Abstract
Currently, collaborative distribution models have not reached the optimal state of carbon emissions. The cost of additional lowcarbon expenditures and the problem of carbon data verification have led to the lack of motivation for reducing emissions among collaborative distribution enterprises. Therefore, how to incentivize them to adopt the lowcarbon model is crucial for achieving lowcarbon goal. By relying on a governmentled digital platform, this paper designs a dual lowcarbon incentive strategy to encourage enterprisealliance to adopt a lowcarbon distribution model. In this paper, we first construct an evolutionary game model of the government, enterprisealliance and endusers; then we explore the conditions of the threeparty equilibrium evolution strategy by solving the model and analyzing the stability; and finally, we conduct simulation validation and results analysis with the help of MATLAB. In summary, we found that government punishment is more effective at regulating enterprisealliance than reward. Endusers’ behavior is affected by the costs they need to bear, and they no longer support enterprisealliance to carry out collaborative lowcarbon distribution above a certain threshold.
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Introduction
In recent years, China's carbon emissions trading market has experienced unprecedented development, and the carbon trading mechanism has entered a new phase. At the same time, consumer attitudes have also changed, shifting from the original affordable to lowcarbon environmental protection. Research shows that consumers' increased preference for lowcarbon makes them pay more attention to the lowcarbon attributes of products when they buy them, and they are willing to pay a high price for them. As a result, the endusers' lowcarbon preference has become an important consideration in logistics distribution decisions^{1}.
Logistics enterprisealliance is a cooperative organization formed by a number of logistics enterprises to improve the overall efficiency and competitiveness of the logistics industry. Logistics enterprise alliance as a kind of enterprise relationship concluded voluntarily between enterprises. It has the characteristics of resource sharing, platform building and result sharing, so it can meet the individualized needs of logistics enterprises and make the relationship between logistics enterprises stable and longterm^{2}. With the goal of "exploring the best opportunities for collective competitive advantage", the Alliance draws on the management philosophy of virtual enterprises and integrates them according to specific needs^{3}. At present, there are enterprise alliances, such as cold chain logistics alliance, searailway logistics alliance, railroad logistics alliance, ecommerce logistics alliance, transportation logistics alliance, crossborder logistics alliance and so on.
Under the traditional model, logistics distribution enterprises have difficulties monitoring and managing carbon emissions, leading to inaccurate data and nontransparent supervision. To solve this problem, a collaborative distribution model with a digital platform has emerged^{4}. Collaborative distribution refers to the modernized distribution method in which logistics enterprise alliance use online platforms to integrate distribution orders from different enterprises and jointly complete distribution tasks. In this way, distribution efficiency can be improved, and distribution costs and carbon emissions can be significantly reduced^{5}. Collaborative distribution becomes more important with the boom of lowcarbon goals and the indepth study of lowcarbon concepts. The digitization platform mentioned in this paper is led by the government and is currently free for distribution enterprises. In the initial stage, the governmentrun digital platform can incentivize distribution enterprises to participate in collaboration, account for carbon emissions in the distribution process, provide data support for lowcarbon incentives, and effectively connect with the carbon trading market^{6} . The flowchart of the enterprisealliance for collaborative distribution in this paper is shown in Fig. 1.
There are many studies on collaborative distribution among enterprises. Rao et al. designed an optimal price discount strategy to incentivize customers to participate in collaborative distribution^{7}. Zhou et al. studied the optimization problem of collaborative distribution using vehicle piggybacking to carry out collaboration^{8}. Qie et al. developed a cost sharing model based on the consensual function to solve the problem of cost allocation among enterprises^{9}. Chu et al. argued that the joint distribution improves the delivery rate of express delivery in the lastmile logistics^{10}. Rao et al. constructed a cost minimization collaborative distribution model to analyze the necessity of coalition splitting^{11}. Improving vehicle utilization through a collaborative distribution approach enables firms to contribute to the three pillars of sustainable development by reducing fuel consumption and emissions, increasing profitability, and improving customer satisfaction^{12}. Rao et al. proposed a vehicle makeup strategy of finding idle vehicles from within the alliance and then leasing vehicles from outside the alliance, and constructed a quantitative model for the cost of multiparty collaborative distribution considering the constraints on the distribution capacity of the enterprises^{13}. Han et al. argued that the distribution of common profit has always been a key obstacle to the effective development of joint distribution in the context of green and lowcarbon, and explored a fairer and more reasonable profit distribution scheme^{14}. Chen et al. develop a cold chain logistics model considering joint distribution and carbon trading mechanism^{15}; Rao et al. design a default recovery and loss compensation mechanism for members' withdrawal from an alliance in collaborative distribution^{16}.
Evolutionary game theory is widely used to analyze the strategy selection problem under the participation of multiple subjects. Li et al. constructed a developergovernmentconsumer threeway evolutionary game model by combining the carbon trading prospect theory^{17}. Based on evolutionary game theory, Zeng et al. explored the influence of platform reward and punishment mechanisms on the lowcarbon strategy selection of manufacturing enterprises^{18}. Lin et al. studied the strategic decisionmaking process of two participants in the context of medical malpractice based on the evolutionary game model of doctors and patients^{19}. Wu et al. developed a game theory model involving packaging suppliers and logistics companies and based on the theory of corporate social responsibility^{20}. Tang et al. constructed a threeparty evolutionary game model to analyze the strategic choices of automobile companies, consumers and the Chinese government^{21}. Gong et al. constructed a threeparty evolutionary game model of cloud manufacturing platform, service provider and service demander, and analyzed the evolutionary stabilization strategy^{22}. Yuan et al. constructed a threeparty evolutionary game model to study the allocation of medical supplies in the rescue environment of public health emergencies under the condition of information incompleteness^{23}. Wang et al. constructed an evolutionary game model of value cocreation in a cloud manufacturing innovation ecosystem, and studied the relationship between platform enterprises and government participation behaviors under dynamic and static reward and punishment mechanisms^{24}. Qiu et al. constructed a threeparty evolutionary game model of ecommerce platforms, merchants, and consumers, and examined the impacts of the influencing factors on the strategic choices of each party^{25}. Chen et al. analyzed the game between the government and polluting enterprises under the environmental tax system through an evolutionary game model^{26}.
