Quantum Game-Based Study on the Incentive Mechanism for the Cooperative Distribution of E-Commerce Logistics Alliance

. Motivating active participation in e-commerce logistics alliances to enhance delivery efciency and customer satisfaction has long been a societal interest. Leveraging the quantum game theory, this paper develops a model for incentivizing collaboration within these alliances. Tis model enables theoretical and numerical analysis of members’ strategies and entanglement levels. Te fndings show that quantum strategies increase members’ profts, achieving Nash equilibriums and Pareto optimal outcomes, out-performing the classical game theory. In addition, the size of quantum entanglement emerges as a critical determinant infuencing members’ active participation in collaborative distribution. Strengthening information sharing and aligning interests can enhance entanglement levels among members, making them more inclined to adopt strategies promoting active involvement in collaborative distribution. Moreover, members can adapt their strategies based on the initial entanglement in collaborative distribution, thereby incentivizing participation and reducing ethical risks. In conclusion, through numerical analysis, we present relevant strategies and recommendations for incentivizing collaborative distribution within e-commerce logistics alliances.


Introduction
Te logistics industry faces rising demand driven by ecommerce growth but noticeable inefciencies and bottlenecks persist.Te fourth-party logistics (4PL) platform functions as a coordination centre derived from the thirdparty logistics (3PL) platform, integrating resources across the supply chain to ofer customers efcient and satisfactory services.However, investing extensively in a single 4PL platform for goods delivery is impractical.To improve business efciency, establishing a close cooperative relationship between the e-commerce platform and the 4PL platform becomes imperative.Tis led to the formation of ecommerce logistics alliances, efectively consolidating customer information and logistics resources among members while improving overall operational efciency [1].Despite these advantages, the practical adoption of the alliances' operational models remains uncommon due to proftfocused members, resulting in issues such as unfair proft distribution, free-riding, and moral hazards [2].Tese challenges, arising from conficts between short-term individual interests and long-term collective interests, are commonly known as social dilemmas.Consequently, motivating active participation in collaborative distribution within e-commerce logistics alliances emerges as a pressing issue in the logistics and e-commerce sectors.
Scholars globally have advanced the study of social dilemmas and supply chain management signifcantly.Addressing social dilemmas, Ariful Kabir et al. [3,4] investigated factors infuencing network reciprocity's impact, exploring their role in promoting cooperative behavior.Rajib Arefn et al. [5] identifed a dual relationship between the dilemma strength and variations in social efciency defcits.In supply chain management, Zhang et al. [6] examined asymmetric information's efects on retailer incentive contract design, considering manufacturer process innovation costs.Du et al. [7] developed an evolutionary game model for cross-border e-commerce platforms and logistics enterprises' information coordination.He et al. [8] compared logistics integration strategies in e-commerce platform service supply chains using game models.Niu et al. [9] constructed a game model for logistics sharing alliances among competitive e-commerce companies.Wang et al. [10] explored government dynamic punishment and incentive mechanisms' impact on trust evolution between platform ecommerce and consumers.Du et al. [11] designed an incentive model for cooperative distribution alliances, considering moral risk.
In collaborative distribution within e-commerce logistics alliances, members oversee and share information to enhance service quality, resource integration, and overall alliance efciency, akin to entanglement in quantum mechanics that denotes correlation between observable values of diferent subsystems [12].Te quantum game theory, situated at the intersection of the quantum information theory and game theory, emerged as a distinct feld with Meyer [13] introducing the concept in 1999.Eisert et al. [14] quantized the prisoner's dilemma model, demonstrating the capability of the quantum game theory to resolve traditional game dilemmas.Subsequently, the quantum game theory has garnered global scholarly attention, with increasing research in the feld.Gender game quantization addresses Nash equilibrium point selection issues [15].Quantum strategies have been found to outperform Nash equilibrium strategies in numerous instances when comparing classical games to quantum games [16].Experimental evidence supports the practical efcacy of quantum games, not merely a theoretical result [17,18].Quantum game models established through EPR-type experiments exhibit a more direct connection with classical games [19].Te stability of the quantum Nash equilibrium increases with the increase of quantum entanglement, as found in the quantization Stackelberg duopoly game model [20].Recently, quantum games have found application in economic investment, management decision-making, and supply chain management, with a particular emphasis on collaborative cooperation.In economic investment, quantum games have been used to address the risk exit dilemma in the fnancial investment market, analyze strategic choices for outward investment by venture capitalists and entrepreneurs, and provide new insights into mechanism design, auction, and contract theory [21][22][23].In management decision-making, scholars have applied quantum games to examine alliance formation in production competition, collaboration in innovation involving industry, academia, and research, and cooperation among diverse governments in environmental governance [24][25][26].Research on supply chain management indicates that quantum game models with distinct characteristics better guide decisionmaking, pricing, and cooperative incentive issues in the supply chain [27][28][29][30].
Based on the analysis of relevant literature, this research, focusing on the e-commerce logistics alliance, extends beyond the classical game theory and adopts the quantum game theory framework.Quantum game theory, distinct from the classical game theory, stands out due to features such as superposition and entanglement and proves more efective in managing cooperation, resolving dilemmas and infuencing equilibrium outcomes.Hence, this paper employs the quantum game theory to explore incentive issues in collaborative distribution within e-commerce logistics alliances.Tis approach is taken to examine the strategic actions of alliance members and assess their expected payofs.In contrast to prior studies, the main contributions of this paper are as follows: (1) Tis paper focuses on e-commerce logistics alliances as its research subject and constructs both classical and quantum game models.Tese models are used to analyze equilibrium solutions and changes in expected payofs for alliance members in both scenarios.
(2) Te paper investigates the efects of diferent initial entanglement values in the quantum game model on alliance members' strategic choices and their corresponding changes in expected payofs, along with an analysis of the corresponding critical conditions.(3) Trough numerical analysis experiments, it examines the impact of diferent quantum strategies and entanglement values on the profts of members within e-commerce logistics alliances.Based on the analytical results, the paper provides recommendations for efectively incentivizing the active participation of alliance members in collaborative distribution.
Te rest of this article is organized as follows.Section 2 introduces the problem description and model assumptions, establishing the classical game model.Section 3 develops the quantum game model, examining the impact of quantum strategies on alliance collaborative distribution in both nonentangled and entangled states.Section 4 carries out numerical analysis, evaluates the parameters of the quantum game model, discusses the research results, and gives corresponding management suggestions.Finally, Section 5 provides a comprehensive summary and identifes directions for future research.

