in Model BF is not predefined. Instead, it is endogenously determined, as the manufacturers expect that the final benefits will be allocated through the cooperative game based on a cost-sharing contract. Consequently, the outcomes of the non-cooperative pricing strategies do not directly represent the final payoffs. Rather, each strategic profile merely establishes a competitive environment that induces the subsequent cooperative game.

The cooperative part: To prompt GPI, the supplier proposes a cost-sharing contract and participates in this part, thereby forming a three-player cooperative game for dividing the benefits of innovation cooperation. To derive this allocation, the characteristic values of all possible coalitions must first be computed, which depend on the GPI investment and the cost-sharing ratios. The Shapley value is then employed to allocate the cooperation benefits. The Shapley value is computationally efficient for supply chains with a limited number of participants, and the ongoing digital transformation of SMEs further improves its practical feasibility [54]. Notably, this allocation is computed under the competitive environment defined by the pricing strategy profile $(p_1^{BF},p_2^{BF})$ from the non-cooperative part. The resulting allocation serves as the objective function for each manufacturer in the non-cooperative part to determine the final equilibrium prices and payoffs.

Based on the decision sequence, we first solve the cooperative game, given the pricing strategy profile $(p_1^{BF},p_2^{BF})$.

3.4.1. Cooperative Game Part

In this part, we use Minimax or Maximin principles to compute the characteristic value for any possible coalition in the three-player cooperative game, which is denoted by $v\left(p_1^{BF},p_2^{BF}\right)\in G^3$ [55]. The detailed solution process and results are in Appendix C.

For the cooperative game $v(p_1^{BF},p_2^{BF})\in G^3$, we adopt the Shapley value to determine a fair allocation. It is essential to verify whether the game, represented in characteristic function form, is convex and super-additive. If these conditions hold, the core of the game is nonempty, and the Shapley value, as a single solution in classical cooperative game theory, lies within the core. Based on the results in Appendix C, we derive proposition 1.

Proposition 1. 

The cooperative game $v(p_1^{BF},p_2^{BF})\in G^3$ is a convex and super-additive.

The proof is provided in Appendix D.

Proposition 1 shows that forming larger coalitions yields greater collective benefits, providing a strong incentive for all players, particularly the manufacturers. It also indicates that the grand coalition is stable under the Shapley value allocation, as the convexity of the game ensures that the Shapley value lies within the core. Under this allocation, no agent has an incentive to deviate, as each receives no less than what they could obtain outside the coalition, making the outcome acceptable to all agents.

Employing the Shapley value formula [56], the profit distribution for the supplier and the two manufacturers is given by Equations (11)–(13).

$$\begin{array}{l}{\widehat{\pi }}_{sp}=(4a{p}_1^{BF}w{\delta }^2\sigma +2a{p}_1^{BF}w{\delta }^2+2a{p}_2^{BF}w{\delta }^2\sigma +2a{p}_2^{BF}w{\delta }^2-3a{w}^2{\delta }^2\sigma -2a{w}^2{\delta }^2\\ +8aw\mu \sigma +4aw\mu +4p_1^{BF}w{\delta }^2\sigma +2p_1^{BF}w{\delta }^2+2p_2^{BF}w{\delta }^2\sigma +2p_{2}w{\delta }^2-8p_2^{BF}w\mu \sigma \\ -4p_2^{BF}w\mu -3w^2{\delta }^2\sigma -2w^2{\delta }^2+8w\mu \sigma +4w\mu )/\left(8a\mu \sigma +4a\mu +8\mu \sigma +4\mu \right)\end{array}$$

(11)

$$\begin{array}{l}{\widehat{\pi }}_{M{f}_1}=\left(2a{p}_1^{BF2}{\delta }^2\sigma -2a{p}_1^{BF}w{\delta }^2\sigma +a{w}^2{\delta }^2\sigma +2p_1^{BF2}{\delta }^2\sigma -4p_1^{BF2}\mu \right.+4p_1^{BF}{p}_2^{BF}\mu \\ -2p_1^{BF}w{\delta }^2\sigma +4p_1^{BF}w\mu +4p_1^{BF}\mu \sigma -4p_2^{BF}w\mu +w^2{\delta }^2\sigma \left.-4w\mu \sigma \right)/\left(4a\mu \sigma +4\mu \sigma \right)\end{array}$$

(12)

$$\begin{array}{l}{\widehat{\pi }}_{M{f}_2}=(2a{p}_2^{BF2}{\delta }^2{\sigma }^2+2a{p}_2^{BF2}{\delta }^2\sigma -2a{p}_2^{BF}w{\delta }^2{\sigma }^2-2a{p}_2^{BF}w{\delta }^2\sigma +8a{p}_2^{BF}\mu {\sigma }^2\\ +4a{p}_2^{BF}\mu \sigma -8aw\mu {\sigma }^2-4aw\mu \sigma +8p_1^{BF}{p}_2\mu \sigma -4p_1^{BF}w\mu +w^2{\delta }^2\sigma +w^2{\delta }^2{\sigma }^2\\ +4p_1^{BF}{p}_2\mu -8p_1^{BF}w\mu \sigma +2p_2^{BF2}{\delta }^2{\sigma }^2+2p_2^{BF2}{\delta }^2\sigma -8p_2^{BF2}\mu {\sigma }^2-12p_2^{BF2}\mu \sigma \\ -4p_2^{BF2}\mu -2p_2^{BF}w{\delta }^2{\sigma }^2-2p_2^{BF}w{\delta }^2\sigma +8p_2^{BF}w\mu {\sigma }^2+12p_2^{BF}w\mu \sigma +4p_2^{BF}w\mu \\ +a{w}^2{\delta }^2{\sigma }^2+a{w}^2{\delta }^2\sigma )/\left(8a\mu {\sigma }^2+4a\mu \sigma +8\mu {\sigma }^2+4\mu \sigma \right)\end{array}$$

(13)

3.4.2. Non-Cooperative Game Part

As previously discussed, the allocation derived in the cooperative game determines the payoff functions in the non-cooperative game. Accordingly, the payoff functions for the two manufacturers are given by Equations (12) and (13), respectively. Since the supplier does not participate in the non-cooperative game, only the two manufacturers make decisions regarding their respective sale prices. In this setting, the manufacturers engage in a Nash game, and their equilibrium strategies and corresponding optimal profits are presented in Proposition 2.

Proposition 2. 

