Marine active defense strategy based on Nap-Stackelberg hierarchical game model

Abstract

The invention discloses a marine active defense strategy based on a Nap-Stackelberg hierarchical game model, which comprises the following steps: constructing a marine defense confrontation simulation scene, modeling a defense power deployment problem as the Nap-Stackelberg hierarchical game model, the Nap-Stackelberg hierarchical game model being used for describing a defense power internal cooperative relationship and a confrontation relationship between a defense party and an attacking party; analyzing the Nash-Stackelberg hierarchical game model, constructing a model solving framework based on a hybrid particle swarm algorithm, optimizing the model solving framework, and solving a game equilibrium solution by iteratively updating particles; and formulating an active defense strategy according to the game equilibrium solution. According to the method, the cooperative relation between defense units and the confrontation relation between the defense party and the attack party are fully considered, the scientificity and the strong confrontation of a computer party deployment scheme in a game are improved, and refined defense power distribution suggestions are provided.

Description

Technical Field

[0001] The invention belongs to the technical field of system engineering methodology and control theory, and relates to a maritime active defense strategy based on a Nash-Stackelberg hierarchical game model. Background Art

[0002] In modern strategy game design, especially games involving defense and confrontation scenarios, the optimization of the strategies of the opposing sides is an important part of game design, and realism and complexity are the key to improving the player experience. Especially in PVE (Player vs. Environment) games of maritime confrontation, players need to reasonably deploy defense forces under limited resource conditions to protect high-value targets, while predicting and responding to the actions of the attacker. This scenario requires the artificial intelligence in the game to have strong dynamic decision-making capabilities to achieve a realistic and challenging confrontation effect.

[0003] However, AI in many current strategy games has certain limitations. It is usually based on simple rules or static optimization models, lacks the ability to deeply learn player behavior patterns, and cannot fully simulate complex and diverse tactical confrontations. This limitation causes players to feel that the computer's defense deployment and response are more mechanical and predictable during the game confrontation, which reduces the fun and challenge of the game.

[0004] Specifically in maritime confrontation games, the game scene settings usually include multiple types of defense units, such as surface ships, aviation forces, air defense forces, etc., and the opposing attackers may launch a variety of threatening behaviors, such as focusing on high-value targets, dispersing and deceiving defense forces, or adopting a more concealed attack path, etc. In this case, how to simulate the collaborative relationship between defense units and the dynamic confrontation relationship between the defender and the attacker is one of the core challenges to improving the intelligence of the game.

[0005] At present, in the field of defense deployment, the existing technologies mainly include the following methods and systems:

[0006] 1) Rule-driven static deployment:

[0007] Rule-driven static deployment is the most basic defense strategy design method. It arranges the positions and ranges of defense units in a fixed manner by presetting a set of rules or parameters. For example, according to the predetermined defense depth principle, defense units are deployed in layers, with interception forces deployed in the front and support units deployed in the rear.

[0008] 2) Optimization method based on heuristic rules:

[0009] This type of method optimizes the defense deployment strategy by summarizing experience or designing heuristic rules. For example, the deployment location is selected based on the rule of maximizing the coverage of the defense area, or the coordination between defense forces is optimized based on the principle of the shortest distance.

[0010] 3) Deployment model based on mathematical optimization:

[0011] Mathematical optimization methods transform defense deployment problems into linear programming, nonlinear programming, or integer programming problems by establishing objective functions and constraints, for example, maximizing defense area coverage and minimizing response time between defense units.

[0012] 4) Intelligent deployment based on artificial intelligence:

[0013] The rapid development of artificial intelligence technology has provided new solutions for defense deployment. For example, strategy generation methods based on deep learning, or dynamic deployment optimization based on reinforcement learning. AI technology can learn more intelligent deployment strategies through a large amount of training data that simulates confrontation scenarios.

[0014] However, the prior art has the following significant disadvantages:

[0015] 1) Lack of dynamic adjustment capabilities:

[0016] Arranging deployments based on fixed rules cannot adapt to dynamic and complex confrontation scenarios. For example, when the attacker changes its strategy, the defender cannot dynamically adjust the deployment. The lack of a response mechanism to the attacker's strategy results in poor adaptability of the defense deployment to unknown threats.

[0017] 2) Limited robustness:

[0018] Although heuristic algorithms and simulated annealing algorithms can provide better solutions, they have high adjustment costs and limited robustness when facing multi-objective or dynamically changing scenarios. When the complexity of the scenario increases or the adaptability of the rules is insufficient, the optimization effect may decrease.

[0019] 3) Unable to reflect the mutual influence between units:

[0020] Models based on mathematical optimization usually focus on optimizing a single objective function and cannot effectively take into account multiple objective requirements. They also have a poor impact on the strategies of players or computer units in a dynamic confrontation environment and cannot reflect the mutual influence between units.

[0021] 4) Limited model adaptability:

[0022] The effectiveness of AI models depends on a large amount of training data and computing resources. It may be challenging to obtain high-quality training data in certain scenarios. AI methods perform well in scenarios with dynamic changes and sufficient data, but may not work well in environments with complex rules and scarce data. In addition, the model training process is complex and time-consuming, which also limits its scope of application and real-time performance.

