Research on decision-making behavior of multi-agent alliance in cross-border electricity market environment:an evolutionary game

0 Introduction

Energy is an important foundation for global economic and social development [1].In recent years,stimulated by the rapid development of the world economy,the scale of electricity demand has been expanding [2,3].However,owing to a single energy source,weak energy base,and lagging resource development in some countries,the contradiction between fast-growing energy demand and lagging local energy infrastructure is becoming increasingly prominent.Energy interconnection and cross-border operations are effective means of resolving this conflict [4].In the cross-border electricity market,the main countries form an alliance that can realize the optimal allocation of resources,enhance their competitiveness,achieve their power balance,and obtain greater revenues.

Game theory is a useful tool in cross-border energy operations to optimize strategies when numerous decisionmakers bid among agents in an alliance.Classical game theory is based on the premise of completely rational game agents and focuses on maximizing self-interest[5].Nevertheless,in the current electricity market,the majority of players are unable to fulfill the criterion of“complete rationality.” The evolving game theory offers a viable solution to the limitations of conventional game theory.The evolutionary game theory asserts that the actors involved possess bounded rationality and that each agent attains equilibrium by continuously adjusting its behavior and strategy during the entire game process [6,7].Shariatzadeh et al.[8] presented a review of demand response (DR),existing applications,and possible implementation strategies for a smart grid environment.Cheng et al.[9] conducted a comprehensive study on the behavioral decision-making challenges in power demand-side response management (DRM) using a multi-population evolutionary game dynamics approach.Their research aimed to address the long-term dynamic interaction concerns in power DRM.Fang et al.[10]established an evolutionary game model of renewable power generation and transmission from the perspective of bounded rationality and a multi-agent game to analyze the necessity and effectiveness of strengthening relevant government regulations.Cheng et al.[11] examined the equilibrium qualities of the long-term strategic bidding concerns in deregulated power generation-side markets (PGMs) with homogeneous and heterogeneous characteristics.We conducted this study using various market-clearing techniques.Cheng et al.[12] focused on general N-population multistrategy evolutionary games and used them to investigate generation-side long-term bidding issues in the electricity market.The above research develops demand response (DR) to make the power grid more efficient,environmentally friendly,and reliable,but also to study the behavioral decision-making problem of power demand-side response management (DRM) from the perspective of multigroup evolutionary dynamics and,to a certain extent,enriches the application scenarios of the evolutionary game model in the power market.Hence,by employing evolutionary game theory,this study constructs a game model that considers the decision-making behavior of cross-border multiagents.

In the field of cross-border cooperation on energy,Mertens et al.[13] evaluated the consequences of an inaccurate reflection of cross-border trade flows in regionally limited long-term planning models.Poplavskaya et al.[14] devised an innovative approach to address congestion in a zonal market using flow-based market coupling.Their goal was to enhance cross-border exchanges by including preemptive redispatch in the dayahead market.Ihlemann et al.[15] introduced a model that simulated the clearing of a joint day-ahead energy and balancing capacity market.The model includes the coordinated purchase and sizing of balancing capacity,as well as the optimum allocation of cross-zonal transmission capacity.Li et al.[16] analysed the driving and supporting factors that affect the application of China’s digital currency electronic payment system in One Belt One Road countries.Bindseil et al.[17] described the current vision of how to find a solution that allows cross-border payments to be immediate,cheap,universal,and settled in a secure settlement medium.

However,most existing literature on cross-border electricity markets treats electrical energy as a commodity at a fixed price and ignores its dynamic trading processes.In addition,with the frequent occurrence of uncertain events in the global economy,avoiding risks in the crossborder trading process is particularly important to ensure that trading agents obtain greater revenue.Therefore,this study introduces the exchange rates of neighboring countries into the cross-border electricity market.In this process,an evolutionary game is used to reflect the change in the decision-making behavior of each agent in the crossborder electricity market.In addition,the factors affecting each agent’s change in decision-making behavior were investigated.

In a real cross-border electricity market,the trading center plays a crucial role in facilitating an alliance’s external transactions and revenue distribution.When agents in an alliance reduce electricity prices,the trading center should adjust the income of agents who are not profiting or incur losses [18].This approach helps enhance positive decision-making and strengthens the unity of the alliance.The goal is to increase the overall revenue of the alliances.In addition,when agents lower electricity prices to cover higher costs,the trading center should adjust the unit price of its services to reduce the relative costs for agents,thereby avoiding economic losses and helping agents increase their revenue [19].

