Algebraic formulation and strategy optimization for a class of evolutionary networked games via semi-tensor product method

Using the semi-tensor product method, this paper investigates the algebraic formulation and strategy optimization for a class of evolutionary networked games with “myopic best response adjustment” rule, and presents a number of new results. First, the dynamics of the evolutionary networked game is converted to an algebraic form via the semi-tensor product, and an algorithm is established to construct the algebraic formulation for the game. Second, based on the algebraic form, the dynamical behavior of evolutionary networked games is discussed, and some interesting results are presented. Finally, the strategy optimization problem is considered by adding a pseudo-player to the game, and a free-type control sequence is designed to maximize the average payoff of the pseudo-player. The study of an illustrative example shows that the new results obtained in this paper work very well.     

Modelling and strategy optimisation for a kind of networked evolutionary games with memories under the bankruptcy mechanism

This paper investigates the evolutionary dynamic and strategy optimisation for a kind of networked evolutionary games whose strategy updating rules incorporate ‘bankruptcy’ mechanism, and the situation that each player's bankruptcy is due to the previous continuous low profits gaining from the game is considered. First, by using semi-tensor product of matrices method, the evolutionary dynamic of this kind of games is expressed as a higher order logical dynamic system and then converted into its algebraic form, based on which, the evolutionary dynamic of the given games can be discussed. Second, the strategy optimisation problem is investigated, and some free-type control sequences are designed to maximise the total payoff of the whole game. Finally, an illustrative example is given to show that our new results are very effective.     

A Matrix Approach to the Modeling and Analysis of Networked Evolutionary Games With Time Delays

Using the semi-tensor product method, this paper investigates the modeling and analysis of networked evolutionary games (NEGs) with finite memories, and presents a number of new results. Firstly, a kind of algebraic expression is formulated for the networked evolutionary games with finite memories, based on which the behavior of the corresponding evolutionary game is analyzed. Secondly, under a proper assumption, the existence of Nash equilibrium of the given networked evolutionary games is proved and a free-type strategy sequence is designed for the convergence to the Nash equilibrium. Finally, an illustrative example is worked out to support the obtained new results.      

Stable degree analysis for strategy profiles of evolutionary networked games网络演化博弈中策略组合的稳定度分析

In this paper, we investigate the stable degree of strategy profile for evolutionary networked games by using the semi-tensor product method, and present a number of new results. First, we propose the concept of k-degree stability for strategy profiles based on a normal evolutionary networked game model. Second, using the semi-tensor product of matrices, we convert the game dynamics with “best imitate” strategy updating rule into an algebraic form. Third, based on the algebraic form of the game, we analyzed the stable degree of strategy profile, and proposed two necessary and sufficient conditions for the k-degree stability of strategy profile. Furthermore, we discuss the computation problem of the transient time within which a disturbed strategy profile can be restored, and also establish an algorithm for the verification of the stable degree of strategy profile. The study of an illustrative example shows that the new results obtained in this paper are very effective.     

Semi-tensor product approach to networked evolutionary games

In this paper a comprehensive introduction for modeling and control of networked evolutionary games （NEGs） via semi-tensor product （STP） approach is presented. First, we review the mathematical model of an NEG, which consists of three ingredients： network graph, fundamental network game, and strategy updating rule. Three kinds of network graphs are considered, which are i） undirected graph for symmetric games; ii） directed graph for asymmetric games, and iii） d-directed graph for symmetric games with partial neighborhood information. Three kinds of fundamental evolutionary games （FEGs） are discussed, which are i） two strategies and symmetric （S-2）; ii） two strategies and asymmetric （A-2）; and iii） three strategies and symmetric （S-3）. Three strategy updating rules （SUR） are introduced, which are i） Unconditional Imitation （UI）; ii） Fermi Rule（FR）; iii） Myopic Best Response Adjustment Rule （MBRA）. First, we review the fundamental evolutionary equation （FEE） and use it to construct network profile dynamics （NPD）of NEGs.
To show how the dynamics of an NEG can be modeled as a discrete time dynamics within an algebraic state space, the fundamental evolutionary equation （FEE） of each player is discussed. Using FEEs, the network strategy profile dynamics （NSPD） is built by providing efficient algorithms. Finally, we consider three more complicated NEGs： i） NEG with different length historical information, ii） NEG with multi-species, and iii） NEG with time-varying payoffs. In all the cases, formulas are provided to construct the corresponding NSPDs. Using these NSPDs, certain properties are explored. Examples are presented to demonstrate the model constructing method, analysis and control design technique, and to reveal certain dynamic behaviors of NEGs.     