From the above review, it can be seen that current researches on the collaborative distribution of logistics enterprise alliance have focused mainly on alliance splitting strategies, cost sharing methods and the mechanism of recovery and compensation for members' withdrawal from the alliance. Researches on collaborative distribution participants has considered only the influence of government regulatory decisions. In this paper, the distribution strategy of logistics enterprise alliance is investigated considering government reward and punishment as well as endusers' lowcarbon preference. Therefore, this paper constructs an evolutionary game model of the government, enterprisealliance and endusers, to consider the impact of the government's reward and punishment mechanism and endusers' lowcarbon preference on the distribution strategy of enterprisealliance. The research questions that are addressed in this paper are as follows: (1) From the analysis of the evolutionary route, what are the differences in the dynamic evolutionary process of the game subjects under different strategy choices of multiple subjects? (2) From the perspective of supplyside government policy analysis, what is the impact of government reward and punishment mechanism on the distribution strategy of logistics enterprisealliance? (3) From the analysis of enduser behavior on the demand side, how does the enduser's lowcarbon preference affect the distribution strategy of logistics enterprisealliance?
Model construction
Description of the problem
The collaborative distribution of logistics enterprisealliance can be categorized into collaborative lowcost distribution and collaborative lowcarbon distribution according to the distribution mode^{27}. The enterprisealliance usually strives to maximize profits, so it prefers to choose collaborative lowcost distribution, while the government is committed to maximizing social benefits, and hopes that the enterprisealliance will choose collaborative lowcarbon distribution. Taxation is a kind of environmental regulation policy of the government, which can improve the environmental protection willingness of enterprisealliance by levying environmental tax. At the same time, from the implementation level, the government needs to implement certain regulatory measures on the collaborative distribution strategy of enterprisealliance. Reward for collaborative distribution enterprisealliance that aim to minimize carbon emissions, and punishment for collaborative distribution enterprisealliance that aim to minimize costs. The government, through taxation and reward and punishment mechanisms, encourages the alliance to optimize the distribution plan with the goal of carbon emissions, so as to achieve the goal of energy saving and emission reduction. Similarly, the government grants distribution subsidies to endusers with low carbon preferences, and imposes a personal environmental tax on endusers without low carbon preferences in order to optimize emissions reduction. Finally, endusers with lowcarbon preferences are willing to pay additional lowcarbon distribution subsidies to enterprisealliance, which provide logistics and distribution services to endusers. Therefore, the mechanism of the threeway game between the government, enterprisealliance and endusers in the distribution process is shown in the following Fig. 2.
Model assumptions and parameter design
In assumption 1 for the natural environment, subject 1 for the government subgroup, subject 2 for the enterprisealliance subgroup, and subject 3 for the enduser subgroup, assuming that the participating subjects have finite rationality, the model of the game parties is in the initial stage, the game strategy selection over time gradually evolves and stabilizes in the optimal strategy and the game process does not take into account the impact of other subjects on the game^{28,29}.
Assumption 2 enterprisealliance: the set of actions available to the enterprisealliance is assumed to be{collaborative lowcarbon distribution, collaborative lowcost distribution}^{30} . Collaborative lowcost distribution refers to an enterprisealliance that minimizes costs as the goal of distribution route design, and collaborative lowcarbon distribution refers to the enterprisealliance that minimizes carbon emissions as the goal of distribution route design. Let the probability of collaborative lowcarbon distribution of enterprisealliance be \(x\) ,the probability of collaborative lowcost distribution is \(1x\) where \(0\le x\le 1\) .
Assumption 3 government: the set of actions available to the government is assumed to be{reward and punishment combination policy, flow in the form of policy}^{31} . The meaning of the policy of combining reward and punishment is that when the enterprisealliance chooses lowcarbon distribution or lowcost distribution, the government, as an external regulator, gives flexible reward or punishment to regulate the market, and the meaning of flow in the form of policy is that the government uses fixed rather than flexible reward or punishment to regulate the market. Let the probability that the government adopts the policy of combining reward and punishment be as follows \(\text{y}\),the probability of adopting the flow in the form of policy is \(1\text{y}\) where \(0\le \text{y}\le 1\).
Assumption 4 endusers: the set of actions available to the endusers is assumed to be{with lowcarbon preference , without lowcarbon preference} . Endusers with lowcarbon preference are willing to spend more money supporting collaborative lowcarbon distribution through enterprisealliance; Endusers without lowcarbon preference are more likely to support collaborative lowcost distribution through enterprisealliance. Let the probability that endusers with lowcarbon preference be \(\text{z}\) , the probability of without lowcarbon preference is \(1\text{z}\) where \(0\le \text{z}\le 1\) . Endusers who support lowcarbon distribution have to pay higher distribution costs. To promote the development of lowcarbon distribution, the government provides endusers with a certain amount of lowcarbon distribution subsidies.
Assumption 5 in a scenario where endusers demand for goods remains constant but the unit price of distribution increases due to endusers' lowcarbon preference, the benefits of collaborative distribution for the enterprisealliance increase accordingly. When an enterprisealliance chooses collaborative lowcost distribution or collaborative lowcarbon distribution strategies, differences in optimized routes lead to differences in distribution costs. Therefore, different collaborative distribution strategies chosen by enterprises will affect the collaborative distribution costs, carbon emissions, and revenues of the enterprisealliance. In addition, the distribution process includes the cost of cargo deployment in addition to the cost generated by the distribution itself^{30} .
