Distributed best response dynamics for Nash equilibrium seeking in potential games

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 result and illustrate the effectiveness of the algorithm.     

Nash equilibrium in differential games and the quasi-strategy formalism

We consider a two-player nonzero-sum differential game in the case where players use nonanticipative strategies. We define the Nash equilibrium in this case and obtain a characterization of Nash equilibrium strategies. We show that a Nash equilibrium solution can be approximately realized by control-with-guide strategies.     

不确定性下非合作博弈强Nash均衡的存在性

在已知不确定参数变化范围的假设下,研究了非合作博弈强Nash均衡的存在性问题.基于经典非合作博弈的强Berge均衡及帕雷托均衡的概念,结合非合作博弈NS均衡,定义了不确定性下非合作博弈的帕雷托强Berge和强Nash均衡的概念,并借助Ky Fan不等式证明其存在性.最后利用算例验证了其可行性和有效性.     

基于SQP局部搜索的蝙蝠优化算法

针对基本蝙蝠算法存在寻优精度不高,后期收敛速度较慢和易陷入局部最优等问题,提出一种基于序贯二次规划（Sequential Quadratic Programming,SQP）的蝙蝠优化算法。该算法应用佳点集理论构造初始种群,增强了初始种群的遍历性;为避免算法陷入早熟收敛,引入柯西变异算子对种群中精英个体进行变异操作,增加种群多样性;在迭代后期,对最优个体进行SQP局部搜索,提高蝙蝠算法的局部深度搜索能力,保证个体在靠近全局最优值时能够寻优到全局最优解,加快种群进化速度。通过仿真实验结果证明,改进后的蝙蝠算法性能优越,具有良好的寻优精度和收敛速度。     

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.     

非光滑聚合博弈纳什均衡的分布式连续时间算法

本文研究多智能体聚合博弈的分布式算法设计.其中,个体的成本函数具有非光滑性.提出一个连续时间分布式算法,使得每个个体仅利用本地数据及局部的信息交互就能达到纳什均衡.利用李雅普诺夫方法,证明了算法的收敛性.在此基础上,进一步研究了带有耦合不等式约束博弈的广义纳什均衡求解.仿真结果验证了方法的有效性.     

Distributed event-triggered generalized Nash equilibrium seeking in multi-coalition noncooperative games with coupling constraints

This paper is concerned with a multi-coalition noncooperative game with coupling equality constraints. Each coalition is a player consisted of multiple agents in noncooperative games and desire to minimize its own objective function based on local information. Each agent as actual decision maker in the same coalition is to optimize the objective function of the coalition cooperately. To seek a generalized Nash equilibrium (GNE) of the multi-coalition game, a distributed continuous-time algorithm is developed. Moreover, to further reduce the communication among agents and coalitions, an event-triggered mechanism (ETM) is introduced for the multi-coalition game. By using ETM, a novel distributed GNE seeking algorithm is proposed, where agents and coalitions are allowed to exchange estimation information with neighbors only when the triggering condition is satisfied. Remarkably, the proposed event-triggered scheme introduces internal variables to regulate its threshold dynamically, which excludes Zeno behavior. By Lyapunov analysis, it is proved that the coalitions' decision variables converge to a GNE in both algorithms. Finally, the effectiveness of the proposed methods is validated by numerical simulations.     

共演化博弈中的反馈机制

本文研究共演化动力学在合作演化中的作用.系统的状态由个体的策略决定.系统状态与个体的属性共同演化,形成一种反馈机制.特别是当个体能够根据博弈的结果调整社会关系时,这种调整势必影响未来的博弈.这种反馈机制在适当的时间尺度下,总是能够促进合作.首先,分析了个体策略与群组属性共演化的情形,在基于个体选择的层面上,结合溯祖理论和演化集合论,给出了区域性利他行为涌现的条件.其次,给出了结构群体中策略选择的参数判据,将两策略判据和适应动力学结合起来,发现通过调整与收益矩阵无关的参数可以实现性能控制.最后,研究了不同的角色分配方案在最后通牒博弈中对公平行为演化的影响,发现当个体的先行者优势或所配置的资源依赖于先前分配结果时,公平的分配方案及对公平的要求得以建立.共演化这种反馈机制在用博弈论解决编队控制、资源配置方案设计等实际问题时显示出广泛的应用前景.     

