Large-scale distributed Nash equilibrium search method and system based on event triggering

Abstract

The invention discloses a distributed Nash equilibrium search method and system based on event triggering, and aims to solve the problem of complex decision in a large-scale dynamic non-cooperative system. By constructing an undirected graph model, defining network participants and communication relationships thereof, and combining with a network average field game theory, macroscopic interaction between individuals and group behaviors is realized. According to the method, three auxiliary variables are introduced, the group aggregation behavior is dynamically estimated on line, and the communication weight between neighbors is optimized through adaptive learning gain, so that the calculation and communication cost is reduced. And an event triggering mechanism is also adopted, so that the information exchange opportunity is effectively controlled, and unnecessary communication overhead is avoided. In this way, players can realize optimization of leader-free consensus and local objective functions under the condition of incomplete information, and decision-making efficiency and expandability of the system are remarkably improved. The method is suitable for various large-scale complex network scenes.

Description

Technical Field

[0001] The present invention relates to the field of distributed control technology, and in particular to a fully distributed Nash equilibrium search method and system based on event triggering in a networked mean field game. Background Art

[0002] In large-scale dynamic non-cooperative systems, individual decisions are significantly affected by the behaviors of other individuals, and solving the resulting complexity has always been a challenge for academic research. Mean Field Games (MFG) theory aims to reveal the characteristics and evolutionary trends of the system by studying the strategies of rational players. The theory assumes that the players are anonymous individuals and simplifies the complexity by utilizing the "smoothing effect" generated by the interactions between a large number of individuals, thereby effectively solving the computational complexity problem in large-scale interactions. In recent years, MFG theory has been widely used in fields such as economics, finance, engineering, and machine learning, providing effective modeling and solution solutions for large-scale complex networks.

[0003] In non-cooperative games, players need to make dynamic decisions in both time and space to optimize the indivisible and nonlinear cumulative rewards. However, it is not realistic to obtain all other players' state information in real time, which leads to the challenge of incomplete information games. To this end, some distributed computing and learning methods based on local information have been proposed in recent years. Among them, a distributed Nash equilibrium (NE) search algorithm based on the leader-follower consensus protocol was proposed to guide players to their own optimal solution.

[0004] In addition, a discrete-time NE search algorithm was developed for aggregate games using dynamic consensus tracking and projected gradient methods, but as the number of players increases, the action and state space grows exponentially, greatly limiting the scalability of multi-player learning. MFG theory helps optimize decision-making for agents in large and complex systems by compressing the problem space, so it is particularly important to study NE search methods in networked mean field games.

[0005] In order to achieve accurate estimation of global information under local interaction, distributed consensus control theory is usually used. However, in actual scenarios, signal transmission between adjacent players consumes a lot of network resources, especially when the players' sensing and communication capabilities are limited. To this end, some studies have proposed event-triggered sampling and extended it to multi-intelligent systems, and designed a distributed adaptive event-triggered protocol that is independent of the network size to avoid continuous monitoring of neighbor status. At the same time, by weighing the errors between itself and its neighbors, the adaptability of players is enhanced.

[0006] Therefore, it is more in line with practical needs to integrate event triggering mechanisms into player strategy design, especially in scenarios involving a large number of players, in order to reduce the communication cost of networked mean-field games.

[0007] In distributed consensus protocols, association weights are used to describe a player's trust in his or her neighbors. In practice, the determination of weights depends on the player's objective function, the communication network topology, and the number of players participating in the game. To this end, adaptive learning gain is introduced to control the estimation error caused by the imbalance of confidence distribution. In this context, the researchers dynamically adjust the weights according to the consensus error, thereby forming a more flexible fully distributed coordination control architecture. Summary of the invention

[0008] In view of the defects of the prior art, the present invention provides a large-scale distributed Nash equilibrium search method and system based on event triggering. A networked mean field game is introduced in a finite time and space domain to simplify the description of distributed online decision-making problems to be suitable for large-scale NE calculations. The present invention also designs three parallel auxiliary variables for dynamically identifying group behaviors, called mean field terms, by integrating estimation theory and consensus theory. Among them, players need to achieve leaderless consensus and optimization of local objective functions at the same time, introduce dynamic event triggering mechanism and adaptive learning gain, and provide an efficient and scalable distributed communication and computing framework for large-scale complex networks.