The main contributions of this paper are as follows:

1) The integration of multi-agent systems in alliance formations was explored,particularly focusing on the incorporation of neighboring countries’ exchange rates in the context of the cross-border electricity trading market.In this case,it can promote the cross-border operation of energy within the alliance,accelerate the efficiency of the agent’s capital arrival and frequency of transactions,and enable the agent to obtain greater trade revenues.

2) To address the complexities of cross-border multiagent energy transactions,a game-theoretical model based on the dynamic evolution game theory was developed.The model is designed to capture the decision-making behaviors of cross-border multiagents,enabling them to dynamically pursue strategies that benefit either themselves or the alliance.

The remainder of this paper is organized as follows.Section 2 describes the assumptions and the construction of the multi-agent evolutionary game model.Section 3 describes a stability analysis of the equilibrium points of the game.Section 4 describes the case study and Section 5 presents the conclusions of this study.

1 Assumptions and construction of multiagent evolutionary game model

1.1 Model assumptions

In recent years,China has maintained a high and stable trade surplus,attracting foreign investment,and thus strong support for the CNY exchange rate [20,21].Therefore,this study prioritizes using the CNY for settlement and uses neighboring countries’ exchange rates to ensure that countries trade fairly in the cross-border market.The“neighboring country exchange rate effect” is a term used to describe the degree to which changes in bilateral exchange rates between China and a specific trading partner are influenced by the exchange rates of the partner’s neighboring countries.This effect represents the cumulative impact of the exchange rates of multiple neighboring countries on China’s bilateral trade with a given partner.This underscores the complex interplay between regional economic factors that shape bilateral trade dynamics.The exchange rates of neighboring countries are given by Eqs.(1).

where HERit represents the neighboring country’s exchange rate,Ti is country i’s trade volume,TC is China’s trade volume,and FX is China’s bilateral exchange rate with that country;that is,the neighboring country’s exchange rate is a percentage of country i’s trade with China multiplied by China’s bilateral real-time exchange rate indexed to each country in that particular country’s neighborhood.A game strategy is the complete set of behaviors or processes that a participant can choose from [22];in an electricity auction game,participants have two systems in each round of bidding,to lower the prices or not to lower the prices.That is,the set of strategic choices for all three countries,based on their national conditions,electricity market demand,and other factors,is {reduction in electricity prices and no change in electricity prices}.All three countries constantly adjust their strategies until they choose the optimal strategy and eventually attain an equilibrium [23,24].

This study proposes the construction of a gametheoretical model designed to analyze the dynamics of decision-making behaviors among alliance agents and identify the factors influencing these behavioral changes over a specific period.To facilitate this analysis,the following assumptions were made:

Assumption 1: Within an alliance,various agents make decisions independently.At the same time,the crossborder trading center in the cross-border electricity market,according to the form of transaction,sets the price cap for the agents in the alliance that is suitable for the electricity market.Setting a price cap can not only ensure that the agents in the alliance in the cross-border power market obtain normal revenues,but also regulate the malicious bidding behavior of the agents to avoid the deterioration of the cross-border electricity market environment and promote peaceful and stable operation and development of the cross-border electricity market.This study assumes that the maximum limit is the initial electricity price for each agent.Reducing electricity prices in the crossborder electricity market can make cross-border electricity transactions less costly,making cross-border transactions more favorable to the agent with reduced electricity prices and facilitating the agent’s better access to the market.The probability that Country A chooses to reduce electricity prices is x,the probability that it decides to keep electricity prices unchanged is 1-x;the probability that Country B chooses to reduce electricity prices is y,and the probability that it decides to keep electricity prices unchanged is 1-y;the probability that Country C chooses to reduce electricity prices is z,and the probability that it decides to keep electricity prices unchanged is 1-z;x,y,z are functions of time and satisfy x,y,z ∈[0,1][25,26].

Assumption 2: Agents’ power systems are interconnected with cross-border trading centers.In this model,it is assumed that the power system of Country A is situated 100 kilometers (LAO=100 km) away from the cross-border trading center.Similarly,the power system of Country B is located at a distance of 120 kilometers (LBO=120 km)from the trading center,and the power system of Country C is positioned 110 kilometers (LCO=110 km) away from the center.The unit price of the cross-border trading center’s services in the electricity transmission process is S (CNY/km).