The two manufacturers’ optimal prices are shown in Equations (14) and (15).

$$\begin{array}{l}{p}_1^{*BF}=(4a{\mu }^2{\sigma }^2+18w{\mu }^2\sigma +12w{\mu }^2{\sigma }^2+2a{\mu }^2\sigma +8{\mu }^2{\sigma }^3+12{\mu }^2{\sigma }^2+4{\mu }^2\sigma \\ -9w{\delta }^2\mu {\sigma }^2-2{\delta }^2\mu {\sigma }^3-9aw{\delta }^2\mu {\sigma }^2-2a{\delta }^2\mu {\sigma }^3-2{\delta }^2\mu {\sigma }^2+FZ)/FM\end{array}$$

(14)

$$\begin{array}{l}{p}_2^{*BF}=(8a{\mu }^2{\sigma }^2+4a{\mu }^2\sigma -4a{\delta }^2\mu {\sigma }^3+8w{\mu }^2{\sigma }^2+16w{\mu }^2\sigma +4{\mu }^2{\sigma }^2+2{\mu }^2\sigma \\ -10w{\delta }^2\mu {\sigma }^2-4a^2{\delta }^2\mu {\sigma }^3-2a^2{\delta }^2\mu {\sigma }^2-10aw{\delta }^2\mu {\sigma }^2+FZ)/FM\end{array}$$

(15)

$$\begin{array}{l}\mathrm{where}FZ=a^{2}w{\delta }^4{\sigma }^3+a^{2}w{\delta }^4{\sigma }^2+2aw{\delta }^4{\sigma }^3+2aw{\delta }^4{\sigma }^2+w{\delta }^4{\sigma }^2-4aw{\delta }^2\mu {\sigma }^3\\ +w{\delta }^4{\sigma }^3-5aw{\delta }^2\mu \sigma -5w{\delta }^2\mu \sigma +6w{\mu }^2-4w{\delta }^2\mu {\sigma }^3-2a{\delta }^2\mu {\sigma }^2\mathrm{and}FM=6{\mu }^2+\\ 2a^2{\delta }^4{\sigma }^3+2a^2{\delta }^4{\sigma }^2+4a{\delta }^4{\sigma }^3+4a{\delta }^4{\sigma }^2-8a{\delta }^2\mu \sigma +2{\delta }^4{\sigma }^3+2{\delta }^4{\sigma }^2-8{\delta }^2\mu {\sigma }^3\\ -16{\delta }^2\mu {\sigma }^2-16a{\delta }^2\mu {\sigma }^2-8{\delta }^2\mu \sigma +20{\mu }^2\sigma -8a{\delta }^2\mu {\sigma }^3+16{\mu }^2{\sigma }^2\end{array}$$

By substituting $p_1^{*BF}$ and $p_2^{*BF}$ into $t_1^{\prime \prime \prime \prime }$ and $t_2^{\prime \prime \prime \prime }$ in Appendix C, the equilibrium GPI investments, $t_1^{*BF}$ and $t_2^{*BF}$, are obtained. Then, by substituting the profit functions ${\widehat{\pi }}_{{Mf}_1}$ and ${\widehat{\pi }}_{{Mf}_2}$, along with the equilibrium prices and GPI investment, into Equations (9) and (10), the equilibrium cost-sharing ratios are determined. Due to the complexity of the green product investment expression, we present ${\lambda }_1^{*BF}$ and ${\lambda }_2^{*BF}$ as illustrative examples in Equations (16) and (17).

$${\lambda }_1^{*BF}=1-{\displaystyle \frac{4p_1^{*BF}{t}_1^{*BF}\delta \mu +2p_1^{*BF}w{\delta }^2-4t_1^{*BF}w\delta \mu -w^2{\delta }^2-2p_1^{*BF2}{\delta }^2}{2t_1^{*BF2}{\mu }^2}}$$

(16)

$$\begin{array}{l}{\lambda }_2^{*BF}=1-(8p_2^{*BF}{t}_2^{*BF}\delta \mu {\sigma }^2-2p_2^{*BF2}{\delta }^2{\sigma }^2-4p_2^{*BF2}{\delta }^2\sigma -2p_2^{*BF2}{\delta }^2\\ +12p_2^{*BF}{t}_2^{*BF}\delta \mu \sigma +4p_2^{*BF}{t}_2^{*BF}\delta \mu +2p_2^{*BF}w{\delta }^2{\sigma }^2+4p_2^{*BF}w{\delta }^2\sigma \\ +2p_2^{*BF}w{\delta }^2-8t_2^{*BF}w\delta \mu {\sigma }^2-12t_2^{*BF}w\delta \mu \sigma -4t_2^{*BF}w\delta \mu -w^2{\delta }^2{\sigma }^2\\ -2w^2{\delta }^2\sigma -w^2{\delta }^2)/\left(8t_2^{*BF2}{\mu }^2{\sigma }^2+8t_2^{*BF2}{\mu }^2\sigma +2t_2^{*BF2}{\mu }^2\right)\end{array}$$

(17)

Since the core of the cooperative game is non-empty, multiple allocation schemes can ensure the stability of the grand coalition. This study adopts the Shapley value because it satisfies the monotonicity property. This property is generally acceptable to all agents, making the Shapley value a suitable basis for deriving the agents’ payoff functions in the non-cooperative game. As a result, the equilibrium solutions $p_1^{*BF},p_2^{*BF}$ obtained from these functions are both reasonable and feasible. Furthermore, these equilibrium outcomes support the applicability of the Shapley value in this context. Overall, the biform game allows us to analytically solve the price competition and innovation cooperation problem under a cost-sharing contract.

4. Analysis

This section presents a series of propositions to compare the equilibrium outcomes under the DF, CF, and BF Models, and derives the following conclusions.

Proposition 3. 

Given $w<a$, $0<k<1$ and $a<1$. Then, we have

(1)

${\displaystyle \frac{\mathit{\partial }{t}_1^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}>{\displaystyle \frac{\mathit{\partial }{t}_2^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}>0$.

(2)

${\displaystyle \frac{\mathit{\partial }{t}_1^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}>0,{\displaystyle \frac{\mathit{\partial }{t}_2^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}<0,$  ${\displaystyle \frac{\mathit{\partial }{t}_1^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}>{\displaystyle \frac{\mathit{\partial }{t}_2^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}$.

(3)

${\displaystyle \frac{\mathit{\partial }{t}_1^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}>{\displaystyle \frac{\mathit{\partial }{t}_2^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}$; ${\displaystyle \frac{\mathit{\partial }{t}_1^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}>0$; ${\displaystyle \frac{\mathit{\partial }{t}_2^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}>0$ if $w<{\displaystyle \frac{a(4{\sigma }^2+12\sigma +6)+2{\sigma }^2+6\sigma +3}{20{\sigma }^2+36\sigma +15}}$, otherwise  ${\displaystyle \frac{\mathit{\partial }{t}_2^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}<0$.

(4)

${\displaystyle \frac{\partial t_1^{\mathrm{*}BF}}{\partial \sigma }}>{\displaystyle \frac{\partial t_1^{\mathrm{*}DF}}{\partial \sigma }}>{\displaystyle \frac{\partial t_1^{\mathrm{*}CF}}{\partial \sigma }}$, ${\displaystyle \frac{\partial t_2^{\mathrm{*}DF}}{\partial \sigma }}>{\displaystyle \frac{\partial t_2^{\mathrm{*}BF}}{\partial \sigma }}>{\displaystyle \frac{\partial t_2^{\mathrm{*}CF}}{\partial \sigma }}$.

The proof is detailed in Appendix E.

Proposition 3(1) shows that under Model DF, greater ICH leads to increased GPI efforts, consistent with the idea that heterogeneity promotes innovation through diffusion pressure [57]. The strong manufacturer invests more than its weak counterpart, as its superior innovation capability yields higher marginal returns and a clearer competitive advantage in GPI. In contrast, the weak manufacturer, facing higher innovation costs and lower returns, has limited incentive to do so. Therefore, strong manufacturers should leverage their innovation advantage by expanding GPI efforts, while weak manufacturers can focus on niche positioning to avoid direct competition.

