Bi-matrix Games with Fuzzy Goals and Fuzzy

A bi-matrix game with fuzzy goal is shown to be equivalent to a (crisp) non-linear programming problem in which the objective as well as all constraint functions are linear except two constraint functions, which are quadratic. This equivalence is further extended to bi-matrix games with fuzzy pay-offs, as well as to bi-matrix games with fuzzy goals and fuzzy payoffs, whose equilibrium strategies are conceptualized by employing a suitable ranking (defuzzification) function.     

基于模糊数的政府与绿色制造商博弈分析

近年来环境问题已经成为人们迫切需要解决的重要问题,促使制造商采取绿色制造模式也是各国政府正在面临的一大难题。因此文章结合三角模糊数与博弈理论建立了政府与绿色制造商的模糊博弈模型,将博弈结果分为纯策略和混合策略两类,并讨论了各种情形下政府与绿色制造商的不同策略以及影响因素,分析了最优博弈结果以及相应的管理策略,为绿色制造模式的顺利实施提供了相关建议。研究结果表明,政府的补贴和惩罚等策略在绿色制造模式的采取方面起着举足轻重的作用。最后利用三角结构元法对算例分纯策略和混合策略两种情况进行分析求解,从而验证了结论的正确性和可行性。     

A Strong Anti-Folk Theorem

We study the properties of finitely complex, symmetric, globally stable, and semi-perfect equilibria. We show that: (1) If a strategy satisfies these properties then players play a Nash equilibrium of the stage game in every period; (2) The set of finitely complex, symmetric, globally stable, semi-perfect equilibrium payoffs in the repeated game equals the set of Nash equilibria payoffs in the stage game; and (3) A strategy vector satisfies these properties in a Pareto optimal way if and only if players play some Pareto optimal Nash equilibrium of the stage game in every stage. Our second main result is a strong anti-Folk Theorem, since, in contrast to what is described by the Folk Theorem, the set of equilibrium payoffs does not expand when the game is repeated.This paper is a revised version of Chapter 3 of my Ph.D. thesis, which has circulated under the title “An Interpretation of Nash Equilibrium Based on the Notion of Social Institutions”.     

A procedure for finding Nash equilibria in bi-matrix games

In this paper we consider the computation of Nash equilibria for noncooperative bi-matrix games. The standard method for finding a Nash equilibrium in such a game is the Lemke-Howson method. That method operates by solving a related linear complementarity problem (LCP). However, the method may fail to reach certain equilibria because it can only start from a limited number of strategy vectors. The method we propose here finds an equilibrium by solving a related stationary point problem (SPP). Contrary to the Lemke-Howson method it can start from almost any strategy vector. Besides, the path of vectors along which the equilibrium is reached has an appealing game-theoretic interpretation. An important feature of the algorithm is that it finds a perfect equilibrium when at the start all actions are played with positive probability. Furthermore, we can in principle find all Nash equilibria by repeated application of the algorithm starting from different strategy vectors.This author is financially supported by the Co-operation Centre Tilburg and Eindhoven Universities, The Netherlands.     

求收益模糊的贝叶斯纳什均衡解的结构元方法

主要目的是利用结构元方法求解收益模糊的贝叶斯纳什均衡.首先,在原有结构元理论基础上,给出了多元模糊值函数的定义及其结构元表示;其次,给出了在混合策略下,收益模糊的贝叶斯纳什均衡的定义,并证明了其存在性定理;然后,利用结构元理论,将该博弈模型等价地转化为一个经典的博弈模型,简化了原问题的求解.最后的应用实例说明了该方法的有效性.     

模糊值二人零和对策的Nash均衡策略

我们将在Ramik定义的模糊最大序关系基础上研究模糊环境中的二人零和对策。在非对称模糊数基础上,引入模糊环境中的几种Nash均衡策略,讨论各种均衡策略存在的充要条件。并引入含参变量确定性矩阵对策及其均衡策略的概念,讨论含参变量确定性矩阵对策的Nash均衡策略和模糊值矩阵对策的均衡策略的关系。     

指派问题的纳什均衡解

为弥补传统指派问题解不符合个体理性的不足,提出指派问题的纳什均衡解,并证明有限指派问题有且仅有纯纳什均衡解。相比传统的指派问题解,纯纳什均衡符合Pareto最优,是个体理性视角下的最优解。在此基础上,给出一个综合考虑个体理性与集体理性的求解方法。     

基于训练和过滤的期望均衡及其实现路径

期望均衡是博弈局中人或局外人对于博弈均衡点的一致期望,强调互利共赢,它要求博弈群体的每个成员对期望均衡点有一个共同的预期.显然,基于纳什均衡的帕累托优化组合策略是一个比纳什均衡更有效的期望均衡.要实现期望均衡,可采用局中人参与的训练与学习使得群体的预期一致,也可采用第三方过滤器来达到期望目标.在期望均衡的概率分布下,个体行为的偏离不能比均衡态取得更多收益.否则,训练就是无效的,第三方过滤器就是不公平的.     

