Lexicographic-based strategy generation method for fuzzy stackelberg game
By using a lexicographically ordered fuzzy Stackelberg game strategy generation method, the problems of fuzziness and uncertainty in complex network games are solved, more accurate protection strategy optimization is achieved, and the practical application of complex network games is expanded.
Patent Information
- Application Number
- CN202410933086.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-07-12
- Publication Date
- 2025-12-19
- Estimated Expiration
- 2044-07-12
AI Technical Summary
Existing research cannot effectively integrate the ambiguity and uncertainty of decision-makers' understanding of complex network game problems, and cannot reflect the subjective judgments of game participants in actual games, resulting in deficiencies in the formulation of critical infrastructure protection strategies.
A lexicographical order-based strategy generation method for fuzzy Stackelberg games is adopted. The fuzzy Stackelberg game model is transformed into a multi-round multi-objective programming problem, and then transformed into a multi-round two-stage single-objective programming problem using the lexicographical order method, thus obtaining the hybrid strategy Nash equilibrium solution.
It effectively solves the problems of ambiguity and uncertainty in complex network games, provides more accurate optimization results for protection strategies, and broadens the practical application of complex network games.
Smart Images
Figure CN118917057B_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application relates to the technical field of complex network game in system engineering, and in particular to a fuzzy Stackelberg game strategy generation method based on lexicographic order. BACKGROUND
[0002] In the current research field of game theory, there is a special network game, in which the network is not an actual existing network such as a computer system, but a key infrastructure such as a train station, an airport, etc. is abstracted into a node in the network topology structure, and the connection between different stations is abstracted into an edge to establish an infrastructure complex network. In the field of security and protection, the key nodes of the infrastructure are vulnerable to attack, which will affect the public security management and normal social life, and the security and protection department needs to protect these nodes. The attack and protection problem of the key nodes in the infrastructure network can be studied by using the complex network game, which is helpful to develop the best protection strategy and explore the importance of the nodes.
[0003] At present, there are some researches on this kind of problem, but the existing researches only give an objective evaluation method based on the network topology structure, such as calculating the payoff matrix of the first mover and the defender by using the network connectivity performance index-maximal connected component size in the complete information static or dynamic game framework, and calculating the corresponding Nash equilibrium strategy. However, in this kind of actual game problem, the understanding of the problem by the two parties of the game is not certain, the information obtained is insufficient, and the decision-making environment is unpredictable. The existing method cannot well integrate the subjective judgment of the decision maker and cannot express the fuzziness and uncertainty of the actual game problem.
[0004] In today's networked society, Stackelberg game on complex networks is an important research direction in the field of game theory, which mainly involves the construction of game model, strategy analysis and solution, and the expansion of application field. The current research mainly focuses on how to design efficient algorithms and tools for analysis and solution, and promotes its application in different fields and further expands its theoretical research, so as to better cope with the challenges of complex systems in reality. In recent years, the research in this field has made many important progress, and provides new ideas and methods for the research and application of game theory. In the field of fuzzy mathematics, Professor Zadeh proposed the fuzzy theory, which provides a reasonable way to solve this kind of problem. The proposal of this theory provides an inspiration for solving more complex game problems. At present, there are few related researches on introducing fuzzy number theory into Stackelberg game, which has important significance. SUMMARY
[0005] The present application aims at at least solving one of the technical problems existing in the prior art. To this end, the present application discloses a lexicographic-based fuzzy Stackelberg game strategy generation method. The method is based on a fuzzy Stackelberg game model, obtains the fuzzy payoff matrix of the first mover and the follower, converts the solution of the fuzzy Stackelberg game model into the solution of a multi-round multi-objective programming problem, uses the lexicographic method to convert the multi-round multi-objective programming problem into a multi-round two-stage single-objective programming problem, and further obtains the mixed strategy Nash equilibrium solution.
