A method, device and storage device for estimating a Nash equilibrium under a time delay

Abstract

The application provides a Nash equilibrium estimation method under time delay, which comprises the following steps: acquiring a system with n intelligent agents, constructing a time delay model of the system, converting the estimation of the deviation of Nash equilibrium under time delay into the upper limit of the deviation of the convergence value of the projection-free distributed gradient method under time delay according to the time delay model, determining the relationship between the perron vectors of the adjacent matrix of the system before and after the time delay is added, and determining the estimation result of the upper limit according to the relationship. The application has the beneficial effects that the upper limit of the deviation of the convergence value of the gradient algorithm under the time delay is estimated, the deviation under different time delays is analyzed according to the upper limit, the loss caused by the time delay is minimized, and more definite information is provided for system optimization and resource allocation.

Description

Technical Field

[0001] This application relates to the field of intelligent agents, and more particularly to a Nash equilibrium estimation method under time delay. Background Technology

[0002] A multi-agent system is a system composed of a group of autonomous, interacting agents. Each agent can perceive the environment, make decisions and perform actions, and cooperate or compete with other agents to achieve the system's goals. Multi-agent systems are widely used in autonomous driving, drone swarming, and robot cooperation.

[0003] Non-cooperative game theory is an important model in game theory that describes situations where participants make their own decisions without cooperation. In recent years, non-cooperative game theory has been widely applied in multi-agent engineering projects such as UAV swarms, microgrid control, and wireless communication.

[0004] In multi-agent systems, solving optimal decision problems often assumes that multiple agents cooperate to achieve a goal. However, in some cases of intelligent transportation networks and computer networks, due to the individual rationality of agents, there are heterogeneities and conflicts of interest among agents. Typically, the Nash equilibrium of non-cooperative games is calculated to solve the multi-agent optimal decision problem in this case.

[0005] In communication networks, delays can occur due to factors such as network congestion and transmission distance, which affects system stability and real-time communication.

[0006] Regarding the stability of time-delay systems, some researchers have proposed a lemma based on adjustable parameters and used it to establish a stability criterion for linear delay systems; others have studied the stability problem of time-varying delay systems with a certain range.

[0007] In other words, how to optimize and simplify the system optimization process remains a major challenge that we need to address in terms of latency system optimization. Summary of the Invention

[0008] The purpose of this application is to address the problem of how to simplify the optimization process of time-delayed systems by providing a Nash equilibrium estimation method under time delay.

[0009] The above-mentioned objective of this application is achieved through the following technical solution:

[0010] S1: Obtain a system with n agents;

[0011] S2: Construct the time delay model of the system;

[0012] S3: Based on the time delay model, the estimation of the deviation of Nash equilibrium under time delay is transformed into the upper bound of the deviation of the convergence value of the unprojected distributed gradient method under time delay;

[0013] S4: Determine the relationship between the perron vectors of the adjacency matrix of the system before and after the time delay is added;

[0014] S5: Determine the estimation result of the upper bound based on the relationship.

[0015] Optionally, step S1 includes:

[0016] The system with n intelligent agents includes:

[0017] The system diagram G A =(I n ,ε);

[0018] Among them, I n ={1,L,n} is the set of points, and ε is the set of arcs;

[0019] Suppose there exists an information transmission path from node j to node i, then (j,i)∈ε;

[0020] Figure G A The adjacency matrix is ​​A∈R n×n The element in the i-th row and j-th column is a. ij When a path exists from node j to node i, a ij >0, otherwise, a ij =0;

[0021] Assume that node i has a self-loop, i.e., a ii >0;

[0022] When A is a random matrix, and G A When A is a strongly connected graph, A has a unique Perron vector associated with the eigenvalue 1. Furthermore, if G... A It is also a weighted balance graph, where A is a double-random matrix.

[0023] Optionally, step S2 includes:

[0024] If the information transmission from node j to node i has T ij The delay is T, which is added between node j and node i. ij Each node, i.e., constructing T ij The delay model;

[0025] The node j, the node i, and T ij The gradient of each node is 0, and the gradients of nodes j, i, and T are... ijThe objective function for each node is 0.

