On modifications of Puu's dynamical duopoly

Puu's economic dynamical model is extended to the following cases: Bounded rationality, differentiated goods and bounded rationality with delay. It is shown that delay increases stability. Hence firms using bounded rationality with delay has a higher chance of reaching Nash equilibrium. Stability and instability (cycles and chaos) conditions for all these cases are determined.     

Study on a 3-dimensional game model with delayed bounded rationality

A nonlinear dynamic triopoly game model is studied based on the theory of nonlinear dynamics and previous researches in this paper. A lagged structure is introduced to the model to study stability conditions of the Nash equilibrium under a local adjustment process when players price their products with delayed bounded rationality. Numerical simulations are provided to demonstrate the complexity of system evolvement and influence of the strategy of delayed bounded rationality on system stability. We find that besides the lagged structure, suitable delayed parameters are also important factors to eliminate chaos or expand the stable region of the system, and various players’ adjustment parameters have different effect on stability of the system.     

有限理性条件下演化博弈行为分析

基于博弈双方有限理性的假设,运用动力系统的相关理论和方法对一般2×2非对称演化博弈过程建立了动态复制方程,并对博弈双方在演化过程中的行为进行了分析,得出博弈双方交互系统均衡点及稳定性相应的结论及其全部动力学行为.     

关于非凸的有限理性的稳定性

关于有限理性方面的文献, 大多数都是在满足凸性条件下研究有限理性的相关性质, 在一定程度上限制了其应用范围. 应用Ekeland变分原理, 减弱了有限理性模型的假设条件, 考虑在不满足凸性条件下的有限理性模型的稳定性问题. 具体给出了非凸的Ky Fan点问题解的稳定性, 非凸非紧的Ky Fan点问题解的稳定性, 非凸向量值函数Ky Fan点解的稳定性和非凸非紧向量值函数Ky Fan点解的稳定性. 作为应用, 还给出了非凸的n人非合作博弈有限理性模型解的稳定性和非凸的多目标博弈有限理性模型解的稳定性.     

Selecting optimal selling format of a product in B2C online auctions with boundedly rational customers

The advancement of Internet technology has enabled new formats for selling products in the B2C online auctions. At present, on the major online auction sites, there exist three popular selling formats, namely, the posted price, pure auction and buy-price auction formats. It is an important decision problem for a firm to select the most profitable format to sell its products through the Internet. The customer behavior is of course a crucial element of the decision process. To the best of our knowledge, most models available today assume that customers are perfectly rational. To better understand the decision process, in this paper, we incorporate the concept of bounded rationality into consideration. We first present a “behavior choice function” to characterize the behavior of the customers with bounded rationality. Then corresponding to each selling format, we construct a revenue model based on the bounded rationality for analysis. Finally, we conduct some elaborate computational experiments to investigate the performance of each revenue model for developing new managerial insights. Our computational results clearly demonstrate how the bounded rationality of customer behavior affects the choice of a preferable selling format for a B2C firm in an online auction.     

Speculative dynamics with bounded rationality learning

We study a linear model for a future market characterized by the presence of different classes of traders. In the market there are three classes of traders: rational traders, feedback traders and fundamentalist traders. Each class of traders is described by a trading strategy and by an information set about the fundamental. The analysis is developed under bounded rationality, rational traders forming expectations do not know the “true” model but believe in a misspecified model. The convergence of the learning activity to the Rational Expectations Equilibria of the model is analyzed. Two different learning mechanisms are studied: the Ordinary Least Squares algorithm and the Least Mean Squares algorithm. The main goal of the study is to analyze how the presence of different classes of traders in the market affects the robustness of the Rational Expectations Equilibria of the model with respect to bounded rationality learning. Moreover we verify the claim that bubbles and erratic behavior in the stock price dynamics may arise because of learning non-convergence to Rational Expectations Equilibria. The results show that if the Ordinary Least Squares algorithm is used by the agents to update beliefs, convergence to one of the two Rational Expectations Equilibria of the model is ensured only if there are positive feedback traders in the market. On the contrary, the Least Mean Squares algorithm guarantees convergence to the Rational Expectations Equilibria given an appropriate initial belief.     