[0006] The purpose of the present application is realized by the following technical scheme. The lexicographic-based fuzzy Stackelberg game strategy generation method comprises the following steps:
[0007] Step 1: determining the players of the game, letting the party that makes the action first be the first mover, and letting the party that makes the action later be the follower;
[0008] Step 2: obtaining the topology structure of the infrastructure network, determining the strategy set of the first mover and the follower, and constructing a fuzzy Stackelberg game model;
[0009] Step 3: determining the fuzzy payoff of the first mover and the follower under each strategy profile in the fuzzy Stackelberg game model through expert decision or experience evaluation, and thus obtaining the fuzzy payoff matrix of the first mover and the follower;
[0010] Step 4: using the solution concept of strong Stackelberg equilibrium to convert the solution of the fuzzy Stackelberg game model into the solution of a multi-round multi-objective programming problem;
[0011] Step 5: using the lexicographic method to convert the multi-round multi-objective programming problem into a multi-round two-stage single-objective programming problem, and further obtaining the mixed strategy Nash equilibrium solution, i.e., obtaining the strategy optimization result of the first mover and the follower;
[0012] The fuzzy payoff matrix can be expressed as wherein is a triangular fuzzy number, m represents the number of strategies of the first mover, n represents the number of strategies of the follower, i represents the i-th strategy of the first mover, and j represents the j-th strategy of the follower, represents the lower bound of the fuzzy number, represents the upper bound of the fuzzy number, represents the median of the fuzzy number.
[0013] Specifically, the first mover is the party that acts first, and there is a selection probability for each strategy of the first mover. The follower is the party that acts later and knows the probability distribution of the commitment of the first mover to all strategies. Usually, the number of selectable strategies of the first mover is m, and i represents the i-th strategy among all strategies. The number of selectable strategies of the follower is n, and j represents the j-th strategy among all strategies.
[0014] Specifically, the fuzzy payoff matrix is divided into a fuzzy payoff matrix of a leader and a fuzzy payoff matrix of a follower, wherein the payoff of the leader when the leader selects strategy i and the follower selects strategy j is a fuzzy number The payoff of the follower when the leader selects strategy i and the follower selects strategy j is a fuzzy number Thus, the fuzzy payoff matrix of the leader in different pure strategy situations is represented as:
[0015]
[0016] wherein m represents the total number of strategies of the leader, n represents the total number of strategies of the follower, and each element in the payoff matrix is a lower bound of the fuzzy number, an upper bound of the fuzzy number, a median of the fuzzy number.
[0017] The fuzzy payoff matrix of the follower in different pure strategy situations is represented as:
[0018]
[0019] wherein m represents the total number of strategies of the leader, n represents the total number of strategies of the follower, and each element in the payoff matrix is a lower bound of the fuzzy number, an upper bound of the fuzzy number, a median of the fuzzy number.
[0020] Further, the fuzzy payoff matrix of the leader and the fuzzy payoff matrix of the follower can be used to obtain the optimal mixed strategy of the leader for each pure strategy t of the follower by solving the following multi-objective programming problem:
[0021]
[0022] wherein, represents an arbitrary mixed strategy probability distribution of the leader; represents an optimal mixed strategy probability distribution of the leader; represents a lower bound of the fuzzy payoff of the leader when the leader selects strategy i and the follower selects strategy t, represents an upper bound of the fuzzy payoff of the leader when the leader selects strategy i and the follower selects strategy t, represents a median of the fuzzy payoff of the leader when the leader selects strategy i and the follower selects strategy t; represents a lower bound of the fuzzy payoff of the follower when the leader selects strategy i and the follower selects strategy t, This represents the upper bound of the fuzzy payoff for followers when the forer chooses strategy i and the follower chooses strategy t. This represents the median of the fuzzy payoffs of followers when the forer chooses strategy i and the followers choose strategy t. It is the forer's optimal hybrid strategy calculated for each follower's pure strategy t. Then, we can calculate the forerunner payoff for each follower under pure policy t. Take the maximum value For the optimal benefit of the first mover, and The globally optimal hybrid strategy for the pioneer. The best response strategy for followers.
