Research on Dynamic Decision-Making of Shared Supply Chain Considering Multi-Agent Altruistic Preferences

Abstract

Shared supply chains (SSCs) represent a critical practice in the field of the sharing economy. With the aid of digital technologies, SSCs improve resource utilization through efficiently re-allocating and sharing. This paper develops a three-tier SSC model comprising a supplier, a retailer, and a sharing platform. The model is based on a Stackelberg dynamic framework and incorporates both resource sharing and altruistic behaviors. Our findings reveal that different altruistic modes significantly alter members’ decision logics: supplier altruism is manifested by lower supply prices and enhanced supply efforts; retailer altruism improves system supply efficiency through increased supply efforts; and the sharing platform, through resource integration and service optimization, comprehensively elevates overall SSC operations. Furthermore, numerical experimental analysis indicated that as altruistic parameters increased, overall system performance improved significantly. Notably, the marginal contribution of platform altruism surpassed that of both the supplier and the retailer, underscoring its pivotal role as a central hub in SSC systems. In addition, joint altruism can achieve a Pareto improvement for both individual members and overall system utility. This study provides an important theoretical basis for designing multi-agent collaborative mechanisms in SSCs.

1. Introduction

In recent years, the global sharing economy has experienced sustained and vigorous growth, with continuous expansion in application fields and increasing diversification of industry participants. The sharing economy is an economic mode in which individuals and organizations share access to underutilized resources (e.g., goods, services, or assets) rather than traditional ownership. Powered by digital technologies, this mode enables efficient peer-to-peer exchanges and collaborative consumption, ultimately maximizing resource utilization and reducing waste. Leading companies like Uber are reshaping global business ecosystems [1]. The sharing economy market is expanding robustly, bolstering economic resilience, stabilizing employment, and achieving remarkable breakthroughs in pivotal sectors such as shared healthcare, transportation, and accommodation. According to Statista, the total value of the global sharing economy is expected to reach USD 794 billion by 2031 [2]. This trend not only reflects the rapid growth in sharing economy business modes in consumer markets but also signifies their gradual penetration into production processes, particularly in supply chain management, where shared models are gaining traction.

Against this backdrop of the rapid development of the sharing economy, shared supply chains (SSC) have emerged as an innovative management model and attracted global corporate attention. SSCs represent a critical implementation of sharing economy principles in supply chain management. By leveraging resource sharing and integration, SSCs optimize traditional operational practices. Taking Alibaba 1688 Tao-Factory as an example, the platform integrated more than 15,000 apparel factories, breaking the bottleneck of traditional production links, and using idle capacity for customized production, which not only improves the efficiency of small orders, but also helps factories maintain profitability in the off-season. This mode of reliance on a sharing platform to achieve high-efficiency production has improved the flexibility and resource utilization of SSCs. Another example is the freight forwarding platform Flexport, which uses digital technologies to integrate data through the coordination of global freight forwarding partners and warehousing providers. Flexport allows small companies to enjoy open and transparent freight prices, greatly improving the efficiency and flexibility of global freight [3]. With these sharing platforms, all parts of SSCs have been able to operate more efficiently and collaboratively, promoting in-depth application of the sharing economy in the production sector.

In both idle-resource-sharing and information-sharing models, SSCs exhibit markedly higher operational efficiency than traditional modes [4,5,6,7,8]. This superior performance is primarily attributable to SSCs’ inherent ability to dismantle conventional supply chain barriers via platform-based operations, thereby integrating previously fragmented manufacturers, suppliers, and retailers. This integration improves the utilization of idle resources and maximizes the overall creation of systemic value [9]. Moreover, sharing platforms serve as central hubs within SSCs, optimizing processes through efficient resource matching and seamless information synchronization, which in turn enhances responsiveness and reduces operational costs. With continuous advancements in digital platforms and the widespread adoption of sharing economy principles, SSCs are well positioned to expand further and may ultimately emerge as the dominant management paradigm.

Within SSC systems, platform-based resource integration mechanisms facilitate the matching of idle resources between upstream and downstream entities, thus reshaping conventional collaboration models among SSC members. This synergistic sharing gives rise to a novel cooperative paradigm: when each participant accurately perceives the demand fluctuations of its partners, the traditional focus on maximizing individual profits shifts to prioritizing overall profit maximization, which ultimately fosters altruistic behavior, as demonstrated by Andreoni’s dictator game experiments [10]. In this context, sharing platforms, serving as the central hub of SSCs, enhance information flow and prompt members to adjust their decision-making processes for a more balanced profit distribution. This behavior aligns with the “inequity aversion” framework proposed by Fehr and Schmidt [11]. Ultimately, driven by altruistic preferences, SSC members engage in reciprocal cooperation to balance individual and collective profits, thereby enhancing system performance.

Although the sharing economy has garnered significant attention across various academic disciplines, research on SSCs remains relatively underdeveloped and exhibits notable limitations. One major shortcoming is the failure to treat sharing platforms as an autonomous decision-making entity, which consequently overlooks their strategic role in supply–demand matching and in facilitating resource sharing. Moreover, while unilateral altruistic behavior in traditional supply chains has been extensively examined, there is a substantial gap in studies addressing multi-agent altruism within SSC contexts.

To address these gaps, this paper introduces an innovative three-tier SSC model, in which the sharing platform (I), the supplier (M), and the retailer (R) are integrated. Firstly, all three members are regarded as autonomous decision-makers. Secondly, we integrate multi-agent altruism into the decision-making framework, and then explore its impact on optimal decisions, members’ profits, and overall system performance. Specifically, we address

• What are the optimal decisions, members’ profits, and overall system performance under different altruistic modes?
• How do altruistic preferences affect the optimal decisions, members’ profits, and overall system performance?
• What similarities and differences exist between single-agent versus multi-agent altruistic modes regarding their impacts?

The remainder of this paper is organized as follows: Section 2 reviews the relevant literature. Section 3 describes the model assumptions and formulates the problem. In Section 4, we solve differential game models under eight altruistic modes and derive corresponding expressions of optimal solutions and system performance. Section 5 presents comparative analyses, while Section 6 details numerical experiments. Finally, in Section 7, we conclude with the main conclusions, managerial insights, and future research.

2. Literature Review

Research themes that are closely related to the core of this paper include shared supply chain and altruistic preferences.

2.1. Shared Supply Chain

In recent years, the rapid development of the sharing economy has introduced both opportunities and challenges to supply chain management, particularly in the field of SSCs. With the advancement of digital platforms, SSC modes are fundamentally reshaping ecosystems and market landscapes [12]. This emerging paradigm integrates resources and supply–demand information across members through platform-based operations, optimizing production, distribution, and supply–demand matching, while breaking down information silos to enhance overall performance [13].

Resource sharing serves as a pivotal driver of supply chain collaboration [14], defined as the strategic integration of capabilities and assets among members to achieve complementary advantages and value co-creation [15,16]. Recent research has emphasized the efficiency-enhancing mechanisms of SSCs, particularly in addressing challenges such as low manufacturing resource utilization and inefficient task execution. Evidence suggests that resource sharing modes significantly improve overall SSC performance through cross-organizational resource integration and process coordination [16].

Although idle resource sharing and optimal allocation have triggered extensive discussions in academia, existing studies predominantly focused on static resource allocation models (e.g., linear programming for idle equipment scheduling), which largely neglect the dynamic interdependencies among stakeholders in SSC ecosystems. For instance, few works have systematically examined how platform-mediated coordination mechanisms affect long-term resource utilization rates. Additionally, the role of behavioral factors (e.g., altruistic preferences, risk aversion) in shaping multi-agent collaboration remains underexplored, limiting the practical applicability of current SSC optimization frameworks.

Future research should develop multi-level theoretical frameworks integrating game theory to optimize resource flows in SSCs. Furthermore, translating these theoretical insights into operational strategies to address real-world supply–demand matching dilemmas will be critical for advancing SSC applications.

2.2. Altruistic Preference

Extensive behavioral experiments have challenged the traditional “rational agent” assumption, revealing the significant role of emotional preferences (e.g., fairness and altruism) in individual and collective decision-making [17]. The pioneering work of Rabin made critical strides by formally integrating fairness considerations into game theory and economic models [18]. His framework laid the theoretical foundation for modern interpretations of altruistic and social preferences. In 2002, Andreoni empirically validated altruistic behaviors through dictator game experiments, formally establishing the theory of altruistic preference [10].

Recent studies have increasingly explored the impact of altruistic preferences on supply chain dynamics. Experimental evidence from Loch et al. highlighted how social preferences systematically influence economic decisions in supply chain transactions [19]. Furthermore, Shi et al. developed a Stackelberg differential game model to analyze altruistic preferences in dual-channel supply chains, revealing that manufacturers’ altruism significantly affects pricing strategies, with higher altruism levels enhancing partners’ profits [20]. Bassi et al. further identified a positive correlation between altruism intensity and overall system efficiency in principal-agent frameworks [21].

Existing research on single-agent altruistic modes has yielded significant insights. For instance, the study by Ge et al. demonstrated that, in supplier-dominated systems, supplier altruism achieves a system efficiency level intermediate between decentralized and centralized decision-making paradigms [22]. Moreover, experimental evidence from Choi and Messinger reveals that altruistic behaviors among supply chain members can override structural power advantages, leading to more equitable profit distribution than predicted by non-cooperative game theory [23]. Another study analyzed the impacts of altruistic preferences in a retailer-led low-carbon supply chain, highlighting that the retailer’s altruism can enhance the profits of small- and medium-sized manufacturers (SMMs), as well as overall system efficiency [24]. For multi-agent altruism, Liu et al. modeled four investment scenarios, where both Logistics Service Integrators (LSIs) and Functional Logistics Service Providers (FLSPs) exhibited altruistic preferences [25]. Their Stackelberg game analysis showed that altruism coefficients must not exceed profit margin rates to ensure minimum profitability, with mutual altruism maximizing system profits compared to unilateral or no-altruism modes.

While studies on SSCs, resource sharing, and altruistic preferences are abundant, few integrated multi-agent altruism with SSCs. Based on Liu’s comparative analysis of altruistic modes [25], this study introduces a sharing platform as an independent decision-maker within a three-tier SSC. We investigate optimal decisions and compare system performance under different altruistic modes.

