Dear Editor,This letter addresses the impulse game problem for a general scope of deterministic,multi-player,nonzero-sum differential games wherein all participants adopt impulse controls.Our objective is to formulate...Dear Editor,This letter addresses the impulse game problem for a general scope of deterministic,multi-player,nonzero-sum differential games wherein all participants adopt impulse controls.Our objective is to formulate this impulse game problem with the modified objective function including interaction costs among the players in a discontinuous fashion,and subsequently,to derive a verification theorem for identifying the feedback Nash equilibrium strategy.展开更多
Dear Editor,This letter presents a solution to the problem of seeking Nash equilibrium(NE)in a class of non-cooperative games of multi-agent systems(MASs)subject to the input disturbance and the networked communicatio...Dear Editor,This letter presents a solution to the problem of seeking Nash equilibrium(NE)in a class of non-cooperative games of multi-agent systems(MASs)subject to the input disturbance and the networked communication.To this end,a novel distributed robust predefined-time algorithm is proposed,which ensures the precise convergence of agent states to the NE within a settling time that can be directly determined by adjusting one or more parameters.The proposed algorithm employs an integral sliding mode strategy to effectively reject disturbances.Additionally,a consensus-based estimator is designed to overcome the challenge of limited information availability,where each agent can only access information from its directly connected neighbors,which conflicts with the computation of the cost function that requires information from all agents.Finally,a numerical example is provided to demonstrate the algorithm's effectiveness and performance.展开更多
This paper designs distributed Nash equilibrium seeking strategies for heterogeneous dynamic cyber-physical systems.In particular, we are concerned with parametric uncertainties in the control channel of the players. ...This paper designs distributed Nash equilibrium seeking strategies for heterogeneous dynamic cyber-physical systems.In particular, we are concerned with parametric uncertainties in the control channel of the players. Moreover, the weights on communication links can be compromised by time-varying uncertainties, which can result from possibly malicious attacks,faults and disturbances. To deal with the unavailability of measurement of optimization errors, an output observer is constructed,based on which adaptive laws are designed to compensate for physical uncertainties. With adaptive laws, a new distributed Nash equilibrium seeking strategy is designed by further integrating consensus protocols and gradient search algorithms.Moreover, to further accommodate compromised communication weights resulting from cyber-uncertainties, the coupling strengths of the consensus module are designed to be adaptive. As a byproduct, the coupling strengths are independent of any global information. With theoretical investigations, it is proven that the proposed strategies are resilient to these uncertainties and players' actions are convergent to the Nash equilibrium. Simulation examples are given to numerically validate the effectiveness of the proposed strategies.展开更多
Dear Editor,This letter addresses the Nash equilibrium seeking problem for games with second-order players subject to unknown input deadzones and denial-of-service(DoS)attacks.By using ideas from the digital twin,a di...Dear Editor,This letter addresses the Nash equilibrium seeking problem for games with second-order players subject to unknown input deadzones and denial-of-service(DoS)attacks.By using ideas from the digital twin,a distributed Nash equilibrium seeking strategy is proposed.In the proposed strategy,the twin players are designed to be second-order integrators,based on which a distributed control law is provided so as to find the Nash equilibrium under DoS attacks.Moreover,adaptive control laws and sliding mode control laws are synthesized for the actual players such that they can track the twin players under unknown input dead-zones.Theoretical investigations show that the proposed strategy is effective to drive the actions of actual players to the Nash equilibrium under the given conditions.A numerical example is provided to verify the effectiveness of the proposed strategy.展开更多
Dear Editor,This letter studies the distributed Nash equilibrium seeking problem of aggregative game,in which the decision of each player obeys second-order dynamics and is constrained by nonidentical convex sets.To s...Dear Editor,This letter studies the distributed Nash equilibrium seeking problem of aggregative game,in which the decision of each player obeys second-order dynamics and is constrained by nonidentical convex sets.To seek the generalized Nash equilibrium(GNE),a projectionbased distributed algorithm via constant step-sizes is developed with linear convergence.In particular,a variable tracking technique is incorporated to estimate the aggregative function,and an event-triggered mechanism is designed to reduce the communication cost.Finally,a numerical example demonstrates the theoretical results.展开更多