[0023] Furthermore, using the lexicographical order method, the multi-objective programming problem is transformed into a two-stage single-objective programming problem. The first stage is:
[0024]
[0025] The optimal strategy for the first stage can be obtained as follows: Phase Two:
[0026]
[0027] The global optimal hybrid strategy for the forerunner can be obtained through the planning problem in the second stage. in, Denotes the probability distribution of any mixed strategies for any first mover; This represents the probability distribution of the mixed strategy for the highest priority mover in the first phase;
[0028] This represents the probability distribution of the mixed strategy of the highest priority mover after two phases. Let represent the lower bound of the fuzzy payoff for the forerunner when the forerunner chooses strategy i and the followers choose strategy t. This represents the upper bound of the fuzzy payoff for the forer when the forer chooses strategy i and the followers choose strategy t. Let represent the median of the fuzzy payoff for the forerunner when the forerunner chooses strategy i and the followers choose strategy t. This represents the lower bound of the fuzzy payoff for followers when the forer chooses strategy i and the follower chooses strategy t. This represents the upper bound of the fuzzy payoff for followers when the forer chooses strategy i and the follower chooses strategy t. Let represent the median of the follower's fuzzy payoff when the forer chooses strategy i and the follower chooses strategy t.
[0029] The forerunner payoff for each follower pure strategy t is described. pass To obtain the optimal benefit for the first mover, and The globally optimal hybrid strategy for the pioneer. The best response strategy of the follower, wherein, represents that in all t, the first satisfies The maximum, secondly satisfies The maximum;
[0030] Finally, from the above multi-round two-stage single-objective planning problem, the global optimal mixed strategy of the leader can be obtained The best response strategy of the follower
[0031] Compared with the prior art, the method has the advantages that the prior research cannot reflect the fuzziness of the understanding of the game problem by the decision maker, the method combines the fuzzy mathematics theory, and proposes a fuzzy Stackelberg game strategy generation method based on lexicographic order, obtains the strategy optimization results of the leader and the follower, and analyzes the results. The uncertainty of the strong Stackelberg game is explained by using the intuitionistic fuzzy theory, so that the application of the complex network game research in practice can be greatly widened. BRIEF DESCRIPTION OF DRAWINGS
[0032] Figure 1 A flowchart of an embodiment of the present application is shown;
[0033] Figure 2 A schematic diagram of an infrastructure network in an embodiment of the present application is shown. DETAILED DESCRIPTION
[0034] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be further described in detail below with reference to the drawings. Obviously, the described embodiments are only some of the embodiments of the present application, but not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the present application.
[0035] It should be understood that the specific embodiments described herein are only used to explain the present application, and are not used to limit the present application.
[0036] In the present embodiment, only one leader and one follower are considered, and both parties are fully aware of the topology information of the existing network. A key infrastructure, such as a railway network, can be abstracted as a simple undirected graph G(V, E), wherein V = {v1, v2,..., v N} represents the set of all nodes in the network, i.e. the stations in the railway network, wherein N = |V| represents the number of nodes in the network. is the set of all edges in the network, i.e. the railway lines in the railway network.
[0037] Only one predecessor and one follower are considered, and both of them are fully aware of the topology information of the existing network. The objects of all attacks and defenses are nodes in the network. The criterion for judging whether a node is successfully attacked is that the node is attacked by the predecessor without being protected by the follower. The two persons play a zero-sum game, and the more important a node is, the higher the cost required for attack or defense is.
[0038] As shown in the method for generating a strong Stackelberg game strategy based on an intuitionistic fuzzy set, the method comprises the following steps: Figure 1
[0039] Step 1, determining players of the game, letting the party that first makes an action be a predecessor, and letting the party that makes an action later be a follower;
[0040] Step 2, obtaining a topology structure of an infrastructure network, determining a strategy set of the predecessor and the follower, and constructing a fuzzy Stackelberg game model;
[0041] Step 3, determining fuzzy revenues of the predecessor and the follower under each strategy profile in the fuzzy Stackelberg game model through expert decision or experience evaluation, and thus obtaining a fuzzy revenue matrix of the predecessor and the follower;
[0042] Step 4, converting solving of the fuzzy Stackelberg game model into solving of a multi-round multi-objective programming problem by using a solution concept of a strong Stackelberg equilibrium;
[0043] Step 5, converting the multi-round multi-objective programming problem into a multi-round two-stage single-objective programming problem by using a lexicographic method, and thus obtaining a mixed strategy Nash equilibrium solution, that is, obtaining an optimization result of a strategy of the predecessor and the follower;
[0044] The fuzzy revenue matrix can be expressed as wherein is a triangular fuzzy number, m represents a number of strategies of the predecessor, n represents a number of strategies of the follower, i represents an i th strategy of the predecessor, and j represents a j th strategy of the follower, represents a lower bound of the fuzzy number, represents an upper bound of the fuzzy number, and represents a median of the fuzzy number.
