A fully distributed Nash equilibrium search method based on quantized communication in directed networks

Abstract

The present invention discloses a fully distributed Nash equilibrium search method based on quantized communication in a directed network, comprising the following steps: according to the communication network structure, a weighted adjacency matrix W is obtained; according to the decision feasible set, each player selects an initial decision and estimates the decisions of all other players; before each iteration, according to the proposed quantization scheme, each player transmits the update information of its decision estimate in the previous step to its neighbors through the communication network, and processes the information sent by the neighbors at the same time to update the estimate of the neighbor's decision estimate; at each iteration, according to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates, and updates its decision estimate accordingly; when the update amount of all players' decision estimates is zero, that is, the algorithm converges, and NE is obtained. The present invention proposes a quantization scheme, and based on this, proposes a fully distributed NE search method, which can converge to the theoretical NE under the limitation of limited network bandwidth.

Description

Technical Field

[0001] The present invention belongs to the field of distributed computing and analysis technology, and relates to, in particular to a fully distributed Nash equilibrium (NE) search method based on quantized communication in a directed network. Background Art

[0002] The distributed Nash equilibrium (NE) search problem is a hot topic in current network research. In recent years, non-cooperative game problems have gained increasing attention due to their wide application in some engineering, economic and social problems. In the game problem, each player can be considered as a selfish decision maker, which only aims to optimize its personal objective function but at the same time depends on the decisions of others. NE calculation is a key problem in non-cooperative games. At present, most traditional NE search algorithms rely on a central node that broadcasts information to all players in the game. However, in practical applications, such a node is basically impossible to exist. Therefore, distributed algorithms are proposed. In distributed algorithms, players can calculate NE only by communicating with their neighbors.

[0003] Communication plays an increasingly important role in distributed algorithms. In non-cooperative games, the objective function of each player depends not only on his own decision, but also on the decisions of other players. However, in practice, players may not be able to directly obtain the decisions of all other players. Therefore, in order to solve the problem of missing information, players have to estimate the decisions of other players by exchanging information with their neighbors.

[0004] Studying the impact of quantization on distributed algorithms has become a new hot topic. With the popularity of large-scale networks and the increase in the number of players participating in the game, the amount of information that needs to be transmitted through the communication network has also increased dramatically. However, in practical applications, the bandwidth of the communication network is often limited and may not be able to transmit such a large amount of information. In this context, people have to consider quantizing the data. Generally speaking, since quantized data will inevitably bring some quantization errors, quantization may reduce the performance of the algorithm to varying degrees.

[0005] There have been some preliminary studies on distributed optimization algorithms using quantized information. However, the distributed NE finding problem in non-cooperative games affected by quantization has not received enough research attention. Existing related studies have discussed the impact of quantization on NE finding algorithms based on discrete time gradients, which assumes that each player has the ability to broadcast its quantized information to all other players in the game. However, in large-scale games, this assumption is difficult to meet, and thus this algorithm may not be applicable. Summary of the invention

[0006] Purpose of the invention: In order to overcome the deficiencies in the prior art, a fully distributed Nash equilibrium (NE) search method based on quantized communication in a directed network is provided, a quantization scheme is proposed, and based on this, a fully distributed NE search method is proposed, which can converge to the theoretical NE under the limitation of limited network bandwidth.

[0007] Technical solution: To achieve the above purpose, the present invention provides a fully distributed Nash equilibrium search method based on quantized communication in a directed network, comprising the following steps:

[0008] S1: Obtain the weighted adjacency matrix W according to the communication network structure;

[0009] S2: Based on the feasible set of decisions, each player selects an initial decision and estimates the decisions of all other players;

[0010] S3: Before each iteration, according to the proposed quantization scheme, each player transmits the updated information of its decision estimate in the previous step to its neighbors through the communication network, and processes the information sent by the neighbors to update the estimate of the neighbor's decision estimate;

[0011] S4: At each iteration, according to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates and updates its decision estimate accordingly;

[0012] S5: When the estimated update amount of all players' decisions is zero, the algorithm converges and NE is obtained.