群体区域搜索算法

借鉴自然界群居生物的搜索行为模式,提出一种群体区域搜索算法。该算法在优化过程中逐步收缩个体搜索半径并进行适度随机调整,引入巡游追随机制,以一种简单而自然的方式有效地实现了算法广域探索能力与局部开发能力之间的平衡。算法结构简单、易实现,易与其他算法相结合。通过6个典型测试函数的实验结果表明,该算法全局优化能力强、收敛精度高、稳定性好、总体性能优,适用于复杂函数优化问题的处理。     

Hybrid arithmetic optimization algorithm with hunger games search for global optimization

Multimedia Tools and Applications - Recently, many population-dependent methods have been proposed. Despite their acceptance in many applications, we are still exploring suggested methods to solve...     

Discovering theorems in game theory: Two-person games with unique pure Nash equilibrium payoffs

In this paper we provide a logical framework for two-person finite games in strategic form, and use it to design a computer program for discovering some classes of games that have unique pure Nash equilibrium payoffs. The classes of games that we consider are those that can be expressed by a conjunction of two binary clauses, and our program re-discovered Kats and Thisse?s class of weakly unilaterally competitive two-person games, and came up with several other classes of games that have unique pure Nash equilibrium payoffs. It also came up with new classes of strict games that have unique pure Nash equilibria, where a game is strict if for both player different profiles have different payoffs.     

Solving the balance problem of massively multiplayer online role-playing games using coevolutionary programming

In massively multiplayer online role-playing games (MMORPGs), each race holds some attributes and skills. Each skill contains several abilities such as physical damage and hit rate. All those attributes and abilities are functions of the character's level, which are called Ability-Increasing Functions (AIFs). A well-balanced MMORPG is characterized by having a set of well-balanced AIFs. In this paper, we propose a coevolutionary design method, including integration with the modified probabilistic incremental program evolution (PIPE) and the cooperative coevolutionary algorithm (CCEA), to solve the balance problem of MMORPGs. Moreover, we construct a simplest turn-based game model and perform a series of experiments based on it. The results indicate that the proposed method is able to obtain a set of well-balanced AIFs more efficiently, compared with the simple genetic algorithm (SGA), the simulated annealing algorithm (SAA) and the hybrid discrete particle swarm optimization (HDPSO) algorithm. The results also show that the performance of PIPE has been significantly improved through the modification works.     

A Nash equilibrium solution in an oligopoly market: The search for Nash equilibrium solutions with replicator equations derived from the gradient dynamics of a simplex algorithm

The present analysis applies continuous time replicator dynamics to the analysis of oligopoly markets. In the present paper, we discuss continuous game problems in which decision-making variables for each player are bounded on a simplex by equalities and non-negative constraints. Several types of problems are considered under conditions of normalized constraints and non-negative constraints. These problems can be classified into two types based on their constraints. For one type, the simplex constraint applies to the variables for each player independently, such as in a product allocation problem. For the other type, the simplex constraint applies to interference among all players, creating a market share problem. In the present paper, we consider a game problem under the constraints of allocation of product and market share simultaneously. We assume that a Nash equilibrium solution can be applied and derive the gradient system dynamics that attain the Nash equilibrium solution without violating the simplex constraints. Models assume that three or more firms exist in a market. Firms behave to maximize their profits, as defined by the difference between their sales and cost functions with conjectural variations. The effectiveness of the derived dynamics is demonstrated using simple data. The present approach facilitates understanding the process of attaining equilibrium in an oligopoly market.     

