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
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
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
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
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
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 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
With the increasing popularity of ecological civilization and sustainable development,enterprises should consider environmental protection measures in their operations in addition to pursue their economic interests.Th...With the increasing popularity of ecological civilization and sustainable development,enterprises should consider environmental protection measures in their operations in addition to pursue their economic interests.This paper establsihes a closed-loop supply chain network model composed of multiple suppliers,manufacturers,retailers,recyclers,and demand markets—regarding their dual goals of the profit maximization and the minimization of carbon emissions.The conditions necessary for establishing overall equilibrium and an equilibrium model of the entire closed-loop supply chain network are determined by applying variational inequality and dual theory.A modified projection contraction algorithm is used to design a model-solving program.Finally,using numerical examples,the paper conducts a comparative static analysis on important parameters such as the weight coefficients of environmental protection objectives and consumers'awareness of low-carbon environmental protection and attains some beneficial enlightenment on management.The results indicate that when the environmental protection objectives of a certain type of enterprise increases,both the economic benefits and environmental protection performance will improve;when the environmental protection objectives of all enterprises increases simultaneously,environmental protection performance improves significantly,but the changes in economic benefits of different enterprises are inconsistent and profit coordination is more complex.Although consumers’awareness of low-carbon preference could improve environmental performance,it reduces the overall profits of network members and the entire closed-loop supply chain network as a whole.The above conclusions can be used as a reference for the government in designing low-carbon environmental protection policy and in closed-loop supply chain research.展开更多
Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium ...Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium in parameter space. However, in real games in business and economics, the input parameters are not given. They are typically observed in several similar occasions of the past. Therefore they have a distribution and the average is used. Even if the averages are in an area of Nash equilibrium, some values may be outside making the entire result meaningless. As the averages are sometimes just guessed, the distribution cannot be known. The main focus of this article is to show this effect in an example, and to explain the surprising result by topological explanations. We give an example of two players having three strategies each (e.g. player and keeper in penalty shooting) where we demonstrate the effect explicitly. As the transition of Nash equilibrium to no equilibrium is sharp, there may be a special form of chaos which we suggest to call topological chaos.展开更多
We define generalized quantum games by introducing the coherent payoff operators and propose a simple scheme to illustrate it.The scheme is implemented with a single spin qubit system and a two-entangled-qubit system....We define generalized quantum games by introducing the coherent payoff operators and propose a simple scheme to illustrate it.The scheme is implemented with a single spin qubit system and a two-entangled-qubit system.The Nash Equilibrium Theorem is proved for the models.展开更多
Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model ...Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model with single supplier and multiples retailers and price dependent demand, where retailers compete on retail prices. Each retailer decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is maximized. Bernstein and Federgruen have proved that the retail prices become a unique Nash equilibrium solution under weak conditions on the price dependent distribution of demand. The authors, however, have not mentioned the numerical values and proprieties on these retail prices, the number of products and their individual and overall profits. In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to theorem of Nash equilibrium solutions, which Bernstein and Federgruen proved, and we show their modified results. Then, we compute numerically Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’ profits, and supply chain optimal retailers’ prices. We also discuss properties on relation between these values and the demand distribution.展开更多
A closed-loop supply chain network equilibrium problem is examined,which consists of mutual competitive manufacturers with production capacity constraints and retailers with fuzzy market demands,mutual competitive col...A closed-loop supply chain network equilibrium problem is examined,which consists of mutual competitive manufacturers with production capacity constraints and retailers with fuzzy market demands,mutual competitive collectors as well as a binding price ceiling of the commodities.By utilizing the credibility measure of fuzzy event,variational inequality and Lagrange dual theory,the optimum behaviors of manufactures,retailers,collectors and consumers are described.Therefore,the supply chain network equilibrium model is established.Numerical examples were given to illustrate the impact of production capacity constraints and price ceiling on the network equilibrium pattern.The results show that the shortage of commodities in consumer market will be more serious,the wholesale price of product and purchase price of waste products rise,the profits of retailers decrease,the profits of manufacturers and collectors increase as the government gradually reduce the binding price ceiling on competitive markets,and the trend will be more obvious in the existence of production capacity constraints at same time.展开更多
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.