3. Model Description and Assumptions

3.1. Model Description

Against the backdrop of the thriving sharing economy, the introduction of sharing platforms has revitalized traditional supply chains. Serving as the hub of SSCs, these platforms leverage information technologies to facilitate supply–demand information sharing and resource matching. Firstly, the platform acts as a data collector, gathering real-time information on customer demand, supplier capacity, and any idle resources (for example, unused machinery or extra warehouse space). Secondly, it serves as a service coordinator, choosing appropriate service levels and matching efforts to keep operating costs under control. The platform optimizes operational efficiency, while maintaining cost-effectiveness, thereby ensuring seamless supplier–retailer connectivity and minimizing lead times. Thirdly, the sharing platform functions as an idle-resource manager. Using cloud-based technologies and resource matching algorithms, it dynamically provides relevant information to demand-side enterprises, enabling demand-driven redistribution that enhances systemic resource utilization rates. Moreover, the overall resilience of the SSC is strengthened. Moreover, SSC operational efficacy is further amplified by member altruistic preferences, which further incentivizes collaboration among SSC members, encouraging them to balance collective benefits, proactively share critical information (e.g., market demand, production capacity, and inventory status), and then enhance the chain’ s responsiveness and flexibility, ultimately optimizing the overall efficiency.

The SSC model constructed in this paper includes three main members: the supplier (M), the retailer (R), and the sharing platform (I). Within this framework, the supplier is positioned upstream, while the retailer is situated downstream, with the sharing platform serving as a pivotal intermediary, like a “bridge” that effectively facilitates supply–demand matching and idle resource sharing.

In this model, we use the Stackelberg game structure for decision-making analysis. The most important variables in this Stackelberg model are divided into two categories: decision variables and state variables. Of these, decision variables include the supplier’s supply effort $A_1$, the supply price $\omega$, the sharing platform’s supply effort Z, the retailer’s supply effort $A_2$, and the retail price p; state variables include the platform’s service level G and the system’s just-in-time supply level Q.

Figure 1 illustrates the decision-making process.

The meanings of key variables and parameters in this paper are summarized and explained in

Table 1. Notations and definitions.

Notations	Definitions
G	Sharing platform’s service level, $G>0$
Q	System’s just-in-time supply level, $Q>0$
$A_1$	Supplier’s supply effort, $A_1>0$
$A_2$	Retailer’s supply effort, $A_2>0$
Z	Sharing platform’s supply effort, $Z>0$
$\omega$	Supply price, $\omega >0$
p	Retail price, $p>0$
D	Market demand, $D>0$
$\beta$	Percentage of the commission received by sharing platform from supplier, $\beta \in (0,1)$
$\delta$	Decay factor of sharing platform’s service level, $\delta >0$
$\tau$	Decay factor of system’s just-in-time supply level, $\tau >0$
${\rho }_1$	Impact factor of sharing platform’s supply effort on its service level, ${\rho }_1>0$
${\rho }_2,{\rho }_3,{\rho }_4$	Impact factor of supplier, retailer and sharing platform’s supply efforts on system’s just-in-time supply level, ${\rho }_2,{\rho }_3,{\rho }_4>0$
${\alpha }_1,{\alpha }_2$	Coefficient of sharing platform’s service level and system’s just-in-time supply level on market demand, ${\alpha }_1,{\alpha }_2>0$
$k_1,k_2,k_3$	Cost coefficient of supplier, sharing platform and retailer $k_1,k_2,k_3>0$
${\lambda }_1,{\lambda }_2,{\lambda }_3$	Altruistic parameter of sharing platform, retailer and supplier ${\lambda }_1,{\lambda }_2,{\lambda }_3\ge 0$

below. At the same time, for clarity and brevity, this paper employs a number of abbreviations, which are compiled in

Table 2. Nomenclature.

Abbreviation	Description
SSC	Shared Supply Chain
M	Supplier
R	Retailer
I	Sharing Platform
HJB	Hamilton-Jacobi-Bellman

.

3.2. Model Assumptions

To ensure the validity and tractability of the proposed differential game model, we adopt the following four assumptions, each of which is well supported by existing research.

Assumption 1. 

In this three-tier SSC, members must exert corresponding efforts to enhance both the system’s just-in-time supply level $Q\left(t\right)$ and the sharing platform’s service level $G\left(t\right)$; additionally, the extent of these efforts increases with higher cost inputs. According to the definition of the SSC model, $G\left(t\right)$ is linearly and positively related to the platform’s supply effort $Z\left(t\right)$ and $Q\left(t\right)$ is linearly and positively related to supply efforts of each member: $A_1\left(t\right)$, $A_2\left(t\right)$ and $Z\left(t\right)$. Based on the classical Nerlove–Arrow model [26] and referring to the construction method of Xu [8], the dynamics of the state variables $G\left(t\right)$ and $Q\left(t\right)$ in the SSC can be governed by differential equations, as follows:

$$\dot{G}\left(t\right)={\rho }_{1}Z\left(t\right)-\delta G\left(t\right),$$

(1)

$$\dot{Q}\left(t\right)={\rho }_2{A}_1\left(t\right)+{\rho }_3{A}_2\left(t\right)+{\rho }_{4}G\left(t\right)-\tau Q\left(t\right).$$

(2)

Assumption 2. 

In recent years, as economic development and living standards have continued to improve, consumers no longer focus solely on price as they did in the past [27], but also consider product quality, service level, and brand goodwill [28]. According to Chiang et al. [29], when consumers are unable to directly assess product quality, they establish an expected reference based on factors such as brand reputation, price, and previous purchasing experiences. In this SSC system, price, service level, and the system’s just-in-time supply level are critical reference factors for consumers. According to Ouardighi and Gavious’s research in 2012 [30], factors influencing market demand $D\left(t\right)$ can be classified into two categories: price-related and non-price determinants. Specifically, market demand can be expressed as the product of these two categories. In this model, the system’s just-in-time supply level and the platform’s service level are considered non-price factors. We assume the market demand expression is as follows:

$$D\left(t\right)=(a-bp\left(t\right))({\alpha }_{1}G\left(t\right)+{\alpha }_{2}Q\left(t\right)).$$

(3)

Assumption 3. 

Drawing on Caulkins’s general cost convexity assumption [31], we utilize the quadratic functions to measure the effort costs [32,33,34,35,36]. Each member continuously increases their own supply effort, in order to improve the just-in-time supply level, and the corresponding cost is described by the quadratic functions of supply efforts, as follows:

$$C_M\left(A_1\right)={\displaystyle \frac{1}{2}}{k}_1{A}_1{\left(t\right)}^2,C_R\left(A_2\right)={\displaystyle \frac{1}{2}}{k}_2{A}_2{\left(t\right)}^2,C_I\left(Z\right)={\displaystyle \frac{1}{2}}{k}_{3}Z{\left(t\right)}^2.$$

(4)

Assumption 4. 

Considering that the system’s just-in-time supply level has an increasingly significant impact on market demand, and that the overall level is jointly determined by the supply efforts of all members, it is natural to associate the coordination and cooperation of SSC members with the critical importance of the healthy operation of the supply chain system. In this context, each member considers their own profits, while also caring about those of the other members and the long-term overall profits of the SSC system. This cooperation coincides with the concept of reciprocal altruism. Therefore, we introduce the assumption of “altruistic preference” into our SSC model; each decision-making agent is endowed with the potential for altruistic behavior.

In the decision-making process, altruistic behavior is manifested by a balanced consideration of self-interest and the profits of other SSC members, coupled with modest self-sacrifices when necessary to enhance the overall performance of the entire supply chain system [37].

Based on the construction of reciprocal altruism utility functions in previous research [19,21,38,39], we define the utility functions under eight altruistic modes from four categories. Firstly, we define the altruistic parameters of the three members: the sharing platform, the retailer, and the supplier as ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$, respectively. We assume ${\lambda }_i\le 1,i=1,2,3$ to ensure that no member values others’ profits more than its own; ${\lambda }_i=0$ means that member i is purely selfish and cares only for their own profit; and ${\lambda }_i=1$ means member i treats partners’ profits as equally important to their own, embodying pure altruism. Based on the definition of altruistic parameters, we define the utility functions under eight altruistic modes from four categories.

• Purely Selfish: In the purely selfish mode $I^{\circ }{M}^{\circ }{R}^{\circ }$, each SSC member is exclusively focused on maximizing their own profit, without regard for the benefits or losses incurred by other members in the decision-making process. Specifically, the profit function for member M (supplier) is formulated as

  $$J_{M^{\circ }}=(1-\beta )\omega \left(t\right)D\left(t\right)-{\displaystyle \frac{1}{2}}{k}_1{A}_1{\left(t\right)}^2,$$

  (5)

  while the profit function for member I (sharing platform) is given by

  $$J_{I^{\circ }}=\beta \omega \left(t\right)D\left(t\right)-{\displaystyle \frac{1}{2}}{k}_{3}Z{\left(t\right)}^2,$$

  (6)

  and for member R (retailer) it is expressed as

  $$J_{R^{\circ }}=\left(p\left(t\right)-\omega \left(t\right)\right)D\left(t\right)-{\displaystyle \frac{1}{2}}{k}_2{A}_2{\left(t\right)}^2.$$

  (7)
• Single-Agent Altruism: Single-agent altruism means exactly one member in the SSC system internalizes the welfare of the others when making decisions, while the remaining members behave purely selfish. Thus, the altruistic actor’s utility function combines their own profit with a weighted sum of other members’s profits, enabling them to steer the overall chain toward greater coordination and performance. In our model, single-agent altruism is partitioned into three distinct modes: mode I, mode R, and mode M.