[0023] Existing technologies have many deficiencies in defense deployment and allocation optimization, and cannot fully meet the needs of modern complex sea and air environment simulation and strategy formulation. These shortcomings significantly affect the flexibility, real-time and overall effectiveness of the defense system, and more advanced optimization solutions are urgently needed to solve these problems. Summary of the invention

[0024] The present invention aims to solve the technical problem that the existing defense force deployment scheme lacks realism and effectiveness in confrontational games, and it is difficult to reflect the cooperation and mutual influence between defense forces. The present invention provides a maritime active defense strategy based on the Nash-Stackelberg hierarchical game model, and the technical solution adopted is:

[0025] The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model includes the following steps:

[0026] S1. Construct a maritime defense confrontation simulation scenario and model the defense force deployment problem as a Nash-Stackelberg hierarchical game model, which is used to describe the internal coordination relationship of the defense force and the confrontation relationship between the defender and the attacker;

[0027] S2, analyzing the Nash-Stackelberg hierarchical game model to find a game equilibrium solution;

[0028] S3. Formulate an active defense strategy based on the game equilibrium solution.

[0029] In one embodiment of the present invention, in step S1, the Nash-Stackelberg hierarchical game model is expressed as:

[0030] G=((L,F),(S L ,S F ),(U L ,U F ),(L,S L ,U L ))

[0031] Among them, (L, F) is the set of participants, L is the set of defense units, and F is the set of attackers; (S L ,S F) is the strategy set of the participants, S L is the strategy set of the defense unit L, S F is the strategy set of the attacker F; (U L ,U F ) is the utility function of the participant under the corresponding strategy combination, U F is the utility function of the defense unit L, U L is the utility function of the attacker F, (L,S L ,U L ) is the inner Nash equilibrium.

[0032] In one embodiment of the present invention, the utility function of the defense unit L is expressed as:

[0033]

[0034] Among them, u i (s) is the total utility function of the i-th defense unit, s represents a pure strategy for the deployment of all defense units, j represents the j-th utility function, is the value of the jth utility function of the ith defense unit, n represents the nth unit, ω nj is the weighting factor of the jth utility function of the nth unit, and η is the total number of utility objectives.

[0035] In one embodiment of the present invention, the utility function of the defense unit L includes the forward defense depth of the defense unit, the overlapping area of ​​the defense zone, the distance of the defense unit, and the interception capability of the defense zone;

[0036] The utility function of the forward defense depth of the defense unit is expressed as:

[0037]

[0038] in, represents the utility function of the i-th defense unit with the forward defense depth as the goal and strategy s, R i represents the defense range of the i-th defense unit, dis min Indicates the minimum distance between the defense unit and the protected target, max Indicates the maximum distance between the defense unit and the protected target, (x i ,y i ) represents the horizontal and vertical coordinates of the i-th defense unit in a Cartesian coordinate system with the protected target as the center origin, due north as the positive direction of the y-axis, and due east as the positive direction of the x-axis;

[0039] The utility function of the overlapping area of ​​the defense zones is expressed as:

[0040]

[0041] in, represents the utility function of the i-th defense unit with the overlapping area of ​​the defense zone as the target and strategy s, Neighbor i represents the set of defense units adjacent to the i-th defense unit, area ij It represents the overlapping area of ​​the defense zone between defense unit i and adjacent defense unit j, which is calculated by the interception radius of the two units and the arc of the overlapping area, as follows:

[0042] area ij =area i +area j -area Δ

[0043] Among them, area i represents the sector area of ​​the overlapping area of ​​defense unit i, area j represents the sector area of ​​the overlapping area of ​​defense unit j, area Δ Represents the common two triangular areas of the sectors occupied by the overlapping areas of defense units i and j;

[0044]

[0045] Among them, R i is the interception range of defending unit i, R j is the interception range of defense unit j, α i is the arc corresponding to the sector area of ​​the defense unit i, α j is the arc corresponding to the sector area of ​​the defense unit j;

[0046] The utility function of the defense unit distance is expressed as:

[0047]

[0048] in, represents the utility function of the i-th defense unit with the distance between defense units as the target and strategy s, dis min Indicates the minimum distance between the defense unit and the protected target, max Indicates the maximum distance between the defense unit and the protected target, (x i ,y i ) represents the coordinates of the i-th defense unit in the above coordinate system, (x j ,y j ) represents the coordinates of the j-th defense unit in the above coordinate system;

[0049] Assume that the minimum interception interval of the i-th defense unit is t i, the incoming target has a velocity of v, and will pass through m units of defense zone in the direction of the incident angle α, and the lengths passed are {l1,l2,...,l M}, then the number of interceptions that the i-th defense unit can make is K i for:

[0050]

[0051] Suppose the interception success rate of the i-th defense unit is p i , then the defense unit's interception success rate for the incoming target is Pr i for:

[0052]

[0053] In the direction of the incident angle α relative to the horizontal polar axis angle, the total success probability p of the incoming target being intercepted by all deployed defense units after passing through the defense zone of m defense units is α for:

[0054]

[0055] Since the closer the defense unit is deployed to the defended target, the larger the range of incident angles it is responsible for defending, and thus the greater its contribution to the defense of the protected target, the utility function of the defense zone interception capability is expressed as:

[0056]

[0057] Among them, θ1 and θ2 are the upper and lower limits of the incident angle of the defense range of the i-th defense unit relative to the horizontal polar axis angle, C i The relative weight of the successful contribution of the i-th defense unit to the protection of the central position can be set manually according to the battlefield situation.

[0058] In one embodiment of the present invention, the utility function of the attacker F is represented by the penetration probability, damage degree, and attack cost of the attack plan:

[0059]

[0060] Among them, Damage is the damage value of the unit in the attack plan, count is the number of units in the attack plan, and Cost is the cost loss value of the attack plan.