[0009] In order to achieve the above invention object, the technical solution adopted by the present invention is as follows:

[0010] A large-scale distributed Nash equilibrium search method based on event triggering, comprising:

[0011] S1: Construct an undirected graph The node set represents the player, the edge set ε describes the communication relationship between players, and defines the adjacency matrix and the Laplace matrix L to describe the connection relationship of the network;

[0012] S2: The network mean field game (NMFG) model is introduced to describe the macroscopic interaction between individuals and groups under local information interaction;

[0013] S3: Design three auxiliary variables to dynamically estimate the aggregation behavior of the group, dynamically track the global behavior and network size through local neighbor information, and realize the estimation and correction of the global state under incomplete information;

[0014] S4: Adaptive learning gain is used to allow players to dynamically adjust the association weights with neighbors based on local estimation errors, ensuring the scalability of the algorithm in fully distributed scenarios;

[0015] S5: Implement leaderless consensus and optimization of local objective functions to reduce communication costs and computational complexity;

[0016] S6: Introduce an event trigger mechanism to determine the timing for players to exchange information with their neighbors and improve decision-making efficiency.

[0017] Furthermore, S1 specifically includes:

[0018] Define a social network of N players And build the node set

[0019] Establish communication relationships and determine edge sets based on the communication capabilities between players

[0020] Define the adjacency matrix as When nodes i and j are connected, the weight is a ij =1, otherwise the weight a ij =0.

[0021] Define a diagonal matrix in a ij represents the degree of node i.

[0022] The Laplacian matrix is ​​calculated as a if and only if (i,j)∈ε and (j,i)∈ε ij =a ji In particular, when When a is an undirected graph ii =0.

[0023] Furthermore, S2 specifically includes:

[0024] Define the set of locally feasible actions for each player i

[0025] Defining the global action space

[0026] Optimize the cost function for each player in the time interval [0,T]:

[0027]

[0028] In the formula, x i,t represents the action vector of player i at time t; F i,t (x i,t ,ρ(x t )) represents the cost function of player i at time t, which depends on the player's action x i,t and the mean field term ρ(x t ).ρ(x t ) represents the mean field term, which is the aggregate information of the action variables of all players at time t. i (x i,t) is the action x of player i i,t Related private functions. i (x i,t ,ρ(x t )) represents the process cost, which is related to the current action of player i and the average field term, reflecting the immediate cost in this state. i (x i,T ) represents the terminal cost, reflecting the cost of player i at time T. N represents the number of players.

[0029] Furthermore, S3 specifically includes:

[0030] The action of player i is updated by projecting the gradient as:

[0031]

[0032] in, represents the action vector of player i at time t, i.e. the updated choice.

[0033] Represents the feasible action set Ω i The projection operation ensures that the updated motion vector Still within the feasible set.

[0034] α represents the step size or learning rate, which is a parameter that controls the update amplitude.

[0035] The cost function F i,t Action vector for player i gradient.

[0036] Denote the cost function F i,t For the mean field term gradient.

[0037] Represents a private function φ i The gradient of a player's actions reflects the impact of the player's choice on his private cost.

[0038] Each player maintains an auxiliary variable y i,t ,z i,t ,w i,t , and dynamically updated to:

[0039]

[0040] Initialized to:

[0041]

[0042] In the formula, represents the fixed neighbor learning gain between players i and j.

[0043] Auxiliary variable representing player i at time t, step k+1, used to dynamically estimate mean field information.

[0044] Auxiliary variable representing player i at time t, step k.

[0045] represents the fixed neighbor learning gain between players i and j, reflecting the efficiency of information transfer between players.

[0046] a ij Represents the strength of the connection between players i and j in an undirected graph, which is 1 if there is a connection, otherwise 0.

[0047] It represents the difference between the private function of player i at step k+1 and step k at time t, reflecting the change of the player's status.

[0048] A second auxiliary variable representing player i at time t, step k+1, used to estimate the total size.

[0049] A second auxiliary variable representing the k-th step of player i at time t.

[0050] A third auxiliary variable representing player i at time t at step k+1, possibly used to adjust the estimated distribution.

[0051] A third auxiliary variable representing the k-th step of player i at time t.

[0052] y (i,0) Represents the initial auxiliary variable for player i, combining the private function values ​​of this player and his neighbors.

[0053] z (i,0) =1+d i represents the initial size estimate of player i, d i is the player's degree (number of neighbors).

[0054] w (i,0) =1 represents the initial weight of player i.

[0055] Furthermore, S4 specifically includes:

[0056] Design adaptive learning gain instead of fixed gain The details are as follows

[0057]

[0058] in, and Three auxiliary variables representing the difference between players i and j at time t, step k+1.