  (1)

  Mode I: Sharing platform altruism

  $$J_M=J_{M^{\circ }},U_I=J_{I^{\circ }}+{\lambda }_1\left(J_{M^{\circ }}+J_{R^{\circ }}\right),J_R=J_{R^{\circ }}.$$

  (8)

  (2)

  Mode R: Retailer altruism

  $$J_M=J_{M^{\circ }},J_I=J_{I^{\circ }},U_R=J_{R^{\circ }}+{\lambda }_2\left(J_{I^{\circ }}+J_{M^{\circ }}\right).$$

  (9)

  (3)

  Mode M: Supplier altruism

  $$U_M=J_{M^{\circ }}+{\lambda }_3\left(J_{I^{\circ }}+J_{R^{\circ }}\right),J_I=J_{I^{\circ }},J_R=J_{R^{\circ }}.$$

  (10)

• Dual-Agent Altruism: Dual-agent altruism means two members in the SSC system internalize a composite utility comprising their own profit plus a weighted sum of their counterparts’s profits, thus coordinating decisions to enhance overall SSC performance. In our model, dual-agent altruism is partitioned into three distinct modes: mode $IR$, mode $MR$, and mode $IM$.

  (1)

  Mode $IR$: Sharing platform and retailer joint altruism

  $$J_M=J_{M^{\circ }},U_I=J_{I^{\circ }}+{\lambda }_1\left(J_{M^{\circ }}+J_{R^{\circ }}\right),U_R=J_{R^{\circ }}+{\lambda }_2\left(J_{I^{\circ }}+J_{M^{\circ }}\right).$$

  (11)

  (2)

  Mode $MR$: Supplier and retailer joint altruism

  $$U_M=J_{M^{\circ }}+{\lambda }_3\left(J_{I^{\circ }}+J_{R^{\circ }}\right),J_I=J_{I^{\circ }},U_R=J_{R^{\circ }}+{\lambda }_2\left(J_{I^{\circ }}+J_{M^{\circ }}\right).$$

  (12)

  (3)

  Mode $IM$: Sharing platform and supplier joint altruism

  $$U_M=J_{M^{\circ }}+{\lambda }_3\left(J_{I^{\circ }}+J_{R^{\circ }}\right),U_I=J_{I^{\circ }}+{\lambda }_1\left(J_{M^{\circ }}+J_{R^{\circ }}\right),J_R=J_{R^{\circ }}.$$

  (13)

• Pure Altruism: In the purely selfish mode $IMR$, all SSC members take full account of each other’s profits in the decision-making process. The utility functions of SSC members are defined as

  $$U_M=J_{M^{\circ }}+{\lambda }_3\left(J_{I^{\circ }}+J_{R^{\circ }}\right),U_I=J_{I^{\circ }}+{\lambda }_1\left(J_{M^{\circ }}+J_{R^{\circ }}\right),U_R=J_{R^{\circ }}+{\lambda }_2\left(J_{I^{\circ }}+J_{M^{\circ }}\right).$$

  (14)

4. Model Solutions

Based on the above assumptions, this paper establishes a Stackelberg game model for different altruistic modes in the context of SSCs. In the subsequent section, the Hamilton–Jacobi–Bellman (HJB) equation is first constructed, and the backward induction method is used to solve for the optimal decisions of the SSC members, and determine the steady-state values of the state variables, including the sharing platform service level and the system’s just-in-time supply level. Then, we plot the evolution trajectories of these variables over time. In addition, this paper also explores the benefits for each member and the long-term benefits of the SSC system.

4.1. Purely Selfish: $I^{\circ }{M}^{\circ }{R}^{\circ }$

In this mode, members are only concerned with their own profits, and the optimal control problem can be expressed as Equation (15):

$$\begin{array}{cc}\hfill J_{R^{\circ }}& =\underset{p(·),A_2(·)}{max}{\int }_0^{\infty }\{{\mathrm{e}}^{-rt}\left[p\left(t\right)D\left(t\right)-\omega \left(t\right)D\left(t\right)-\frac{1}{2}{k}_2{A}_2{\left(t\right)}^2\right]dt,\hfill \\ \hfill J_{I^{\circ }}& =\underset{Z(·)}{max}{\int }_0^{\infty }{\mathrm{e}}^{-rt}\left[\beta \omega \left(t\right)D\left(t\right)-\frac{1}{2}{k}_{3}Z{\left(t\right)}^2\right]dt,\hfill \\ \hfill J_{M^{\circ }}& =\underset{\omega (·),A_1(·)}{max}{\int }_0^{\infty }{\mathrm{e}}^{-rt}\left[(1-\beta )\omega \left(t\right)D\left(t\right)-\frac{1}{2}{k}_1{A}_1{\left(t\right)}^2\right]dt,\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \dot{G}\left(t\right)={\rho }_{1}Z\left(t\right)-\delta G\left(t\right),G\left(0\right)=G_0;\hfill \\ \hfill & \dot{Q}\left(t\right)={\rho }_2{A}_1\left(t\right)+{\rho }_3{A}_2\left(t\right)+{\rho }_{4}G\left(t\right)-\tau Q\left(t\right),Q\left(0\right)=Q_0.\hfill \end{array}$$

(15)

Theorem 1. 

The state variable time evolution paths are as follows:

$$\begin{array}{cc}& G\left(t\right)=G_{\infty }+(G_0-G_{\infty }){\mathrm{e}}^{-\delta t},\hfill \\ & Q\left(t\right)=Q_{\infty }+{\displaystyle \frac{{\rho }_4(G_0-G_{\infty })}{r-\delta }}{\mathrm{e}}^{-\delta t}+(Q_0-Q_{\infty }-{\displaystyle \frac{{\rho }_4(G_0-G_{\infty })}{r-\delta }}){\mathrm{e}}^{-\delta t},\hfill \end{array}$$

where

$$G_{\infty }={\displaystyle \frac{{\rho }_{1}Z}{\delta }},Q_{\infty }={\displaystyle \frac{{\rho }_2{A}_1+{\rho }_3{A}_2+\frac{{\rho }_1{\rho }_4}{\delta }Z}{r}}.$$

Theorem 2. 

The optimal decisions of SSC members in mode $I^{\circ }{M}^{\circ }{R}^{\circ }$ are as follows:

$$\begin{array}{cc}& \omega ={\displaystyle \frac{a}{2b}},p={\displaystyle \frac{3a}{4b}},A_1={\displaystyle \frac{{\rho }_2}{k_1}}·{\displaystyle \frac{a^2(1-\beta ){\alpha }_2}{8b(r+\tau )}},A_2={\displaystyle \frac{{\rho }_3}{k_3}}·{\displaystyle \frac{a^2{\alpha }_2}{16b(r+\tau )}},\hfill \\ & Z={\displaystyle \frac{{\rho }_1}{k_2}}·{\displaystyle \frac{a^2\beta }{8b(\delta +r)}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

Steady-state values of state variables in this mode are as follows:

$$\begin{array}{cc}& G_{\infty }={\displaystyle \frac{{\rho }_1^2}{k_2\delta }}·{\displaystyle \frac{a^2\beta }{8b(\delta +r)}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right),\hfill \\ & Q_{\infty }={\displaystyle \frac{{\rho }_2^2}{k_1\tau }}·{\displaystyle \frac{a^2(1-\beta ){\alpha }_2}{8b(r+\tau )}}+{\displaystyle \frac{{\rho }_3^2}{k_3\tau }}·{\displaystyle \frac{a^2{\alpha }_2}{16b(r+\tau )}}+{\displaystyle \frac{{\rho }_4{\rho }_1^2}{\delta \tau k_2}}·{\displaystyle \frac{a^2\beta }{8b(r+\delta )}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

The complete proof of the theorem is provided in the Supplementary Materials.

4.2. Single-Agent Altruism

We classified the single-agent altruistic modes into three categories, I altruism, R altruism, and M altruism, and solved for the optimal decisions and the steady-state values of the state variables, respectively.

Theorem 3. 

The optimal decisions of the SSC members under I altruism are given by the following expressions:

$$\begin{array}{cc}\hfill \omega & ={\displaystyle \frac{a}{2b}},p={\displaystyle \frac{3a}{4b}},A_1={\displaystyle \frac{{\rho }_2}{k_1}}·{\displaystyle \frac{a^2(1-\beta ){\alpha }_2}{8b(r+\tau )}},\hfill \\ \hfill A_2& ={\displaystyle \frac{{\rho }_3}{k_3}}·{\displaystyle \frac{a^2{\alpha }_2}{16b(r+\tau )}},Z={\displaystyle \frac{{\rho }_1}{k_2}}·{\displaystyle \frac{a^2\left[2\beta +{\lambda }_1(3-2\beta )\right]}{16b(\delta +r)}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

The steady-state values of state variables in this mode are

$$\begin{array}{cc}\hfill G_{\infty }& ={\displaystyle \frac{{\rho }_1^2}{k_2\delta }}·{\displaystyle \frac{a^2}{16b(\delta +r)}}\left[2\beta +{\lambda }_1(3-2\beta )\right]\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right),\hfill \\ \hfill Q_{\infty }& ={\displaystyle \frac{{\rho }_2^2}{k_1\tau }}·{\displaystyle \frac{a^2(1-\beta ){\alpha }_2}{8b(r+\tau )}}+{\displaystyle \frac{{\rho }_3^2}{k_3\tau }}·{\displaystyle \frac{a^2{\alpha }_2}{16b(r+\tau )}}+{\displaystyle \frac{{\rho }_4{\rho }_1^2}{\delta \tau k_2}}·{\displaystyle \frac{a^2\left[2\beta +{\lambda }_1(3-2\beta )\right]}{16b(r+\delta )}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

Theorem 4. 

The optimal decisions of the SSC members under R altruism are given by the following expressions:

$$\begin{array}{cc}\hfill \omega & ={\displaystyle \frac{a}{2b(1-{\lambda }_2)}},p={\displaystyle \frac{3a}{4b}},A_1={\displaystyle \frac{{\rho }_2}{k_1}}·{\displaystyle \frac{a^2(1-\beta ){\alpha }_2}{8b(r+\tau )(1-{\lambda }_2)}},\hfill \\ \hfill A_2& ={\displaystyle \frac{{\rho }_3}{k_3}}·{\displaystyle \frac{a^2{\alpha }_2}{16b(r+\tau )}},Z={\displaystyle \frac{{\rho }_1}{k_2}}·{\displaystyle \frac{a^2\beta }{8b(\delta +r)(1-{\lambda }_2)}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

The steady-state values of the state variables in this mode are

$$\begin{array}{cc}\hfill G_{\infty }& ={\displaystyle \frac{{\rho }_1^2}{k_2\delta }}·{\displaystyle \frac{a^2\beta }{8b(\delta +r)(1-{\lambda }_2)}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right),\hfill \\ \hfill Q_{\infty }& ={\displaystyle \frac{{\rho }_2^2}{k_1\tau }}·{\displaystyle \frac{a^2(1-\beta ){\alpha }_2}{8b(r+\tau )(1-{\lambda }_2)}}+{\displaystyle \frac{{\rho }_3^2}{k_3\tau }}·{\displaystyle \frac{a^2{\alpha }_2}{16b(r+\tau )}}+{\displaystyle \frac{{\rho }_4{\rho }_1^2}{\tau \delta k_2}}·{\displaystyle \frac{\beta a^2}{8b(r+\delta )(1-{\lambda }_2)}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

Theorem 5. 