Research Background and Fundamental Assumptions
During e-commerce logistics alliances' cooperative distribution, parties' efort levels remain challenging to accurately observe due to implicit investments.For example, gauging e-commerce platforms' costs in customer and merchant maintenance proves difcult, while the 4PL platform struggles to discern the e-commerce platform operators' eforts in handling customer inquiries.Hence, the collaborative distribution process is not a deterministic "full efort-no efort" binary strategy set game; efort levels should be treated as a continuous variable.Tis concept resembles the quantum mechanics notion of superposition, prompting the adoption of a quantum game analysis framework for studying collaborative distribution in e-commerce logistics alliances.In this cooperative distribution game, the following hypotheses are posited: 2 Discrete Dynamics in Nature and Society Hypothesis 1. the e-commerce logistics alliance is a complete ecosystem where members exhibit bounded rationality and possess learning capabilities.Tey aim to maximize their interests by selecting and modifying strategies.
Hypothesis 2. this study only considers the benefts and common costs generated by the cooperative allocation process between the alliance parties.
Hypothesis 3. the cooperative distribution benefts and total cooperative distribution costs in the e-commerce logistics alliance remain constant.Alliance parties share a fxed net proft and common costs based on specifc allocation coefcients.
Hypothesis 4. the study does not account for the infuence of entities outside the e-commerce logistics alliance, which are not the subject of this research.
During the collaborative distribution process within ecommerce logistics alliances, the e-commerce platform is denoted as E, and the 4PL platform is denoted as F. Tis paper introduces variables e E , e F as the efort degree of the ecommerce platform and 4PL platform to participate in collaborative distribution tasks, where e i � 0, 1 (i � E, F, 0 indicates no efort, and 1 indicates full efort).Assuming that the fnal revenue function of the alliance is the Cobb Douglas type [31], i.e., where A is the output coefcient of cooperative distribution, and α(0 < α < 1) and 1 − α represent the elasticity of efort utility of e-commerce platforms and 4PL platforms, respectively, which are used to measure the contributions of both parties' eforts.ε is the random disturbance term subject to normal distribution.Te total cost of cooperative distribution is recorded as C E and C F , and the cost coefcients of cooperative distribution are ω E and ω F , respectively.As the efort degree increases, the cost also increases.Here, we use the quadratic function model, that is, the cost function is proportional to the square of the degree of efort. ( Te profts obtained by both parties during the cooperative distribution process are recorded as R E and R F , and the distribution of profts is in a linear form, that is, R E � (1 − β)π and R F � βπ, where β is the proft distribution coefcient.θ i (i � E, F, θ i � 0, 1) is viewed as the level of efort of e-commerce platforms and 4PL platform, and the corresponding relationship with e i given by ( Te expected payof function of alliance members (e-commerce platform and 4PL platform) is as follows: Within the classical game theory framework, this study investigates the game between an e-commerce platform and a 4PL platform based on the aforementioned assumptions.Te payof matrix is presented in Table 1, while Table 2 provides the parameters and symbols for the game model between the two parties within the alliance.
From the payof matrix, it is evident that the "full efort" strategy adopted by both parties constitutes the only Pareto optimal outcome in this game.However, there exist two pure strategy Nash equilibrium points in which both parties opt for either "full efort" or "no efort."If one party exerts complete efort while the other party does not exert any efort, the former not only bears the cost of their own efort but also must face a situation where the latter gains no proft due to "betrayal."Tis poses a great risk for the exerting party, especially in projects that require signifcant investment (when C E and C F are large).Te challenge for this paper is to fnd a solution that enables alliance members to achieve Pareto optimality while avoiding the risk of potential betrayal for the exerting party.