相位阻尼信道条件下量子斗鸡博弈模型均衡解分析

利用量子博弈的相关理论,以噪音强度和记忆强度为参量,建立了相位阻尼信道条件下的量子斗鸡博弈模型,求出了模型的量子纳什均衡解,讨论了两参量对均衡解稳定性的影响,得出在无记忆相位阻尼信道条件下,当噪音强度小于阈值0.24时,纳什均衡仍然为帕累托最优解,当噪音强度大于0.24时,均衡解演变为另2个均衡解,不再是帕累托最优;在有记忆相位阻尼信道条件下,当噪音强度小于0.24,且记忆强度大于0.5时,均衡解是稳定的.特殊地,当信道是完全记忆时,均衡解的稳定性与噪音强度无关.     

具有模糊支付的Moran过程演化博弈动态

以往对演化博弈的研究都假设个体从博弈中获得的支付是确定的并以精确的数来表示。然而由于受环境中各种不确定因素的影响,个体博弈时所获得的支付并不是一个精确的数值,而需要用一个模糊数来表示。本文研究模糊支付下2×2的对称博弈, 利用模糊数的运算, 分析具有模糊支付的有限种群Moran过程演化动态。在弱选择下以梯形模糊数和三角模糊数表示博弈支付,计算策略的模糊扎根概率,分析自然选择有利于策略扎根及策略成为模糊演化稳定策略的条件。将经典博弈推广到模糊环境中丰富了演化博弈理论,更具有现实意义。     

Juche versus Sadae: Game-theoretic approach

We argue that to some degree Juche is represented by the concept of Nash equilibrium, and Sadae by Thompson and Faith's truly perfect information equilibrium. We characterize the latter, and show that for a Pareto optimal Nash equilibrium, Juche is as good as, or better than Sadae. This includes the game of brinkmanship.     

An experimental study of constant-sum centipede games

In this paper, we report the results of a series of experiments on a version of the centipede game in which the total payoff to the two players is constant. Standard backward induction arguments lead to a unique Nash equilibrium outcome prediction, which is the same as the prediction made by theories of “fair” or “focal” outcomes. We find that subjects frequently fail to select the unique Nash outcome prediction. While this behavior was also observed in McKelvey and Palfrey (1992) in the “growing pie” version of the game they studied, the Nash outcome was not “fair”, and there was the possibility of Pareto improvement by deviating from Nash play. Their findings could therefore be explained by small amounts of altruistic behavior. There are no Pareto improvements available in the constant-sum games we examine. Hence, explanations based on altruism cannot account for these new data. We examine and compare two classes of models to explain these data. The first class consists of non-equilibrium modifications of the standard “Always Take” model. The other class we investigate, the Quantal Response Equilibrium model, describes an equilibrium in which subjects make mistakes in implementing their best replies and assume other players do so as well. One specification of this model fits the experimental data best, among the models we test, and is able to account for all the main features we observe in the data.     

Finding a Nash equilibrium in noncooperativeN-person games by solving a sequence of linear stationary point problems

In this paper we present an algorithm for finding a Nash equilibrium in a noncooperative normal formN-person game. More generally, the algorithm can be applied for solving a nonlinear stationary point problem on a simplotope, being the Cartesian product of several simplices. The algorithm solves the problem by solving a sequence of linear stationary point problems. Each problem in the sequence is solved in a finite number of iterations. Although the overall convergence cannot be proved, the method performs rather well. Computational results suggest that this algorithm performs at least as good as simplicial algorithms do.For the special case of a bi-matrix game (N=2), the algorithm has an appealing game-theoretic interpretation. In that case, the problem is linear and the algorithm always finds a solution. Furthermore, the equilibrium found in a bi-matrix game is perfect whenever the algorithm starts from a strategy vector at which all actions are played with positive probability.This research is part of the VF-program Co-operation and Competition, which has been approved by the Netherlands Ministery of Education and Sciences.     