[0061] In one embodiment of the present invention, the game equilibrium solution obtained by solving the Nash-Stackelberg hierarchical game model satisfies the following conditions:

[0062]

[0063] Among them, N is the total number of defense units in the inner game model, represents a pure strategy of the nth defense unit in the inner game model, represents the set of pure strategies of the nth defense unit in the inner game model, represents a pure strategy combination of all units except the nth defense unit in the inner game model. In the outer game model, the attacker, i.e. the follower, takes A pure strategy selected after combining the strategies of In the outer game model, the attacker takes The optimal pure strategy after the strategy combination, It represents a Nash equilibrium solution of the inner game under the premise that the attacker of the outer game takes the best, U L and U F It is a utility function with the strategy combination of the participants as variables.

[0064] In one embodiment of the present invention, step S2 comprises:

[0065] S21: Construct a model solving framework based on hybrid particle swarm algorithm;

[0066] S22: Optimize the model solving framework and solve the game equilibrium solution by iteratively updating particles.

[0067] In one embodiment of the present invention, step S21 includes:

[0068] Initialize the population and randomly generate an initial population of N particles. Each particle represents a deployment plan of the defense unit or an attack plan of the attacker, and the speed of the particle is also randomly initialized.

[0069] Entering the fitness value calculation stage, a fitness value is calculated for each particle through a preset utility function, and the fitness value reflects the quality of the allocation scheme represented by the particle;

[0070] Update the individual optimal value and global optimal value of the particle population and record them;

[0071] The genetic algorithm is introduced for optimization. A roulette wheel selection strategy is used to select a number of individuals from the current population for subsequent genetic operations. A new generation of individuals is generated through single-point crossover or two-point crossover, and gene mutation operations are performed on the new individuals.

[0072] The new generation of individuals optimized by the particle swarm algorithm and the genetic algorithm form a new population to replace the old population, and repeat the above iterative process until the preset termination conditions are met, such as reaching the maximum number of generations, time limit or convergence of the solution, and output the optimized results to obtain a high-quality game equilibrium solution.

[0073] In one embodiment of the present invention, the step S22 includes: inner layer Nash game optimization and outer layer Stackelberg game optimization;

[0074] For the optimization solution of the inner layer Nash game, the element parameters and the leader's random strategy are initialized, and the optimal strategy of the corresponding player is obtained by maximizing the fitness function. The strategies of the remaining leaders are adjusted based on the optimal strategy recorded after each iteration and the group dynamics of the particle swarm algorithm-genetic algorithm. In the inner layer optimization process, the best strategy is generated for all units of the leader. All participants obtain the Nash equilibrium, and no participant can obtain higher benefits by changing their strategy;

[0075] For the optimization solution of the outer Stackelberg game, initialize the element parameters and the random strategies of the followers, run the optimization algorithm of the inner game mentioned above, determine the strategy of a leader, adjust the follower strategy according to the algorithm group dynamics, and terminate the optimization process when the maximum number of iterations is exceeded. Count all the best response strategy combinations of the followers, maximize the leader's effect function, and obtain a Nash-Stackelberg game equilibrium solution.

[0076] In one embodiment of the present invention, the inner layer Nash game optimization step includes:

[0077] Step 1: Initialize the parameters of the hybrid particle swarm algorithm, including population size, maximum number of iterations, inertia weight, individual and global learning factors;

[0078] Step 2: Set the current number of iterations to 0 and generate a leader’s random strategy set as the population particle;

[0079] Step 3: Record the current optimal population particles;

[0080] Step 4: Start the iteration cycle and confirm the optimal particle of the previous iteration;

[0081] Step 5: Update the new round of population particles according to the optimal particles and calculate the corresponding utility function;

[0082] Step 6: Record and update the best population particles in the current iteration round;

[0083] Step 7: Repeat the above iterations and finally obtain a Nash equilibrium solution for the leader;

[0084] The outer Stackelberg game optimization steps include:

[0085] Step 1: Initialize the parameters of the hybrid particle swarm algorithm, including population size, maximum number of iterations, inertia weight, individual and global learning factors;

[0086] Step 2: Set the current number of iterations to 0 and generate a set of random strategies for followers as population particles;

[0087] Step 3: Start loop iteration;

[0088] Step 4: Run the above algorithm to determine a leader’s strategy;

[0089] Step 5: Update the population particles of the new round of followers according to the optimal particle and calculate the corresponding fitness function;

[0090] Step 6: Record and update the best population particles in the current iteration round;

[0091] Step 7: Repeat the above iterations to finally obtain the best response strategy combination for all followers;

[0092] Step 8: Calculate the leader’s best response strategy among these strategy combinations and obtain the Nash-Stackelberg game equilibrium solution.

[0093] Beneficial effects of the present invention:

[0094] 1. The present invention fully considers the cooperative relationship between defense units and the confrontational relationship between the defender and the attacker by constructing the Nash-Stackelberg hierarchical game model, which improves the scientificity and strong confrontation of the computer deployment plan in the game. Moreover, it can effectively weigh the multi-objective requirements such as defense depth, overlapping area of ​​defense zones, distance between defense units and interception capability, and provide refined defense force distribution suggestions;

[0095] 2. The present invention combines a hybrid particle swarm algorithm to solve the Nash-Stackelberg hierarchical game model, and utilizes the global search capability and fast convergence characteristics of the particle swarm algorithm to efficiently obtain the game equilibrium solution in multi-variable, multi-objective complex problems;

[0096] 3. The present invention utilizes the leader-follower mechanism of the Stackelberg game. The present invention fully considers the response behavior and strategy adjustment of the attacker, so that the defender can predict potential threats and formulate targeted defense plans. BRIEF DESCRIPTION OF THE DRAWINGS

[0097] Figure 1 It is a flow chart of a maritime active defense strategy based on a Nash-Stackelberg hierarchical game model provided by an embodiment of the present invention;

[0098] Figure 2 It is a structural diagram of the Nash-Stackelberg hierarchical game model provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0099] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0100] In the game environment, the computer's defense mission at sea faces many challenges in response to the player's attack methods, including the multi-directional, multi-wave, and high-intensity characteristics of incoming threats, limited defense force resources (such as ammunition, ships, and sensor coverage), and the need for rapid decision-making in an uncertain and dynamic environment to achieve optimized collaboration.