Quantum Game Models
Based on the preceding discussion and analysis, this section aims to examine the distinctive features of quantum games compared to classical games.Specifcally, the Eisert-Wilkens-Levenstein (EWL) quantum game scheme will be employed to investigate the infuence of quantum entanglement on the earnings of alliance members and to highlight the disparities between quantum strategies and classical strategies.

Model Construction.
Under the theoretical framework of quantum computing, the participant game process is described as the process of accepting, manipulating, and measuring quantum bits, which is actually the information processing process in the game, i.e., the state transition process of the game.Alliance members continuously adjust their own strategies based on the observed situations during the game between the two parties to better achieve an Discrete Dynamics in Nature and Society evolutionary stable state.Te main idea of the EWL scheme is as follows [14], and the specifc quantization process is shown in Figure 1: Within the framework of EWL quantum games, each member can be described as a qubit in a two-dimensional Hilbert space, represented by state vectors |0〉 � (1, 0) T and |1〉 � (0, 1) T .Initially, each member is in the state represented by |0〉.Subsequently, the general entanglement gate  J is applied to achieve the following: (1) Quantumizing classical game problems, within the framework of the quantum game theory, the two classical game strategies "full efort" and "no efort at all" correspond, respectively, to two polarized states |0〉 and |1〉 in a two-dimensional Hilbert space (i.e., θ i � 0 and θ i � 1 in classical games).Te initial strategies of the game are expressed through tensor product states of quantum bits, denoted as |00〉, |01〉, |10〉, |11〉, representing four possible combinations (the frst digit representing the ecommerce platform, and the second digit representing the 4PL).Tis paper assumes that both parties initially adopt the strategy of "full efort," denoted as |0〉, which means that the initial quantum states for both sides are represented as , where ⊗ signifes the tensor product.(2) Tis paper discusses the EWL quantum game model in the two-parameter case as shown in Figure 1.Tat is, the strategies of the e-commerce platform and 4PL platform are unitary operator U E and U F .Te quantum strategies selected by each member of the cooperative distribution alliance are shown as follows: Te strategy space is composed of a two-parameter set ) is the efort degree parameter and φ i is the cooperative distribution capability parameter displayed of each member.e iφ i is the complex phase, cos(θ i /2) is the amplitude, and the product of the two gives the probability amplitude of the quantum strategy.In this context, it can be understood as the probability amplitude of benefts obtained by alliance members in the cooperative distribution process when they choose the quantum strategy.Strategies  U(0, 0) and  U(π, 0) are referred to as "full efort" strategies and "no efort" strategies.Te general quantum strategy is denoted by  U(θ, φ) when 0 < φ < π/2.(3) Suppose that the default entanglement operator of alliance members is  J, Table 1: Te proft matrix from the perspective of classical game.