The optimal decisions of the SSC members under M altruism are given by the following expressions:

$$\begin{array}{cc}\hfill \omega & ={\displaystyle \frac{a(1-{\lambda }_3)(1-\beta )}{2b[1-\beta +{\lambda }_3(\beta -\frac{1}{2})]}},p={\displaystyle \frac{3a}{4b}}{\displaystyle \frac{(1-\beta )+\left(\beta -\frac{2}{3}\right){\lambda }_3}{(1-\beta )+\left(\beta -\frac{1}{2}\right){\lambda }_3}},\hfill \\ \hfill A_1& ={\displaystyle \frac{{\rho }_2}{k_1}}·{\displaystyle \frac{a^2{\alpha }_2(1-\beta +{\lambda }_3\beta )}{(r+\tau )}}·{\displaystyle \frac{\left[\beta (2\beta -1){\lambda }_3^2+(1-\beta )(4\beta -1){\lambda }_3+2{(1-\beta )}^2\right]}{4b{\left[2-2\beta +{\lambda }_3(2\beta -1)\right]}^2}},\hfill \\ \hfill A_2& ={\displaystyle \frac{{\rho }_3}{k_3}}·{\displaystyle \frac{a^2{\alpha }_2{(1-\beta +{\lambda }_3\beta )}^2}{(r+\tau )4b{\left[2-2\beta +{\lambda }_3(2\beta -1)\right]}^2}},\hfill \\ \hfill Z& ={\displaystyle \frac{{\rho }_1}{k_2}}·{\displaystyle \frac{a^2\beta (1-{\lambda }_3)(1-\beta )(1-\beta -{\lambda }_3\beta )}{2b(\delta +r){\left[2-2\beta +{\lambda }_3(2\beta -1)\right]}^2}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

The steady-state values of the state variables in this mode are

$$\begin{array}{cc}\hfill G_{\infty }& ={\displaystyle \frac{{\rho }_1^2}{k_2\delta }}·{\displaystyle \frac{a^2\beta (1-{\lambda }_3)(1-\beta )(1-\beta -{\lambda }_3\beta )}{2b(\delta +r){\left[2-2\beta +{\lambda }_3(2\beta -1)\right]}^2}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right),\hfill \\ \hfill Q_{\infty }& ={\displaystyle \frac{{\rho }_2^2}{k_1\tau }}·{\displaystyle \frac{a^2{\alpha }_2(1-\beta +{\lambda }_3\beta )}{(r+\tau )}}·{\displaystyle \frac{\left[\beta (2\beta -1){\lambda }_3^2+(1-\beta )(4\beta -1){\lambda }_3+2{(1-\beta )}^2\right]}{4b{\left[2-2\beta +{\lambda }_3(2\beta -1)\right]}^2}}\hfill \\ & +{\displaystyle \frac{{\rho }_3^2}{k_3\tau }}·{\displaystyle \frac{a^2{\alpha }_2{(1-\beta +{\lambda }_3\beta )}^2}{(r+\tau )4b{\left[2-2\beta +{\lambda }_3(2\beta -1)\right]}^2}}\hfill \\ & +{\displaystyle \frac{{\rho }_4{\rho }_1^2}{k_2\tau \delta }}·{\displaystyle \frac{a^2\beta (1-{\lambda }_3)(1-\beta )(1-\beta +{\lambda }_3\beta )}{2b(r+\delta ){\left[2-2\beta +{\lambda }_3(2\beta -1)\right]}^2}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

4.3. Dual-Agent Altruism

We classified the dual-agent altruistic modes into three categories, $IR$ altruism, $MR$ altruism, and $IM$ altruism, and solved for the optimal decisions and steady-state value of each state variable, respectively.

Theorem 6. 

The optimal decisions of the SSC members under $IR$ altruism are given by the following expressions:

$$\begin{array}{cc}\hfill \omega & ={\displaystyle \frac{a}{2b(1-{\lambda }_2)}},p={\displaystyle \frac{3a}{4b}},A_1={\displaystyle \frac{{\rho }_2}{k_1}}·{\displaystyle \frac{a^2(1-\beta ){\alpha }_2}{8b(r+\tau )(1-{\lambda }_2)}},A_2={\displaystyle \frac{{\rho }_3}{k_3}}·{\displaystyle \frac{a^2{\alpha }_2}{16b(r+\tau )}},\hfill \\ \hfill Z& ={\displaystyle \frac{{\rho }_1}{k_2}}·{\displaystyle \frac{a^2\left[3{\lambda }_1(1-{\lambda }_2)+2(1-{\lambda }_1)\beta \right]}{16b(\delta +r)(1-{\lambda }_2)}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

The steady-state values of state variables in this mode are

$$\begin{array}{cc}\hfill G_{\infty }& ={\displaystyle \frac{{\rho }_1^2}{k_2\delta }}·{\displaystyle \frac{a^2\left[3{\lambda }_1(1-{\lambda }_2)+2(1-{\lambda }_1)\beta \right]}{16b(\delta +r)(1-{\lambda }_2)}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right),\hfill \\ \hfill Q_{\infty }& ={\displaystyle \frac{{\rho }_2^2}{k_1\tau }}·{\displaystyle \frac{a^2(1-\beta ){\alpha }_2}{8b(1-{\lambda }_2)(r+\tau )}}+{\displaystyle \frac{{\rho }_3^2}{k_3\tau }}·{\displaystyle \frac{a^2{\alpha }_2}{16b(r+\tau )}}\hfill \\ & +{\displaystyle \frac{{\rho }_4{\rho }_1^2}{\delta \tau k_2}}·{\displaystyle \frac{a^2\left[3{\lambda }_1(1-{\lambda }_2)+2(1-{\lambda }_1)\beta \right]}{16b(1-{\lambda }_2)(r+\delta )}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

Theorem 7. 

The optimal decisions of the SSC members under $MR$ altruism are given by the following expressions:

$$\begin{array}{cc}\hfill \omega & ={\displaystyle \frac{a(1-{\lambda }_3)(1-\beta )}{2b(1-{\lambda }_2)[1-\beta +{\lambda }_3(\beta -\frac{1+{\lambda }_2}{2})]}},p={\displaystyle \frac{3a}{4b}}·{\displaystyle \frac{(1-\beta )+\left(\beta -\frac{2+{\lambda }_2}{3}\right){\lambda }_3}{(1-\beta )+\left(\beta -\frac{1+{\lambda }_2}{2}\right){\lambda }_3}},\hfill \\ \hfill A_1& ={\displaystyle \frac{{\rho }_2}{k_1}}·{\displaystyle \frac{a^2{\alpha }_2\left(1-\beta +{\lambda }_3(\beta -{\lambda }_2)\right)}{4b(1-{\lambda }_2)(r+\tau ){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}}\hfill \\ & ·\left\{\left[3{\lambda }_3(1-\beta )+{\lambda }_3^2(3\beta -2-{\lambda }_2)\right](1-{\lambda }_2)+2{(1-{\lambda }_3)}^2{(1-\beta )}^2\right\},\hfill \\ \hfill A_2& ={\displaystyle \frac{{\rho }_3}{k_3}}·{\displaystyle \frac{a^2{\alpha }_2{\left[1-\beta +{\lambda }_3(\beta -{\lambda }_2)\right]}^2}{4b(r+\tau ){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}},\hfill \\ \hfill Z& ={\displaystyle \frac{{\rho }_1}{k_2}}·{\displaystyle \frac{a^2\beta (1-\beta )(1-{\lambda }_3)\left[1-\beta +{\lambda }_3(\beta -{\lambda }_2)\right]}{2b(r+\delta )(1-{\lambda }_2){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

The steady-state values of state variables in this mode are

$$\begin{array}{cc}\hfill G_{\infty }& ={\displaystyle \frac{{\rho }_1^2}{k_2\delta }}·{\displaystyle \frac{a^2\beta (1-\beta )(1-{\lambda }_3)\left[1-\beta +{\lambda }_3(\beta -{\lambda }_2)\right]}{2b(r+\delta )(1-{\lambda }_2){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right),\hfill \\ \hfill Q_{\infty }& ={\displaystyle \frac{{\rho }_2^2}{k_1\tau }}·{\displaystyle \frac{a^2{\alpha }_2(1-\beta +{\lambda }_3(\beta -{\lambda }_2))}{4b(1-{\lambda }_2)(r+\tau ){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}}\hfill \\ & ·\left\{\left[3{\lambda }_3(1-\beta )+{\lambda }_3^2(3\beta -2-{\lambda }_2)\right](1-{\lambda }_2)+2{(1-{\lambda }_3)}^2{(1-\beta )}^2\right\}\hfill \\ & +{\displaystyle \frac{{\rho }_3^2}{k_3\tau }}·{\displaystyle \frac{a^2{\alpha }_2}{4b\left(r+\tau \right)}}·{\displaystyle \frac{{[1-\beta +{\lambda }_3(\beta -{\lambda }_2)]}^2}{{\left[2-2\beta +{\lambda }_3\left(2\beta -{\lambda }_2-1\right)\right]}^2}}\hfill \\ & +{\displaystyle \frac{{\rho }_4{\rho }_1^2}{k_2\tau \delta }}·{\displaystyle \frac{a^2\beta (1-\beta )(1-{\lambda }_3)(1-\beta +{\lambda }_3(\beta -{\lambda }_2))}{2b(r+\delta )(1-{\lambda }_2){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

Theorem 8. 