Proposition 3(2) indicates that under Model CF, greater ICH leads to increased GPI efforts by the strong manufacturer, while reducing that of the weak manufacturer. In the centralized framework, decisions are made from the perspective of maximizing collective profits. As ICH rises, the system reallocates innovation resources toward the strong manufacturer to improve overall efficiency. This resource reallocation is based on cost–benefit considerations, reflecting the centralized decision maker’s preference for optimizing marginal returns.

Proposition 3(3) demonstrates that under Model BF, when the weak manufacturer operates in a market environment characterized by high demand and low wholesale prices, the favorable conditions provide sufficient incentives for increased GPI efforts as ICH rises. Conversely, when market demand fails to offset the high pricing pressure, the weak manufacturer reduces input to avoid inefficient resource allocation. The strong manufacturer, in this case, exhibits the same behavior as under Model DF. Weak manufacturers should dynamically adjust their GPI investments based on demand and pricing conditions, taking advantage of favorable market conditions to increase investment and reducing investment under adverse pressures.

Proposition 3(4): compared to Models DF, Model BF makes manufacturers’ efforts in GPI more sensitive to the ICH when w is large. Specifically, the strong manufacturer invests more in GPI as ICH increases under Model BF. In contrast, the weak manufacturer reduces investment, as ICH rises when w is large. These differing responses highlight the interplay between resource endowments and competitive pressures in distinct decision-making model. Manufacturers should monitor fluctuations in the wholesale price and adjust their GPI strategies accordingly to mitigate the risks of over- or under-investment.

Proposition 4. 

If $w<a$, $0<k<1$ and $a<1$. Then, we have

(1)

${\displaystyle \frac{\mathit{\partial }{p}_1^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}>0$, ${\displaystyle \frac{\mathit{\partial }{p}_2^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}>0$, ${\displaystyle \frac{\mathit{\partial }{p}_1^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}>{\displaystyle \frac{\mathit{\partial }{p}_2^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}$.

(2)

${\displaystyle \frac{\mathit{\partial }{p}_1^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}>0$; ${\displaystyle \frac{\mathit{\partial }{p}_2^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}<0$ if $a>4{\sigma }^2+4\sigma$; otherwise  ${\displaystyle \frac{\mathit{\partial }{p}_2^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}>0$; ${\displaystyle \frac{\mathit{\partial }{p}_1^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}>{\displaystyle \frac{\mathit{\partial }{p}_2^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}$.

(3)

${\displaystyle \frac{\mathit{\partial }{p}_1^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}>0$, ${\displaystyle \frac{\mathit{\partial }{p}_2^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}>0$, ${\displaystyle \frac{\mathit{\partial }{p}_1^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}>{\displaystyle \frac{\mathit{\partial }{p}_2^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}$.

(4)

${\displaystyle \frac{\partial p_i^{\mathrm{*}BF}}{\partial \sigma }}>{\displaystyle \frac{\partial p_i^{\mathrm{*}DF}}{\partial \sigma }}>{\displaystyle \frac{\partial p_i^{\mathrm{*}CF}}{\partial \sigma }}$.

The proof is detailed in Appendix F.

Proposition 4(1) demonstrates that, under Model DF, greater ICH leads to higher selling prices, with the price increase for the strong manufacturer exceeding that of the weak manufacturer. This reflects the pricing transmission effect of ICH in market competition: the strong manufacturers can leverage their differentiation advantage to raise prices, while weaker ones face greater limitations.

Proposition 4(2) indicates that under Model CF, an increase in ICH leads to a higher selling price for the strong manufacturer. In contrast, the pricing strategy of the weak manufacturer is contingent upon the interplay between market size and ICH. Specifically, when the market size is relatively large, the weak manufacturer is inclined to lower its price to capitalize on economies of scale. Conversely, when the market size is limited, the weak manufacturer tends to raise its price to offset its innovation disadvantage. In such cases, constrained market scale reduces the potential to compensate for weak innovation through sales volume, making price elevation a primary strategy for sustaining profitability. This outcome underscores the significant role of the interaction between ICH and market size in shaping pricing behavior.

Proposition 4(3) shows that the role of ICH on pricing under Model BF aligns with the findings under Model DF. However, as Proposition 4(4) reveals, the influence of ICH on pricing is most pronounced in Model BF. This stems from its hybrid structure that integrates both cooperative and competitive elements, thereby amplifying the strategic impact of ICH. Manufacturers should leverage their respective positions under Mode BF: the strong manufacturer can consolidate its market share by capitalizing on price advantages, while the weak manufacturer can enhance profitability through cooperation to support moderate price increases.

Proposition 5. 

Given that $\mu$ is sufficiently large, the following is derived.

(1)

${\displaystyle \frac{\mathit{\partial }{q}_1^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}<0$, ${\displaystyle \frac{\mathit{\partial }{q}_2^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}<0$, $\left|{\displaystyle \frac{\mathit{\partial }{q}_1^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}\right|>\left|{\displaystyle \frac{\mathit{\partial }{q}_2^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}\right|$, ${\displaystyle \frac{\mathit{\partial }{Q}^{\mathrm{*}DF}}{\mathit{\partial }\sigma }}<0$.

(2)

${\displaystyle \frac{\mathit{\partial }{q}_1^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}>0$; ${\displaystyle \frac{\mathit{\partial }{q}_2^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}<0$; when  $a>4{\sigma }^2+4\sigma$, $\left|{\displaystyle \frac{\mathit{\partial }{q}_1^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}\right|<\left|{\displaystyle \frac{\mathit{\partial }{q}_2^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}\right|$, ${\displaystyle \frac{\mathit{\partial }{Q}^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}<0$; when  $a<4{\sigma }^2+4\sigma$, $\left|{\displaystyle \frac{\mathit{\partial }{q}_1^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}\right|>\left|{\displaystyle \frac{\mathit{\partial }{q}_2^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}\right|$, ${\displaystyle \frac{\mathit{\partial }{Q}^{\mathrm{*}CF}}{\mathit{\partial }\sigma }}>0$.

(3)

${\displaystyle \frac{\mathit{\partial }{q}_1^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}<0$, ${\displaystyle \frac{\mathit{\partial }{q}_2^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}<0$, $\left|{\displaystyle \frac{\mathit{\partial }{q}_1^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}\right|>\left|{\displaystyle \frac{\mathit{\partial }{q}_2^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}\right|$, ${\displaystyle \frac{\mathit{\partial }{Q}^{\mathrm{*}BF}}{\mathit{\partial }\sigma }}<0$.

(4)

$\left|{\displaystyle \frac{\partial q_1^{\mathrm{*}BF}}{\partial \sigma }}\right|>\left|{\displaystyle \frac{\partial q_1^{\mathrm{*}DF}}{\partial \sigma }}\right|>{\displaystyle \frac{\partial q_1^{\mathrm{*}CF}}{\partial \sigma }}$, $\left|{\displaystyle \frac{\partial q_2^{\mathrm{*}BF}}{\partial \sigma }}\right|>\left|{\displaystyle \frac{\partial q_2^{\mathrm{*}DF}}{\partial \sigma }}\right|>\left|{\displaystyle \frac{\partial q_2^{\mathrm{*}CF}}{\partial \sigma }}\right|$, $\left|{\displaystyle \frac{\partial Q^{\mathrm{*}BF}}{\partial \sigma }}\right|>\left|{\displaystyle \frac{\partial Q^{\mathrm{*}DF}}{\partial \sigma }}\right|>\left|{\displaystyle \frac{\partial Q^{\mathrm{*}CF}}{\partial \sigma }}\right|$.