Strategies 4PL platform Full efort 0
No efort 1 E-commerce platform Full efort 0 Table 2: Main parameters and their meanings.
Parameters meanings e i Alliance member's eforts level in cooperative distribution 0 Te degree of quantum strategy adopted by alliance member's Discrete Dynamics in Nature and Society Here, is a variant of the Pauli-x matrix, I is the 4 × 4 identity matrix, and c represents the entanglement degree between the two players (c ∈ [0, π/2]).When c � 0, the state is unentangled; in other words, both parties are completely unafected by each other when they play the game.When c � π/2, the entanglement is maximized, that is, the strategy chosen by both parties and the action information transparent to each other.By solving for the entanglement operator  J, we can obtain the initial state |ψ 0 〉 as follows: After one round of the game, the state becomes (U E ⊗ U F )  J |00〉.(4) Te antientanglement operator  J † can be solved according to the entanglement operator  J: Te fnal state is obtained by the antientanglement operator According to the collapse property of quantum measurement, the fnal state |ψ f 〉 is observed, and it will randomly collapse into one of the four basis vectors |00〉, |01〉, |10〉, |11〉.Te probabilities of each result are as follows: which is calculated as P 00 + P 01 + P 10 + P 11 � 1.According to the above results, combined with equations (4) and ( 5), the expected revenue function of the ecommerce platform and 4PL platform can be expressed in the following form: . (11)

Nonentangled State.
Under the nonentangled state (c � 0), the expected payof of the e-commerce platform and the 4PL platform are as follows: From equation (12), it is clear that the e-commerce platform and the 4PL platform's expected benefts rely solely on the parameter θ, implying that the cooperation eforts of each platform directly afect their expected benefts.Te following propositions outline how each platform chooses its strategy based on the cooperative eforts of the opponent: Proposition 5.If the alliance members are in a nonentangled state, the following can be inferred: , the expected revenue ER E for the e-commerce platform decreases with θ E , while Proof.See Appendix A.1.Proposition 5 shows that insufcient efort from one party does not harm its own benefts but does not improve them either.Positive correlation between efort and benefts only occurs when one party's efort is substantial.In a nonentangled state, an alliance member might engage in free riding for benefts.Table 3 illustrates this with four strategies and expected benefts for both platforms.Entanglement is introduced to address this, and the discussion in the following explores its impact on the game process.□ 3.3.Entangled State.In this situation (0 < c < π/2), the members of the e-commerce logistics alliance are in an entangled state.For the convenience of mathematical computation, this paper considers the case of maximum entanglement, denoted as c � π/2.

Proposition . Under the condition of maximum entanglement c � π/2, if the e-commerce platform adopts a nonquantum strategy 􏽢
U E (θ E , 0), the sufcient and necessary condition for ER E will decrease with θ E , that is, 0 holds simultaneously with sin 2 φ F cos 2 θ F /2 ≥ 0 and neither takes the value of " � " at the same time; in other words, ER E increases with the efort level e E .Currently, the e-commerce platform's optimal strategy is to make full eforts θ E � 0. Similarly, if the 4PL platform adopts a nonquantum strategy  U F (θ F , 0), the sufcient and necessary condition for ER F will decrease with θ F , that is, (βAε − ω F /2)cos 2 φ E cos 2 θ E /2 − ω F /2 sin 2 θ E /2 ≥ 0 holds simultaneously with sin 2 φ E cos 2 θ E /2 ≥ 0 and neither takes the value of " � " at the same time; in other words, ER F increases with the efort level e E .Currently, the optimal strategy for the 4PL platform is to make full eforts θ F � 0.
Proposition 6 indicates that within the collaborative distribution alliance, when the 4PL platform does not make eforts, the e-commerce platform must exhibit specifc cooperative distribution capabilities and invest eforts to incentivize the 4PL platform.□ Proposition 7.Under the condition of maximum entanglement c � π/2, if the e-commerce platform adopts a fully quantum strategy  U E (θ E , π/2), then the sufcient and necessary condition for ER E increase with e E is sin φ F cos 2 θ E /2 > 0, and the optimal strategy for the ecommerce platform is to exert full efort  U E (0, π/2).Similarly, if the 4PL platform adopts a fully quantum strategy  U F (θ F , π/2), then the sufcient and necessary condition for ER F increase with e F is sin φ E cos 2 θ F /2 > 0, and the optimal strategy for the 4PL platform is also to exert full efort  U F (0, π/2).
Proposition 7 shows that when both e-commerce and 4PL platforms collaborate actively within the alliance, and member interests rise with improved collaborative distribution capabilities and eforts.To visually illustrate the impact of quantum strategies on the expected returns of alliance members under entanglement, we further analyze four specifc strategies as presented in Table 4.
From Table 4, it can be seen that among the six Nash equilibrium points, only (  U E (0, 0),  U F (0, 0)) and (  U E (0, π/2),  U F (0, π/2)) can bring payofs to both parties, and the strategies  U E (0, π/2) and  U F (0, π/2) are the Pareto optimal in the scenario of maximum entangled state, that is, the situation where both parties adopt "complete quantum strategy for fully efort."If the e-commerce platform adopts the strategy  U(0, π/2) in the entangled state, no concern is required regarding the passive stance of the 4PL platform, as it can eliminate the betrayal risk resulting from the no-eforts party.It signifes that the 4PL platform will bear its own losses.Tis implies that when both platforms employ the quantum strategy in an entangled state, free-riding conduct can be efciently averted, and alliance members can be encouraged to actively engage in cooperative distribution, resulting in mutually benefcial cooperation.