[0013] Furthermore, the weighted adjacency matrix W in step S1 is obtained in the following manner:

[0014] A1: For a directed network, if node i in the network can receive information from node j, then set A ij =1, j∈N i , where N i Indicates the incoming neighbor of node i. If it cannot be received, set A ij =0, In this way, we get the adjacency matrix of the communication network A = [A ij ];

[0015] A2: Given its adjacency matrix A, if A ij =1, then set w ij >0, if A ij =0, then set w ij = 0, select an appropriate distributed iterative strategy to calculate the weight distribution, and obtain a double random weighted adjacency matrix with self-loops W = [w ij ], that is, it satisfies the following properties:

[0016] wii > 0, for all nodes i in the communication network; (I)

[0017]

[0018] Furthermore, the step S2 is specifically as follows:

[0019] For each player participating in the game, the decision at the initial moment is selected according to its feasible decision set, that is, it satisfies:

[0020]

[0021] in, represents the set of players participating in the game, x i，i and x i，-i Represents the decision of player i and the estimation of other people’s decisions, Ω i represents the feasible set of decisions for player i.

[0022] Furthermore, the step S3 is specifically as follows:

[0023] B1: Define a standard consistent quantizer with 2K+1 quantization levels, which is defined as follows:

[0024]

[0025] in,

[0026] when When q[γ i ]=-q[-γ i ] (VI)

[0027] According to the above definition, we can get When , the quantization error is:

[0028]

[0029] It is said that the quantizer is non-saturated at this time;

[0030] B2: Based on the above consistent quantizer, a quantization scheme for encoding and decoding is proposed:

[0031] For any agent i and its neighbor j, i passes through the encoder F i→j The information is quantized and transmitted to neighbor j, who passes through the decoder To process the received quantization information, the encoder and decoder are defined as follows:

[0032]

[0033]

[0034] Where s(k) is the introduced global proportional function, which satisfies that when k→∞, s(k)→0; represents j's decision estimate x for i i estimates;

[0035] B3: According to the above definition, ξ i (k) and With the same update method and initial value, we can get:

[0036]

[0037] Furthermore, the step S4 is specifically as follows:

[0038] C1: According to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates, that is,

[0039]

[0040] C2: Based on the weighted average of all players’ decision estimates, each player updates his or her own decision using the projected gradient method, and uses the estimate of other players’ decisions in the weighted average as the updated decision estimate, namely:

[0041]

[0042]

[0043] Among them, α represents the step size, J i (x) represents the cost function for player i, which is a function of all players' decisions x.

[0044] Furthermore, the step S5 is specifically as follows:

[0045] When the estimated update amount of all players' decisions is 0, that is:

[0046]

[0047] The algorithm can be considered to have converged and reached NE.

[0048] Furthermore, in step S5, to ensure the convergence of the algorithm and converge to the theoretical NE, denoted as x * :

[0049] The following assumptions are made: Ω i is a non-empty convex compact set; given x -i , J i(x i , x -i ) is about x i Continuously differentiable convex function; define pseudo gradient function respectively and the extended pseudo gradient function Assuming that F is μ-strongly monotone and l0-Lipschitz continuous, we can prove that there exists μ≤l≤l0, F is l-Lipschitz continuous; the communication network is strongly connected; let the second largest eigenvalue of W be σ N-1 (W), then there exists There exist positive constants M0 and M * ≤M0,||x i，i (0)|| ∞ ≤M0,||x * || ∞ ≤M * ; α>0, and satisfies the inequality and

[0050] Furthermore, the selection of the global proportional function s(k) and the parameter k in step B2 is as follows:

[0051]

[0052]

[0053] Where M = M0 + M * , ρ α is a matrix spectral radius.

[0054] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0055] 1. Based on the standard consistent quantizer, the present invention proposes a dynamic quantization scheme for encoding and decoding, introduces a global proportional function, controls the quantization error while meeting the bandwidth limitation, and ensures that both parties transmitting information can obtain the same information, which is of great significance for eliminating the influence of quantization error on the algorithm in algorithm design.