Assumption 6 revenues^{32}: revenues from collaborative distribution by the enterprisealliance \(\left({\varphi }_{1},{\varphi }_{2}\right)\), lowcarbon distribution support \(\beta {\mathcal{V}}\) that the enterprisealliance receives from endusers; environmental taxes levied by the government on the enterprisealliance \(\left({\mathcalligra{m}}{\mathcal{E}}_{1},{\mathcalligra{m}}{\mathcal{E}}_{2}\right)\) and personal environmental taxes levied by the government on the endusers \(\left(\gamma L,{D}_{2}\right)\), and penalties \(A\) imposed by the government on enterprisealliance for lowcarbon distribution; endusers with lowcarbon preference can receive the government's distribution subsidy \(\left(\alpha {\mathcal{W}},{D}_{1}\right)\).
Assumption 7 costs^{32}: the cost of collaborative distribution by the enterprisealliance \(\left({\mathcal{C}}_{x1},{\mathcal{C}}_{x2}\right)\), the deployment cost of collaborative distribution by the enterprisealliance \(\uptheta\); the regulatory costs invested by the government \(\left({\mathcal{C}}_{y1},{\mathcal{C}}_{y2}\right)\), the government's bonus to the enterprisealliance when it distributes low carbon \(S\), and distribution subsidies for endusers with lowcarbon preferences \(\left(\alpha {\mathcal{W}},{D}_{1}\right)\); the endusers' costs are mainly the time cost \(\updelta\) needed to support lowcarbon distribution in terms of the lowcarbon distribution support \(\beta {\mathcal{V}}\) they are willing to pay to the enterprisealliance.
Based on the above assumptions, the parameters and their significance required to comprehensively set up the threeparty evolutionary game model are shown below:
Enterprisealliance:
\({\mathcal{C}}_{x1}\): Distribution costs of collaborative lowcarbon distribution \({\mathcal{C}}_{x1}>0\)
\({\mathcal{C}}_{x2}\): Distribution costs for collaborative lowcost distribution \({\mathcal{C}}_{x2}>0\)
\(\uptheta\): Redeployment costs for collaborative distribution \(\uptheta >0\)
\({\mathcal{E}}_{1}\): Carbon emissions when collaborating on lowcarbon distribution \({\mathcal{E}}_{1}>0\)
\({\mathcal{E}}_{2}\): Carbon emissions when collaborating on lowcost distribution \({\mathcal{E}}_{2}>0\)
\({\varphi }_{1}\): Revenues of collaborative lowcarbon distribution \({\varphi }_{1}>0\)
\({\varphi }_{2}\): Revenues of collaborative lowcost distribution \({\varphi }_{2}>0\)
Governments:
\({\mathcalligra{m}}\): Government's unit penalty factor for carbon emissions from corporate alliances (environmental tax rate^{30})\(0\le {\mathcalligra{m}}\le 1\)
\(S\): Government reward for enterprisealliance when collaborating on lowcarbon distribution \(S>0\)
\(A\): Government punishment when enterprisealliance collaborate on lowcost distribution \(A>0\)
\({\mathcal{C}}_{y1}\): Regulatory costs incurred by the government in adopting the policy of reward and punishment combination \({\mathcal{C}}_{y1}>0\)
\({\mathcal{C}}_{y2}:\) Regulatory costs incurred by governments in adopting the policy of flow in the form \({\mathcal{C}}_{y2}>0\)
\(\omega\): Environmental revenues to governments when enterprisealliance collaborate on lowcarbon distribution \(\omega >0\)
\(\alpha{ \mathcal{W}}\): Distribution subsidies for endusers with lowcarbon preference when the government adopts a policy of reward and punishment.\({\mathcal{W}}\) is the cap of government reward
\(\gamma L\): Individual environmental taxes on endusers without lowcarbon preferences when the government adopts a policy of reward and punishment,^{32} \(L\) is the upper limit of the government's tax revenue
\({D}_{1}\): Distribution subsidies for endusers with lowcarbon preference when the government adopt flow in the form policy
\({D}_{2}\): Individual environmental taxes levied on endusers without lowcarbon preferences when governments adopt flow in the form policy.
Endusers:
\(\beta {\mathcal{V}}\): Endusers with lowcarbon preference are willing to pay for lowcarbon distribution support \({\mathcal{V}}\) is the upper limit of what endusers are willing to pay
\(\updelta\): Cost of time for endusers to support lowcarbon distribution.
Game model construction and analysis
When logistics enterprise alliance choose distribution strategies, their behaviors are constrained by a variety of factors. To analyze the influencing factors of the behavior of each participating subject, evolutionary game theory can be used to construct an evolutionary game model^{33}. First, according to the assumptions, the benefits of enterprisealliance, endusers and government under different behavioral strategies are listed, and then their average expected benefits are calculated, and the replicated dynamic equations of each party are finally derived. The three parties, the enterprisealliance, enduser and government obtain the benefits according to the benefit payment matrix in Tables 1 and 2 below , where the benefits of each subject is equal to the revenues of each subject minus its corresponding costs (the equations in rows 1–3 of each decision combination in the payment matrix represent the expected benefits of the government, the enterprisealliance and the enduser under the decisions of the corresponding parties).
Enterprisealliance, endusers, and government behaviors are interactive, and the three parties utilize continuous evolution to achieve mutual benefits. In the following, the replicated dynamic equations are developed and solved for evolutionary stabilization strategies.