[0059] and Three auxiliary variables representing the relationship between players i and j at time t at step k.

[0060] and represents the learning gain between players i and j, and three auxiliary variables that control the update speed.

[0061] and Represents the square norm of the differences in the three auxiliary variables between players j and i, measuring the degree of difference in their states.

[0062] The symmetry of the learning gain is set so that the node in-degree is equal to the out-degree to ensure the stability and consistency of the estimation.

[0063] Furthermore, S6 specifically includes:

[0064] Defining the trigger instant sequence To decide when to exchange information, it means the mth trigger moment of player i at time t is used to decide when to exchange information.

[0065] Update broadcast variables:

[0066]

[0067] In the formula, The broadcast variable at step k+1 at time t represents the updated state of player i.

[0068] The broadcast variable at step k at time t represents the current state of player i.

[0069] Auxiliary variable representing the information difference between players i and j at time t, step k.

[0070] a ij : The connection weight between players i and j, indicating the strength of information transmission.

[0071] Represents the state change at the kth step at time t.

[0072] Define the dynamics of a quantized variable:

[0073]

[0074] In the formula, The quantized variable at the k+1th step at time t represents a certain state quantity after the update of player i.

[0075] The quantized variable at the kth step at time t represents the current state of player i.

[0076] β i : Decay factor for player i, controlling how much the quantized variables are retained during updates.

[0077] δ i Adjustment factor for player i, affecting the update rate of quantitative variables.

[0078] σ i The standard deviation of player i is usually related to the volatility of the system.

[0079] An estimate of a quantity at the kth step at time t.

[0080] Represents the degree of self-influence or self-learning of player i.

[0081] The trigger moment is defined as:

[0082]

[0083] In the formula, Φ is used to evaluate the constraints in the current state, involving errors Quantitative variables and

[0084] The error at step k at time t represents the deviation in the state update made by player i.

[0085] The update at the m+1th triggering moment at time t indicates the time of the next information exchange.

[0086] The present invention also discloses a large-scale distributed Nash equilibrium search method system, which can be used to implement the above-mentioned large-scale distributed Nash equilibrium search method, specifically including:

[0087] Social network building blocks: constructing undirected graphs, defining node sets, building adjacency matrices and Laplacian matrices

[0088] Network Mean Field Game Model Module: Introduces the Network Mean Field Game (NMFG) model to describe the macroscopic interactions between individuals and groups and to handle the impact of local information interactions;

[0089] Aggregation behavior estimation module: Design auxiliary variables, dynamically estimate the aggregation behavior of the group, track the global state through local neighbor information, and realize the estimation and correction of the global state;

[0090] Adaptive Learning Gain Module: Introduces an adaptive learning gain mechanism that allows players to adjust neighbor association weights based on local estimation errors, ensuring the scalability of the algorithm in fully distributed scenarios;

[0091] Optimization module: realize leaderless consensus, optimize local objective functions, and reduce communication costs and computational complexity;

[0092] Event trigger mechanism module: introduce event trigger mechanism to determine the timing of information exchange between players and neighbors and improve decision-making efficiency;

[0093] Information exchange module: handles information transmission between players to ensure real-time update and sharing of information;

[0094] Results display module: Displays key indicators such as player strategy changes, convergence status, and network status through charts and graphs.

[0095] Update the player's status and global balance results in real time, allowing users to intuitively understand the system operation status.

[0096] Generate system operation reports, including performance evaluation, convergence speed and summary of final results.

[0097] Provides a user-friendly interface that allows users to view and analyze results, and make parameter adjustments and settings.

[0098] The present invention also discloses a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the above-mentioned large-scale distributed Nash equilibrium search method when executing the program.

[0099] The present invention also discloses a computer-readable storage medium on which a computer program is stored. When the program is executed by a processor, the large-scale distributed Nash equilibrium search method is implemented.

[0100] Compared with the prior art, the advantages of the present invention are:

[0101] 1. Application of Networked Mean Field Game Theory: By introducing the NMFG theory, an effective distributed Nash equilibrium (NE) search communication and computing framework is provided for large-scale group decision-making problems, which can achieve effective decision-making within a limited time.

[0102] 2. Design of auxiliary estimation variables: Combining estimation theory and consensus theory, three auxiliary estimation variables are designed, which can track group aggregation behavior online in a distributed manner, and avoid continuous reception and broadcasting of communications through public dynamic event triggering strategies, thereby improving the communication efficiency of the system.