This paper presents a comprehensive overview of distributed Nash equilibrium(NE)seeking algorithms in non-cooperative games for multiagent systems(MASs),with a distinct emphasis on the dynamic control perspective.It s...This paper presents a comprehensive overview of distributed Nash equilibrium(NE)seeking algorithms in non-cooperative games for multiagent systems(MASs),with a distinct emphasis on the dynamic control perspective.It specifically focuses on the research addressing distributed NE seeking problems in which agents are governed by heterogeneous dynamics.The paper begins by introducing fundamental concepts of general non-cooperative games and the NE,along with definitions of specific game structures such as aggregative games and multi-cluster games.It then systematically reviews existing studies on distributed NE seeking for various classes of MASs from the viewpoint of agent dynamics,including first-order,second-order,high-order,linear,and Euler-Lagrange(EL)systems.Furthermore,the paper highlights practical applications of these theoretical advances in cooperative control scenarios involving autonomous systems with complex dynamics,such as autonomous surface vessels,autonomous aerial vehicles,and other autonomous vehicles.Finally,the paper outlines several promising directions for future research.展开更多
In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with...In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrixcan be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.展开更多
The fuzzy non-cooperative game with fuzzy payoff function is studied. Based on fuzzy set theory with game theory, the fuzzy Nash equilibrium of fuzzy non-cooperative games is proposed. Most of researchers rank fuzzy n...The fuzzy non-cooperative game with fuzzy payoff function is studied. Based on fuzzy set theory with game theory, the fuzzy Nash equilibrium of fuzzy non-cooperative games is proposed. Most of researchers rank fuzzy number by its center of gravity or by the real number with its maximal membership. By reducing fuzzy number into a real number, we lose much fuzzy information that should be kept during the operations between fuzzy numbers. The fuzzy quantities or alternatives are ordered directly by Yuan＇s binary fuzzy ordering relation. In doing so, the existence of fuzzy Nash equilibrium for fuzzy non-cooperative games is shown based on the utility function and the crisp Nash theorem. Finally, an illustrative example in traffic flow patterns of equilibrium is given in order to show the detailed calculation process of fuzzy Nash equilibrium.展开更多
This paper explores the problem of distributed Nash equilibrium seeking in games, where players have limited knowledge on other players' actions. In particular, the involved players are considered to be high-order...This paper explores the problem of distributed Nash equilibrium seeking in games, where players have limited knowledge on other players' actions. In particular, the involved players are considered to be high-order integrators with their control inputs constrained within a pre-specified region. A linear transformation for players' dynamics is firstly utilized to facilitate the design of bounded control inputs incorporating multiple saturation functions. By introducing consensus protocols with adaptive and time-varying gains, the unknown actions for players are distributively estimated. Then, a fully distributed Nash equilibrium seeking strategy is exploited, showcasing its remarkable properties: (1) ensuring the boundedness of control inputs;(2) avoiding any global information/parameters;and (3) allowing the graph to be directed. Based on Lyapunov stability analysis, it is theoretically proved that the proposed distributed control strategy can lead all the players' actions to the Nash equilibrium. Finally, an illustrative example is given to validate effectiveness of the proposed method.展开更多
This paper is concerned with anti-disturbance Nash equilibrium seeking for games with partial information.First,reduced-order disturbance observer-based algorithms are proposed to achieve Nash equilibrium seeking for ...This paper is concerned with anti-disturbance Nash equilibrium seeking for games with partial information.First,reduced-order disturbance observer-based algorithms are proposed to achieve Nash equilibrium seeking for games with firstorder and second-order players,respectively.In the developed algorithms,the observed disturbance values are included in control signals to eliminate the influence of disturbances,based on which a gradient-like optimization method is implemented for each player.Second,a signum function based distributed algorithm is proposed to attenuate disturbances for games with secondorder integrator-type players.To be more specific,a signum function is involved in the proposed seeking strategy to dominate disturbances,based on which the feedback of the velocity-like states and the gradients of the functions associated with players achieves stabilization of system dynamics and optimization of players'objective functions.Through Lyapunov stability analysis,it is proven that the players'actions can approach a small region around the Nash equilibrium by utilizing disturbance observerbased strategies with appropriate control gains.Moreover,exponential(asymptotic)convergence can be achieved when the signum function based control strategy(with an adaptive control gain)is employed.The performance of the proposed algorithms is tested by utilizing an integrated simulation platform of virtual robot experimentation platform(V-REP)and MATLAB.展开更多
