Evolutionary games

This paper deals with a mathematical game. As the name implies, the game concept is formulated with biological evolution in mind. An evolutionary game differs from the usual game concepts in that the players cannot choose their strategies. Rather, the strategies used by the players are handed down from generation to generation. It is the survival characteristics of a strategy that determine the outcome of the evolutionary game. Players interact and receive payoffs according to the strategies they are using. These interactions, in turn, determine the fitness of players using a given strategy. The survival characteristics of strategy are determined directly from the fitness functions. Necessary conditions for determining an evolutionarily stable strategy are developed here for a continuous game. Results are illustrated with an example.Dedicated to G. LeitmannThis work was supported by NSF Grant No. INT-82-10803 and The University of Western Australia (Visiting Fellowship, Department of Mathematics, 1983).     

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.     

外卖食品安全问题的演化博弈分析

运用演化博弈论的方法对外卖商家之间,订餐平台监管部门和外卖商家之间相互作用时策略选择行为进行分析.结果表明:当外卖商家生产不合格的外卖收益大于生产合格外卖的收益、订餐平台监督部门对生产不合格外卖的商家处罚力度过轻或对外卖合格性的检测成本过高时,不合格外卖生产必然发生.要确保外卖商家生产合格外卖,就必须对生产不合格外卖的商家进行严惩,降低生产不合格外卖的预期收益.同时,要加强订餐平台建设,降低监管成本,保证监管部门严格执法.     

A dynamical characterization of evolutionarily stable states

Evolutionary stability, the central solution concept in evolutionary game theory, is closely related to local asymptotic stability in a certain nonlinear dynamical system operating on the state space, the so-called "replicator dynamics". However, a purely dynamical characterization of evolutionary stability is not available in an elementary manner. This characterization can be achieved by investigating so-called "derived games" which consist of mixed strategies corresponding to successful states in the original game. Using well-known facts, several characterization results are obtained within this context. These also may shed light on the extremality properties of evolutionary stability.     

Discounting-sensitivity in symmetric repeated games: An example

I present an example of a symmetric repeated game in which an arbitrarily small change in the discount factor can lead to a discontinuous change in the level of symmetry that can be sustained in equilibrium.     

On the asymptotic stability in functional differential equations

Consider a system of functional differential equations where is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional and negative definite functional . In applications one can construct a positive definite functional , whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional in functional differential equations is autonomous ( does not depend on ), and N. N. Krasovskii created such criterion for the case where the functional is periodic in . For the general case of the non-autonomous functional V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when is almost periodic in . This case is a particular case of the class of non-autonomous functionals.

     

Oscillation and global asymptotic stability in a discrete epidemic model

In this paper, we study the oscillation, global asymptotic stability, and other properties of the positive solutions of the difference equation

     

Guaranteed asymptotic stability for a class of uncertain linear dynamical systems

We consider a class of linear dynamical systems with bounded, Lebesgue-measurable uncertainties in the system and input matrices as well as in the input itself. A state feedback control is derived, which guarantees global, uniform asymptotic stability of the zero state; this control is continuous, except at the zero state.This paper is based in part on research supported by the National Science Foundation.     

On the saddle-point stability for a class of dynamic games

In this paper, we study the stability properties of the class of capital accumulation games introduced by Fershtman and Muller (Ref. 1). Both discrete and continuous time versions are discussed. It is shown that the open-loop Nash equilibrium solutions for both games are characterized by a general saddle-point property, a result best known from the turnpike literature in optimal growth theory. In the case of zero discount rates, an even stronger result can be derived: As long as the Hessian matrix of the instantaneous profit functions has a quasidominant diagonal, no pure imaginary roots are possible.The authors thank J. Boyd III, G. Feichtinger, S. Jørgensen, and G. Schwann for helpful comments. The first author acknowledges financial support from the Natural Science and Engineering Research Council of Canada, Grant No. OGP-0037342.     

GLOBAL ASYMPTOTIC STABILITY OF THE PERIODIC LOTKA-VOLTERRA SYSTEM WITH TWO-PREDATORS AND ONE-PREY

The three species Lotka-Volterra periodic model with two predators and one prey is considered. A set of easily verifiable sufficient conditions is obtained. Finally, an example is given to illustrate the feasibility of these conditious.     

Global asymptotic stability of a family of difference equations

We consider the family of difference equations of the form

     

Preservation of stability and asymptotic behaviour of perturbed integrodifferential equations in a Banach space

In this paper stability and asymptotic behaviour of solutions of the integrodifferential system in a Banach space is related to that of the integrodifferential system in a Banach space. The results obtained constitute a generalization of similar results for ordinary differential equations in a Banach space, which motivate the approach and proofs.     

Stability,structural stability,and games: Toward a theory of differential hypergames

Differential games (DG's) are investigated from a stability point of view. Several resemblances between the theory of optimal control and that of structural stability suggest a differential game approach in which the operators have conflicting interests regarding the stability of the system only. This qualitative approach adds several interesting new features. The solution of a differential game is defined to be the equilibrium position of a dynamical system in the framework of a given stability theory: this is the differential hypergame (DHG). Three types of DHG are discussed: abstract structural DHG, Liapunov DHG, and Popov DHG. The first makes the connection between DG and the catastrophe theory of Thom; the second makes the connection between the value function approach and Liapunov theory; and the third provides invariant properties for DG's. To illustrate the fact that the theory sketched here may find interesting applications, the up-to-date problem of the world economy is outlined.This research was supported by the National Research Council of Canada.     

Uniform asymptotic stability of impulsive discrete systems with time delay

This paper studies impulsive discrete systems with time delay. Some novel criteria on uniform asymptotic stability are established by using the method of Lyapunov functions and the Razumikhin-type technique. Examples are presented to illustrate the criteria.     

Global asymptotic stability of a higher order rational difference equation

In this note, we consider the following rational difference equation:

     

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form

     

非线性时滞差分议程的全局渐近稳定性

In this paper,a sufficient condition for the global asymptotic stability of the solutions of the following nonlinear delay difference equation is obtained,xn 1=xn xn-1xn-2 a/xmxm-1 xn-2 a,n=0,1…,where a∈(0,∞) and the initial values x-2,x-1,x0∈(0,∞).As a special case,a conjecture by Ladas is confirmed.     

New explicit global asymptotic stability criteria for higher order difference equations

New explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained. These criteria can be applied to autonomous and nonautonomous equations. The celebrated Clark asymptotic stability criterion is improved. Also, applications to models from mathematical biology and macroeconomics are given.