The proof is detailed in Appendix G.

Proposition 5(1) shows that under Model DF, an increase in ICH reduces the demand for both manufacturers, with a more pronounced decline for the strong manufacturer. GPI efforts lead to higher production costs and, consequently, increased product prices. However, the demand-boosting effect of GPI is insufficient to offset the demand loss resulting from higher prices, leading to an overall decline in market demand. Furthermore, when the strong manufacturer charges a higher price, consumers may shift their purchases to the lower-priced product offered by the weaker manufacturer.

Proposition 5(2) shows that under Model CF, an increase in ICH leads to higher market demand for the strong manufacturer, while demand for the weak counterpart declines. The overall market demand depends on the interplay between the weak manufacturer’s market share and the level of ICH. When the weak manufacturer holds a relatively small market share, the decline in its demand has limited impact on the total market, and the increased demand for the strong manufacturer may dominate, leading to overall demand growth. However, if the weak manufacturer holds a large market share, the negative impact of its demand decline may offset the gains of the strong manufacturer. This weakens the overall effectiveness of GPI in expanding market demand and may hinder its broader diffusion. This dynamic resembles a Gresham-like effect in product markets, where low-innovative offerings crowd out higher-quality green innovations, ultimately discouraging long-term sustainability efforts.

Proposition 5(3) indicates that under Model BF, the effect of ICH on market demand is consistent with that under Model DF. However, Proposition 5(4) shows that this effect is most pronounced in Model BF, where complex interactions and pricing competition intensify the impact of ICH on market positioning and consumer preferences, leading to greater demand fluctuations. In contrast, Model DF exhibits milder effects due to the absence of cooperation, while Model CF, characterized by unified control, suppresses competitive dynamics, limiting the transmission of heterogeneity to market outcomes. Accordingly, manufacturers may adopt appropriate strategies to enhance market demand.

Proposition 6. 

Given that $\mu$ is large enough, we have ${\displaystyle \frac{\partial {CS}^{*DF}}{\partial \sigma }}<0$; ${\displaystyle \frac{\partial {CS}^{*BF}}{\partial \sigma }}<0$; ${\displaystyle \frac{\partial {CS}^{*CF}}{\partial \sigma }}>0$ if $a^2+a<4{\sigma }^2+4\sigma +1$, otherwise ${\displaystyle \frac{\partial {CS}^{*CF}}{\partial \sigma }}<0$.

The proof is detailed in Appendix H.

Proposition 6 shows that under the DF and BF Models, increased ICH leads to a decline in CS, primarily due to higher selling prices that narrow the gap between consumers’ willingness to pay and actual prices. Manufacturers should adopt marketing strategies, such as branding and promotions, to raise consumers’ willingness to pay, thereby offsetting the negative impact of higher ICH on CS. Under Model CF, the effect depends on the market size of the weak manufacturer. When the weak manufacturer holds a large market share, increased ICH widens the price gap between manufacturers, allowing the weak manufacturer to attract more consumers and enhance overall CS. In contrast, when the weak manufacturer’s market share is small, rising ICH leads to higher average prices, thereby reducing CS. These results highlight the interaction between ICH, pricing strategies, and market structure.

Proposition 7. 

Given $\mu$ is large enough, $t_i^{*CF}>t_i^{*BF}>t_i^{*DF}$ holds.

The proof is detailed in Appendix I.

Proposition 7 shows that GPI investment is highest under Model CF, followed by BF, and lowest under DF. In Model CF, resources across the entire AlBSC are centrally planned and allocated to maximize overall benefit. By reducing unnecessary competition and resource fragmentation, Model CF efficiently integrates capital and technology, thereby maximizing GPI investment. This makes Model CF the ideal framework for fostering GPI. Model BF combines features of both CF and DF. It allows partial central allocation of resources while leaving some decisions to individual participants. As a result, although there is some degree of coordination and resource sharing, residual competition among manufacturers limits integration efficiency compared to Model CF. Thus, GPI investment under BF is lower than in CF but higher than in DF. In Model DF, manufacturers act independently to maximize individual profits. The absence of systemic resource access and coordination leads to cautious investment behavior, especially under market uncertainty. This fragmented decision making results in the lowest level of GPI investment. In light of its potential to improve the efficiency of GPI investment through enhanced coordination and increased resource sharing while retaining operational flexibility, Model BF may represent a preferable option for the AlBSC.

Proposition 8. 

Given $\mu$ is large enough, $p_i^{*CF}>p_i^{*BF}>p_i^{*DF}$ when $w<{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3a}{6+4\sigma }}$; $p_i^{*BF}>p_i^{*DF}>p_i^{*CF}$ generally holds when $w>{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3a+2a\sigma }{6+6\sigma }}$; $p_1^{*BF}>p_1^{*DF}>p_1^{*CF}$ and $p_2^{*CF}>p_2^{*BF}>p_2^{*DF}$ when ${\displaystyle \frac{1}{2}}+{\displaystyle \frac{3a}{6+4\sigma }}<w<{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3a+2a\sigma }{6+6\sigma }}$.

The proof is detailed in Appendix J.

Proposition 8 reveals the dynamic nature of selling price ordering across different models as $w$ varies. Specifically, when $w$ is relatively low, Model CF yields the highest selling prices, reflecting the stronger profit integration capability of centralized decision-making. In contrast, when $w$ exceeds the upper threshold, Model BF results in the highest prices, while Model CF yields the lowest, as fully centralized coordination tends to suppress selling prices to mitigate demand loss under high $w$. Interestingly, within the intermediate range, the price hierarchy diverges between the two manufacturers: for the strong one, Model BF generates the highest price, whereas for the weak one, Model CF prevails. These findings indicate that fluctuations in the wholesale price not only trigger regime shifts in price ordering across decision models but also induce asymmetric pricing effects between competing manufacturers. Therefore, firms should establish flexible contractual and collaborative frameworks to align stakeholder interests effectively in dynamic market environments, thereby enhancing overall supply chain coordination.

Proposition 9. 

Given $\mu$ is large enough, $q_i^{*BF}>q_i^{*DF}>q_i^{*CF}$ when $w<{\displaystyle \frac{1}{2}}+{\displaystyle \frac{a}{4(\sigma +1)}}$, otherwise, $q_1^{*BF}>q_1^{*DF}>q_1^{*CF}$ and ${q_2^{*CF}>q}_2^{*BF}>q_2^{*DF}$ generally hold. $Q^{*BF}>Q^{*DF}>Q^{*CF}$ if $w<{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3a}{2(2\sigma +3)}}$, otherwise, $Q^{*CF}>Q^{*BF}>Q^{*DF}$ generally holds.

The proof is detailed in Appendix K.

Proposition 9 demonstrates that market demand is generally the highest under Model BF. Although Model DF tends to induce lower prices and thereby higher demand due to intensified competition, Model CF is designed to maximize total supply chain profit, which typically leads to higher prices and reduced demand. These results align with extant literature [29,58]. Notably, Model BF achieves the highest demand in most cases due to its hybrid structure combining cooperation and competition. This arrangement supports greater GPI investment than Model DF and more competitive pricing than Model CF, thereby enhancing perceived value and expanding the consumer base. However, when the wholesale price is sufficiently high, the demand under Model CF surpasses that of BF and DF. In such cases, the centralized coordination in CF helps mitigate demand loss despite higher prices, resulting in relatively greater market demand. Model BF is generally advantageous for expanding market share, and firms should proactively explore and implement this flexible form of cooperation.