[0056] 2. Based on the proposed quantization scheme, the present invention proposes a fully distributed NE search method, which can converge to the theoretical NE while meeting the limited network bandwidth constraints; NE is a strategy combination, under which no agent can expand its own interests by changing its own strategy alone, that is, when the theoretical NE is reached, no agent needs to worry about others damaging their own interests by changing their strategies. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1It is a schematic diagram of the communication network;

[0058] Figure 2 It is a schematic diagram of the convergence process;

[0059] Figure 3 This is a graph showing the change in relative error during the convergence process. DETAILED DESCRIPTION

[0060] The present invention is further explained below in conjunction with the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent forms of modifications to the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0061] The present invention provides a fully distributed Nash equilibrium search method based on quantized communication in a directed network. For ease of understanding, the terms involved in the method are explained below, specifically as follows:

[0062] Nash equilibrium (NE): Nash equilibrium, also known as non-cooperative game equilibrium. If any player's strategy is optimal when all other players' strategies are determined, then this decision combination is defined as Nash equilibrium.

[0063] Directed network: A network is composed of several nodes and links connecting them, representing many objects and the connections between them. Depending on whether the links connecting these nodes have a certain direction, we can divide the network into directed networks and undirected networks.

[0064] Quantization: Quantization means dividing the continuously changing range of input data into a finite number of non-overlapping sub-intervals (i.e., quantization levels). Each sub-interval is represented by a certain number within the interval, and the input data falling into it will be output with this value, thereby converting continuous input data into quantized data with a finite number of discrete values.

[0065] Referring to the above explanation, the present invention provides a fully distributed Nash equilibrium search method based on quantized communication in a directed network, which specifically includes the following steps:

[0066] S1: According to the communication network structure, a weighted adjacency matrix W is obtained, which includes the following steps A1-A2:

[0067] A1: For a directed network, if node i in the network can receive information from node j, then set A ij =1, j∈N i , where N i Indicates the incoming neighbor of node i. If it cannot be received, set A ij =0, In this way, we get the adjacency matrix of the communication network A = [A ij ];

[0068] A2: Given its adjacency matrix A, if A ij =1, then set w ij >0, if A ij =0, then set w ij = 0, select an appropriate distributed iterative strategy to calculate the weight distribution, and obtain a double random weighted adjacency matrix with self-loops W = [w ij ], that is, it satisfies the following properties:

[0069] w ii > 0, for all nodes i in the communication network; (I)

[0070]

[0071] S2: Based on the feasible set of decisions, each player selects an initial decision and estimates the decisions of all other players (hereinafter referred to as decision estimation):

[0072] For each player participating in the game, the decision at the initial moment is selected according to its feasible decision set, that is, it satisfies:

[0073]

[0074] in, represents the set of players participating in the game, x i，i and x i，-i Represents the decision of player i and the estimation of other people’s decisions, Ω i represents the feasible set of decisions for player i.

[0075] S3: Before each iteration, according to the proposed quantization scheme, each player transmits the updated information of its decision estimate in the previous step to its neighbors through the communication network, and processes the information from the neighbors to update the estimate of the neighbor's decision estimate; it includes the following steps B1-B3:

[0076] B1: Define a standard consistent quantizer with 2K+1 quantization levels, which is defined as follows:

[0077]

[0078] in,

[0079] when When q[γ i ]=-q[-γ i ] (VI)

[0080] According to the above definition, we can get When , the quantization error is:

[0081]

[0082] It is said that the quantizer is non-saturated at this time;

[0083] B2: Based on the above consistent quantizer, a quantization scheme for encoding and decoding is proposed:

[0084] For any agent i and its neighbor j, i passes through the encoder F i→j The information is quantized and transmitted to neighbor j, who passes through the decoder To process the received quantization information, the encoder and decoder are defined as follows:

[0085]

[0086]

[0087] Where s(k) is the introduced global proportional function, which satisfies that when k→∞, s(k)→0; represents j's decision estimate x for i i estimates;

[0088] B3: According to the above definition, ξ i (k) and With the same update method and initial value, we can get:

[0089]

[0090] S4: At each iteration, according to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates and updates its decision estimate accordingly;

[0091] It includes the following steps C1-C2:

[0092] C1: According to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates, that is,

[0093]

[0094] C2: Based on the weighted average of all players’ decision estimates, each player updates his or her decision using the projected gradient method, and uses the estimate of other players’ decisions in the weighted average as the updated decision estimate, namely:

[0095]

[0096]

[0097] Among them, α represents the step size, J i (x) represents the cost function for player i, which is a function of all players' decisions x.