The solvability of the coupled Riccati differential equations appearing in the differential game approach to the formation control problem is vital to the finite horizon Nash equilibrium solution.These equations(if so...The solvability of the coupled Riccati differential equations appearing in the differential game approach to the formation control problem is vital to the finite horizon Nash equilibrium solution.These equations(if solvable)can be solved numerically by using the terminal value and the backward iteration.To investigate the solvability and solution of these equations the formation control problem as the differential game is replaced by a discrete-time dynamic game.The main contributions of this paper are as follows.First,the existence of Nash equilibrium controls for the discretetime formation control problem is shown.Second,a backward iteration approximate solution to the coupled Riccati differential equations in the continuous-time differential game is developed.An illustrative example is given to justify the models and solution.展开更多
This paper is concerned with distributed Nash equi librium seeking strategies under quantized communication. In the proposed seeking strategy, a projection operator is synthesized with a gradient search method to achi...This paper is concerned with distributed Nash equi librium seeking strategies under quantized communication. In the proposed seeking strategy, a projection operator is synthesized with a gradient search method to achieve the optimization o players' objective functions while restricting their actions within required non-empty, convex and compact domains. In addition, a leader-following consensus protocol, in which quantized informa tion flows are utilized, is employed for information sharing among players. More specifically, logarithmic quantizers and uniform quantizers are investigated under both undirected and connected communication graphs and strongly connected digraphs, respec tively. Through Lyapunov stability analysis, it is shown that play ers' actions can be steered to a neighborhood of the Nash equilib rium with logarithmic and uniform quantizers, and the quanti fied convergence error depends on the parameter of the quan tizer for both undirected and directed cases. A numerical exam ple is given to verify the theoretical results.展开更多
The generalized Nash equilibrium problem （GNEP） is a generalization of the standard Nash equilibrium problem （NEP）, in which both the utility function and the strategy space of each player depend on the strategies...The generalized Nash equilibrium problem （GNEP） is a generalization of the standard Nash equilibrium problem （NEP）, in which both the utility function and the strategy space of each player depend on the strategies chosen by all other players. This problem has been used to model various problems in applications. However, the convergent solution algorithms are extremely scare in the literature. In this paper, we present an incremental penalty method for the GNEP, and show that a solution of the GNEP can be found by solving a sequence of smooth NEPs. We then apply the semismooth Newton method with Armijo line search to solve latter problems and provide some results of numerical experiments to illustrate the proposed approach.展开更多
In this paper,we consider distributed Nash equilibrium(NE)seeking in potential games over a multi-agent network,where each agent can not observe the actions of all its rivals.Based on the best response dynamics,we des...In this paper,we consider distributed Nash equilibrium(NE)seeking in potential games over a multi-agent network,where each agent can not observe the actions of all its rivals.Based on the best response dynamics,we design a distributed NE seeking algorithm by incorporating the non-smooth finite-time average tracking dynamics,where each agent only needs to know its own action and exchange information with its neighbours through a communication graph.We give a sufficient condition for the Lipschitz continuity of the best response mapping for potential games,and then prove the convergence of the proposed algorithm based on the Lyapunov theory.Numerical simulations are given to verify the resultandillustrate the effectiveness of the algorithm.展开更多
This paper deals with an extension of the one-period model in non-life insurance markets (cf. [1]) by using a transition probability matrix depending on some economic factors. We introduce a multi-period model and in ...This paper deals with an extension of the one-period model in non-life insurance markets (cf. [1]) by using a transition probability matrix depending on some economic factors. We introduce a multi-period model and in each period the solvency constraints will be updated. Moreover, the model has the inactive state including some uninsured population. Similar results on the existence of premium equilibrium and sensitivity analysis for this model are presented and illustrated by numerical results.展开更多
