In this paper, we consider multiobjective two-person zero-sum games with vector payoffs and vector fuzzy payoffs. We translate such games into the corresponding multiobjective programming problems and introduce the pe...In this paper, we consider multiobjective two-person zero-sum games with vector payoffs and vector fuzzy payoffs. We translate such games into the corresponding multiobjective programming problems and introduce the pessimistic Pareto optimal solution concept by assuming that a player supposes the opponent adopts the most disadvantage strategy for the self. It is shown that any pessimistic Pareto optimal solution can be obtained on the basis of linear programming techniques even if the membership functions for the objective functions are nonlinear. Moreover, we propose interactive algorithms based on the bisection method to obtain a pessimistic compromise solution from among the set of all pessimistic Pareto optimal solutions. In order to show the efficiency of the proposed method, we illustrate interactive processes of an application to a vegetable shipment problem.展开更多
Nowadays,China is the largest developing country in the world,and the US is the largest developed country in the world.Sino-US economic and trade relations are of great significance to the two nations and may have apr...Nowadays,China is the largest developing country in the world,and the US is the largest developed country in the world.Sino-US economic and trade relations are of great significance to the two nations and may have aprominent impact on the stability and development of the global economy.展开更多
There are a few studies that focus on solution methods for finding a Nash equilibrium of zero-sum games. We discuss the use of Karmarkar’s interior point method to solve the Nash equilibrium problems of a zero-sum ga...There are a few studies that focus on solution methods for finding a Nash equilibrium of zero-sum games. We discuss the use of Karmarkar’s interior point method to solve the Nash equilibrium problems of a zero-sum game, and prove that it is theoretically a polynomial time algorithm. We implement the Karmarkar method, and a preliminary computational result shows that it performs well for zero-sum games. We also mention an affine scaling method that would help us compute Nash equilibria of general zero-sum games effectively.展开更多
In this paper,an accelerated value iteration(VI)algorithm is established to solve the zero-sum game problem with convergence guarantee.First,inspired by the successive over relaxation theory,the convergence rate of th...In this paper,an accelerated value iteration(VI)algorithm is established to solve the zero-sum game problem with convergence guarantee.First,inspired by the successive over relaxation theory,the convergence rate of the iterative value function sequence is accelerated significantly with the relaxation factor.Second,the convergence and monotonicity of the value function sequence are analyzed under different ranges of the relaxation factor.Third,two practical approaches,namely the integrated scheme and the relaxation function,are introduced into the accelerated VI algorithm to guarantee the convergence of the iterative value function sequence for zero-sum games.The integrated scheme consists of the accelerated stage and the convergence stage,and the relaxation function can adjust the value of the relaxation factor.Finally,including the autopilot controller,the fantastic performance of the accelerated VI algorithm is verified through two examples with practical physical backgrounds.展开更多
Missile interception problem can be regarded as a two-person zero-sum differential games problem,which depends on the solution of Hamilton-Jacobi-Isaacs(HJI)equa-tion.It has been proved impossible to obtain a closed-f...Missile interception problem can be regarded as a two-person zero-sum differential games problem,which depends on the solution of Hamilton-Jacobi-Isaacs(HJI)equa-tion.It has been proved impossible to obtain a closed-form solu-tion due to the nonlinearity of HJI equation,and many iterative algorithms are proposed to solve the HJI equation.Simultane-ous policy updating algorithm(SPUA)is an effective algorithm for solving HJI equation,but it is an on-policy integral reinforce-ment learning(IRL).For online implementation of SPUA,the dis-turbance signals need to be adjustable,which is unrealistic.In this paper,an off-policy IRL algorithm based on SPUA is pro-posed without making use of any knowledge of the systems dynamics.Then,a neural-network based online adaptive critic implementation scheme of the off-policy IRL algorithm is pre-sented.Based on the online off-policy IRL method,a computa-tional intelligence interception guidance(CIIG)law is developed for intercepting high-maneuvering target.As a model-free method,intercepting targets can be achieved through measur-ing system data online.The effectiveness of the CIIG is verified through two missile and target engagement scenarios.展开更多
This paper will present an approximate/adaptive dynamic programming(ADP) algorithm,that uses the idea of integral reinforcement learning(IRL),to determine online the Nash equilibrium solution for the two-player zerosu...This paper will present an approximate/adaptive dynamic programming(ADP) algorithm,that uses the idea of integral reinforcement learning(IRL),to determine online the Nash equilibrium solution for the two-player zerosum differential game with linear dynamics and infinite horizon quadratic cost.The algorithm is built around an iterative method that has been developed in the control engineering community for solving the continuous-time game algebraic Riccati equation(CT-GARE),which underlies the game problem.We here show how the ADP techniques will enhance the capabilities of the offline method allowing an online solution without the requirement of complete knowledge of the system dynamics.The feasibility of the ADP scheme is demonstrated in simulation for a power system control application.The adaptation goal is the best control policy that will face in an optimal manner the highest load disturbance.展开更多