[0098] S5: When the estimated update amount of all players' decisions is zero, the algorithm converges and NE is obtained.

[0099] When the estimated update amount of all players' decisions is 0, that is:

[0100]

[0101] The algorithm can be considered to have converged and reached NE.

[0102] In order to ensure the convergence of the algorithm and converge to the theoretical NE, the present invention is denoted as x * :

[0103] The following assumptions are made: Ω i is a non-empty convex compact set; given x -i , J i (x i , x -i ) is about x i Continuously differentiable convex function; define pseudo gradient function respectively and the extended pseudo gradient function Assuming that F is μ-strongly monotone and l0-Lipschitz continuous, we can prove that there exists μ≤l≤l0, F is l-Lipschitz continuous; the communication network is strongly connected; let the second largest eigenvalue of W be σ N-1 (W), then there exists There exist positive constants M0 and M * ≤M0,||x i，i (0)|| ∞ ≤M0,||x * || ∞ ≤M * ; α>0, and satisfies the inequality and

[0104] The selection of global scale function s(k) and parameter k are as follows:

[0105]

[0106]

[0107] Where M = M0 + M * , ρ α is a matrix spectral radius.

[0108] This embodiment also provides a fully distributed Nash equilibrium search system based on quantized communication in a directed network, the system comprising a network interface, a memory and a processor; wherein the network interface is used to realize the reception and transmission of signals in the process of sending and receiving information between other external network elements; the memory is used to store computer program instructions that can be run on the processor; the processor is used to execute the steps of the above-mentioned consensus method when running the computer program instructions.

[0109] The present embodiment also provides a computer storage medium, which stores a computer program, and the method described above can be implemented when the processor executes the computer program. The computer readable medium can be considered to be tangible and non-temporary. Non-limiting examples of non-temporary tangible computer-readable media include non-volatile memory circuits (such as flash memory circuits, erasable programmable read-only memory circuits or mask read-only memory circuits), volatile memory circuits (such as static random access memory circuits or dynamic random access memory circuits), magnetic storage media (such as analog or digital tapes or hard drives) and optical storage media (such as CDs, DVDs or Blu-ray discs), etc. The computer program includes processor executable instructions stored on at least one non-temporary tangible computer-readable medium. The computer program may also include or rely on stored data. The computer program may include a basic input / output system (BIOS) that interacts with the hardware of a special-purpose computer, a device driver that interacts with a specific device of a special-purpose computer, one or more operating systems, user applications, background services, background applications, etc.

[0110] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented in one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that include computer-usable program code.

[0111] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes 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 generate 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.

[0112] 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 including 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.

[0113] 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.

[0114] Based on the above scheme, this embodiment applies the above scheme as an example, and selects five companies producing homogeneous products as five nodes, namely The links between them indicate whether they can transmit information. The specific communication network is as follows: Figure 1 shown.

[0115] The specific steps include:

[0116] S1: According to the communication network structure, a weighted adjacency matrix W is obtained, which includes the following steps A1-A2:

[0117] A1: For a directed network, if node i in the network can receive information from node j, then set A ij =1, j∈N i , where N i Indicates the incoming neighbor of node i. If it cannot be received, set A ij =0, In this way, we get the adjacency matrix of the communication network A = [A ij ];

[0118] A2: Given its adjacency matrix A, if A ij =1, then set w ij >0, if A ij =0, then set w ij = 0, select an appropriate distributed iterative strategy to calculate the weight distribution, and obtain a double random weighted adjacency matrix with self-loops W = [w ij ], that is, it satisfies the following properties:

[0119] w ii > 0, for all nodes i in the communication network; (I)

[0120]

[0121] According to the communication network structure of the five companies in this example, the weighted matrix W can be obtained as follows:

[0122]

[0123] S2: Based on the feasible set of decisions, each player chooses an initial decision and estimates the decisions of all other players:

[0124] For each player participating in the game, the decision at the initial moment is selected according to its feasible decision set, that is, it satisfies:

[0125]

[0126] in, represents the set of players participating in the game, x i，i and x i，-i Represents the decision of player i and the estimation of other people’s decisions, Ω i represents the feasible set of decisions for player i.