Weighted Nash equilibrium of incomplete-profile networked evolutionary games with multiple payoffs

This paper investigates the existence and convergence of weighted Nash equilibrium for incomplete-profile networked evolutionary games with multiple payoffs. First, the incomplete-profile networked evolutionary game under probabilistic myopic best response adjustment rule is transformed into an algebraic form based on the semi-tensor product of matrices. Second, a method for calculating weighted Nash equilibrium is presented, and the relationship between weighted Nash equilibrium and positive-probability fixed point is derived. Furthermore, a criterion is provided to verify whether the profiles in the feasible profile set can converge to the set of weighted Nash equilibriums with probability one. Finally, an illustrative example is given to support the new results obtained in this paper.     

多值逻辑网络的输出跟踪牵制控制

本文利用代数状态空间方法,研究了多值逻辑控制网络的输出跟踪牵制控制.首先利用矩阵的半张量积给出了带牵制控制的多值逻辑控制网络的代数表示.其次基于该代数表示,定义了一组合适的能达集,并建立了多值逻辑控制网络输出跟踪牵制控制器的设计方法.再次,利用多值逻辑哑算子的性质,给出了多值逻辑控制网络分布式输出跟踪控制问题可解的充要条件.最后将所得的理论结果应用于网络演化博弈的演化行为分析.     

概率级联布尔网络的集镇定及其应用

随着系统生物学和医学的迅速发展,基因调控网络已经成为一个热点研究领域.布尔网络作为研究生物系统和基因调控网络的一种重要模型,近年来引起了包括生物学家和系统科学家在内的很多学者的广泛关注.本文利用代数状态空间方法,研究了概率级联布尔网络的集镇定问题.首先给出概率级联布尔网络集镇定的定义,并利用矩阵的半张量积给出了概率级联布尔网络的代数表示.其次基于该代数表示,定义了一组合适的概率能达集,并给出了概率级联布尔网络集镇定问题可解的充要条件.最后将所得的理论结果应用于概率级联布尔网络的同步分析及n人随机级联演化布尔博弈的策略一致演化行为分析.     

Dynamics and stability for a class of evolutionary games with time delays in strategies一类具有时滞策略的演化博弈的动力学和稳定性

This paper investigates the modeling and stability of a class of finite evolutionary games with time delays in strategies. First, the evolutionary dynamics of a sequence of strategy profiles, named as the profile trajectory, is proposed to describe the strategy updating process of the evolutionary games with time delays. Using the semi-tensor product of matrices, the profile trajectory dynamics with two kinds of time delays are converted into their algebraic forms respectively. Then a sufficient condition is obtained to assure the stability of the delayed evolutionary potential games at a pure Nash equilibrium.     

A Matrix Approach to the Analysis and Control of Networked Evolutionary Games with Bankruptcy Mechanism

This paper investigates the evolutionary dynamic and control problem for a kind of networked evolutionary games with bankruptcy mechanism by using semi‐tensor product of matrices, and presents a number of new results. First, this kind of games are expressed as logical dynamic networks and converted into their algebraic forms, based on which, the evolutionary dynamics of the given games can be discussed. Second, the control problem is considered, and a control sequence is designed to guarantee that none of players goes bankrupt as the control target requires. Finally, an illustrative example is given to show the effectiveness of our main results.     

Dynamics and stability of potential hyper-networked evolutionary games

This paper considers the modeling and convergence of hyper-networked evolutionary games (HNEGs). In an HNEG the network graph is a hypergraph, which allows the fundamental network game to be a multi-player one. Using semi-tensor product of matrices and the fundamental evolutionary equation, the dynamics of an HNEG is obtained and we extend the results about the networked evolutionary games to show whether an HNEG is potential and how to calculate the potential. Then we propose a new strategy updating rule, called the cascading myopic best response adjustment rule (MBRAR), and prove that under the cascading MBRAR the strategies of an HNEG will converge to a pure Nash equilibrium. An example is presented and discussed in detail to demonstrate the theoretical and numerical results.     

基于矩阵方法的Banzhaf值的计算及应用

在合作博弈中,Banzhaf值提供了每个参与者形成大联盟的预期边际贡献,因此Banzhaf值的求解是一个重要的研究内容.本文首先回顾合作博弈及Banzhaf值的定义,并且运用矩阵半张量积,给出合作博弈特征函数的代数表示.然后给出了Banzhaf值的等价的代数形式和简捷的计算方法.最后将所得结果应用于生物网络中,用Banzhaf值度量遗传疾病基因相关性的可能性,确定与遗传疾病发病高度相关的基因.     

Two kinds of optimal controls for probabilistic mix-valued logical dynamic networks

This paper addresses two kinds of optimal control problems of probabilistic mix-valued logical control networks by using the semi-tensor product of matrices, and presents a number of new results on the optimal finite-horizon control and the first-passage model based control problems, respectively. Firstly, the probabilistic mix-valued logical control network is expressed in an algebraic form by the semi-tensor product method, based on which the optimal finite-horizon control problem is studied and a new algorithm for choosing a sequence of control actions is established to minimize a given cost functional over finite steps. Secondly, the first-passage model of probabilistic mix-valued logical networks is given and a new algorithm for designing the optimal control scheme is proposed to maximize the corresponding probability criterion. FinMly, an illustrative example is studied to support our new results/algorithms.     