In this paper we study zero-sum stochastic games. The optimality criterion is the long-run expected average criterion, and the payoff function may have neither upper nor lower bounds. We give a new set of conditions f...In this paper we study zero-sum stochastic games. The optimality criterion is the long-run expected average criterion, and the payoff function may have neither upper nor lower bounds. We give a new set of conditions for the existence of a value and a pair of optimal stationary strategies. Our conditions are slightly weaker than those in the previous literature, and some new sufficient conditions for the existence of a pair of optimal stationary strategies are imposed on the primitive data of the model. Our results are illustrated with a queueing system, for which our conditions are satisfied but some of the conditions in some previous literatures fail to hold.展开更多
In this paper,a zero-sum game Nash equilibrium computation problem with a common constraint set is investigated under two time-varying multi-agent subnetworks,where the two subnetworks have opposite payoff function.A ...In this paper,a zero-sum game Nash equilibrium computation problem with a common constraint set is investigated under two time-varying multi-agent subnetworks,where the two subnetworks have opposite payoff function.A novel distributed projection subgradient algorithm with random sleep scheme is developed to reduce the calculation amount of agents in the process of computing Nash equilibrium.In our algorithm,each agent is determined by an independent identically distributed Bernoulli decision to compute the subgradient and perform the projection operation or to keep the previous consensus estimate,it effectively reduces the amount of computation and calculation time.Moreover,the traditional assumption of stepsize adopted in the existing methods is removed,and the stepsizes in our algorithm are randomized diminishing.Besides,we prove that all agents converge to Nash equilibrium with probability 1 by our algorithm.Finally,a simulation example verifies the validity of our algorithm.展开更多
In this paper,a zero-sum game Nash equilibrium computation problem with event-triggered communication is investigated under an undirected weight-balanced multi-agent network.A novel distributed event-triggered project...In this paper,a zero-sum game Nash equilibrium computation problem with event-triggered communication is investigated under an undirected weight-balanced multi-agent network.A novel distributed event-triggered projection subgradient algorithm is developed to reduce the communication burden within the subnetworks.In the proposed algorithm,when the difference between the current state of the agent and the state of the last trigger time exceeds a given threshold,the agent will be triggered to communicate with its neighbours.Moreover,we prove that all agents converge to Nash equilibrium by the proposed algorithm.Finally,two simulation examples verify that our algorithm not only reduces the communication burden but also ensures that the convergence speed and accuracy are close to that of the time-triggered method under the appropriate threshold.展开更多
This paper studies the policy iteration algorithm(PIA)for zero-sum stochastic differential games with the basic long-run average criterion,as well as with its more selective version,the so-called bias criterion.The sy...This paper studies the policy iteration algorithm(PIA)for zero-sum stochastic differential games with the basic long-run average criterion,as well as with its more selective version,the so-called bias criterion.The system is assumed to be a nondegenerate diffusion.We use Lyapunov-like stability conditions that ensure the existence and boundedness of the solution to certain Poisson equation.We also ensure the convergence of a sequence of such solutions,of the corresponding sequence of policies,and,ultimately,of the PIA.展开更多
文摘In this paper, we consider multiobjective two-person zero-sum games with vector payoffs and vector fuzzy payoffs. We translate such games into the corresponding multiobjective programming problems and introduce the pessimistic Pareto optimal solution concept by assuming that a player supposes the opponent adopts the most disadvantage strategy for the self. It is shown that any pessimistic Pareto optimal solution can be obtained on the basis of linear programming techniques even if the membership functions for the objective functions are nonlinear. Moreover, we propose interactive algorithms based on the bisection method to obtain a pessimistic compromise solution from among the set of all pessimistic Pareto optimal solutions. In order to show the efficiency of the proposed method, we illustrate interactive processes of an application to a vegetable shipment problem.
文摘Nowadays,China is the largest developing country in the world,and the US is the largest developed country in the world.Sino-US economic and trade relations are of great significance to the two nations and may have aprominent impact on the stability and development of the global economy.
文摘There are a few studies that focus on solution methods for finding a Nash equilibrium of zero-sum games. We discuss the use of Karmarkar’s interior point method to solve the Nash equilibrium problems of a zero-sum game, and prove that it is theoretically a polynomial time algorithm. We implement the Karmarkar method, and a preliminary computational result shows that it performs well for zero-sum games. We also mention an affine scaling method that would help us compute Nash equilibria of general zero-sum games effectively.
基金Supported by National High Technology Research and Development Program of China （863 Program） （2006AA04Z183）, National Natural Science Foundation of China （60621001, 60534010, 60572070, 60774048, 60728307）, Program for Changjiang Scholars and Innovative Research Groups of China （60728307, 4031002）
基金supported in part by the National Natural Science Foundation of China under Grant 62222301,Grant 61890930-5,and Grant 62021003the National Science and Technology Major Project under Grant 2021ZD0112302 and Grant 2021ZD0112301the Beijing Natural Science Foundation under Grant JQ19013.