The payment matrices in Tables 1 and 2 show the expected benefits of collaborative lowcarbon distribution and collaborative lowcost distribution for enterprisealliance \({E}_{11}\) and \({E}_{12}\):
The average expected return of the enterprisealliance \({E}_{1}\) is:
This leads to the following equation for the replication dynamics of the enterprisealliance \({F}_{\left(x\right)}\) as^{34}:
Similarly, it follows that the expected returns to the government's adoption of the policy of reward and punishment combination and the policy of flow in the form \({E}_{21}\) and \({E}_{22}\) respectively:
The average expected return to the government \({E}_{2}\) is:
This leads to the following equation for the replication dynamics of the government \({F}_{\left(y\right)}\) as:
Similarly, the expected returns of the endusers with and without lowcarbon preference are \({E}_{31}\) and \({E}_{32}\) respectively:
The average expected return to the endusers \({E}_{3}\) is:
This leads to the following equation for the dynamics of endusers replication \({F}_{\left(z\right)}\) as:
Game model solving and analysis
Analysis of the stability of single subject strategies
Strategic stability analysis of enterprisealliance
From the above, the replication dynamic equation of the corporate alliance game strategy is Eq. (4):
At this point the first order derivative of \(x\) the firstorder derivative of \({\mathcalligra{d}}\left({F}_{\left(x\right)}\right)/{\mathcalligra{d}}x=\left(12x\right){H}_{\left(\text{y}\right)}\)

(1)
When \(\text{y}=\widehat{\text{y}}\) and \({F}_{\left(x\right)}=0\), regardless of whether \(x\) takes any value, the enterprisealliance ' strategy tends to stabilize, which indicates that the enterprisealliance is a stable strategy regardless of what strategy it chooses^{34}, as shown in Fig. 3a.

(2)
When \(\text{y}\ne \widehat{\text{y}}\), order \({F}_{\left(x\right)}=0\) , there exist two equilibrium points:\({x}_{1}^{*}=0\) and \({x}_{2}^{*}=1\) .According to the stability theorem of differential equations, realizing the stabilization needs to satisfy both \({F}_{\left(x\right)}=0\) and \({\mathcalligra{d}}\left({F}_{\left(x\right)}\right)/{\mathcalligra{d}}x<0\).\({\mathcalligra{d}}{H}_{\left(\text{y}\right)}/{\mathcalligra{d}}\text{y}=\text{S}\text{A}>0\) and \({H}_{\left(\text{y}\right)}\) is an increasing function.
When \(0<\text{y}<\widehat{\text{y}}\) and \({H}_{\left(\text{y}\right)}<0\),\({\mathcalligra{d}}\left({F}_{\left(x\right)}\right)/{\mathcalligra{d}}x{}_{{x}_{1}^{*}=0}<0\),\(\left({F}_{\left(x\right)}\right)/{\mathcalligra{d}}x{}_{{x}_{1}^{*}=1}>0\).Therefore \({x}_{1}^{*}=0\) is an evolutionarily stable strategy. The adoption of collaborative lowcost distribution by the enterprisealliance is the only evolutionary stable strategy globally, as shown in Fig. 3b. Similarly, when \(\widehat{\text{y}}<\text{y}<1\),\(\left({F}_{\left(x\right)}\right)/{\mathcalligra{d}}x{}_{{x}_{2}^{*}=1}<0\) , and \({x}_{2}^{*}=1\) is the stable strategy, the enterprisealliance adopts collaborative lowcarbon distribution as the global unique evolutionary stable strategy as shown in Fig. 3c.
Strategic stability analysis of government
From the above, the replication dynamic equation for the government game strategy is:
At this point the first order derivative of \(\text{y}\) the firstorder derivative of \({\mathcalligra{d}}\left({F}_{\left(\text{y}\right)}\right)/{\mathcalligra{d}}\text{y}=\left(12\text{y}\right){G}_{\left(\text{z}\right)}\)

(1)
When \(\text{z}=\widehat{\text{z}}\) and \({F}_{\left(\text{y}\right)}=0\) ,regardless of whether \(\text{y}\) takes any value, the government's strategy tends to stabilize, which indicates that the government's strategy is stable regardless of what it chooses, as shown in Fig. 4a.

(2)
When \(\text{z}\ne \widehat{\text{z}}\), order \({F}_{\left(\text{y}\right)}=0\) , there exist two equilibrium points:\({\text{y}}_{1}^{*}=0\) and \({\text{y}}_{2}^{*}=1\).According to the stability theorem of differential equations, realizing the strategy stabilization needs to satisfy both \({F}_{\left(\text{y}\right)}=0\) and \({\mathcalligra{d}}\left({F}_{\left(\text{y}\right)}\right)/{\mathcalligra{d}}\text{y}<0\).\({\mathcalligra{d}}{G}_{\left(\text{z}\right)}/{\mathcalligra{d}}\text{z}={D}_{1}+{D}_{2}\alpha \mathcal{W}\gamma L<0\) and \({G}_{\left(\text{z}\right)}\) is a decreasing function.
When \(0<\text{z}<\widehat{\text{z}}\) and \({G}_{\left(\text{z}\right)}>0\),\({\mathcalligra{d}}\left({F}_{\left(\text{y}\right)}\right)/{\mathcalligra{d}}\text{y}{}_{{\text{y}}_{1}^{*}=0}>0\),\(\left({F}_{\left(\text{y}\right)}\right)/{\mathcalligra{d}}\text{y}{}_{{\text{y}}_{2}^{*}=1}<0\), therefore \({\text{y}}_{2}^{*}=1\) is an evolutionarily stabilization strategy. The government adopts the flow in the form policy as the only evolutionary stabilization strategy in the whole world, as shown in Fig. 4b. Similarly, when \(\widehat{\text{z}}<\text{z}<1\) and \(\left({F}_{\left(\text{y}\right)}\right)/{\mathcalligra{d}}\text{y}{}_{{\text{y}}_{1}^{*}=0}<0\),\({\text{y}}_{1}^{*}=0\) is an evolutionarily stable strategy. The government adopts the policy of reward and punishment combination as the only evolutionary stable strategy in the whole situation, as shown in Fig. 4c.