Assumption 3: Country A’s revenue from electricity sales in the electricity trading market when electricity prices remain unchanged is n1.When it chooses to reduce electricity prices,the cost to Country A is c1,considering the size of the country’s electricity system,technology,and so on.The additional revenue that Country A receives after a unilateral price reduction that results in a gain for the alliance is a1.

Assumption 4: Country B’s revenue from electricity sales in the electricity trading market when electricity prices remain unchanged is n2.When it chooses to reduce electricity prices,the cost to Country B is c2,considering the size of the country’s electricity system,technology,etc.The additional revenue that Country B receives after a unilateral price reduction that results in a gain for the alliance is a2.

Assumption 5: Country C’s revenue from electricity sales in the electricity trading market when electricity prices remain unchanged is n3.When it chooses to reduce electricity prices,the costs to Country C are c3,considering the size of the Country’s electricity system,technology,and so on.The additional revenue that Country C receives after a unilateral price reduction resulting in a gain for the alliance is a3.

Assumption 6: When countries A and B choose to lower their electricity prices while Country C’s electricity prices remain unchanged,the alliance’s additional revenues from the price advantage are L;when countries A and C choose to lower their electricity prices while Country B’s electricity prices remain unchanged,the alliance’s additional revenues from the price advantage are M;and when countries B and C choose to lower their electricity prices while Country A’s electricity prices remain unchanged,the alliance’s additional revenues from the price advantage are N When countries A,B,and C choose to reduce their electricity prices,the alliance’s additional revenues due to the price advantage are O1.

Based on the decision-making choices of each agent in the alliance,the strategy mix of each agent can be obtained as shown in Table A2 in Appendix A.

1.2 Model construction

Based on the above assumptions,a mixed-strategy game matrix for each alliance agent is obtained,as shown in Table 1.

Table 1 Mixed strategy game matrix of agent

2 Stability analysis of equilibrium points of game system

2.1 Replication dynamic equation and their analysis in Country A

The expected returns when Country A reduces electricity prices or when electricity prices remain unchanged and the mathematical expression of the average expected returns(E11,E12,) is as follows (2):

The replication dynamic equation for Country A’s decision-making behavior is given by Eq.(3).

For ease of calculation,let A=1/[(c1+LAO)+(c2+LBO)],B=1/[(c1+LAO)+(c3+LCO)],C=1/[(c2+LBO)+(c3+LCO)],and D=1/[(c1+LAO)+(c2+LBO)+(c3+LCO)].

The first-order derivatives of x and G(y) are expressed by Eqs.(4) and (5),respectively.

For the probability of being in a steady state when Country A’s decision-making behavior is to reduce electricity prices,it is necessary to satisfy F(x)=0 and d(F(x))/dx＜0.Of these

When G(y)=0,the value of y is given by (7).

The replication dynamic equations and their analyses for Countries B and C were similar to those for Country A,as shown in Appendix B.

2.2 Stability analysis of equilibrium points of an evolutionary game system

Based on the above analysis,the nine equilibrium points of the alliance can be obtained by making F(x)=0,F(y)=0,and F(z)=0,which are as follows: E1(0,0,0),E2(0,1,0),E3(0,0,1),E4(0,1,1),E5(1,0,0),E6(1,1,0),E7(1,0,1),E8(1,1,1),and E9(x*,y*,z*),where the expression for point E9 is expressed in Eq.(8).

Based on an analytical method for judging the stability of the equilibrium points of a multiagent evolutionary game system,the local stability analysis of the Jacobian matrix of the system can obtain the evolutionary stabilization strategy [27,28].Taking the partial derivatives of the replication dynamic equations of the decision-making behaviors of each alliance agent concerning x,y,and z,the Jacobian matrix of the threeparty evolutionary game system of the cross-border alliance can be obtained,as shown in Eq.(9).