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.
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 (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.
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.
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.
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 Humanity and Social Science Foundation of Ministry of Education of China[Grant number 17YJA630130].
文摘With the increasing popularity of ecological civilization and sustainable development,enterprises should consider environmental protection measures in their operations in addition to pursue their economic interests.This paper establsihes a closed-loop supply chain network model composed of multiple suppliers,manufacturers,retailers,recyclers,and demand markets—regarding their dual goals of the profit maximization and the minimization of carbon emissions.The conditions necessary for establishing overall equilibrium and an equilibrium model of the entire closed-loop supply chain network are determined by applying variational inequality and dual theory.A modified projection contraction algorithm is used to design a model-solving program.Finally,using numerical examples,the paper conducts a comparative static analysis on important parameters such as the weight coefficients of environmental protection objectives and consumers'awareness of low-carbon environmental protection and attains some beneficial enlightenment on management.The results indicate that when the environmental protection objectives of a certain type of enterprise increases,both the economic benefits and environmental protection performance will improve;when the environmental protection objectives of all enterprises increases simultaneously,environmental protection performance improves significantly,but the changes in economic benefits of different enterprises are inconsistent and profit coordination is more complex.Although consumers’awareness of low-carbon preference could improve environmental performance,it reduces the overall profits of network members and the entire closed-loop supply chain network as a whole.The above conclusions can be used as a reference for the government in designing low-carbon environmental protection policy and in closed-loop supply chain research.
文摘Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium in parameter space. However, in real games in business and economics, the input parameters are not given. They are typically observed in several similar occasions of the past. Therefore they have a distribution and the average is used. Even if the averages are in an area of Nash equilibrium, some values may be outside making the entire result meaningless. As the averages are sometimes just guessed, the distribution cannot be known. The main focus of this article is to show this effect in an example, and to explain the surprising result by topological explanations. We give an example of two players having three strategies each (e.g. player and keeper in penalty shooting) where we demonstrate the effect explicitly. As the transition of Nash equilibrium to no equilibrium is sharp, there may be a special form of chaos which we suggest to call topological chaos.
文摘We define generalized quantum games by introducing the coherent payoff operators and propose a simple scheme to illustrate it.The scheme is implemented with a single spin qubit system and a two-entangled-qubit system.The Nash Equilibrium Theorem is proved for the models.
文摘Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model with single supplier and multiples retailers and price dependent demand, where retailers compete on retail prices. Each retailer decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is maximized. Bernstein and Federgruen have proved that the retail prices become a unique Nash equilibrium solution under weak conditions on the price dependent distribution of demand. The authors, however, have not mentioned the numerical values and proprieties on these retail prices, the number of products and their individual and overall profits. In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to theorem of Nash equilibrium solutions, which Bernstein and Federgruen proved, and we show their modified results. Then, we compute numerically Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’ profits, and supply chain optimal retailers’ prices. We also discuss properties on relation between these values and the demand distribution.
基金study by the National Natural Science Foundation of China,Shandong province by the National Science Foundation Project,Project number:71202142,ZR2012GM002
文摘A closed-loop supply chain network equilibrium problem is examined,which consists of mutual competitive manufacturers with production capacity constraints and retailers with fuzzy market demands,mutual competitive collectors as well as a binding price ceiling of the commodities.By utilizing the credibility measure of fuzzy event,variational inequality and Lagrange dual theory,the optimum behaviors of manufactures,retailers,collectors and consumers are described.Therefore,the supply chain network equilibrium model is established.Numerical examples were given to illustrate the impact of production capacity constraints and price ceiling on the network equilibrium pattern.The results show that the shortage of commodities in consumer market will be more serious,the wholesale price of product and purchase price of waste products rise,the profits of retailers decrease,the profits of manufacturers and collectors increase as the government gradually reduce the binding price ceiling on competitive markets,and the trend will be more obvious in the existence of production capacity constraints at same time.