Numerical Simulations
In this section, numerical analysis was performed using MATLAB R2022b software to investigate the infuence of quantum strategies and entanglement on the expected returns of the alliance members and (A, β i , ε, ω i ) � (70, 0.6, 0.3, 6).Specifcally, the parameter ε was varied to analyze its impact on the members' returns.It was found that when ε � 0.3, the resulting graph provided the clearest depiction and efectively illustrated the underlying dynamics.

Te Impact of θ on the Alliance Members' Profts Given φ.
In this section, we analyze the infuence of θ on the alliance members' profts when φ � 0, π/2. Figure 2 shows the infuence of θ on the profts of both platforms in the nonentangled state, while Figure 3 presents the impact of θ on the members' profts in the entangled state.

Nonentangled State.
Under the nonentangled state (c � 0), the members' returns in the alliance are solely dependent on their respective levels of efort, considering the case of the e-commerce platform, as shown in Figure 2. Figure 4 shows the two-dimensional profle of the impact on the payofs of alliance members when the efort degree θ i reaches θ i � 0 to θ i � π, fve diferent values in the nonentangled state.Figure 5 further details the impact on alliance members when the efort degree is θ i � 0 and θ i � π.
It can be seen from Figures 2, 3, and 5 that (1) in the process of θ F increasing and approaching π/2, the maximum value ER E decreases from 5.4 to 1.2 and decreases with θ E , that is, the payof of the e-commerce platform increases with its own eforts; (2) in the process of θ F increasing and approaching from 3π/4 to π, the minimum value of ER E decreases from about −1.77 to −3 and increases with θ E ; in other words, the proft of the e-commerce platform will decrease with its own eforts; (3) when θ F is approaching π, ER E decreases with θ F , that is, the payof of the e-commerce platform increases with the efort of 4PL platform; (4) the trend shown in Figures 4 and 5 accords with the hypothesis of Proposition 5, where the critical point θ * F is between π/2 and 3π/4.In other words, before the critical point, the revenue of the e-commerce platform increases with its efort level.After reaching the critical point, the revenue of the e-commerce platform decreases with its efort level.Tis critical point is primarily infuenced by the efort level of the 4PL platform.Te same is true of the relationship between ER F and θ F .