Strategy stability analysis for endusers
From the above, the replication dynamic equation for the enduser game strategy is:
At this point the first order derivative of \(\text{z}\) the firstorder derivative of \({\mathcalligra{d}}\left({F}_{\left(\text{z}\right)}\right)/{\mathcalligra{d}}\text{z}=\left(12\text{z}\right){R}_{\left(x\right)}\)

(1)
When \(x=\widehat{x}\) and \({F}_{\left(\text{z}\right)}=0\) , regardless of whether \(\text{z}\) takes any value, the endusers' strategy tends to stabilize, which indicates that the endusers' strategy is a stable strategy regardless of the strategy chosen, as shown in Fig. 5a.

(2)
When \(x\ne \widehat{x} ,\text{order}{F}_{\left(\text{z}\right)}=0\), there exist two equilibrium points:\({\text{ z}}_{1}^{*}=0\) and \({\text{z}}_{2}^{*}=1\). According to the stability theorem of differential equations, realizing strategy stabilization needs to satisfy both \({F}_{\left(\text{z}\right)}=0\) and \({\mathcalligra{d}}\left({F}_{\left(\text{z}\right)}\right)/{\mathcalligra{d}}\text{z}<0\) . Since \({\mathcalligra{d}} {R}_{\left(x\right)}/{\mathcalligra{d}}x=\left(\beta \mathcal{V}+\delta \right)<0\), \({R}_{\left(x\right)}\) is a decreasing function.
When \(0<x<\widehat{x}\) and \({R}_{\left(x\right)}>0\) ,\({\mathcalligra{d}}\left({F}_{\left(\text{z}\right)}\right)/{\mathcalligra{d}}\text{z}{}_{{\text{z}}_{1}^{*}=0}>0\) ,\(\left({F}_{\left(\text{z}\right)}\right)/{\mathcalligra{d}}\text{z}{}_{{\text{z}}_{2}^{*}=1}<0\)
Therefore \({\text{z}}_{2}^{*}=1\) is an evolutionarily stable strategy. Endusers without lowcarbon preference is the only evolutionarily stable strategy globally, as shown in Fig. 5b. Similarly, when \(\widehat{x}<x<1\) and \(\left({F}_{\left(\text{z}\right)}\right)/{\mathcalligra{d}}\text{z}{}_{{\text{z}}_{1}^{*}=0}<0\) , the \({\text{z}}_{1}^{*}=0\) is an evolutionarily stable strategy. Endusers with lowcarbon preference is the global unique evolutionary stable strategy, as shown in Fig. 5c.
Stability analysis of the equilibrium point of the tripartite evolutionary game system
Threeparty game model ESS solving
The previous section analyzed the equilibrium conditions for each game subject to reach the stabilization strategy from the perspective of a single game subject, but in essence, the achievement of the final stable state of the system requires the joint action of the three parties^{35}. Therefore, in this section, the asymptotic stability of the equilibrium point is determined by constructing a Jacobi matrix and solving for the eigenvalues, taking into account the existing studies^{36,37}. According to the replicated dynamic equations \({F}_{\left(x\right) },{F}_{\left(y\right)}\) and \({F}_{\left(\text{z}\right)}\) ,the of \(x\),\(y\) and \(\text{z}\) derivation yield the Jacobian matrix^{38} . In fact, let the replicated dynamic equation of the three parties of interest be 0. Eight pure strategy Nash equilibrium points can be obtained:\(\left(0,0,0\right)\left(0,0,1\right)\left(0,1,0\right)\left(1,0,0\right)\left(1,1,0\right)\left(1,0,1\right)\left(0,1,1\right)\left(1,1,1\right)\).The Jacobi matrix is obtained from the above replicated dynamic equation as shown in Eq. (19):
Among them:
Equilibrium point stability analysis
From evolutionary game theory^{39} , the stability of the equilibrium point of the system can be judged by the eigenvalues of the Jacobi matrix proposed by Friedman^{40}. According to Lyapunov’s^{41,42} first method, when the eigenvalues are negative, the equilibrium point is stable; when the eigenvalues are positive, the equilibrium point is unstable. The above eight equilibrium points are substituted into the Jacobi matrix to obtain the corresponding eigenvalues, as shown in the following table:
The local equilibrium points in the table are stable points if the three corresponding eigenvalues are not greater than zero. By further solving the eigenvalues in Table 3, we find that there are six stable points in the evolutionary game, and the specific scenarios are as follows:
Scenario 1 \(\left(1,1,1\right)\) : When \(\left(SA\right)+\left({\mathcal{C}}_{x2}{\mathcal{C}}_{x1}\right)+{\mathcalligra{m}}\left({\mathcal{E}}_{2}{\mathcal{E}}_{1}\right)+\beta \mathcal{V}>{\varphi }_{2}{\varphi }_{1}\) ,\({\mathcal{C}}_{y1}{\mathcal{C}}_{y2}<{D}_{1}\alpha \mathcal{W}S\) and \(\beta \mathcal{V}+\delta <\alpha \mathcal{W}+\gamma L\) ,the stable strategy of the evolutionary game is (collaborative lowcarbon distribution, reward and punishment combination, with lowcarbon preference). In this scenario, the benefits of the enterprisealliance when it engages in lowcarbon distribution are supplemented by the government and endusers, and the benefits of collaborative lowcarbon distribution are much greater than the benefits of collaborative lowcost distribution, so the enterprisealliance group will choose to engage in collaborative lowcarbon distribution. The net benefit^{43} to the government of adopting the policy of reward and punishment combination is greater than the net benefit of adopting flow in the form of policy, so the government group will choose to adopt policy of reward and punishment combination to guide the enterprisealliance and the endusers, and the government can obtain the greatest environmental benefits. The benefits to endusers with lowcarbon preference outweigh the costs, so the endusers will gradually support lowcarbon distribution through enterprisealliance.