[0127] In this example, represents the quantity of the product that the i-th company decides to produce, Ω i =[0,100],x i，-i It represents the estimation vector of the i-th company on the quantity of the product that other companies decide to produce except itself. Assuming that the companies are not aware of each other's production at the beginning, the following data can be obtained:

[0128]

[0129] S3: Before each iteration, according to the proposed quantization scheme, each player transmits the updated information of its decision estimate in the previous step to its neighbors through the communication network, and processes the information from the neighbors to update the estimate of the neighbor's decision estimate; it includes the following steps B1-B3:

[0130] B1: Define a standard consistent quantizer with 2K+1 quantization levels, which is defined as follows:

[0131]

[0132] in,

[0133] when When q[γ i ]=-q[-γ i ] (VI)

[0134] According to the above definition, we can get When , the quantization error is:

[0135]

[0136] It is said that the quantizer is non-saturated at this time;

[0137] B2: Based on the above consistent quantizer, a quantization scheme for encoding and decoding is proposed:

[0138] For any agent i and its neighbor j, i passes through the encoder F i→j The information is quantized and transmitted to neighbor j, who passes through the decoder To process the received quantization information, the encoder and decoder are defined as follows:

[0139]

[0140]

[0141] Where s(k) is the introduced global proportional function, which satisfies that when k→∞, s(k)→0; represents j's decision estimate x for i i estimates;

[0142] B3: According to the above definition, ξ i (k) and With the same update method and initial value, we can get:

[0143]

[0144] S4: At each iteration, according to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates and updates its decision estimate accordingly;

[0145] It includes the following steps C1-C2:

[0146] Cl: According to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates, that is,

[0147]

[0148] C2: Based on the weighted average of all players’ decision estimates, each player updates his or her decision using the projected gradient method, and uses the estimate of other players’ decisions in the weighted average as the updated decision estimate, namely:

[0149]

[0150]

[0151] Among them, α represents the step size, J i (x) represents the cost function for player i, which is a function of all players' decisions x.

[0152] In this example, where u i (x i，i )=[100+50(x i，i )]x i，i ,

[0153] S5: When the estimated update amount of all players' decisions is zero, the algorithm converges and NE is obtained.

[0154] When the estimated update amount of all players' decisions is 0, that is:

[0155]

[0156] The algorithm can be considered to have converged and reached NE.

[0157] Under the assumptions of this example, it can be verified that all assumptions are satisfied, and it can be found through theoretical methods that NE exists and is unique, x * =[10.35 9.06 7.56 5.67 2.67] T .

[0158] In order to verify the above results, this embodiment is simulated by MATLAB, and it can be seen that the algorithm proposed in this embodiment can converge to the theoretical NE. Figure 2 The convergence process is shown. Figure 3 It describes the change of relative error during the convergence process, from Figure 2 It can be seen that all five nodes converge to the theoretical NE; Figure 3 It can be seen that the relative error gradually decreases during the convergence process, and when the convergence is completed, the relative error is basically eliminated.