Networked noncooperative games are investigated,where each player(or agent) plays with all other players in its neighborhood. Assume the evolution is based on the fact that each player uses its neighbors current infor...Networked noncooperative games are investigated,where each player(or agent) plays with all other players in its neighborhood. Assume the evolution is based on the fact that each player uses its neighbors current information to decide its next strategy. By using sub-neighborhood, the dynamics of the evolution is obtained. Then a method for calculating Nash equilibriums from mixed strategies of multi-players is proposed.The relationship between local Nash equilibriums based on individual neighborhoods and global Nash equilibriums of overall network is revealed. Then a technique is proposed to construct Nash equilibriums of an evolutionary game from its one step static Nash equilibriums. The basic tool of this approach is the semi-tensor product of matrices, which converts strategies into logical matrices and payoffs into pseudo-Boolean functions, then networked evolutionary games become discrete time dynamic systems.展开更多
In this work,we study a Nash equilibrium(NE)seeking problem for strongly monotone non-cooperative games with prescribed performance.Unlike general NE seeking algorithms,the proposed prescribed-performance NE seeking l...In this work,we study a Nash equilibrium(NE)seeking problem for strongly monotone non-cooperative games with prescribed performance.Unlike general NE seeking algorithms,the proposed prescribed-performance NE seeking laws ensure that the convergence error evolves within a predefined region.Thus,the settling time,convergence rate,and maximum overshoot of the algorithm can be guaranteed.First,we develop a second-order Newton-like algorithm that can guarantee prescribed performance and asymptotically converge to the NE of the game.Then,we develop a first-order gradient-based algorithm.To remove some restrictions on this first-order algorithm,we propose two discontinuous dynamical system-based algorithms using tools from non-smooth analysis and adaptive control.We study the special case in optimization problems.Then,we investigate the robustness of the algorithms.It can be proven that the proposed algorithms can guarantee asymptotic convergence to the Nash equilibrium with prescribed performance in the presence of bounded disturbances.Furthermore,we consider a second-order dynamical system solution.The simulation results verify the effectiveness and efficiency of the algorithms,in terms of their convergence rate and disturbance rejection ability.展开更多
Generalized Nash equilibrium problem （GNEP） is an important model that has many applications in practice. However, a GNEP usually has multiple or even infinitely many Nash equilibrium points and it is not easy to ch...Generalized Nash equilibrium problem （GNEP） is an important model that has many applications in practice. However, a GNEP usually has multiple or even infinitely many Nash equilibrium points and it is not easy to choose a favorable solution from those equilibria. This paper considers a class of GNEP With some kind of separability. We first extend the so-called normalized equilibrium concept to the stationarity sense and then, we propose an approach to solve the normalized stationary points by reformulating the GNEP as a single optimization problem. We further demonstrate the proposed approach on a GNEP model in similar product markets.展开更多
In this paper,we consider a networked game with coupled constraints and focus on variational Nash equilibrium seeking.For distributed algorithm design,we eliminate the coupled constraints by employing local Lagrangian...In this paper,we consider a networked game with coupled constraints and focus on variational Nash equilibrium seeking.For distributed algorithm design,we eliminate the coupled constraints by employing local Lagrangian functions and construct exact penalty terms to attain multipliers'optimal consensus,which yields a set of equilibrium conditions without any coupled constraint and consensus constraint.Moreover,these conditions are only based on strategy and multiplier variables,without auxiliary variables.Then,we present a distributed order-reduced dynamics that updates the strategy and multiplier variables with guaranteed convergence.Compared with many other distributed algorithms,our algorithm contains no auxiliary variable,and therefore,it can save computation and communication.展开更多
基金supported in part by the National Natural Science Foundation of China(62173051)the Fundamental Research Funds for the Central Universities(2024CDJCGJ012,2023CDJXY-010)+1 种基金the Chongqing Technology Innovation and Application Development Special Key Project(CSTB2022TIADCUX0015,CSTB2022TIAD-KPX0162)the China Postdoctoral Science Foundation(2024M763865)
文摘Dear Editor,This letter addresses the impulse game problem for a general scope of deterministic,multi-player,nonzero-sum differential games wherein all participants adopt impulse controls.Our objective is to formulate this impulse game problem with the modified objective function including interaction costs among the players in a discontinuous fashion,and subsequently,to derive a verification theorem for identifying the feedback Nash equilibrium strategy.