The optimal decisions of the SSC members under $IM$ altruism are given by the following expressions:

$$\begin{array}{cc}\hfill \omega & ={\displaystyle \frac{a(1-{\lambda }_3)(1-\beta )}{2b[(1-{\lambda }_3)(1-\beta )+\frac{{\lambda }_3}{2}]}},p={\displaystyle \frac{3a}{4b}}·{\displaystyle \frac{(1-\beta )+\left(\beta -\frac{2}{3}\right){\lambda }_3}{(1-\beta )+\left(\beta -\frac{1}{2}\right){\lambda }_3}},\hfill \\ \hfill A_1& ={\displaystyle \frac{{\rho }_2}{k_1}}·{\displaystyle \frac{a^2{\alpha }_2\left[2{(1-{\lambda }_3)}^2{(1-\beta )}^2+3(1-\beta ){\lambda }_3+(3\beta -2){\lambda }_3^2\right]}{4b{\left[2(1-{\lambda }_3)(1-\beta )+{\lambda }_3\right]}^2(r+\tau )}}(1-\beta +{\lambda }_3\beta ),\hfill \\ \hfill A_2& ={\displaystyle \frac{{\rho }_3}{k_3}}·{\displaystyle \frac{a^2{\alpha }_2{(1-\beta +{\lambda }_3\beta )}^2}{4b(r+\tau ){\left[2-2\beta +{\lambda }_3(2\beta -1)\right]}^2}},\hfill \\ \hfill Z& ={\displaystyle \frac{{\rho }_1{a}^2}{k_2}}·{\displaystyle \frac{2\beta (1-{\lambda }_1)(1-{\lambda }_3)(1-\beta )+3{\lambda }_1(1-\beta )+(3\beta -2){\lambda }_3{\lambda }_1}{4b{\left[2(1-{\lambda }_3)(1-\beta )+{\lambda }_3\right]}^2(r+\delta )}}\hfill \\ & ·\left(1-\beta +{\lambda }_3\beta \right)\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

The steady-state values of the state variables in this mode are

$$\begin{array}{cc}\hfill G_{\infty }& ={\displaystyle \frac{{\rho }_1^2}{k_2\delta }}·{\displaystyle \frac{a^2[2\beta (1-{\lambda }_1)(1-{\lambda }_3)(1-\beta )+3{\lambda }_1(1-\beta )+(3\beta -2){\lambda }_3{\lambda }_1]}{4b{\left[2(1-{\lambda }_3)(1-\beta )+{\lambda }_3\right]}^2(r+\delta )}}\hfill \\ & ·\left(1-\beta +{\lambda }_3\beta \right)\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right),\hfill \\ \hfill Q_{\infty }& ={\displaystyle \frac{{\rho }_2^2}{k_1\tau }}·{\displaystyle \frac{a^2{\alpha }_2\left[2{(1-{\lambda }_3)}^2{(1-\beta )}^2+3(1-\beta ){\lambda }_3+(3\beta -2){\lambda }_3^2\right]}{4b{\left[2(1-{\lambda }_3)(1-\beta )+{\lambda }_3\right]}^2(r+\tau )}}(1-\beta +{\lambda }_3\beta )\hfill \\ & +{\displaystyle \frac{{\rho }_3^2}{k_3\tau }}·{\displaystyle \frac{a^2{\alpha }_2{(1-\beta +{\lambda }_3\beta )}^2}{4b{\left[2(1-{\lambda }_3)(1-\beta )+{\lambda }_3\right]}^2(r+\tau )}}\hfill \\ & +{\displaystyle \frac{{\rho }_4{\rho }_1^2}{\tau k_2\delta }}·{\displaystyle \frac{a^2[2\beta (1-{\lambda }_1)(1-{\lambda }_3)(1-\beta )+3{\lambda }_1(1-\beta )+(3\beta -2){\lambda }_3{\lambda }_1]}{4b{\left[2(1-{\lambda }_3)(1-\beta )+{\lambda }_3\right]}^2(r+\delta )}}\hfill \\ & ·\left(1-\beta +{\lambda }_3\beta \right)\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right).\hfill \end{array}$$

4.4. Purely Altruistic: $IMR$

In this mode, members take full account of each other’s profits in decision-making.

Theorem 9. 

The optimal decisions of the SSC members under $IMR$ altruism are given by the following expressions:

$$\begin{array}{cc}\hfill \omega & ={\displaystyle \frac{a(1-{\lambda }_3)(1-\beta )}{2b(1-{\lambda }_2)[1-\beta +{\lambda }_3(\beta -\frac{1+{\lambda }_2}{2})]}},p={\displaystyle \frac{3a}{4b}}·{\displaystyle \frac{(1-\beta )+\left(\beta -\frac{2+{\lambda }_2}{3}\right){\lambda }_3}{(1-\beta )+\left(\beta -\frac{1+{\lambda }_2}{2}\right){\lambda }_3}},\hfill \\ \hfill A_1& ={\displaystyle \frac{{\rho }_2}{k_1}}·{\displaystyle \frac{a^2{\alpha }_2}{4b(1-{\lambda }_2)(r+\tau ){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}}\left(1-\beta +{\lambda }_3(\beta -{\lambda }_2)\right)\hfill \\ & ·\left\{\left[3{\lambda }_3(1-\beta )+{\lambda }_3^2(3\beta -2-{\lambda }_2)\right](1-{\lambda }_2)+2{(1-{\lambda }_3)}^2{(1-\beta )}^2\right\},\hfill \\ \hfill A_2& ={\displaystyle \frac{{\rho }_3}{k_3}}·{\displaystyle \frac{a^2{\alpha }_2{\left[1-\beta +{\lambda }_3(\beta -{\lambda }_2)\right]}^2}{4b(r+\tau ){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill Z& ={\displaystyle \frac{{\rho }_1}{k_2}}·{\displaystyle \frac{a^2\left(1-\beta +(\beta -{\lambda }_2){\lambda }_3\right)}{4b(1-{\lambda }_2){\left[2-2\beta +(2\beta -1-{\lambda }_2){\lambda }_3\right]}^2(r+\delta )}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right)\hfill \\ & ·[{\lambda }_1(1-\beta )(3-{\lambda }_3)(1-{\lambda }_2)+{\lambda }_1{\lambda }_3(2\beta -1-{\lambda }_2)(1-{\lambda }_2)\hfill \\ & +2\beta (1-\beta )(1-{\lambda }_3)(1-{\lambda }_1)].\hfill \end{array}$$

The steady-state values of the state variables in this mode are

$$\begin{array}{cc}\hfill G_{\infty }& ={\displaystyle \frac{{\rho }_1^2}{k_2\delta }}·{\displaystyle \frac{a^2\left(1-\beta +(\beta -{\lambda }_2){\lambda }_3\right)}{4b(1-{\lambda }_2){\left[2-2\beta +(2\beta -1-{\lambda }_2){\lambda }_3\right]}^2(r+\delta )}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right)\hfill \\ & ·[{\lambda }_1(1-\beta )(3-{\lambda }_3)(1-{\lambda }_2)+{\lambda }_1{\lambda }_3(2\beta -1-{\lambda }_2)(1-{\lambda }_2)\hfill \\ & +2\beta (1-\beta )(1-{\lambda }_3)(1-{\lambda }_1)],\hfill \\ \hfill Q_{\infty }& ={\displaystyle \frac{{\rho }_2^2}{k_1\tau }}·{\displaystyle \frac{a^2{\alpha }_2(1-\beta +{\lambda }_3(\beta -{\lambda }_2))}{4b(1-{\lambda }_2)(r+\tau ){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}}\hfill \\ & ·\left\{\left[3{\lambda }_3(1-\beta )+{\lambda }_3^2(3\beta -2-{\lambda }_2)\right](1-{\lambda }_2)+2{(1-{\lambda }_3)}^2{(1-\beta )}^2\right\}\hfill \\ & +{\displaystyle \frac{{\rho }_3^2}{k_3\tau }}·{\displaystyle \frac{a^2{\alpha }_2{[1-\beta +{\lambda }_3(\beta -{\lambda }_2)]}^2}{4b(r+\tau ){\left[2-2\beta +{\lambda }_3(2\beta -{\lambda }_2-1)\right]}^2}}\hfill \\ & +{\displaystyle \frac{{\rho }_4{\rho }_1^2}{\tau k_2\delta }}·{\displaystyle \frac{a^2\left(1-\beta +(\beta -{\lambda }_2){\lambda }_3\right)}{4b(1-{\lambda }_2){\left[2-2\beta +(2\beta -1-{\lambda }_2){\lambda }_3\right]}^2(r+\delta )}}\left({\alpha }_1+{\displaystyle \frac{{\rho }_4{\alpha }_2}{r+\tau }}\right)\hfill \\ & ·[{\lambda }_1(1-\beta )(3-{\lambda }_3)(1-{\lambda }_2)+{\lambda }_1{\lambda }_3(2\beta -1-{\lambda }_2)(1-{\lambda }_2)\hfill \\ & +2\beta (1-\beta )(1-{\lambda }_3)(1-{\lambda }_1)].\hfill \end{array}$$

5. Model Analysis

In the previous chapter, we computed the expressions for the optimal decision variables in different altruistic modes. Now, we proceed to conduct a systematic comparative analysis of the optimal decision variables across eight altruistic modes. All results are presented in tabular form, and it is worth mentioning that the comparative analysis in this chapter is carried out solely from a theoretical perspective. In order to facilitate the comparative analysis, we assume that all altruistic parameters take the same value, i.e., ${\lambda }_1={\lambda }_2={\lambda }_3=\lambda$.

5.1. Comparative Analysis of Supply Price

A comparative analysis of supply prices under the eight altruistic modes is presented in

Table 3. Comparative analysis of $\omega$.