Proposition 10. 

Given $\mu$ is large enough, if $w<{\displaystyle \frac{1}{2}}+{\displaystyle \frac{a}{4(\sigma +1)}}$, ${CS}^{*BF}>{CS}^{*DF}>{CS}^{*CF}$ holds.

The proof is detailed in Appendix L.

Proposition 10 shows that CS is the highest under Model BF when $w$ is not excessively high. Typically, intense competition in decentralized decision making leads to lower selling prices, resulting in higher CS in Model DF compared to Model CF. Model CF entails the highest GPI investment but sets a higher sales price that suppresses demand, resulting in lower CS compared to Model BF. This suggests that price competition between manufactures benefits consumers. However, in Model BF, the combination of cooperation and competition increases market demand and, consequently, CS. The interplay of price competition and innovation cooperation enables consumers to obtain greater value at lower prices than in Model CF, while benefiting from higher market demand, thus further enhancing CS. Hence, it is recommended that the supply chain adopt Model BF, balancing cooperation and competition to enhance market efficiency and customer satisfaction.

5. Extension

As demonstrated above, Model BF achieves a more effective balance across environmental and CS dimensions compared to Model DF. However, given that the supplier actively bears part of the costs associated with downstream GPI, its profitability may be adversely affected [11]. In addition, GPI often incurs high upfront costs and may not generate sharable returns in the short term, rendering revenue-sharing contracts ineffective.

To address this issue, a promotion compensation contract is introduced to offset the supplier’s potential losses. Instead of directly transferring profits to the supplier, the two manufacturers increase their investment in the promotion of green products to stimulate market demand. This demand expansion indirectly enhances the supplier’s profitability, thereby achieving compensation. Let $y_1$ and $y_2$ denote the promotion efforts of the two manufacturers, respectively, with a unit promotion cost normalized to 1 for simplicity. Let $b$ denote the demand-boosting effect of promotion. We consider three decision-making scenarios for $y_1$ and $y_2$ within a biform game framework: (1) exogenous (Model PBF), (2) endogenous under a cooperative strategy (Model CPBF), and (3) endogenous under a competitive strategy (Model DPBF). In the following, we model each scenario. The profit functions for the three agents are given by Equations (18)–(20).

$$\begin{array}{c}{\pi }_S=w\left({\displaystyle \frac{1+a-p_2}{1+a}}+\delta t_1+\delta t_2+b{y}_1+b{y}_2\right)-{\displaystyle \frac{\mu }{2}}{t}_1^2{\lambda }_1-{\displaystyle \frac{\mu }{2}}\left(1+{\displaystyle \frac{\sigma }{1+\sigma }}\right)t_2^2{\lambda }_2\end{array}$$

(18)

$$\begin{array}{c}{\pi }_{M{f}_1}=\left(p_1-w\right)\left({\displaystyle \frac{1}{1+a}}\left(1-{\displaystyle \frac{p_1-p_2}{\sigma }}\right)+\delta t_1+b{y}_1\right)-{\displaystyle \frac{1}{2}}{y}_1^2-{\displaystyle \frac{\mu }{2}}{t}_1^2\left(1-{\lambda }_1\right)\end{array}$$

(19)

$$\begin{array}{c}\begin{array}{l}{\pi }_{M{f}_2}=\left(p_2-w\right)\left({\displaystyle \frac{1}{1+a}}\left(a-p_2+{\displaystyle \frac{p_1-p_2}{\sigma }}\right)+\delta t_2+b{y}_2\right)\\ -{\displaystyle \frac{\mu }{2}}\left(1+{\displaystyle \frac{\sigma }{1+\sigma }}\right)t_2^2\left(1-{\lambda }_2\right)-{\displaystyle \frac{1}{2}}{y}_2^2\end{array}\end{array}$$

(20)

When $y_1$ and $y_2$ are exogenous variables influenced by the external environment or industry conditions, the calculation process, similar to that of Model BF, is omitted for brevity.

When the promotion strategies are treated as endogenous variables, the biform game framework allows $y_1$ and $y_2$ to serve as cooperative strategies in the cooperative game part or as competitive strategies in the non-cooperative game part. As the calculation process is similar to Model BF, it is not elaborated further.

6. Numerical Results

To reinforce the empirical validity of the findings, this study conducts further analysis using the aluminum alloy building materials supply chain as a case, comprising one aluminum ingot supplier upstream and two aluminum alloy manufacturers downstream.

Model parameters are set based on publicly available macro-level industry statistics and firm-level data from the aluminum sector. After reasonable simplifications consistent with model assumptions, the simulations capture the behaviors of key supply chain agents and market conditions. Given the difficulty of obtaining complete micro-level data, this approach offers a practical way to demonstrate the model’s applicability and decision-making relevance in real supply chain contexts [59].

6.1. Parameter Setting

The parameters used in the simulation are estimated based on publicly available industry data, patent databases, and relevant literature. To ensure representativeness, aluminum alloy building material is selected as a case study. Although the demand enhancement effect of GPI cannot be directly measured, the growth rate of the Al Alloy sector [60] and related studies [59,61] inform the setting of the demand enhancement coefficient $\delta =0.2$. The wholesale price of aluminum ingots is based on the mode of the London Metal Exchange aluminum closing prices from 1 May 2024, to 1 May 2025, after min–max normalization. The wholesale price is set at 0.15, corresponding to the lower end of the normalized price range and representing a low-price market condition. Additionally, to comprehensively capture wholesale price fluctuations, very low (0.05), medium (0.3), and high (0.6) values are included in the sensitivity analysis to verify the robustness of the model’s conclusions across different market scenarios.

The innovation capability data are sourced from the CSMAR database, which is widely used in finance and industrial research and covers financial, market, and industry-related information of listed companies in China. The data preparation involved the following steps:

Step 1: Sample selection. Given the focus on aluminum building material manufacturers, A-share listed companies in the aluminum rolling and processing industry (Industry Code: C3252) from 2001 to 2021 were selected. A total of 26 companies primarily engaged in the production of architectural aluminum profiles were identified as the sample.

Step 2: Data collection. The number of green patents held by each sample firm was collected to measure their innovation capability.

Step 3: Grouping and calculation. Firms were categorized into high- and low-innovation groups using a threshold of 10 green patents. This grouping does not affect the main findings, as sensitivity analyses on ICH have been conducted to ensure the robustness of results. The ICH was calculated as the ratio of the patent difference between groups to the number in the higher group, with $\sigma =0.6$ and $\sigma ϵ [0.1,1]$.

According to the market concentration of China’s aluminum processing industry, with a CR10 of 57.46% [62], the market share of the weaker manufacturer is assumed to be 0.4, resulting in the parameter $a=2/3$. Based on this baseline, lower (0.25) and higher (0.75) values of $a$ are selected to examine the model’s robustness against market structure variations.