Claims

1. A fully distributed Nash equilibrium search method based on quantized communication in a directed network, characterized in that: The steps include: S1: Obtain the weighted adjacency matrix W according to the communication network structure; S2: Based on the feasible set of decisions, each player selects an initial decision and estimates the decisions of all other players; S3: Before each iteration, according to the proposed quantization scheme, each player transmits the updated information of its decision estimate in the previous step to its neighbors through the communication network, and processes the information sent by the neighbors to update the estimate of the neighbor's decision estimate; S4: At each iteration, according to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates and updates its decision estimate accordingly; S5: When the update amount of all players' decision estimates is zero, the algorithm converges and NE is obtained; Step S3 is specifically as follows: B1: Define a standard consistent quantizer with 2K+1 quantization levels, which is defined as follows: in, When , q[γ i = -q[-γ i (VI) According to the above definition, we can get When , the quantization error is: It is said that the quantizer is non-saturated at this time; B2: Based on the above consistent quantizer, a quantization scheme for encoding and decoding is proposed: For any agent i and its neighbor j, i passes through the encoder F i→j The information is quantized and transmitted to neighbor j, who passes through the decoder To process the received quantization information, the encoder and decoder are defined as follows: Where s(k) is the introduced global proportional function, which satisfies that when k→∞, s(k)→0; represents j's decision estimate x for i i estimates; B3: According to the above definition, ξ i (k) and With the same update method and initial value, we can get:

2. According to claim 1, a fully distributed Nash equilibrium search method based on quantized communication in a directed network is characterized in that: The specific method of obtaining the weighted adjacency matrix W in step S1 is: A1: For a directed network, if node i in the network can receive information from node j, then set A ij =1,j∈N i , where N i Indicates the incoming neighbor of node i. If it cannot be received, set A ij =0, In this way, we get the adjacency matrix of the communication network A = [A ij ]; A2: Given its adjacency matrix A, if A ij =1, then set w ij >0, if A ij =0, then set w ij = 0, select the distributed iterative strategy to calculate the weight distribution, and obtain the double random weighted adjacency matrix W with self-loops = [w ij ], that is, it satisfies the following properties: w ii >0, for all nodes i in the communication network; (I) 3. According to claim 1, a fully distributed Nash equilibrium search method based on quantized communication in a directed network is characterized in that: Step S2 is specifically as follows: For each player participating in the game, the decision at the initial moment is selected according to its feasible decision set, that is, it satisfies: in, represents the set of players participating in the game, x i,i and x i,-i Represents the decision of player i and the estimation of other people’s decisions, Ω i represents the feasible set of decisions for player i.

4. According to claim 1, a fully distributed Nash equilibrium search method based on quantized communication in a directed network is characterized in that: Step S4 is specifically as follows: C1: According to the set weighted adjacency matrix W, each player calculates the weighted average of all players' decision estimates, that is, k=0,1,…: C2: Based on the weighted average of all players’ decision estimates, each player updates his or her decision using the projected gradient method, and uses the estimate of other players’ decisions in the weighted average as the updated decision estimate, namely: k=0,1,…: Among them, α represents the step size, J i (x) represents the cost function for player i, which is a function of all players' decisions x.

5. According to claim 1, a fully distributed Nash equilibrium search method based on quantized communication in a directed network is characterized in that: Step S5 is specifically as follows: When the estimated update amount of all players' decisions is 0, that is: The algorithm can be considered to have converged and reached NE.

6. The method for searching for a fully distributed Nash equilibrium based on quantized communication in a directed network according to claim 5, characterized in that: In step S5, to ensure the convergence of the algorithm and converge to the theoretical NE, denoted as x * : The following assumptions are made: Ω i is a non-empty convex compact set; given x -i , J i (x i ,x -i ) is about x i Continuously differentiable convex functions; Define pseudo gradient functions separately and the extended pseudo gradient function Assuming that F is μ-strongly monotone and l0-Lipschitz continuous, we can prove that there exists μ≤l≤l0, F is l-Lipschitz continuous; the communication network is strongly connected; let the second largest eigenvalue of W be σ N-1 (W), then there exists There exist positive constants M0 and M * ≤M0,||x i,i (0)|| ∞ ≤M0,||x * || ∞ ≤M * ; α>0, and satisfies the inequality and 7. The method for searching for a fully distributed Nash equilibrium based on quantized communication in a directed network according to claim 1, characterized in that: The selection of the global proportional function s(k) and the parameter k in step B2 are as follows: Where M = M0 + M * , ρ α is a matrix spectral radius.

Images