基金supported by the National Natural Science Foundation of China(62373162,U24A20268,624B2055).
文摘Dear Editor,This letter presents a solution to the problem of seeking Nash equilibrium(NE)in a class of non-cooperative games of multi-agent systems(MASs)subject to the input disturbance and the networked communication.To this end,a novel distributed robust predefined-time algorithm is proposed,which ensures the precise convergence of agent states to the NE within a settling time that can be directly determined by adjusting one or more parameters.The proposed algorithm employs an integral sliding mode strategy to effectively reject disturbances.Additionally,a consensus-based estimator is designed to overcome the challenge of limited information availability,where each agent can only access information from its directly connected neighbors,which conflicts with the computation of the cost function that requires information from all agents.Finally,a numerical example is provided to demonstrate the algorithm's effectiveness and performance.
基金supported by the National Key R&D Program of China(2022ZD0119604)the National Natural Science Foundation of China(NSFC)(62173181,62222308,62221004)the Natural Science Foundation of Jiangsu Province(BK20220139)
文摘This paper designs distributed Nash equilibrium seeking strategies for heterogeneous dynamic cyber-physical systems.In particular, we are concerned with parametric uncertainties in the control channel of the players. Moreover, the weights on communication links can be compromised by time-varying uncertainties, which can result from possibly malicious attacks,faults and disturbances. To deal with the unavailability of measurement of optimization errors, an output observer is constructed,based on which adaptive laws are designed to compensate for physical uncertainties. With adaptive laws, a new distributed Nash equilibrium seeking strategy is designed by further integrating consensus protocols and gradient search algorithms.Moreover, to further accommodate compromised communication weights resulting from cyber-uncertainties, the coupling strengths of the consensus module are designed to be adaptive. As a byproduct, the coupling strengths are independent of any global information. With theoretical investigations, it is proven that the proposed strategies are resilient to these uncertainties and players' actions are convergent to the Nash equilibrium. Simulation examples are given to numerically validate the effectiveness of the proposed strategies.
基金supported by the National Natural Science Foundation of China(NSFC)(62222308,62173181,62221004)the Natural Science Foundation and Maojiao Ye of Jiangsu Province(BK20220139).
文摘Dear Editor,This letter addresses the Nash equilibrium seeking problem for games with second-order players subject to unknown input deadzones and denial-of-service(DoS)attacks.By using ideas from the digital twin,a distributed Nash equilibrium seeking strategy is proposed.In the proposed strategy,the twin players are designed to be second-order integrators,based on which a distributed control law is provided so as to find the Nash equilibrium under DoS attacks.Moreover,adaptive control laws and sliding mode control laws are synthesized for the actual players such that they can track the twin players under unknown input dead-zones.Theoretical investigations show that the proposed strategy is effective to drive the actions of actual players to the Nash equilibrium under the given conditions.A numerical example is provided to verify the effectiveness of the proposed strategy.
基金supported by the National Natural Science Foundation of China(62473048,61925303,62088101,62273195,U19B2029).
文摘Dear Editor,This letter studies the distributed Nash equilibrium seeking problem of aggregative game,in which the decision of each player obeys second-order dynamics and is constrained by nonidentical convex sets.To seek the generalized Nash equilibrium(GNE),a projectionbased distributed algorithm via constant step-sizes is developed with linear convergence.In particular,a variable tracking technique is incorporated to estimate the aggregative function,and an event-triggered mechanism is designed to reduce the communication cost.Finally,a numerical example demonstrates the theoretical results.