[0103] 3. Fully distributed network element search algorithm: The proposed three-parallel iterative algorithm adjusts the communication weight through adaptive learning gain, eliminating the dependence on global prior knowledge, making the algorithm still effective in an incomplete information environment and improving the flexibility and adaptability of the system.

[0104] 4. Improve decision-making efficiency and stability: The system can respond to changes quickly, ensure stability and robustness in a dynamic environment, and enhance decision-making efficiency.

[0105] 5. Reduce communication costs and complexity: Through event triggering mechanism and distributed structure, unnecessary information exchange is reduced, and communication costs and computational complexity are reduced. BRIEF DESCRIPTION OF THE DRAWINGS

[0106] Figure 1 It is a schematic diagram of a randomly generated connected communication network according to an embodiment of the present invention.

[0107] Figure 2 is an evolution diagram of the actions of the embodiment of the present invention, (a) The first-dimensional evolution diagram of , (b) The second-dimensional evolution diagram of , (c) 's trajectory diagram.

[0108] Figure 3 is an evolution diagram of the estimated variables of the embodiment of the present invention. (a) The first-dimensional evolution diagram of , (b) The second-dimensional evolution diagram of , (c) Evolution diagram.

[0109] Figure 4 is the evolution of the adaptive learning gain in the embodiment of the present invention. (a) Evolution diagram of , (b) Evolution diagram of , (c) Evolution diagram.

[0110] Figure 5 This is a diagram for verifying the event triggering mechanism of an embodiment of the present invention. (a) Evolution diagram of, (b) communication triggering ratio.

[0111] Figure 6 It is a schematic diagram of the framework of the network mean field game model of an embodiment of the present invention. DETAILED DESCRIPTION

[0112] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and examples.

[0113] The present invention provides a large-scale distributed Nash equilibrium search method based on event triggering, the main steps of which are as follows:

[0114] 1) Construct an undirected graph to describe the topological structure of the communication network, and establish a network mean field game (NMFG) to describe the macroscopic interaction between individuals and groups under local information interaction, so as to solve the decision-making complexity problem in large-scale group games, such as Figure 6 shown.

[0115] 2) In order to update its own action vector based on gradient calculation, three auxiliary estimation variables are introduced to dynamically track and estimate the global group behavior and network size based on local neighbor information, so as to realize the estimation and correction of the global state under incomplete local information.

[0116] 3) The introduction of adaptive learning gain allows each player to dynamically adjust the association weight with its neighbors based on the local estimation error, ensuring the scalability of the algorithm in a fully distributed scenario.

[0117] 4) Design a common event trigger mechanism to give a parallel iteration formula for auxiliary estimation variables to reduce communication costs and computational complexity.

[0118] The method for constructing an undirected graph and establishing a network mean field game in step 1) is as follows:

[0119] The present invention uses an undirected graph To describe the topology of a communication network, where the node set

[0120] Edge Set The adjacency matrix of all nodes is defined as When nodes i and j are connected, the weight is a ij =1, otherwise the weight a ij =0. Represents the set of neighbor nodes of point i. Define the diagonal matrix in a ij represents the degree of node i. Correspondingly, The Laplace matrix of a if and only if (i,j)∈ε and (j,i)∈ε ij =a ji . In particular ii =0 when is an undirected graph.

[0121] Consider a set of rational players Each player From its local feasible action set Select its action vector represents the global action space, In the non-cooperative network game scenario considered in the present invention, it is assumed that each player Optimize its own cost function in a pre-specified time interval [0,T]:

[0122]

[0123] in It is called the mean field (MF) term, which encapsulates the aggregate information of all action variables, private function Only player i knows, are process cost and terminal cost respectively. Generally speaking, G i Designed to force certain specific choices on individual players.

[0124] In (1), the cost function of each player is coupled to the overall evolution of the system, which means that the complex micro-individual behaviors of the group are simplified and abstracted into macro-dynamic decision variables. In addition, under the limited perception of the networked mean field game, each player needs to maintain a local estimate of the MF term to capture the interaction rules in the large-scale individual evolution system. The rigorous mathematical definition of NE is given below to analyze the results of the players' adaptive behaviors.

[0125] Definition 1: If for all and There is a set of joint actions So that the following inequality holds

[0126]

[0127] in Then it is called It is the NE of N player NMFG.

[0128] Due to the dependence of the local loss function on the MF term, let Respectively represent F i,t (x i,t ,ρ(x t )) for x i,t and ρ(x t ). Define the pseudo gradient vector of all players as

[0129]

[0130] Assumption 1:

[0131] (a) It is strongly connected.