To obtain the stability results of the nine local equilibrium points,firstly,all the eigenvalues λ of the equilibrium points are calculated based on Lyapunov’s first method and the Jacobian matrix J,and the sign is judged to be positive or negative.The trace of the Jacobian matrix is zero due to E9(x*,y*,z*);the point is a centroid,a saddle point,and will not be an Evolutionarily Stable Strategy(ESS).Except for E9,an equilibrium point is an ESS if all the eigenvalues of the Jacobian matrix at that point are negative.The eigenvalues of the Jacobian matrix for each equilibrium point are calculated as shown in Table A3 of Appendix A,where Q1=c1+LAO,Q2=c2+LBO,and Q3=c3+LCO.

Analyze the stability of the equilibrium points more efficiently and ensure general results.It is assumed that the revenues gained by agents A,B,and C in countries A,B,and C when they choose to reduce electricity prices are more significant than when they choose to keep electricity prices unchanged.At this point,D(c1+SLAO)O1-(c1+SLAO)＞0,D(c2+SLBO)O1-(c2+SLBO)＞0,and D(c3+SLCO)O1-(c3+SLCO)＞0.In this study,the equilibrium points were analyzed in the following two scenarios;the eigenvalue results and stability analysis in the two scenarios are shown in Table A4 of Appendix A.

Scenario 1: If an individual agent within the alliance opts to reduce electricity prices and the resultant additional revenues exceed the associated costs,the following conditions are observed: a1＞c1+SLAO for Country A,a2＞c2+SLBO for Country B,and a3＞c3+SLCO for Country C.Furthermore,when two agents decide to lower electricity prices while one maintains unchanged prices,and the additional revenues generated under this scenario are greater than the costs incurred,the model is described by several conditions:A(c1+SLAO)L＞c1+SLAO,B(c1+SLAO)M＞c1+SLAO,C(c2+SLBO)N＞c2+SLBO,A(c2+SLBO)L＞c2+SLBO,B(c3+SLCO)M＞c3+SLCO,and C(c3+SLCO)N＞c3+SLCO.Table 4 shows that in this case,only the equilibrium point E8(1,1,1) corresponding to the eigenvalues of the Jacobian matrix are negative.Therefore,in this case,the model has only one ESS,which corresponds to the final evolutionary stabilization strategy of {reduce electricity prices,reduce electricity prices,reduce electricity prices};E1(0,0,0) is the saddle point,and the rest of the equilibrium points are the destabilization points.

Scenario 2: In the scenario where only a single agent within the alliance opts to reduce electricity prices and the resultant additional revenues exceed the costs incurred,the following conditions apply: a1 ＞ c1+SLAO for Country A,a2 ＞ c2+SLBO for Country B,and a3 ＞ c3 +SLCO for Country C.Conversely,when two agents in the alliance decide to reduce electricity prices and the third agent chooses to maintain unchanged prices,the additional revenues generated by the alliance are less than the costs incurred: This scenario is represented by the following conditions: A(c1+SLAO)L＜c1+SLAO,B(c1+SLAO)M＜c1+SLAO,C(c2+SLBO)N＜c2+SLBO,A(c2+SLBO)L＜c2+SLBO,B(c3+SLCO)M＜c3+SLCO,and C(c3+SLCO)N＜c3+SLCO.From Table 4,the equilibrium points that satisfy the eigenvalues of the Jacobian matrix are all negative and four exist in this case.Specifically,E2(0,1,0),E3(0,0,1),E5(1,0,0),and E8(1,1,1),and the remaining four equilibrium points are saddle points,and there are no destabilization points.

Based on the constructed three-party evolutionary game model and parameter constraints for the two scenarios,the parameters were set as listed in Table A5 in Appendix A.

3 Simulation analysis

To more conveniently analyze the influence of the change in each parameter-that is,the initial willingness value,costs of payment,and the alliance’s additional revenues-on the decision-making behavior of each agent,the model was assigned values,and the evolutionary process was simulated and analyzed using MATLAB software.When assigning a value to a parameter,the rest of the parameters should be set to satisfy the basic assumption in the model described above,i.e.,D (c1+SLAO)O1 -(c1+SLAO) ＞ 0 and O1 ＞ c1+SLAO+c2+SLBO+c3+SLCO,at that point,each agent’s decision-making behavior is such that the additional revenues gained from lowering the electricity prices are more significant than that gained from the electricity prices remaining unchanged;and assume that the initial willingness value of the three agents’ decisionmaking is unbiased,x,y,and z are set to be 0.5.The control variable method was used to change the values of the three factors and to simulate and analyze the evolutionary process.