基金National Natural Science Foundation of China(62325304).
文摘This paper presents a comprehensive overview of distributed Nash equilibrium(NE)seeking algorithms in non-cooperative games for multiagent systems(MASs),with a distinct emphasis on the dynamic control perspective.It specifically focuses on the research addressing distributed NE seeking problems in which agents are governed by heterogeneous dynamics.The paper begins by introducing fundamental concepts of general non-cooperative games and the NE,along with definitions of specific game structures such as aggregative games and multi-cluster games.It then systematically reviews existing studies on distributed NE seeking for various classes of MASs from the viewpoint of agent dynamics,including first-order,second-order,high-order,linear,and Euler-Lagrange(EL)systems.Furthermore,the paper highlights practical applications of these theoretical advances in cooperative control scenarios involving autonomous systems with complex dynamics,such as autonomous surface vessels,autonomous aerial vehicles,and other autonomous vehicles.Finally,the paper outlines several promising directions for future research.
基金This work was supported by the National Natural Science Foundation of China(No.60474029)China Postdoctoral Science Foundation (No.2005038558)
文摘In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrixcan be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.
基金supported by the National Natural Science Foundation of China (70771010)
文摘The fuzzy non-cooperative game with fuzzy payoff function is studied. Based on fuzzy set theory with game theory, the fuzzy Nash equilibrium of fuzzy non-cooperative games is proposed. Most of researchers rank fuzzy number by its center of gravity or by the real number with its maximal membership. By reducing fuzzy number into a real number, we lose much fuzzy information that should be kept during the operations between fuzzy numbers. The fuzzy quantities or alternatives are ordered directly by Yuan＇s binary fuzzy ordering relation. In doing so, the existence of fuzzy Nash equilibrium for fuzzy non-cooperative games is shown based on the utility function and the crisp Nash theorem. Finally, an illustrative example in traffic flow patterns of equilibrium is given in order to show the detailed calculation process of fuzzy Nash equilibrium.
基金supported by the National Natural Science Foundation of China(62222308,62173181,62073171,62221004)the Natural Science Foundation of Jiangsu Province(BK20220139,BK20200744)+3 种基金Jiangsu Specially-Appointed Professor(RK043STP19001)the Young Elite Scientists Sponsorship Program by China Association for Science and Technology(CAST)(2021QNRC001)1311 Talent Plan of Nanjing University of Posts and Telecommunicationsthe Fundamental Research Funds for the Central Universities(30920032203)。
文摘This paper explores the problem of distributed Nash equilibrium seeking in games, where players have limited knowledge on other players' actions. In particular, the involved players are considered to be high-order integrators with their control inputs constrained within a pre-specified region. A linear transformation for players' dynamics is firstly utilized to facilitate the design of bounded control inputs incorporating multiple saturation functions. By introducing consensus protocols with adaptive and time-varying gains, the unknown actions for players are distributively estimated. Then, a fully distributed Nash equilibrium seeking strategy is exploited, showcasing its remarkable properties: (1) ensuring the boundedness of control inputs;(2) avoiding any global information/parameters;and (3) allowing the graph to be directed. Based on Lyapunov stability analysis, it is theoretically proved that the proposed distributed control strategy can lead all the players' actions to the Nash equilibrium. Finally, an illustrative example is given to validate effectiveness of the proposed method.