[0132] (b) For all t∈[0,T], In Ω i The top is convex.

[0133] (c) and They satisfy l1-Lipschitz and l2-Lipschitz continuity, that is,

[0134]

[0135] For any x,x′∈Ω, Where l1 and l2 are positive constants.

[0136] (d) and They are all bounded by the constant C>0.

[0137] Note: Assuming the existence of a unique NE, this means If and only if Established at that time.

[0138] The specific method for updating the auxiliary vector in step 2) is as follows:

[0139] Because F i,t It is gradually revealed. For each k-th iteration at each time step, the action of player i is dynamically updated based on the projected gradient as follows:

[0140]

[0141] The step size is 0<α≤1. This projection can restrict x i,t Stay in the tight set Ω i In this way, unsafe behaviors can be corrected to a reasonable range.

[0142] Note that the MF term in (3) and the size of the player group N are global information. In a networked game, a single player cannot obtain this information, especially when only partial decision-making information is available. Unbalanced distribution, each node i needs to calculate the scaled global sum and the scaled number of sensor nodes b i d i , where b i is an arbitrary non-zero scalar. The MF estimation problem can be solved by replacing ρ(x t ) is decomposed into a local state vector and an additional scalar state Therefore, each player maintains an auxiliary variable y i,t ,z i,t ,w i,t , and update them completely in parallel to dynamically estimate and track ρ(x t )and Expressed as

[0143]

[0144] Initialized to:

[0145]

[0146] in represents the fixed neighbor learning gain between players i and j.

[0147] thereby and The update can be written as

[0148]

[0149] The specific method of the adaptive learning gain design in step 3) is as follows:

[0150] The associated coupling weight a in NE calculation ij describes the strength of the connection between a network player and his neighbors. The purpose of the neighbor learning gain is to eliminate the uncertainty in the trust estimate between connected players. However, since the choice of gain depends on the Lipschitz constant of the gradient associated with the player and the centralized Laplacian matrix associated with the network topology, it is challenging to explicitly calculate the bounds to ensure the stability of the gradient solution. By introducing an additional stabilization term, the present invention designs an adaptive learning gain to replace the one in formula (4) The details are as follows

[0151]

[0152] Among them, the normal number and Used to adjust the update rate of the estimator without affecting the convergence analysis.

[0153] Assumption 2: Constant Satisfies the symmetry, that is, for and

[0154] Lemma 1: For an undirected network when Estimated Value of All Players and The average value of remains unchanged, that is for

[0155] Remark 2: Assumption 2 and Lemma 1 ensure that within a time interval unit, the change in density mass is equal to the inflow mass minus the outflow mass, thus achieving average consistency.

[0156] Note 3: Unlike fixed constant gain, adaptive gain is fully distributed and scalable, does not require global information of the network, and does not depend on the size of the network. Therefore, it avoids losing information from non-neighbor players through indirect multi-hop communication paths in large-scale networks.

[0157] In the presence of network errors, each player can estimate and asymptotically reconstruct the MF terms and scaling coefficients in non-stationary games. Another issue in distributed algorithm design is how to balance communication complexity and computational complexity. Event-driven thinking improves adaptability to uncertainty and reduces response time, leading to more efficient decision-making. Action-oriented strategies triggered by dynamic events are closer to real behavioral logic. Therefore, the distributed NE search strategy will be designed based on the dynamic event triggering mechanism.

[0158] The specific method of reconstructing the estimated variables through the event triggering mechanism in step 4) is as follows

[0159] Specifically, the present invention defines the discrete trigger instant sequence as It determines when individual players exchange information with their neighbors. The most recently received broadcast variable Updated in zero-order hold mode as The jump dynamics at the trigger moment are:

[0160]

[0161] Compared with the classic continuous transmission strategy, the event trigger mechanism increases the tolerance to inaccurate local estimation information and subsequent gradient information. In order to dynamically adjust the threshold and determine the common trigger time of the three variables, the quantization variable is defined. Dynamically defined based on something like (7)

[0162]

[0163] where χ i,0 >0,β i >0,δ i ∈[0,1],σ i ∈[0,1), and

[0164]

[0165] Given an allowable constant θ i and relative trigger conditions, the dynamic event generation function is defined as

[0166]

[0167] Further, the trigger time series is obtained

[0168]

[0169] Note 4: Attention is time-varying, and its update depends on self-expectation, relative error with neighbors, and negative self-feedback. Therefore, the event triggering strategy described in equations (8) and (10) depends on the actual state of each player, rather than the sampled state information of its neighbors. This eliminates the need to continuously monitor the state of neighbors and obtain global parameters in advance in distributed deployments, thereby avoiding continuous interaction between neighbors.