3.1 Impact of the initial willingness value on evolutionary results

The differences in the initial willingness values of the alliance agents produce different evolutionary outcomes,as shown in Fig.1.

Fig.1 The impact of initial willingness value on evolutionary results

When the willingness value of each agent in the alliance is greater than or equal to 0.5,this indicates that each agent operates profitably.Under such circumstances,agents are driven by the psychology of maximizing their interests,which ultimately influences an alliance’s collective decision to reduce electricity prices.Specifically,the final ESS was(1,1,1).Furthermore,the higher the willingness value,the more rapidly the alliance converges towards this equilibrium state of unity.When the willingness value of each agent of the alliance is lower than or equal to 0.4,the decision of the alliance is ultimately the unchanged electricity price;that is,the final ESS is (0,0,0),and the smaller the willingness value,the faster the alliance reaches equilibrium and converges to zero.

In this case,because the parameter setting of Country A is lower,Country A appears to be more hesitant and has slower decision-making behavior.However,to preserve the stability of the alliance,Country A ultimately aligns its strategy with those of the other agents.The simulation results show that the agent country with higher revenue gained in the parameter setting has a faster decision-making speed.

When the willingness value of different agents changes,it also affects the remaining agents.Taking Country A as an example,a reduction in its initial willingness value leads to a notable observation from the simulation results:the convergence speed of the other agents in the alliance increases significantly.The observed reduction in the time required to reach equilibrium suggests that the agents within the alliance are interdependent,rather than operating independently.This dynamic interdependence is evident,as the decision-making behavior of one agent influences the collective decision-making process of the alliance.Consequently,to ensure the stability of an alliance,it is imperative to coordinate and balance the actions of all participating agents effectively.

Any two agents of the alliance with a willingness value below 0.5,and another agent with a willingness value above 0.5,can also make a difference in alliance decision-making behavior.When the willingness values of A,B,and C are 0.7,0.4,and 0.3,respectively,the alliance will eventually refuse to reduce the electricity prices;at this time,the alliance is less sticky.When the willingness values of A,B,and C are 0.4,0.4,and 0.7,respectively,the alliance’s decision-making behavior at this time is to reduce the electricity prices.

3.2 Impact of costs on evolutionary results

As shown in Fig.2,when the costs of the alliance agents to reduce prices gradually decrease,this study takes the cost reductions of two agents as an example,and the results obtained when the costs of a single agent and three agents are reduced similarly.The simulation results indicate that when two agents decrease the costs associated with reducing electricity prices,all three agents experience an accelerated convergence speed,with their willingness values ultimately converging to one.Notably,Country A,which initially reduced electricity prices as a compromise for the overall interests of the alliance,exhibited a significant increase in the motivation to reduce prices.This change in Country A’s attitude was marked by a notable reduction in hesitation.This shows that after cost reduction,the decision-making behavior of each agent is more inclined to reduce electricity prices to maximize the revenues they receive.However,when the costs increase to a specific range of values and the revenues of the alliance agents cannot be balanced with the costs of payment,the agents will refuse to reduce the electricity prices,and the alliance’s willingness value will ultimately converge to 0,and the larger the costs,the faster the speed of convergence to 0.In addition,when the costs of Country B increase,the speed of convergence of Country B’s willingness to converge to zero surpasses that of Country A to become first.By contrast,country C pays the highest costs when it reduces its prices and receives the most significant additional revenue.Hence,its convergence to zero is the slowest when electricity prices remain unchanged.This shows that cost is one of the critical factors affecting revenue in the operation of the cross-border electricity trading market,and every small increase has a considerable impact on the revenue of the alliance agent.Cost also plays a crucial role in influencing alliance stability.Excessively high costs can lead all members of the alliance to opt for maintaining unchanged electricity prices,resulting in the lowest level of alliance stickiness,which is detrimental to the alliance’s stability.In the cross-border electricity market,a strategic approach that cross-border trading centers can adopt is to moderately reduce the unit price of services during electricity transportation.This reduction in the relative costs of agents within the alliance indirectly improves their additional revenue when they decide to lower their electricity prices.Such a measure not only enhances the stickiness of the alliance but also contributes to its overall stability.