基金supported by the National Natural Science Foundation of China(NSFC)(62222308,62173181,62073171,62221004)the Natural Science Foundation of Jiangsu Province(BK20200744,BK20220139)+3 种基金Jiangsu Specially-Appointed Professor(RK043STP19001)1311 Talent Plan of Nanjing University of Posts and Telecommunicationsthe Young Elite Scientists SponsorshipProgram by CAST(2021QNRC001)the Fundamental Research Funds for the Central Universities(30920032203)。
文摘This paper is concerned with anti-disturbance Nash equilibrium seeking for games with partial information.First,reduced-order disturbance observer-based algorithms are proposed to achieve Nash equilibrium seeking for games with firstorder and second-order players,respectively.In the developed algorithms,the observed disturbance values are included in control signals to eliminate the influence of disturbances,based on which a gradient-like optimization method is implemented for each player.Second,a signum function based distributed algorithm is proposed to attenuate disturbances for games with secondorder integrator-type players.To be more specific,a signum function is involved in the proposed seeking strategy to dominate disturbances,based on which the feedback of the velocity-like states and the gradients of the functions associated with players achieves stabilization of system dynamics and optimization of players'objective functions.Through Lyapunov stability analysis,it is proven that the players'actions can approach a small region around the Nash equilibrium by utilizing disturbance observerbased strategies with appropriate control gains.Moreover,exponential(asymptotic)convergence can be achieved when the signum function based control strategy(with an adaptive control gain)is employed.The performance of the proposed algorithms is tested by utilizing an integrated simulation platform of virtual robot experimentation platform(V-REP)and MATLAB.
文摘The solvability of the coupled Riccati differential equations appearing in the differential game approach to the formation control problem is vital to the finite horizon Nash equilibrium solution.These equations(if solvable)can be solved numerically by using the terminal value and the backward iteration.To investigate the solvability and solution of these equations the formation control problem as the differential game is replaced by a discrete-time dynamic game.The main contributions of this paper are as follows.First,the existence of Nash equilibrium controls for the discretetime formation control problem is shown.Second,a backward iteration approximate solution to the coupled Riccati differential equations in the continuous-time differential game is developed.An illustrative example is given to justify the models and solution.
基金supported by the National Natural Science Foundation of China (NSFC)(62222308, 62173181, 62073171, 62221004)the Natural Science Foundation of Jiangsu Province (BK20200744, BK20220139)+3 种基金Jiangsu Specially-Appointed Professor (RK043STP19001)the Young Elite Scientists Sponsorship Program by CAST (2021QNRC001)1311 Talent Plan of Nanjing University of Posts and Telecommunicationsthe Fundamental Research Funds for the Central Universities (30920032203)。
文摘This paper is concerned with distributed Nash equi librium seeking strategies under quantized communication. In the proposed seeking strategy, a projection operator is synthesized with a gradient search method to achieve the optimization o players' objective functions while restricting their actions within required non-empty, convex and compact domains. In addition, a leader-following consensus protocol, in which quantized informa tion flows are utilized, is employed for information sharing among players. More specifically, logarithmic quantizers and uniform quantizers are investigated under both undirected and connected communication graphs and strongly connected digraphs, respec tively. Through Lyapunov stability analysis, it is shown that play ers' actions can be steered to a neighborhood of the Nash equilib rium with logarithmic and uniform quantizers, and the quanti fied convergence error depends on the parameter of the quan tizer for both undirected and directed cases. A numerical exam ple is given to verify the theoretical results.
文摘The generalized Nash equilibrium problem （GNEP） is a generalization of the standard Nash equilibrium problem （NEP）, in which both the utility function and the strategy space of each player depend on the strategies chosen by all other players. This problem has been used to model various problems in applications. However, the convergent solution algorithms are extremely scare in the literature. In this paper, we present an incremental penalty method for the GNEP, and show that a solution of the GNEP can be found by solving a sequence of smooth NEPs. We then apply the semismooth Newton method with Armijo line search to solve latter problems and provide some results of numerical experiments to illustrate the proposed approach.
基金This work was supported by the Shanghai Sailing Program(No.20YF1453000)the Fundamental Research Funds for the Central Universities(No.22120200048).
文摘In this paper,we consider distributed Nash equilibrium(NE)seeking in potential games over a multi-agent network,where each agent can not observe the actions of all its rivals.Based on the best response dynamics,we design a distributed NE seeking algorithm by incorporating the non-smooth finite-time average tracking dynamics,where each agent only needs to know its own action and exchange information with its neighbours through a communication graph.We give a sufficient condition for the Lipschitz continuity of the best response mapping for potential games,and then prove the convergence of the proposed algorithm based on the Lyapunov theory.Numerical simulations are given to verify the resultandillustrate the effectiveness of the algorithm.