Scenario 2 \(\left(0,1,1\right)\) : When \(\left(SA\right)+\left({\mathcal{C}}_{x2}{\mathcal{C}}_{x1}\right)+{\mathcalligra{m}}\left({\mathcal{E}}_{2}{\mathcal{E}}_{1}\right)+\beta \mathcal{V}<{\varphi }_{2}{\varphi }_{1}\) , \({\mathcal{C}}_{y1}{\mathcal{C}}_{y2}<A{D}_{1}\alpha \mathcal{W}\) , the evolutionary game stabilizes the strategy as (collaborative lowcost distribution, rewardpunishment combination policy, with lowcarbon preference). In this scenario, the enterprisealliance group will prefer collaborative lowcost distribution because the revenue subsidies to the enterprisealliance from the government and endusers are smaller than the revenue difference between the enterprisealliance performing lowcost distribution and lowcarbon distribution. The government will prefer reward and punishment combination policy because the government receives a penalty for lowcarbon distribution, which is greater than the difference between the government's costs of adopting the two policies.
Scenario 3 \(\left(1,0,1\right)\) : When \(\left({\mathcal{C}}_{x2}{\mathcal{C}}_{x1}\right)+{\mathcalligra{m}}\left({\mathcal{E}}_{2}{\mathcal{E}}_{1}\right)+\beta \mathcal{V}>{\varphi }_{2}{\varphi }_{1}\),\(S+{D}_{1}\alpha \mathcal{W}<{\mathcal{C}}_{y1}{\mathcal{C}}_{y2}\) and \(\beta \mathcal{V}+\delta <{D}_{1}+{D}_{2}\),the stable strategy of the evolutionary game is (collaborative lowcarbon distribution, flow in the form of policy, with lowcarbon preference). In this scenario, although the enterprisealliance does not have government incentives, the endusers with lowcarbon preference is willing to pay a certain distribution cost. The income of the enterprisealliance's collaborative lowcarbon distribution is greater than income of the collaborative lowcost distribution, so the enterprisealliance group still chooses the collaborative lowcarbon distribution. When the government adopts flow in the form of policy, the incentives for endusers are greater than the costs that endusers with lowcarbon preference need to pay, so the endusers group will gradually have lowcarbon preference and support the enterprisealliance in collaborating on lowcarbon distribution.
Scenario 4 \(\left(1,1,0\right)\) : When \(\left(\text{S}\text{A}\right)+\left({\mathcal{C}}_{x2}{\mathcal{C}}_{x1}\right)+{\mathcalligra{m}}\left({\mathcal{E}}_{2}{\mathcal{E}}_{1}\right)<{\varphi }_{2}{\varphi }_{1}\),\(S+\gamma L{D}_{2}>{\mathcal{C}}_{y1}{\mathcal{C}}_{y2}\),\(\text{and }\alpha \mathcal{W}+\gamma L<\beta \mathcal{V}+\delta\) , the stable strategy of the evolutionary game is (collaborative lowcarbon distribution, reward and punishment combination policy, without lowcarbon preference). In this scenario, the reward received by the endusers are less than the costs they need to bear to support lowcarbon distribution, so the endusers group's lowcarbon preference will gradually decrease until it disappears. Although the enterprisealliance does not support of endusers, the benefit of collaborative lowcarbon distribution is still greater than benefit of collaborative lowcost distribution under the government subsidy, so the enterprisealliance group will choose collaborative lowcarbon distribution. Similarly, the benefit to the government of adopting reward and punishment combination policy is greater than benefit of adopting flow in the form of policy, so the government group will choose to adopt policy of reward and punishment combination.
Scenario 5 \(\left(1,0,0\right)\) : When \(\left({\mathcal{C}}_{x2}{\mathcal{C}}_{x1}\right)+{\mathcalligra{m}}\left({\mathcal{E}}_{2}{\mathcal{E}}_{1}\right)>{\varphi }_{2}{\varphi }_{1}\),\(\gamma LS{D}_{2}<{\mathcal{C}}_{y1}{\mathcal{C}}_{y2}\),\(\text{and }{D}_{1}+{D}_{2}<\beta \mathcal{V}+\delta\) , the evolutionary game stabilizes the strategy (collaborative lowcarbon distribution, flow in the form of policy, without low carbon preference). In this scenario, the government subsidy to endusers is less than their cost of supporting lowcarbon distribution, so the endusers group's lowcarbon preference fades. The benefit to the enterprisealliance of undertaking collaborative lowcarbon distribution is greater than benefit of undertaking collaborative lowcost distribution, so the enterprisealliance group will undertake collaborative lowcarbon distribution.
Scenario 6 \(\left(0,0,1\right)\) : When \(\left({\mathcal{C}}_{x2}{\mathcal{C}}_{x1}\right)+{\mathcalligra{m}}\left({\mathcal{E}}_{2}{\mathcal{E}}_{1}\right)+\beta \mathcal{V}<{\varphi }_{2}{\varphi }_{1}\),\(A+{D}_{1}\alpha \mathcal{W}<{\mathcal{C}}_{y1}{\mathcal{C}}_{y2}\),the stable strategy of the evolutionary game is (collaborative lowcost distribution, flow in the form of policy, with lowcarbon preference). In this scenario, although the endusers have lowcarbon preference, the cost they are willing to pay is lower, and the enterprisealliance adopts the flow in the form of policy, therefore the enterprisealliance group will choose the collaborative lowcost distribution based on the principle of profit maximization. The benefits to the government of adopting reward and punishment combination policy are lower than those of adopting flow in the form of policy, so the government group will also choose to adopt flow in the form of policy.