Full-parameter vision navigation based on scene matching for aircrafts

Vision navigation based on scene matching between real-time images and a reference image has many advantages over the commonly used inertial navigation system （INS）, such as no cumulative measurement errors for long-endurance flight. Recent developments in vision navigation are mainly used for partial navigation parameters measurements, such as the position and the relative velocity, which cannot meet the requirements of completely autonomous navigation. We present the concept, principle and method of full-parameter vision navigation （FPVN） based on scene matching. The proposed method can obtain the three-dimensional （3D） position and attitude angles of an aircraft by the scene matching for multiple feature points instead of a single point in existing vision navigations. Thus, FPVN can achieve the geodetic position coordinates and attitude angles of the aircraft and then the velocity vector, attitude angular velocity and acceleration can be derived by the time differentials, which provide a full set of navigation parameters for aircrafts and unmanned aerial vehicles （UAVs）. The method can be combined with the INS and the cumulative errors of the INS can be corrected using the measurements of FPVN rather than satellite navigation systems. The approach provides a completely autonomous and accurate navigation method for long-endurance flight without the help of satellites.     

演化博弈的鲁棒稳定与镇定

本文研究了干扰影响下演化博弈的稳定与镇定问题. 首先, 文章给出了干扰博弈、控制–干扰博弈以及鲁棒 Nash均衡等概念, 并在此基础上提出了干扰演化博弈与控制–干扰演化博弈鲁棒稳定与镇定的定义. 其次, 利用矩 阵半张量积工具, 得到了干扰演化博弈与控制–干扰演化博弈的代数状态空间表示, 将鲁棒稳定与镇定问题转化为 一个辅助系统的集合稳定与集合镇定问题. 紧接着, 文章建立了干扰演化博弈与控制–干扰演化博弈鲁棒稳定与镇 定的充分必要条件, 并进一步设计了状态反馈控制器. 最后, 通过两个例子验证了所得结论的有效性.     

State feedback control design to avoid players going bankrupt

This paper investigates the state feedback control design problem to avoid players going bankrupt for a class of networked evolutionary games. First, an algebraic expression is formulated for the given networked evolutionary games by using the semi‐tensor product of matrices. Then, based on the algebraic form, a necessary and sufficient condition for the existence of state feedback control, which can guarantee that no players will go bankrupt, is presented. Finally, an illustrative example demonstrates the effectiveness of the obtained results.     

Simultaneous stabilization for a set of Boolean control networks

This paper investigates simultaneous stabilization of a collection of Boolean control networks (BCNs) by using the semi-tensor product method, and presents a number of new results. First, an algebraic expression of the BCNs is obtained by the semi-tensor product, based on which some necessary and sufficient conditions are presented to solve the simultaneous stabilization problem by a free control sequence, a state-feedback control, and an output-feedback control, respectively. Second, using the column stacking form of matrices, a new procedure is established to design both state-feedback and output-feedback controllers for the simultaneous stabilization problem. The study of two illustrative examples shows that the new results obtained in this paper are very effective in solving simultaneous stabilization of a collection of BCNs.     

局部信息约束下网络演化博弈的动力学与优化

网络演化博弈的优化问题是混合值逻辑网络的一个自然推广.本文研究了一类网络演化博弈的优化控制问题,其中每个控制个体在极大化自己的收益时只能获取到邻域信息.首先,利用矩阵的半张量积,将局部信息约束下控制网络演化博弈的动力学转化为相应的代数形式.然后得到了局部信息约束下确定型网络演化博弈的最优控制序列.最后,基于动态规划的解,研究了局部信息约束下概率型网络演化博弈的优化控制问题,得到了最优控制序列的简单计算公式.两个数值例子验证了本文的理论结果.     

Optimal control problem of singular Boolean control networks

In this paper, the optimal control problem of singular Boolean control networks is considered via semi-tensor product. Using an analogous needle variation, for multi-input case, a necessary condition for the existence of optimal control is provided based on the algebraic form of singular Boolean control networks, and the result is specialized to the single-input case. Then, an algorithm is presented to calculate an optimal control. Illustrative examples, including the single-input case, are given to show the feasibility of the theoretical results.     

Output controllability and optimal output control of state-dependent switched Boolean control networks

In the present paper, we investigate the output-controllability and optimal output control problems of a state-dependent switched Boolean control network. By using the semi-tensor product, the algebraic form of the system is obtained. Then, output-controllability problems of the system are discussed and some necessary and sufficient conditions are given. Next, the Mayer-type optimal output control issue is considered and an algorithm is provided to find out the control sequence. At last, an example is given to show the effectiveness of the main results.