Analysis And Control Of Networked Games With Applications Via The Semi-tensor Product Method

In recent years, with the rapid development of complex networks, the networked game has become a hot topic in the study of game theory. Compared with the tra-ditional game, networked games use the nodes of the network topology to represen-t players, and use edges to represent the game relationship between two adjacent players. Each player only plays games with his neighbors and gains the cumulative payoff. This is more consistent with the reality. Nowadays, the networked game theory has been applied to many practical fields such as biology, economics, inter-national relations, computer science, military strategy and so on. Therefore, it has an extremely important theoretical value and research significance. In finite evo-lutionary networked games, each player has finite strategies to choose. The game dynamic evolution process has a natural connection with logic network dynamics. As the powerful tool to deal with the dynamics on finite sets, the semi-tensor prod-uct of matrices can also be used to study finite evolutionary networked games. By this method, the game dynamics can be converted into an algebraic form. A rigorous mathematical framework is established, which is helpful for people to get a more ac-curate and more in-depth understanding for the game phenomenon in reality, and can guide people to predict and control the game. Due to the diversity of evolutionary networked game problems, the theory of semi-tensor product of matrices still has a wide application space in this field, which deserves further research.By using the semi-tensor product method, this paper investigates the existence of Nash equilibrium, the modeling, analysis and control of evolutionary networked games under fixed and time-variant network topology, and applies the obtained re-sults to fashion game strategy choice and the vaccination control of epidemic dy-namics. The main contents of this paper are listed as follows:1. The existence of pure strategy Nash equilibrium for static games is stud-ied. By establishing the structural matrix, the payoff function of each player is ex- pressed into an algebraic form. Based on the algebraic form of the pseudo-Boolean derivative, necessary and sufficient conditions are obtained for the existence of Nash equilibrium in the multi-player binary choice game with complete and incomplete information, respectively. Two effective algorithms are established on how to find Nash equilibrium for the multi-player multi-choice games. Moreover, a unified nec-essary and sufficient condition is given for the existence of Nash equilibrium of bina-ry choice game and multi-choice game. The obtained results are applied to the Nash equilibrium strategy choice of fashion game. Two kinds of optimization problems, that is, the social welfare and normalized satisfaction degree optimization problems are investigated.2. The algebraic formulation and strategy optimization for a class of evolution-ary networked games is studied under fixed network topology. Based on the myopic best response adjustment rule, an algorithm is established to construct the algebraic formulation of the game. Moreover, the final game dynamical behaviors are dis-cussed. By the control of the pseudo-player, the strategy optimization problem is considered, and a free-type control strategy sequence is designed to maximize the average payoff of players.3. The stable degree of strategy profile for evolutionary networked games under fixed network topology is investigated. The game dynamics with the best imitation strategy updating rule is converted into an algebraic form. Based on the proposed concept of stable degree for strategy profiles, some necessary and sufficient con-ditions are obtained for the k-degree stability of strategy profile. Furthermore, a computation method is given to calculate the transient time within which a disturbed strategy profile can be restored. An event-triggered control design method is estab-lished to make the given strategy profile achieve the expected stable degree.4. For evolutionary networked games under time-variant network topology, the algebraic formulation is established and the evolutionary results are analyzed. The obtained results are applied to investigate epidemic dynamics over dynamic network-s. Based on a class of determinate co-evolutionary rule, the matrix expressions are established for the dynamics of individual states and network topologies, respective-ly. All possible final spreading equilibria are obtained for any given initial epidemic state and network topology. A sufficient and necessary condition of the existence of state feedback vaccination control is presented to make every individual susceptible.