Structural stability and robustness to bounded rationality for non-compact cases

We study the model M consisting of “general games” with noncompact action space, together with an associated abstract rationality function. We prove that M is structurally stable and robust to ϵ-equilibria for “almost all” parameters. As applications, we investigate structural stability and robustness to bounded rationality for noncooperative games, multiobjective optimizations and fixed point problems satisfying existence and some continuity conditions. Specifically, we introduce concrete rationality functions for such three kinds of problems with both payoffs and strategy sets, objective functions and domain spaces, and correspondence and domain spaces as parameters, respectively, and show the generic structural stability and robustness to bounded rationality for the corresponding model Ms.     

On stability of perfect equilibrium points

We consider stability of Selten's perfect equilibrium point against slight imperfections of rationality of players. As its stability is not sufficient, we strengthen the perfectness concept and define astrictly perfect equilibrium point. We provide sufficient conditions for this equilibrium point.     

Analysis of the dynamics of Cournot team-game with heterogeneous players

In this paper, we study a dynamical system of a two-team Cournot game played by a team consisting of two firms with bounded rationality and a team consisting of one firm with naive expectation. The equilibrium solutions and the conditions of their locally asymptotic stability are studied. It is demonstrated that, as some parameters in the model are varied, the stability of the equilibrium will get lost and many such complex behaviors as the period bifurcation, chaotic phenomenon, periodic windows, strange attractor and unpredictable trajectories will occur. The great influence of the model parameters on the speed of convergence to the equilibrium is also shown with numerical analysis.     

Global asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays

In the Lotka–Volterra competition system with N-competing species if the effect of dispersion and time-delays are both taken into consideration, then the densities of the competing species are governed by a coupled system of reaction–diffusion equations with time-delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to the competing rate constants to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the competing system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system.     

分享经济环境下制造业产能分享的三群体演化博弈分析

制造业产能分享的兴起,改变了传统的制造模式,打破了原有制造业系统的平衡。为研究制造业产能分享参与群体的决策行为,在产能分享平台具有网络外部性、制造企业具有接入成本且博弈群体均具有有限理性的假设条件下,构建了“产能分享平台-制造业产能所有企业-制造业产能需求企业”三个博弈群体的演化博弈模型。运用演化博弈理论分析了模型的演化稳定策略,探讨了制造企业接入成本、产能分享平台的网络外部性系数等对此动态系统稳定性的影响,给出了演化稳定策略的经济和管理意义。最后,用数值仿真对比分析不同参数变化对演化结果的影响,为产能分享平台和制造企业的行为决策提供理论参考依据。     

Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale

A repeated, discrete time, heterogeneous Cournot duopoly game with bounded rational and adaptive players adjusting the quantities of production is subject of investigation. Linear inverse demand function and quadratic cost functions reflecting decreasing returns to scale are assumed. The game is modeled with a system of two difference equations. Evolution of outputs over time is obtained by iteration of a two dimensional nonlinear map. Existing equilibria and their stability are analyzed. In face of diseconomies of scale, bounded rational and adaptive duopolists are shown to experience a decrease in the latitude of their output adjustment decisions with respect to the market stability compared to constant returns to scale and ceteris paribus. Chaotic dynamics is confirmed to depend mainly on the adjustment behavior of the bounded rational player, who if overshoots leaves the adaptive player with limited opportunities to stabilize the market again, hence industries facing diseconomies of scale are found to be less stable than those with constant marginal costs. Complexity of the dynamical system is examined by means of numerical simulations, where the paper extends the results of other authors who considered analogous games assuming linear cost functions. Intermittent transition to chaos and attractor merging crisis are shown among others.     

The Asymptotic Behavior and the Quasineutral Limit forthe Bipolar Euler-Poisson System withBoundary Effects and a Vacuum

In this paper, a one-dimensional bipolar Euler-Poisson system(a hydrodynamic model) from semiconductors or plasmas with boundary efects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy’s law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary efects and a vacuum.     

The condition on the stability of stationary lines in a curvature flow in the whole plane

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t−1/2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.     