Guided by the Green Manufacturing Project Implementation Guidelines issued by the Ministry of Industry and Information Technology of the People’s Republic of China in 2016, this study sets 2015 as the base year for traditional product innovation (TPI), with its unit innovation cost normalized to 1. The unit cost multiplier for GPI is then measured for 2016 and subsequent years. Drawing on data from A-share listed companies in the aluminum smelting and rolling processing industry that consistently disclosed R&D investment from 2007 to 2024, this study incorporates China’s core inflation rate into the empirical analysis to perform time discounting, thereby controlling for the effect of time. According to data from 2008 to 2025, the average annual core inflation rate is 1.19% [63]. Based on this, it is estimated that the actual unit cost multiplier of GPI relative to TPI falls within the range of $[1.1,3.7]$, with an average value of 2.1. This range is also consistent with previous studies [59,61], which set the cost multiplier at 2 or 2.5, serving as a representative value within this interval. Accordingly, we set $\mu =2.1$. Sensitivity analysis considers lower (1.5) and higher (3.5) values of $\mu$, representing typical GPI unit costs in the aluminum building materials sector.

Next, we perform sensitivity analyses on the performance of the coordination mechanism, which incorporates the innovative promotion compensation contract, and examine the impact of ICH on the corresponding equilibrium solutions.

6.2. The Impact of ICH on GPI Decisions

The relevant conclusions are obtained by propositional derivation in Section 4. This section is verified by numerical analysis. Figure 3 illustrates the impact of ICH on GPI investment, which corroborates Propositions 3 and 7. As shown in Figure 3a, the difference between ${\displaystyle \frac{\partial t_i^{*BF}}{\partial \sigma }}$ and ${\displaystyle \frac{\partial t_i^{*DF}}{\partial \sigma }}$ is minor. However, compared with Model CF, Model BF significantly enhances how sensitive both manufacturers’ GPI investment is to ICH.

Figure 4 supports Propositions 4 and 8. The empirical results further indicate that the manufacturers’ selling prices under Models BF and DF are nearly identical. However, when $\delta$ is relatively high ($\delta =0.9$), as shown in Figure 4b, Model BF tends to yield slightly higher selling prices than Model DF. Figure 4c further reveals that when the wholesale price is high and within a narrow range, the selling price under Model CF may fall between those of the BF and DF models. This indicates a mild transitional effect near the high-price threshold. Although this does not affect the overall conclusions, it underscores the need for firms to carefully calibrate pricing and cooperation strategies within this sensitive range.

Figure 5 confirms Proposition 5, while Figure 5 supports Proposition 9. As shown in Figure 5a, the demand under Model BF is slightly higher than that under Model DF. Moreover, Figure 6 shows that within certain low wholesale price intervals, demand under Model CF may fall between that of Models BF and DF. Overall, Model BF demonstrates higher demand across most wholesale price ranges, whereas Model CF outperforms the others only under extremely high wholesale prices.

Figure 7 lends support to Propositions 6 and 10. Empirical findings reveal that Model BF results in a marginally higher CS than Model DF, but significantly higher than Model CF.

6.3. The Impact of ICH on Profit Allocation and Profits

Given the complexity of the expressions, we examine the impact of ICH on the cost-sharing ratio and profits through numerical simulations.

Regarding the cost-sharing ratio, Figure 8 reveals the following key findings: (1) The cost-sharing ratios are positive, indicating that Model BF can incentivize suppliers to share the cost of GPI with manufacturers in the early stages [11]. (2) The supplier shares a higher ratio with the weak manufacturer than with the strong one. This is mainly because the weak manufacturer has limited resource endowments, making the cost of GPI exceed its own capability. (3) Greater ICH leads to lower cost-sharing ratios. For the strong manufacturer, this typically implies less dependence on the supplier. In contrast, the weak manufacturer tends to rely more on supplier support to remain competitive. However, as ICH increases, the supplier may perceive diminishing returns from supporting the weak manufacturer and thus becomes less willing to share the innovation costs. (4) A higher unit cost of innovation leads to an increase in the cost-sharing ratio, but the magnitude of this increase is relatively small. This occurs because the supplier considers the diminishing marginal returns and increased investment risks associated with higher innovation costs, making them less willing to significantly raise the cost-sharing ratio. Therefore, when ICH among downstream manufacturers is high, it is advisable for the supplier to adopt a relatively low cost-sharing ratio. Moreover, as unit innovation costs increase, the supplier should exercise greater prudence in cost-sharing decisions and consider providing technical support to mitigate potential risks.

Concerning profits, Figure 9 presents the following key findings: (1) The demand-boosting effect of GPI moderates the impact of ICH on the supplier’s profit, with distinct mechanisms observed under the DF and BF Models. Specifically, when GPI yields limited returns, ICH amplifies competitive imbalance: strong manufacturers are unable to fully capitalize on their capabilities, while the weaker one struggle to remain competitive, leading to weakened realization of green premiums and suppressed supplier profitability. In contrast, when GPI generates substantial market returns, ICH facilitates effective differentiation and market segmentation under Model DF, enhancing channel vitality and enabling the supplier to benefit from a diversified downstream structure. Under Model BF, however, ICH contributes to supplier profit growth only when both ICH and the demand-enhancing effect of GPI are sufficiently high, highlighting the importance of coordinated market strength and innovation capability for realizing upstream profitability.

(2) An increase in ICH leads to increase in the profits of the manufacturers and the supply chain. As the ICH increases, the strong manufacturer can capitalize on its innovation outcomes to strengthen product differentiation, thereby attracting greater market demand. The enhanced competitive advantage enables both manufacturers to command higher retail prices, thereby improving their profitability. However, despite the reduced cost-sharing ratios due to increased heterogeneity, the overall market demand contracts slightly. Since the supplier’s wholesale price remains unchanged, the decline in demand ultimately diminishes the supplier’s profit.

(3) The profits of the two manufacturers and the supply chain are marginally higher in Model BF than in Model DF, indicating that Model BF enhances benefits through cooperation in GPI. This finding aligns with the results of literature [11], further validating the effectiveness of Model BF.

(4) ICH significantly affects the relative advantage of supplier profits under Models BF and DF. When ICH is low, the cooperative nature of Model BF leads to slightly higher supplier profits through more balanced profit sharing. However, when ICH is high, supplier profits under Model DF surpass those in Model BF, as greater ICH creates natural barriers that reduce competition. The competitive strategy in Model DF allows suppliers to capture more profits from differentiation. This indicates that the effectiveness of cooperative versus competitive strategies depends on the degree of ICH, which directly impacts supplier profitability. When incentivizing downstream manufacturers to engage in GPI, suppliers should assess the level of manufacturers’ ICH. Under low ICH, collaboration through Model BF may lead to a more balanced profit distribution. However, when ICH is high, suppliers should carefully evaluate the potential profit constraints imposed by Model BF and consider implementing compensation contracts or adjusting cooperation terms to mitigate possible losses. (5) The profit of the supply chain is maximized under Model CF, which is a well-established result in the existing literature, as it eliminates double marginalization [61].

To further validate and enrich the robustness of the findings, sensitivity analyses were conducted on parameters $a$, $w$, and $\mu$. The results (see Appendix M) support the main conclusions, while also revealing that when $w$ is relatively high and ICH is low, supplier profits under Model BF exceed those under Model DF. This occurs because higher wholesale prices allocate a greater share of supply chain profits to the supplier. Consequently, even when the supplier bears part of the downstream GPI costs, its own profit is not adversely affected.

