Algebraic Modeling And Analysis Of Finite Games Based On Semi-tensor Product Of Matrices

Since its birth,game theory has been attracting the attention of researchers in different fields.With the joint efforts of all of them,game theory has been continuously enriched and improved,and now it has been widely used in many fields such as biology,economics,computer science,etc.It has become a powerful tool for people to analyze and solve problems,so the current research on game theory has extremely important theoretical value and practical significance.The semi-tensor product of matrices is an emerging mathematical tool proposed by Professor Cheng Diezhan,which has been proved to be an efficient tool for dealing with finite games and can systematically portray the dynamics of players in the game process.In this paper,two more important types of games,network evolutionary games and potential games,are studied based on the semi-tensor product of matrices.There is a certain time delay in the transmission of players’ strategy information,which is an important factor affecting the network evolutionary game.The existing studies usually treat the time delay in the game as a unified whole,however,different types of players in the real world may have different time delays in the transmission of information and different impacts on the game,so this paper divides the time delay in the network evolution game into two categories,control time delay and state time delay,according to the source,and discusses the modeling and control problems of the game.Potential games are widely used in various fields because of their special nature.In order to meet different needs,potential games are being extended,such as weighted potential games,state potential games,group-based potential games,multipotential games,etc.Different players in real life have different degrees of influence on the game,and different external environments have different influences on the game.Taking these two factors into account,this paper introduces weights and states respectively on the basis of multipotential games,proposes two new types of potential games,and gives the corresponding determination methods.The main research content and innovative results of the thesis are summarized as follows:1)The time delay in the network evolution game is classified into control time delay and state time delay according to the source;we discuss how to model the network evolution game with control time delay and state time delay by using the semi-tensor product of matrices;for different values of control time delay and state time delay,we use the reachable set method to construct the feedback control in the corresponding cases and obtain the sufficient conditions for the game system to converge to the expected state.2)Introduce weights and states into multipotential games,propose two new games,q-weighted potential game and q-state potential game,and expand the types of potential games;discuss how to determine whether a given game is a weighted potential game or a state potential game based on the semi-tensor product of matrices;obtain the determination methods of q-weighted potential game and q-state potential game based on the semi-tensor product of matrices.