[0045] Specifically, the predecessor is the party that first makes an action, and there is a selection probability for each strategy of the predecessor. The follower is the party that makes an action later and knows a probability distribution of a commitment of the predecessor to all strategies. Usually, a number of selectable strategies of the predecessor is m, and i represents an i th strategy in all strategies. A number of selectable strategies of the follower is n, and j represents a j th strategy in all strategies.
[0046] Specifically, the fuzzy payoff matrix is divided into a fuzzy payoff matrix of a leader and a fuzzy payoff matrix of a follower, wherein the payoff of the leader when the leader selects strategy i and the follower selects strategy j is a fuzzy number The payoff of the follower when the leader selects strategy i and the follower selects strategy j is a fuzzy number Therefore, the fuzzy payoff matrix of the leader in different pure strategy situations is represented as:
[0047]
[0048] wherein m represents the total number of strategies of the leader, n represents the total number of strategies of the follower, and each element in the payoff matrix is represents the lower bound of the fuzzy number, represents the upper bound of the fuzzy number, represents the median of the fuzzy number.
[0049] The fuzzy payoff matrix of the follower in different pure strategy situations is represented as:
[0050]
[0051] wherein m represents the total number of strategies of the leader, n represents the total number of strategies of the follower, and each element in the payoff matrix is represents the lower bound of the fuzzy number, represents the upper bound of the fuzzy number, represents the median of the fuzzy number.
[0052] Further, the fuzzy payoff matrix of the leader and the follower can obtain the optimal mixed strategy of the leader for each pure strategy t of the follower By solving the following multi-objective programming problem:
[0053]
[0054] wherein, represents an arbitrary mixed strategy probability distribution of the leader; represents the optimal mixed strategy probability distribution of the leader; represents the lower bound of the fuzzy payoff of the leader when the leader selects strategy i and the follower selects strategy t, represents the upper bound of the fuzzy payoff of the leader when the leader selects strategy i and the follower selects strategy t, represents the median of the fuzzy payoff of the leader when the leader selects strategy i and the follower selects strategy t; represents the lower bound of the fuzzy payoff of the follower when the leader selects strategy i and the follower selects strategy t, represents the lower bound of the fuzzy payoff of the follower when the leader chooses strategy i and the follower chooses strategy t, represents the median of the fuzzy payoff of the follower when the leader chooses strategy i and the follower chooses strategy t, and the leader's optimal mixed strategy corresponding to each follower's pure strategy t is calculated After that, the leader's payoff under each follower's pure strategy t can be calculated Take the maximum value is the optimal payoff of the leader, and is the global optimal mixed strategy of the leader, is the best response strategy of the follower.
[0055] Further, the multi-round multi-objective programming problem solving method uses the lexicographic method to convert the multi-objective programming problem into a two-stage single-objective programming problem, the first stage:
[0056]
[0057] The optimal strategy of the leader in the first stage is The second stage:
[0058]
[0059] The global optimal mixed strategy of the leader can be obtained through the second stage programming problem where, represents the probability distribution of any leader mixed strategy; represents the probability distribution of the optimal leader mixed strategy in the first stage;
[0060] represents the probability distribution of the optimal leader mixed strategy after two stages; represents the lower bound of the fuzzy payoff of the leader when the leader chooses strategy i and the follower chooses strategy t, represents the upper bound of the fuzzy payoff of the leader when the leader chooses strategy i and the follower chooses strategy t, represents the median of the fuzzy payoff of the leader when the leader chooses strategy i and the follower chooses strategy t; represents the lower bound of the fuzzy payoff of the follower when the leader chooses strategy i and the follower chooses strategy t, represents the upper bound of the fuzzy payoff of the follower when the leader chooses strategy i and the follower chooses strategy t, represents the median of the fuzzy payoff of the follower when the leader chooses strategy i and the follower chooses strategy t;
[0061] The leader's payoff under each follower's pure strategy t is Through the optimal payoff of the leader is obtained, and the globally optimal mixed strategy for the leader, the best response strategy for the follower, where, denotes that among all t, the first one to satisfy is the maximum, and the second one to satisfy is the maximum;
[0062] Finally, from the above multi-round two-stage single-objective planning problem, the globally optimal mixed strategy for the leader the best response strategy for the follower
[0063] Those skilled in the art will appreciate that embodiments of the application can be provided as methods, systems or computer program products. Accordingly, the application can take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the application can take the form of a computer program product on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROMs, optical storage devices, etc.) embodying computer readable program code thereon for use by or in connection with an instruction execution system.