[0101] The present invention provides a maritime active defense strategy based on the Nash-Stackelberg hierarchical game model. Figure 1 , including the following steps:

[0102] S1. Construct a maritime defense confrontation simulation scenario and model the defense force deployment problem as a Nash-Stackelberg hierarchical game model. The Nash-Stackelberg hierarchical game model is used to describe the internal coordination relationship of the defense force and the confrontation relationship between the defender and the attacker, so as to optimize the deployment strategy.

[0103] S2. Analyze the Nash-Stackelberg hierarchical game model and find the game equilibrium solution.

[0104] S3. Formulate active defense strategies based on the game equilibrium solution.

[0105] On the one hand, since different types of defense forces usually have independent resources and capabilities (such as the firepower coverage of ships, the interception probability of missiles, and the search capability of drones), there is a relationship of cooperation and competition, and their decision-making goal is to maximize their respective defense effectiveness while avoiding resource waste or conflict. This non-cooperative and mutually influential decision-making relationship is suitable for description through the Nash game model.

[0106] On the other hand, the defender tends to pre-allocate resources before launching an attack, while the attacker will flexibly adjust its attack strategy according to the defender's deployment. In this scenario, the defender plays the role of a leader, taking the lead in formulating and implementing a deployment plan; accordingly, the attacker, as a follower, will choose the path or target of the attack based on the defender's deployment strategy. This decision-making order coincides with the basic assumptions of the Stackelberg game model.

[0107] The present invention models the defense force deployment problem as a Nash-Stackelberg hierarchical game model. The Nash-Stackelberg hierarchical game model can clearly describe complex multi-agent decision-making problems in a hierarchical manner. The Nash game models the internal collaboration or competition of defense forces. By defining a utility function for each defense force, it can characterize the performance of its optimal strategy in global equilibrium. The utility function usually includes factors such as interception effectiveness, coverage, and resource consumption, and can flexibly describe the decision-making characteristics of the defense force. The Stackelberg game models the interaction between the defender and the attacker. By designing a dominant decision utility function for the defender (such as maximizing interception effectiveness) and a response decision utility function for the attacker (such as maximizing destruction effectiveness), the model can enable the defender to dynamically predict the behavior of the attacker and optimize the defense deployment strategy.

[0108] The Nash-Stackelberg hierarchical game model of the present invention is expressed as:

[0109] G=((L,F),(S L ,S F ),(U L ,U F ),(L,S L ,U L ))

[0110] Among them, (L, F) is the set of participants, L is the set of defense units, and F is the set of attackers; (S L ,S F ) is the strategy set of the participants, S L is the strategy set of the defense unit L, S F is the strategy set of the attacker F; (U L ,U F ) is the utility function of the participant under the corresponding strategy combination, U F is the utility function of the defense unit L, U L is the utility function of the attacker F, (L,S L ,U L) is the inner Nash equilibrium, which is also a non-cooperative game model. In the game, participants choose their own strategies from the strategy set based on their own goals and the strategy choices of others. Each strategy combination is mapped to benefits through the utility function, which affects the decision of each participant.

[0111] The Nash-Stackelberg hierarchical game model includes participants, strategy sets, utility functions, and inner Nash equilibrium. Participants are divided into two categories: leaders and followers. Leaders have the first-mover advantage in the game and can formulate strategies by predicting the reactions of followers; followers choose their own optimal strategies based on the leader's decisions after the leader determines the strategy. In addition, the leaders usually need to meet the Nash equilibrium conditions, that is, each leader chooses a strategy that maximizes its own utility. The decision-making behaviors of leaders influence each other, and the decision-making behaviors of leaders and followers influence each other, which together constitute the structure of the hierarchical game model. The structural diagram of the Nash-Stackelberg hierarchical game model is shown in the attached figure. Figure 2 As shown in Figure 2. The strategy set defines the feasible decision space of each participant. The strategy set of the follower may depend on the decision of the leader, that is, this dependency reflects that the response strategy of the follower is constrained by the leader's behavior. In terms of utility function, the leader and the follower each have clear goals, but the utility function of all participants is affected by the strategy of each participant.

[0112] The utility function of the defender's defense unit deployment is analyzed. Since the utility function may have multiple target factors, the utility function of the defense unit L is expressed as:

[0113]

[0114] Among them, u i (s) is the total utility function of the i-th defense unit, s represents a pure strategy for the deployment of all defense units, j represents the j-th utility function, is the value of the jth utility function of the ith defense unit, n represents the nth unit, ω nj is the weighting factor of the jth utility function of the nth unit, and η is the total number of utility objectives.

[0115] The utility function of the defense unit L of the present invention includes the forward defense depth of the defense unit, the overlapping area of ​​the defense zone, the defense unit distance and the defense zone interception capability.