Entangled State. Under quantum entanglement
(c � π/2), member's profts are related to their efort level and cooperative distribution capabilities.As in the entangled state, Figures 3, 6, and 7 also show the relationship between θ and φ in the entangled state.
From Figures 3, 6, and 7, it can be observed that (1) in the entangled state, as the efort degree increases from θ F � 0 to θ F � π, ER E decreases with θ E , and the decreasing amplitude of ER E tends to be gentle with the increase of θ F ; (2) in the process of θ F � 0 increasing to θ F � π, ER E from the maximum 5.4 gradually decreased to about 1 × 10 −32 , that is, the payof of the e-commerce platform not only increases with its own eforts but also increases with the eforts of 4PL platform; (3) the e-commerce platform's payof increases with its own efort, and this increase is positively correlated with the efort of the 4PL platform.In the maximum entangled state, the e-commerce platform is not burdened with the cost of the 4PL platform's lack of efort.Te same is true of the relationship between ER F and θ F .
Figure 8 reveals the following observations: (1) when the e-commerce platform adopts the strategy  U E � (θ E , 0) (cooperative distribution capacity is 0), the optimal strategy for the 4PL platform is  U F � (0, π/2).In this scenario, as θ E approaches 0, the payof of the e-commerce platform decreases with θ F .In other words, the revenue of the ecommerce platform increases with the efort level of the e-commerce platform and 4PL platform.At this point, the optimal strategy for the 4PL platform is to "fully exert efort and demonstrate the maximum collaborative distribution capability; " (2) when the e-commerce platform adopts the strategy  U E � (θ E , π/2) and the 4PL platform chooses the strategy  U F � (θ F , 0), the payof of the e-commerce platform remains unafected with the efort level of the e-commerce platform increasing or decreasing; (3) when the e-commerce platform adopts the strategy  U E � (θ E , π/2) and 4PL platform chooses the strategy  U F � (θ F , π/2), as θ F approaches 0, the payof of the e-commerce platform will decrease with θ E .In other words, the income of the e-commerce platform will increase with the degree of efort.Terefore, the optimal strategy for the 4PL platform is  U F � (0, π/2).Likewise, the optimal strategy for the e-commerce platform is  U E � (0, π/2).

Te Impact of φ on the Alliance Members' Profts Given θ.
Tis section considers the impact of parameters φ on the revenue of alliance members when θ i � 0, π, as shown in Figure 9. Since when (θ E , θ F ) � (π, π), the income of both members is 0, the discussion of this case is omitted in this paper.It can be seen from Figure 9 that (1) when both members adopt strategies  U � (0, φ), if one member chooses strategy  U � (0, 0), the other member's returns will decrease with φ; and if one member chooses strategy  U � (0, π/2), the other member's returns will increase with φ. (2) Specifcally, when the e-commerce platform chooses strategy  U E � (0, φ), the revenue of the e-commerce platform will increase with φ E as φ F approaches π/2.In this scenario, the optimal strategy for the 4PL platform is "fully exert efort and demonstrate the maximum collaborative distribution capability  U F � (0, π/2)."(3) If the e-commerce platform chooses strategy  U E � (π, φ), φ E does not have an infuence on the revenue of the e-commerce platform as φ F approaches π/2.At this point, the optimal strategy for the 4PL platform is  U F � (0, π/2).Similarly, it can be concluded that the optimal strategy for the e-commerce platform is  U E � (0, π/2).
Figure 4: Te efect of θ i � 0 to θ i � π on the payofs of alliance members in the nonentangled state.(a) Te impact of variations in the efort level of the e-commerce platform on its revenue under the nonentangled state, considering diferent efort levels of the 4PL platform (represented by curves with distinct colors and shapes corresponding to values 0, π/4, π/2, 3π/4, π).In (b), under the nonentangled state, the infuence of changes in the efort level of the 4PL platform on its revenue is illustrated, while keeping the efort level of the e-commerce platform constant (with values 0, π/4, π/2, 3π/4, π, represented by curves with diferent colors and shapes).

Discrete Dynamics in Nature and Society
As can be seen from Figures 8 and 9, if a member chooses the strategy  U � (0, φ), when φ tends to 0, it means that the member's attitude towards cooperative distribution is "eforts but insufcient ability."In general, regardless of the opponent's strategy, the optimal strategy for alliance members is  U � (0, π/2).Tis means that in the entangled  (c-d), the specifc depiction is provided for the infuence of variations in the efort level of the 4PL platform on its revenue under the nonentangled state, with the e-commerce platform values set at 0 and π, respectively.
Figure 6: Te efect of θ � 0 to θ i � π on the payofs of alliance members in the entangled state.(a) Te impact of changes in the efort level of the e-commerce platform on its revenue under the entangled state, with a given efort level for the 4PL platform (represented by curves with distinct colors and shapes corresponding to values 0, π/4, π/2, 3π/4, π).In (b), the illustration focuses on the infuence of variations in the efort level of the 4PL platform on its revenue under the entangled state, with a given efort level for the e-commerce platform (with values 0, π/4, π/2, 3π/4, π, represented by curves with diferent colors and shapes).
10 Discrete Dynamics in Nature and Society state, the Nash equilibrium of the quantum game is (  U E � (0, π/2),  U F � (0, π/2)) which is also the Pareto optimum in this game.