Data simulation
To verify the validity of the evolutionary stability analysis, more intuitively present the evolutionary stability process and explore the influence of relevant factors on the strategic behavior of each subject, this paper uses MATLAB R2021 (Full name of software and Version number: Matlab R2021a Win64 Crack. URL link: https://matlab.waltzsy.com/) software to carry out sensitivity analysis and numerical simulation on the equilibrium strategies of each subject^{44}. At the same time, the parameters involved in the replicated dynamic equation system assumed to meet the corresponding conditions of different evolutionary stable states, to make them as close as possible to the actual case situation and to representatively present the model's evolution of the game laws and characteristics^{45,46}.
First, assume that array 1 satisfies Case 1: \({\mathcal{C}}_{x1}\)=50，\({\mathcal{C}}_{x2}\)=40，\({\mathcal{E}}_{1}=50\) ，\({\mathcal{E}}_{2}=35\)，\({\varphi }_{1}=300\)，\({\varphi }_{2}=250\)，\({\mathcalligra{m}}=0.5\)，\(S=5\)，\(A=1\)，\({\mathcal{C}}_{y1}=15\)，\({\mathcal{C}}_{y2}=20\)，\(\alpha =0.1\)，\(\mathcal{W}=10\)，\(\gamma =0.7\)，\(L=10\)，\({D}_{1}=4\)，\({D}_{2}=1\)，\(\beta =0.5\)，\(\mathcal{V}=6\)，\(\delta =1\). The array 1 \(,x\),\(y\) and \(\text{z}\) evolutionary simulation routes under different initial value states are explored to verify the validity of the parameters. The initial values of \(x\),\(y\) and \(\text{z}\) are set to 0.2, 0.5, and 0.7, respectively, and the results are shown in Figs. 6, 7 and 8: according to the model results, the initial values of \(x\),\(y\),\(\text{z}\) converge to 1 in the end, which is consistent with the evolutionary equilibrium point of case 1 \(\left(1,1,1\right)\) , therefore the model and parameters are valid.
Analysis of the impact of different strategy choices of multiple subjects on the outcome of the evolutionary game
The data simulation of the six evolutionary stabilization scenarios proposed in “Game model solving and analysis” is shown in Figs. 9, 10, 11, 12, 13 and 14: from the evolutionary results, it can be seen that array 1 satisfies the conditions of scenario 1, and at this time, there exists an evolutionary equilibrium point of the system \(\left(1,1,1\right)\), the combination of the strategies of the enterprisealliance, the government, and the enduser is (collaborative lowcarbon distribution, reward and punishment combination policy, with lowcarbon preference). By changing the value of \({\varphi }_{1}\) the value of the array 2 is adjusted to obtain array 2 satisfies the following case 2 conditions:\({\mathcal{C}}_{x1}\)=50, \({\mathcal{C}}_{x2}\)=40, \({\mathcal{E}}_{1}=50\) , \({\mathcal{E}}_{2}=35\), \({\varphi }_{1}=255\), \({\varphi }_{2}=250\), \({\mathcalligra{m}}=0.5\), \(S=5\), \(A=1\), \({\mathcal{C}}_{y1}=15\), \({\mathcal{C}}_{y2}=20\), \(\alpha =0.1\), \(\mathcal{W}=10\), \(\gamma =0.7\), \(L=10\), \({D}_{1}=4\), \({D}_{2}=1\), \(\beta =0.5\), \({\mathcal{V}}=6\), \(\delta =1\) . According to the simulation results Fig. 9, the evolutionary equilibrium point of the system at this time becomes \(\left(0,1,1\right)\), and the strategy combinations of enterprisealliance, government and endusers are (collaborative lowcost distribution, reward and punishment combination policy, with lowcarbon preference). Therefore, when the revenue difference between collaborative lowcost distribution and collaborative lowcarbon distribution of enterprisealliance is relatively small, it will tend to choose collaborative lowcarbon distribution, on the contrary, if the revenue difference is large, enterprisealliance will tend to choose collaborative lowcost distribution. This result suggests that enterprisealliance are not only seeking to maximize economic benefits, but are also willing to take part of the social responsibility of reducing carbon emissions. However, when the difference between the two revenues is large, the enterprisealliance will still choose collaborative lowcost distribution by focusing on economic benefits. The simulation analysis is consistent with the conclusions of the stability analysis of each party's strategy and has the validity, which is of practical significance in guiding the choice of distribution strategy for enterprise alliances (other cases will not be verified for reasons of space).
The impact of supplyside government policies on enterprisealliance distribution strategies
Since the combination of strategies in Case 1 (collaborative lowcarbon distribution, reward and punishment combination policy, with lowcarbon preference) is a desirable state of social operation, based on Case 1, the impact of government reward on enterprisealliance distribution strategies is further discussed. When \(S=(1,5,10),\) the results of 100 simulations of the replicable dynamic equation system evolving over time are shown in Fig. 15: in the process of the system evolving to a stable point, an increase in the government reward can accelerate the evolution of the enterprisealliance to carry out collaborative lowcarbon distribution. Enterprisealliance will engage in collaborative lowcarbon distribution in order to maintain their reputation with the government and endusers because of the rewards they receive from the government. The stronger the incentives from the government, the more stable the strategy of the enterprisealliance in choosing collaborative lowcarbon distribution. But with an increase in the government's financial burden, the government's financial burden increases, so the evolution of the government's policy of combining reward and punishment slows down when \(S=10\) the financial burden is too heavy, and the government will turn to flow in the form of policy by reducing the cost of regulation or reducing incentives.