[0026] Optionally, add T between node j and node i. ij The steps for each node include:

[0027] Let the system be a third-order system. Then the adjacency matrix of the third-order system is as follows:

[0028]

[0029] Introduce a delay, let T 12 =1,T 23 =2,T 31 =1,T 32 =1 transforms the adjacency matrix as follows:

[0030]

[0031] The node addition order is T. 12 T 13 T 21 T 23 T 31 T 32 There is no time delay in the self-loop.

[0032] Optionally, step S3 includes:

[0033] S31: The distributed gradient descent method of the system is as follows:

[0034]

[0035] Where, x i (k+1) represents the state of the i-th agent at time k+1; x j (k) represents the state of the j-th agent at time k; d i (k) is the gradient, i.e. d i (k) has an upper bound L; if α k If α is too large, it will fluctuate around the optimal solution and fail to converge to the optimal solution; if α k If it's too small, it won't converge to the optimal solution, let and

[0036] S32: When solving for Nash equilibrium in a non-cooperative game, each agent has a corresponding set of policy constraints; the distributed gradient descent method is combined with projection so that the policy of each agent always lies within its policy constraint set X:

[0037]

[0038] PX (·) denotes the projection onto the set X. The algorithm converges to... The solution, i.e., the Nash equilibrium point;

[0039] S33: Let the objective function... Given a convex policy constraint set X, the communication network graph G... A (k) is a weighted, strongly connected graph, G A The adjacency matrix of (k) is a random matrix, and G A (k) elements greater than 0 have a lower bound λ;

[0040] After adding a time delay, we get a new point set I′ n ,have

[0041]

[0042]

[0043] Where the element in the i-th row and j-th column of the new point set is b. ij ;

[0044] For projection algorithms, there are

[0045]

[0046] Optionally, step S4 includes:

[0047] If the system is a third-order system, then the adjacency matrix of the third-order system is as follows:

[0048]

[0049] The Perron vector of the adjacency matrix of the third-order system is:

[0050] v(k)=[v1(k),v2(k),v3(k)]′, there is (v(k))′A(k)=(v(k))′, that is:

[0051]

[0052] Adding the time delay T, we obtain a new adjacency matrix B(k), and the Perron vector of B(k) is v′(k)=[v′1(k),v′2(k),v′3(k),L,v′ 3+T (k)]′, we have (v′(k))′B(k)=(v′(k))′, that is

[0053] (1-a 11 (k))v′1(k)=v′4(k)+v′ 4+T12 (k)

[0054]

[0055]

[0056] a 21 (k)v′2(k)=v′4(k)

[0057]

[0058]

[0059]

[0060]

[0061]

[0062] The above equation shows that before the time delay is added, we have v1(k) = c1(k)v3(k), v2(k) = c2(k)v3(k) and v1(k) + v2(k) + v3(k) = 1, where c1(k) and c2(k) are positive constants. After the time delay is added, we have v′1(k) = c1(k)v′3(k), v′2(k) = c2(k)v′3(k) and v′1(k) + v′2(k) + v′3(k) < 1.

[0063] Generalizing to an nth-order system, for any k∈N time step, before the time delay is introduced, we have a Perron vector v = [v1, v2, ..., v...]. n ]′, satisfying v1=c1v n v2=c2v n ,L,v n-1 =c n-1 v n , where c i ∈R + ,i∈I n-1 After adding a time delay, we have a Perron vector v′=[v′1,v′2,L,v′] n ,L,v′ n+T ]′, satisfying v′1=c1v′ n v′2=c2v′ n ,L,v′ n-1 =c n-1 v′ n The difference is that v1+v2+L+v n =1, and v′1+v′2+L+v′ n <1.

