Equilibrium Study Of Weighted Congestion Games And Bayesian Games Via The Semi-tensor Product Method

Game theory mainly studies the interaction between incentive structures,which is widely used in economics,engineering and other fields.Nash equilibrium,one of the most important concepts in the game,has very important theoretical value and research significance.For finite games,the semi-tensor product of matrices as a powerful tool to deal with the dynamics of finite sets can be used to study such game problems.This method could be used to transform the game dynamics into algebraic form and establish a rigorous mathematical framework,which would help people to have a deeper understanding of the game phenomenon in real life and guide people predict and control the game.This paper applies the semi-tensor product theory of matrix and its related properties to transform the weighted congestion games and bayesian games into equivalent algebraic forms.On the basis of this theory,the existence and seeking of Nash equilibrium of these two kinds of static games are discussed.Firstly,the related concepts of semi-tensor product theory and the correlation analysis prop-erties of semi-tensor product are introduced,as well as the general representation of the mapping and finite game on finite set.Secondly,combined with the actual congestion problem,this paper proposes a new weight-ed congestion game and gives the corresponding weighted congestion game.By resorting to semi-tensor product of matrices,we obtain the structures and algebraic representation of result-ed games.We also prove that the resulted game is a weighted potential game.Moreover,an algorithm is provided to search the Nash equilibrium of resulted games.Furthermore,the problem of Nash equilibrium seeking for Bayesian potential games is con-sidered.By constructing a novel transformation of Bayesian games,the Bayesian Nash equilib-rium seeking is converted into Nash equilibrium seeking for the new agent games.Compared with the existing transformations,the new proposed transformation can grantee that the result-ing new agent game of Bayesian potential game still keeps potential.Furthermore,the algebraic representation of Bayesian game is presented.Leveraging on the special structure of the payoff in the resulting new agent game,a reduced-complexity potential equation is established and an algorithm is provided to seek Bayesian Nash equilibrium.