Similarly, when enterprisealliance collaborate on lowcost distribution, and government punishment \(A=(1,10,20)\), the simulation results of replicating the system of dynamic equations evolving over time 100 times are shown in Fig. 16. During the evolution of the system to the stable point, an increase of government punishment can accelerate the evolution of the enterprisealliance to carry out collaborative lowcarbon distribution, and the enterprisealliance will continue choosing collaborative lowcarbon distribution in this case as long as the government's punishment for collaborative lowcost distribution is \(A>0\).When the government punishment collaborative lowcost distribution, the enterprisealliance will always choose collaborative lowcarbon distribution in this case. This suggests that government punishment have a greater impact on enterprisealliance's strategy choices than reward, because government punishment affect enterprisealliance 's reputation in the industry as well as its credit rating.
The impact of demandside endusers’ behavior on the distribution strategies of enterprisealliance
Additionally, based on Case 1, we analyze the impact of endusers behavior on the distribution strategy of the enterprisealliance, and first explore the impact of endusers with lowcarbon preference who are willing to pay the lowcarbon distribution subsidy coefficient \(\beta\).The influence of the value of the subsidy coefficient of lowcarbon preference is investigated \(\beta =0.5\) .On the basis of \(\beta =0.1\) and \(\beta =1.2\) , the simulation results of 100 repetitions of the dynamic equation system evolving over time are shown in Fig. 17. According to the evolution results, when \(\beta =0.1\text{and}0.5\), the endusers will tend to choose the lowcarbon preference; when the subsidy coefficient \(\beta\) is lower, the endusers with lowcarbon preference consider it acceptable to spend extra distribution costs. However, when \(\beta =1.2\), endusers no longer support the enterprise alliance in lowcarbon distribution from the perspective of maximizing their own interests because the lowcarbon distribution subsidy they have to pay is too high. It follows that the behavior of endusers is influenced by the costs they need to bear, and above a certain threshold they will no longer support enterprisealliance for collaborative lowcarbon distribution.
Similarly, the simulation results of replicating the dynamic system of equations with time evolution 100 times when the time cost δ = (1, 3,6) that the endusers need to pay to support lowcarbon distribution are shown in Fig. 18. From the evolution results, it can be seen that, when δ = 1 and 3, the endusers will tend to choose to support the enterprisealliance in carrying out lowcarbon distribution; But when δ = 6, the endusers will not support the enterprisealliance in carrying out lowcarbon distribution due to the high time cost that they need to pay. The current enduser requirements for the speed of logistics and distribution are increasingly high, if the enterprisealliance in the collaboration of lowcarbon distribution at the same time, logistics efficiency has not been improved, it will lead to the enduser's lowcarbon preference is reduced until there is none.
Conclusions and implications
This study aims to explore the dual effects of government reward and punishment mechanisms and endusers lowcarbon preference on the choice of distribution strategies of enterprisealliance. Through the governmentled digitalization platform, the accurate accounting of cost data and carbon data of each distribution enterprise is realized, and the carbon trading market is effectively docked. Lossofbenefit and carbon emission quantification models of collaborative lowcarbon distribution modes and collaborative lowcost distribution modes are established, a dual incentive strategy considering government reward and punishment and enduser subsidies is proposed, and the effectiveness and feasibility of the models and strategies are verified through numerical experiments The main conclusions of this paper are as follows.

(1)
From the viewpoint of supplyside policy, the government should macroscopically control the reward and punishment mechanism according to the actual situation of the market^{47}. Blindly increasing the reward will impose a heavy financial burden to the government, and with the development of the carbon trading market and increase in the carbon price, the government will gradually withdraw from the reward mechanism and play a major role in punishment. Compared with reward, strict punishment have a more obvious effect on the regulation of enterprisealliance, so the government should consider local actuality, adopt a policy combination of dynamic reward and static punishment, further improve the construction of the carbon market, guide enterprises to actively participate in carbon trading, and give full play to the role of the carbon market.

(2)
Regarding demandside endusers’ behavior, the enduser's support for the lowcarbon distribution of the enterprisealliance has a threshold, and after exceeding the threshold, the endusers will no longer support the enterprisealliance in carrying out lowcarbon distribution from the perspective of maximizing selfinterest, so the enterprisealliance should reasonably formulate the cost of lowcarbon distribution, and guide the endusers to participate in the construction of the carbon market^{48}.
The research presented in this paper further improves standard emissions and designs reasonable reward and punishment mechanisms^{49}. However, the choice of enterprisealliance strategies in real situations is often influenced by other constraints, while the evolutionary game model in this study is based on assumptions and the choice of parameters is idealized. Future research directions should combine the actual situation of distribution development and comprehensively consider the multiple factors involved in enterprisealliance distribution.
Data availability
The datasets used and analysed during the current study available from the corresponding author on reasonable request.
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Wenxue Ran: Conceptualization, Funding acquisition, Project administration, Dandan He: Conceptualization, Methodology, Software, Writingoriginal draft, Zhaoxia Li: Data curation, Writingoriginal draft, Yun Xue: Data curation, Writingoriginal draft, Zhenzhen He: Methodology, Software, Writingoriginal draft, Aravinda Dananjaya Basnayaka Basnayaka Gunarathnage: Writingoriginal draft, Resources.
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Ran, W., He, D., Li, Z. et al. Research on distribution strategy of logistics enterprise alliance based on threeparty evolution game. Sci Rep 14, 14894 (2024). https://doi.org/10.1038/s41598024657239
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DOI: https://doi.org/10.1038/s41598024657239
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