	${\mathit{I}}^{\circ }{\mathit{M}}^{\circ }{\mathit{R}}^{\circ }$	I	R	M	$\mathit{IR}$	$\mathit{MR}$	$\mathit{IM}$	$\mathit{IMR}$
$I^{\circ }{M}^{\circ }{R}^{\circ }$	=	=	≤	≥	≤	when $\beta \le 0.5$, ≤	≥	when $\beta \le 0.5$, ≤
I		=	≤	≥	≤	when $\beta \le 0.5$, ≤	≥	when $\beta \le 0.5$, ≤
R			=	≥	=	≥	≥	≥
M				=	≤	when $\beta \le 0.5$, ≤	=	when $\beta \le 0.5$, ≤
$IR$					=	≥	≥	≥
$MR$						=	when $\beta \le 0.5$, ≥	=
$IM$							=	when $\beta \le 0.5$, ≤
$IMR$								=

below. For clarity, we employ simple notations in the tables to represent the results of the comparative analysis. For example, in Table 3, “$I\ge M$” indicates that the supply price in altruistic mode I is greater than or equal to that in altruistic mode M; “$I\le R$” indicates that the supply price in altruistic mode I is less than or equal to that in altruistic mode R; “$IR=R$” indicates that the supply prices in altruistic modes $IR$ and R are identical; and “*” indicates that the comparison is uncertain.

As shown in Table 3, we found substantial differences in supply prices under different altruistic modes. Taking the optimal supply price in the purely selfish mode $I^{\circ }{M}^{\circ }{R}^{\circ }$ as a benchmark, when the retailer has an altruistic preference (modes R and $IR$), the supply prices are the same, higher than the benchmark; when the supplier has an altruistic preference (modes M and $IM$), the supply prices are the same, lower than the benchmark; and when only the sharing platform has an altruistic preference, i.e., in the modes I, the supply price does not change, being no different from $I^{\circ }{M}^{\circ }{R}^{\circ }$.

The above phenomenon occurs because the retailer’s altruism manifests itself in the form of an increased supply effort $A_2$, which improves the system’s just-in-time supply level and then drives the growth in market demand. Moreover, the supplier, the leader of the Stackelberg game model, can anticipate the growth in market demand when setting the supply price $\omega$. With lower elasticity of market demand, the supplier can capture more channel margins by increasing $\omega$ without fear of the retailer passing on costs through lower retail prices. On the other hand, supplier altruism is manifested by lowering the supply price directly, reducing downstream cost pressures, stimulating market demand, and achieving Pareto improvements in the long run. However, although the sharing platform’s altruism is manifested by increasing its own supply effort Z, this has minimal impact on the pricing strategy, thus $\omega$ remains constant.

5.2. Comparative Analysis of Retail Price

By calculating the expressions of retail prices under different altruistic modes, the results of the comparative analysis are summarized in the following

Table 4. Comparative analysis of p.

	${\mathit{I}}^{\circ }{\mathit{M}}^{\circ }{\mathit{R}}^{\circ }$	I	R	M	$\mathit{IR}$	$\mathit{MR}$	$\mathit{IM}$	$\mathit{IMR}$
$I^{\circ }{M}^{\circ }{R}^{\circ }$	=	=	=	≥	=	≥	≥	≥
I		=	=	≥	=	≥	≥	≥
R			=	≥	=	≥	≥	≥
M				=	≤	≤	=	≤
$IR$					=	≥	≥	≥
$MR$						=	≥	=
$IM$							=	≤
$IMR$								=

.

Taking the retail price p in $I^{\circ }{M}^{\circ }{R}^{\circ }$ as a benchmark, we found the result below:

$$I^{\circ }{M}^{\circ }{R}^{\circ }=I=R=IR\ge MR=IMR\ge M=IM.$$

Under supplier altruism, p is less than the benchmark. But retailer altruism and sharing platform altruism have no significant impact on p. This occurs because supplier altruism is manifested in lowering supply price to reduce the cost pressure on the retailer, then stimulating downstream expansion of market demand. At the same time, it improves supply efforts to enhance the overall efficiency, indirectly reducing the unit operating cost of the retailer. This dual drive allows the retailer to lower p while maintaining higher margins and achieving long-term good co-operation in the SSC system.

5.3. Comparative Analysis of Supplier’s Supply Effort

By calculating the expressions of the supplier’s supply efforts under different altruistic modes, the results of the comparative analysis are as summarized in the

Table 5. Comparative analysis of $A_1$.

	${\mathit{I}}^{\circ }{\mathit{M}}^{\circ }{\mathit{R}}^{\circ }$	I	R	M	$\mathit{IR}$	$\mathit{MR}$	$\mathit{IM}$	$\mathit{IMR}$
$I^{\circ }{M}^{\circ }{R}^{\circ }$	=	=	≤	≤	≤	≤	≤	≤
I		=	≤	≤	≤	≤	≤	≤
R			=	∗	=	when $\lambda$ is small, ≤	∗	when $\lambda$ is small, ≤
M				=	∗	∗	=	∗
$IR$					=	when $\lambda$ is small, ≤	∗	when $\lambda$ is small, ≤
$MR$						=	∗	=
$IM$							=	∗
$IMR$								=

below.

Taking the supply effort $A_1$ level in $I^{\circ }{M}^{\circ }{R}^{\circ }$ as a benchmark, comparative analyses of different altruistic modes revealed significant regularities. When only the sharing platform exhibits altruistic behavior, its increased supply effort enhances the overall performance but produces little change in $A_1$. In contrast, when the retailer has an altruistic preference, it increases its own supply effort and reduces the supplier’s transactional costs. This encourages the supplier to reallocate resources and invest more in timely supply, thereby enhancing $A_1$. Additionally, the supplier’s altruism directly improves $A_1$, further contributing to the overall performance.

5.4. Comparative Analysis of Retailer’s Supply Effort

By calculating the expressions of the retailer’s supply efforts under the different altruistic modes, the results of a comparative analysis were as summarized in the

Table 6. Comparative analysis of $A_2$.

	${\mathit{I}}^{\circ }{\mathit{M}}^{\circ }{\mathit{R}}^{\circ }$	I	R	M	$\mathit{IR}$	$\mathit{MR}$	$\mathit{IM}$	$\mathit{IMR}$
$I^{\circ }{M}^{\circ }{R}^{\circ }$	=	=	=	≤	=	≤	≤	≤
I		=	=	≤	=	≤	≤	≤
R			=	≤	=	≤	≤	≤
M				=	≥	≥	=	≥
$IR$					=	≤	≤	≤
$MR$						=	≤	=
$IM$							=	≥
$IMR$								=

below.

Taking the supply effort level of the retailer $A_2$ in mode $I^{\circ }{M}^{\circ }{R}^{\circ }$ as a benchmark, the comparative analysis of different altruistic modes revealed significant regularities:

$$I^{\circ }{M}^{\circ }{R}^{\circ }=I=R=IR\le MR=IMR\le M=IM.$$

When altruism is introduced, the dynamics of supply efforts exhibit asymmetric characteristics: the analysis confirmed that, when only the supplier exhibits altruistic behavior, the retailer’s supply effort $A_2$ is maximized. In contrast, when altruism is exhibited solely by the platform or the retailer, $A_2$ does not exceed the benchmark. The underlying logic is that supplier altruism directly modifies the core parameter $(p-\omega )$ in the retailer’s profit function by adjusting supply prices, thereby expanding marginal revenue per unit. This creates incentives for the retailer to increase $A_2$ under identical cost constraints to capture amplified benefits.

Notably, in joint altruistic modes, e.g., in mode $MR$ or $IMR$, $A_2$ is lower than that in mode M or $IM$. This implies that the retailer reduces its supply effort investment, creating counterintuitive outcomes. The paradoxical decline in $A_2$ under joint altruistic modes stems from resource allocation synergies and diminishing marginal returns. When the supplier and the platform jointly offer concessions, the retailer experiences dual cost reductions that substantially expand the resource constraint boundaries. Consequently, the marginal returns on supply effort investment diminish relative to the scale of demand expansion. While alternative activities (e.g., market expansion or technological innovation) exhibit relatively higher marginal contributions, the retailer dynamically reallocates resources and achieves systemic Pareto improvements.

Overall, the analysis provides robust evidence that supplier-driven altruism is the dominant factor in elevating the retailer’s supply effort, while the effects of platform or retailer altruism, either individually or in conjunction with supplier altruism, are comparatively less pronounced. Moreover, the change under joint altruistic modes is characterized by diminishing marginal benefits. This suggests that, in the joint altruistic mode, the retailer optimizes its own resource allocation and rationally deploys its inputs to achieve a more efficient operation of the entire supply chain system.

5.5. Comparative Analysis of Sharing Platform’s Supply Effort

Expressions for the sharing platform’s supply effort under different altruistic modes were derived, and the comparative analysis is summarized in

Table 7. Comparative analysis of Z.

	${\mathit{I}}^{\circ }{\mathit{M}}^{\circ }{\mathit{R}}^{\circ }$	I	R	M	$\mathit{IR}$	$\mathit{MR}$	$\mathit{IM}$	$\mathit{IMR}$
$I^{\circ }{M}^{\circ }{R}^{\circ }$	=	≤	≤	≥	≤	when $\beta \le {\displaystyle \frac{1}{2}}$, ≤	when $\beta \le {\displaystyle \frac{1}{2}}$, ≤	when $\beta \le {\displaystyle \frac{2}{9}}$, ≤
I		=	when $\beta \le {\displaystyle \frac{3}{5}}$, ≥	≥	≤	when $\lambda$ is small, ≥	when $\beta \le {\displaystyle \frac{1}{2}}$, ≤	∗
R			=	≥	when $\beta \le {\displaystyle \frac{3}{5}}$, ≤	≥	∗	∗
M				=	≤	when $\beta \le {\displaystyle \frac{1}{2}}$, ≤	when $\beta \le {\displaystyle \frac{1}{2}}$, ≤	when $\beta \le {\displaystyle \frac{2}{9}}$, ≤
$IR$					=	∗	∗	∗
$MR$						=	∗	∗
$IM$							=	∗
$IMR$								=

.

Taking the supply effort level of the sharing platform in mode $I^{\circ }{M}^{\circ }{R}^{\circ }$ as a benchmark, the comparative analysis of different altruistic modes showed significant regularities. The altruism of individual SSC members exerts distinctly different effects on the sharing platform’s supply effort Z, leading to structural adjustments in the platform’s decision-making and resource allocation. Specifically, under I altruism, the platform notably increases Z; and under R altruism, the retailer’s supply effort increases, which boosts the overall cooperation confidence and further motivates the platform to ramp up its supply input; consequently, Z also increases. However, under M altruism, Z decreases.