[0064] Optionally, step S5 includes:

[0065] S51: Let the transition matrix be as follows:

[0066] P(k,s)=A(k)A(k-1)LA(s),k≥s

[0067] Where k, k-1, L, s represent time;

[0068] Conclusion 1:

[0069] For any s, we have in, Let P(k,s) be a positive random vector. For all k and s satisfying k≥s, when the element in the i-th row and j-th column of P(k,s) is not 0, we have [P(k,s)]. ij ≥λ k-s+1 ;

[0070] S52: Estimate the upper bound based on conclusion 1:

[0071] No time delay, has:

[0072]

[0073] Sometimes there is a delay, such as:

[0074]

[0075] When there are the same initial values, we have:

[0076]

[0077] Among them, I T ={n+1,L,n+T}; Define ξ = [P(k,r+1)] ij -[P′(k,r+1)] ij Let I be the set of agents that satisfy ξ>0. 1 The set of agents satisfying ξ≤0 is I. 2 When j∈I 1 When ξ ≤ [P(k,r+1)] ij -λ k-r Since A(k) is a strongly connected graph, we have 0 < η < 1 such that... Existing:

[0078]

[0079] From the above formula, we can obtain that When j∈I 2 When -ξ≤1-λ k-r <1, there exists 0<σ<1 such that

[0080] From the above formula, we can obtain the first formula:

[0081]

[0082] S53: Conclusion 2 is obtained as follows:

[0083] Let 0 < ρ < 1, {a k} is a positive sequence, if So

[0084] S54: Conclusion 3 is obtained as follows:

[0085] {A(k)} is a sequence of random matrices if, for k ≥ 0, A(k) has a common Perron vector v with respect to eigenvalue 1, and its associated graph sequence {G} A(k) If} is always strongly connected, then for any s, we have

[0086] S54: Based on conclusions 1 and 2, taking the limit of both sides of the first formula, since x approaches infinity as k→∞... i (k+1) will reach its Nash equilibrium. have to:

[0087]

[0088] S55: When the adjacency matrices A(k) of G(k) have a common Perron vector, according to conclusion 3, we get

[0089]

[0090] S56: Based on the above steps, determine the estimation result of the upper bound:

[0091] For a time-varying graph G(k), when the adjacency matrix A(k) of G(k) has no common Perron vector, we have

[0092]

[0093] in and They are vectors and The j-th element;

[0094] When the adjacency matrices A(k) of G(k) have a common Perron vector, we have

[0095]

[0096]

[0097] Among them, v j and These are the common Perron vectors v and v', respectively. 1 The j-th element.

[0098] A storage device that stores instructions and data to implement a Nash equilibrium estimation method under time delay.

[0099] A time-delayed Nash equilibrium estimation device includes: a processor and a storage device; the processor loads and executes instructions and data in the storage device to implement a time-delayed Nash equilibrium estimation method.

[0100] The beneficial effects of the technical solution provided in this application are:

[0101] A time-delay model of a multi-agent system is constructed. The estimation of Nash equilibrium deviation under time delay is transformed into an upper bound of the convergence deviation of the unprojected distributed gradient method under time delay. The upper bound estimation result of the system after adding time delay is calculated. Based on the time delay, an upper bound of its impact on the algorithm's convergence value is estimated. Using this upper bound, the magnitude of deviation under different time delays can be analyzed, thereby minimizing the loss caused by time delay and providing more explicit information for system optimization and resource allocation. Attached Figure Description

[0102] The present application will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0103] Figure 1 This is a flowchart illustrating the steps of the Nash equilibrium estimation method under time delay in the embodiments of this application;

[0104] Figure 2 This is a schematic diagram of the hardware device working in the embodiments of this application;

[0105] Figure 3 This is a diagram showing the state changes of the hardware device before and after the delay in operation, as described in this application embodiment.

[0106] Figure 4 This is a graph showing the change in the state error of the hardware device in the embodiments of this application. Detailed Implementation

[0107] To provide a clearer understanding of the technical features, objectives, and effects of this application, the specific embodiments of this application will now be described in detail with reference to the accompanying drawings.