6.4. The Impact of ICH on SW

According to ref. [64], social welfare (SW) is defined as $SW={\pi }_s+CS$. As shown in Figure 10, the main conclusions can be drawn and are discussed as follows. (1) The inequality ${SW}^{*CF}>{SW}^{*BF}>{SW}^{*DF}$ indicates that cooperation contributes positively to SW. Model CF primarily enhances overall SW from the perspective of the supply chain, while Model BF better balances CS. For the aluminum industry, this suggests a greater degree of supply chain integration, such as enhanced collaboration between upstream and downstream agents, which can reduce inefficiencies, improve incentive alignment, and ultimately generate higher overall value. (2) With the increase in ICH, SW gradually improves. This suggests that greater heterogeneity promotes a more efficient allocation of resources by directing them toward more innovative and productive entities, which in turn enhances SW. Therefore, the AlBSC incorporates Model BF, thereby enhancing CS and SW. Meanwhile, ICH promotes efficient resource allocation, achieving mutual benefits for both supply chain value and social welfare.

To verify the robustness of the above conclusions, sensitivity analyses were conducted on key parameters $\delta$, $a$, $w$, and $\mu$. The results indicate that the conclusions remain consistent, thereby demonstrating robustness (see Appendix M). Only when $w$ is relatively high does SW under Model BF potentially exceed that under Model CF.

6.5. The Coordination Effect of the Promotion Compensation Contract

6.5.1. Exogenous Promotion Compensation

To investigate the effect of the strong manufacturer’s compensation strategy on profits, we set $y_2=0$. Against the backdrop of a projected 6.21% CAGR for the aluminum alloy industry from 2024 to 2034 [65], it is assumed that promotional investment will lead to a 10% demand enhancement effect. Considering the diminishing marginal returns of promotion efforts, the strong manufacturer, assuming rational behavior, would not increase its promotional investment indefinitely. Based on the analysis of annual reports from the non-ferrous light metal industry for the period 2022–2024, the maximum observed ratio of selling expenses to profits is approximately 11.33%. Given the previously established parameter values, the strong manufacturer’s profit is estimated to be around 0.17. According to the cost function, the maximum value of $y_1$ is thus inferred to be approximately 0.2. Taking into account that ${\lambda }_i>0$ and the presence of loss aversion among manufacturers, the parameter $y_1$ is set within the interval $(0,0.1)$.

As illustrated in Figure 11, the range of the compensation coefficient can be divided into four distinct zones. ${\Omega }_1$: Ineffective zone; the supplier’s profit is lower than that in Models BF and DF. ${\Omega }_2$: BF-Dominant Pareto Optimal Zone; both the collective and individual profits exceed those under Model BF. ${\Omega }_3$: Strong Pareto Optimal Zone; both the collective and individual profits exceed those under Models BF and DF. ${\Omega }_4$: DF-Dominant Pareto Optimal Zone; both the collective and individual profits exceed those under Model DF. Therefore, manufacturers can adopt a low-promotion investment strategy to improve their own profitability as well as that of the supply chain and supplier.

Through a sensitivity analysis of the above conclusions (see Appendix M), the results indicate the following: (1) the core conclusions remain consistent; (2) when ICH and $\delta$ are at lower levels, the findings still hold, although the required promotional compensation ratio for achieving Pareto improvements decreases as ICH declines; (3) when ICH is higher, the exogenous mechanism continues to yield Pareto improvements over Model BF, but achieving improvements over both BF and DF requires a higher compensation ratio; and (4) when $\delta$ is high, Pareto improvements can only be attained relative to Model BF.

As shown in Figure 12, in all compensation zones, both the level of GPI investment and the SW under Model PBF exceed those under Model BF. While promotion compensation generally enhances investment in GPI, it does not always lead to higher SW. This is because excessively high compensation coefficients can reduce the overall profit of the supply chain, thereby diminishing SW.

6.5.2. Endogenous Promotion Compensation

The following key findings can be drawn from Figure 13: (1) GPI’s demand-enhancing effect significantly moderates the influence of ICH on supplier profitability, aligning with the findings in Section 6.2 and holding under both promotional and non-promotional conditions. (2) Endogenous strategies can yield equilibrium solutions and, under certain conditions, achieve Pareto improvements. Specifically, equilibria exist under both cooperative and competitive strategies; when cooperation is adopted and the demand enhancement effect is relatively weak, the system can realize a Pareto improvement. (3) Cooperative strategies benefit the supplier, while competitive strategies favor the manufacturer. Specifically, the supplier earns slightly more under Model CPBF than Model BF, while Model DPBF results in lower profits than BF. For the manufacturer, endogenous promotional compensation contracts better coordinate the supply chain and increase both manufacturers’ profits, with Model DPBF yielding higher manufacturer profits than Model CPBF. This is attributed to the higher promotional investment under cooperation compared to competition. Firms should adjust contract strategies based on market conditions. Under high market risk, they should prioritize CPBF to share risks and promote collaborative innovation. Conversely, when increased manufacturer profits favor cooperation, firms should adopt DPBF while balancing overall supply chain performance.

Figure 14 further illustrates the following key finding: decisions based on biform game, with or without promotional compensation mechanisms, are beneficial to the environment and society. Specifically, compared to Models DF and BF, CPBF and DPBF lead to increased GPI investment and higher SW. Moreover, the difference between cooperative and competitive strategies in their impact on GPI and SW is marginal.

6.5.3. Comparison of the Coordination Results Across Different Decision-Making Mode

We further compare the coordination outcomes across the three decision-making modes.

Table 4. Coordination outcomes under different decision-making modes.

How decisions are made	Coordinate the results
Exogenous	Pareto improvement over Models BF or DF (${\Omega }_2$: Manufacturer-biased, ${\Omega }_3$: Balanced, ${\Omega }_4$: Supplier-biased)
Endogenous cooperation	Pareto improvement over Models BF or DF (Supplier-biased)
Endogenous competition	No Pareto improvement (Manufacturer-biased)

presents three key findings: (1) The promotion-compensation contracts based on biform game are characterized by multi-perspective coordination, as variations in their design can strategically favor different agents. (2) Exogenous decision making provides higher coordination effectiveness, as it is not influenced by the strategic interactions between agents, whereas endogenous decision making may lead to a weakening or loss of coordination due to game dynamics. This approach bypasses the equilibrium dilemma, achieving system-wide optimal coordination. (3) In endogenous games, cooperation facilitates coordination, while competition exacerbates divergences, particularly disadvantaging the supplier. Supply chain agents should adopt a multi-perspective promotional compensation contract that balances the interests of all parties and facilitates effective coordination. Priority should be given to exogenous decision making mechanisms to avoid coordination failures caused by strategic games.

7. Conclusions

This paper develops an AlBSC model involving a supplier and two manufacturers with ICH investing in GPI. We introduce a novel biform game that jointly optimizes both the GPI investment cooperation strategy and the profit-maximizing strategy to develop a decision-making model, and compare it with the DF and CF Models. Furthermore, we introduce a promotion-compensation contract based on the biform game considering the supplier’s profit loss under Model BF, which incorporates three decision-making modes, enabling multi-perspective coordination paradigm to achieve effective supply chain coordination.