Considering that the altruistic behavior of the supplier is mainly reflected in lowering the supply price $\omega$, this directly eases the cost pressure on the retailer and indirectly expands the market size $D\left(t\right)$. Since both $\omega$ and $D\left(t\right)$ are key variables influencing the platform’s profit function, M altruism may lead the platform into a “Free Rider” trap, thereby reducing its own incentive to optimize supply efforts. Moreover, a decrease in $\omega$ can enhance the bargaining power of the retailer, prompting the platform to reallocate resources from Z to relationship coordination, which also results in diminishing marginal returns on supply efforts Z.

Overall, our analysis revealed that the altruistic behaviors of different SSC members play distinct roles in optimal decisions-making. Specifically, the altruistic behavior of the sharing platform is directly reflected in increased supply efforts and enhanced service levels; retailer altruism is primarily manifested through a boost in supply input rather than through adjustments in retail pricing; and the supplier’s altruistic actions are characterized by both an elevated supply input and a reduction in supply price. Moreover, it is essential to account for the unique characteristics inherent in certain altruistic modes. Notably, under joint altruistic modes (e.g., $IR$ or $MR$), Z may exhibit nonlinear changes. For instance, when both the platform and the retailer display altruistic behaviors, their combined incentive effects can elevate supply efforts beyond those observed under isolated altruism, whereas the involvement of supplier altruism may introduce an inhibitory effect that partially counterbalances these incentives. This finding underscores the necessity of aligning the altruistic objectives of all SSC members to prevent “incentive dilution” and the resulting strategic conflicts or resource misallocation.

6. Numerical Analysis

In this section, we performed numerical experiments to better understand the effects of critical system parameters on the theoretical results. The numerical experiments are organized as follows: Firstly, we plot the trajectories of state variables over time under different altruistic modes. Secondly, we present trajectory plots which show the steady values of state variables and optimal decisions as functions of the altruistic parameter, thereby further validating and analyzing the conclusions drawn in the previous model analysis chapter. Finally, we calculate the utility and profit values for each member and for the overall system under different modes and under varying altruistic parameters. We illustrate these results in the form of line charts.

This section combines insights from previous research [27,40,41,42,43] with practical operations to set the key parameter values. Guided by the non-negativity of state variables and decision variables, we restricted the altruistic parameters to $[0,\frac{1}{3}]$. In addition, the settings of the other key parameters were grounded in a comprehensive review of earlier studies [27,41,44,45,46], and were seamlessly integrated into our experimental design. In detail, $k_i$ is the supply cost coefficient of the supplier, the sharing platform, and the retailer, $k_1=k_2=k_3=5.0$ [31,32,33,34]. The market demand parameters a and b were set referring to Pietro [47]. The impact factor of the state variables on market demand was ${\alpha }_i$ (i = 1, 2). In this context, we assume $G\left(t\right)$ is more influential, thus ${\alpha }_1=1.0>{\alpha }_2=0.8$ [27,40]. $\delta =0.1$ and $\tau =0.2$ are the decay factors of $G\left(t\right)$ and $Q\left(t\right)$, if there are no efforts, the service level and overall supply level will decay over time, with the value setting referring to Jørgensen et al. [32]. ${\rho }_i$(i = 1, 2, 3) is the impact factor of supply efforts on state variables, we assume members have the same influence, thus ${\rho }_1={\rho }_2={\rho }_3=1.0$. Moreover, ${\rho }_4$ is the impact factor of $G\left(t\right)$ on $Q\left(t\right)$, also set at ${\rho }_4=0.1$ [40]. And r is the discount rate, set at $r=0.1$; the sharing platform receives a commission proportion of $\beta =0.2$ from the supplier [40].

Firstly, the trajectory of state variables over time is plotted, as shown in Figure 2. In all eight altruistic modes, the platform service level and the system’s just-in-time supply level both increase over time and eventually stabilize, thereby verifying the global asymptotic stability of the state variables.

Taking the mode $I^{\circ }{M}^{\circ }{R}^{\circ }$ as a benchmark, all altruistic modes except mode M were higher than the benchmark value; and for both state variables, mode $IMR$ produced the highest steady-state value.

In response to the above results, this paper examined the impacts of altruistic behaviors on state variables from several perspectives. Research has shown that altruistic behaviors can effectively mitigate the double marginal effect in supply chains and increase consumer surplus [40]. In models where members exhibit altruistic preferences, each participant not only pursues their own profits but also considers the well-being of other members and the overall efficiency of the SSC system. To sustain long-term cooperation, the sharing platform enhances its supply level and improves service quality, while suppliers and retailers adjust their pricing strategies and increase supply inputs to ensure timely supplies.

Of particular interest is the altruistic behavior of the supplier, a key player in the SSC system (such as a core raw provider or a major component manufacturer), whose actions are mainly directed at ensuring product quality and supply reliability. This focus on core functions, however, tends to limit the overall effect on performance enhancement across other segments. As a result, the supplier’s performance does not exceed the steady-state value observed in other altruistic modes, remaining comparable to the benchmark.

Furthermore, the purely altruistic mode appears to be the optimal behavior pattern. In this mode, the profits accrued by all members yield a higher overall performance in the entire SSC system, leading to maximized operational efficiency and optimal outcomes for both the platform’s service level and the just-in-time supply level.

6.1. Effect Analysis of Altruistic Parameters

The effects of the altruistic parameters on state variables, market demand, and optimal decisions under different altruistic modes are explored in this subsection. We adjusted the altruistic parameters, while keeping the other parameters fixed. It should be noted that the following analysis was conducted from the perspective of system steady-state equilibrium, thereby reducing the complexity associated with time-dependent dynamics and focusing on the influence of altruistic parameters.

6.1.1. Effects of Altruistic Parameters on Steady-State Values of State Variables and Market Demand

In Section 6.1.1, we assume that all altruistic parameters take the same value, i.e., ${\lambda }_1={\lambda }_2={\lambda }_3=\lambda$. As shown in Figure 3, the sharing platform’s service level, the system’s just-in-time supply level, and the market demand have similar trends as the altruistic parameter $\lambda$ varies under the different altruistic modes. Specially, in mode M, both the state variables and market demand decrease with increasing $\lambda$; whereas in all other altruistic modes, the state variables increase with $\lambda$.

6.1.2. Effects of Altruistic Parameters on Optimal Decisions

In this subsection, we explore the effects of altruistic parameters on optimal decisions.

Figure 4 presents the numerical simulation results of optimal decisions varying with the altruistic parameter $\lambda$ under different modes, and provides an in-depth analysis of the corresponding variable trends.

As shown in Figure 4a, under the supplier-driven altruistic modes M and $IM$, the supply price $\omega$ decreases as $\lambda$ increases, indicating that supplier altruistic behavior is primarily reflected in lowering the supply price. In contrast, under I altruism, $\omega$ remains constant at ${\displaystyle \frac{a}{2b}}$, independently of $\lambda$; in the other modes, $\omega$ increases with $\lambda$, and $\omega$ in modes R and $IR$ is always higher than that in modes $MR$ and $IMR$. Figure 4b shows that the retail price p is unaffected by $\lambda$ in modes I, R, and $IR$; however, in the remaining modes, p decreases as $\lambda$ increases, with p in modes M and $IM$ consistently lower than that in modes $MR$ and $IMR$. This suggests that the retailer’s pricing strategy is largely driven by external factors such as market competition and cost structure.

Figure 4c indicates that the supplier’s supply effort $A_1$ remains unchanged with variations in $\lambda$ in modes I, whereas, in other modes, $A_1$ increases with $\lambda$. This is because in mode I, the supplier’s effort is primarily determined by its own production capacity. Figure 4d demonstrates that the retailer’s supply effort $A_2$ remains stable in modes I, R and $IR$, while it increases with $\lambda$ in the other modes, reaching the highest levels in the modes M and $IM$. This may reflect that the retailer’s effort is more strongly influenced by supplier altruism. Finally, Figure 4e reveals that in mode M, the sharing platform’s supply effort Z decreases as $\lambda$ increases, whereas in other modes, Z increases with $\lambda$. This phenomenon might be explained by the supplier’s increased investment in product quality and timely supply, which reduces the platform’s pressure and consequently diminishes its own supply effort.

In summary, these simulation results demonstrate that the responses of the decision variables to the altruistic parameter $\lambda$ exhibit distinct patterns in different modes. This not only reflects the underlying logic of individual behaviors but also highlights the complexity of system-wide collaborative effects, providing a solid theoretical foundation for a deeper understanding of cooperative mechanisms in SSCs and their impact on overall performance.

6.1.3. Joint Effects of Altruistic Parameters and Commission Percentage on Optimal Decisions

Recall that $\beta$ denotes the commission rate that the sharing platform extracts from the supplier’s revenue, appearing directly in the supplier’s and platform’s profit functions:

$$J_{M^{\circ }}=(1-\beta )\omega \left(t\right)D\left(t\right)-\frac{1}{2}{k}_1{A}_1{\left(t\right)}^2,J_{I^{\circ }}=\beta \omega \left(t\right)D\left(t\right)-\frac{1}{2}{k}_{3}Z{\left(t\right)}^2.$$

As shown in Figure 5a, the supply price $\omega$ remains nearly constant in the modes I, R, and $IR$; declines moderately with $\beta$ in the modes $MR$ and $IMR$; and falls more sharply in the modes M and $IM$. Economically, a higher commission rate acts like a “tax” on the supplier’s margin. Facing downward-sloping demand, the supplier lowers prices to boost volume and sustain channel cooperation. When the altruistic weight $\lambda$ increases, this tax-like pressure intensifies, especially in the modes MR and IMR, reflecting the combined influence of commission and altruism on pricing strategy.

Figure 5b shows that the retail price p is insensitive to $\beta$ in the modes I, R, and $IR$, but decreases steadily in the modes M, $IM$, $MR$, and $IMR$. Since the retailer’s profit does not depend explicitly on $\beta$, price adjustments occur through supplier price cuts and demand shifts. From a management standpoint, higher commissions compress channel margins, compelling retailers to reduce prices to remain competitive, an effect that is amplified under greater altruism.