[0108] The embodiments of this application provide a Nash equilibrium estimation method under time delay.

[0109] Please refer to Figure 1 , Figure 1 This is a flowchart illustrating the steps of a Nash equilibrium estimation method under time delay in an embodiment of this application, specifically including the following steps:

[0110] S1: Obtain a system with n agents;

[0111] S2: Construct the time delay model of the system;

[0112] S3: Based on the time delay model, the estimation of the deviation of Nash equilibrium under time delay is transformed into the upper bound of the deviation of the convergence value of the unprojected distributed gradient method under time delay;

[0113] S4: Determine the relationship between the perron vectors of the adjacency matrix of the system before and after the time delay is added;

[0114] S5: Determine the estimation result of the upper bound based on the relationship.

[0115] Specifically, based on the unprojected distributed gradient method, the upper bound of the Nash equilibrium deviation of the system in the presence of time delay is estimated, i.e., the upper bound of the deviation of the algorithm's convergence value. The upper bound is related to the time delay and the initial value, simplifying the optimization process of the time-delayed system and providing more explicit and intuitive information for the system's resource allocation. In addition, the relationship between the Perron vectors of the system's adjacency matrix before and after the addition of time delay is determined.

[0116] Step S1 includes:

[0117] The system with n intelligent agents includes:

[0118] The system diagram G A =(I n ,ε);

[0119] Among them, I n ={1,L,n} is the set of points, and ε is the set of arcs;

[0120] Suppose there exists an information transmission path from node j to node i, then (j,i)∈ε;

[0121] Figure G A The adjacency matrix is ​​A∈R n×n The element in the i-th row and j-th column is a. ij When a path exists from node j to node i, a ij >0, otherwise, a ij =0;

[0122] Assume that node i has a self-loop, i.e., a ii >0;

[0123] When A is a random matrix, and G A When A is a strongly connected graph, A has a unique Perron vector associated with the eigenvalue 1. Furthermore, if G... A It is also a weighted balance graph, where A is a double-random matrix.

[0124] Step S2 includes:

[0125] If the information transmission from node j to node i has T ij The delay is T, which is added between node j and node i. ij Each node, i.e., constructing T ij The delay model;

[0126] The node j, the node i, and T ij The gradient of each node is 0, and the gradients of nodes j, i, and T are... ij The objective function for each node is 0.

[0127] Add T between node j and node i ij The steps for each node include:

[0128] Let the system be a third-order system. Then the adjacency matrix of the third-order system is as follows:

[0129]

[0130] Introduce a delay, let T 12 =1,T 23 =2,T 31 =1,T 32 =1 transforms the adjacency matrix as follows:

[0131]

[0132] The node addition order is T. 12 T 13 T 21 T 23 T 31 T 32 There is no time delay in the self-loop.

[0133] Step S3 includes:

[0134] S31: The distributed gradient descent method of the system is as follows:

[0135]

[0136] Where, x i (k+1) represents the state of the i-th agent at time k+1; x j (k) represents the state of the j-th agent at time k; d i (k) is the gradient, i.e. d i (k) has an upper bound L; if α k If α is too large, it will fluctuate around the optimal solution and fail to converge to the optimal solution; if αk If it's too small, it won't converge to the optimal solution, let and

[0137] S32: When solving for Nash equilibrium in a non-cooperative game, each agent has a corresponding set of policy constraints; the distributed gradient descent method is combined with projection so that the policy of each agent always lies within its policy constraint set X:

[0138]

[0139] P X (·) denotes the projection onto the set X. The algorithm converges to... The solution, i.e., the Nash equilibrium point;

[0140] S33: Let the objective function... Given a convex policy constraint set X, the communication network graph G... A (k) is a weighted, strongly connected graph, G A The adjacency matrix of (k) is a random matrix, and G A (k) elements greater than 0 have a lower bound λ;

[0141] After adding a time delay, we get a new point set I′ n ,have

[0142]

[0143]