两阶段主从博弈均衡解的存在性

首先给出带参数的纳什均衡问题Γ(x),在此基础上给出了具有带参数的纳什均衡约束的两阶段主从博弈问题G.可以证明带参数的纳什均衡点是存在的,即无论领导者选择何种策略,跟随者的最佳回应集都是非空的.最后推出了关于两阶段主从博弈均衡点的存在性定理.     

Generalising diagonal strict concavity property for uniqueness of Nash equilibrium

In this paper, we extend the notion of diagonally strictly concave functions and use it to provide a sufficient condition for uniqueness of Nash equilibrium in some concave games. We then provide an alternative proof of the existence and uniqueness of Nash equilibrium for a network resource allocation game arising from the so-called Kelly mechanism by verifying the new sufficient condition. We then establish that the equilibrium resulting from the differential pricing in the Kelly mechanism is related to a normalised Nash equilibrium of a game with coupled strategy space.     

Pareto efficiency,simple game stability,and social structure in finitely repeated games

Simple game (sensu Brown and Vincent, 1987) evolutionary theory, when coupled with social structure measured as non‐random encounter of strategy “clones”, often permits equilibrium refinement leading to Pareto superior outcomes (e.g., Axelrod, 1981; Myerson et al., 1991), a foundational goal of economic game theory (Myerson, 1991: 370–375). This conclusion, derived from analyses of one‐shot and infinitely repeated games, fails for finitely repeated games. While mutant cluster invasion enhances Pareto efficiency of equilibria in the former, it can depress Pareto efficiency in the latter. Cooperative equilibria of finitely repeated games (under economic analysis) can be susceptible to cluster‐invasion by even more Pareto efficient strategies which are not themselves evolutionarily stable. Evolutionary (simple) game theory's ability to eliminate Pareto inferior Nash equilibrium strategies induces vulnerabilities foreign to economic analysis. Simple game analysis of finitely repeated games suggests that social structure, modeled as perennial invasion by mutant‐clusters, can induce cyclic invasion, saturation, and loss of cooperation.     

A class of games having Pareto optimal Nash equilibria

A simple two-person exchange model is considered first as a cooperative game without side payments, then as a non-cooperative game. It is shown that the Nash equilibrium solution of the latter agrees with the von Neumann-Morgenstern and core solutions of the former. This is made possible by the Pareto optimality of the Nash equilibrium, which is the main theoretical result presented.     

The Hotelling bi-matrix game

We study the pure equilibrium set for a specific symmetric finite game in strategic form, referred to as the Hotelling bi-matrix game. General results that guarantee non-emptiness of this set (for all parametric values) do not seem to exist. We prove non-emptiness by determining the pure equilibrium set. In this proof so-called demi-modality properties of the conditional payoff functions play an important role.     

Nondominated equilibrium solutions of a multiobjective two-person nonzero-sum game in extensive form and corresponding mathematical programming problem

In most of studies on multiobjective noncooperative games, games are represented in normal form and a solution concept of Pareto equilibrium solutions which is an extension of Nash equilibrium solutions has been focused on. However, for analyzing economic situations and modeling real world applications, we often see cases where the extensive form representation of games is more appropriate than the normal form representation. In this paper, in a multiobjective two-person nonzero-sum game in extensive form, we employ the sequence form of strategy representation to define a nondominated equilibrium solution which is an extension of a Pareto equilibrium solution, and provide a necessary and sufficient condition that a pair of realization plans, which are strategies of players in sequence form, is a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. Finally, giving a numerical example, we demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.     

Platform modelling and scheduling game with multiple intelligent cloud-computing pools for big data

We develop a generic game platform that can be used to model various real-world systems with multiple intelligent cloud-computing pools and parallel-queues for resources-competing users. Inside the platform, the software structure is modelled as Blockchain. All the users are associated with Big Data arrival streams whose random dynamics is modelled by triply stochastic renewal reward processes (TSRRPs). Each user may be served simultaneously by multiple pools while each pool with parallel-servers may also serve multi-users at the same time via smart policies in the Blockchain, e.g. a Nash equilibrium point myopically at each fixed time to a game-theoretic scheduling problem. To illustrate the effectiveness of our game platform, we model the performance measures of its internal data flow dynamics (queue length and workload processes) as reflecting diffusion with regime-switchings (RDRSs) under our scheduling policies. By RDRS models, we can prove our myopic game-theoretic policy to be an asymptotic Pareto minimal-dual-cost Nash equilibrium one globally over the whole time horizon to a randomly evolving dynamic game problem. Iterative schemes for simulating our multi-dimensional RDRS models are also developed with the support of numerical comparisons.