This paper develops an AlBSC model involving a supplier and two manufacturers with ICH, both of whom invest in GPI. A novel biform game is proposed to jointly determine the GPI cooperation strategies and profit-maximizing decisions, and its outcomes are compared with those under DF and CF Models. Furthermore, a promotion-compensation contract is designed based on the biform game to address the supplier’s potential profit loss under Model BF. This framework incorporates three decision-making modes and provides a multi-perspective coordination mechanism to achieve effective AlBSC coordination.

7.1. Key Findings

We summarize the core findings as follows.

Table 5. The impact of ICH on decisions and coordination.

Model	Environment	Economics				Society
	GPI	Prices	Supplier’s Profit	Manufacturers’ Profit	Supply Chain’s Profit	SW
DF	↑	↑	↓↑	↑	↑	↑
CF	${\mathrm{M}\mathrm{f}}_1$↑, ${\mathrm{M}\mathrm{f}}_2$↓	${\mathrm{M}\mathrm{f}}_1$↑, ${\mathrm{M}\mathrm{f}}_2$↓↑	-	-	↑	↑
BF	${\mathrm{M}\mathrm{f}}_1$↑, ${\mathrm{M}\mathrm{f}}_2$↓↑	↑	↓↑	↑	↑	↑

Note: ↑ monotonic increase; ↓ monotonic decrease.

addresses RQ1. As ICH increases, strong manufacturers consistently increase GPI investment. In contrast, weak manufacturers’ GPI investment rises under Model DF but declines under Model CF. In Model BF, GPI investment may either increase or decrease, depending on the interaction among wholesale prices, market size, and ICH. Selling prices rise under both Models DF and BF. Under Model CF, prices for strong manufacturers increase, while those for weak manufacturers may either increase or decrease, depending on the relationship between market size and ICH. Overall, while greater ICH promotes supply chain coordination and delivers economic and social benefits, it complicates the alignment of supplier profits. Nevertheless, higher ICH may cause a decline in supplier profit but does not diminish its willingness to cooperate, as collaboration helps avoid greater losses resulting from inaction.

As for RQ2, Model BF consistently outperforms Model DF across all key indicators, though it does not surpass Model CF. Notably, BF yields higher market demand and CS than CF, suggesting greater market responsiveness and enhanced consumer welfare. As full centralization is often impractical due to organizational and contractual constraints, BF provides a more feasible and effective alternative. By balancing coordination and competitive autonomy, it offers a practical compromise between theoretical optimality and real-world applicability.

While Model BF improves the overall supply chain performance compared to Model DF, it fails to achieve a Pareto improvement, as it leads to supplier losses. This finding is consistent with the results reported by ref. [11]. To address this, we propose a promotion compensation contract based on the biform game framework, which facilitates a multi-perspective coordination paradigm. Specifically, when the compensation coefficient is treated as an exogenous variable, different intervals of its value lead to coordination outcomes that favor either the manufacturer or the supplier, or achieve a globally balanced result. Endogenous cooperative strategies can yield a Pareto improvement, under specific conditions, relative to the outcomes of Models BF and DF. Notably, endogenous compensation reveals the principle of ‘cooperation to benefit the supplier, competition to benefit the manufacturer.’ In uncertain environments, this contract offers a more flexible and resilient mechanism, enhancing adaptability to market fluctuations and policy changes.

7.2. Managerial Implications

Our results unveil the following managerial insights, which are synthesized and visually presented in Figure 15. Based on this summary, the key managerial implications are as follows.

(1) Suppliers are advised to adopt differentiated cooperation strategies based on downstream manufacturers’ innovation capabilities and market-driving power. Rather than passively relying on manufacturers’ GPI efforts or exclusively favoring dominant partners, suppliers should proactively identify those able to convert innovation into market performance. Investment in green collaboration is justified only when both technological and commercial potential are evident. Moreover, the supplier can motivate downstream manufacturers by designing a win–win promotional compensation contract. When contract parameters are exogenous, the supplier should aim to secure the highest possible compensation within a reasonable range; when parameters are endogenous, choosing a cooperative decision will lead to more favorable outcomes for the themselves. In practice, tools such as customer analytics, innovation records, and third-party green certifications can assist in partner evaluation. For the AlBSC, adopting Model BF is recommended, as it generally leads to more optimal environmental, economic, and social outcomes compared to Model DF.

(2) For manufacturers, forming partnerships is the optimal decision, as appropriate collaboration supports their green transition. While compensating suppliers may result in some profit loss, as long as compensation is not excessive, the manufacturer’s profit will always exceed that of non-cooperation. Regarding supply chain coordination mechanisms, it is recommended that AlBSC adopt a multi-perspective promotion compensation contract based on biform game. To effectively implement the promotional compensation contract, manufacturers may utilize diversified marketing channels, such as participating in large-scale trade fairs and industry exhibitions. Promotion investment strategies should be based on actual conditions. Specifically, firms can flexibly adjust incentive structures and cost-sharing ratios in response to external changes, thereby maintaining the effectiveness of both coordination and incentives. Moreover, when certain environmental factors (e.g., unfavorable policies or intensified market competition) disadvantage one party, the contract can be adapted to favor the disadvantaged party through multi-promotion compensation strategies to maintain stable cooperation. This flexibility helps stabilize cooperation and fosters more resilient, sustainable supply chain collaborations.

(3) In addition, this study offers valuable insights for policymaking. Given manufacturers’ ICH, policymakers should adopt differentiated strategies tailored to firms’ innovation capacity and market transformation potential. For manufacturers with strong innovation capabilities and commercialization prospects, targeted incentives such as R&D subsidies, tax relief, or technical support can accelerate the adoption and diffusion of GPI. For suppliers whose profits may be adversely affected by participating in GPI collaboration, supportive policies should facilitate fairness. Furthermore, policy interventions should be dynamically adjusted based on demand-side conditions. When green product demand is strong, supply-side support (e.g., innovation grants or production incentives) should be prioritized. Conversely, under weak demand, greater emphasis should be placed on consumer-side measures such as subsidies, public awareness campaigns, or green public procurement to stimulate market development and prevent resource misallocation.

(4) Although this study assumes a relatively stable policy environment, in reality, carbon tax policies, green subsidies, and environmental regulations often experience significant fluctuations and uncertainties. Sudden policy changes may substantially affect firms’ GPI investment and pricing strategies. Specifically, when environmental policies become more stringent, such as through increased carbon taxes, supply chain agents face greater external pressures. Models CF and BF can more effectively foster upstream–downstream collaboration, enabling firms to share policy-related risks, enhance GPI investment, and improve overall supply chain performance. Conversely, when environmental policies are relaxed, external pressure for GPI weakens. Nevertheless, under Model BF, the AlBSC can still maintain a certain level of collaboration, strengthening green resilience and its capacity for continuous improvement.

7.3. Limitations and Possible Directions

This study is not exempt from limitations. It is based on a simplified supply chain structure, which facilitates broad applicability of the results. However, real-world supply chains are often more complex, involving multiple tiers and diverse actors. Future research could extend the model to multi-tier supply chains to improve the generalizability of the findings. Moreover, bounded rationality and information asymmetry, which commonly occur in practice, may affect GPI coordination and could be incorporated into future models. Finally, this paper analyzes scenarios under known market demand conditions, while future studies could consider uncertainty in supply and demand by incorporating stochastic variables.