[0144] Where the element in the i-th row and j-th column of the new point set is b. ij ;

[0145] For projection algorithms, there are

[0146]

[0147] Step S4 includes:

[0148] If the system is a third-order system, then the adjacency matrix of the third-order system is as follows:

[0149]

[0150] The Perron vector of the adjacency matrix of the third-order system is:

[0151] v(k)=[v1(k),v2(k),v3(k)]′, there is (v(k))′A(k)=(v(k))′, that is:

[0152]

[0153] Adding the time delay T, we obtain a new adjacency matrix B(k), and the Perron vector of B(k) is v′(k)=[v1′(k),v2′(k),v3′(k),L,v3′ +T (k)]′, we have (v′(k))′B(k)=(v′(k))′, that is

[0154]

[0155]

[0156]

[0157] a 21 (k)v2′(k)=v4′(k)

[0158]

[0159]

[0160]

[0161]

[0162]

[0163] The above equation shows that before the time delay is added, we have v1(k) = c1(k)v3(k), v2(k) = c2(k)v3(k) and v1(k) + v2(k) + v3(k) = 1, where c1(k) and c2(k) are positive constants. After the time delay is added, we have v′1(k) = c1(k)v′3(k), v′2(k) = c2(k)v′3(k) and v′1(k) + v′2(k) + v′3(k) < 1.

[0164] Generalizing to an nth-order system, for any k∈N time step, before the time delay is introduced, we have a Perron vector v = [v1, v2, ..., v...]. n ]′, satisfying v1=c1v n v2=c2v n ,L,v n-1 =c n-1 v n , where c i ∈R + ,i∈I n-1 After adding a time delay, we have a Perron vector v′=[v′1,v′2,L,v′] n ,L,v′ n+T ]′, satisfying v′1=c1v′ n v′2=c2v′ n ,L,v′n-1 =c n-1 v′ n The difference is that v1+v2+L+v n =1, and v′1+v′2+L+v′ n <1.

[0165] Step S5 includes:

[0166] S51: Let the transition matrix be as follows:

[0167] P(k,s)=A(k)A(k-1)LA(s),k≥s

[0168] Where k, k-1, L, s represent time;

[0169] Conclusion 1:

[0170] For any s, we have in, Let P(k,s) be a positive random vector. For all k and s satisfying k≥s, when the element in the i-th row and j-th column of P(k,s) is not 0, we have [P(k,s)]. ij ≥λ k-s+1 ;

[0171] S52: Estimate the upper bound based on conclusion 1:

[0172] No time delay, has:

[0173]

[0174] Sometimes there is a delay, such as:

[0175]

[0176] When there are the same initial values, we have:

[0177]

[0178] Among them, I T ={n+1,L,n+T}; Define ξ = [P(k,r+1)] ij -[P′(k,r+1)] ij Let I be the set of agents that satisfy ξ>0. 1 The set of agents satisfying ξ≤0 is I. 2 When j∈I 1 When ξ ≤ [P(k,r+1)] ij -λ k-r Since A(k) is a strongly connected graph, we have 0 < η < 1 such that... Existing:

[0179]

[0180] From the above formula, we can obtain that When j∈I 2 When -ξ≤1-λ k-r <1, there exists 0<σ<1 such that

[0181] From the above formula, we can obtain the first formula:

[0182]

[0183] S53: Conclusion 2 is obtained as follows:

[0184] Let 0 < ρ < 1, {a k} is a positive sequence, if So

[0185] S54: Conclusion 3 is obtained as follows:

[0186] {A(k)} is a sequence of random matrices if, for k ≥ 0, A(k) has a common Perron vector v with respect to eigenvalue 1, and its associated graph sequence {G} A(k) If} is always strongly connected, then for any s, we have

[0187] S54: Based on conclusions 1 and 2, taking the limit of both sides of the first formula, since x approaches infinity as k→∞... i (k+1) will reach its Nash equilibrium. have to:

[0188]

[0189] S55: When the adjacency matrices A(k) of G(k) have a common Perron vector, according to conclusion 3, we get

[0190]

[0191] S56: Based on the above steps, determine the estimation result of the upper bound:

[0192] For a time-varying graph G(k), when the adjacency matrix A(k) of G(k) has no common Perron vector, we have

[0193]

[0194] in and They are vectors and The j-th element;

[0195] When the adjacency matrices A(k) of G(k) have a common Perron vector, we have

[0196]

[0197]

[0198] Among them, v j and These are the common Perron vectors v and v', respectively. 1 The j-th element.