Fig.2 The impact of costs on evolution results

3.3 Impact of additional revenues on evolutionary results

As depicted in Fig.3,a scenario is observed in which two agents opt to reduce their electricity prices,whereas one agent maintains unchanged prices.However,when the additional revenue (denoted as L,M,and N) increased,all three agents shifted their strategies to reduce electricity prices.This collective decision results in the alliance’s strategic combination converging to {reduce electricity prices,electricity prices,and electricity prices},represented by (1,1,1).These outcomes demonstrate how varying perspectives can influence the decision-making behavior of alliance agents.For example,when L increases,the willingness of Country C to reduce electricity prices lags behind the “positive” situation because it is countries A and B that reduce electricity prices to increase the alliance’s revenues,but because of the overall interests of the alliance,it will also make the same decision.The alliance can negotiate the distribution of revenues and appropriately increase the interests of the agent whose willingness to be“active” lags to make the alliance more stable.

Fig.3 The impact of additional revenues on evolutionary results under Scenario 1 condition

As illustrated in Fig.4,a scenario emerges when an alliance’s additional revenue begins to decline.T Taking n as an example,a decrease i in n leads to insufficient costs being covered by Countries B and C,making it unfeasible for them to seek revenue through reduced prices.Consequently,Countries B and C did not opt to lower their electricity prices.By contrast,Country A,benefiting from lower costs in its parameter settings for reducing electricity prices,elects to reduce its prices.This divergence in strategies results in the alliance’s final strategic combination being {reduce electricity prices,no change in electricity prices,and no change in electricity prices}.That is,the willingness value converges to (1,0,0),which leads to low viscosity and weak stability of the alliance,and even to the breakdown of the alliance.

Fig.4 The impact of additional revenues on evolutionary results under Scenario 2 condition

The above findings indicate that additional revenue is one of the critical factors affecting the stability of the alliance;therefore,the alliance should ensure that the distribution of revenue is balanced so that its stickiness increases.

3.4 Scenarios simulation Validation

The two scenarios began with different initial strategy combinations and evolved 50 times,as shown in Figs.5 and 6.

From Fig.5,the simulation results show that only one stable strategy combination exists in the system at this point,that is,{reduce the electricity prices,reduce the electricity prices,reduce the electricity prices},which is consistent with the results.The decision-making behavior of the agents in the alliance to reduce electricity prices when the agents are profitable can increase the alliance revenue and promote the stability of the alliance.

Fig.5 The result of case one evolving 50 times

As shown in Fig.6,there are four evolutionary stability points in the system,namely (0,1,0),(0,0,1),(1,0,0),(1,1,1),(1,1,1).When different combinations of decision behavior strategies bring different revenues to the alliance and the agents,the decision behavior of the agents within the alliance is diversified.This indicates that when the degree of profitability is small or not profitable,the decision-making behavior of the agent tends to have an unchanged electricity price,at which point only the profitable agent chooses to reduce the price.The stable points of alliance evolution are(0,1,0),(0,0,1),and (1,0,0).However,when all agents in the alliance are profitable,each agent’s decision-making behavioral strategies converge to (1,1,1).

Fig.6 The result of case two evolving 50 times

These results indicate that the decision-making behavior of alliance agents is closely related to the revenue received.The simulation results match the stability analysis of the strategies of each agent,providing practical guidance for multiple countries to conduct cross-border transactions in the electricity trading market.

3.5 Advantages of the evolutionary game approach

In this study,we take the example of Country A to reduce the electricity price alone and compare it with the alliance agents without an evolutionary game;the resultant comparison graph is shown in Fig.7.

Fig.7 Comparison chart of results

In the absence of an evolutionary game within the alliance,the decision-making behavior of Country A remains static,leading to the trading of electricity at a fixed price in the cross-border electricity market.As shown in Fig.7,under these conditions,Country A’s revenue is consistently fixed at n1:

When the decision-making behavior of Country A can be changed dynamically based on evolutionary game theory,the decision-making behavior of Country A is divided into two types: reduction in electricity prices and no change in electricity prices,which correspond to 1 and 0,respectively.Country A’s decision-making behavior tends to align with strategies that are mutually beneficial to both the alliance and itself.When faced with the choice to reduce electricity prices to bear higher costs or gain less additional revenue,Country A opts against lowering prices,thereby maintaining the status quo in electricity pricing.In such scenarios,Country A’s decision-making behavior converges to a strategy of no change,as indicated by ‘0.’ As shown in Fig.7,under these circumstances,the revenue for Country A is fixed at n1: Consequently,the alliances’ stable strategic combinations at this point are (0,1,0) and (0,0,1),respectively.When Country A is faced with the option of reducing electricity prices,either to incur lower costs or to achieve greater additional revenue,it opts for the strategy of lowering electricity prices.In this scenario,Country A’s decisionmaking behavior converges to ‘1.’ As illustrated in Fig.7,under these conditions,the revenue of Country A is calculated as n1-c1+a1.Consequently,the stable strategic combination for the alliance at this point is represented by (1,0,0).

The above results show that dynamic changes in the decision-making behavior of cross-border multiagents based on evolutionary game theory cause the agents in the alliance to converge on strategies beneficial to themselves or the alliance.The convergence performance of the agents in the alliance reflects the dynamic selection process of the agents to maintain their revenues,which is conducive to avoiding economic losses and increasing the alliance and the revenues of the agents.

4 Conclusions

In this study,we explore the formation of a multiagent alliance in a cross-border electricity trading market by integrating the exchange rates of neighboring countries.We construct a cooperative evolutionary game system that encompasses agents from Countries A,B,and C.This framework facilitates analysis of the strategic stability of each agent and examines the factors influencing the evolution of their decision-making behaviors.

The conclusions of this study are as follows:

1) In the context of a cross-border electricity market,the willingness values of alliance agents are intricately linked and mutually influential.A change in the willingness value of one agent,whether it increases or decreases,prompts a corresponding adjustment among the other agents.

2) A reduction in costs or an increase in additional benefits significantly influences agents’ decision-making processes,leading to an increase in their initial willingness values.This adjustment is crucial for enhancing the agents’motivation to positively alter their decision-making behaviors.

3) In practice,as the international power energy demand increases,small power energy countries do not have sufficient budgets and capacity;they will cooperate with other countries in cross-border transactions and alliances in the form of multiple agents to reduce the price of electricity,and energy can provide better access to the market to obtain an advantage in the cross-border power trading market.

Appendix A

Table A1 Nomenclature of variables

Table A2 The policy combinations of the agents

Table A3 The calculation results of the Jacobian eigenvalue λ at different equilibrium points

Table A4 The sign of the eigenvalue λ and evolutionary stability points

Table A5 Initial values of each parameter (CNY)

Appendix B

B.1 Replication dynamic equation and their analysis in Country B

The expected returns when Country B reduces electricity prices or when electricity prices remain unchanged and the mathematical expression of the average expected returns (E 21,E 22,E2) is as follows (B1).

The replication dynamic equation for Country B’s decision-making behavior is given by Eq.(B2).

The first-order derivatives of y and G(z) are expressed by Eqs.(B3) and (B4):

For the probability of being in a steady state when Country B’s decision-making behavior is to reduce electricity prices,it is necessary to satisfy F ( y )=0 and d ( F ( y )) / d y＜0.

The expression for ∂G (z)/∂z is presented in Eq.(B5).

When G(z)=0,z is given by Eq.(B6).

B.2 Replication eynamic equation and their analysis in Country C

The expected returns when Country C reduces electricity prices or when electricity prices remain unchanged and the mathematical expression of the average expected returns (E31,E32,) is as follows (B7):

The replication dynamic equation for Country C’s decision-making behavior is given by Eq.(B8).

The first-order derivatives of z and G(x) are expressed by Eqs.(B9) and (B10),as follows:

For the probability of being in a steady state when Country C’s decision-making behavior is to reduce electricity prices,it is necessary to satisfy F ( z ) = 0 and d ( F ( z )) / d z ＜ 0.

The expression for ∂G (x)/∂x is shown in Eq.(B11).

When G(x)=0,the value of x is calculated using Eq.(B12).

Acknowledgments

This study was supported by the National Key R&D Program of China (Grant No.2022YFB2703500),the National Natural Science Foundation of China (Grant No.52277104),the National Key R&D Program of Yunnan Province (202303AC100003),the Applied Basic Research Foundation of Yunnan Province (202301AT070455,202101AT070080),and the Revitalizing Talent Support Program of Yunnan Province (KKRD202204024).

Declaration of Competing Interest

We declare that we have no conf lict of interest.