Claims
1. A lexicographic-based strategy generation method for a fuzzy Stackelberg game, characterized in that, The method comprises: Step 1, determining the players of the game, taking the party making the action first as the leader, and taking the party making the action later as the follower; Step 2, acquiring the topology of the infrastructure network, determining the strategy set of the leader and the follower, and constructing a fuzzy Stackelberg game model; Step 3, determining the fuzzy benefits of the leader and the follower under each strategy profile in the fuzzy Stackelberg game model through expert decision or experience evaluation, thereby obtaining the fuzzy benefit matrix of the leader and the follower; Step 4, using the solution concept of strong Stackelberg equilibrium, converting the solution of the fuzzy Stackelberg game model into the solution of a multi-round multi-objective programming problem; Step 5, using the lexicographic method, converting the multi-round multi-objective programming problem into a multi-round two-stage single-objective programming problem, and then obtaining the mixed strategy Nash equilibrium solution, that is, obtaining the strategy optimization result of the leader and the follower; The fuzzy payoff matrix is represented as wherein is represented as a triangular fuzzy number, m represents the number of strategies of the leader, n represents the number of strategies of the follower, i represents the ith strategy of the leader, j represents the jth strategy of the follower, represents the lower bound of the triangular fuzzy number, represents the upper bound of the triangular fuzzy number, represents the middle value of the triangular fuzzy number; The fuzzy payoff matrix is divided into a fuzzy payoff matrix of a leader and a fuzzy payoff matrix of a follower, and the win of the leader when the leader selects strategy i and the follower selects strategy j is a fuzzy number And the win of the follower when the leader selects strategy i and the follower selects strategy j is a fuzzy number Therefore, the fuzzy payoff matrix of the leader in different pure strategy situations is represented as: where each element in the payoff matrix is denotes the lower bound of the fuzzy number, denotes the upper bound of the fuzzy number, denotes the middle value of the fuzzy number; The fuzzy benefit matrix of the follower under different pure strategy situations is expressed as: where each element in the payoff matrix is denotes the lower bound of the fuzzy number, denotes the upper bound of the fuzzy number, denotes the middle value of the fuzzy number; Based on the preceding leader and follower's fuzzy payoff matrices described in Step 3, for each pure strategy t of the follower, the optimal mixed strategy of the leader is obtained by solving the following round of multi-objective programming problem wherein, represents an arbitrary leader mixed strategy probability distribution, represents the optimal leader mixed strategy probability distribution; represents the lower bound of the leader's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, represents the upper bound of the leader's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, represents the median of the leader's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t; represents the lower bound of the follower's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, represents the upper bound of the follower's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, represents the median of the follower's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, after computing the leader's optimal mixed strategy for each follower's pure strategy t taking the maximum optimal payoff for the leader, and a globally optimal mixed strategy for the leader, a best response strategy for the follower.
2. The lexicographic-based fuzzy Stackelberg game strategy generation method according to claim 1, characterized in that, Using the lexicographic method, the multi-objective programming problem is converted into a two-stage single-objective programming problem, the first stage: The optimal strategy for the first stage of the leader is Second stage: the global optimal mixed strategy of the leader through the second stage planning problem where, denotes the arbitrary leader mixed strategy probability distribution, denotes the first stage optimal leader mixed strategy probability distribution, denotes the two stage post optimal leader mixed strategy probability distribution; denotes the lower bound of the leader's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, denotes the upper bound of the leader's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, denotes the median of the leader's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t; denotes the lower bound of the follower's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, denotes the upper bound of the follower's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t, denotes the median of the follower's fuzzy payoff when the leader chooses strategy i and the follower chooses strategy t; the leader optimal payoff for each follower pure strategy t by obtained, and is the global optimal mixed strategy for the leader, is the best response strategy for the follower, where, denotes that among all t, first satisfies maximizes, second satisfies maximizes; Finally, the global optimal mixed strategy of the leader is obtained The optimal response strategy of the follower