Te Impact of c on the Alliance Members' Profts Given
Some Specifc Strategies.In this section, the impact of entanglement on the profts of alliance members is examined under specifc strategies.By analyzing Table 5, it is evident that when one platform adopts the strategy "fully exert efort and demonstrate the maximum collaborative distribution capability  U � (0, π/2)" while the other platform does not choose same strategy, the proft of the latter platform decreases with increasing levels of entanglement.

Discussion and Managerial
Insights.Tis section unveils intriguing discoveries.First, in the classical game scenario, where one party contributes maximum efort while the other does not, the fully committed party not only bears its own efort costs but also faces an unproftable outcome due to the other party's "betrayal."In addition, within quantum nonentanglement scenarios, the strategy space for e-commerce platforms and 4PL platforms expands.While results align in both classical and quantum nonentanglement scenarios, they lay the groundwork for analyzing the quantum maximum entanglement state.Consequently, under the quantum maximum entanglement state, the party refraining from efort bears the cost itself rather than shifting it to the fully committed party.Te risk of nonefort-based betrayal can be entirely avoided, efectively reducing free-rider behavior.Ultimately, all alliance members tend to choose the "fully committed complete quantum strategy," benefting both parties and leading to a win-win situation.In contrast, our research indicates that the quantum game theory can yield optimal results.Tis approach posits that the states of the game players are continually evolving, expanding the strategy space for both parties, rendering it more aligned with practical scenarios.Consequently, we conclude that quantum gaming holds certain advantages over classical gaming, as its strategy set is shaped by its unique characteristics of superposition and entanglement.
Based on these fndings, some managerial insights are summarized as follows: Figure 8: Te efect of θ i on the payofs of alliance members in specifc φ i .Under the entangled state, given the values of φ (represented by 0, π/2), Figure 8 illustrates the 3D impact of variations in θ on the alliance members' revenue.In (a-b), under the entangled state, when the ecommerce platform exhibits a collaborative distribution capacity of 0 and the 4PL platform exhibits a collaborative distribution capacity of 0 and π/2, and the diferent efort levels of the e-commerce platform and the 4PL platform, respectively, infuence the revenue of both parties.In (c-d), under the entangled state, when the e-commerce platform exhibits a collaborative distribution capacity of π/2 and the 4PL platform exhibits a collaborative distribution capacity of 0 and π/2, the diferent efort levels of the e-commerce platform and the 4PL platform, respectively, infuence the revenue of both parties.12 Discrete Dynamics in Nature and Society Figure 9: Te infuence of φ on the payofs of alliance members in specifc θ i .(a-b) Te impact of varying collaborative distribution capacities exhibited by the e-commerce platform and the 4PL platform on their respective revenues under the entangled state when (θ E , θ F ) � (0, 0).In (c-d), under the entangled state, when (θ E , θ F ) � (0, 0) and (θ E , θ F ) � (π, 0), the diverse collaborative distribution capacities demonstrated by the e-commerce platform and the 4PL platform afect their respective revenues.
Note. ↑(↓) represents the increase or decrease of each member of the e-commerce logistics alliance with c.
(1) Within the collaborative distribution process of ecommerce logistics alliances, establishing several observable and quantifable evaluation metrics or delegating a third-party institution to defne assessment criteria, such as order completion volume, delivery time, and customer satisfaction rate, can transform implicit eforts into tangible indicators.Tis reduces information asymmetry among alliance members, minimizing the potential occurrence of bilateral moral hazards and reinforcing mutual trust.(2) In addressing the incentive problem of collaborative distribution within e-commerce logistics alliances through the quantum game theory, the key lies in whether due consideration has been given to the efort levels, collaborative distribution capabilities, and entanglement among all alliance members.To ensure sufcient efort and efcient collaboration, an "entanglement contract" can be implemented before the collaborative distribution process.Tis protocol binds the interests of members, enhancing their interconnectedness.However, practical applications also require consideration of other factors, including trust levels among members and the prevailing market conditions, to formulate more comprehensive and rational quantitative metrics and "entanglement contract."