Auswertungsverfahren für Polynome in mehreren Variablen

A general class of evaluation schemes for polynomials in one or several variables is discussed. By the same concept, error bounds are obtained for various methods, for instance Horner's scheme and Clenshaw's method, which are strict in some cases with a loss of a factor logn at best. For multivariable polynomials, a new family of evaluation schemes is suggested which generalizes a modification of Clenshaw's method and is therefore expected to have a favorable stability behavior with respect to round-off.     

基于多期有限理性的可再生资源寡头博弈模型与分析

本文讨论多期有限理性条件下可再生资源开发的寡头博弈模型。在每个经济时期以实现多个相邻时期的总贴现利润最大化为目的而得到双寡头决策者资源开发的最优策略组合,并以各期最优策略组合构建资源开发决策的动力系统。通过对系统的动力学分析,讨论了系统中各参数对系统的稳定性与稳定状态的作用与影响,并给出了相应的经济解释。分析了反馈控制系统的稳定性。最后通过对系统的数值模拟,分析了各参数的取值变化与系统稳定速度之间的关系。本文通过系统的稳定性分析获得的主要结论是,资源市场竞争的稳定与可持续发展与决策者注重后期经济利益的决策理性密切相关。     

Nonlinear dynamics and stability analysis of piezo-visco medium nanoshell resonator with electrostatic and harmonic actuation

In this paper, nonlinear dynamics, vibration and stability analysis of piezo-visco medium nanoshell resonator (PVM-NSR) based on functionally graded (FG) cylindrical nanoshell integrated with two piezoelectric layers subjected to visco-pasternak medium, electrostatic and harmonic excitations is investigated. Nonclassical method of the electro-elastic Gurtin–Murdoch surface/interface theory with von-Karman–Donnell's shell model as well as Hamilton's principle, the assumed mode method combined with Lagrange–Euler's are considered. Complex averaging method combined with arc-length continuation is used to achieve a numerical solution for the steady state vibrations of the system. The stability analysis of the steady state response is performed. The parametric studies such as the effects of different boundary conditions, different geometric ratios, structural parameters, electrostatic and harmonic excitation on the nonlinear frequency response and stability analysis are studied. The results indicate that near the natural frequency of the nanoshell, it will lead to resonance and will have large motion amplitude and near the resonant frequency, the nanoshell shows a softening type of nonlinear behavior, and the nanoshell bandwidth increases due to nonlinear factors. In this range, nanoshell has three different ranges of motion, of which two are stable and the other unstable, and so the jump phenomenon and saddle-node bifurcation are visible in the behavior of the system. Also piezoelectric voltage influences on static deformation and resonant frequency but has no significant effect on nonlinear behavior and bandwidth and also system very sensitive to the damping coefficient and due to decrease of nano shell stiffness, natural frequency decreases. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system's instability, nonlinear behavior and bandwidth.     

Solvability of linear equations in the Besicovitch and Bohr classes of almost periodic functions

An example is given of a finite-dimensional equation u′+A(t)u=f(t), where A(t) andf(t) are Bohr almost periodic elements, having bounded solutions but not almost periodic solutions (the question of a similar example was already posed and discussed in Favard's original papers). On the other hand, solvability in the Besicovitch class does not require subtle separability or stability conditions. General theorems of such a kind are provided in this note.     

A robust approach for modeling limited observability in bilevel optimization

Many applications of bilevel optimization contain a leader facing a follower whose reaction deviates from the one expected by the leader due to some kind of bounded rationality. We consider bilinear bilevel problems with follower's response uncertainty due to limited observability regarding the leader's decision and exploit robust optimization to model the decision making of the follower. We show that the robust counterpart of the lower level allows to tackle the problem via the lower level's KKT conditions.     

Switching Stability of Automotive Roll Dynamics Subject to Interval Uncertainty

In this paper we apply some recent results on the stability of switched systems with interval uncertainty for analyzing the roll stability of automotive vehicles subject to bounded parametric uncertainties and switching. The application is motivated by the fact that the roll dynamics of an automotive vehicle is affected in a significant and a nonlinear fashion due to possible sudden switches in the vehicle's center of gravity (CG) height, as well as the uncertainty in the suspension parameters. Utilizing a recent stability result, it is possible to model such parametric variations as bounded interval uncertainties for this problem, and obtain easily verifiable conditions for analyzing the stability of the resulting switched/uncertain system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)