[0199] Specifically, for example, verification can be performed through numerical simulation, as follows:

[0200] The system is set as a third-order system, and the communication network diagram of the third-order system is G. A (k) does not change over time, G A The adjacency matrix of (k) is as follows:

[0201]

[0202] The objective function is f1(x) = 4x 2 , f2(x)=2|x| and f3(x)=(x-2) 4 .

[0203] Let T 13 =1,T 21 =1,T 23 =2, the resulting adjacency matrix is:

[0204]

[0205] Given initial values ​​x1(0) = 0.5, x2(0) = 0.2, x3(0) = -0.3, and setting the initial values ​​of all delay nodes to 0, the feasible constraint set is X = [-1, 1]. Let... The state changes before and after the time delay obtained through numerical simulation are shown in the figure below. Figure 3 As shown in the figure, the change in state error is as follows: Figure 4 As shown. The calculated upper bound is 0.1111, and the deviation is 0.0094. Since 0.0094 < 0.1111, the conclusion is proved.

[0206] Specifically, in multi-agent time-delay systems, by calculating the impact of different time delays on the system based on the above results, determining the estimated upper bound, and comparing the magnitude of the impact, the direction of system optimization can be more clearly identified, providing clearer information for system optimization and greatly simplifying the optimization process. For example, in unmanned aerial vehicle (UAV) systems, communication delays between UAVs can lead to system decision-making errors. During optimization, the impact of different time delays can be calculated, and the impact of each delay can be compared to determine which has the greater impact. The more impactful delay portion can then be optimized primarily. In terms of resource allocation, more resources can be allocated to UAVs less affected by time delays, while fewer resources can be allocated to UAVs more affected by time delays.

[0207] A time-delayed Nash equilibrium estimation device 401: A time-delayed Nash equilibrium estimation device 401 implements a time-delayed Nash equilibrium estimation method.

[0208] Processor 402: Processor 402 loads and executes instructions and data in storage device 403 to implement a Nash equilibrium estimation method under time delay.

[0209] Storage device 403: Storage device 403 stores instructions and data; storage device 403 is used to implement a Nash equilibrium estimation method under time delay.

[0210] The above are merely preferred embodiments of this application and are not intended to limit this application. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A Nash equilibrium estimation method under time delay, characterized in that, The method includes the following steps: S1: Obtain A system of intelligent agents; Step S1 includes: The above has A system of one intelligent agent includes: The system diagram ; in, For point set, For arc sets; Set nodes To the node If there is an information transmission path, then ; picture The adjacency matrix is , of which Line number The elements of the column are When the node To the node When a path exists, ,otherwise, ; Assuming the node There is a self-loop, that is ; when It is a random matrix, and When it is a strongly connected graph, There is a unique Perron vector associated with the eigenvalue 1. Furthermore, if It is also a weighted balance diagram. It is a double random matrix; S2: Construct the time delay model of the system; Step S2 includes: If the node To the node Information transmission exists The delay at the node To the node Add between Each node, i.e., the construction The delay model; The node The node as well as The gradient of each node is 0. The node as well as The objective function for each node is 0; S3: Based on the time delay model, the estimation of the deviation of Nash equilibrium under time delay is transformed into the upper bound of the deviation of the convergence value of the unprojected distributed gradient method under time delay; Step S3 includes: S31: The distributed gradient descent method of the system is as follows: in, express +1 moment The state of each agent; express Time of the first The state of each agent; in The step size for descent along the gradient direction at any given time is ; For gradient, i.e. , There is an upper realm ;like If the value is too large, it will fluctuate around the optimal solution and will not converge to the optimal solution; if If it's too small, it won't converge to the optimal solution, let and ; S32: When solving for Nash equilibrium in a non-cooperative game, each agent has a corresponding set of policy constraints; the distributed gradient descent method is combined with projection so that the policy of each agent always lies within its policy constraint set. Inside: Indicates in set The projection on the surface converges to the algorithm. The solution, i.e., the Nash equilibrium point; S33: Let the objective function... and policy constraint set A convex communication network diagram For a weighted, strongly connected graph, The adjacency matrix of is a random matrix, and Elements greater than 0 have a lower bound. ; After adding a time delay, a new point set is obtained. ,have Among them, the first point set on the new point set Line number The elements of the column are ; For projection algorithms, there are ； S4: Determine the relationship between the perron vectors of the adjacency matrix of the system before and after the time delay is added; S5: Determine the estimation result of the upper bound based on the relationship.

