Robust mixed strategies in fuzzy non-cooperative Nash games

Game theory has traditionally used real-valued utility functions in decision-making problems. However, the real information available to assess these utility functions is normally uncertain, suggesting the use of uncertainty distributions for a more realistic modelling. In this sense, utilities results or pay-offs have been normally modelled with probability distributions, assuming random uncertainty. However, when statistical information is unavailable, probability may not be the most adequate paradigm, and can lead to very large execution times when some real complex problems are addressed. In this article possibility distributions are used to model the uncertainty of utility functions when the strategies are probability distributions (mixed strategies) over a set of original and discrete strategies (pure strategies). Two dual approaches to solve the resulting non-cooperative fuzzy games are proposed: modelling players’ risk aversion, and thus providing realistic conservative strategies. Two examples show the robustness of the strategies obtained with the proposed approaches.     

Aspiration dynamics of multi-player games in finite populations

On studying strategy update rules in the framework of evolutionary game theory, one can differentiate between imitation processes and aspiration-driven dynamics. In the former case, individuals imitate the strategy of a more successful peer. In the latter case, individuals adjust their strategies based on a comparison of their pay-offs from the evolutionary game to a value they aspire, called the level of aspiration. Unlike imitation processes of pairwise comparison, aspiration-driven updates do not require additional information about the strategic environment and can thus be interpreted as being more spontaneous. Recent work has mainly focused on understanding how aspiration dynamics alter the evolutionary outcome in structured populations. However, the baseline case for understanding strategy selection is the well-mixed population case, which is still lacking sufficient understanding. We explore how aspiration-driven strategy-update dynamics under imperfect rationality influence the average abundance of a strategy in multi-player evolutionary games with two strategies. We analytically derive a condition under which a strategy is more abundant than the other in the weak selection limiting case. This approach has a long-standing history in evolutionary games and is mostly applied for its mathematical approachability. Hence, we also explore strong selection numerically, which shows that our weak selection condition is a robust predictor of the average abundance of a strategy. The condition turns out to differ from that of a wide class of imitation dynamics, as long as the game is not dyadic. Therefore, a strategy favoured under imitation dynamics can be disfavoured under aspiration dynamics. This does not require any population structure, and thus highlights the intrinsic difference between imitation and aspiration dynamics.     

On the structure of the set of perfect equilibria in bimatrix games

Summary In this paper attention is focussed on the structure of the set of perfect equilibria. It turns out that the structure of this set resembles the structure of the Nash equilibrium set. Maximal Selten subsets are introduced to take the role of maximal Nash subsets. It is found that the set of perfect equilibria is the finite union of maximal Selten subsets. Furthermore it is shown that the dimension relation for maximal Nash subsets can be extended to faces of such sets. As a result a dimension relation for maximal Selten subsets is derived.     

用Normal Form直接法研究壁板的热颤振与控制

建立了二维受热壁板在超音速气流中的颤振方程。运用分岔理论求得了系统的Hopf分岔点,应用Normal Form直接法计算得到系统Hopf分岔Normal Form系数。引入wash-out滤波器技术对壁板热颤振进行了主动控制,延迟系统Hopf分岔的产生而不改变分岔类型。最后采用数值模拟验证了理论分析。     

研究两自由度强非线性振动系统的规范形方法

传统的规范形理论常用于研究弱非线性振动问题,对于非线性项不再是小量的强非线性振动系统则并不适用.为进一步拓展这一理论的适用范围,基于研究单自由度强非线振动问题的待定瞬时固有频率法,提出了可用来求解两自由度强非线性振动系统的改进规范形方法.首先引入了复数形式的一阶方程并且利用新的未知瞬态基频替换系统原有的固有频率,再依照规范形理论计算了一类两自由度强非线性Duffing-Van der Pol振子的5阶传统规范形.最后求解平均方程获得了此类系统的瞬时频率、振幅以及相应的稳态渐近解.通过对比算例中本文方法、原有规范形理论及数值仿真的结果,证明了改进的规范形理论对于多自由度强非线性振动问题的适用性.     

Equilibrium on line method (ELM) for imposition of Neumann boundary conditions in the finite point method (FPM)

Equilibrium on line method (ELM) for imposition of Neumann boundary conditions in the finite point method (FPM) is presented. In contrary to weak‐form‐based methods, strong‐form‐based methods such as the FPM are often unstable and less accurate, especially for problems governed by partial differential equations with Neumann (derivative) boundary conditions. In this paper, a truly meshless approach for imposition of Neumann boundary conditions in the FPM is proposed and adopted for 2D elasticity analyses. In the proposed method, equilibrium on lines on the Neumann boundary conditions is satisfied as Neumann boundary condition equations. Numerical studies show that this method for imposition of Neumann boundary is simple to implement and computationally efficient and also leads to more stable and accurate results. Copyright © 2006 John Wiley & Sons, Ltd.