Conclusion
Tis paper investigated the incentive problem in collaborative distribution within an e-commerce logistics alliance.First, by analyzing the costs and benefts of alliance members in the context of collaborative distribution, the quantum game theory was introduced to quantumize the classical game model, achieving Pareto optimality in collaborative distribution within the e-commerce logistics alliance, thus reducing bilateral moral risks.Second, numerical simulations discussed the impact of diferent levels of quantum strategies and various quantum entanglement states on alliance members' strategic choices, providing critical conditions for the quantum game system.Finally, according to the abovementioned analysis, the research results were discussed about previous studies, and some management opinions were put forward.Based on the research content of this paper, the following main conclusions are drawn: (1) Te quantum game theory enhances the classical game theory by expanding the binary strategy sets, introducing quantum entanglement and potentially increasing the earnings of alliance members.It effectively addresses the "prisoner's dilemma" issue within the alliance, achieving consistency between Nash equilibrium and Pareto optimality.Te benefts acquired by both parties in the game are superior when employing quantum strategies compared to classical game.Consequently, alliance members are more motivated to adopt quantum strategies to maximize their individual gains.
(2) According to the simulation results, as entanglement emerges, the likelihood of choosing complete efort strategies increases and also enhancing returns based on efort levels.It is evident that the profts obtained by alliance members in quantum entanglement states during collaborative distribution vary with the levels of efort and collaborative distribution capability.Tis signifcantly mitigates the "free-rider" issue and bilateral moral hazard.Tese fndings highlight the importance of entanglement in promoting cooperative behavior and the advantages of quantum strategies in e-commerce logistics alliances.
While the quantum game model developed in this paper efectively promotes active participation in collaborative distribution among e-commerce logistics alliance participants, it does have certain limitations.First, as highlighted by the research conducted by Khoobkar et al. [32], a comprehensive analysis of stability equilibrium in game theory studies is essential, as it can unveil signifcant advancements of the proposed method over other approaches.Due to the constraints of our study, this paper temporarily examines the infuence of diferent parameters in the quantum game model on participants' interests and decisions.However, a detailed numerical refnement and analysis of stability equilibrium are still part of our forthcoming series of research.Second, within the context of collaborative distribution in e-commerce logistics alliances, exploring alternative quantum game mechanisms can provide a more comprehensive assessment of the performance of quantum games in the collaborative distribution process.

Figure 2 :
Figure 2: Te efect of θ on the payofs of alliance members in the nonentangled state.(a) Te impact of diferent efort levels of the e-commerce platform and the 4PL platform on the revenue of the e-commerce platform under the nonentangled state.(b) Te infuence of diferent efort levels of the e-commerce platform and the 4PL platform on the revenue of the 4PL platform under the nonentangled state.In the fgures, the size of member revenue can be observed from the color scale (shifting from blue to yellow indicates an increase in the value).

Figure 3 :
Figure 3: Te efect of θ on the payofs of alliance members in the entangled state.(a) Te impact of diferent efort levels of the e-commerce platform and the 4PL platform on the revenue of the e-commerce platform under the entangled state.(b) Te infuence of diferent efort levels of the e-commerce platform and the 4PL platform on the revenue of the 4PL platform under the entangled state.In the fgures, the size of member revenue can be observed from the color scale (shifting from blue to yellow indicates an increase in the value).

Figure 5 :
Figure5: Te efect of θ i � 0 and θ i � π on the returns of alliance members in the nonentangled state.(a-b) Te impact of changes in the efort level of the e-commerce platform on its revenue under the nonentangled state, considering 4PL platform values of 0 and π, respectively.In (c-d), the specifc depiction is provided for the infuence of variations in the efort level of the 4PL platform on its revenue under the nonentangled state, with the e-commerce platform values set at 0 and π, respectively.

Figure 7 :
Figure 7: Te efect of θ i � 0 and θ i � π on the payofs of alliance members in the entangled state.(a-b) A detailed showcase of the impact of changes in the efort level of the e-commerce platform on its revenue under the entangled state, considering the 4PL platform with values of 0 and π.In (c-d), the specifc demonstration focuses on the infuence of variations in the efort level of the 4PL on its revenue under the entangled state, considering the e-commerce platform with values of 0 and π.

Table 3 :
Te proft matrix of alliance members under four strategies in the nonentangled state.

Table 4 :
Te proft matrix of alliance members under four strategies in the entangled state.

Table 5 :
Te proft matrix of alliance members under four strategies in the entangled state.