2. The Nash equilibrium estimation method under time delay as described in claim 1, characterized in that, At the node To the node Add between The steps for each node include: Let the system be a third-order system. Then the adjacency matrix of the third-order system is as follows: Introducing delay, making , , , The adjacency matrix is ​​transformed as follows: The order in which nodes are added is as follows , , , , , There is no time delay in the self-loop.

3. The Nash equilibrium estimation method under time delay as described in claim 2, characterized in that, Step S4 includes: If the system is a third-order system, then the adjacency matrix of the third-order system is as follows: The Perron vector of the adjacency matrix of the third-order system is: ,have ,Right now: Add the aforementioned delay This yields a new adjacency matrix. , The Perron vector is ,have ,Right now The above equation shows that before the time delay is added, we have , and ,in , As a positive integer, after the delay is added, we have , and ; Promoted to For any order system, At any given moment, the time delay is introduced before the Perron vector. ,satisfy ,in After adding a time delay, there is a Perron vector. ,satisfy The difference is that, ,and .

4. The Nash equilibrium estimation method under time delay as described in claim 3, characterized in that, Step S5 includes: S51: Let the transition matrix be as follows: in For time; Conclusion 1: For any ,have ,in, Let be a positive random vector; for all vectors satisfying of and ,when The Line number When the elements of a column are not 0, we have ; S52: Estimate the upper bound based on conclusion 1: No time delay, has: Sometimes there is a delay, such as: When there are the same initial values, we have: in, ;definition ,satisfy The set of intelligent agents is ,satisfy The set of intelligent agents is ;when Sometimes, ;because It is a strongly connected graph, therefore it has make There are: From the above formula, we can obtain that ;when hour, ,have make ; From the above formula, we obtain the first formula: S53: Conclusion 2 is obtained as follows: make , If it is a positive sequence, ,So ; S54: Conclusion 3 is obtained as follows: It is a sequence of random matrices, if in hour, A common Perron vector relating to the eigenvalue 1 And the graph sequence associated with it It is always strongly connected, so for any ,have ; S54: Based on conclusions 1 and 2, taking the limit of both sides of the first formula, since when hour It will reach its Nash equilibrium. ,have to: S55: When adjacency matrix When there is a common Perron vector, according to conclusion 3, we get S56: Based on the above steps, determine the estimation result of the upper bound: For time-varying graphs ,when adjacency matrix When there is no common Perron vector, we have ， , ,in and They are vectors and The One element; when adjacency matrix When there is a common Perron vector, we have ， ； in, and These are the common Perron vectors. and The Each element.

5. A storage device, characterized in that: The storage device stores instructions and data to implement any of the Nash equilibrium estimation methods under any of the time delays described in claims 1 to 4.

6. A Nash equilibrium estimation device under time delay, characterized in that: include: Processor and storage device; the processor loads and executes instructions and data in the storage device to implement the Nash equilibrium estimation method under any one of the time delays of claims 1 to 4.

Images