Model And Strategy Design Of Networked Evolutionary Game—the Semi-tensor Product Method

In the last few years,because of the wide background in biological systems,economical systems,and social systems etc.,the evolutionary game on graphs,called the networked evolutionary game(NEG),has been a very appealing research topic.The most challenging problem,among the investigation of networked evolutionary games,is to analyze each player's behavior when the evolution proceeds.However,because of a lack of proper mathematic tools,the most existed results are mainly based on the experiment or computer simulation method.This paper presents a new theoretical framework to analyze and adjust these players'behaviours.The main contents of this paper are listed as follows:1.The algebraic formulation and optimization control for a class of networked evolutionary games with switched topologies are studied.Via the semi-tensor prod-uct method,an algebraic expression is formulated for the given networked evolu-tionary games,based on which,the behavior of corresponding evolutionary games is analyzed.Then,under some certain assumptions,the existence of fixed points for the given systems is proved and a free-type control sequence is designed to guarantee the best strategy profile reachable globally.2.The modeling and analysis of networked evolutionary games with finite memories is investigated,and a number of new results are presented.Using the semi-tensor product method,a kind of algebraic expression is formulated for the networked evolutionary games with finite memories,based on which the behavior of the corresponding evolutionary game is analyzed.Under a proper assumption,the existence of Nash equilibrium of the given networked evolutionary games is proved and a free-type strategy sequence is designed for the convergence to the Nash equilibrium.3.The algebraic formulation and optimization control for a class of dynamic games with random entrance is investigated.The given dynamic game is considered as a kind of networked evolutionary games with switch networks,based on which,it is formulated as a Markov processes to analyze.Using receding horizon control method,the given game' s optimization problem is solved by a state feedback con-troller,when the major player is considered as a control.4.Invertibility of higher order k-valued logical control networks is investigat-ed.First,the higher order k-valued logical control networks are considered as the mappings from the space of input trajectories to the space of output trajectories,based on which the continuity,injectivity,and surjectivity of higher order k-valued logical control networks are analyzed via the theory of symbolic dynamics.As the concept for invertibility of higher order k-valued logical control networks is defined,an equivalent test criterion for invertibility and a method of constructing the inverse system are given via the semi-tensor product method.As the concept for trajecto-ry controllability of higher order k-valued logical control networks is defined,the invertibility is proved to be a sufficient condition for the trajectory controllability.5.Repeated games with a certain payoff function are studied,and the cor-responding best strategy is designed.Firstly,many inherent differences are given between two kinds of repeated games with different payoff functions.Secondly,a concise algorithm and its proof are offered.