[0170] The complete online distributed algorithm is shown in Algorithm 1 in the schematic diagram of this patent solution, which provides a fully distributed framework for NE search and decision mechanism analysis of NFMG. In each iteration, each player corrects its action based on the current gradient information, then updates the estimated variables from three different scales, and then updates the adaptive gain.

[0171] The present invention verifies the theoretical results through numerical examples. Consider the NMFG of a 100-player network in two-dimensional space, with the initial action x i,0 Randomly distributed in [-4,6]×[-4,6]. Assuming the connected edge density is 0.04, the random generation is as follows Figure 1 The connected communication network shown.

[0172] The cost function for each player is expressed as follows

[0173]

[0174] In this way, the players' main target decisions can be made within a limited time while maintaining the aggregation of group decisions.

[0175] We set K = 50 iterations in each time interval and select parameter α = 0.015. For all At the same time, select the dynamic event trigger policy parameters β i =0.25,δ i =0.01,σ i =0.05,θ i =5.

[0176] The simulation results of Algorithm 1 in this embodiment are as follows:

[0177] Figure 2 The evolution process of the action is shown. It can be seen from the figure that the proposed strategy can make the player's actions converge to the final Nash equilibrium 0 after aggregation, where Respectively two dimensions.

[0178] Figure 3 The estimated variables for each player are shown and The red line represents the true value of the set. We can see that the error decays to 0, and the estimated variable gradually approaches the true value. For node 4 with 4 connected edges (4,28), (4,38), (4,95), and (4,100), the adaptive learning gain The convergence of Figure 4 As shown, this also indirectly illustrates the convergence of the corresponding estimation error.

[0179] also, Figure 5 (a) describes the quantitative variables In order to verify the effectiveness of the dynamic event triggering mechanism, we conducted a comparative experiment between the time triggering strategy and the proposed event triggering strategy. Figure 5 (b) gives the triggering ratio of each player in the specified time period, which shows a significant reduction in the frequency of proposed strategy communications.

[0180] In yet another embodiment of the present invention, a large-scale distributed Nash equilibrium search method system is provided. The system can be used to implement the above-mentioned large-scale distributed Nash equilibrium search method, specifically including:

[0181] Network building blocks: construct undirected graphs, define node sets, build adjacency matrices and Laplacian matrices

[0182] Network Mean Field Game Model Module: Introduces the Network Mean Field Game (NMFG) model to describe the macroscopic interactions between individuals and groups and to handle the impact of local information interactions;

[0183] Aggregation behavior estimation module: Design auxiliary variables, dynamically estimate the aggregation behavior of the group, track the global state through local neighbor information, and realize the estimation and correction of the global state;

[0184] Adaptive Learning Gain Module: Introduces an adaptive learning gain mechanism that allows players to adjust neighbor association weights based on local estimation errors, ensuring the scalability of the algorithm in fully distributed scenarios;

[0185] Optimization module: realize leaderless consensus, optimize local objective functions, and reduce communication costs and computational complexity;

[0186] Event trigger mechanism module: introduce event trigger mechanism to determine the timing of information exchange between players and neighbors and improve decision-making efficiency;

[0187] Information exchange module: handles information transmission between players to ensure real-time update and sharing of information;

[0188] Results display module: Displays key indicators such as player strategy changes, convergence status, and network status through charts and graphs.

[0189] Update the player's status and global balance results in real time, allowing users to intuitively understand the system operation status.

[0190] Generate system operation reports, including performance evaluation, convergence speed and summary of final results.

[0191] Provides a user-friendly interface that allows users to view and analyze results, and make parameter adjustments and settings.

[0192] In another embodiment of the present invention, a terminal device is provided, the terminal device includes a processor and a memory, the memory is used to store a computer program, the computer program includes program instructions, and the processor is used to execute the program instructions stored in the computer storage medium. The processor may be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSP), application-specific integrated circuits (ASIC), field-programmable gate arrays (FPGA) or other programmable logic devices, discrete gates or transistor logic devices, discrete hardware components, etc., which are the computing core and control core of the terminal, which are suitable for implementing one or more instructions, and are specifically suitable for loading and executing one or more instructions to implement the corresponding method flow or corresponding functions; the processor described in the embodiment of the present invention can be used for the operation of a large-scale distributed Nash equilibrium search method.