基于事件驱动控制的混杂动态博弈系统的纳什均衡分析

本文研究了基于事件驱动控制的混杂动态博弈系统的纳什均衡分析问题. 首先, 分析了事件驱动机制对混 杂动态博弈过程的影响, 进而, 在进行状态空间描述的基础上, 给出了混杂动态博弈的纳什均衡的定义, 并建立了对 应博弈系统的策略型模型. 其次, 结合Lanchester方程, 分别讨论了两类混杂动态博弈系统的均衡问题, 包括事件驱 动策略设计和固定的情况, 获得了均衡解存在的必要条件. 最后, 通过数值模拟进行了应用分析, 验证了所取得结果 的合理性和科学性, 并总结了混杂动态博弈研究的未来工作.     

Hybrid genetic algorithm with adaptive local search scheme

This paper proposes a hybrid genetic algorithm (a-hGA) with adaptive local search scheme. For designing the a-hGA, a local search technique is incorporated in the loop of genetic algorithm (GA), and whether or not the local search technique is used in the GA is automatically determined by the adaptive local search scheme. Two modes of adaptive local search schemes are developed in this paper. First mode is to use the conditional local search method that can measure the average fitness values obtained from the continuous two generations of the a-hGA, while second one is to apply the similarity coefficient method that can measure a similarity among the individuals of the population of the a-hGA. These two adaptive local search schemes are included in the a-hGA loop, respectively. Therefore, the a-hGA can be divided into two types: a-hGA1 and a-hGA2. To prove the efficiency of the a-hGA1 and a-hGA2, a canonical GA (cGA) and a hybrid GA (hGA) with local search technique and without any adaptive local search scheme are also presented. In numerical example, all the algorithms (cGA, hGA, a-hGA1 and a-hGA2) are tested and analyzed. Finally, the efficiency of the proposed a-hGA1 and a-hGA2 is proved by various measures of performance.     

Uniqueness and efficiency of Nash equilibrium in a family of randomly generated repeated games

This paper describes the results of an analysis of the Nash equilibrium in randomly generated repeated games. We study two families of games: symmetric bimatrix games G(A, B) with B = A^⊤ and nonsymmetric bimatrix games (the first includes the classical games of prisoner dilemma, battle of the sexes, and chickens). We use pure strategies, implemented by automata of size two, and different strategy domination criteria. We observe that, in this environment, the uniqueness and efficiency of equilibria outcomes is the typical result.     

N人合作博弈的Nash及演化均衡稳定策略分析

在有限理性的基础上,对N人合作博弈的对称Nash均衡进行了分析,并引入演化博弈理论分析了参与人的演化均衡稳定策略,得到了不同策略选择下的均衡点。进而应用生物复制动态理论对离散时间及连续时间下的复制动态稳定集进行了研究。最后通过实例说明了该方法在博弈均衡选择上的有效性。     

A coevolutionary differential evolution with harmony search for reliability-redundancy optimization

Solving reliability-redundancy optimization problems via meta-heuristic algorithms has attracted increasing attention in recent years. In this paper, an effective coevolutionary differential evolution with harmony search algorithm (CDEHS) is proposed to solve the reliability-redundancy optimization problem by dividing the problem into a continuous part and an integer part. In CDEHS, two populations evolve simultaneously and cooperatively, where one population for the continuous part evolves by means of differential evolution while another population for the integer part evolves by means of harmony search. After half of the whole evolving process, the integer part stops evolving and provides the best solution to the other part for further evolving with differential evolution. Simulations results based on three typical problems and comparisons with some existing algorithms show that the proposed CDEHS is effective, efficient and robust for solving the reliability-redundancy optimization problem.     

Comparison of semantic-based local search methods for multiobjective genetic programming

We report a series of experiments that use semantic-based local search within a multiobjective genetic programming (GP) framework. We compare various ways of selecting target subtrees for local search as well as different methods for performing that search; we have also made comparison with the random desired operator of Pawlak et al. using statistical hypothesis testing. We find that a standard steady state or generational GP followed by a carefully-designed single-objective GP implementing semantic-based local search produces models that are mode accurate and with statistically smaller (or equal) tree size than those generated by the corresponding baseline GP algorithms. The depth fair selection strategy of Ito et al. is found to perform best compared with other subtree selection methods in the model refinement.