文摘In this paper,an accelerated value iteration(VI)algorithm is established to solve the zero-sum game problem with convergence guarantee.First,inspired by the successive over relaxation theory,the convergence rate of the iterative value function sequence is accelerated significantly with the relaxation factor.Second,the convergence and monotonicity of the value function sequence are analyzed under different ranges of the relaxation factor.Third,two practical approaches,namely the integrated scheme and the relaxation function,are introduced into the accelerated VI algorithm to guarantee the convergence of the iterative value function sequence for zero-sum games.The integrated scheme consists of the accelerated stage and the convergence stage,and the relaxation function can adjust the value of the relaxation factor.Finally,including the autopilot controller,the fantastic performance of the accelerated VI algorithm is verified through two examples with practical physical backgrounds.
文摘Missile interception problem can be regarded as a two-person zero-sum differential games problem,which depends on the solution of Hamilton-Jacobi-Isaacs(HJI)equa-tion.It has been proved impossible to obtain a closed-form solu-tion due to the nonlinearity of HJI equation,and many iterative algorithms are proposed to solve the HJI equation.Simultane-ous policy updating algorithm(SPUA)is an effective algorithm for solving HJI equation,but it is an on-policy integral reinforce-ment learning(IRL).For online implementation of SPUA,the dis-turbance signals need to be adjustable,which is unrealistic.In this paper,an off-policy IRL algorithm based on SPUA is pro-posed without making use of any knowledge of the systems dynamics.Then,a neural-network based online adaptive critic implementation scheme of the off-policy IRL algorithm is pre-sented.Based on the online off-policy IRL method,a computa-tional intelligence interception guidance(CIIG)law is developed for intercepting high-maneuvering target.As a model-free method,intercepting targets can be achieved through measur-ing system data online.The effectiveness of the CIIG is verified through two missile and target engagement scenarios.
基金supported by the National Science Foundation (No.ECCS-0801330)the Army Research Office (No.W91NF-05-1-0314)
文摘This paper will present an approximate/adaptive dynamic programming(ADP) algorithm,that uses the idea of integral reinforcement learning(IRL),to determine online the Nash equilibrium solution for the two-player zerosum differential game with linear dynamics and infinite horizon quadratic cost.The algorithm is built around an iterative method that has been developed in the control engineering community for solving the continuous-time game algebraic Riccati equation(CT-GARE),which underlies the game problem.We here show how the ADP techniques will enhance the capabilities of the offline method allowing an online solution without the requirement of complete knowledge of the system dynamics.The feasibility of the ADP scheme is demonstrated in simulation for a power system control application.The adaptation goal is the best control policy that will face in an optimal manner the highest load disturbance.
文摘In this paper we study zero-sum stochastic games. The optimality criterion is the long-run expected average criterion, and the payoff function may have neither upper nor lower bounds. We give a new set of conditions for the existence of a value and a pair of optimal stationary strategies. Our conditions are slightly weaker than those in the previous literature, and some new sufficient conditions for the existence of a pair of optimal stationary strategies are imposed on the primitive data of the model. Our results are illustrated with a queueing system, for which our conditions are satisfied but some of the conditions in some previous literatures fail to hold.
文摘In this paper,a zero-sum game Nash equilibrium computation problem with a common constraint set is investigated under two time-varying multi-agent subnetworks,where the two subnetworks have opposite payoff function.A novel distributed projection subgradient algorithm with random sleep scheme is developed to reduce the calculation amount of agents in the process of computing Nash equilibrium.In our algorithm,each agent is determined by an independent identically distributed Bernoulli decision to compute the subgradient and perform the projection operation or to keep the previous consensus estimate,it effectively reduces the amount of computation and calculation time.Moreover,the traditional assumption of stepsize adopted in the existing methods is removed,and the stepsizes in our algorithm are randomized diminishing.Besides,we prove that all agents converge to Nash equilibrium with probability 1 by our algorithm.Finally,a simulation example verifies the validity of our algorithm.
文摘In this paper,a zero-sum game Nash equilibrium computation problem with event-triggered communication is investigated under an undirected weight-balanced multi-agent network.A novel distributed event-triggered projection subgradient algorithm is developed to reduce the communication burden within the subnetworks.In the proposed algorithm,when the difference between the current state of the agent and the state of the last trigger time exceeds a given threshold,the agent will be triggered to communicate with its neighbours.Moreover,we prove that all agents converge to Nash equilibrium by the proposed algorithm.Finally,two simulation examples verify that our algorithm not only reduces the communication burden but also ensures that the convergence speed and accuracy are close to that of the time-triggered method under the appropriate threshold.
文摘This paper studies the policy iteration algorithm(PIA)for zero-sum stochastic differential games with the basic long-run average criterion,as well as with its more selective version,the so-called bias criterion.The system is assumed to be a nondegenerate diffusion.We use Lyapunov-like stability conditions that ensure the existence and boundedness of the solution to certain Poisson equation.We also ensure the convergence of a sequence of such solutions,of the corresponding sequence of policies,and,ultimately,of the PIA.