[0193] In another embodiment of the present invention, a storage medium is provided, specifically a computer-readable storage medium (Memory), which is a memory device in a terminal device for storing programs and data. It is understandable that the computer-readable storage medium here can include both the built-in storage medium in the terminal device and the extended storage medium supported by the terminal device. The computer-readable storage medium provides a storage space, which stores the operating system of the terminal. In addition, one or more instructions suitable for being loaded and executed by the processor are also stored in the storage space, and these instructions can be one or more computer programs (including program codes). It should be noted that the computer-readable storage medium here can be a high-speed RAM memory, or a non-volatile memory (non-volatile memory), such as at least one disk storage.

[0194] One or more instructions stored in a computer-readable storage medium can be loaded and executed by a processor to implement the corresponding steps of the large-scale distributed Nash equilibrium search method in the above embodiment; one or more instructions in the computer-readable storage medium are loaded and executed by the processor.

[0195] Those skilled in the art will appreciate that embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0196] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0197] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0198] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0199] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the implementation methods of the present invention, and should be understood that the protection scope of the present invention is not limited to such special statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A large-scale distributed Nash equilibrium search method based on event triggering, characterized in that: include: S1: Construct an undirected graph The node set represents the player, the edge set ε describes the communication relationship between players, and defines the adjacency matrix and the Laplace matrix L to describe the connection relationship of the network; S2: Introduce the network mean field game (NMFG) model to describe the macroscopic interaction between individuals and groups under local information interaction; S3: Design three auxiliary variables to dynamically estimate the aggregation behavior of the group, dynamically track the global behavior and network size through local neighbor information, and realize the estimation and correction of the global state under incomplete information; S4: Adaptive learning gain is used to allow players to dynamically adjust the association weights with neighbors based on local estimation errors, ensuring the scalability of the algorithm in fully distributed scenarios; S5: Implement leaderless consensus and optimization of local objective functions to reduce communication costs and computational complexity; S6: Introduce an event trigger mechanism to determine the timing for players to exchange information with their neighbors and improve decision-making efficiency.

2. The large-scale distributed Nash equilibrium search method according to claim 1, characterized in that: S1 specifically includes: Define the number of players N, the social network composed of N players, and construct a node set Establish communication relationships and determine edge sets based on the communication capabilities between players Define the adjacency matrix as When nodes i and j are connected, the weight is a ij =1, otherwise the weight a ij =0; Define a diagonal matrix in a ij represents the degree of node i; The Laplacian matrix is ​​calculated as a if and only if (i,j)∈ε and (j,i)∈ε ij =a ji ; especially a ii =0 when is an undirected graph.

3. The large-scale distributed Nash equilibrium search method according to claim 1, characterized in that: S2 specifically includes: Define the set of locally feasible actions for each player i Defining the global action space Optimize the cost function for each player in the time interval [0,T]: In the formula, x i,t represents the action vector of player i at time t; F i,t (x i,t ,ρ(x t )) represents the cost function of player i at time t, which depends on the player's action x i,t and the mean field term ρ(x t );ρ(x t ) represents the mean field term, which is the aggregate information of the action variables of all players at time t; φ i (x i,t ) is the action x of player i i,t Related private functions; L i (x i,t ,ρ(x t )) represents the process cost, which is related to the current action of player i and the average field term, reflecting the immediate cost in this state; G i (x i,T ) represents the terminal cost, reflecting the cost of player i at time T; N represents the number of players.

4. The large-scale distributed Nash equilibrium search method according to claim 1, characterized in that: S3 specifically includes: The action of player i is updated by projecting the gradient as: in, represents the action vector of player i at time t, i.e. the updated choice; Represents the feasible action set Ω i The projection operation ensures that the updated motion vector Still within the feasible set; α represents the step size or learning rate, which is a parameter that controls the update amplitude; The cost function F i,t Action vector for player i The gradient of Denote the cost function F i,t For the mean field term The gradient of Represents a private function φ i The gradient of a player's actions reflects the impact of the player's choice on his private cost; Each player maintains an auxiliary variable y i,t ,z i,t ,w i,t , and dynamically updated to: Initialized to: In the formula, represents the fixed neighbor learning gain between players i and j; Auxiliary variables representing player i at time t, step k+1, used to dynamically estimate mean field information; Auxiliary variables representing player i at step k at time t; represents the fixed neighbor learning gain between players i and j, reflecting the efficiency of information transfer between players; a ij Represents the connection strength between players i and j in an undirected graph, which is 1 if there is a connection, otherwise 0; The difference between the private function of player i at step k+1 and step k at time t reflects the change in the player's state; The second auxiliary variable representing player i at time t, step k+1, is used to estimate the total size; A second auxiliary variable representing player i at step k at time t; A third auxiliary variable representing player i at time t, step k+1, which may be used to adjust the estimated distribution; The third auxiliary variable representing player i at step k at time t; y (i,0) represents the initial auxiliary variable for player i, combining the private function values ​​of this player and his neighbors; z (i,0) =1+d i represents the initial size estimate of player i, d i is the player's degree (number of neighbors); w (i,0) =1 represents the initial weight of player i.