[0116] In one embodiment of the present invention, the utility function of the forward defense depth of the defense unit is expressed as:

[0117]

[0118] in, represents the utility function of the i-th defense unit with the forward defense depth as the goal and strategy s, R i represents the defense range of the i-th defense unit, dis min Indicates the minimum distance between the defense unit and the protected target, max Indicates the maximum distance between the defense unit and the protected target, (x i ,y i ) represents the horizontal and vertical coordinates of the i-th defense unit in a Cartesian coordinate system with the protected target as the center origin, due north as the positive direction of the y-axis, and due east as the positive direction of the x-axis;

[0119] In one embodiment of the present invention, the utility function of the overlapping area of ​​the defense zones is expressed as:

[0120]

[0121] in, represents the utility function of the i-th defense unit with the overlapping area of ​​the defense zone as the target and strategy s, Neighbor i represents the set of defense units adjacent to the i-th defense unit, area ij It represents the overlapping area of ​​the defense zone between defense unit i and adjacent defense unit j, which is calculated by the interception radius of the two units and the arc of the overlapping area, as follows:

[0122] area ij =area i +area j -area Δ

[0123] Among them, area i represents the sector area of ​​the overlapping area of ​​defense unit i, area j represents the sector area of ​​the overlapping area of ​​defense unit j, area Δ Represents the common two triangular areas of the sectors occupied by the overlapping areas of defense units i and j;

[0124]

[0125]

[0126] Among them, R i is the interception range of defending unit i, R j is the interception range of defense unit j, α i is the arc corresponding to the sector area of ​​the defense unit i, α j is the arc corresponding to the sector area of ​​the defense unit j;

[0127] In one embodiment of the present invention, the utility function of the defense unit distance is expressed as:

[0128]

[0129] in, represents the utility function of the i-th defense unit with the distance between defense units as the target and strategy s, dis min Indicates the minimum distance between the defense unit and the protected target, max Indicates the maximum distance between the defense unit and the protected target, (x i ,y i ) represents the coordinates of the i-th defense unit in the above coordinate system, (x j ,y j ) represents the coordinates of the j-th defense unit in the above coordinate system;

[0130] Assume that the minimum interception interval of the i-th defense unit is t i , the incoming target has a velocity of v, and will pass through m units of defense zone in the direction of the incident angle α, and the lengths passed are {l1,l2,...,l M}, then the number of interceptions that the i-th defense unit can make is K i for:

[0131]

[0132] Suppose the interception success rate of the i-th defense unit is p i , then the defense unit's interception success rate for the incoming target is Pr i for:

[0133]

[0134] In the direction of the incident angle α relative to the horizontal polar axis angle, the total success probability p of the incoming target being intercepted by all deployed defense units after passing through the defense zone of m defense units is α for:

[0135]

[0136] Since the closer the defense unit is deployed to the defended target, the larger the range of incident angles it is responsible for defending, and thus the greater its contribution to the defense of the protected target, the utility function of the defense zone interception capability is expressed as:

[0137]

[0138] Among them, θ1 and θ2 are the upper and lower limits of the incident angle of the defense range of the i-th defense unit relative to the horizontal polar axis angle, C i The relative weight of the successful contribution of the i-th defense unit to the protection of the central position can be set manually according to the battlefield situation.

[0139] By comprehensively considering these target factors, defenders can effectively optimize resource allocation, improve the overall defense capabilities of the system, and provide a reasonable basis for predicting the strategy adjustment of the attacker. This multi-objective collaborative optimization strategy design can not only improve defense effectiveness, but also enhance the system's adaptability and robustness to complex threats.

[0140] The utility function of the attacker F is expressed by the penetration probability, damage degree, and attack cost of the attack plan:

[0141]

[0142] Among them, Damage is the damage value of the unit in the attack plan, count is the number of units in the attack plan, and Cost is the cost loss value of the attack plan.

[0143] The Nash-Stackelberg hierarchical game model is constructed. The Nash-Stackelberg hierarchical game model is solved and the obtained game equilibrium solution meets the conditions:

[0144]

[0145] Among them, N is the total number of defense units in the inner game model, represents a pure strategy of the nth defense unit in the inner game model, represents the set of pure strategies of the nth defense unit in the inner game model, represents a pure strategy combination of all units except the nth defense unit in the inner game model. In the outer game model, the attacker, i.e. the follower, takes A pure strategy selected after combining the strategies of In the outer game model, the attacker takes The optimal pure strategy after the strategy combination, It represents a Nash equilibrium solution of the inner game under the premise that the attacker of the outer game takes the best, U L and U F It is a utility function with the strategy combination of the participants as variables.

[0146] This is actually a multi-level optimization problem. In the game model of the present invention, Stackelberg game and Nash equilibrium are combined to form a hierarchical and mutually feedback optimization structure. The main body of this optimization structure is to maximize the leader's utility function. The optimal solution has the following three constraints, which are also optimization conditions: First, the follower's strategy is the optimal response of the leader's strategy, that is, Within the leader, each participant’s strategy is the optimal response to the combination of strategies of other participants, that is, The reverse is also true. Ultimately, the leader achieves global equilibrium by maximizing its own utility function and combining the responses of followers and the strategies of other leaders. As follows:

[0147]

[0148] Next, to solve the Nash-Stackelberg hierarchical game model, we can build a model solving framework based on the hybrid particle swarm algorithm, optimize the model solving framework, and solve the game equilibrium solution by iteratively updating particles.