From Figure 5c, the supplier’s effort $A_1$ decreases with $\beta$ across all modes. High commission rates erode the supplier’s marginal returns, dampening investment in capacity and quality. Moreover, with a larger altruistic weight $\lambda$, the supplier further sacrifices its own returns to enhance system welfare, reinforcing the decline in $A_1$.

Figure 5d illustrates that the retailer’s effort $A_2$ remains flat in the modes I, R, and $IR$, but rises with $\beta$ under the modes M, $IM$, $MR$, and $IMR$. Under supplier altruism, a lower supply price widens the retailer’s net margin, incentivizing greater effort; increased platform service weighting further boosts demand and strengthens this positive relationship.

Finally, Figure 5e reveals that the platform’s service effort Z increases monotonically with $\beta$ in all modes. As $\beta$ rises, the marginal payoff of platform effort grows, and with a higher altruistic weight, the platform balances its own revenue with system welfare, leading to even greater service investment.

6.2. Comparison of Supply Chain Profits and Utilities in Different Altruistic Modes

Taking profits and utilities in mode $I^{\circ }{M}^{\circ }{R}^{\circ }$ as a benchmark, note that when the altruism parameters were all set to 0, the altruistic mode was indistinguishable from $I^{\circ }{M}^{\circ }{R}^{\circ }$. In order to ensure the non-negativity of profits and utilities, the values of the altruism parameters were restricted to $0<{\lambda }_i\le 0.3$.

Figure 6 illustrates the utility performance under various modes, highlighting the impact of variations in the values of altruistic parameters.

A close examination of Figure 6 reveals that all altruistic modes yielded substantial improvements in the overall utility of SSC system relative to the benchmark case. Both the system-level and the individual-level utilities increase monotonically with the altruism parameter, except in mode M.

As Figure 6c shows, in mode M, however, the platform’s utility declines slightly as $\lambda$ rises. Meanwhile, the supplier, the retailer, and overall system utilities remain below those observed in the other six altruistic modes. In contrast, when only the sharing platform itself adopts altruistic behavior, its deliberate trade-off of personal profit for systemic gains drives the highest enhancement in overall performance between the three members, showing superior idle-resource-sharing and information-integration efficiencies.

In brief, as the altruistic parameter increases, the degree of trust and cooperation among SSC members is enhanced, and the ability to share supply–demand information, as well as to integrate and allocate resources, is improved, thereby enhancing overall utility. Moreover, the utilities of both the supplier and the retailer also improve, fully reflecting the positive influence of altruistic behavior on the profits of all members in the SSC.

Similarly, Figure 7 illustrates the profit performance of the SSC in different modes, each characterized by a unique combination of altruistic parameters.

By analyzing Figure 7, we can derive the following insights regarding the impact of altruistic behaviors on the SSC system and the profits of its individual members:

Under different altruistic modes, an increase in the altruism parameter consistently enhances the steady-state value of the SSC system’s overall profits. Notably, joint altruistic modes outperform single-agent altruistic modes in terms of enhancing system-wide profits. This superiority arises because greater altruism fosters deeper cooperation among SSC members. In joint altruistic modes, the members share both responsibilities and profits, leading to stronger synergy effects and a more substantial enhancement in overall performance.

However, in single-agent altruistic modes, the altruistic member’s own profit tends to decline. This effect is particularly evident for the retailer, whose steady-state benefit decreases most noticeably with greater $\lambda$ increases. This occurs due to the inherent trade-off in altruistic behavior: the altruism of a single member often involves self-sacrifice to enhance the profits of the other members. As the degree of altruism intensifies, the decline in self-interest becomes more pronounced.

Among the dual-member altruistic modes, mode $MR$ does not perform as well as modes $IR$ and $IM$ in enhancing overall system profits. Interestingly, the altruistic modes $IR$ and $IM$ yield system profits comparable to those in the purely altruistic mode $IMR$. This observation suggests that sharing platform altruism plays a key role in driving higher system profits, making it a central factor in optimizing SSC performance under altruistic settings.

In summary, altruistic behaviors introduce intricate dynamics in balancing overall system profits and individual profits. While joint altruism generally leads to superior performance, single-agent altruism may result in self-interest losses. Notably, the altruistic behavior of the sharing platform emerges as a key determinant of overall profits.

7. Conclusions

7.1. Main Conclusions

This study developed a three-tier shared supply chain model comprising a supplier, a retailer, and a sharing platform. We focused on dynamic decision-making under multi-agent altruistic behaviors. By constructing a Stackelberg differential game model and solving for equilibrium solutions through backward induction, the research combined theoretical analysis with numerical experiments to compare optimal decisions and system performance under various altruistic modes. Additionally, the effects of altruistic parameters were analyzed using numerical experiments. We obtained the main conclusions as follows:

(1)

Under R altruism, the retailer exhibits altruistic behavior, increasing its own supply effort and encouraging the sharing platform to enhance its effort. Under M altruism, the supplier proactively reduces supply prices to alleviate the retailer’s cost pressures and jointly increases its own supply effort to foster long-term cooperation, thereby expanding market demand and boosting the retailer’s supply effort. However, when only the sharing platform exhibits altruism, manifested in improved service levels and increased supply efforts, although supply–demand matching efficiency is enhanced, the optimal decisions of the other members are relatively weakly affected.

(2)

In all altruistic modes, both the platform’s service level and the just-in-time supply level progressively improve and eventually stabilize over time. However, under the mode M, the state variables consistently remain below the benchmark values observed in the purely selfish mode. Moreover, as the altruism parameter increases, the steady-state values of state variables generally rise in all modes except for the mode M. This anomaly may be attributed to the supplier’s distinct upstream role, where cost investments are primarily allocated to production and resource optimization, with relatively less emphasis on aspects such as product transportation, compared with those of the other SSC members.

(3)

Compared to the purely selfish mode, the utilities of the overall system are significantly improved under other altruistic modes. The altruism of the sharing platform significantly surpasses that of the other two members in enhancing overall system utility. This highlights the sharing platform’s crucial role as a facilitator of resource sharing and supply–demand matching, as its altruism effectively boosts service levels and supply efforts, thereby improving the overall system efficiency.

(4)

As the altruism parameter increases, the steady-state values of individual profits for suppliers and retailers under single-agent altruistic modes typically fall below the benchmark values. This occurs because altruistic behavior requires members to willingly self-sacrifice, leading to short-term profit declines. Conversely, in joint altruistic modes, collaborative altruism among multiple agents can stimulate stronger long-term cooperation, significantly enhancing both system-wide and individual member profits.

In summary, this study reveals the impact of different altruistic modes on member decision-making and overall system performance within the SSC. It highlights that, while individual altruistic behaviors may boost short-term profits for specific members, collaborative altruism among multiple members offers greater advantages in improving the overall efficiency of SSC systems. Furthermore, within the SSC context, the sharing platform plays a crucial role in optimizing resource sharing and supply–demand matching, thereby promoting stable and effective collaboration across the entire supply chain.

7.2. Managerial Implications

In SSC management practice, enhancing overall efficiency, optimizing member collaboration, and designing effective incentive mechanisms have become key research priorities. By analyzing the decision behaviors of SSC members, this study revealed the profound impact of altruistic preferences on supply chain operations.Building on the analytical framework developed herein, we distill our findings into three actionable insights. The following recommendations aim to bridge theory and practice for SSC decision–makers.

(1)

Leverage the critical role of the sharing platform. Our results demonstrate that platform altruistic behavior is instrumental in driving system–wide performance improvements. Managers should prioritize the design and continuous enhancement of platform services and supply capabilities, investing in infrastructure that fosters resource integration and demand matching, thereby securing sustainable long–term growth.

(2)

Foster collaborative altruism and reciprocal cooperation. The findings indicate that joint altruistic actions by multiple agents are particularly effective in achieving Pareto improvements. Rather than focusing solely on short-term individual gains, managers ought to establish robust cooperation frameworks among all SSC members and formalize long-term partnership agreements.

(3)

Optimize pricing strategies and cooperative incentives. Altruistic preferences significantly shape both pricing decisions and supply efforts across SSC members. Managers must develop pricing schemes that reflect interdependencies, and in multi-agent altruistic contexts, carefully balance profit allocation and incentives to prevent value erosion from unilateral concessions, ultimately boosting overall efficiency.

7.3. Future Research

This study employed differential game methods to explore the dynamic decision-making processes within a three-level SSC model. It presented a comprehensive comparative analysis of optimal decision-making and system performance under various altruistic modes. Both the theoretical framework and the numerical experiments offered valuable insights that can significantly inform future research on SSCs.

While this study provides valuable insights into dynamic decision-making within a three-level SSC model, certain limitations should be acknowledged to contextualize the results appropriately. One notable constraint lies in the assumptions embedded within the differential game framework. Although these assumptions facilitate analytical tractability, they may not fully capture the complexity and unpredictability of real-world supply chain interactions. Factors such as unforeseen disruptions, behavioral inconsistencies among agents, and evolving market conditions could lead to deviations from the theoretical predictions. Additionally, numerical experiments provide controlled insights into system behavior under various conditions, yet their applicability to real-world scenarios remains contingent upon accurate parameter estimations. The extent to which these results can be extrapolated depends on the alignment between theoretical assumptions and empirical observations.

In more complex real-world market environments, our future research will first evaluate the applicability of the primary findings derived from our SSC model, with a particular emphasis on verifying the counterintuitive outcomes observed in mode M. Subsequently, we will propose an advanced SSC model that incorporates additional supply chain members, such as distributors or customers, thereby extending the scope of our investigation. Furthermore, by exploring intricate parameter configurations and incorporating more complex state equations and demand function, we aim to fine-tune both the curvature and sensitivity of our model. Ultimately, by connecting our theoretical insights with real-world applications, we seek to enhance the overall relevance and impact of our work, thus paving the way for meaningful advancements in the field. Future studies could build upon these findings by incorporating elements such as incomplete information, external shocks, additional supply chain members, and broader economic factors, e.g., policy changes and technological innovations. This would enable a deeper investigation into how altruistic behaviors impact the long-term stability and profitability of SSCs in intricate market scenarios. Moreover, considering that practical supply chain management often involves more sophisticated game models, subsequent research should aim to refine dynamic game frameworks by integrating a wider array of strategic choices and decision-making processes.