5. The large-scale distributed Nash equilibrium search method according to claim 1, characterized in that: S4 specifically includes: Design adaptive learning gain instead of fixed gain The details are as follows in, and Three auxiliary variables representing the difference between players i and j at time t, step k+1; and Three auxiliary variables representing the relationship between players i and j at step k at time t; and Represents the learning gain between players i and j, and three auxiliary variables that control the update speed; and The square norm of the difference between the three auxiliary variables of players j and i, which measures the degree of difference between their states; The symmetry of the learning gain is set so that the node in-degree is equal to the out-degree to ensure the stability and consistency of the estimation.

6. The large-scale distributed Nash equilibrium search method according to claim 1, characterized in that: S6 specifically includes: Defining the trigger instant sequence To decide when to exchange information, it means the mth trigger moment of player i at time t is used to decide when to exchange information; Update broadcast variables: In the formula, The broadcast variable at step k+1 at time t represents the updated state of player i; The broadcast variable at time t, step k, represents the current state of player i; Auxiliary variable between players i and j at time t, step k, representing the information difference; a ij : The connection weight between players i and j, indicating the strength of information transmission; represents the state change at the kth step at time t; Define the dynamics of a quantized variable: In the formula, The quantized variable at step k+1 at time t represents a state quantity after player i updates; The quantized variable at the kth step at time t represents the current state of player i; β i : The decay factor of player i, which controls the degree to which the quantitative variables are retained during updates; δ i The adjustment factor of player i affects the update rate of the quantitative variables; σ is the standard deviation of player i, which is usually related to the volatility of the system; The estimated value of a quantity at the kth step at time t; represents the degree of self-influence or self-learning of player i; The trigger moment is defined as: In the formula, φ is used to evaluate the constraints in the current state, involving the error Quantitative variables and The error at step k at time t represents the deviation caused by player i in the state update; The update at the m+1th triggering moment at time t indicates the time of the next information exchange.

7. A large-scale distributed Nash equilibrium search method system, characterized in that: The system can be used to implement the large-scale distributed Nash equilibrium search method described in any one of claims 1 to 6, specifically including: Social network building blocks: constructing undirected graphs, defining node sets, building adjacency matrices and Laplacian matrices Network Mean Field Game Model Module: Introduces the Network Mean Field Game (NMFG) model to describe the macroscopic interactions between individuals and groups and to handle the impact of local information interactions; Aggregation behavior estimation module: Design auxiliary variables, dynamically estimate the aggregation behavior of the group, track the global state through local neighbor information, and realize the estimation and correction of the global state; Adaptive Learning Gain Module: Introduces an adaptive learning gain mechanism that allows players to adjust neighbor association weights based on local estimation errors, ensuring the scalability of the algorithm in fully distributed scenarios; Optimization module: realize leaderless consensus, optimize local objective functions, and reduce communication costs and computational complexity; Event trigger mechanism module: introduce event trigger mechanism to determine the timing of players exchanging information with neighbors and improve decision-making efficiency; Information exchange module: handles information transmission between players to ensure real-time update and sharing of information; Results display module: displays key indicators such as player strategy changes, convergence status, and network status through charts and graphs; Update the player's status and global balance results in real time, allowing users to intuitively understand the system operation status; Generate system operation reports, including performance evaluation, convergence speed and summary of final results; Provides a user-friendly interface that allows users to view and analyze results, and make parameter adjustments and settings.

8. A computer device, characterized in that: The invention comprises a memory, a processor and a computer program stored in the memory and executable on the processor, wherein when the processor executes the program, the large-scale distributed Nash equilibrium search method as claimed in any one of claims 1 to 6 is implemented.

9. A computer-readable storage medium, characterized in that: A computer program is stored thereon, and when the program is executed by a processor, the large-scale distributed Nash equilibrium search method described in one of claims 1 to 6 is implemented.

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