[0149] The brief steps of the hybrid particle swarm algorithm are as follows:

[0150] First, the population is initialized and an initial population of N particles is randomly generated. Each particle represents a deployment plan of the defense unit or an attack plan of the attacker, and the speed of the particle is also randomly initialized.

[0151] Then, the fitness value calculation phase is entered, and a fitness value is calculated for each particle through a preset utility function. The fitness value reflects the quality of the allocation scheme represented by the particle.

[0152] Next, the individual optimal value and global optimal value of the particle population are updated and recorded. Since the particle swarm algorithm is an evolutionary algorithm that simulates the foraging habits of bird flocks, the speed and position of each particle are updated according to the individual optimal value and global optimal value of each particle, in the hope that the particles can gradually approach the optimal solution.

[0153] Next, the genetic algorithm is introduced for optimization. A roulette wheel selection strategy is used to select several individuals from the current population for subsequent genetic operations. A new generation of individuals is generated through single-point crossover or two-point crossover, and gene mutation operations are performed on the new individuals.

[0154] Finally, the new generation of individuals optimized by the particle swarm algorithm (PSO) and genetic algorithm (GA) form a new population to replace the old population, and the above iterative process is repeated until the preset termination conditions are met, such as reaching the maximum number of generations, time limit or convergence of the solution, and the optimized results are output to obtain a high-quality game equilibrium solution.

[0155] In the whole process, by utilizing the fast convergence of particle swarm optimization and the global search capability of genetic algorithm, a better equilibrium solution can be obtained.

[0156] In the framework of the game model, the pseudocode of each optimization level is as follows:

[0157] Step 1: To solve a Nash equilibrium strategy for the leader, determine a deployment strategy for the defender.

[0158] Step 2: Starting from the leader’s first defensive unit, calculate the best response under the deployment strategy of other units and update the strategy;

[0159] Step 3: Perform Step 2 on the subsequent defensive units;

[0160] Step 4: Repeat Step 2 and Step 3 until the leader's strategy no longer changes, and obtain the Nash equilibrium strategy of the upper leader of the model;

[0161] Step 5: Calculate the pure strategy of the follower's best response based on the leader's strategy;

[0162] Step 6: Repeat Step 1 to Step 5 until all the best response strategy combinations of followers are calculated;

[0163] Step 7: Calculate the best response strategy of the leader among all the best response strategy combinations of the followers to obtain the equilibrium solution of the Stackelberg model. Through Stackelberg game analysis, we can predict some offensive plans and strategies of the attacker, analyze the weak links of the computer side, i.e. the defender, and use them to make countermeasure plans.

[0164] According to the optimization level of Nash-Stackelberg game equilibrium design, a hybrid particle swarm algorithm is developed to achieve and obtain the optimal solution. The implementation process includes the following two steps: inner Nash game optimization and outer Stackelberg game optimization.

[0165] For the optimization solution of the inner layer Nash game, the element parameters and the leader's random strategy are initialized, and the optimal strategy of the corresponding player is obtained by maximizing the fitness function. The strategies of the remaining leaders are adjusted based on the optimal strategy recorded after each iteration and the group dynamics of the particle swarm algorithm-genetic algorithm. In the inner layer optimization process, the best strategy is generated for all units of the leader. All participants obtain the Nash equilibrium, and no participant can obtain higher benefits by changing their strategy.

[0166] The inner Nash game optimization steps include:

[0167] Step 1: Initialize the parameters of the hybrid particle swarm algorithm, including population size, maximum number of iterations, inertia weight, individual and global learning factors;

[0168] Step 2: Set the current number of iterations to 0 and generate a leader’s random strategy set as the population particle;

[0169] Step 3: Record the current optimal population particles;

[0170] Step 4: Start the iteration cycle and confirm the optimal particle of the previous iteration;

[0171] Step 5: Update the new round of population particles according to the optimal particles and calculate the corresponding utility function;

[0172] Step 6: Record and update the best population particles in the current iteration round;

[0173] Step 7: Repeat the above iterations and finally obtain a Nash equilibrium solution for the leader.

[0174] For the optimization solution of the outer Stackelberg game, initialize the element parameters and the random strategies of the followers, run the optimization algorithm of the inner game mentioned above, determine the strategy of a leader, adjust the follower strategy according to the algorithm group dynamics, and terminate the optimization process when the maximum number of iterations is exceeded. Count all the best response strategy combinations of the followers, maximize the leader's effect function, and obtain a Nash-Stackelberg game equilibrium solution.

[0175] The outer Stackelberg game optimization steps include:

[0176] Step 1: Initialize the parameters of the hybrid particle swarm algorithm, including population size, maximum number of iterations, inertia weight, individual and global learning factors;

[0177] Step 2: Set the current number of iterations to 0 and generate a set of random strategies for followers as population particles;

[0178] Step 3: Start loop iteration;

[0179] Step 4: Run the above algorithm to determine a leader’s strategy;

[0180] Step 5: Update the population particles of the new round of followers according to the optimal particle and calculate the corresponding fitness function;

[0181] Step 6: Record and update the best population particles in the current iteration round;

[0182] Step 7: Repeat the above iterations to finally obtain the best response strategy combination for all followers;

[0183] Step 8: Calculate the leader's best response strategy among these strategy combinations, and obtain the Nash-Stackelberg game equilibrium solution.

[0184] Because the leader's strategy is optimized internally and covers the optimal strategies of all followers given the leader's strategy externally, the convergence result eventually presents a global Nash-Stackelberg equilibrium.