文摘This paper deals with an extension of the one-period model in non-life insurance markets (cf. [1]) by using a transition probability matrix depending on some economic factors. We introduce a multi-period model and in each period the solvency constraints will be updated. Moreover, the model has the inactive state including some uninsured population. Similar results on the existence of premium equilibrium and sensitivity analysis for this model are presented and illustrated by numerical results.
文摘Networked noncooperative games are investigated,where each player(or agent) plays with all other players in its neighborhood. Assume the evolution is based on the fact that each player uses its neighbors current information to decide its next strategy. By using sub-neighborhood, the dynamics of the evolution is obtained. Then a method for calculating Nash equilibriums from mixed strategies of multi-players is proposed.The relationship between local Nash equilibriums based on individual neighborhoods and global Nash equilibriums of overall network is revealed. Then a technique is proposed to construct Nash equilibriums of an evolutionary game from its one step static Nash equilibriums. The basic tool of this approach is the semi-tensor product of matrices, which converts strategies into logical matrices and payoffs into pseudo-Boolean functions, then networked evolutionary games become discrete time dynamic systems.
基金supported by the RIE2020 Industry Alignment Fund-Industry Collaboration Projects(IAF-ICP)Funding Initiative,as well as cash and in-kind contribution from the industry partner(s).
文摘In this work,we study a Nash equilibrium(NE)seeking problem for strongly monotone non-cooperative games with prescribed performance.Unlike general NE seeking algorithms,the proposed prescribed-performance NE seeking laws ensure that the convergence error evolves within a predefined region.Thus,the settling time,convergence rate,and maximum overshoot of the algorithm can be guaranteed.First,we develop a second-order Newton-like algorithm that can guarantee prescribed performance and asymptotically converge to the NE of the game.Then,we develop a first-order gradient-based algorithm.To remove some restrictions on this first-order algorithm,we propose two discontinuous dynamical system-based algorithms using tools from non-smooth analysis and adaptive control.We study the special case in optimization problems.Then,we investigate the robustness of the algorithms.It can be proven that the proposed algorithms can guarantee asymptotic convergence to the Nash equilibrium with prescribed performance in the presence of bounded disturbances.Furthermore,we consider a second-order dynamical system solution.The simulation results verify the effectiveness and efficiency of the algorithms,in terms of their convergence rate and disturbance rejection ability.
基金Supported by the National Natural Science Foundation of China(Grant No.11071028)
文摘Generalized Nash equilibrium problem （GNEP） is an important model that has many applications in practice. However, a GNEP usually has multiple or even infinitely many Nash equilibrium points and it is not easy to choose a favorable solution from those equilibria. This paper considers a class of GNEP With some kind of separability. We first extend the so-called normalized equilibrium concept to the stationarity sense and then, we propose an approach to solve the normalized stationary points by reformulating the GNEP as a single optimization problem. We further demonstrate the proposed approach on a GNEP model in similar product markets.
基金supported in part by the National Key Research and Development Program of China under grant 2022YFA1004700in part by the Natural Science Foundation of China under grant 72171171in part by Shanghai Municipal Science and Technology Major Project under grant 2021SHZDZX0100.
文摘In this paper,we consider a networked game with coupled constraints and focus on variational Nash equilibrium seeking.For distributed algorithm design,we eliminate the coupled constraints by employing local Lagrangian functions and construct exact penalty terms to attain multipliers'optimal consensus,which yields a set of equilibrium conditions without any coupled constraint and consensus constraint.Moreover,these conditions are only based on strategy and multiplier variables,without auxiliary variables.Then,we present a distributed order-reduced dynamics that updates the strategy and multiplier variables with guaranteed convergence.Compared with many other distributed algorithms,our algorithm contains no auxiliary variable,and therefore,it can save computation and communication.