[0185] Formulate active defense strategies based on the game equilibrium solution. The core logic of formulating active defense strategies is to predict the optimal attack plan that the attacker may take based on the game equilibrium solution, and make defense plans in advance accordingly. After obtaining the equilibrium solution, the defender can analyze the attacker's optimal response, and thus secretly deploy defenses in key areas and set up concealed defense forces to form an effective deterrent to the attacker. At the same time, the defender can also implement deception defense in certain directions and targets, and deliberately expose secondary targets or set up disguised baits by designing deception strategies to mislead the attacker's decision-making and make him choose an attack path or target that is not conducive to his own benefits. In this case, under the constraints of rationality and equilibrium solutions, the attacker will concentrate attack resources on the false targets or secondary targets carefully set by the defender, thereby effectively protecting the real important targets and achieving the purpose of active defense.

[0186] The Nash-Stackelberg equilibrium strategy can be obtained through the solution of the present invention. From the perspective of Stackelberg game, the leader, i.e. the defender, acts as a forerunner in the decision-making process and formulates a defense plan by analyzing the equilibrium solution. The core idea is to predict the best response strategy of the follower, i.e. the attacker, and combine its own resource constraints and objective function to achieve global optimization and improvement of defense deployment.

[0187] The Nash-Stackelberg equilibrium solution reveals the attacker's optimal strategic response path, and the defender can use this information to predict the strategy. For example, if the equilibrium solution shows that a certain area may become the focus of attack, the defender can make significant deployment in that area, and at the same time, through covert force allocation or virtual target setting, induce the attacker, the player, to change strategy, or the computer to make a counter-plan in advance, thereby further weakening the player's attack effectiveness and enhancing the authenticity and confrontation of the game.

[0188] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by any technician familiar with the technical field within the technical scope disclosed by the present invention and within the spirit and principle of the present invention should be covered by the protection scope of the present invention.

Claims

1. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model is characterized by: Includes steps: S1. Construct a maritime defense confrontation simulation scenario and model the defense force deployment problem as a Nash-Stackelberg hierarchical game model, which is used to describe the internal coordination relationship of the defense force and the confrontation relationship between the defender and the attacker; S2, analyzing the Nash-Stackelberg hierarchical game model to find a game equilibrium solution; S3. Formulate an active defense strategy based on the game equilibrium solution.

2. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 1 is characterized in that: In step S1, the Nash-Stackelberg hierarchical game model is expressed as: G=((L,F),(S L ,S F ),(U L ,U F ),(L,S L ,U L )) Among them, (L, F) is the set of participants, L is the set of defense units, and F is the set of attackers; (S L ,S F ) is the strategy set of the participants, S L is the strategy set of the defense unit L, S F is the strategy set of the attacker F; (U L ,U F ) is the utility function of the participant under the corresponding strategy combination, U F is the utility function of the defense unit L, U L is the utility function of the attacker F, (L,S L ,U L ) is the inner Nash equilibrium.

3. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 2 is characterized in that: The utility function of the defense unit L is expressed as: Among them, u i (s) is the total utility function of the i-th defense unit, s represents a pure strategy for the deployment of all defense units, j represents the j-th utility function, is the value of the jth utility function of the ith defense unit, n represents the nth unit, ω nj is the weighting factor of the jth utility function of the nth unit, and η is the total number of utility objectives.

4. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 3 is characterized in that: The utility function of the defense unit L includes the forward defense depth of the defense unit, the overlapping area of ​​the defense zone, the distance of the defense unit and the interception capability of the defense zone; The utility function of the forward defense depth of the defense unit is expressed as: in, represents the utility function of the i-th defense unit with the forward defense depth as the goal and strategy s, R i represents the defense range of the i-th defense unit, dis min Indicates the minimum distance between the defense unit and the protected target, max Indicates the maximum distance between the defense unit and the protected target, (x i ,y i ) represents the horizontal and vertical coordinates of the i-th defense unit in a Cartesian coordinate system with the protected target as the center origin, due north as the positive direction of the y-axis, and due east as the positive direction of the x-axis; The utility function of the overlapping area of ​​the defense zones is expressed as: in, represents the utility function of the i-th defense unit with the overlapping area of ​​the defense zone as the target and strategy s, Neighbor i represents the set of defense units adjacent to the i-th defense unit, area ij It represents the overlapping area of ​​the defense zone between defense unit i and adjacent defense unit j, which is calculated by the interception radius of the two units and the arc of the overlapping area, as follows: area ij =area i +area j -Has Δ Among them, area i represents the sector area of ​​the overlapping area of ​​defense unit i, area j represents the sector area of ​​the overlapping area of ​​defense unit j, area Δ Represents the common two triangular areas of the sectors occupied by the overlapping areas of defense units i and j; Among them, R i is the interception range of defending unit i, R j is the interception range of defense unit j, α i is the arc corresponding to the sector area of ​​the defense unit i, α j is the arc corresponding to the sector area of ​​the defense unit j; The utility function of the defense unit distance is expressed as: in, represents the utility function of the i-th defense unit with the distance between defense units as the target and strategy s, dis min Indicates the minimum distance between the defense unit and the protected target, max Indicates the maximum distance between the defense unit and the protected target, (x i ,y i ) represents the coordinates of the i-th defense unit in the above coordinate system, (x j ,y j ) represents the coordinates of the j-th defense unit in the above coordinate system; Assume that the minimum interception interval of the i-th defense unit is t i , the incoming target has a velocity of v, and will pass through m units of defense zone in the direction of the incident angle α, and the lengths passed are {l1,l2,...,l M }, then the number of interceptions that the i-th defense unit can make is K i for: Suppose the interception success rate of the i-th defense unit is p i , then the defense unit's interception success rate for the incoming target is Pr i for: In the direction of the incident angle α relative to the horizontal polar axis angle, the total success probability p of the incoming target being intercepted by all deployed defense units after passing through the defense zone of m defense units is α for: Since the closer the defense unit is deployed to the defended target, the larger the range of incident angles it is responsible for defending, and thus the greater its contribution to the defense of the protected target, the utility function of the defense zone interception capability is expressed as: Among them, θ1 and θ2 are the upper and lower limits of the incident angle of the defense range of the i-th defense unit relative to the horizontal polar axis angle, C i The relative weight of the successful contribution of the i-th defense unit to the protection of the central position can be set manually according to the battlefield situation.

5. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 4 is characterized in that: The utility function of the attacker F is expressed by the penetration probability, damage degree, and attack cost of the attack plan: Among them, Damage is the damage value of the unit in the attack plan, count is the number of units in the attack plan, and Cost is the cost loss value of the attack plan.

6. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 1 is characterized in that: The game equilibrium solution obtained by solving the Nash-Stackelberg hierarchical game model satisfies the following conditions: Among them, N is the total number of defense units in the inner game model, represents a pure strategy of the nth defense unit in the inner game model, represents the set of pure strategies of the nth defense unit in the inner game model, represents a pure strategy combination of all units except the nth defense unit in the inner game model. In the outer game model, the attacker, i.e. the follower, takes A pure strategy selected after combining the strategies of In the outer game model, the attacker takes The optimal pure strategy after the strategy combination, It represents a Nash equilibrium solution of the inner game under the premise that the attacker of the outer game takes the best, U L and U F It is a utility function with the strategy combination of the participants as variables.

7. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 1 is characterized in that: The step S2 comprises: S21: Construct a model solving framework based on hybrid particle swarm algorithm; S22: Optimize the model solving framework and solve the game equilibrium solution by iteratively updating particles.

8. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 7 is characterized in that: The step S21 comprises: Initialize the population and randomly generate an initial population of N particles. Each particle represents a deployment plan of the defense unit or an attack plan of the attacker, and the speed of the particle is also randomly initialized. Entering the fitness value calculation stage, a fitness value is calculated for each particle through a preset utility function, and the fitness value reflects the quality of the allocation scheme represented by the particle; Update the individual optimal value and global optimal value of the particle population and record them; The genetic algorithm is introduced for optimization. A roulette wheel selection strategy is used to select a number of individuals from the current population for subsequent genetic operations. A new generation of individuals is generated through single-point crossover or two-point crossover, and gene mutation operations are performed on the new individuals. The new generation of individuals optimized by the particle swarm algorithm and the genetic algorithm form a new population to replace the old population, and repeat the above iterative process until the preset termination conditions are met, such as reaching the maximum number of generations, time limit or convergence of the solution, and output the optimized results to obtain a high-quality game equilibrium solution.

9. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 7 is characterized in that: The step S22 includes: inner layer Nash game optimization and outer layer Stackelberg game optimization; For the optimization solution of the inner layer Nash game, the element parameters and the leader's random strategy are initialized, and the optimal strategy of the corresponding player is obtained by maximizing the fitness function. The strategies of the remaining leaders are adjusted based on the optimal strategy recorded after each iteration and the group dynamics of the particle swarm algorithm-genetic algorithm. In the inner layer optimization process, the best strategy is generated for all units of the leader. All participants obtain the Nash equilibrium, and no participant can obtain higher benefits by changing their strategy; For the optimization solution of the outer Stackelberg game, initialize the element parameters and the random strategies of the followers, run the optimization algorithm of the inner game mentioned above, determine the strategy of a leader, adjust the follower strategy according to the algorithm group dynamics, and terminate the optimization process when the maximum number of iterations is exceeded. Count all the best response strategy combinations of the followers, maximize the leader's effect function, and obtain a Nash-Stackelberg game equilibrium solution.

10. The active maritime defense strategy based on the Nash-Stackelberg hierarchical game model according to claim 9 is characterized in that: The inner Nash game optimization steps include: Step 1: Initialize the parameters of the hybrid particle swarm algorithm, including population size, maximum number of iterations, inertia weight, individual and global learning factors; Step 2: Set the current number of iterations to 0 and generate a leader’s random strategy set as the population particle; Step 3: Record the current optimal population particles; Step 4: Start the iteration cycle and confirm the optimal particle of the previous iteration; Step 5: Update the new round of population particles according to the optimal particles and calculate the corresponding utility function; Step 6: Record and update the best population particles in the current iteration round; Step 7: Repeat the above iterations and finally obtain a Nash equilibrium solution for the leader; The outer Stackelberg game optimization steps include: Step 1: Initialize the parameters of the hybrid particle swarm algorithm, including population size, maximum number of iterations, inertia weight, individual and global learning factors; Step 2: Set the current number of iterations to 0 and generate a set of random strategies for followers as population particles; Step 3: Start loop iteration; Step 4: Run the above algorithm to determine a leader’s strategy; Step 5: Update the population particles of the new round of followers according to the optimal particle and calculate the corresponding fitness function; Step 6: Record and update the best population particles in the current iteration round; Step 7: Repeat the above iterations to finally obtain the best response strategy combination for all followers; Step 8: Calculate the leader’s best response strategy among these strategy combinations and obtain the Nash-Stackelberg game equilibrium solution.

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